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4,416	Historical Test-time Prompt Tuning for Vision Foundation Models Jingyi Zhang1, Jiaxing Huang1, Xiaoqin Zhang2, Ling Shao3, Shijian Lu1∗ 1 College of Computing and Data Science, Nanyang Technological University, Singapore 2 College of Computer Science and Technology, Zhejiang University of Technology, China 3 UCAS-Terminus AI Lab, University of Chinese Academy of Sciences, China Abstract Test-time prompt tuning, which learns prompts online with unlabelled test samples during the inference stage, has demonstrated great potential by learning effective prompts on-the-fly without requiring any task-specific annotations. However, its performance often degrades clearly along the tuning process when the prompts are continuously updated with the test data flow, and the degradation becomes more severe when the domain of test samples changes continuously. We propose HisTPT, a Historical Test-time Prompt Tuning technique that memorizes the useful knowledge of the learnt test samples and enables robust test-time prompt tuning with the memorized knowledge. HisTPT introduces three types of knowledge banks, namely, local knowledge bank, hard-sample knowledge bank, and global knowledge bank, each of which works with different mechanisms for effective knowledge memorization and test-time prompt optimization. In addition, HisTPT features an adaptive knowledge retrieval mechanism that regularizes the prediction of each test sample by adaptively retrieving the memorized knowledge. Extensive experiments show that HisTPT achieves superior prompt tuning performance consistently while handling different visual recognition tasks (e.g., image classification, semantic segmentation, and object detection) and test samples from continuously changing domains. 1 Introduction Vision Foundation Models (VFMs) [1, 2, 3] have demonstrated impressive zero-shot generalization capabilities over various downstream tasks at the cost of domain expertise for crafting appropriate task-specific prompts [4, 5, 6]. To circumvent this limitation, prompt learning [4], which aims to adapt VFMs to fit downstream tasks by optimizing prompts as learnable vectors with few-shot task training samples, has been extensively explored recently. However, existing prompt tuning methods generally suffer from two constraints: 1) they require labelled training data for each downstream task which can be tedious and laborious to collect [7, 8], and 2) the learnt prompts tend to overfit to the few-shot training samples, leading to degraded generalization toward downstream tasks [9, 10, 11]. Test-time prompt tuning [7] instead learns prompts with a online flow of unlabelled test samples during the inference stage. It has attracted increasing attention recently as it allows learning effective prompts on-the-fly without requiring any task-specific annotations as illustrated in Fig. 1 (a). Existing test-time prompt tuning methods usually start with an initial template prompt like “a photo of a [class]" and optimize it with a self-supervised objective over test images together with their model predictions [7, 8]. However, these methods often experience a clear performance degradation along the tuning process when the prompts are continuously updated with the test data flow, largely due to the lack of test-sample annotations as illustrated in Fig. 1 (b). Specifically, these methods ∗Corresponding author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Test Data Flow Pre-Trained Large Vision Models … … [class] 🔥 ❄ Learnable Prompts Online Update Current Sample Learnt Samples Incoming Samples (a) (b) Test Domain #1 Test Domain #2 Test Domain #3 Data Flow: Self-Supervised Objective Prediction Figure 1: (a) Test-time Prompt Tuning learns and optimizes prompts from a continuous flow of unlabelled test samples during the inference stage. (b) Most existing test-time prompt tuning methods such as TPT [7] and DiffTPT [8] tend to ‘forget’ historical knowledge learnt from previous test samples when the prompts are continuously updated with the test data flow. They learn effective prompts at early tuning stage, but the learnt prompts degrade gradually along the tuning process. This phenomenon becomes more apparent when the domain of test samples changes continuously. The curves are derived from 100 runs over 3 different domains [16, 17]. In each run, the order of the 3 domains as well as the samples within each domain is randomly shuffled to simulate continuously changing test domains. learn prompts well at the early test-time tuning stage, and the learnt prompt outperforms the initial template prompts clearly. However, while the tuning continues, the learnt prompts deteriorate and gradually perform even worse than the initial template prompt especially when the test domain changes continuously. These results show that existing methods [7, 8] learn effective prompts via self-supervised objectives at the early training stage, but tend to forget the useful knowledge learnt from previous test samples, and the forgetting is largely due to the accumulation of prediction errors over the unlabelled test samples along the tuning process [12, 13]. Inspired by prior studies [14, 15] in memory-based learning, we propose Historical Test-time Prompt Tuning (HisTPT) that introduces three types of knowledge banks to help memorize the previously learnt useful knowledge to mitigate the knowledge ‘forgetting’ problem. The three types of knowledge banks are local knowledge bank, hard-sample knowledge bank and global knowledge bank, each of which stores complementary historical information and works with different mechanisms. Specifically, local knowledge bank buffers fresh information from the recent batches of test images, capturing up-to-date distribution changes. Hard-sample knowledge bank identifies and stores the features of hard samples from local knowledge bank, capturing difficult and rare corner cases along the tuning process. Global knowledge bank stores global information by accumulating the features from the local knowledge bank and hard-sample knowledge bank, leading to comprehensive memorization that captures representative features. In addition, HisTPT introduces an adaptive knowledge retrieval mechanism which retrieves memorized knowledge adaptively for each test image for prediction regularization and prompt optimization. To this end, HisTPT builds up comprehensive memorization that preserves useful knowledge from previous test samples, mitigating the knowledge forgetting and enabling robust test-time prompt tuning as illustrated in Fig. 1 (b). The contributions of this work can be summarized in three aspects. First, we design HisTPT, a general test-time prompt tuning framework that explores memory learning to learn effective prompts on-the-fly. To the best of our knowledge, this is the first work that explores memory learning for test-time prompt tuning. Second, HisTPT constructs three types of knowledge banks that store complementary historical information and introduces an adaptive knowledge retrieval mechanism that retrieves memorized knowledge adaptively for each test image, mitigating the ‘forgetting’ of learnt useful knowledge along the prompt tuning process and ultimately leading to robust prompt learning with unlabelled test samples. Third, extensive experiments over multiple benchmarks show that HisTPT achieves superior performance consistently across different visual recognition tasks such as image classification, semantic segmentation, and object detection, especially when the domain of test images continuously changes. 2 2 Related Work Test-time Adaptation, which is a type of domain adaptation technique [18, 19, 20, 21], aims for designing the technique to improve model generalization over test samples [22, 23, 24]. Early studies such as test-time training (TTT) and its variants [22, 23], introduce auxiliary tasks (e.g., rotation prediction task [25]) into the supervised training objective to improve the model generalization at the training stage, and then adapt the pre-trained model to test samples via self-supervised objectives at the inference stage. Differently, recent studies [24, 20, 26, 27, 28, 29, 30, 31] generally focuses on fully test-time adaptation, where the model is adapted to test samples only during the inference stage, without introducing any auxiliary task into the training phase. For example, TENT [24] minimizes the batch-wise prediction entropy for test images while MEMO [27] enforces the prediction consistency between different augmentations of each test sample. With the advent of vision foundation models (VFMs), test-time prompt tuning [7, 8] has recently been explored for adapting pre-trained VFMs toward downstream tasks via prompt tuning at the inference stage. Prompt Learning of Vision Foundation Models (VFMs) [1, 2, 3] has been studied extensively as VFMs despite their impressive zero-shot generalization capabilities over various downstream tasks often require to design appropriate task-specific prompts for optimal adaptation. Inspired by the “prompt learning” in NLP [32], one typical prompt learning approach for VFMs [4, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41] learns to optimize prompts as learnable vectors with few-shot labelled samples of downstream tasks. Despite its effectiveness, it requires to label task-specific training data which is often laborious with poor scalability [7]. In addition, the learnt prompts tend to overfit to few-shot task samples, and this often degrades the generalization of VFMs while adapting toward various downstream tasks [7]. Different from prompt learning, test-time prompt tuning [7, 8] explores a new prompt learning setup that learns prompts on-the-fly with an online flow of unlabelled test images during the inference stage. Test-time Prompt Tuning (TPT) aims to learn prompts on-the-fly using the test samples at inference. It has attracted increasing attention recently [7, 8, 42, 43, 44, 45] as it can learn effective prompts online with unlabelled test samples flow continuously. Most existing test-time prompt tuning studies focus on image classification tasks [7, 8, 42, 43, 44, 45]. For example, TPT [7] optimizes prompts by minimizing the prediction entropy between each test sample and its augmented views. DiffTPT [8] improves the TPT by introducing the pre-trained diffusion model [46] to produce multiple diverse and informative augmented views. Different from these studies [7, 8, 42, 43, 44, 45], HisTPT aims to mitigate the knowledge ‘forgetting’ problem in test-time prompt tuning when the text tokens are continuously updated with the test data flow. HisTPT achieves it by constructing comprehensive memorization capturing useful historical knowledge. In addition, HisTPT achieves superior performance across various visual recognition tasks consistently, and it can effectively handle the challenging scenario where the domain of test samples changes continuously. Memory-based Learning has been studied extensively in computer vision [12, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], such as semi-supervised learning [51, 58], long-term video understanding [15, 59] and domain adaptation [60, 61, 14]. For the adaptation of vision foundation models (VFMs), several studies employ memory for improving the performance on downstream tasks [62, 63, 64, 65, 66]. For instance, [66] tackles image captioning challenge by memorizing visual-related sentences which helps VFMs to generate high-quality captions with fewer hallucinations. [65] replaces text features by identity-specific sequence features extracted by CLIP, which effectively facilitates video-based person re-identification. [64] and [62] enable efficient training-free VFMs adaptation by caching categoryspecific data features. Different from these studies, HisTPT designs three types of knowledge banks for memorizing useful knowledge learnt from previously test samples and introduces an adaptive knowledge retrieval mechanism that retrieves memorized knowledge for each test sample adaptively, aiming for mitigating the knowledge ‘forgetting’ problem in test-time prompt tuning. 3 Method 3.1 Preliminaries and Task Definition Preliminaries of Vision Foundation Models (VFMs). We denote a pre-trained VFM by F = {F I, F T }, where F I and F T are image encoder and text encoder respectively. Given a test image x ∈Xtest and the names of its possible belonged classes yc ∈Ytest = {yc}C c=1, the VFM image 3 encoder and text encoder can produce image features and category-wise text features, respectively , i.e., v = F I(x) and uc = F T (yc). The predictions can be obtained by calculating the similarity between the image features and the category-wise text features: ˆc = arg max c pc, pc = exp (cos(uc, v))/ τ PC j=1 exp (cos(uj, v))/ τ , (1) where cos(·) denotes the cosine similarity, and τ is a temperature hyper-parameter that controls the density of the encoded feature. Instead of directly obtaining text features using the raw class names, certain hand-crafted template prompts, e.g., “a photo of a [class]”, are often adopted for generating task-related textual descriptions. However, designing appropriate prompts for each downstream task is a non-trivial task which often requires domain expertise. To this end, prompt learning [4, 9] has been extensively studied, aiming to adapt VFMs to fit downstream tasks by optimizing prompts as learnable text tokens with few-shot task samples. Specifically, M learnable text tokens are adopted to append the raw class names, i.e., t = {t1, t2, ..., tM} each being a vector of dimension D (e.g., D = 512). Thus, the textual description for class c becomes (t; yc). The learnable text tokens t are optimized with a task-related loss (e.g., cross-entropy loss) over the few-shot labelled training samples. Task Definition. Different from conventional prompt learning, this work focuses on continual testtime prompt tuning that adapts VFMs via prompt tuning with unlabelled test images. The objective of test-time prompt tuning is to optimize the text tokens t for test image x with certain self-supervised training losses Lself that can be formulated by: t∗= arg min t Lself(F, t, x). (2) Note that the test data is presented in a continuous flow, where the text tokens are continuously updated with the test data flow. 3.2 Historical Test-time Prompt Tuning We design three types of knowledge banks to help memorize the useful knowledge learnt from the previous test samples and adaptively exploit the memorized knowledge for regularizing the prediction of the current test samples. As illustrated in Fig. 2, local knowledge bank buffers features of the recent test images, capturing up-to-date distribution changes along the tuning process. Hardsample knowledge bank actively identifies and stores hard samples from the local knowledge bank, which helps to capture difficult and corner features. Global knowledge bank maintains global and representative information along the whole prompt tuning process by accumulating all the features from the local knowledge bank and hard-sample knowledge bank. In addition, HisTPT introduces an adaptive knowledge retrieval mechanism that adaptively retrieves relevant memorized knowledge for prediction regularization and prompt optimization for each test image. Given a continuous flow of N test samples Xtest = {xn}N n=1, we take the time step n as an example to describe the knowledge bank construction with the previous test sample xn−1 and the prompt optimization of the current sample xn with the memorized knowledge. Knowledge Bank Construction. HisTPT comes with three types of knowledge banks for capturing fresh and representative knowledge during the test-time prompt tuning with previous test samples. Local Knowledge Bank captures and stores fresh and up-to-date knowledge by buffering the features of the recent test samples. It works as a FIFO queue with a fixed size of L, where the features of the oldest test sample will be dequeued and the features of the most recent test sample will be enqueued to update the local knowledge bank, i.e, Mlocal = {ul local, pl local}L l=1 on the flow. Specifically, for the latest test sample xn−1 and its learnt text tokens tn−1, local knowledge bank enqueues its text feature un−1 and prediction probability pn−1, i.e., un−1 = {uc n−1}C c=1 where uc n−1 = F T ((tn−1; yc)), and pn−1 = {pc n−1}C c=1 where pc n−1 is calculated via Eq. 1. Note that the size of local knowledge bank L is much smaller than the total number of test samples N since local knowledge bank aims to capture fresh information and up-to-date distribution changes of test samples along the test-time prompt tuning process. Hard-sample Knowledge Bank identifies hard samples from local knowledge bank for capturing difficult and corner information. We identify hard samples by those having high classification 4 𝑥! 𝑥!"# 𝑥!"$ … 𝐭!"# Store Global Knowledge Bank Local Knowledge Bank Hard-Sample Knowledge Bank 𝐭!"$ ℒ!"#$ Adaptive Knowledge Retrieval Vision Foundation Models Prediction Regularized Prediction Online Update ❄ 𝐭! Previous Test Images with the learnt text tokens Current Test Image Adaptive Category-wise Prototypes Figure 2: Overview of the proposed HisTPT. HisTPT features three types of knowledge banks, namely, local knowledge bank, hard-sample knowledge bank, and global knowledge bank, which learn and memorize up-to-date, difficult and representative knowledge, respectively, from previous test samples (e.g., xn−2 and xn−1) and their learnt text tokens (e.g., tn−2 and tn−1) along the test-time prompt tuning process. For the current test sample xn, HisTPT regularizes its prediction by retrieving the memorized knowledge via an adaptive knowledge retrieval mechanism, enabling prompt optimization for xn with the self-supervised loss Lself. uncertainty, where the uncertainty is measured by their prediction entropy which can be computed from their prediction probability as stored in the local knowledge bank: E(ul local) = − C X c=1 p(l,c) local log p(l,c) local, (3) where the first K samples with the highest entropy are selected and stored in the hard-sample knowledge bank. To enable robust memorization, we first compact the features of K selected samples via category-wise average and store the compacted feature in the hard-sample knowledge bank. Similar to the local knowledge bank, hard-sample knowledge bank also works as a FIFO queue with a fixed size of H, i.e., Mhard = {uh hard}H h=1. Global Knowledge Bank stores global and representative knowledge the whole prompt tuning process by accumulating all the features from the local knowledge and hard-sample knowledge banks. Specifically, we compact the features uglobal and uhard dequeued from the local and hard-sample knowledge banks to generate category-wise feature prototype δglobal = {δc global}C c=1, where δc global = 1/2 (uc local+uc hard). To facilitate stable and sustainable global memorization along the tuning process, we update the global knowledge bank with compacted feature prototype in a momentum way: δglobal ←(1 −γ) δglobal + γ δglobal, (4) where δglobal denotes the old global feature prototype and γ is a coefficient for smooth feature update in the global knowledge bank. Prompt Optimization with the Constructed Knowledge Banks. With the built comprehensive memorization, HisTPT introduces an Adaptive Knowledge Retrieval Mechanism that enables adaptive retrieval of memorized knowledge for prediction regularization and prompt optimization of each test sample. Given the test sample xn and the text tokens learnt at time step n −1, i.e., tn−1, the category-wise prediction probability pn = {pc n}C c=1 can be obtained by measuring the similarity between the image feature vn = F I(xn) and category-wise text feature uc n = F T ((tn−1; yc)) via Eq.1. The prediction pn can be enhanced via regularization with the three types of knowledge banks. For temporary knowledge in the local and hard-sample knowledge banks, we first compact the stored features into category-wise feature prototypes, i.e., δlocal and δhard, via an average operation: δlocal = {δc local}C c=1, δhard = {δc hard}C c=1 where δc local = 1 L L X 1 u(l,c) local, δc hard = 1 H H X 1 u(h,c) hard. (5) 5 The new prediction for xt can thus be obtained based on the derived prototypes δlocal, δhard, and δglobal. Take the local prototype δlocal as an example. The prediction regularization of xn can be obtained with the local knowledge bank plocal by plocal = {pc local}C c=1, pc local = exp (cos(δc local, vn))/ τ PC j=1 exp (cos(δj local, vn))/ τ . (6) The prediction regularization by the hard-sample and global knowledge banks can be obtained in a similar way. Generally, the prediction with higher confidence (i.e., lower entropy) means that the corresponding feature prototype is better aligned with the current test sample in feature space, and it should contribute more to the final prediction ˆpn that can be obtained as follows: ˆpn = X i wi pi, wi = Softmax( C X c=1 p(c) i log p(c) i ), (7) where i ∈{local, hard, global}. The softmax operation is performed across the entropy of different predictions. With the regularized prediction probability ˆpn, the text tokens tn−1 can be optimized for the current test sample xn with the self-supervised loss defined as follows: Lself = l(pn, ˆpn) (8) where l(·) denotes a task-related loss, e.g., the standard cross-entropy loss for image classification. 4 Experiments This section presents experiments including datasets, implementation details, benchmarking with the state-of-the-art, as well as discussion of our designs. 4.1 Datasets We evaluate HisTPT over multiple datasets across three widely studied visual recognition tasks: Semantic Segmentation: We benchmark HisTPT over 6 image segmentation datasets with pixelwise annotations, including Cityscapes [16], BDD100K [67], Mapillary [68], ADE20K [69], Pascal Content [70] and ACDC [17]. Image Classification: We benchmark HisTPT over 10 classification datasets, including Flowers102 [71], DTD [72], Oxford-Pets [73], StanfordCars [74], UCF101 [75], Caltech101 [76], Food101 [77], SUN397 [78], Aircraft [79] and EuroSAT [80]. Object Detection: We benchmark HisTPT over 4 object detection datasets, including Cityscapes [16], BDD100K [67], ADE20K [69] and ACDC [17]. 4.2 Implementation Details Semantic Segmentation: Following [81], we adopt SEEM [3] with two vision backbones including Focal-Tiny [82] and Davit-Large [83] as the segmentation foundation models. In training, we employ AdamW optimizer [84] with a weight decay of 0.05, and set the initial learning rate as 0.0001. Image Classification: Following [7, 8], we use CLIP [1] with two backbones, i.e., ResNet-50 [85] and ViT-B/16 [86], as the classification foundation models. In training, we adopt AdamW optimizer [84] with a weight decay of 0.01, and set the initial learning rate as 0.005. Object Detection: For object detection task, we adopt SEEM [3] with two vision backbones including Focal-Tiny [82] and Davit-Large [83] as the detection foundation models. In training, we employ AdamW optimizer [84] with a weight decay of 0.05, and set the initial learning rate as 0.0001. For all experiments, the prompt is initialized as “a photo of a” and the corresponding 4 tokens (i.e., M = 4) of dimension D = 512 are optimized as in [7, 8]. Unless otherwise specified, we set the size of the local knowledge bank and hard-sample knowledge bank at L = H = 32 and the number of the selected hard-sample features K at 16. We set the update coefficient γ of the global knowledge bank at 0.99. Following [7], we set the optimization step in test-time prompt tuning at 1 by default. All the experiments are conducted on one NVIDIA Tesla V100 GPU with batch size 1. 6 Table 1: Test-time prompt tuning on semantic segmentation over 6 widely adopted datasets. mIoU is reported. Method Cityscapes BDD Mapillary ADE Pascal ACDCF og ACDCNight ACDCRain ACDCSnow Mean SEEM-Tiny 39.2 37.4 14.7 14.6 45.1 34.6 20.7 33.1 35.8 30.5 TPT [7] 42.3 38.9 15.4 16.1 46.8 35.2 21.4 34.9 36.5 31.9 TPT [7] + HisTPT 45.1 41.8 17.5 17.6 49.4 37.2 22.9 37.2 37.8 34.0 DiffTPT [8] 42.9 39.6 15.8 16.3 47.1 35.7 21.6 35.3 36.6 32.3 DiffTPT [8] + HisTPT 45.4 42.1 16.7 17.9 49.2 47.6 22.7 37.7 38.1 35.2 HisTPT 44.7 41.2 17.2 17.3 48.7 36.8 22.1 36.7 37.1 33.5 SEEM-Large 49.3 44.6 18.7 15.2 37.1 48.1 32.0 47.4 45.0 37.4 TPT [7] 50.1 45.2 19.1 15.7 40.2 48.7 32.4 47.9 45.7 38.3 TPT [7] + HisTPT 52.1 47.4 21.3 17.1 45.8 52.1 33.4 49.4 48.8 40.8 DiffTPT [8] 50.4 45.7 19.3 16.1 41.2 49.1 32.2 48.2 46.3 38.7 DiffTPT [8] + HisTPT 52.4 47.8 21.1 17.4 46.3 52.4 33.6 49.7 49.1 41.0 HisTPT 51.9 47.3 20.1 16.9 45.7 51.6 33.1 49.1 48.5 40.4 Table 2: Test-time prompt tuning on image classification over 10 widely adopted datasets. Top-1 classification accuracy is reported. Method Flower DTD Pets Cars UCF101 Caltech101 Food101 SUN397 Aircraft EuroSAT Mean CLIP-RN50 61.7 40.3 83.5 55.7 58.8 85.8 73.9 58.8 15.6 23.6 55.8 TPT [7] 62.2 40.1 83.9 58.3 60.3 86.3 74.4 60.9 16.7 27.4 57.1 DiffTPT [8] 63.1 39.7 82.9 60.1 62.1 86.4 78.3 62.4 17.3 39.3 59.2 HisTPT 67.6 41.3 84.9 61.3 64.1 87.2 81.3 63.5 18.1 42.5 61.2 CLIP-ViT-B/16 67.4 44.2 88.2 65.4 65.1 93.3 83.6 62.5 23.6 42.0 63.5 TPT [7] 68.2 47.3 87.1 66.5 67.7 93.7 84.2 65.1 24.3 42.1 64.6 DiffTPT [8] 69.4 46.3 87.9 66.4 68.1 92.3 86.5 65.3 25.1 42.8 65.0 HisTPT 71.2 48.9 89.1 69.2 70.1 94.5 89.3 67.2 26.9 49.7 67.6 4.3 Comparisons with State of the Arts Semantic Segmentation. We evaluate and benchmark HisTPT over 6 semantic segmentation datasets. Since there is little prior study on test-time prompt tuning on semantic segmentation, we benchmark HisTPT by reproducing methods [7, 8], which are designed for image classification task, on semantic segmentation task. Table 1 shows experimental results. We can observe that HisTPT achieves superior segmentation performance, largely due to its comprehensive memorization that helps to regularize the predictions of test samples and mitigates the knowledge forgetting problem in test-time prompt tuning. In addition, HisTPT is complementary to existing methods and produces clear and consistent performance boosts. This is attributed to the proposed HisTPT which can effectively mitigate the knowledge forgetting existing methods. Image Classification. Following [7, 8], we evaluate HisTPT over 10 image classification tasks. To suit the setup in this work, we reproduce methods [7, 8] by keeping their prompts continuously updated during the test-time adaptation. As shown in Table 2, HisTPT outperforms state-of-the-art methods consistently over different classification tasks such as classic classification on Flowers102 [71], texture classification on DTD [72] and human action recognition on UCF101 [75]. This demonstrates the superior generalization ability while HisTPT faces diverse downstream data. Object Detection. We evaluate and benchmark HisTPT over 4 object detection datasets. Similar to semantic segmentation benchmarking, we benchmark HisTPT by reproducing methods [7, 8] (designed for image classification task) on the object detection task. As shown in Table 3, HisTPT achieves superior detection performance and can well handle a wide range of detection tasks including detection under various weather conditions [17] across different scenes [16, 69]. The superior detection performance is largely attributed to the knowledge banks in HisTPT which effectively help generate more accurate predictions and learn better prompts for test samples. 4.4 Ablation Studies We examine the proposed HisTPT by performing ablation study over Cityscapes semantic segmentation task. As shown in Table 4, the three types of knowledge banks can work well alone and improve 7 Table 3: Test-time prompt tuning on object detection over 4 widely adopted datasets. mAP50 is reported. Method Cityscapes BDD ADE ACDCF og ACDCNight ACDCRain ACDCSnow Mean SEEM-Tiny 30.5 26.1 15.7 44.2 22.3 25.9 33.9 28.3 TPT [7] 30.9 27.0 16.2 44.8 23.1 26.3 34.4 28.9 DiffTPT [8] 31.2 27.4 16.8 45.1 23.3 26.7 34.6 29.3 HisTPT 31.9 28.3 17.5 46.2 24.7 27.2 35.6 30.2 SEEM-Large 31.4 31.8 18.3 55.2 31.4 34.8 43.7 35.2 TPT [7] 31.8 32.2 18.5 55.6 31.9 35.1 44.2 35.6 DiffTPT [8] 32.5 32.3 18.9 56.1 32.3 35.4 44.8 36.0 HisTPT 33.2 33.4 19.4 56.9 33.1 36.4 45.2 36.8 Table 4: Ablation study of the proposed HisTPT over Cityscapes semantic segmentation task. Method Histrocial Knowledge Banks Adaptive knowledge retrieval mIoU local knowledge bank hard-sample knowledge bank global knowledge bank SEEM-Tiny 39.2 ✓ 41.1 ✓ 40.9 ✓ 41.7 ✓ ✓ 42.2 ✓ ✓ 42.8 ✓ ✓ 42.5 ✓ ✓ ✓ 43.6 HisTPT ✓ ✓ ✓ ✓ 44.7 the performance consistently, indicating that all the stored historical knowledge is helpful in prompt tuning. In addition, the three types of knowledge banks are complementary to each other, largely because the three knowledge banks store different types of knowledge, i.e., local knowledge bank stores fresh information, hard-sample knowledge bank stores difficult corner case information, and global knowledge bank stores the global and representative features. On top of the three types of knowledge, including the proposed adaptive knowledge retrieval improves the performance further. This shows that adaptively retrieving different types of memorized information for each test image could generate more accurate prediction and ultimately lead to better test-time prompt tuning. 4.5 Discussion Complementarity to Prompt Learning Methods. As a test-time tuning technique, the proposed HisTPT is complementary to prompt learning methods that learn prompts at the training stage. We examine this feature by setting the learnt prompts by prompt learning [4, 9] as the initial prompts of HisTPT. As Table 5 shows, equipping HisTPT with the learnt prompts improves the performance clearly, indicating that HisTPT as a plug-in can greatly enhance existing prompt learning methods. Figure 3: HisTPT with multiple optimization steps. Optimization Steps. We examined how the optimization step affects HisTPT by increasing it from 1 to 10. Figure 3 shows the mean mIoU over 6 semantic segmentation datasets with SEEMTiny. We can observe that increasing the optimization step improves segmentation consistently. Nevertheless, the performance gain becomes marginal after 6-8 optimization steps. The actual optimization step can be set by balancing the inference efficiency and the inference accuracy. Continuously Changing Test Domains. As discussed in Section 1, HisTPT can handle challenging scenarios when the domain of test samples changes continuously. We examine this feature over semantic segmentation data that were collected under normal weather [16] and various adverse weathers [17, 87] (fog, night, rain and snow). As Table 6(a) shows, the performance of existing test-time prompt tuning methods TPT [7] and DiffTPT [8] degrades gradually along the tuning process when the weather changes from normal to adverse, largely due to increasing error accumulation and ‘forgetting’ while the test domain changes continuously. As a 8 Table 5: Complementarity to state-of-the-art prompt learning methods CoOp [4] and CoCoOp [9]. The mean top-1 accuracy across 10 image classification datasets is reported, and CoOp and CoCoOp are supervised with 16-shot labelled training data per category. Method CLIP-RN50 CoOp CoCoOp HisTPT HisTPT + CoOp HisTPT + CoCoOp Mean Accuracy 55.8 56.1 57.2 61.2 62.4 63.1 Table 6: Test-time prompt tuning on semantic segmentation across continuously changing test domains. mIoU is reported. Test Order (→) Normal Fog Night Rain Snow SEEM-Tiny 39.2 34.6 20.7 33.1 35.8 TPT 42.3(+3.1) 34.8(+0.2) 20.1(-0.6) 31.7(-1.4) 30.6(-5.2) DiffTPT 42.9(+3.7) 35.2(+0.6) 20.3(-0.4) 32.0(-1.1) 31.4(-4.4) HisTPT 44.7(+5.5) 36.9(+2.3) 23.6(+2.9) 37.3(+4.2) 38.1(+2.3) (a) Test Order (→) Snow Rain Night Fog Normal SEEM-Tiny 35.8 33.1 20.7 34.6 39.2 TPT 36.5(+0.7) 34.1(+1.0) 20.1(-0.6) 32.7(-1.9) 35.8(-3.4) DiffTPT 36.6(+0.8) 34.7(+1.6) 20.5(-0.2) 32.9(-1.7) 36.1(-3.1) HisTPT 37.1(+1.3) 36.8(+3.7) 22.1(+1.4) 37.0(+2.4) 44.9(+5.7) (b) comparison, HisTPT improves the performance consistently across different weathers, and this is largely due to two factors: 1) HisTPT effectively preserves representative and up-to-date knowledge from past test samples along the tuning process; 2) HisTPT retrieves relevant memorized knowledge for each test sample, mitigating the ‘forgetting’ and leading to more robust test-time prompt tuning. Similar results are obtained when the test domain changes from adverse weather to normal weather as shown in Table 6(b), further verifying HisTPT’s effectiveness and robustness while facing changing test domains. Comparisons to Existing Memory-based Learning Methods. We examine how the proposed HisTPT performs as compared with existing memory-based learning techniques. We benchmark it with two categories of memory-based learning techniques: 1) memory-based learning in traditional network training [60, 61, 14] and 2) memory-based learning with vision foundation models [66, 65, 62]. Table 7 shows experimental results on the task of semantic segmentation on Cityscapes with SEEM-Tiny. It can be seen that HisTPT outperforms all existing memory learning techniques [60, 61, 14, 66, 65, 62] with clear margins. The superior performance is largely attributed to two factors: 1) HisTPT memorizes comprehensive knowledge of previous test samples on the fly along the prompt tuning process and 2) HisTPT features a retrieval mechanism that adaptively retrieves the memorized knowledge to learn specific prompts for each current test sample. Table 7: Comparison with existing memory-based learning methods over Cityscapes semantic segmentation task on SEEM-Tiny. mIoU is reported. Method HCL [60] MeGA [61] BiMem [14] MeaCap [66] TF-Clip [65] TDA [62] HisTPT mIoU 40.3 40.7 41.2 41.9 41.4 42.6 44.7 5 Conclusion This paper introduces Historical Test-time Prompt Tuning (HisTPT), a general test-time prompt tuning framework that aims to mitigate the ‘knowledge forgetting’ problem across various visual recognition tasks. HisTPT introduces three types of knowledge banks, including local knowledge bank, hardsample knowledge bank and global knowledge bank, each of which works with different mechanisms for memorizing useful knowledge. With the three knowledge banks, HisTPT builds up comprehensive memorization that preserves useful knowledge from previous test samples, mitigating the knowledge forgetting and enabling robust test-time prompt tuning. In addition, HisTPT comes with an adaptive knowledge retrieval mechanism that regularizes the prediction of the current test sample by adaptively retrieving the memorized knowledge. Extensive experiments show that HisTPT achieves superior performance consistently across various vision tasks. In addition, HisTPT can effectively handle the challenging scenario where the domain of test samples changes continuously. Moving forwards, we will further investigate memory-based learning for adaptation of vision foundation models. Acknowledgement. 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These datasets have rich diversity as shown in table 8. Specifically, the 10 image classification datasets involves a wide range of visual recognition tasks from fine-grained classification, to human action recognition and texture classification. Similarly, the images of the semantic segmentation and object detection datasets are also in rich diversity, spinning from street scene images collected from various cities with different weather conditions, to images collected under indoor scenes such as office and kitchen. Table 8: Details of the datasets used for benchmarking HisTPT. Datasets Test Images Classes Description Image Classification Flower102 [71] 2,463 102 Flower images with various sizes and illumination environments. DTD [72] 1,692 47 A dataset of textural images for image recognition. Oxford-IIIT PETS [73] 3,669 37 A dataset for pet recognition with cat and dog images of 37 breeds. Stanford Cars [74] 8,041 196 Car images for fine-grained recognition. UCF101 [75] 3,783 101 A video dataset for human action recognition. Caltech101 2,465 101 A dataset for common object recognition. Food-101 [77] 30,300 101 Food images for fine-grained recognition. SUN397 [78] 19,850 397 Indoor and outdoor scene images for fine-grained recognition. Aircraft [79] 3,333 100 A dataset of 100 aircraft model variants for aircraft model recognition. EuroSAT [80] 8,100 10 A dataset of satellite images for land use and land cover recognition. Semantic Segmentation Cityscapes [16] 500 19 Scene images collected in different cities for street scene understanding. BDD100K [67] 1,000 19 Street scene images collected at different times of the day. Mapillary [68] 2,000 65 A dataset of street-level images with high resolution. ADE20K [69] 2,000 150 A large-scale dataset of images collected from outdoor and indoor scenes. Pascal Content [70] 5101 59 An extension of PASCAL VOC 2010 dataset with pixel-wise annotations. ACDC [17] 406 19 Scene images with adverse weather conditions, i.e., fog, night, rain, snow. Object Detection Cityscapes [16] 500 8 Scene images collected in different cities for street scene understanding. BDD100K [67] 1,000 8 Street scene images collected at different times of the day. ADE20K [69] 2,000 100 A large-scale dataset of images collected from outdoor and indoor scenes. ACDC [17] 406 8 Scene images with adverse weather conditions, i.e., fog, night, rain, snow. B Parameter Analysis We study the size of the local knowledge bank and hard-sample knowledge bank (L and H), the parameter K used in hard-sample knowledge bank update, and the update coefficient γ used in Eq. 4 for global knowledge bank, over the semantic segmentation task Cityscapes with SEEM-Tiny. Size of the local knowledge bank L. As discussed in the main text, the size of local knowledge bank L is much smaller than the total number of test samples, since local knowledge bank aims to buffer fresh information from recent previous test samples. Here we study how it affects the test-time prompt tuning. As shown in Table 9 (a), HisTPT yields robust performance when L is relatively small (from 8 to 64), while the performance drops slightly when it becomes too large. This show that the local knowledge bank with relatively small size could effectively capture fresh information and up-to-date distribution changes along the tuning process. Size of the hard-sample knowledge bank H. Hard-sample knowledge bank stores the features of hard-samples, capturing different and rare corner cases during the test-time prompt tuning process. Table 9 (b) show that HisTPT is quite robust when H is between 8 to 128. Hence, we simply set it as the same as the size of the local knowledge bank, i.e., H = L = 32. The number of selected hard-sample features K. As discussed in the main text, hard-sample identifies and stores K hard-sample features from the local knowledge bank. Here we study the sensitivity of K by increasing it from 8 to 24 with a step of 4. As shown in Table 9(c), the performance is quite tolerant to the parameter N and the best performance is obtained when K = 16. Update coefficient γ. The update coefficient γ in Eq. 4 determines the update speed of global knowledge bank, where the larger update coefficient results in the slower update of global knowledge bank. From Table 9 (d), we can observe that HisTPT is robust when γ is large enough (i.e., from 0.9 to 0.999) while the performance of HisTPT drops slightly when γ becomes too small. This demonstrates that a large update coefficient, ensuring 15 smooth and gradual updates, facilitates stable global memorization. Conversely, a too small update coefficient leads to rapid updates of the global knowledge bank, resulting in unstable memorization and less effective test-time prompt tuning. Table 9: Parameter analysis of HisTPT over semantic segmentation task Cityscapes with SEEM-Tiny. L 8 16 32 64 128 512 HisTPT 44.5 44.7 44.7 44.6 44.2 43.9 (a) The size of local knowledge bank L. H 8 16 32 64 128 512 HisTPT 44.7 44.6 44.7 44.5 44.6 43.5 (b) The size of hard-sample knowledge bank H. K 8 12 16 20 24 HisTPT 44.6 44.5 44.7 44.6 44.6 (c) The number of hard-sample features K. γ 0.1 0.5 0.9 0.99 0.999 HisTPT 43.1 43.9 44.5 44.7 44.6 (d) The update coefficient γ. C More Discussion about the Design of Historical Knowledge Banks Update of the hard-sample knowledge bank. As discussed in the main text, hard-sample knowledge bank works as an FIFO queue with a fixed size, and it is updated using the hard-sample features selected from local knowledge bank with an average compaction operation. Here we provide more discussion about the different update ways of hard-sample knowledge bank, including 1) directly update using the selected features and 2) update using the compacted features with an average operation. From Table 10, we can observe that updating hard-sample knowledge bank using the selected features with average compaction operation performs better, which is largely due to that the compacted features enabling to filter out some noises and results in more robust memorization of difficult and corner-case information. Table 10: Comparison of different update ways of hard-sample knowledge bank over semantic segmentation task Cityscapes with SEEM-Tiny. Method Directly update Update with average operation mIoU 43.9 44.7 Update of the global knowledge bank. As described in the main text, we update the global knowledge bank using the features dequeued from both the local knowledge bank and hard-sample knowledge bank. Here we study its effectiveness with different update ways of global knowledge bank, including 1) update global knowledge bank with only the features dequeued from local knowledge bank; 2) update global knowledge bank with only the features dequeued from hard-sample knowledge bank and 3) update global knowledge bank with the features dequeued from the local knowledge bank and hard-sample knowledge bank. Table 11 shows the experimental results. It can be observed that updating global knowledge bank with the features dequeued from both the local knowledge bank and hard-sample knowledge bank performs the best, which indicates that the features stored in local knowledge bank and hard-sample knowledge bank are complementary to each other, working together to help build a more comprehensive and representative global memorization. Table 11: Comparison of different update ways of global knowledge bank over semantic segmentation task Cityscapes with SEEM-Tiny. Method local knowledge bank hard-sample knowledge bank global& hard-sample knowledge banks mIoU 44.2 43.8 44.7 D More Comparisons with Memory-based Learning Methods We provide more comparisons with existing memory-based learning methods [31, 88, 89, 28]. Our HisTPT differs in two major aspects: Memory Types - HisTPT designs three types of knowledge banks for capturing and storing both fresh and representative features; Memory Retrieval - HisTPT designs an Adaptive Knowledge Retrieval Mechanism for retrieving the memorized information adaptively for each test image. Due to the very different designs, HisTPT outperforms [31, 88, 89, 28] clearly as shown in Table 12. 16 Table 12: Comparison with existing memory-based learning methods over Cityscapes semantic segmentation task on SEEM-Tiny. mIoU is reported. Method T3A [31] TAST [88] RoTTA [89] FAU [28] HisTPT mIoU 41.8 42.0 41.9 42.2 44.7 E Pseudo Codes of HisTPT We provide the pseudo codes of the proposed historical test-time prompt tuning (HisTPT), as shown in Algorithm 1. We initialize the three knowledge banks with the features of the first test sample and then gradually update them as in Lines 3-7 along the test-time prompt tuning process. Note that, for the first test sample, we skip the prediction regularization in Line 10 and optimize the tokens for it with the vanilla self-training objective since the knowledge banks have not been constructed at that time. Algorithm 1 Historical Test-Time Prompt Tuning. Require: Online optimized text tokens t, a pre-trained vision foundation model F = {F I, F T }, a continuous flow of test samples Xtest = {xn}N n=1 and their possible belonged class names Ytest = {yc}C c=1 1: Initialization: Initialize t as t0 2: for n = 1 to N do 3: Knowledge bank construction with xn−1 and tn−1: 4: Encode xn−1: un−1 = F T (tn−1; Ytest) 5: Update local knowledge bank: dequeue old feature ulocal and enqueue un−1 6: Update hard-sample knowledge bank: dequeue old feature uhard and enqueue new feature selected by Eq. 3 7: Update global knowledge bank: generate new category-wise feature prototype using ulocal and uhard, and update the global knowledge bank by Eq. 4 8: Prompt optimization for xn with the constructed knowledge banks: 9: Generate prediction pn for xn with tn−1 via Eq. 1 10: Generate the regularized prediction ˆpn by adaptively retrieving the memorized knowledge as in Eqs. 5-7 11: Optimize the text token for xn, i.e., tn ←tn−1, by Eq. 8 12: end for F Quantification of the Forgetting Mitigation Ability of HisTPT Following prior study [29], we measure the forgetting by randomly selecting one of the five datasets in Table 6 as the reference domain and perform continual adaptation toward the other four datasets. During the continuous adaptation process, we evaluate HisTPT’s ability of preserving the knowledge of vision foundation models by measuring its performance on the reference domain. As shown in the Figure 4, HisTPT shows less performance degradation on the reference domain consistently, demonstrating its effectiveness in preserving the knowledge of vision foundation models and mitigating forgetting during the adaptation process. G Further Analysis of the Three Knowledge Banks We analyse the three knowledge banks by visualizing their stored features along the test-time adaptation process. Three points can be drawn as illustrated in Figure 5: 1) the global prototypes exhibit slow and gradual shift from the initial feature prototypes, preserving the knowledge of pre-trained vision foundation models and facilitating stable test-time adaptation; 2) the features in the local knowledge bank change rapidly, validating their effectiveness in capturing fresh and up-to-date distribution changes along the test-time adaptation process; 3) most features in the hard-sample knowledge bank lies around inter-category boundary, indicating their effectiveness in capturing difficult and rare corner cases along the tuning process. With the three types of complementary knowledge, HisTPT enables adaptive regularization for the prediction of current test samples. 17 Fog Night Rain Snow Fog Night Rain Snow Adaptation Order ( >) 34 35 36 37 38 39 40 mIoU (%) Performance on Normal Weather (Citycsapes) SEEM-T TPT HisTPT(Ours) Normal Night Rain Snow Normal Night Rain Snow Adaptation Order ( >) 31 32 33 34 35 36 mIoU (%) Performance on Foggy Weather (ACDC-Fog) SEEM-T TPT HisTPT(Ours) Normal Fog Rain Snow Normal Fog Rain Snow Adaptation Order ( >) 17 18 19 20 21 22 mIoU (%) Performance on Night Weather (ACDC-Night) SEEM-T TPT HisTPT(Ours) Normal Fog Night Snow Normal Fog Night Snow Adaptation Order ( >) 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 mIoU (%) Performance on Rainy Weather (ACDC-Rain) SEEM-T TPT HisTPT(Ours) Figure 4: Comparison of preventing forgetting on continual test-time adaptation task with SEEMTiny. For each experiment, one dataset is selected as the reference domain, and then we perform the continual adaptation on the other datasets. We record the performance change on the reference domain for measuring the forgetting during the continual adaptation process. Our HisTPT shows clearly less performance degradation on the reference domain, demonstrating the effectiveness of HisTPT in mitigating forgetting during the adaptation process. Category Truck: (a) 100-th test sample (a) 200-th test sample (a) 400-th test sample Local prototype Samples of local knowledge bank Hard-sample prototype Samples of hard-sample knowledge bank Global prototype Initial prototype Category Car: Local prototype Samples of local knowledge bank Hard-sample prototype Samples of hard-sample knowledge bank Global prototype Initial prototype Figure 5: T-SNE visualization of the features stored in each knowledge bank with Cityscapes semantic segmentation task on SEEM-Tiny. For clear illustration, we select two categories (i.e., car and truck) for visualization. T-SNE visualization shows that 1) global prototype shifts slowly from the initial prototype, preserving the original knowledge of pre-trained vision foundation models; 2) local knowledge bank updates rapidly, capturing fresh information and reflecting real-time distribution changes and 3) hard-sample knowledge bank captures challenging and rare cases situated near decision boundaries. H Analysis with Error Bars In experiments, we observe negligible variance on the results between multiple random runs. Nevertheless, we provide the error bar with 5 random runs to analyze the proposed HisTPT on semantic segmentation task with SEEM-Tiny, image classification task with CLIP-RN50 and object detection with SEEM-Tiny, respectively. From Table 13, we can observe that our proposed HisTPT performs well consistently over multiple random runs. Table 13: Analysis of our proposed HisTPT with error bars. Method Semantic segmentation task (Mean) Image classification task (Mean) Object detection task (Mean) HisTPT 33.5 ±0.2 61.2 ±0.1 30.2 ±0.2 18 I Qualitative Results We present qualitative illustrations and comparisons over semantic segmentation task on Cityscapes. As shown in Fig. 6, HisTPT yields the best segmentation consistently which is well aligned with the quantitative results. Original Image SEEM-Tiny [3] TPT [7] HisTPT(Ours) Ground Truth Figure 6: Qualitative comparison of HisTPT with the baseline model (SEEM-Tiny) [3] and TPT [7] over semantic segmentation task on Cityscapes. J Broader Impacts and Limitations Broader Impacts. This work explores a novel pipeline for transfer learning with vision foundation models, namely, test-time prompt tuning. Our proposed method offers great advantages by eliminating the need for labelled task-specific data and allowing learning prompts from test samples on-the-fly. It thus makes a very valuable contribution to the computer vision research community by providing a novel and efficient transfer learning pipeline. The feature of requiring no labelled task-specific training data enables efficient adoption of vision foundation models in various downstream tasks, broadening the applicability of vision foundation models significantly. Limitations. As discussed in Section 4.2 of the main text, HisTPT offers a general framework that can perform well across different computer vision tasks. It enables effective test-time prompt tuning with the generic text prompt that is universally applicable across all vision foundation models (VFMs), thus avoiding the complexity of task-specific designs in VFM adaptation. At the other end, task-specific designs allow incorporating task-relevant knowledge which often helps improve performance. For instance, the incorporation of specific visual prompts, such as points and bounding boxes, in segmentation or detection foundation models often lead to more precise segmentation masks and bounding boxes. We will investigate how to incorporate task-specific prompt tuning in our future work. 19 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The abstract and introduction accurately describe the paper’s contributions and scope. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,417	Boosting Transferability and Discriminability for Time Series Domain Adaptation Mingyang Liu1, Xinyang Chen1B, Yang Shu2B, Xiucheng Li1B, Weili Guan3, Liqiang Nie1 1School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen) 2School of Data Science and Engineering, East China Normal University 3School of Electronics and Information Engineering, Harbin Institute of Technology (Shenzhen) mingyangliu1024@gmail.com, yshu@dase.ecnu.edu.cn {chenxinyang,lixiucheng,guanweili,nieliqiang}@hit.edu.cn Abstract Unsupervised domain adaptation excels in transferring knowledge from a labeled source domain to an unlabeled target domain, playing a critical role in time series applications. Existing time series domain adaptation methods either ignore frequency features or treat temporal and frequency features equally, which makes it challenging to fully exploit the advantages of both types of features. In this paper, we delve into transferability and discriminability, two crucial properties in transferable representation learning. It’s insightful to note that frequency features are more discriminative within a specific domain, while temporal features show better transferability across domains. Based on the findings, we propose Adversarial CO-learning Networks (ACON), to enhance transferable representation learning through a collaborative learning manner in three aspects: (1) Considering the multi-periodicity in time series, multi-period frequency feature learning is proposed to enhance the discriminability of frequency features; (2) Temporalfrequency domain mutual learning is proposed to enhance the discriminability of temporal features in the source domain and improve the transferability of frequency features in the target domain; (3) Domain adversarial learning is conducted in the correlation subspaces of temporal-frequency features instead of original feature spaces to further enhance the transferability of both features. Extensive experiments conducted on a wide range of time series datasets and five common applications demonstrate the state-of-the-art performance of ACON. Code is available at https://github.com/mingyangliu1024/ACON. 1 Introduction Time series classification has achieved significant success in the deep learning era by leveraging discriminative features learned from extensive labeled data [18]. However, the presence of distribution shift may arise when deploying the model, potentially impeding the generalization ability of deep models [31]. Unsupervised domain adaptation [13], offering the potential to transfer knowledge from a labeled source domain to an unlabeled target domain, emerges as a promising solution. Existing domain adaptation methods tailored for time series primarily focus on learning domaininvariant temporal features [30, 41, 29], yielding promising results. Recently, the significance of frequency features for enhancing domain-invariant representation has also been recognized [16]. However, frequency features and temporal features are treated equally, and their distinct properties are overlooked, leading to the inability to fully leverage both types of features to boost transfer learning. In this paper, we analyze the two most important properties of features in transfer learning: transferability and discriminability, to investigate the characteristics of the frequency features and temporal 38th Conference on Neural Information Processing Systems (NeurIPS 2024). features. We find that under the premise of adopting advanced backbones in state-of-the-art works [31, 16], frequency features are more discriminative within a specific domain, while temporal features show better transferability across domains. Based on the findings, we propose Adversarial CO-learning Networks (ACON) to maximize the potential of temporal features and frequency features in terms of both transferability and discriminability in a collaborative learning manner. Firstly, to fully leverage the properties of multi-periodicity in time series, we propose multi-period frequency feature learning to further enhance the discriminability of frequency features. Secondly, we propose temporal-frequency domain mutual learning to enhance the discriminability of temporal features in the source domain and improve the transferability of frequency features in the target domain. Specifically, to harness the potent discriminability of frequency features within the domain, we enable the transfer of knowledge from frequency features to temporal features within the source domain via knowledge distillation. To leverage the strong transferability of temporal features across domains, we facilitate the transfer of knowledge from temporal features to frequency features in the target domain through knowledge distillation. Thirdly, we propose to learn transferable representations via domain adversarial learning in temporal-frequency correlation subspace instead of the original temporal feature space. The temporal-frequency correlation subspace not only possesses the properties of the original temporal feature space and original frequency feature space but also incorporates the correlation between the two types of features. Learning transferable representations in the temporal-frequency correlation subspace can further enhance the transferability of features. Our main contributions can be summarized as follows: • We uncover the characteristics wherein temporal features and frequency features cannot be equally treated in transfer learning. Specifically, we observe that frequency features are more discriminative within a specific domain, while temporal features show better transferability across domains through empirical findings. • We design ACON, which enhances UDA in three key aspects: a multi-period feature learning module to enhance the discriminability of frequency features, a temporal-frequency domain mutual learning module to enhance the discriminability of temporal features in the source domain and improve the transferability of frequency features in the target domain, and a domain adversarial learning module in temporal-frequency correlation subspace to further enhance transferability of features. • Experiments conducted on a wide range of time series datasets and five common applications verify the effectiveness of ACON. 2 Related Work General Unsupervised Domain Adaptation Methods Unsupervised domain adaptation leverages the labeled source domain to predict the labels of a different but related, unlabeled target domain. It finds wide applications in computer vision [46, 15, 8] and natural language processing [40, 39, 44]. Existing UDA methods can be classified into three categories: (1) Methods based on adversarial training aim to learn domain-invariant representations via the game between the feature extractor and the domain discriminator. Widely used methods include DANN [13], CDAN [26] and DIRT-T [34]. (2) Methods based on statistical divergence aim to reduce the domain discrepancy by minimizing domain discrepancy in a latent feature space. Widely used methods include DAN [25], DeepCoral [36] and HoMM [5]. (3) Methods based on self-training produce pseudo-labels on unlabeled data and use confident pseudo-labels together with the labeled data to train the model. Widely used methods include PFAN[6], CST [22] and AdaMatch [2]. However, these methods are generally designed and do not fully leverage the properties of time series. Although these methods can be applied to time series through tailored feature extractors, they often obtain suboptimal performance and UDA algorithm specially designed for time series is needed. Unsupervised Domain Adaptation for Time Series To date, a few methods have been tailored to unsupervised domain adaptation for time series data. VRADA [30] is the first UDA method for multivariate time series that uses adversarial learning for reducing domain discrepancy. In VRADA, a variational recurrent neural network (VRNN) [10] is trained in an adversarial way to learn domain-invariant temporal features. CoDATS [41] builds upon VRADA but uses a convolutional neural network for the feature extractor, proposing a solution for multi-source domain adaptation in 2 time series classification. SASA [3] adopts LSTM [33] as feature extractors to capture the domaininvariant association, and aligns sparse associative structure between source and target domain via the minimization of maximum mean discrepancy (MMD) [38]. AdvSKM [23] modifies MMD to make it more suitable for time series data. CLUDA [29] learns contextual representation via contrastive learning, and aligns features between source and target domain via adversarial training. RAINCOAT [16] is the first to introduce frequency features into domain adaptation, aligning temporal features and frequency features respectively via Sinkhorn divergence. Research gap In general, in terms of representation learning, most methods only focus on the temporal domain or assume that the temporal domain and the frequency domain are independent of each other, hindering the full utilization of two types of features. In terms of feature adaptation, existing works only focus on aligning temporal features or adopting simple statistical divergence to align frequency features, ignoring the different properties of the temporal features and frequency features in transfer learning. In terms of evaluation, the existing evaluations are conducted on several datasets of limited scale in a few specific tasks, and more general evaluations are needed. 3 Transferability and Discriminability in Time Series 3.1 Problem setup In this paper, we study the UDA problem for time series classification. In time series classification problem, the model receives a set of n labeled samples {(xi, yi)}n i=1, where i-th sample xi ∈RC×T contains observation of C variates over T time steps. We allow for both univariate and multivariate time series. In UDA setup, we are given ns labeled samples from a source domain ˆP = {(xs i, ys i )}ns i=1 and nt unlabeled samples from a target domain ˆQ = {(xt i)}nt i=1, which are sampled from different distributions P and Q. Superscripts s and t are adopted to distinguish the source domain and the target domain. UDA for time series aims to learn a time series classification model with labeled source data ˆP and unlabeled target data ˆQ, which can make accurate predictions on the target domain. In addition to the source domain and target domain in UDA, time series naturally can be represented in the temporal domain and frequency domain. By Fast Fourier Transform (FFT), the raw time series input xi in the temporal domain can be transformed to corresponding frequency input vi in the frequency domain: vi = FFT (xi) , (1) where the complex variable vi ∈CC×⌊T 2 ⌋contains observation of C variates over ⌊T 2 ⌋different frequencies. Due to the conjugacy of frequency domain, we only consider the frequencies within {1, ..., ⌊T 2 ⌋}. 3.2 Discriminability of frequency feature + !** $ ! #%'$ #"&)% ( !%& !%& !' (a) Temporal data vs. Frequency data % $!& ""#"$#" ! ! $& !' (b) Source classification !' &#( $$%"!$& %$"!#% "&#"! ( "&#"! ( !' (c) Target classification Figure 1: Discriminability of frequency feature: (a) The Electroencephalography (EEG) signal and corresponding frequency data of two classes in the CAP dataset: Wake and Rapid Eye Movement (REM). (b) Classification on the source domain: Temporal domain vs. Frequency domain. (c) Source-only and DANN: Temporal domain vs. Frequency domain. As Figure 1(a) presented, compared to the uniform distribution of temporal data for different classes, the frequency data for different classes shows distinct differences in the dominant frequencies and peaks, which holds more discriminative information. To further investigate the discriminability of 3 frequency features, we perform the single data domain classification task in the frequency domain and temporal domain respectively on all five data domains of the CAP [37, 14] dataset. In order to minimize the impact of specific model structures, we adopt 3-layer 1D-CNN, a generic structure as the temporal feature extractor, and 1-layer linear as the frequency feature extractor, which have both widely validated for their effectiveness in existing time series analysis methods [23, 16, 42, 43]. We only retain the low-frequency data to ensure that the temporal feature extractor and the frequency feature extractor have comparable parameter quantities. For the classifiers, we uniformly use 1-layer linear. As Figure 1(b) shown, with a simple feature extractor, the frequency classification outperforms the temporal classification, demonstrating that the frequency features have better discriminability. More analysis results on different datasets are included in Appendix C.1. 3.3 Transferability of temporal feature Another key criterion that characterizes the performance of domain adaptation is transferability [7]. Transferability indicates the ability to learn invariant features across domains. Since the frequency features have better discriminability within the source domain, it is natural to raise the question: Will the frequency features also have better discriminability in the target domain? We investigate this problem starting with the comparison of four methods: (1) Source-only-F, a model trained in the frequency domain without UDA. (2) Source-only-T, a model trained in the temporal domain without UDA. (3) DANN-F, a model aligning the source features and the target features in the frequency domain via DANN. (4) DANN-T. a model aligning the source features and the target features in the temporal domain via DANN. Figure 1(c) shows the accuracy in the target domains of four source-target domain pairs from the CAP dataset. Compared with Figure 1(b), the frequency classification, which has better discriminability performance in the source domain, actually slightly underperforms in the target domain. It indicates that better discriminability in the source domain does not necessarily imply better discriminability in the target domain. Compared with Source-only methods, the gap between DANN-F and DANN-T is further exacerbated. This suggests that the temporal feature extractor more easily learns domain-invariant features. More analysis results on different datasets are included in Appendix C.2. The above analysis reveals two insights for time series domain adaptation: With better discriminability but worse transferability, domain adaptation in the frequency domain obtains suboptimal performance; while with better transferability, domain adaptation in the temporal domain has the potential to achieve superior performance under the guidance of more discriminative information. 4 Approach Based on the above observations, our motivation is to simultaneously leverage the strong discriminability of frequency features and the strong transferability of temporal features to enhance domain adaptation. This inspires us to learn domain-invariant temporal and frequency features in a collaborative learning manner. Figure 2 illustrates the overall structure of our Adversarial CO-learning Networks (ACON). To avoid confusion, subscripts T and F are adopted to distinguish the temporal domain and the frequency domain. Specifically, in the temporal domain, we have a temporal feature extractor with temporal input f = ψT (x) and a temporal classifier ˆyT = gT (f); while in the frequency domain, we have a frequency feature extractor with frequency input z = ψF (v) and a frequency classifier ˆyF = gF (z). Additionally, we have a domain discriminator gD, which is trained to distinguish the source feature and the target feature. In the following, we will introduce three main contributions in ACON: multi-period frequency feature learning in Section 4.1, temporal-frequency domain mutual learning in Section 4.2, and domain adversarial learning in temporal-frequency correlation subspace in Section 4.3. 4.1 Multi-period frequency feature learning The real-world time series usually present multi-periodicity, which is reflected in the frequency domain as the presence of a few dominant frequencies with significantly larger amplitudes. Data from different periods can have different discriminative patterns. Based on this, before performing FFT, we segment the raw time series according to the top-k significant periods, enhancing the 4 ˆys T B QomQtv1tVJaWV1bXqu1jc2t7R1zd68r4pQj3ExjXk/gAJTwnBHEklxP+EYRgHFvWByVfi9B8wFidmdzBLsRXDESEgQlFryzQN3DKVyIyjHQaiyPL9XIvfVde6bdbthT2EtEqckdVCi7Ztf7jBGaYSZRBQKMXDsRHoKckQxXnNTQVOIJrAER5oymCEhaemH+TWsVaGVhzXUxaU/X3hIKREFkU6M7iVDHvFeJ/3iCV4YWnCEtSiRmaLQpTasnYKuKwhoRjJGmCUSc6FstNIYcIqlDq+kQnPmXF0n3tOE0G83 bs3rsoyjCg7BETgBDjgHLXAD2qADEHgEz+AVvBlPxovxbnzMWitGObMP/sD4/AHtz5fc</latexit>ˆys F LCT LCF gT b QhrLZbtulm03YnQgl9Dd48aCIV3+QN/+N2zYHbX0w8Hhvhpl5YSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoaeJUM+6zWMa6HVLDpVDcR4GStxPNaRK3grHtzO/9cS1EbF6xEnCg4gOlRgIRtFK/rCX3U175Ypbdecgq8TLSQVyNHrlr24/ZmnEFTJjel4boJBRjU KJvm01E0NTygb0yHvWKpoxE2QzY+dkjOr9Mkg1rYUkrn6eyKjkTGTKLSdEcWRWfZm4n9eJ8XBdZAJlaTIFVsGqSYExmn5O+0JyhnFhCmRb2VsJGVFOGNp+SDcFbfnmVNC+qXq1ae7is1G/yOIpwAqdwDh5cQR3uoQE+MBDwDK/w5ijnxXl3PhatBSefOY/cD5/ ANm5jrs=</latexit>gF LD D T F L OZNgmt9aZWl5ZXWtul7b2Nza3tF397oySgShHRLxSPQ9LClnIe0A07saA48DjteZOrwu89UCFZFN5BGlMnwKOQ+YxgUJKrH9hjDJkdYBh7fpbm+X0GuZtd565eNxvmFMYisUpSRyXarv5lDyOSBDQEwrGUA8uMwcmwAEY4zWt2ImMyQSP6EDREAdUOtn0g9w4VsrQ8COhKgRjqv6eyHAgZRp4qrM4Vc57hfifN0jAv3AyFsYJ0JDMFvkJNyAyijiMIROUAE8VwUQwdatBxlhgAiq0mgrBmn95kXRPG1az0bw 9q7cuyziq6BAdoRNkoXPUQjeojTqIoEf0jF7Rm/akvWjv2sestaKVM/voD7TPH+9Yl90=</latexit> ˆyt F J M81urKyurW9UN2tb2zu7e/r+QU9GiSC0SyIeiYGHJeUspF1gwOkgFhQHqd9b3pT+P0HKiSLwg6kMXUCPA6ZzwgGJbn6kT3BkNkBhonZ2me32eQu1knd/W62TBnMJaJVZI6KtF29S97FJEkoCEQjqUcWmYMToYFMJpXrMTSWNMpnhMh4qGOKDSyWYf5MapUkaGHwlVIRgz9fdEhgMp08BTncWpctErxP+8YQL+lZOxME6AhmS+yE+4AZFRxGMmKAEeKoIJoKpWw0ywQITUKHVAjW4svLpHfesJqN5t1FvXVdxlFx +gEnSELXaIWukVt1EUEPaJn9IretCftRXvXPuatFa2cOUR/oH3+AStl+s=</latexit> ˆyt T FFT Average Period 1 Period k DKL(ˆyt T \|\|ˆyt F ) DKL(ˆys F \|\|ˆys T ) . . . . . . . . . . . . . . . . . . FFT Average . . . . . . . . . . . . FFT Figure 2: The architecture of ACON. ACON models temporal data (blue) and frequency data (green) simultaneously. Left part: Segment raw frequency data by period to capture different discriminative patterns. Middle part: Align distributions in temporal-frequency correlation subspace via adversarial training. Right part: Mutual learning between the temporal domain and frequency domain. discriminability of the frequency domain. Additionally, by period-based segmentation, the noises brought by meaningless high frequencies are effectively filtered out [4, 45]. To capture the overall multi-periodicity, before training, we randomly sample mini-batches from the training set to perform FFT and select the frequencies with the top-k amplitudes {f1, . . . , fk}. Given the frequency fj, the corresponding period is pj = ⌈T fj ⌉. For each selected period pj in {p1, . . . , pk} and frequency fj in the corresponding {f1, . . . , fk}, we perform the following transform on input xi: Xj i = Reshapepj(xi), j ∈{1, . . . , k}, vj i = Avg FFT Xj i . (2) where Xj i ∈RC×fj×pj, vj i ∈CC×⌊ pj 2 ⌋is averaged from fj dimensions by Avg(·). In other words, we perform FFT on each segment obtained by segmenting xi with period pj, and average the FFT results across segments to obtain the distribution vj i over the frequencies within {1, . . . , ⌊pj 2 ⌋}. In this way, we obtain the overall frequency pattern for each period. To keep the discriminative patterns derived from different periods, we concatenate the different vj i , obtaining vi as the frequency input corresponding to the temporal input xi: vi = v1 i ⊕... ⊕vk i , j ∈{1, . . . , k}. (3) We extend the source sample set ˆP and the target sample set ˆQ to the frequency domain: ˆP = {(xs i, vs i , ys i )}ns i=1 and ˆQ = {(xt i, vt i)}nt i=1. To learn features in both real part and imaginary part of complex frequency data, we adopt a complex-valued linear layer as the frequency feature extractor ψF . Since the phase generally does not provide strong discriminative information, we only retain the amplitudes of each frequency to construct the frequency domain feature zi: zi = Amp (ψF (vi)) , (4) where Amp(·) denotes the calculation of amplitude values. For multivariate time series, we convert vi into a single-channel vector by concatenating across different variates. 4.2 Temporal-frequency domain mutual learning Discriminability and transferability are two key criteria that characterize the goodness of feature representations to enable domain adaptation. In Section 3, we reveal that the frequency features are more discriminative within the source domain, while the temporal features are more transferable across domains. Based on this discovery, we propose temporal-frequency domain mutual learning, aiming to leverage the respective advantages of the temporal domain and frequency domain. The essence of domain mutual learning relies on how to transfer knowledge between the temporal domain and frequency domain. Inspired by model distillation, where the knowledge is transferred by matching the predictions between the teacher and student via the Kullback Leibler (KL) divergence [17], we focus mutual learning on the alignment between the temporal predictions and the frequency predictions. The KL divergence between two predictions p1 and p2 is formulated as: DKL(p1\|\|p2) = C X m=1 pm 1 logpm 1 pm 2 . (5) 5 The KL divergence is asymmetric, that is, DKL(p1\|\|p2) emphasizes aligning p2 to p1, while DKL(p2\|\|p1) emphasizes aligning p1 to p2. Based on the asymmetry, we use different alignment strategies in the source domain and target domain. Specifically, in the source domain, the frequency model serves as a more discriminative teacher, helping the temporal model make more accurate predictions; conversely, in the target domain, the temporal model acts as a more transferable teacher, assisting the frequency model in learning domain-invariant representations. We achieve temporalfrequency domain mutual learning by minimizing the KL Divergence. Formally, domain mutual learning is formulated as: LMs(ψT , gT ) = E(xs i ,vs i )∼ˆ P [DKL(ˆys F \|\|ˆys T )], LMt(ψF , gF ) = E(xt i,vt i)∼ˆ Q[DKL(ˆyt T \|\|ˆyt F )], (6) where ˆys F and ˆys T refer to the frequency prediction and temporal prediction in the source domain respectively; while ˆyt F and ˆyt T refer to the frequency prediction and temporal prediction in the target domain respectively. By aligning ˆys T to ˆys F , the training of the temporal feature extractor and classifier is guided with more discriminative information; by aligning ˆyt F to ˆyt T , the transferable knowledge contained in the temporal features is transferred to frequency domain. 4.3 Domain adversarial learning in temporal-frequency correlation subspace Domain adversarial learning [13] is one of the most popular transferable representation learning methods, and it can be employed to learn transferable representation in time series. The key to the effectiveness of the method lies in how to fully utilize two types of features to learn transferable representations. Given time series in temporal domain and frequency domain, domain adversarial learning can be formulated as a minimax optimization problem with three competitive loss terms: (a) LCT on the temporal feature extractor ψT and classifier gT , which is minimized to guarantee lower source risk of the temporal classifier; (b) LCF on the frequency feature extractor ψF and classifier gF , which is minimized to guarantee lower source risk of the frequency classifier; (c) LD on the temporal feature extractor ψT , the frequency feature extractor ψF and the domain discriminator gD, which is minimized over gD but maximized over ψT and ψF : LCT (ψT , gT ) = E(xs i ,ys i )∼ˆ P [ℓ(gT (ψT (xs i)) , ys i )], LCF (ψF , gF ) = E(vs i ,ys i )∼ˆ P [ℓ(gF (ψF (vs i )) , ys i )], LD(ψT , ψF , gD) = −E(xs i ,vs i )∼ˆ P log[gD (ψT (xs i) , ψF (vs i ))] −E(xt i,vt i)∼ˆ Qlog[1 −gD ψT xt i , ψF vt i ], (7) where ℓdenotes cross-entropy loss. Different from standard domain adversarial learning, where there is only one type of feature, domain adversarial learning in time series needs to consider the temporal features and frequency features simultaneously. A simple strategy is to concatenate the temporal feature f and the frequency feature z. However, with the concatenation strategy, the adversarial game between the domain discriminator and the feature extractors can be viewed as two independent components: the game between gD and ψT and the game between gD and ψF . With the worse transferability, z provides gD with rich domain-label relevant information. In this case, gD only needs to focus on the game with ψF , ignoring the domain adversarial learning in the temporal domain. To achieve co-alignment in the temporal domain and frequency domain, we propose domain adversarial learning in temporal-frequency correlation subspace. The temporal-frequency correlation subspace not only possesses statistical characteristics of the original temporal feature subspace and original frequency feature subspace but also reflects the correlation between temporal features and frequency features. Reducing the discrepancy of the temporal-frequency correlation subspace not only reduces the discrepancy in the cross-domain temporal and frequency features but also decreases the differences in cross-domain temporal-frequency correlations. Formally, the vectors in temporal-frequency correlation subspace can be calculated as the outer product ⊗between the temporal feature f and the frequency feature z: f ⊗z = [z [1] · f, z [2] · f, . . . , z [l] · f] , (8) where z ∈R1×l. By adjusting the order of dimensions, z ⊗f is equivalent to f ⊗z. Considering the sparsity of the frequency domain and the modeling of long-length time series, the direct outer product 6 leads to dimension explosion and the sparsity in temporal-frequency feature subspace. We address the problem by performing average pooling over z. Average pooling, which calculates the average value for the amplitudes of neighboring frequencies, yields dense frequency features with smaller dimensions. With outer product and average pooling, the adversarial loss LD is formulated as: h(xi, vi) =ψT (xi) ⊗P (ψF (vi)) , LD(ψT , ψF , gD) = −E(xs i ,vs i )∼ˆ P log[gD (h (xs i, vs i ))] −E(xt i,vt i)∼ˆ Qlog[1 −gD h xt i, vt i ], (9) where h(·) denotes the mapping from the inputs of the overall model to the inputs of the domain discriminator, and P (·) denotes the calculation of average pooling. 4.4 Overview During alignment, our method trains the temporal feature extractor ψT and classifier gT by minimizing the loss LCT and trains the frequency feature extractor ψF and classifier gF by minimizing the loss LCF using the source sample set ˆP. Additionally, our method promotes mutual learning between the temporal domain and frequency domain by minimizing the loss LMs on the source domain and the loss LMt on the target domain. Meanwhile, our method aligns distributions of the source domain and target domain in the temporal-frequency correlation subspace. With two gradient reversal layers between the two feature extractors and the domain discriminator, the adversarial training is achieved by minimizing the loss LD. To simplify notation, we denote θF as parameters containing ψF and gF , and θT is parameters containing ψT and gT . The minimax optimization problem is formulated as: min θF ,θT LCF (θF ) + LCT (θT ) + LMs(θT ) + LMt(θF ) −LD(ψT , ψF , gD), min gD LD(ψT , ψF , gD). (10) 5 Experiments 5.1 Setup Datasets We conduct extensive experiments using a wide range of time series datasets. (1) Experiments using benchmark datasets in sensor-based human activity recognition (HAR) task: UCIHAR [1], HHAR [35] and WISDM[20]. For HHAR, we first split domains from the perspective of participants, denoted as HHAR-P [16, 31] dataset. Then, we split domains from the perspective of devices, denoted as HHAR-D [12] datasets. (2) Experiments using the benchmark dataset in sleep stage classification (SSC) task: CAP [14, 37]. (3) Experiments using EMG [24, 27] dataset in gesture recognition (GR) task. (4) Experiments using PCL [32, 9, 21, 19] dataset in motor imagery classification (MIC) task. (5) Experiments using FD [31] dataset in machine fault diagnosis (MFD) task. For each dataset, following the existing DA methods on time series [2, 16], we randomly sample 10 source-target domain pairs for evaluation. If the dataset has less than 10 pairs, we evaluate all available domain pairs. Further details, processing and domain splits are included in Appendix A. Baselines (1) We report the performance of a model without UDA (Source-only) in the temporal domain to show the overall contribution of UDA methods. (2) We implement the following stateof-the-art baselines for UDA of time series data: CODATS[41], AdvSKM[23], CLUDA[29] and RAINCOAT[16]. (3) We additionally implement general unsupervised DA methods: CDAN [26], DeepCoral [36], AdaMatch [2], HoMM [5] and DIRT-T [34]. Evaluation We report accuracy and Macro-F1 Score calculated using target test datasets. Accuracy is computed by dividing the number of correctly classified samples by the total number of samples. Macro-F1 Score is calculated using the unweighted mean of all the per-class F1 scores. Implementation We adopt the implementation of AdaTime [31] as a benchmarking suite for domain adaptation on time series data, using 1D-CNN as the temporal feature extractor and 1-lyer complex-valued linear as the frequency feature extractor. We use the same feature extractor across all algorithms, ensuring a fair comparison. In all experiments, we use the prediction of the temporal classifier to calculate accuracy and Macro-F1 Score. More experimental details are provided in Appendix B. 7 Table 1: Average Accuracy (%) on Eight Datasets and Five Applications for UDA. Task GR MFD MI HAR SSC Dataset EMG FD PCL UCIHAR HHAR-P WISDM HHAR-D CAP Source-only 76.24 70.04 60.95 75.12 54.25 65.78 46.82 55.86 CDAN 79.89 90.56 63.36 85.78 68.73 70.05 54.94 67.33 DeepCoral 78.71 84.10 63.51 82.01 68.03 70.80 52.55 64.88 AdaMatch 80.69 82.11 57.78 76.07 65.91 69.79 53.84 65.12 HoMM 78.74 85.72 63.83 80.99 65.01 67.26 52.33 65.67 DIRT-T 79.27 88.08 61.02 83.26 64.99 69.62 56.14 70.42 CLUDA 75.62 84.99 54.69 85.53 68.73 67.04 53.84 65.79 AdvSKM 78.81 83.37 63.58 83.26 66.41 66.97 52.80 64.39 CoDATS 80.60 87.20 64.18 75.54 68.71 70.66 56.27 68.23 RAINCOAT 79.93 86.75 58.99 94.43 74.21 76.60 49.07 69.13 Ours 82.91 91.74 65.02 97.02 81.74 84.80 65.04 74.08 Improve(%) 2.75 1.30 1.31 2.74 10.15 10.70 15.85 5.20 5.2 Results Table 1 shows the average accuracy of each method on all datasets and tasks. Overall, our method has won 5 out of 5 tasks and 8 out of 8 datasets (2 metrics). Specifically, our method improves accuracy by 2.75% on GR task, 5.20% on SSC task, 1.31% on MI task, 9.86% on HAR task and 1.30% on MFD task over the advanced baseline on each dataset respectively. Due to the limited pages, we report the results for selected source-target domain pairs with metric accuracy on the representative datasets EMG (GR task), CAP (SSC task) and HHAR-P (HAR task). More accuracy results are given in Table 11-15. Average macro-f1 score results are given in Table 16. Full macro-f1 score results are given in Table 17-24. Table 2: Accuracy (%) on CAP for unsupervised domain adaptation. Method 0→1 0→3 0→4 1→0 1→4 2→3 3→0 3→1 4→1 4→3 Avg Source-only 42.44 75.75 66.09 57.98 63.26 50.75 69.47 26.88 33.90 72.09 55.86 CDAN 68.04 77.47 72.84 62.66 67.29 53.93 72.64 63.58 58.17 76.63 67.33 DeepCoral 67.44 77.34 72.33 59.18 67.71 58.09 70.66 53.02 49.12 73.94 64.88 AdaMatch 60.28 77.67 75.55 68.90 63.17 37.33 73.52 59.29 60.26 75.22 65.12 HoMM 69.89 77.11 72.27 60.36 67.89 57.96 71.58 57.61 47.52 74.49 65.67 DIRT-T 72.16 79.21 76.04 64.18 68.69 57.75 75.47 69.91 64.59 76.16 70.42 CLUDA 67.67 75.77 58.82 70.31 70.23 53.94 74.91 53.62 58.30 74.29 65.79 AdvSKM 63.88 77.04 72.17 60.61 66.09 58.00 70.93 55.26 46.64 73.32 64.39 CoDATS 70.54 78.64 70.40 67.89 72.05 57.32 76.08 53.79 60.43 75.17 68.23 RAINCOAT 70.58 72.80 73.47 65.34 69.62 56.08 71.34 70.86 70.47 70.70 69.13 Ours 75.32 80.14 76.58 70.68 73.25 57.48 77.75 75.17 75.67 78.74 74.08 Table 2 presents the results on CAP dataset. CAP contains over 40,000 samples of 3000 time steps, so adaptation on it is more challenging. Our method outperforms general DA and time series DA methods on 9 out of 10 source-target domain pairs, achieving an average improvement of 5.20% over the advanced baseline, DIRT-T. Table 3 presents the results on HHAR-P dataset. Our method significantly outperforms RAINCOAT, the state-of-the-art DA method for time series, by 10.15%. Table 3: Accuracy (%) on HHAR-P for unsupervised domain adaptation. Method 0→2 1→6 2→4 4→0 4→5 5→1 5→2 7→2 7→5 8→4 Avg Source-only 64.51 70.63 45.42 32.81 78.32 90.63 25.67 32.37 39.26 62.92 54.25 CDAN 76.19 92.57 52.57 29.09 97.27 96.16 35.04 37.05 75.26 96.11 68.73 DeepCoral 84.23 90.14 47.08 28.13 90.49 89.91 38.39 34.45 55.73 76.88 68.03 AdaMatch 84.78 92.31 54.50 36.45 78.45 94.20 41.96 37.65 63.80 64.69 65.91 HoMM 75.67 90.79 52.83 36.61 87.66 90.78 37.23 37.32 61.29 79.88 65.01 DIRT-T 77.83 88.54 50.69 32.22 93.16 91.86 38.62 38.10 72.46 65.83 64.99 CLUDA 79.84 93.40 45.90 38.84 94.08 95.57 33.93 37.80 77.57 96.52 69.35 AdvSKM 78.94 87.91 52.57 33.49 92.64 92.71 36.53 39.95 65.49 83.75 66.41 CoDATS 79.61 90.90 60.07 21.80 97.66 97.66 41.44 38.54 58.15 97.01 68.71 RAINCOAT 87.72 93.33 63.75 46.46 98.05 98.25 42.63 43.32 84.17 93.75 74.21 Ours 86.65 93.45 79.01 53.53 97.15 98.32 65.80 65.71 88.59 89.17 81.74 8 Table 4: Ablation studies: Average Accuracy (%) on UCIHAR, HHAR-P and WISDM. LCT LCF Period LMs LMt LD UCIHAR HHAR-P WISDM Average 1 ✓ 75.12 54.25 65.78 65.05 2 ✓ 66.88 51.08 56.47 58.14 3 ✓ ✓ 73.47 53.16 59.10 61.91 4 ✓ ✓ ✓ ✓ 94.05 79.49 74.19 82.58 5 ✓ ✓ ✓ ✓ ✓ 90.83 57.83 67.77 72.14 6 ✓ ✓ ✓ ✓ ✓ ✓ 97.02 81.74 84.80 87.85 5.3 Analysis Ablation Study We conduct ablation experiments on three datasets, UCIHAR, HHAR-P and WISDM. For each datasets, we select the same 10 source-target domain pairs as mentioned in Section 5.1. The ablation results (average accuracy of 10 domain pairs) are presented in Table 4. We can observe that all learning modules in the proposed method are effective. Further discussions are included in Appendix C.3. 0.6 0.8 1.0 1.2 1.4 95 96 97 Accuracy (%) Sensitivity analysis on UCIHAR ACON-λD ACON-λMt ACON-λMs RAINCOAT (a) Sensitivity analysis on UCIHAR 0.6 0.8 1.0 1.2 1.4 76 79 82 Accuracy (%) Sensitivity analysis on HHAR-P ACON-λD ACON-λMt ACON-λMs RAINCOAT (b) Sensitivity analysis on HHAR 0.6 0.8 1.0 1.2 1.4 78 82 86 Accuracy (%) Sensitivity analysis on WISDM ACON-λD ACON-λMt ACON-λMs RAINCOAT (c) Sensitivity analysis on WISDM Figure 3: Sensitivity Analysis on three different datasets: (a) UCIHAR (b) HHAR-P (c) WISDM. RAINCOAT: The advanced baseline that achieves suboptimal performance on the three datasets. Sensitivity Analysis It’s worth noting that our total loss in Equation (10) does not include any hyperparameters. In UDA setup, how to search the optimal trade-offs without access to labeled target samples is still an open problem. Considering that, we choose to set all the trade-offs to 1, as it is the most intuitive choice. Without tuning the trade-offs, our proposed ACON still achieves significant improvements. To further investigate the sensitivity of ACON, we update Equation (10) as: min θF ,θT LCF (θF ) + LCT (θT ) + λMsLMs(θT ) + λMtLMt(θF ) −λDLD(ψT , ψF , gD), min gD λDLD(ψT , ψF , gD), (11) where hyperparameters λD, λMs and λMt control the contribution of each component. We investigate the sensitivity of the model to the hyperparameters λD, λMs and λMt. ACON-λD refers that the currently investigated hyperparameter is λD, and others are analogous. From Figure 3, we observe that the performance of ACON is quite stable to the hyperparameters in Equation (11). Although setting all the trade-offs to 1 may not achieve the optimal performance, ACON still significantly outperforms the advanced baseline. This implies that ACON can achieve superior performance on a wider range of datasets without the need for careful hyperparameter tuning. 6 Conclusion In this paper, the phenomenon is revealed——that temporal features exhibit better transferability across domains, whereas frequency features tend to be more discriminative within a specific domain. Based on the findings, Adversarial CO-learning Networks (ACON) is proposed to boost the transferability and discriminability in a collaborative learning manner. Specifically, multi-period feature learning is proposed to enhance the discriminability of frequency features; temporal-frequency domain mutual learning is proposed to enhance the discriminability of temporal features in the source domain and improve the transferability of frequency features in the target domain; domain adversarial learning in temporal-frequency correlation subspace is proposed to further enhance transferability of features. ACON achieves state-of-the-art performance on a wide range of time series datasets. 9 Acknowledgements This work was supported by the National Natural Science Foundation of China (62306085, 62206074, 62406112, 62476071, 62236003), Shenzhen College Stability Support Plan (GXWD20231130151329002, GXWD20220811173233001, GXWD20220817144428005). References [1] Davide Anguita, Alessandro Ghio, Luca Oneto, Xavier Parra, Jorge Luis Reyes-Ortiz, et al. A public domain dataset for human activity recognition using smartphones. In Esann, 2013. [2] David Berthelot, Rebecca Roelofs, Kihyuk Sohn, Nicholas Carlini, and Alexey Kurakin. Adamatch: A unified approach to semi-supervised learning and domain adaptation. In ICLR, 2021. [3] Ruichu Cai, Jiawei Chen, Zijian Li, Wei Chen, Keli Zhang, Junjian Ye, Zhuozhang Li, Xiaoyan Yang, and Zhenjie Zhang. Time series domain adaptation via sparse associative structure alignment. In AAAI, 2021. [4] Chris Chatfield and Haipeng Xing. 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In ECCV, 2018. 12 A Dataset A.1 Detailed Statistics We conduct extensive experiments using a wide range of time series datasets. The detailed statistics for each dataset is included in Table 5. For EMG dataset, we use the processed version released by DIVERSIFY [28]. For PCL, CAP and HHAR-D datasets, we use the processed versions released by WOODS [11]. For UCIHAR, HHAR-P, WISDM and FD datasets, we use the processed versions released by AdaTime [31]. Table 5: Summary of datasets. Dataset Subjects Channels Length Class Total Task EMG 4 8 200 6 6883 GR FD 4 1 5120 3 10916 FD PCL 3 48 750 2 22598 MIC CAP 5 19 3000 6 40387 SSC UCIHAR 30 9 128 6 3290 HAR HHAR-P 9 3 128 6 17934 HAR WISDM 30 3 128 6 2070 HAR HHAR-D 5 6 500 6 13674 HAR A.2 Dataset Processing Each domain of datasets is randomly divided into 80% training, and 20% testing. We follow [31], apply Z-score normalization to both the training and testing splits of the data, using the following equation: xnormalize i = xi −xmean xstd , i = 1, 2, . . . , N (12) where N = Ns for the source domain data and N = Nt for the target domain data. Note that both the training and testing splits are normalized based on the training set statistics only. B Experimental Details The experiments were conducted on a single NVIDIA GeForce696 RTX 4090 with 24GiB of memory. As shown in Figure 3, without tuning the trade-offs of training loss, ACON still achieves significant improvements. Here we report other key hyperparameters for ACON in Table 6. Additional hyperparameters can be found in our code. In all experiments, we adopt 3-layer 1D-CNN as the temporal feature extractor (the specific structure is kept consistent with the existing works [31, 16]). For frequency feature extraction, we adopt a 1-layer complex-valued linear as the frequency feature extractor. Table 6: Key hyperparameters for ACON. Hyperparameter EMG FD PCL CAP UCIHAR HHAR-P WISDM HHAR-D Epoch 50 50 50 50 50 50 50 50 Batch Size 32 32 32 32 32 32 32 32 Learning Rate 0.001 0.01 0.001 0.001 0.01 0.001 0.003 0.01 C Further Analysis C.1 Discriminability of Frequency Feature In Table 7, we report the average accuracy of classification experiments under the setting of Section 3.2 using five different datasets: UCIAHR[1], HHAR-P[35], WISDM[20], CAP[14, 37] and FD[31]. For each dataset, we collect all the domains involved in the selected 10 domain pairs as mentioned in Section 5.1, and perform the classification task on them. 13 Table 7: Classification Accuracy (%) in the source domain: Temporal domain vs. Frequency domain. Dataset UCIHAR HHAR-P WISDM CAP FD Temporal domain 86.18 97.01 95.63 81.63 97.99 Frequency domain 95.69 97.77 98.01 82.73 98.83 C.2 Transferability of Temporal Feature In Table 8, we report the average accuracy of classification experiments under the setting of Section 3.3 using three different datasets: UCIAHR[1], HHAR-P[35], WISDM[20], CAP[14, 37] and FD[31]. For each dataset, we perform the domain adaptation and classification task on the selected 10 domain pairs as mentioned in Section 5.1. Table 8: Classification Accuracy (%) in the target domain: Temporal domain vs. Frequency domain. Dataset UCIHAR HHAR-P WISDM CAP FD Source-only-T 75.12 54.25 65.78 70.14 70.04 Source-only-F 66.88 51.08 56.47 67.15 69.53 DANN-T 88.30 72.57 71.54 75.49 86.21 DANN-F 80.64 68.73 60.99 70.76 81.46 C.3 Ablation Study C.3.1 Ablation Study on different modules We conduct ablation experiments on three datasets, UCIHAR, HHAR-P and WISDM. For each datasets, we select the same 10 source-target domain pairs as mentioned in Section 5.1. The ablation results (average accuracy of 10 domain pairs) are presented in Table 9. We verify the effectiveness of all learning modules in the proposed method by answering the following questions. Can multi-period frequency feature learning enhance the discriminability of frequency feature?By comparing the 2nd and 3rd rows, we can observe that with multi-period frequency feature learning, the model makes more accurate predictions on the target domain. Meanwhile, compared with 1st row, even with multi-period frequency feature learning, the performance on the target domain is still inferior to the classification on the temporal domain, which is consistent with our conclusion in Section 3.3 that the temporal features have better transferability. Can aligning distributions in temporal-frequency subspace effectively learn domain-invariant features? By comparing the 1st, 3rd and 4th rows, we can observe that distribution alignment significantly improves the performance. It indicates that by aligning the distributions in the temporalfrequency subspace, the model learns more domain-invariant features. Can temporal-frequency domain mutual learning leverage the respective advantages? By comparing the 1st, 3rd and 5th rows, we can observe that the model with domain mutual learning outperforms the model using only the temporal domain or only the frequency domain. It demonstrates that via domain mutual learning, the temporal domain and frequency domain successfully transfer meaningful knowledge, leveraging their respective advantages. Can the different modules mutually promote each other? By comparing the 3rd, 4th, 5th and 6th rows, we can observe that with all modules, the model achieves the optimal performance. Specifically, with multi-period frequency feature learning, the frequency domain transfers more discriminative knowledge to the temporal domain; with aligning distribution in temporal-frequency subspace, the temporal domain transfers more transferable knowledge to the frequency domain; with aligning distribution in temporal-frequency subspace and the temporal domain as a more tranferbale teacher, both the temporal domain and frequency domain learn domain-invariant features; with domain mutual learning, the frequency domain and the temporal domain enhance each other in the transfer progress, achieving a synergistic effect. 14 Table 9: Ablation study on different modules: Average Accuracy (%) on UCIHAR, HHAR-P and WISDM. LCT LCF Period LMs LMt LD UCIHAR HHAR-P WISDM Average 1 ✓ 75.12 54.25 65.78 65.05 2 ✓ 66.88 51.08 56.47 58.14 3 ✓ ✓ 73.47 53.16 59.10 61.91 4 ✓ ✓ ✓ ✓ 94.05 79.49 74.19 82.58 5 ✓ ✓ ✓ ✓ ✓ 90.83 57.83 67.77 72.14 6 ✓ ✓ ✓ ✓ ✓ ✓ 97.02 81.74 84.80 87.85 D Limitations Although ACON boosts transferability and discriminability for time series Domain adaptation, like existing DA methods, it is still unstable enough in time series with relatively large variances. This is a problem that needs to be solved urgently in the future. E Broader Impacts We investigate how to boost Transferability and discriminability for domain adaptation in time series classification. We reveal that the frequency features are more discriminative, while the temporal features are more transferable. Upon this, we propose multi-period frequency feature learning, domain mutual learning, and distribution alignment in temporal-frequency feature subspace. The purpose of our research is to advance the research progress in the relevant community without any negative social impact. F Full Resluts We present all experimental results in this section. Notably, our model achieves superior performance, yielding improvements of more than 6% in terms of accuracy and 4% in terms of Macro-F1 across 8 cross-domain time series datasets and 5 common applications on average. Table 10: Accuracy (%) on EMG for unsupervised domain adaptation. Method 0→1 0→2 0→3 1→2 1→3 2→0 2→1 2→3 3→1 3→2 Avg Source-only 84.94 74.38 73.38 74.38 73.88 73.88 82.16 73.69 79.38 72.31 76.24 CDAN 87.84 76.63 77.63 77.44 81.63 73.94 87.10 75.13 83.98 77.63 79.89 DeepCoral 87.50 76.44 76.19 77.63 77.63 74.69 84.72 75.50 81.93 74.88 78.71 AdaMatch 89.03 75.94 79.38 76.94 80.00 76.31 89.94 81.31 84.26 73.81 80.69 HoMM 87.61 76.50 75.75 77.00 77.94 73.94 84.89 75.88 82.61 75.31 78.74 DIRT-T 89.77 75.25 78.69 75.88 80.06 70.63 84.77 77.69 83.30 76.69 79.27 CLUDA 78.18 75.00 76.75 74.75 74.19 75.94 79.43 70.00 76.88 75.13 75.62 AdvSKM 86.42 75.94 76.25 77.25 78.00 74.88 85.06 77.25 81.76 75.31 78.81 CoDATS 88.24 77.44 78.31 78.44 81.81 73.75 86.65 78.88 84.43 78.06 80.60 RAINCOAT 89.60 77.00 78.56 78.25 83.13 73.06 85.68 76.88 83.13 74.00 79.93 Ours 92.50 79.06 81.75 80.13 83.13 77.94 90.91 79.75 85.11 78.88 82.91 15 Table 11: Accuracy (%) on PCL for unsupervised domain adaptation. Method 0→1 0→2 1→0 1→1 2→0 2→1 Avg Source-only 65.68 57.79 59.65 60.65 56.51 65.46 60.95 CDAN 68.37 59.77 62.79 62.44 57.50 69.27 63.36 DeepCoral 67.66 60.08 62.59 63.58 57.76 69.36 63.51 AdaMatch 66.11 54.92 58.30 58.73 53.75 54.88 57.78 HoMM 68.28 60.27 62.80 63.75 58.71 69.17 63.83 DIRT-T 61.69 57.29 60.77 62.13 56.79 67.47 61.02 CLUDA 56.60 53.46 60.13 57.79 49.78 50.35 54.69 AdvSKM 67.62 59.90 63.06 64.15 58.07 68.68 63.58 CoDATS 70.52 57.83 65.10 64.17 57.83 69.62 64.18 RAINCOAT 58.46 54.04 59.88 60.81 57.63 63.14 58.99 Ours 70.63 60.58 63.42 64.63 60.32 70.53 65.02 Table 12: Accuracy (%) on FD for unsupervised domain adaptation. Method 0→1 0→2 0→3 1→0 1→2 2→0 2→1 2→3 3→0 3→2 Avg Source-only 62.21 53.71 62.41 63.91 73.95 64.08 93.17 95.54 57.08 74.31 70.04 CDAN 91.29 71.83 90.13 96.50 90.09 83.10 99.38 99.98 95.40 87.95 90.56 DeepCoral 75.54 71.79 76.03 89.13 83.55 76.34 98.84 98.55 87.50 83.71 84.10 AdaMatch 67.81 55.38 62.88 92.21 98.57 79.08 89.96 90.40 87.23 97.57 82.11 HoMM 81.54 71.63 78.17 89.89 84.78 76.03 98.71 99.55 90.94 85.96 85.72 DIRT-T 75.94 70.85 76.36 98.10 90.27 81.92 100.0 99.98 97.06 90.29 88.08 CLUDA 90.47 82.63 88.68 89.06 92.23 61.92 93.91 90.80 82.01 78.17 84.99 AdvSKM 74.71 66.05 73.30 87.86 86.29 76.85 98.66 99.38 84.89 85.74 83.37 CoDATS 81.79 73.26 83.15 89.22 88.68 81.43 99.89 100.0 85.47 89.00 87.20 RAINCOAT 85.18 79.40 89.04 78.84 90.11 81.43 95.18 96.81 77.39 94.08 86.75 Ours 86.52 69.00 86.96 97.92 99.80 84.29 98.62 98.93 97.72 97.66 91.74 Table 13: Accuracy (%) on UCIHAR for unsupervised domain adaptation. Method 2→11 6→23 7→13 9→18 12→16 13→19 18→21 20→6 23→13 24→12 Avg Source-only 76.56 67.36 83.68 24.65 61.11 88.89 100.0 94.10 71.18 83.68 75.12 CDAN 85.42 87.50 92.01 58.86 66.67 96.52 100.0 95.13 82.64 93.40 85.78 DeepCoral 90.63 84.38 87.50 46.88 65.28 95.49 100.0 95.49 69.79 87.50 82.01 AdaMatch 75.00 80.20 85.76 56.59 49.65 94.79 100.0 84.37 68.75 70.83 76.07 HoMM 74.06 82.71 81.88 73.96 70.21 96.67 98.75 73.33 77.71 80.63 80.99 DIRT-T 80.21 74.31 82.99 59.03 67.01 99.30 98.61 92.36 74.72 94.27 83.26 CLUDA 81.77 92.01 99.31 67.71 65.28 94.44 98.96 97.22 72.92 99.31 85.53 AdvSKM 98.96 88.54 92.71 74.65 69.44 93.05 100.0 85.41 79.51 96.87 83.26 CoDATS 68.23 74.31 77.43 63.89 66.32 94.09 99.65 70.49 56.25 82.81 75.54 RAINCOAT 100.0 95.83 100.0 75.69 86.52 100.0 100.0 93.41 86.52 93.75 94.43 Ours 100.0 96.25 99.16 91.66 85.63 100.0 100.0 97.50 100.0 100.0 97.02 Table 14: Accuracy (%) on WISDM for unsupervised domain adaptation. Method 2→32 4→15 7→30 12→7 12→19 18→20 20→30 21→31 25→29 26→2 Avg Source-only 81.16 79.86 89.32 71.53 54.29 83.74 67.96 21.29 26.11 82.52 65.78 CDAN 89.37 65.97 84.79 70.48 51.01 88.62 77.02 46.58 44.33 83.33 70.05 DeepCoral 87.92 62.50 91.26 79.86 51.77 64.23 81.88 54.62 53.89 77.44 70.80 AdaMatch 74.39 78.47 89.64 73.26 55.30 75.20 74.76 31.32 57.78 87.20 69.79 HoMM 77.10 74.58 78.64 68.13 50.61 71.22 72.82 56.39 57.00 66.10 67.26 DIRT-T 77.78 70.83 90.61 70.20 51.51 85.36 71.84 54.41 60.04 66.46 69.62 CLUDA 73.91 67.36 86.40 65.97 49.24 83.74 72.49 49.97 35.00 86.47 67.04 AdvSKM 70.83 95.85 93.85 77.08 47.47 81.30 21.28 44.45 74.79 74.95 66.97 CoDATS 77.29 70.83 83.20 70.17 47.47 76.01 82.85 52.61 53.89 83.29 70.66 RAINCOAT 79.71 97.91 91.28 89.80 85.00 92.23 91.66 59.09 82.97 83.50 76.60 Ours 89.86 86.25 98.06 98.13 77.73 83.66 91.26 63.61 60.00 99.51 84.80 16 Table 15: Accuracy (%) on HHAR-D for unsupervised domain adaptation. Method 0→1 0→2 0→3 0→4 1→0 1→3 1→4 2→1 3→4 4→1 Avg Source-only 65.48 33.59 31.71 39.79 34.69 44.83 49.54 38.17 86.17 44.23 46.82 CDAN 69.86 48.28 38.22 48.42 48.75 60.48 51.33 47.84 87.33 48.89 54.94 DeepCoral 68.94 42.88 40.67 47.96 35.63 55.31 56.21 44.71 87.25 45.96 52.55 AdaMatch 71.78 39.60 39.74 47.50 52.50 55.48 58.33 46.49 85.83 41.15 53.84 HoMM 69.66 40.51 39.16 50.42 35.94 55.02 57.13 42.36 86.79 46.35 52.33 DIRT-T 68.37 42.14 47.21 52.92 41.25 60.14 55.63 46.73 92.25 54.81 56.14 CLUDA 71.78 39.60 39.74 47.50 52.50 55.48 58.33 46.49 85.83 41.15 53.84 AdvSKM 67.93 40.71 40.19 47.33 37.19 55.65 59.54 42.69 87.46 49.33 52.80 CoDATS 72.50 43.35 50.79 45.50 58.44 62.24 54.54 40.14 89.63 45.53 56.27 RAINCOAT 74.47 36.52 48.82 35.29 51.25 41.49 41.50 34.28 88.58 38.46 49.07 Ours 77.50 61.36 54.69 65.46 69.38 71.30 62.13 50.10 93.63 44.86 65.04 Table 16: Average Macro-F1 Score on Eight Datasets and Five Applications for UDA. Task GR MFD MI HAR SSC Dataset EMG FD PCL UCIHAR HHAR-P WISDM HHAR-D CAP Source-only 0.76 0.65 0.60 0.73 0.50 0.52 0.43 0.52 CDAN 0.80 0.92 0.63 0.86 0.68 0.54 0.53 0.62 DeepCoral 0.79 0.81 0.63 0.82 0.62 0.52 0.49 0.59 AdaMatch 0.81 0.78 0.56 0.76 0.62 0.54 0.51 0.57 HoMM 0.79 0.81 0.64 0.79 0.64 0.49 0.49 0.60 DIRT-T 0.79 0.88 0.61 0.81 0.64 0.54 0.53 0.64 CLUDA 0.75 0.82 0.48 0.86 0.67 0.57 0.51 0.59 AdvSKM 0.79 0.80 0.63 0.87 0.65 0.55 0.49 0.59 CoDATS 0.81 0.88 0.63 0.72 0.63 0.56 0.55 0.62 RAINCOAT 0.80 0.89 0.59 0.93 0.75 0.74 0.47 0.59 Ours 0.83 0.93 0.65 0.97 0.80 0.74 0.62 0.67 Table 17: Macro-F1 Score on EMG for unsupervised domain adaptation. Method 0→1 0→2 0→3 1→2 1→3 2→0 2→1 2→3 3→1 3→2 Avg Source-only 0.85 0.74 0.74 0.74 0.75 0.75 0.82 0.74 0.78 0.72 0.76 CDAN 0.88 0.77 0.78 0.78 0.82 0.74 0.87 0.76 0.84 0.78 0.80 DeepCoral 0.87 0.76 0.76 0.78 0.78 0.75 0.84 0.76 0.82 0.75 0.79 AdaMatch 0.89 0.76 0.79 0.77 0.80 0.76 0.90 0.81 0.84 0.74 0.81 HoMM 0.87 0.77 0.76 0.77 0.78 0.74 0.84 0.76 0.82 0.75 0.79 DIRT-T 0.90 0.75 0.79 0.76 0.80 0.71 0.84 0.78 0.83 0.77 0.79 CLUDA 0.78 0.75 0.77 0.75 0.74 0.76 0.79 0.70 0.75 0.75 0.75 AdvSKM 0.86 0.76 0.76 0.77 0.78 0.76 0.85 0.77 0.81 0.75 0.79 CoDATS 0.88 0.77 0.78 0.79 0.82 0.74 0.86 0.79 0.84 0.78 0.81 RAINCOAT 0.89 0.77 0.79 0.78 0.83 0.73 0.85 0.77 0.83 0.74 0.80 Ours 0.92 0.79 0.82 0.80 0.83 0.78 0.91 0.80 0.85 0.79 0.83 Table 18: Macro-F1 Score on CAP for unsupervised domain adaptation. Method 0→1 0→3 0→4 1→0 1→4 2→3 3→0 3→1 4→1 4→3 Avg Source-only 0.39 0.71 0.61 0.50 0.54 0.45 0.63 0.30 0.33 0.69 0.52 CDAN 0.62 0.73 0.66 0.58 0.59 0.48 0.68 0.61 0.55 0.73 0.62 DeepCoral 0.61 0.73 0.65 0.54 0.58 0.53 0.66 0.44 0.44 0.70 0.59 AdaMatch 0.52 0.73 0.64 0.58 0.55 0.29 0.66 0.52 0.51 0.67 0.57 HoMM 0.62 0.73 0.65 0.56 0.59 0.54 0.66 0.50 0.48 0.71 0.60 DIRT-T 0.65 0.75 0.69 0.57 0.59 0.50 0.69 0.67 0.59 0.71 0.64 CLUDA 0.58 0.71 0.51 0.63 0.61 0.44 0.67 0.50 0.55 0.68 0.59 AdvSKM 0.58 0.73 0.65 0.55 0.59 0.53 0.66 0.48 0.41 0.69 0.59 CoDATS 0.64 0.75 0.65 0.61 0.63 0.51 0.70 0.51 0.54 0.71 0.62 RAINCOAT 0.58 0.65 0.61 0.55 0.56 0.50 0.62 0.63 0.60 0.61 0.59 Ours 0.64 0.76 0.68 0.62 0.63 0.54 0.71 0.72 0.70 0.74 0.67 17 Table 19: Macro-F1 Score on PCL for unsupervised domain adaptation. Method 0→1 0→2 1→0 1→1 2→0 2→1 Avg Source-only 0.65 0.57 0.58 0.60 0.55 0.64 0.60 CDAN 0.68 0.59 0.62 0.62 0.57 0.69 0.63 DeepCoral 0.68 0.60 0.62 0.63 0.57 0.69 0.63 AdaMatch 0.66 0.53 0.58 0.57 0.53 0.51 0.56 HoMM 0.68 0.60 0.63 0.63 0.58 0.69 0.64 DIRT-T 0.61 0.57 0.61 0.62 0.56 0.67 0.61 CLUDA 0.55 0.49 0.59 0.56 0.33 0.36 0.48 AdvSKM 0.67 0.60 0.63 0.63 0.58 0.69 0.63 CoDATS 0.70 0.55 0.65 0.64 0.57 0.69 0.63 RAINCOAT 0.58 0.54 0.59 0.61 0.57 0.63 0.59 Ours 0.71 0.60 0.63 0.64 0.60 0.71 0.65 Table 20: Macro-F1 Score on FD for unsupervised domain adaptation. Method 0→1 0→2 0→3 1→0 1→2 2→0 2→1 2→3 3→0 3→2 Avg Source-only 0.41 0.33 0.41 0.65 0.77 0.64 0.95 0.97 0.59 0.78 0.65 CDAN 0.91 0.76 0.90 0.95 0.92 0.86 1.00 1.00 0.94 0.91 0.92 DeepCoral 0.61 0.62 0.62 0.90 0.87 0.77 0.99 0.99 0.89 0.88 0.81 AdaMatch 0.50 0.45 0.46 0.91 0.98 0.80 0.93 0.93 0.87 0.97 0.78 HoMM 0.61 0.52 0.62 0.91 0.88 0.78 0.99 1.00 0.91 0.89 0.81 DIRT-T 0.80 0.62 0.70 0.97 0.93 0.84 1.00 1.00 0.96 0.93 0.88 CLUDA 0.84 0.80 0.79 0.88 0.93 0.50 0.95 0.90 0.84 0.80 0.82 AdvSKM 0.55 0.54 0.57 0.89 0.89 0.76 0.99 1.00 0.87 0.89 0.80 CoDATS 0.80 0.69 0.87 0.90 0.92 0.86 1.00 1.00 0.87 0.92 0.88 RAINCOAT 0.89 0.84 0.92 0.81 0.92 0.85 0.96 0.98 0.81 0.94 0.89 Ours 0.86 0.75 0.89 0.96 1.00 0.88 0.99 0.99 0.96 0.98 0.93 Table 21: Macro-F1 Score on UCIHAR for unsupervised domain adaptation. Method 2→11 6→23 7→13 9→18 12→16 13→19 18→21 20→6 23→13 24→12 Avg Source-only 0.69 0.63 0.84 0.17 0.58 0.91 1.00 0.94 0.71 0.84 0.73 CDAN 0.85 0.88 0.91 0.61 0.64 0.97 1.00 0.95 0.82 0.92 0.86 DeepCoral 0.91 0.81 0.87 0.44 0.65 0.95 1.00 0.95 0.70 0.88 0.82 AdaMatch 0.73 0.81 0.86 0.55 0.48 0.94 1.00 0.84 0.67 0.70 0.76 HoMM 0.73 0.78 0.81 0.69 0.69 0.96 0.99 0.71 0.75 0.78 0.79 DIRT-T 0.81 0.68 0.82 0.58 0.62 0.99 0.98 0.92 0.74 0.93 0.81 CLUDA 0.81 0.92 0.99 0.67 0.64 0.94 0.99 0.98 0.71 0.99 0.86 AdvSKM 0.99 0.87 0.92 0.73 0.68 0.93 1.00 0.84 0.77 0.96 0.87 CoDATS 0.66 0.71 0.78 0.60 0.64 0.93 0.99 0.65 0.54 0.81 0.72 RAINCOAT 1.00 0.96 1.00 0.76 0.86 1.00 1.00 0.94 0.86 0.94 0.93 Ours 1.00 0.97 0.99 0.91 0.86 1.00 1.00 0.98 1.00 1.00 0.97 Table 22: Macro-F1 Score on HHAR-P for unsupervised domain adaptation. Method 0→2 1→6 2→4 4→0 4→5 5→1 5→2 7→2 7→5 8→4 Avg Source-only 0.60 0.64 0.32 0.29 0.78 0.90 0.19 0.31 0.36 0.58 0.50 CDAN 0.70 0.93 0.52 0.27 0.98 0.98 0.35 0.32 0.76 0.97 0.68 DeepCoral 0.86 0.91 0.45 0.26 0.90 0.90 0.36 0.32 0.50 0.73 0.62 AdaMatch 0.83 0.93 0.46 0.32 0.76 0.94 0.40 0.37 0.60 0.61 0.62 HoMM 0.70 0.91 0.45 0.37 0.88 0.91 0.34 0.40 0.61 0.79 0.64 DIRT-T 0.76 0.86 0.51 0.30 0.93 0.90 0.36 0.34 0.73 0.64 0.64 CLUDA 0.82 0.94 0.44 0.40 0.94 0.96 0.37 0.36 0.65 0.84 0.67 AdvSKM 0.72 0.88 0.44 0.33 0.93 0.92 0.35 0.41 0.64 0.83 0.65 CoDATS 0.73 0.90 0.46 0.20 0.96 0.94 0.41 0.36 0.59 0.95 0.63 RAINCOAT 0.87 0.93 0.59 0.45 0.98 0.98 0.41 0.44 0.86 0.94 0.75 Ours 0.86 0.93 0.74 0.52 0.97 0.98 0.62 0.65 0.89 0.89 0.80 18 Table 23: Macro-F1 Score on WISDM for unsupervised domain adaptation. Method 2→32 4→15 7→30 12→7 12→19 18→20 20→30 21→31 25→29 26→2 Avg Source-only 0.68 0.52 0.77 0.53 0.36 0.81 0.56 0.10 0.15 0.69 0.52 CDAN 0.72 0.44 0.70 0.50 0.31 0.87 0.64 0.31 0.23 0.71 0.54 DeepCoral 0.71 0.42 0.85 0.67 0.35 0.63 0.67 0.27 0.25 0.64 0.52 AdaMatch 0.59 0.54 0.76 0.67 0.38 0.66 0.54 0.16 0.24 0.74 0.54 HoMM 0.63 0.42 0.62 0.55 0.39 0.63 0.60 0.30 0.26 0.54 0.49 DIRT-T 0.65 0.41 0.78 0.56 0.39 0.67 0.65 0.28 0.21 0.54 0.54 CLUDA 0.64 0.61 0.81 0.59 0.41 0.70 0.70 0.27 0.26 0.75 0.57 AdvSKM 0.61 0.55 0.84 0.53 0.35 0.71 0.61 0.28 0.28 0.55 0.55 CoDATS 0.66 0.41 0.75 0.62 0.37 0.76 0.72 0.30 0.30 0.70 0.56 RAINCOAT 0.68 0.98 0.86 0.72 0.78 0.92 0.87 0.43 0.44 0.75 0.74 Ours 0.81 0.65 0.99 1.00 0.63 0.76 0.87 00.36 0.28 1.00 0.74 Table 24: Macro-F1 Score on HHAR-D for unsupervised domain adaptation. Method 0→1 0→2 0→3 0→4 1→0 1→3 1→4 2→1 3→4 4→1 Avg Source-only 0.61 0.27 0.25 0.33 0.44 0.43 0.46 0.32 0.85 0.38 0.43 CDAN 0.67 0.42 0.35 0.42 0.66 0.57 0.50 0.44 0.88 0.44 0.53 DeepCoral 0.65 0.34 0.33 0.40 0.48 0.53 0.53 0.39 0.86 0.41 0.49 AdaMatch 0.69 0.36 0.36 0.41 0.60 0.49 0.56 0.41 0.86 0.36 0.51 HoMM 0.66 0.33 0.31 0.41 0.47 0.52 0.53 0.37 0.86 0.42 0.49 DIRT-T 0.66 0.38 0.40 0.44 0.52 0.60 0.53 0.39 0.93 0.49 0.53 CLUDA 0.69 0.36 0.36 0.41 0.60 0.49 0.56 0.41 0.86 0.36 0.51 AdvSKM 0.63 0.32 0.31 0.38 0.46 0.54 0.56 0.36 0.86 0.44 0.49 CoDATS 0.71 0.38 0.44 0.39 0.70 0.61 0.53 0.38 0.90 0.44 0.55 RAINCOAT 0.72 0.32 0.42 0.32 0.56 0.39 0.38 0.31 0.89 0.35 0.47 Ours 0.76 0.53 0.49 0.56 0.81 0.67 0.59 0.44 0.93 0.41 0.62 19 NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. 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4,418	Non-asymptotic Global Convergence Analysis of BFGS with the Armijo-Wolfe Line Search Qiujiang Jin ECE, UT Austin qiujiangjin0@gmail.com Ruichen Jiang ECE, UT Austin rjiang@utexas.edu Aryan Mokhtari ECE, UT Austin mokhtari@austin.utexas.edu Abstract In this paper, we present the first explicit and non-asymptotic global convergence rates of the BFGS method when implemented with an inexact line search scheme satisfying the Armijo-Wolfe conditions. We show that BFGS achieves a global linear convergence rate of (1−1 κ)t for µ-strongly convex functions with L-Lipschitz gradients, where κ = L µ represents the condition number. Additionally, if the objective function’s Hessian is Lipschitz, BFGS with the Armijo-Wolfe line search achieves a linear convergence rate that depends solely on the line search parameters, independent of the condition number. We also establish a global superlinear convergence rate of O(( 1 t )t). These global bounds are all valid for any starting point x0 and any symmetric positive definite initial Hessian approximation matrix B0, though the choice of B0 impacts the number of iterations needed to achieve these rates. By synthesizing these results, we outline the first global complexity characterization of BFGS with the Armijo-Wolfe line search. Additionally, we clearly define a mechanism for selecting the step size to satisfy the Armijo-Wolfe conditions and characterize its overall complexity. 1 Introduction In this paper, we focus on solving the following unconstrained convex minimization problem min x∈Rd f(x), (1) where f : Rd →R is strongly convex and twice differentiable. Quasi-Newton methods are among the most popular algorithms for solving this class of problems due to their simplicity and fast convergence. Like gradient descent-type methods, they require only gradient information for implementation, while they aim to mimic the behavior of Newton’s method by using gradient information to approximate the curvature of the objective function. There are several variations of quasi-Newton methods, primarily distinguished by their update rules for the Hessian approximation matrices. The most well-known among these include the Davidon-Fletcher-Powell (DFP) method [1, 2], the BroydenFletcher-Goldfarb-Shanno (BFGS) method [3–6], the Symmetric Rank-One (SR1) method [7, 8], and the Broyden method [9]. Apart from these classical methods, other variants have also been proposed in the literature, including randomized quasi-Newton methods [10–14], greedy quasi-Newton methods [13–16], and those based on online learning techniques [17, 18]. In this paper, we mainly focus on the global analysis of the BFGS method, arguably the most successful quasi-Newton method in practice. The classic analyses of BFGS, including [19–28], primarily focused on demonstrating local asymptotic superlinear convergence without addressing an explicit global convergence rate when BFGS is deployed with a line-search scheme. While attempts have been made to establish global convergence for quasi-Newton methods using line search or trust-region techniques in previous studies [8, 29–33], these efforts provided only asymptotic convergence guarantees without explicit global convergence rates, thus not fully characterizing the global convergence rate of classical quasi-Newton methods. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). In recent years, there have been efforts to characterize the explicit convergence rate of BFGS within a local neighborhood of the solution, establishing a superlinear convergence rate of the form ( 1 √ t)t; see, for example, [34–37]. However, these results focus solely on local convergence analysis of BFGS under conditions where the stepsize is consistently set to one, the iterate remains close to the optimal solution, and the initial Hessian approximation matrix meets certain necessary conditions. Consequently, these analyses do not extend to providing a global convergence guarantee. For more details on this subject, we refer the reader to the discussion section in [38]. To the best of our knowledge, only few papers are closely related to our work and establish a global non-asymptotic guarantee for BFGS. In [39], it was shown that BFGS with exact line search achieves a global linear rate of (1 −2µ3 L3 (1 + µ Tr(B−1 0 ) t )−1(1 + Tr(B0) Lt )−1)t, where µ is the strong convexity parameter, L is the Lipschitz constant of the gradient, B0 is the initial Hessian approximation matrix, and Tr(·) denotes the trace of a matrix. After t = O(d) iterations, this rate approaches (1 −2µ3 L3 )t, which is significantly slower than the convergence rate of gradient descent. Additionally, a recent draft in [40] studied the global convergence of BFGS under an inexact line search. While this work establishes a local superlinear rate, it only shows a global linear rate of the form (1 −µ2 L2 )t. Hence, both these results fail to prove any global advantage for BFGS over gradient descent. In [38], the authors improved upon [39] by showing a better global linear convergence rate and a faster superlinear rate for BFGS with exact line search. Specifically, for an L-Lipschitz and µ-strongly convex function, BFGS initialized with B0 = LI achieves a global linear rate of (1 −µ3/2 L3/2 )t for t ≥1, while BFGS with B0 = µI achieves the same rate after d log κ iterations. With the additional assumption that the objective’s Hessian is Lipschitz, an improved linear rate of (1 −µ L)t is achieved after O(κ) iterations when B0 = LI and after O(d log κ + κ) when B0 = µI, matching the rate of gradient descent. A superlinear rate of (1/ √ t)t was also shown when the number of iterations exceeds specific thresholds. Contributions. In this paper, we analyze the BFGS method combined with the Armijo-Wolfe line search, the most commonly used line search criteria in practical BFGS applications; see, e.g., [41]. For minimizing an L-smooth and µ-strongly convex function, we present a global convergence rate of (1−µ L)t. To the best of our knowledge, this is the first result demonstrating a global linear convergence rate for BFGS that matches the rate of gradient descent under these assumptions. Furthermore, we show that if the objective function’s Hessian is Lipschitz continuous, BFGS with the Armijo-Wolfe line search converges at a linear rate determined solely by the line search parameters and not the problem’s condition number, κ = L/µ, when the number of iterations is sufficiently large. Finally, we prove a global non-asymptotic superlinear convergence rate of (h(d,κ,C0)/t)t, where h(d, κ, C0) depends on the condition number κ, the dimension d, and the weighted distance between the initial point x0 and the optimal solution x∗, denoted by C0. We summarize our results in Table 1. By combining these convergence results, we establish the total iteration complexity of BFGS with the Armijo-Wolfe line search. We also specify the line search complexity by investigating a bisection algorithm for choosing the step size that satisfies the Armijo-Wolfe conditions. Our result is one of the first non-asymptotic analysis characterizing the global convergence complexity of the BFGS quasi-Newton method with an inexact line search. Notation. We denote the ℓ2-norm by ∥· ∥, the set of d × d symmetric positive definite matrices by Sd ++, and use A ⪯B to mean B −A is symmetric positive semi-definite. The trace and determinant of a matrix A are represented as Tr(A) and Det(A), respectively. 2 Preliminaries In this section, we present the assumptions, notations, and intermediate results useful for the global convergence analysis. First, we state the following assumptions on the objective function f. Assumption 2.1. The function f is twice differentiable and strongly convex with parameter µ > 0. Assumption 2.2. The gradient of f is Lipschitz continuous with parameter L > 0. These assumptions are common in the convergence analysis of quasi-Newton methods. Under these, we show a global linear convergence rate of O((1 −µ L)t). To achieve a faster linear convergence rate that is independent of the problem condition number, and a global superlinear rate, we require an additional assumption that the objective function Hessian is Lipschitz continuous, as stated next. Assumption 2.3. The Hessian of f is Lipschitz continuous with parameter M > 0, i.e., for x, y ∈Rd, we have ∥∇2f(x) −∇2f(y)∥≤M∥x −y∥. 2 Initial Matrix Convergence Phase Convergence Rate Starting moment B0 Linear phase I 1 −1 κ t Ψ( ¯ B0) B0 Linear phase II 1 −1 3 t Ψ( ˜ B0) + C0Ψ( ¯ B0) + C0κ B0 Superlinear phase Ψ( ˜ B0)+C0Ψ( ¯ B0)+C0κ t t Ψ( ˜ B0) + C0Ψ( ¯ B0) + C0κ LI Linear phase I 1 −1 κ t 1 LI Linear phase II 1 −1 3 t dκ + C0κ LI Superlinear phase dκ+C0κ t t dκ + C0κ µI Linear phase I 1 −1 κ t d log κ µI Linear phase II 1 −1 3 t (1 + C0)d log κ + C0κ µI Superlinear phase (1+C0)d log κ+C0κ t t (1 + C0)d log κ + C0κ Table 1: Summary of our results for (i) an arbitrary positive definite B0, (ii) B0 = LI, and (iii) B0 = µI. Here, Ψ(A) := Tr(A) −d −log Det(A), ¯B0 = 1 LB0 and ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 . The last column shows the number of iterations required to achieve the corresponding linear or superlinear convergence phase. For brevity, the absolute constants are dropped. Note that the above regularity condition on the Hessian assumption is also common for establishing the superlinear convergence rate of quasi-Newton methods [19–28]. BFGS Update. Next, we state the general update rule of BFGS. If we denote xt as the iterate at time t, the vector gt = ∇f(xt) as the objective function gradient at xt, and Bt as the Hessian approximation matrix at step t, then the update is given by xt+1 = xt + ηtdt, dt = −B−1 t gt, (2) where ηt > 0 is the step size and dt is the descent direction. By defining the variable difference st := xt+1 −xt and the gradient difference yt := ∇f(xt+1) −∇f(xt), we can present the Hessian approximation matrix update for BFGS as follows: Bt+1 = Bt −Btsts⊤ t Bt s⊤ t Btst + yty⊤ t s⊤ t yt . (3) To avoid the costly operation of inverting the matrix Bt, one can define the inverse Hessian approximation matrix as Ht := B−1 t and apply the Sherman-Morrison-Woodbury formula to obtain Ht+1 := I −sty⊤ t y⊤ t st Ht I −yts⊤ t s⊤ t yt + sts⊤ t y⊤ t st . (4) It is well-known that for a strongly convex objective function, the Hessian approximation matrices Bt remain symmetric and positive definite if the initial matrix B0 is symmetric positive definite [41]. Therefore, all matrices Bt and Ht are symmetric positive definite throughout this paper. As mentioned earlier, establishing a global convergence guarantee for BFGS requires pairing it with a line search scheme to select the stepsize ηt. This paper focuses on implementing BFGS with the Armijo-Wolfe line search, detailed in the following subsection. Armijo-Wolfe Line Search. We consider a stepsize ηt >0 that satisfies the Armijo-Wolfe conditions f(xt + ηtdt) ≤f(xt) + αηt∇f(xt)⊤dt, (5) ∇f(xt + ηtdt)⊤dt ≥β∇f(xt)⊤dt, (6) where α and β are the line search parameters, satisfying 0 < α < β < 1 and 0 < α < 1 2. The condition in (5) is the Armijo condition, ensuring that the step size ηt provides a sufficient decrease in the objective function f. The condition in (6) is the curvature condition, which guarantees that the slope ∇f(xt + ηtdt)⊤dt at ηt is not strongly negative, indicating that further movement along dt would significantly decrease the function value. These conditions provide upper and lower bounds on the admissible step size ηt. In some references, the Armijo-Wolfe line search conditions are known as the weak Wolfe conditions [42, 43]. The procedure for finding ηt that satisfies these conditions is described in Section 7. Next lemma presents key properties of the Armijo-Wolfe conditions. 3 Lemma 2.1. Consider the BFGS method with Armijo-Wolfe inexact line search, where the step size satisfies the conditions in (5) and (6). Then, for any initial point x0 and any symmetric positive definite initial Hessian approximation matrix B0, the following results hold for all t ≥0: f(xt) −f(xt+1) −g⊤ t st ≥α, y⊤ t st −g⊤ t st ≥1 −β, and f(xt+1) ≤f(xt). (7) Remark 2.1. While in this paper we only focus on the Armijo-Wolfe line search, our results are also valid for some other line search schemes that require stricter conditions. For instance, in the strong Wolfe line search, given 0 < α < β < 1 and 0 < α < 1 2, the required conditions for the step size are f(xt + ηtdt) ≤f(xt) + αηt∇f(xt)⊤dt, \|∇f(xt + ηtdt)⊤dt\| ≤β∇f(xt)⊤dt, Indeed, if ηt satisfies the strong Wolfe conditions, it also satisfies the Armijo-Wolfe conditions. Another commonly employed line search scheme is Armijo–Goldstein, which imposes the conditions −c1ηt∇f(xt)⊤dt ≤f(xt) −f(xt + ηtdt) ≤−c2ηt∇f(xt)⊤dt, with 0 < c1 ≤c2 < 1. The lower bound on f(xt) −f(xt + ηtdt) in the Armijo–Goldstein line search indicates that ηt satisfies the sufficient decrease condition in (5) required for the Armijo-Wolfe conditions, with α = c1. Moreover, given the convexity of f, the upper bound on f(xt)−f(xt +ηtdt) in the Armijo–Goldstein line search suggests −ηt∇f(xt + ηtdt)⊤dt ≤f(xt) −f(xt + ηtdt) ≤ −c2ηt∇f(xt)⊤dt. Thus, ηt also meets the curvature condition in (6) required in the Armijo-Wolfe conditions with β = c2. Hence, all our results derived under the Armijo-Wolfe line search are also valid for both the strong Wolfe line search and the Armijo–Goldstein line search. 3 Convergence Analysis In this section, we present our theoretical framework for analyzing the global linear convergence rates of BFGS with the Armijo-Wolfe line search scheme. To start, we introduce some necessary definitions and notations. We define the average Hessian matrices Jt and Gt as Jt := Z 1 0 ∇2f(xt + τ(xt+1 −xt))dτ, Gt := Z 1 0 ∇2f(xt + τ(x∗−xt))dτ. (8) Further, for measuring the suboptimality of the iterates we define the sequence Ct as Ct := M µ 3 2 p 2(f(xt) −f(x∗)), ∀t ≥0, (9) where M is the Lipschitz constant of the Hessian defined in Assumption 2.3 and µ is the strong convexity parameter introduced in Assumption 2.1.To analyze the dynamics of the Hessian approximation matrices {Bt}+∞ t=0, we use the function Ψ(A) Ψ(A) := Tr(A) −d −log Det(A), (10) well-defined for any A ∈Sd ++. It was introduced in [32] to capture the discrepancy between A and the identity matrix I. Note that Ψ(A) ≥0 for any A ∈Sd ++ and Ψ(A) = 0 if and only if A = I. Before we start convergence analysis, given any weight matrix P ∈Sd ++, we define the weighted versions of the vectors gt, st, yt, dt and the matrix Bt, Jt as ˆgt = P −1 2 gt, ˆst = P 1 2 st, ˆyt = P −1 2 yt, ˆdt = P 1 2 dt. (11) ˆBt = P −1 2 BtP −1 2 , ˆJt = P −1 2 JtP −1 2 . (12) Note that these weighted matrices and vectors preserve many properties of their unweighted counterparts. For instance, two of these main properties are ˆg⊤ t ˆst = g⊤ t st and ˆy⊤ t ˆst = y⊤ t st. Similarly, the update for the weighted version of Hessian approximation matrices closely mirrors the update of their unweighted counterparts, as noted in the following expression: ˆBt+1 = ˆBt − ˆBtˆstˆs⊤ t ˆBt ˆs⊤ t ˆBtˆst + ˆytˆy⊤ t ˆs⊤ t ˆyt , ∀t ≥0. (13) Finally, we define a crucial quantity, ˆθt, which measures the angle between the weighted descent direction and the negative of the weighted gradient direction, satisfying cos(ˆθt) = −ˆg⊤ t ˆst ∥ˆgt∥∥ˆst∥. (14) 4 3.1 Intermediate Results In this section, we present our framework for analyzing the convergence of BFGS with an inexact line search. We first characterize the relationship between the function value decrease at each iteration and key quantities, including the angle ˆθt defined in (14). Proposition 3.1. Let {xt}t≥0 be the iterates generated by BFGS. Recall the definitions of weighted vectors in (11). Then, for any weight matrix P and for all t ≥1, we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 − t−1 Y i=0 ˆpiˆqiˆni cos2(ˆθi) ˆmi 1 t t . (15) where ˆpt, ˆqt, ˆmt and ˆnt are defined as ˆpt := f(xt) −f(xt+1) −ˆg⊤ t ˆst , ˆqt := ∥ˆgt∥2 f(xt) −f(x∗), ˆmt := ˆy⊤ t ˆst ∥ˆst∥2 , ˆnt = ˆy⊤ t ˆst −ˆg⊤ t ˆst . (16) This result shows the convergence rate of BFGS with Armijo-Wolfe line search depends on four products: Qt−1 i=0 ˆpi, Qt−1 i=0 ˆqi, Qt−1 i=0 ˆni, and Qt−1 i=0 cos2(ˆθi) ˆmi . To establish an explicit rate, we need lower bounds on these products. Lemma 2.1 shows that the lower bounds for Qt−1 i=0 ˆpi and Qt−1 i=0 ˆni depend on the inexact line search parameters α and β. We will further prove that if the unit step size ηt = 1 satisfies the Armijo-Wolfe conditions, better lower bounds can be obtained for these products. The lower bounds for Qt−1 i=0 ˆqi and Qt−1 i=0 cos2(ˆθi) ˆmi were established in previous work [38] as presented in Appendix D. Specifically, the bounds for Qt−1 i=0 ˆqi depend on the choice of the weight matrix, which varies in different sections of the paper, requiring separate bounds for each case. However, the bound for Qt−1 i=0 cos2(ˆθi) ˆmi does not require separate treatment. This is explicitly established in Proposition D.1, a classical result, as discussed in [41, Section 6.4]. We build all our linear and superlinear results by establishing different bounds on the terms in (15). 4 Global Linear Convergence Rates Building on the tools introduced in Section 3, we establish explicit global linear convergence rates for BFGS with the Armijo-Wolfe line search, requiring only the strong convexity and gradient Lipschitz conditions from Assumptions 2.1 and 2.2. Our proof leverages the fundamental inequality in (15) from Proposition 3.1 and lower bounds on the terms that appear in the contraction factor. Here, we set the weight matrix P to P = LI and hence define the initial weighted matrix ¯B0 as ¯B0 = 1 LB0. The following theorem presents our first global linear convergence rate of BFGS for any B0 ∈Sd ++. Theorem 4.1. Suppose Assumptions 2.1 and 2.2 hold. Let {xt}t≥0 be the iterates generated by BFGS, where the step size satisfies the Armijo-Wolfe conditions in (5) and (6). For any initial point x0 ∈Rd and any initial Hessian approximation matrix B0 ∈Sd ++, we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −e−Ψ( ¯ B0) t 2α(1 −β) κ t , ∀t ≥1. (17) Remark 4.1. In [38], the authors analyzed BFGS with exact line search and established a global linear rate of (1 −e−Ψ( ¯ B0) t 1 κ(1+√κ))t. In comparison, our result in (17) achieves a faster linear rate by eliminating the √κ factor in the denominator. This improvement arises from using the Armijo-Wolfe conditions. Specifically, under these conditions, we show f(xt)−f(xt+1) −g⊤ t st ≥α as shown in Lemma 2.1, where α ∈(0, 1/2) is a line search parameter. In contrast, using exact line search, the authors in [38] proved that f(xt)−f(xt+1) −g⊤ t st ≥ 2 √κ+1, thus leading to the extra √κ factor in their rate. From Theorem 4.1, we observe that the linear convergence rate is determined by the quantity Ψ( ¯B0) Thus, to simplify our bounds, we consider two different initializations: B0 = LI and B0 = µI. Corollary 4.2. Suppose Assumptions 2.1 and 2.2 hold, {xt}t≥0 are generated by BFGS with step size satisfying the Armijo-Wolfe conditions in (5) and (6), and x0 ∈Rd is an arbitrary initial point. 5 • If the initial Hessian approximation matrix is set as B0 = LI, then for any t ≥1 f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −2α(1 −β) κ t . (18) • If the initial Hessian approximation matrix is set as B0 = µI, then for any t ≥1 we have f(xt)−f(x∗) f(x0)−f(x∗) ≤(1 −e−d log κ t 2α(1−β) κ )t. Moreover, for t ≥d log κ, we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −2α(1 −β) 3κ t . (19) Corollary 4.2 shows that when initialized with B0 = LI, BFGS achieves a linear rate of O((1 −1 κ)t) from the first iteration, matching the rate of gradient descent. It also indicates that initializing with B0 = µI achieves a similar rate but after d log κ iterations. While this suggests a preference for initializing with B0 = LI, subsequent analysis reveals that with enough iterations, BFGS with either initialization can attain a faster linear rate independent of κ. In some cases, starting with B0 = µI may lead to fewer total iterations to achieve this faster rate. We will explore this trade-off later. 5 Condition Number Independent Linear Convergence Rates In this section, we improve the previous results and establish a non-asymptotic, condition numberfree global linear convergence rate for BFGS with the Armijo-Wolfe line search. This requires the additional assumption that the Hessian is Lipschitz continuous. Our analysis builds on the previous methodology but uses P = ∇2f(x∗) instead of P = LI to prove the condition number-independent global linear rate. Thus, the weighted initial matrix ˜B0 is ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 . Next, we present a general global convergence bound for any initial Hessian approximation B0 ∈Sd ++. Proposition 5.1. Suppose Assumptions 2.1, 2.2 and 2.3 hold. Let {xt}t≥0 be the iterates generated by BFGS with the step size satisfying the Armijo-Wolfe conditions in (5) and (6). Recall the definition of Ct in (9) and Ψ(·) in (10). For any initial point x0 ∈Rd and any initial Hessian approximation matrix B0 ∈Sd ++, the following result holds: f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −2α(1 −β)e− Ψ( ˜ B0)+3 Pt−1 i=0 Ci t t , ∀t ≥1. Proposition 5.1 demonstrates that the convergence rate of BFGS with the Armijo-Wolfe line search is influenced by Ψ( ˜B0) and the sum Pt−1 i=0 Ci. The first term Ψ( ˜B0) is a constant that depends on our choice of the initial Hessian approximation matrix B0. The second term Pt−1 i=0 Ci can also be upper bounded using the non-asymptotic global linear convergence rate provided in Theorem 4.1. Theorem 5.2. Suppose Assumptions 2.1, 2.2 and 2.3 hold, and let {xt}t≥0 be the iterates generated by BFGS with the Armijo-Wolfe line search in (5) and (6). Then, for any initial point x0 ∈Rd and any initial Hessian approximation B0 ∈Sd ++, if t ≥Ψ( ˜B0) + 3C0Ψ( ¯B0) + 9 α(1−β)C0κ, we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −2α(1 −β) 3 t . (20) This result shows that when the number of iterations meets t ≥Ψ( ˜B0) + 3C0Ψ( ¯B0) + 9 α(1−β)C0κ, BFGS with Armijo-Wolfe conditions achieves a condition number-independent linear rate. The choice of B0 is critical as it influences the required iterations through ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 and ¯B0 = 1 LB0. Different choices of B0 affect Ψ( ˜B0) + 3C0Ψ( ¯B0) and thus the number of iterations needed for condition-free linear convergence. While optimizing B0 to minimize Ψ( ˜B0) + 3C0Ψ( ¯B0) is possible, we focus on two practical initialization schemes: B0 = LI and B0 = µI. Corollary 5.3. Suppose that Assumptions 2.1, 2.2 and 2.3 hold. Let {xt}t≥0 be the iterates generated by the BFGS method, where the step size satisfies the Armijo-Wolfe conditions in (5) and (6), and x0 ∈Rd as an arbitrary initial point. Then, given the result in Theorem 5.2, we have • If we set B0 = LI, the rate in (20) holds for t ≥dκ + 9 α(1−β)C0κ, • If we set B0 = µI, the rate in (20) holds for t ≥(1 + 3C0)d log κ + 9 α(1−β)C0κ. Based on Corollary 5.3, if C0 ≪κ, or equivalently f(x0)−f(x∗) ≪L2µ M 2 , then BFGS with B0 = µI requires less iterations to achieve the condition number-independent linear convergence rate. 6 6 Global Superlinear Convergence Rates In this section, we present our global superlinear result. Consider the definition ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 as well as the definition of ρt which is given by ρt := −g⊤ t dt ∥˜dt∥2 , ˜dt := ∇2f(x∗) 1 2 dt, ∀t ≥0. (21) To motivate, let us briefly discuss why we are only able to show a linear convergence rate instead of a superlinear rate in Theorem 5.2. By inspecting the proof, we observe that the bottleneck is due to the lower bounds on ˆpt and ˆnt: we used ˆpt ≥α and ˆnt ≥1 −β from Lemma 2.1, which leads to the constant factor α(1 −β) in the final linear rate in Theorem 5.2. Thus, to show a superlinear convergence rate, we need to establish tighter lower bounds for ˆpt and ˆnt. In the following lemma, we show that if the step size ηt = 1, we are able to establish such tighter lower bounds. Lemma 6.1. Recall ˆpt = f(xt)−f(xt+1) −ˆg⊤ t ˆst and ˆnt = ˆy⊤ t ˆst −ˆg⊤ t ˆst defined in (16). If the unit step size ηt = 1 satisfies the Armijo-Wolfe conditions (5) and (6), then we have ˆpt ≥1 −1 + Ct 2ρt , ˆnt ≥ 1 (1 + Ct)ρt . (22) In contrast to the constant lower bounds in Lemma 2.1, the lower bounds in (22) depend on Ct and ρt. Later, we show Ct →0 and ρt →1. Hence, the lower bounds in (22) approach 1 as the number of iterations increases, enabling us to prove a superlinear rate. That said, the lower bounds in Lemma 6.1 hold only when ηt = 1. To complete the picture, we need to quantify when and how often the unit step size is selected during BFGS execution. This is addressed in the next lemmas. Lemma 6.2. Suppose Assumptions 2.1, 2.2, and 2.3 hold and define the constants δ1 :=min 1 6, p 2(1 −α) −1, 1 √1 −β −1 , δ2 := max{7 8, 1 p 2(1 −α) }, δ3 := 1 √1 −β , (23) which satisfy 0 < δ1 < δ2 < 1 < δ3. If Ct ≤δ1 and δ2 ≤ρt ≤δ3, then ηt = 1 satisfies the Armijo-Wolfe conditions (5) and (6). Lemma 6.2 shows that when Ct ≤δ1 and ρt falls within the interval [δ2, δ3], the step size ηt = 1 is admissible and meets the Armijo-Wolfe conditions. Note that by the linear convergence result in Theorem 4.1, the first condition on Ct will be satisfied when t is sufficiently large. Additionally, using Proposition G.2 in the Appendix, we can show that the second condition on ρt is violated only for a finite number of iterations. These observations are formally presented in the following lemma. Lemma 6.3. Suppose Assumptions 2.1, 2.2, and 2.3 hold and the iterates {xt}t≥0 are generated by the BFGS method with step size satisfying the Armijo-Wolfe conditions in (5) and (6). Recall Ct defined in (9), Ψ(·) defined in (10), {δi}3 i=1 defined in (23) and ¯B0 = 1 LB0. We have Ct ≤δ1 when t ≥t0 := max Ψ( ¯B0), 3κ α(1 −β) log C0 δ1 . (24) Moreover, if we define ω(x) = x −log(1 + x), the size of the set I = {t : ρt /∈[δ2, δ3]} is at most \|I\| ≤δ4 Ψ( ˜B0) + 2C0Ψ( ¯B0) + 6C0κ α(1 −β) , where δ4 := 1 min{ω(δ2 −1), ω(δ3 −1)}. (25) Lemma 6.3 implies that conditions Ct ≤δ1 and ρt ∈[δ2, δ3] will be satisfied for all but a finite number of iterations. Thus, if the line search always starts by testing the unit step size (as shown in Section 7), we will choose ηt = 1, and accordingly, the tighter lower bound in Lemma 6.1 will apply for all but a finite number of iterations. By applying these lower bounds along with (15) from Proposition 3.1, we can prove a global superlinear convergence rate, as presented next. Remark 6.1. Lemmas 6.2 and 6.3 are inspired by the analysis in [40]. Specifically, Lemma 5.10 of [40] characterized the conditions on Ct and ρt under which η = 1 satisfies the Armijo condition (5), and further bounded the number of iterations where these conditions are violated. However, our Lemma 6.2 addresses both the Armijo condition in (5) and the curvature condition in (6), and the arguments appear simpler. Additionally, our proof for the superlinear convergence rate differs from [40]. Their approach analyzed the Dennis-Moré ratio and measured “local” superlinear convergence using the distance ∥∇f(x∗) 1 2 (xt −x∗)∥. In contrast, our “global” result is based on the unified framework in Proposition 3.1 and uses the function value gap as a measure of convergence. 7 Theorem 6.4. Suppose Assumptions 2.1, 2.2, and 2.3 hold and the iterates {xt}t≥0 are generated by BFGS with step size satisfying the Armijo-Wolfe conditions in (5) and (6). Recall the definition of Ct in (9), Ψ(·) in (10), ¯B0 := 1 LB0, ˜B0 := ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 , and δ1, δ2, δ3, δ4 in (23) and (25). Then, for any x0 ∈Rd and any B0 ∈Sd ++, the following global superlinear result holds: f(xt) −f(x∗) f(x0) −f(x∗) ≤ δ7Ψ( ˜B0) + (δ6 + δ8C0)Ψ( ¯B0) + ( 3δ6 α(1−β) log C0 δ1 + 3δ8 α(1−β)C0)κ t !t , (26) where {δi}8 i=5 defined below are constants that only depend on line search parameters α and β, δ5 := max{2 + 2 δ2 , 4δ3} 2δ2 −1 −δ1 , δ6 :=log 1 2α(1−β), δ7 :=1+δ4δ6+δ5, δ8 :=1+2δ7+ 2δ2−δ1−log δ2 2δ2−1−δ1 . The above result shows a global superlinear convergence rate of the form O(( C′ t )t), where C′ depends on the condition number κ, the initial weighted distance C0, and the initial Hessian approximation matrix B0. To simplify the expression, we report the above bound for B0 = LI and B0 = µI. Corollary 6.5. Suppose Assumptions 2.1, 2.2, and 2.3 hold and the iterates {xt}t≥0 are generated by the BFGS method with step size satisfying the Armijo-Wolfe conditions in (5) and (6), and x0 ∈Rd as an arbitrary initial point. Then, given the result in Theorem 6.4, the following results hold: • If we set B0 = LI, then we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ δ7dκ + ( 3δ6 α(1−β) log C0 δ1 + 3δ8 α(1−β)C0)κ t !t . (27) • If we set B0 = µI, then we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ (δ6 + δ7 + δ8C0)d log κ + ( 3δ6 α(1−β) log C0 δ1 + 3δ8 α(1−β)C0)κ t !t . (28) This result shows that BFGS with B0 = LI achieves a global superlinear rate of O(( dκ+C0κ t )t), while BFGS with the initialization B0 = µI converges at a global superlinear rate of O(( C0d log κ+C0κ t )t). Hence, the superlinear result for B0 = µI outperforms the rate for B0 = LI when C0 log κ ≪κ. Remark 6.2. We chose B0 = LI and B0 = µI as two specific cases since they lead to explicit upper bounds in terms of the dimension d and the condition number κ in various theorems, simplifying the interpretation of our results. In practice, however, we often set B0 = cI, where c = s⊤y ∥s∥2 , with s = x2 −x1, y = ∇f(x2) −∇f(x1), and x1, x2 as two randomly selected vectors. This choice ensures c ∈[µ, L], and in the following numerical experiments, the performance of B0 = cI is similar to that of B0 = µI. The complexity of BFGS with this initialization is reported in Appendix H. 7 Complexity Analysis Discussions on the iteration complexity. Using the three established convergence results in Theorems 4.1, 5.2 and 6.4, we can characterize the total number of iterations required for the BFGS method with the Armijo-Wolfe line search to find a solution with function suboptimality less than ϵ. However, as discussed above, the choice of the initial Hessian approximation B0 heavily influences the number of iterations required to observe these rates. To simplify our discussion, we focus on two specific initializations: B0 = LI and B0 = µI. The case of B0 = LI: The overall iteration complexity of BFGS with B0 = LI is given by O  min   κ log 1 ϵ , (d + C0)κ + log 1 ϵ , log 1 ϵ log 1 2 + q 1 4 + 1 dκ+C0κ log 1 ϵ     . The case of B0 = µI: The overall iteration complexity of BFGS with B0 = µI is given by O  min   d log κ + κ log 1 ϵ , C0(d log κ + κ) + log 1 ϵ , log 1 ϵ log 1 2 + q 1 4 + 1 C0(d log κ+κ) log 1 ϵ     . 8 We remark that the comparison between these two complexity bounds depends on the relative values of κ, d, C0, and ϵ, and neither is uniformly better than the other. It is worth noting that for BFGS with B0 = LI, we achieve a complexity that is consistently superior to the O κ log 1 ϵ complexity of gradient descent. Moreover, in scenarios where C0 = O(1) and d ≪κ, BFGS with B0 = µI could result in an iteration complexity of O κ + log 1 ϵ , which is much more favorable than that of gradient descent. The proof of these complexity bounds can be found in Appendix I. Discussions on the line search complexity. We present the log bisection algorithm to choose the step size ηt at iteration t satisfying the Armijo-Wolfe conditions (5) and (6) in Algorithm 1 in Appendix J. We define ηmin and ηmax as the lower and upper bounds of the “slicing window” containing the trial step size ηt, respectively. We start with the initial trial step size ηt = 1 and keep enlarging or decreasing it depending on whether the Armijo condition (5) or the curvature condition (6) is satisfied. Then, we dynamically update ηmin, ηmax and shrink the size of this “slicing window” (ηmin, ηmax). We pick the trial step size η as the geometric mean of ηmin and ηmax, i.e., log η = (log ηmax + log ηmax)/2, which is the reason why we call this algorithm “log bisection”. Note that in each loop of Algorithm 1, we query the function value and gradient at most once to check the Armijo-Wolfe conditions at Lines 2 and 9. The next theorem characterizes the average number of function value and gradient evaluations per iteration in Algorithm 1 after t iterations, denoted by Λt, which is equivalent to the average number of loops per iterations. Theorem 7.1. Suppose Assumptions 2.1, 2.2 and 2.3 hold. Let {xt}t≥0 be generated by BFGS with step size satisfying the Armijo-Wolfe conditions in (5) and (6) and is chosen by Algorithm 1. If we define σ := (Ψ( ¯B0) + 3 α(1−β)κ)C0, then for any initial point x0 ∈Rd and initial Hessian approximation B0 ∈Sd ++, the average number of the function value and gradient evaluations per iteration in Algorithm 1 after t iterations satisfies Λt ≤2+log2 1+ 1 −β β −α +2(1 −β) β −α σ t +2 log2 log2 16(1−α)+log2 1+σ t )+6Ψ( ˜B0) + 12σ t . The above result shows that when we run BFGS for N iterations, the total number of function and gradient evaluations is O N + N log(1 + σ N ) + N log(1 + Ψ( ˜ B0)+σ N ) . Thus, the total line search complexity can always be bounded by O(N log(Ψ( ˜B0) + σ)) = O(N max{log d, log κ, log C0}). Furthermore, notice that when N is sufficiently large such that we reach the superlinear convergence stage, i.e., N = Ω(Ψ( ˜B0) + σ), the total line search complexity becomes O(N), which means the average number of function and gradient evaluations per iteration is a constant O(1). We report the line search complexity results of different B0 = LI and B0 = µI in Appendix K.4. 8 Numerical Experiments We conduct numerical experiments on a cubic objective function defined as f(x) = α 12 d−1 X i=1 g(v⊤ i x −v⊤ i+1x) −βv⊤ 1 x ! + λ 2 ∥x∥2, (29) and g : R →R is defined as g(w) = 1 3\|w\|3 \|w\| ≤∆, ∆w2 −∆2\|w\| + 1 3∆3 \|w\| > ∆, (30) where α, β, λ, ∆∈R are hyper-parameters and {vi}n i=1 are standard orthogonal unit vectors in Rd. We focus on this objective function because it is used in [26] to establish a tight lower bound for second-order methods. We compare the convergence paths of BFGS with an inexact line search step size ηt that satisfies the Armijo-Wolfe conditions (5) and (6) for various initialization matrices B0: specifically, B0 = LI, B0 = µI, B0 = I, and B0 = cI where c is defined in Remark 6.2. It is easily verified that c ∈[µ, L]. We also compare the performance of BFGS methods to the gradient descent (GD) method with backtracking line search, using α = 0.1 in condition (5) and β = 0.9 in condition (6). Step size ηt is chosen at each iteration via log bisection in Algorithm 1. Empirical results are compared across various dimensions d and condition numbers κ, with the x-axis representing the number of iterations t and the y-axis showing the ratio f(xt)−f(x∗) f(x0)−f(x∗). 9 0 300 600 900 1200 1500 1800 2100 2400 2700 3000 10-20 10-15 10-10 10-5 100 (a) d = 100, κ = 100. 0 300 600 900 1200 1500 1800 2100 2400 2700 3000 10-20 10-15 10-10 10-5 100 (b) d = 100, κ = 1000. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10-20 10-15 10-10 10-5 100 (c) d = 300, κ = 100. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10-20 10-15 10-10 10-5 100 (d) d = 300, κ = 1000. 0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 10-20 10-15 10-10 10-5 100 (e) d = 600, κ = 100. 0 1200 2400 3600 4800 6000 7200 8400 9600 10800 12000 10-20 10-15 10-10 10-5 100 (f) d = 600, κ = 1000. Figure 1: Convergence curves of BFGS with inexact line search of different B0 and gradeint descent with backtracking line search. First, we observe that BFGS with B0 = LI initially converges faster than BFGS with B0 = µI in most plots, aligning with our theoretical findings that the linear convergence rate of BFGS with B0 = LI surpasses that of B0 = µI in Corollary 4.2. In Corollary 4.2, we show that BFGS with B0 = LI could achieve the linear rate of (1 −1/κ) from the first iteration while BFGS with B0 = µI needs to run d log κ to reach the same linear rate. Second, the transition to superlinear convergence for BFGS with B0 = µI typically occurs around t ≈d, as predicted by our theoretical analysis. Although BFGS with B0 = LI initially converges faster, its transition to superlinear convergence consistently occurs later than for B0 = µI. Notably, for a fixed dimension d = 600, the transition to superlinear convergence for B0 = LI occurs increasingly later as the problem condition number rises, an effect not observed for B0 = µI. This phenomenon indicates that the superlinear rate for B0 = LI is more sensitive to the condition number κ, which corroborates our results in Corollary 6.5. In Corollary 6.5, we present that BFGS with B0 = LI needs dκ steps to reach the superlinear convergence stage while this is improved to d log κ for BFGS with B0 = µI. Moreover, the performance of BFGS with B0 = I and B0 = cI is similar to BFGS with B0 = µI. Notice that the initializations of B0 = I and B0 = cI are two commonly-used practical choices of the initial Hessian approximation matrix B0. 9 Conclusions, Limitations, and Future Directions In this paper, we analyzed the global non-asymptotic convergence rates of BFGS with ArmijoWolfe line search. We showed for an objective function that is µ-strongly convex with an LLipschitz gradient, BFGS achieves a global convergence rate of (1 −1/κ)t, where κ = L/µ. Additionally, assuming the Hessian is M-Lipschitz, we showed BFGS achieves a linear convergence rate determined solely by the line search parameters, independent of the condition number. Under similar assumptions, we also established a global superlinear convergence rate. Given these bounds, we determined the overall iteration complexity of BFGS with the Armijo-Wolfe line search and specified this complexity for initial Hessian approximations B0 = LI and B0 = µI. One limitation of this paper is that the analysis only applies to strongly convex functions. Developing an analysis for the general convex setting is still unsolved. Another drawback is that we focus solely on the BFGS method. Extending our theoretical results to the entire convex Broyden’s class of quasi-Newton methods, including both BFGS and DFP, is a natural next step. 10 Acknowledgments The research of Q. Jin, R. Jiang, and A. 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SIAM Review 13.2 (1971), pp. 185–188 (page 3). [44] Y. Nesterov. Lectures on convex optimization. Springer Optimization and Its Applications (SOIA, volume 137), 2018 (page 20). 12 Appendix A Some Results on the Connections between Different Hessian Matrices Lemma A.1. Suppose Assumptions 2.1, 2.2, and 2.3 hold, and recall the definitions of the matrices Jt and Gt in (8), and the quantity Ct in (9). Then, the following statements hold: (a) Suppose that f(xt+1) ≤f(xt) for any t ≥0, we have that 1 1 + Ct ∇2f(x∗) ⪯Jt ⪯(1 + Ct)∇2f(x∗). (31) (b) Suppose that f(xt+1) ≤f(xt) for any t ≥0 and ˆτ ∈[0, 1], we have that 1 1 + Ct ∇2f(x∗) ⪯∇2f(xt + ˆτ(xt+1 −xt)) ⪯(1 + Ct)∇2f(x∗). (32) (c) For any t ≥0, we have that 1 1 + Ct ∇2f(x∗) ⪯∇2f(xt) ⪯(1 + Ct)∇2f(x∗). (33) (d) For any t ≥0, we have that 1 1 + Ct ∇2f(x∗) ⪯Gt ⪯(1 + Ct)∇2f(x∗). (34) (e) For any t ≥0 and ˜τ ∈[0, 1], we have that 1 1 + Ct Gt ⪯∇2f(xt + ˜τ(x∗−xt)) ⪯(1 + Ct)Gt. (35) (f) For any t ≥0 and ˜τ, ˆτ ∈[0, 1], suppose that f(xt+1) ≤f(xt). Then, we have that 1 1 + 2Ct ∇2f(xt + ˆτst) ⪯∇2f(xt + ˜τst) ⪯(1 + 2Ct)∇2f(xt + ˆτst). (36) Proof. (a) Recall the definition of Jt in (8). Using the triangle inequality, we have that ∥∇2f(x∗) −Jt∥= Z 1 0 ∇2f(x∗) −∇2f(xt + τ(xt+1 −xt)) dτ ≤ Z 1 0 ∥∇2f(x∗) −∇2f(xt + τ(xt+1 −xt))∥dτ. Moreover, it follows from Assumption 2.3 that ∥∇2f(x∗) −∇2f(xt + τ(xt+1 −xt))∥≤ M∥(1 −τ)(x∗−xt) + τ(x∗−xt+1)∥for any τ ∈[0, 1]. Thus, we can further apply the triangle inequality to obtain ∥∇2f(x∗) −Jt∥≤ Z 1 0 M∥(1 −τ)(x∗−xt) + τ(x∗−xt+1)∥dτ ≤M∥xt −x∗∥ Z 1 0 (1 −τ)dτ + M∥xt+1 −x∗∥ Z 1 0 τdτ = M 2 (∥xt −x∗∥+ ∥xt+1 −x∗∥). Since f is strongly convex, by Assumption 2.1 and f(xt+1) ≤f(xt), we have µ 2 ∥xt − x∗∥2 ≤f(xt) −f(x∗), which implies that ∥xt −x∗∥≤ p 2(f(xt) −f(x∗))/µ. Similarly, since f(xt+1) ≤f(xt), it also holds that ∥xt+1 −x∗∥≤ p 2(f(xt+1) −f(x∗))/µ ≤ p 2(f(xt) −f(x∗))/µ. Hence, we obtain that ∥∇2f(x∗) −Jt∥≤M √µ p 2(f(xt) −f(x∗)). (37) 13 Moreover, notice that by Assumption 2.1, we also have Jt ⪰µI and ∇2f(x∗) ⪰µI. Hence, (37) implies that ∇2f(x∗) −Jt ⪯∥∇2f(x∗) −Jt∥I ⪯M µ 3 2 p 2(f(xt) −f(x∗))Jt = CtJt, Jt −∇2f(x∗) ⪯∥Jt −∇2f(x∗)∥I ⪯M µ 3 2 p 2(f(xt) −f(x∗))∇2f(x∗) = Ct∇2f(x∗). where we used the definition of Ct in (9). By rearranging the terms, we obtain (31). (b) Similar to the arguments in (a), for any ˆτ ∈[0, 1], we have that ∇2f(xt + ˆτ(xt+1 −xt)) −∇2f(x∗) ≤M∥(1 −ˆτ)(xt −x∗) + ˆτ(xt+1 −x∗)∥ ≤M (1 −ˆτ)∥xt −x∗∥+ ˆτ∥xt+1 −x∗∥ ≤M (1 −ˆτ) r 2 µ(f(xt) −f(x∗)) + ˆτ r 2 µ(f(xt+1) −f(x∗)) ≤M r 2 µ(f(xt) −f(x∗)) Moreover, notice that by Assumption 2.1, we also have ∇2f(xt + ˆτ(xt+1 −xt)) ⪰µI and ∇2f(x∗) ⪰µI. The rest follows similarly as in the proof of (a) and we prove (32). (c) Similar to the arguments in (a), we have that ∇2f(x∗) −∇2f(xt) ≤M∥xt −x∗∥≤M √µ p 2(f(xt) −f(x∗)). Moreover, notice that by Assumption 2.1 we also have ∇2f(xt) ⪰µI and ∇2f(x∗) ⪰µI. The rest follows similarly as in the proof of (a) and we prove (33). (d) Recall the definition of Gt in (8). Similar to the arguments in (a), we have that ∥∇2f(x∗) −Gt∥= Z 1 0 ∇2f(x∗) −∇2f(xt + τ(x∗−xt)) dτ ≤ Z 1 0 ∥∇2f(x∗) −∇2f(xt + τ(x∗−xt))∥dτ ≤M Z 1 0 ∥(1 −τ)(x∗−xt)∥dτ = M∥xt −x∗∥ Z 1 0 (1 −τ)dτ = M 2 ∥xt −x∗∥≤M √µ p 2(f(xt) −f(x∗)). Moreover, notice that by Assumption 2.1 we also have Gt ⪰µI and ∇2f(x∗) ⪰µI. The rest follows similarly as in the proof of (a) and we prove (34). (e) Recall the definition of gt in (8). Similar to the arguments in (a), for any ˜τ ∈[0, 1], we have that ∇2f(xt + ˜τ(x∗−xt)) −Gt = Z 1 0 ∇2f(xt + ˜τ(x∗−xt)) −∇2f(xt + τ(x∗−xt)) dτ ≤ Z 1 0 ∇2f(xt + ˜τ(x∗−xt)) −∇2f(xt + τ(x∗−xt)) dτ ≤ Z 1 0 M\|˜τ −τ\|∥xt −x∗∥dτ ≤1 2M∥xt −x∗∥≤M √µ p 2(f(xt) −f(x∗)). Moreover, notice that by Assumption 2.1, we also have ∇2f(xt + ˜τ(x∗−xt)) ⪰µI and Gt ⪰µI. The rest follows similarly as in the proof of (a) and we prove (35). 14 (f) Similar to the arguments in (a), for any ˜τ, ˆτ ∈[0, 1], we have that ∇2f(xt + ˜τst) −∇2f(xt + ˆτst) ≤M\|˜τ −ˆτ\|∥st∥≤M∥st∥≤M(∥xt+1 −x∗∥+ ∥xt −x∗∥) ≤M r 2 µ(f(xt) −f(x∗)) + r 2 µ(f(xt+1) −f(x∗)) ≤2M r 2 µ(f(xt) −f(x∗)) Moreover, notice that by Assumption 2.1, we also have ∇2f(xt + ˜τst) ⪰µI and ∇2f(xt + ˆτst) ⪰µI. The rest follows similarly as in the proof of (a) and we prove (36). B Proof of Lemma 2.1 Recall that gt = ∇f(xt). Given the condition in (5) and the fact that st = ηtdt, we have f(xt+1) ≤f(xt) + αg⊤ t st. Moreover, since Bt is symmetric positive definite, we have −g⊤ t st = ηtg⊤ t B−1 t gt > 0 (unless gt = 0 and we are at the optimal solution). This further leads to the first claim, which is f(xt) −f(xt+1) −g⊤ t st ≥α. Similarly, the above argument implies that αg⊤ t st < 0 and as a result f(xt+1) ≤f(xt) and the last claim also follows. To prove the second claim, we leverage the condition in (6). Specifically, if we subtract g⊤ t dt from both sides of that condition, we obtain that (gt+1 −gt)⊤dt ≥(β −1)g⊤ t dt Next, using the fact that st = ηtdt, by multiplying both sides by ηt and use the simplification yt = gt+1 −gt we obtain that y⊤ t st ≥(β −1)g⊤ t st = −g⊤ t st(1 −β). Again using the argument that −g⊤ t st is positive (if we are not at the optimal solution), we can divide both sides of the above inequality by −g⊤ t st, leading to the second claim. C Proof of Proposition 3.1 First, we note that ˆg⊤ t ˆst = g⊤ t st and ˆy⊤ t ˆst = y⊤ t st. Using the definition of ˆpt in (16), we have that f(xt) −f(xt+1) = ˆpt −ˆg⊤ t ˆst ∥ˆgt∥2 ∥ˆgt∥2. (38) Hence, using the definition of ˆθt in (14) and the definition of ˆmt, ˆnt in (16), it follows that −ˆg⊤ t ˆst ∥ˆgt∥2 = (ˆg⊤ t ˆst)2 ∥ˆgt∥2∥ˆst∥2 ∥ˆst∥2 −ˆg⊤ t ˆst = (ˆg⊤ t ˆst)2 ∥ˆgt∥2∥ˆst∥2 ∥ˆst∥2 ˆy⊤ t ˆst ˆy⊤ t ˆst −ˆg⊤ t ˆst = ˆnt cos2(ˆθt) ˆmt . Furthermore, we have ∥ˆgt∥2 = ˆqt(f(xt)−f(x∗)) from the definition of ˆqt in (16). Thus, the equality in (38) can be rewritten as f(xt) −f(xt+1) = ˆptˆqtˆnt cos2(ˆθt) ˆmt (f(xt) −f(x∗)). 15 By rearranging the term in the above equality, we obtain f(xt+1) −f(x∗) = 1 −ˆptˆqtˆnt cos2(ˆθt) ˆmt (f(xt) −f(x∗)), (39) To prove the inequality in (15), note that for any t ≥1, we have f(xt) −f(x∗) f(x0) −f(x∗) = t−1 Y i=0 f(xi+1) −f(x∗) f(xi) −f(x∗) = t−1 Y i=0 1 −ˆpiˆqiˆni cos2(ˆθi) ˆmi ! , where the last equality is due to (39). Note that all the terms of the form 1 −ˆpiˆqiˆni cos2(ˆθi) ˆmi are non-negative, for any i ≥0. Thus, by applying the inequality of arithmetic and geometric means twice, we obtain t−1 Y i=0 1 −ˆpiˆqiˆni cos2(ˆθi) ˆmi ! ≤ " 1 t t−1 X i=0 1 −ˆpiˆqiˆni cos2(ˆθi) ˆmi !#t = " 1 −1 t t−1 X i=0 ˆpiˆqiˆni cos2(ˆθi) ˆmi #t ≤  1 − t−1 Y i=0 ˆpiˆqiˆni cos2(ˆθi) ˆmi ! 1 t   t . This completes the proof. D Results from [38] In this section, we summarize some results that we use from [38] to establish a lower bound on Qt−1 i=0 cos2(ˆθi) ˆmi and ˆqt. Proposition D.1 ([38, Proposition 2]). Let {Bt}t≥0 be the Hessian approximation matrices generated by the BFGS update in (3). For a given weight matrix P ∈Sd ++, recall the weighted vectors defined in (11) and the weighted matrix in (12). Then, we have Ψ( ˆBt+1) ≤Ψ( ˆBt) + ∥ˆyt∥2 ˆy⊤ t ˆst −1 + log cos2 ˆθt ˆmt , ∀t ≥0, where ˆmt is defined in (16) and cos(ˆθt) is defined in (14). As a corollary, we have, t−1 X i=0 log cos2(ˆθi) ˆmi ≥−Ψ( ˆB0) + t−1 X i=0 1 −∥ˆyi∥2 ˆy⊤ i ˆsi , ∀t ≥1. (40) If we take exponentiation on both sides of the above inequality (40) in Proposition D.1, we can obtain a lower bound for the product Qt−1 i=0 cos2(ˆθi) ˆmi with the sum Pt−1 i=0 ∥ˆyi∥2 ˆs⊤ i ˆyi and Ψ( ˆB0). This classical inequality describing the relationship between the ratio cos2(ˆθt) ˆmt and the potential function Ψ(.) plays a critical role in the following convergence analysis. In the following two lemmas, we provide bounds on the quantities ˆqt and ∥ˆyt∥2/ˆs⊤ t ˆyt respectively by directly citing results from Lemma 4 and Lemma 5 in [38] again. Notice that both ˆqt and ∥ˆyt∥2/ˆs⊤ t ˆyt depend on different choices of the weight matrix P. Lemma D.2 ([38, Lemma 4]). Recall the definition ˆqt = ∥ˆgt∥2 f(xt)−f(x∗) in (16). Suppose Assumptions 2.1, 2.2, and 2.3 hold. Then we have the following results: (a) If we choose P = LI, then ˆqt ≥2/κ. (b) If we choose P = ∇2f(x∗), then ˆqt ≥2/(1 + Ct)2. Lemma D.3 ([38, Lemma 5]). Let {xt}t≥0 be the iterates generated by the BFGS algorithm with inexact line search satisfying (5) and (6). Suppose Assumptions 2.1, 2.2, and 2.3 hold. Then we have the following results: (a) If we choose P = LI, then ∥ˆyt∥2 ˆs⊤ t ˆyt ≤1. (b) If we choose P = ∇2f(x∗), then ∥ˆyt∥2 ˆs⊤ t ˆyt ≤1 + Ct. 16 E Proofs in Section 4 E.1 Proof of Theorem 4.1 Recall that we choose P = LI throughout the proof. Note that given this weight matrix, it can be easily verified that ∥ˆyt∥2 ˆs⊤ t ˆyt ≤1 for any t ≥0 by using Lemma D.3 (a). Hence, we use (40) in Proposition D.1 to obtain t−1 X i=0 log cos2(ˆθi) ˆmi ≥−Ψ( ¯B0) + t−1 X i=0 1 −∥ˆyi∥2 ˆs⊤ i ˆyi ≥−Ψ( ¯B0), which further implies that t−1 Y i=0 cos2(ˆθi) ˆmi ≥e−Ψ( ¯ B0). Moreover, for the choice P = LI, it can be shown that ˆqt = ∥gt∥2 L(f(xt)−f(x∗)) ≥2 κ by using Lemma D.2 (a). From Lemma 2.1, we know ˆpt ≥α and ˆnt ≥1 −β, which lead to t−1 Y i=0 ˆpiˆniˆqi ˆmi cos2(ˆθi) ≥ t−1 Y i=0 ˆpi t−1 Y i=0 ˆqi t−1 Y i=0 ˆni t−1 Y i=0 cos2(ˆθi) ˆmi ≥ 2α(1 −β) κ t e−Ψ( ¯ B0). Thus, it follows from Proposition 3.1 that f(xt) −f(x∗) f(x0) −f(x∗) ≤  1 − t−1 Y i=0 ˆpiˆqiˆni ˆmi cos2(ˆθi) ! 1 t   t ≤ 1 −e−Ψ( ¯ B0) t 2α(1 −β) κ t . This completes the proof. E.2 Proof of Corollary 4.2 Notice that in the first case where B0 = LI, we have Ψ( ¯B0) = 0 and thus it achieves the best linear convergence results according to Theorem 4.1. On the other hand, for B0 = µI, we have Ψ( ¯B0) = Ψ( µ LI) = d( 1 κ −1 + log κ) ≤d log κ. We complete the proof by combining these conditions with the inequality (17) in Theorem 4.1. Notice that e−x ≥e−1 ≥1 3 for x ≤1. F Proofs in Section 5 F.1 Proof of Proposition 5.1 Recall that we choose the weight matrix as P = ∇2f(x∗) throughout the proof. Similar to the proof of Theorem 4.1, we start from the key inequality in (15), but we apply different bounds on the ˆqt and cos2(ˆθt) ˆmt . Specifically, by using Lemma D.3 (b), we have ∥ˆyt∥2 ˆs⊤ t ˆyt ≤1 + Ct for any t ≥0. Hence, we use (40) in Proposition D.1 to obtain t−1 X i=0 log cos2(ˆθi) ˆmi ≥−Ψ( ˜B0) + t−1 X i=0 1 −∥ˆyi∥2 ˆs⊤ i ˆyi ≥−Ψ( ˜B0) − t−1 X i=0 Ci, which further implies that t−1 Y i=0 cos2(ˆθi) ˆmi ≥e−Ψ( ˜ B0)−Pt−1 i=0 Ci. (41) Moreover, since ˆqt ≥ 2 (1+Ct)2 for any t ≥0 by using Lemma D.2 (b), we get t−1 Y i=0 ˆqi ≥ t−1 Y i=0 2 (1 + Ci)2 ≥2t t−1 Y i=0 e−2Ci = 2te−2 Pt−1 i=0 Ci, (42) 17 where we use the inequality 1 + x ≤ex for any x ∈R. From Lemma 2.1, we know ˆpt ≥α and ˆnt ≥1 −β, which lead to t−1 Y i=0 ˆpiˆni ≥αt(1 −β)t. (43) Combining (41), (42), (43) and (15) from Proposition 3.1, we prove that f(xt) −f(x∗) f(x0) −f(x∗) ≤  1 − t−1 Y i=0 ˆpiˆqiˆni ˆmi cos2(ˆθi) ! 1 t   t ≤ 1 −2α(1 −β)e− Ψ( ˜ B0)+3 Pt−1 i=0 Ci t t . This completes the proof. F.2 Proof of Theorem 5.2 When we have t ≥Ψ( ˜B0) + 3 Pt−1 i=0 Ci, Proposition 5.1 implies the condition that f(xt)−f(x∗) f(x0)−f(x∗) ≤ 1 −2α(1−β) e t ≤ 1 −2α(1−β) 3 t , which leads to the linear rate in (20). Hence, it is sufficient to establish an upper bound on Pt−1 i=0 Ci. Recall that Ci = M µ 3 2 p 2(f(xi) −f(x∗)) defined in (9). We decompose the sum into two parts: P⌈Ψ( ¯ B0)⌉−1 i=0 Ci and Pt i=⌈Ψ( ¯ B0)⌉Ci. For the first part, note that since f(xi+1) ≤f(xi) by Lemma 2.1, we also have Ci+1 ≤Ci for i ≥0. Hence, we have P⌈Ψ( ¯ B0)⌉−1 i=0 Ci ≤C0⌈Ψ( ¯B0)⌉≤C0(Ψ( ¯B0) + 1). Moreover, by Theorem 4.1, when t ≥Ψ( ¯B0) we have f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −e−Ψ( ¯ B0) t 2α(1 −β) κ t ≤ 1 −2α(1 −β) eκ t ≤ 1 −2α(1 −β) 3κ t . Hence, this further implies that t X i=⌈Ψ( ¯ B0)⌉ Ci = C0 t X i=⌈Ψ( ¯ B0)⌉ s f(xi) −f(x∗) f(x0) −f(x∗) ≤C0 t X i=⌈Ψ( ¯ B0)⌉ 1 −2α(1 −β) 3κ i 2 ≤C0 ∞ X i=1 1 −2α(1 −β) 3κ i 2 ≤C0 3κ α(1 −β) −1 , where we used the fact that P∞ i=1(1 −ρ) i 2 = √1−ρ 1−√1−ρ = √1−ρ+1−ρ ρ ≤2 ρ −1 for any ρ ∈(0, 1). Hence, by combining both inequalities, we have t−1 X i=0 Ci = ⌈Ψ( ¯ B0)⌉−1 X i=0 Ci + t X i=⌈Ψ( ¯ B0)⌉ Ci ≤C0Ψ( ¯B0) + 3C0κ α(1 −β). (44) Hence, this proves that (20) is satisfied when t ≥Ψ( ˜B0) + 3C0Ψ( ¯B0) + 9C0κ α(1−β). F.3 Proof of Corollary 5.3 For B0 = LI, we have ¯B0 = 1 LB0 = I and ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 = L∇2f(x∗)−1. Thus, it holds that Ψ( ¯B0) = Ψ(I) = 0. Moreover, by Assumptions 2.1 and 2.2, we have 1 LI ⪯ ∇2f(x∗)−1 ⪯1 µI, which implies that I ⪯˜B0 ⪯κI. Thus, we further have Ψ( ˜B0) ≤Tr(κI) −d −log Det(I) = dκ −d ≤dκ. Combining these two results, the threshold for transition time can be bounded by Ψ( ˜B0) + 3C0Ψ( ¯B0) + 9 α(1−β)C0κ ≤dκ + 9 α(1−β)C0κ. Hence, by Theorem 5.2, the linear rate in (20) is achieved when t ≥dκ + 9 α(1−β)C0κ. 18 For B0 = µI, we have ¯B0 = 1 LB0 = 1 κI and ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 = µ∇2f(x∗)−1. Thus, it holds that Ψ( ¯B0) = Ψ( 1 κI) = d κ −d + d log κ ≤d log κ. Moreover, by Assumptions 2.1 and 2.2, we have 1 κI ⪯˜B0 ⪯I. This implies that Ψ( ˜B0) = Tr( ˜B0) −d −log Det( ˜B0) ≤Tr(I) −d −log Det( 1 κI) = d log κ. Combining these two results, the threshold for the transition tume can be bounded by Ψ( ˜B0) + 3C0Ψ( ¯B0) + 9 α(1−β)C0κ ≤(1 + 3C0)d log κ + 9 α(1−β)C0κ. Hence, by Theorem 5.2, the linear rate in (20) is satisfied when t ≥(1 + 3C0)d log κ + 9 α(1−β)C0κ. G Intermediate Results and Proofs in Section 6 G.1 Intermediate Results To present our result we first introduce the following function ω(x) := x −log (x + 1), (45) which is defined for x > −1. Further In the next result, we present some basic properties of the function ω(x) defined in (45). Lemma G.1. Recall the definition of function ω(x) in (45), we have that (a) ω(x) is increasing function for x > 0 and decreasing function for −1 < x < 0. Moreover, ω(x) ≥0 for all x > −1. (b) When x ≥0, we have that ω(x) ≥ x2 2(1+x). (c) When −1 < x ≤0, we have that ω(x) ≥ x2 2+x. Proof. Notice that ω′(x) = x 1+x, we know that when x > 0, ω′(x) > 0 and when −1 < x < 0, ω′(x) < 0, ω′(x) < 0. Therefore, ω(x) is increasing function for x > 0 and ω(x) is decreasing function for −1 < x < 0. Hence, ω(x) ≥ω(0) = 0 for all x > −1. ω(x) ≥ x2 2(1+x) is equivalent to ω1(x) := 2(1 + x)ω(x) −x2 ≥0. Since ω′ 1(x) = 2x − 2 log (1 + x) = 2ω(x) ≥0 for all x > −1, we know that ω1(x) is increasing function for x > −1 and hence, ω1(x) ≥ω1(0) = 0 for x ≥0. ω(x) ≥ x2 2+x is equivalent to ω2(x) := (2+x)ω(x)−x2 ≥0. Since ω′ 2(x) = x 1+x −log (1 + x) ≤0 for all x > −1, we know that ω2(x) is decreasing function for x > −1 and hence, ω2(x) ≥ω2(0) = 0 for x ≤0. Proposition G.2. Let {Bt}t≥0 be the Hessian approximation matrices generated by the BFGS update in (3). Suppose Assumptions 2.1, 2.2, and 2.3 hold and f(xt+1) ≤f(xt) for any t ≥0. Recall the definition of Ψ(.) in (10) and Ct in (9), we have that t−1 X i=0 ω(ρi −1) ≤Ψ( ˜B0) + 2 t−1 X i=0 Ci, ∀t ≥1, (46) Proof. First, taking the trace and determinant on both sides of the equation (13) for any weight matrix P ∈Sd ++ and using results from Lemma 6.2 of [34], we show that Tr( ˆBt+1) = Tr( ˆBt) −∥ˆBtˆst∥2 ˆs⊤ t ˆBtˆst + ∥ˆyt∥2 ˆs⊤ t ˆyt , Det( ˆBt+1) = Det( ˆBt) ˆs⊤ t ˆyt ˆs⊤ t ˆBtˆst . Taking the logarithm on both sides of the second equation, we obtain that log ˆs⊤ t ˆyt ˆs⊤ t ˆBtˆst = log Det( ˆBt+1) −log Det( ˆBt). 19 Thus, we obtain that Ψ( ˆBt+1) −Ψ( ˆBt) = Tr( ˆBt+1) −Tr( ˆBt) + log Det( ˆBt) −log Det( ˆBt+1) = ∥ˆyt∥2 ˆs⊤ t ˆyt −∥ˆBtˆst∥2 ˆs⊤ t ˆBtˆst −log ˆs⊤ t ˆyt ˆs⊤ t ˆBtˆst = ∥ˆyt∥2 ˆs⊤ t ˆyt −∥ˆBtˆst∥2 ˆs⊤ t ˆBtˆst −log ˆs⊤ t ˆyt ∥ˆst∥2 −log ∥ˆst∥2 ˆs⊤ t ˆBtˆst , which leads to ∥ˆBtˆst∥2 ˆs⊤ t ˆBtˆst −log ˆs⊤ t ˆBtˆst ∥ˆst∥2 −1 = Ψ( ˆBt) −Ψ( ˆBt+1) + ∥ˆyt∥2 ˆs⊤ t ˆyt −1 + log ∥ˆst∥2 ˆs⊤ t ˆyt . Notice that ˆBtˆst = −ηtˆgt, ˆs⊤ t ˆBtˆst = −η2 t ˆg⊤ t ˆdt and ∥ˆst∥2 = η2 t ∥ˆdt∥2, we have that ∥ˆgt∥2 −ˆg⊤ t ˆdt −log −ˆg⊤ t ˆdt ∥ˆdt∥2 −1 = Ψ( ˆBt) −Ψ( ˆBt+1) + ∥ˆyt∥2 ˆs⊤ t ˆyt −1 + log ∥ˆst∥2 ˆs⊤ t ˆyt . Note that given the fact that −ˆg⊤ t ˆdt = ˆg⊤ t ˆB−1 t ˆgt > 0, by using the Cauchy–Schwarz inequality we obtain ∥ˆgt∥2 −ˆg⊤ t ˆdt ≥−ˆg⊤ t ˆdt ∥ˆdt∥2 . Hence, we can write −ˆg⊤ t ˆdt ∥ˆdt∥2 −log −ˆg⊤ t ˆdt ∥ˆdt∥2 −1 ≤Ψ( ˆBt) −Ψ( ˆBt+1) + ∥ˆyt∥2 ˆs⊤ t ˆyt −1 + log ∥ˆst∥2 ˆs⊤ t ˆyt . Now, by selecting the weight matrix as P = ∇2f(x∗), many expressions get simplified and we have −ˆg⊤ t ˆdt ∥ˆdt∥2 = −g⊤ t dt ∥˜dt∥2 = ρt, ρt −log ρt −1 = ω(ρt −1), and ˆBt = ˜Bt = ∇2f(x∗)−1 2 Bt∇2f(x∗)−1 2 . Hence, we have ω(ρt −1) ≤Ψ( ˜Bt) −Ψ( ˜Bt+1) + ∥ˆyt∥2 ˆs⊤ t ˆyt −1 + log ∥ˆst∥2 ˆs⊤ t ˆyt . (47) Notice that ∥ˆyt∥2 ˆs⊤ t ˆyt ≤1 + Ct for any t ≥0 by using Lemma D.3 (b) with P = ∇2f(x∗) and log ∥ˆst∥2 ˆs⊤ t ˆyt = log ∥ˆst∥2 ˆs⊤ t ˆ Jtˆst ≤log(1 + Ct) ≤Ct for any t ≥0 by using (31) from Lemma A.1. Leveraging these conditions with the inequality (47), we obtain that ω(ρt −1) ≤Ψ( ˜Bt) −Ψ( ˜Bt+1) + 2Ct. Summing both sides of the above inequality from i = 0 to t −1, we prove the conclusion t−1 X i=0 ω(ρi −1) ≤Ψ( ˜B0) −Ψ( ˜Bt) + 2 t−1 X i=0 Ci ≤Ψ( ˜B0) + 2 t−1 X i=0 Ci, where the last inequality holds since Ψ( ˜Bt) ≥0. Lemma G.3. Suppose Assumptions 2.1, 2.2, and 2.3 hold and Ct ≤1 6 and ρt ≥7 8 at iteration t, then we have f(xt + dt) ≤f(xt). (48) Proof. Since assumption 2.3 hold, using Lemma 1.2.4 in [44], we have that \|f(y) −f(x) −∇f(x)⊤(y −x) −1 2(y −x)⊤∇2f(x)(y −x)\| ≤M 6 ∥y −x∥3, ∀x, y ∈Rd. Setting x = xt and y = xt + dt, we have that f(xt + dt) −f(xt) ≤g⊤ t dt + 1 2d⊤ t ∇2f(xt)dt + M 6 ∥dt∥3. (49) Notice that using (33) from Lemma A.1 and the definition of ρt in (21), we have that d⊤ t ∇2f(xt)dt ≤(1 + Ct)d⊤ t ∇2f(x∗)dt = −g⊤ t dt(1 + Ct) ∥˜dt∥2 −g⊤ t dt = −g⊤ t dt 1 + Ct ρt . (50) 20 Applying Assumption 2.1 with the definition ˜dt = ∇2f(x∗) 1 2 dt, we obtain that ∥dt∥3 ≤1 µ 3 2 ∥˜dt∥3 = −g⊤ t dt µ 3 2 ∥˜dt∥2 −g⊤ t dt ∥˜dt∥= −g⊤ t dt µ 3 2 1 ρt ∥˜dt∥. Since −˜g⊤ t ˜dt ≤∥˜gt∥∥˜dt∥by Cauchy–Schwarz inequality where ˜gt = ∇2f(x∗)−1 2 gt, we obtain ∥˜dt∥= ∥˜gt∥∥˜dt∥ ∥˜gt∥≤∥˜gt∥∥˜dt∥2 −˜g⊤ t ˜dt = 1 ρt ∥˜gt∥, which leads to ∥dt∥3 ≤−g⊤ t dt µ 3 2 1 ρt ∥˜dt∥≤−g⊤ t dt µ 3 2 1 ρ2 t ∥˜gk∥. (51) By applying Taylor’s theorem with Lagrange remainder, there exists ˜τt ∈[0, 1] such that f(xt) = f(x∗) + ∇f(x∗)⊤(xt −x∗) + 1 2(xt −x∗)⊤∇2f(xt + ˜τt(x∗−xt))(xt −x∗) = f(x∗) + 1 2(xt −x∗)⊤∇2f(xt + ˜τt(x∗−xt))(xt −x∗), (52) where we used the fact that ∇f(x∗) = 0 in the last equality. Moreover, by the fundamental theorem of calculus, we have ∇f(xt) −∇f(x∗) = Z 1 0 ∇2f(xt + τ(x∗−xt))(xt −x∗) dτ = Gt(xt −x∗), where we use the definition of Gt in (8). Since ∇f(x∗) = 0 and we denote gt = ∇f(xt), this further implies that xt −x∗= G−1 t (∇f(xt) −∇f(x∗)) = G−1 t gt. (53) Combining (52) and (53) leads to f(xt) −f(x∗) = 1 2g⊤ t G−1 t ∇2f(xt + ˜τt(x∗−xt))G−1 t gt. (54) Based on (35) in Lemma A.1, we have ∇2f(xt + ˜τt(x∗−xt)) ⪰ 1 1+Ct Gt, which implies that G−1 t ∇2f(xt + ˜τt(x∗−xt))G−1 t ⪰ 1 1 + Ct G−1 t . Moreover, it follows from (34) in Lemma A.1 that Gt ⪯(1 + Ct)∇2f(x∗), which implies that G−1 t ⪰ 1 1 + Ct (∇2f(x∗))−1. Combining the above two conditions, we obtain that G−1 t ∇2f(xt + ˜τt(x∗−xt))G−1 t ⪰ 1 (1 + Ct)2 (∇2f(x∗))−1, and hence g⊤ t G−1 t ∇2f(xt + ˜τt(x∗−xt))G−1 t gt ≥ 1 (1 + Ct)2 g⊤ t (∇2f(x∗))−1gt = 1 (1 + Ct)2 ∥˜gt∥2. (55) Combining (54) and (55) leads to ∥˜gk∥≤(1 + Ct) p 2(f(xt) −f(x∗)). (56) Combining (51) and (56) leads to ∥dt∥3 ≤−g⊤ t dt µ 3 2 1 ρ2 t ∥˜gk∥≤−g⊤ t dt µ 3 2 1 ρ2 t (1 + Ct) p 2(f(xt) −f(x∗)). (57) 21 Leveraging (49), (50) and (57) with the definition of Ct in (9), we have that f(xt + dt) −f(xt) ≤g⊤ t dt + 1 2d⊤ t ∇2f(xt)dt + M 6 ∥dt∥3 = −g⊤ t dt(−1 + 1 + Ct 2ρt + M 6 1 µ 3 2 1 ρ2 t (1 + Ct) p 2(f(xt) −f(x∗))) = −g⊤ t dt(−1 + 1 + Ct 2ρt + Ct(1 + Ct) 6ρ2 t ). (58) Notice that −g⊤ t dt = −g⊤ t B−1 t gt > 0 and when Ct ≤1 6 and ρt ≥7 8, we can verify that 1 + Ct 2ρt + Ct(1 + Ct) 6ρ2 t < 1. Therefore, (58) implies the conclusion that f(xt + dt) −f(xt) ≤0. G.2 Proof of Lemma 6.1 Since ηt = 1 satisfies Armijo-Wolfe conditions, we know that ηt is chosen to be one at iteration t and xt+1 = xt + dt. We have f(xt+1) ≤f(xt) from Lemma 2.1. Using Taylor’s expansion, we have that f(xt+1) = f(xt) + g⊤ t dt + 1 2d⊤ t ∇2f(xt + ˆτ(xt+1 −xt))dt, where ˆτ ∈[0, 1]. Hence, we have that ˆpt = f(xt) −f(xt+1) −g⊤ t dt = −g⊤ t dt −1 2d⊤ t ∇2f(xt + ˆτ(xt+1 −xt))dt −g⊤ t dt = 1 −1 2 d⊤ t ∇2f(xt + ˆτ(xt+1 −xt))dt −g⊤ t dt ≥1 −1 + Ct 2 d⊤ t ∇2f(x∗)dt −g⊤ t dt = 1 −1 + Ct 2ρt , where we apply the (32) from Lemma A.1 since f(xt+1) ≤f(xt) and recall the definition of ρt in (21). Similarly, using (31) from Lemma A.1 since f(xt+1) ≤f(xt), we have that ˆnt = y⊤ t st −g⊤ t st = s⊤ t Jtst −g⊤ t st = d⊤ t Jtdt −g⊤ t dt ≥ 1 1 + Ct d⊤ t ∇2f(x∗)dt −g⊤ t dt = 1 (1 + Ct)ρt , where we use the fact that yt = Jtst with Jt defined in (8) and st = xt+1 −xt = dt. Therefore, we prove the conclusions. G.3 Proof of Lemma 6.2 Denote ¯xt+1 = xt + dt and ¯st = ¯xt+1 −xt = dt. Since δ1 ≤1 6 and δ2 ≥7 8, we have f(¯xt+1) ≤ f(xt) from Lemma G.3. Using Taylor’s expansion, we have that f(¯xt+1) = f(xt) + g⊤ t dt + 1 2d⊤ t ∇2f(xt + ˆτ(¯xt+1 −xt))dt, where ˆτ ∈[0, 1]. Hence, we have f(xt) −f(¯xk+1) −g⊤ t dt = −g⊤ t dt −1 2d⊤ t ∇2f(xt + ˆτ(¯xt+1 −xt))dt −g⊤ t dt = 1 −1 2 d⊤ t ∇2f(xt + ˆτ(¯xt+1 −xt))dt −g⊤ t dt ≥1 −1 + Ct 2 d⊤ t ∇2f(x∗)dt −g⊤ t dt = 1 −1 + Ct 2ρt , where we apply the (32) from Lemma A.1 since f(¯xt+1) ≤f(xt). Therefore, when Ct ≤δ1 ≤ p 2(1 −α) −1 and ρt ≥δ2 ≥ 1 √ 2(1−α), we obtain that f(xt)−f(¯xk+1) −g⊤ t dt ≥1 −1+Ct 2ρt ≥α and unit step size ηt = 1 satisfies the sufficient condition (5). Similarly, using (31) from Lemma A.1 since f(¯xt+1) ≤f(xt) and denote ¯gk+1 = ∇f(¯xt+1), ¯yt = ¯gk+1 −gt, we have that ¯y⊤ t ¯st −g⊤ t ¯st = ¯s⊤ t Jt¯st −g⊤ t ¯st = d⊤ t Jtdt −g⊤ t dt ≥ 1 1 + Ct d⊤ t ∇2f(x∗)dt −g⊤ t dt = 1 (1 + Ct)ρt . 22 Therefore, when Ct ≤δ1 ≤ 1 √1−β −1 and ρt ≤δ3 = 1 √1−β , we obtain that ¯y⊤ t ¯st −g⊤ t ¯st ≥ 1 (1+Ct)ρt ≥ 1 −β, which indicates that ¯g⊤ t+1dt = ¯g⊤ t+1¯st = ¯y⊤ t ¯st + g⊤ t ¯st ≥−g⊤ t ¯st(1 −β) + g⊤ t ¯st = βg⊤ t ¯st = βg⊤ t dt. Hence, unit step size ηt = 1 satisfies the curvature condition (6). Therefore, we prove that when Ct ≤δ1 and δ2 ≤ρt ≤δ3, step size ηt = 1 satisfies the Armijo-Wolfe conditions (5) and (6). G.4 Proof of Lemma 6.3 Since in Theorem 4.1, we already prove that f(xt) −f(x∗) f(x0) −f(x∗) ≤ 1 −e−Ψ( ¯ B0) t 2α(1 −β) κ t . This implies that Ct ≤ 1 −e−Ψ( ¯ B0) t 2α(1 −β) κ t 2 C0. When t ≥Ψ( ¯B0), we obtain that Ct ≤ 1 −2α(1 −β) 3κ t 2 C0. When t ≥ 3κ α(1−β) log C0 δ1 , we obtain that Ct ≤ 1 −2α(1 −β) 3κ t 2 C0 ≤δ1. Therefore, the first claim in (24) follows. Now define I1 = {t : ρt < δ2} and I2 = {t : ρt > δ3}, we know that \|I\| = \|I1\| + \|I2\|. Notice that for t ∈I1, we have that ρt −1 < δ2 −1 < 0 since δ2 < 1 and the function ω(x) defined in (45) is decreasing for −1 < x < 0 from (a) in Lemma G.1. Hence, we have that P i∈I1 ω(ρi −1) ≥P i∈I1 ω(δ2 −1) = ω(δ2 −1)\|I1\|. Similarly, we have that for t ∈I2, we have that ρi −1 > δ3 −1 > 0 since δ3 > 1 and the function ω(x) is increasing for x > 0 from (a) in Lemma G.1. Hence, we have that P i∈I2 ω(ρi −1) ≥P i∈I2 ω(δ3 −1) = ω(δ3 −1)\|I2\|. Using (46) from Proposition G.2, we have that Pt−1 i=0 ω(ρi −1) ≤Ψ( ˜B0) + 2 Pt−1 i=0 Ci ≤Ψ( ˜B0) + 2 P+∞ i=0 Ci for any t ≥1. Therefore, we obtain that Ψ( ˜B0) + 2 +∞ X i=0 Ci ≥ +∞ X i=0 ω(ρi −1) ≥ X i∈I1 ω(βi −1) + X i∈I2 ω(βi −1) ≥ω(δ2 −1)\|I1\| + ω(δ3 −1)\|I2\| ≥min{ω(δ2 −1), ω(δ3 −1)}(\|I1\| + \|I2\|), which leads to the result \|I\| = \|I1\| + \|I2\| ≤ Ψ( ˜B0) + 2 P+∞ i=0 Ci min{ω(δ2 −1), ω(δ3 −1)} = δ4 Ψ( ˜B0) + 2 +∞ X i=0 Ci ! , (59) where δ4 := 1 min{ω(δ2−1),ω(δ3−1)}. Using the upper bound of P+∞ i=0 Ci ≤C0Ψ( ¯B0) + 3C0κ α(1−β) in (44), we prove the second claim in (25). G.5 Proof of Theorem 6.4 First, we prove that for any initial point x0 ∈Rd and any initial Hessian approximation matrix B0 ∈Sd ++, the following result holds: f(xt) −f(x∗) f(x0) −f(x∗) ≤ δ6t0 + δ7Ψ( ˜B0) + δ8 P+∞ i=0 Ci t !t , ∀t > t0, 23 where t0 is defined in (24). We choose the weight matrix as P = ∇2f(x∗) throughout the proof. Using results (41) and (42) from the proof of Proposition 5.1, we obtain that t−1 Y i=0 cos2(ˆθi) ˆmi ≥e−Ψ( ˜ B0)−Pt−1 i=0 Ci ≥e−Ψ( ˜ B0)−P+∞ i=0 Ci. (60) t−1 Y i=0 ˆqi ≥2te−2 Pt−1 i=0 Ci ≥2te−2 P+∞ i=0 Ci. (61) Recall the definition of the set I = {t : ρt /∈[δ2, δ3]}. Notice that for t ≥t0, define I3 = {t : t ≥ t0, ρt /∈[δ2, δ3]} and I4 = {t : t ≥t0, ρt ∈[δ2, δ3]}. Then, we have that t−1 Y i=0 ˆpiˆni = t0−1 Y i=0 ˆpiˆni t−1 Y i=t0 ˆpiˆni = t0−1 Y i=0 ˆpiˆni Y i∈I3 ˆpiˆni Y i∈I4 ˆpiˆni. (62) From Lemma 2.1, we know ˆpt ≥α and ˆnt ≥1 −β for any t ≥0, which lead to t0−1 Y i=0 ˆpiˆni ≥αt0(1 −β)t0 = 1 2t0 e−t0 log 1 2α(1−β) . (63) Y i∈I3 ˆpiˆni ≥ Y i∈I3 α(1 −β) = 1 2\|I3\| e−\|I3\| log 1 2α(1−β) ≥ 1 2\|I3\| e−\|I\| log 1 2α(1−β) ≥ 1 2\|I3\| e −δ4 Ψ( ˜ B0)+2 P+∞ i=0 Ci log 1 2α(1−β) , (64) where the second inequality holds since \|I3\| ≤\|I\|, log 1 2α(1−β) > 0 and the last inequality holds since (59) from the proof of Lemma 6.3 in Appendix G.4. Notice that when index i ∈I4, we have Ci ≤δ1 from Lemma 6.3 and ρi ∈[δ2, δ3]. Applying Lemma 6.1 and Lemma 6.2, we know that for i ∈I4, ηi = 1 satisfies the Armijo-Wolfe conditions (5), (6) and we have ˆpi ≥1 −1+Ci 2ρi > 0 (since Ci ≤δ1 ≤1 6, ρi ≥δ2 ≥7 8) and ˆni ≥ 1 (1+Ci)ρi from (22). Hence, we obtain that Y i∈I4 ˆpiˆni ≥ 1 2\|I4\| Y i∈I4 (2 −1 + Ci ρi ) 1 (1 + Ci)ρi ≥ 1 2\|I4\| e−P i∈I4 Ci Y i∈I4 (2 −1 + Ci ρi ) 1 ρi , (65) where the last inequality holds since 1 1+Ci ≥e−Ci. Using the fact that log x ≥1 −1 x, we obtain Y i∈I4 (2 −1 + Ci ρi ) 1 ρi = Y i∈I4 elog (2−1+Ci ρi )−log ρi ≥ Y i∈I4 e 1− 1 2−1+Ci ρi −log ρi = Y i∈I4 e ρi−1−Ci 2ρi−1−Ci −log ρi = Y i∈I4 e ρi−1−log ρi+2(1−ρi) log ρi−(1−log ρi)Ci 2ρi−1−Ci = Y i∈I4 e ω(ρi−1)+2(1−ρi) log ρi−(1−log ρi)Ci 2ρi−1−Ci ≥ Y i∈I4 e −2(ρi−1) log ρi−(1−log ρi)Ci 2ρi−1−Ci = Y i∈I4 e−2(ρi−1) log ρi+(1−log ρi)Ci 2ρi−1−Ci ≥ Y i∈I4 e−2(ρi−1) log ρi+(1−log δ2)Ci 2δ2−1−δ1 , (66) where the second inequality holds since ω(ρi −1) ≥0 and the third inequality holds since ρi ≥δ2 due to i ∈I4 and Ci ≤δ1 due to i ≥t0 and Lemma 6.3. Notice that 2ρi −1−Ci ≥2δ2 −1−δ1 > 0 for all i ∈I4 since Ci ≤δ1 ≤1 6 and ρi ≥δ2 ≥7 8. When ρi ≥1, using log ρi ≤ρi −1, (b) in Lemma G.1 and ρi ≤δ3 due to i ∈I4, we have that (ρi −1) log ρi ≤(ρi −1)2 ≤2ρiω(ρi −1) ≤2δ3ω(ρi −1). (67) Similarly, when ρi < 1, using log ρi ≥1 −1 ρi , (c) in Lemma G.1 and ρi ≥δ2 due to i ∈I4, we have (ρi −1) log ρi ≤(ρi −1)2 ρi ≤ρi + 1 ρi ω(ρi −1) ≤(1 + 1 δ2 )ω(ρi −1). (68) 24 Combining (66), (67) and (68), we obtain that Y i∈I4 (2 −1 + Ci ρi ) 1 ρi ≥ Y i∈I4 e−2(ρi−1) log ρi+(1−log δ2)Ci 2δ2−1−δ1 = Y i∈I4 e−2(ρi−1) log ρi 2δ2−1−δ1 Y i∈I4 e−(1−log δ2)Ci 2δ2−1−δ1 = Y i∈I4,ρi<1 e−2(ρi−1) log ρi 2δ2−1−δ1 Y i∈I4,ρi≥1 e−2(ρi−1) log ρi 2δ2−1−δ1 Y i∈I4 e−(1−log δ2)Ci 2δ2−1−δ1 ≥ Y i∈I4,ρi<1 e− 2(1+ 1 δ2 )ω(ρi−1) 2δ2−1−δ1 Y i∈I4,ρi≥1 e−4δ3ω(ρi−1) 2δ2−1−δ1 Y i∈I4 e−(1−log δ2)Ci 2δ2−1−δ1 = e− 2+ 2 δ4 2δ2−1−δ1 P i∈I2,ρi<1 ω(ρi−1)− 4δ3 2δ2−1−δ1 P i∈I4,ρi≥1 ω(ρi−1)− 1−log δ2 2δ2−1−δ1 P i∈I4 Ci ≥e −δ5 P i∈I4,ρi<1 ω(ρi−1)+P i∈I4,ρi≥1 ω(ρi−1) − 1−log δ2 2δ2−1−δ1 P i∈I4 Ci = e−δ5 P i∈I4 ω(ρi−1)− 1−log δ2 2δ2−1−δ1 P i∈I4 Ci (69) where δ5 = max{ 2+ 2 δ2 2δ2−1−δ1 , 4δ3 2δ2−1−δ1 }. Combining (65) and (69), we obtain that Y i∈I4 ˆpiˆni ≥ 1 2\|I4\| e−P i∈I4 Ci Y i∈I4 (2 −1 + Ci ρi ) 1 ρi ≥ 1 2\|I4\| e−δ5 P i∈I4 ω(ρi−1)−(1+ 1−log δ2 2δ2−1−δ1 ) P i∈I4 Ci ≥ 1 2\|I4\| e−δ5 P+∞ i=0 ω(ρi−1)−2δ2−δ1−log δ2 2δ2−1−δ1 P+∞ i=0 Ci ≥ 1 2\|I4\| e −δ5 Ψ( ˜ B0)+2 P+∞ i=0 Ci −2δ2−δ1−log δ2 2δ2−1−δ1 P+∞ i=0 Ci, (70) where the last inequality is due to (46) from Lemma G.1. Combining (62), (63), (64) and (70), we obtain that t−1 Y i=0 ˆpiˆni = t0−1 Y i=0 ˆpiˆni Y i∈I3 ˆpiˆni Y i∈I4 ˆpiˆni ≥1 2t e − t0 log 1 2α(1−β) +(δ4 log 1 2α(1−β) +δ5)Ψ( ˜ B0)+(2δ4 log 1 2α(1−β) +2δ5+ 2δ2−δ1−log δ2 2δ2−1−δ1 ) P+∞ i=0 Ci . (71) Leveraging (60), (61), (71) with (15) from Proposition 3.1, we prove that f(xt) −f(x∗) f(x0) −f(x∗) ≤  1 − t−1 Y i=0 ˆpiˆqiˆni cos2(ˆθi) ˆmi ! 1 t   t =  1 − t−1 Y i=0 ˆpiˆni t−1 Y i=0 ˆqi t−1 Y i=0 cos2(ˆθi) ˆmi ! 1 t   t ≤ 1 −e− t0 log 1 2α(1−β) +(1+δ4 log 1 2α(1−β) +δ5)Ψ( ˜ B0)+(3+2δ4 log 1 2α(1−β) +2δ5+ 2δ2−δ1−log δ2 2δ2−1−δ1 ) P+∞ i=0 Ci t !t = 1 −e− δ6t0+δ7Ψ( ˜ B0)+δ8 P+∞ i=0 Ci t t ≤ δ6t0 + δ7Ψ( ˜B0) + δ8 P+∞ i=0 Ci t !t , where the inequality is due to the fact that 1 −e−x ≤x for any x ∈R and δ6, δ7, δ8 are defined in Theorem 6.4. Hence, we prove that f(xt) −f(x∗) f(x0) −f(x∗) ≤ δ6t0 + δ7Ψ( ˜B0) + δ8 P+∞ i=0 Ci t !t , ∀t > t0. (72) 25 Using (44) from the proof of Theorem 5.2 in Appendix F.2, we have that +∞ X i=0 Ci ≤C0Ψ( ¯B0) + 3C0κ α(1 −β). (73) Notice that from (24) in Lemma 6.3, we have that t0 = max{Ψ( ¯B0), 3κ α(1 −β) log C0 δ1 } ≤Ψ( ¯B0) + 3κ α(1 −β) log C0 δ1 . (74) Leveraging (72), (73) and (74), we prove the conclusion. G.6 Proof of Corollary 6.5 Using the fact that for B0 = LI, we have Ψ( ¯B0) = 0 and Ψ( ˜B0) ≤dκ, and for the case that B0 = µI, we have Ψ( ¯B0) ≤d log κ, and Ψ( ˜B0) ≤d log κ, we obtain the corresponding superlinear results for these two conditions. G.7 Specific Values of {δi}8 i=1 As we stated before, all the {δi}8 i=1 are universal constants that only depend on line search parameters α and β. We can choose specific values of α and β to make definitions of {δi}8 i=1 more clear. If we pick α = 1 4 and β = 3 4, we have that δ1 = 1 6, δ2 = 7 8, δ3 = 2, δ4 = 118, δ5 = 14, δ6 = log 8, δ7 = 260, δ8 = 524. H Complexity of BFGS with the Initialization B0 = cI Recall that c ∈[µ, L] by our choice of c in Remark 6.2. If we choose B0 = cI, then Ψ( ¯B0) = Ψ( c LI) = c Ld −d + d log L c . Moreover, we have Ψ( ˜B0) = Ψ(c∇2f(x∗)−1) = cTr(∇2f(x∗)−1) − d −log Det(c∇2f(x∗)−1), which is determined by the Hessian matrix ∇2f(x∗)−1. In this case, one can use the upper bounds Ψ( ¯B0) = d( c L −1 + log L c ) and Ψ( ˜B0) = Tr(c∇2f(x∗)−1) −d − log Det(c∇2f(x∗)−1) ≤d( c µ −1 + log L c ) to simplify the expressions. Applying these values of Ψ( ¯B0) and Ψ( ˜B0) to our linear convergence result in Theorem 4.1 and the superlinear convergence result in Theorem 6.4, we can obtain the following convergence guarantees for B0 = cI: • For t ≥d( c L −1 + log L c ), we have f(xt)−f(x∗) f(x0)−f(x∗) ≤ 1 −2α(1−β) 3κ t ; • For t = Ω(d( c µ −1 + log L c ) + C0d( c L −1 + log L c ) + C0κ), we have f(xt)−f(x∗) f(x0)−f(x∗) ≤ O d( c µ −1+log L c )+C0d( c L −1+log L c )+C0κ t t. Moreover, we can derive similar iteration complexity bounds following the same arguments as in Section I. We also include the performance of BFGS with B0 = cI in our numerical experiments as presented in Figure 1. We observe that the performance of BFGS with B0 = cI is very similar to the convergence curve of BFGS with B0 = µI in our numerical experiments. I Proof of Iteration Complexity When B0 = LI, if we regard the line search parameters α and β as absolute constants, the first result established in Corollary 4.2 leads to a global complexity of O(κ log 1 ϵ ), which is on par with gradient descent. Moreover, the first result in Corollary 5.3 implies a complexity of O (d + C0)κ + log 1 ϵ , where the first term represents the number of iterations required to attain the linear rate in (20), and the second term represents the additional number of iterations needed to achieve the desired accuracy 26 ϵ from the condition number-independent linear rate. For the analysis of the superlinear convergence rate, we denote that ΩL = dκ + C0κ. From the first result in Corollary 6.5, we have that f(xt) −f(x∗) f(x0) −f(x∗) ≤(ΩL t )t Let T∗be the number such that the inequality ( ΩL t )t ≤ϵ above becomes equality. we have log 1 ϵ = T∗log T∗ ΩL ≤T∗( T∗ ΩL −1), T∗≥ ΩL + q Ω2 L + 4ΩL log 1 ϵ 2 . Hence, we have that log 1 ϵ = T∗log T∗ ΩL ≥T∗log ΩL + q Ω2 L + 4ΩL log 1 ϵ 2ΩL ≥T∗log  1 2 + s 1 4 + log 1 ϵ ΩL  , T∗≤ log 1 ϵ log 1 2 + q 1 4 + log 1 ϵ ΩL . Hence, to reach the accuracy of ϵ, we need the number of iterations t to be at least t ≥ log 1 ϵ log 1 2 + q 1 4 + 1 ΩL log 1 ϵ . Therefore, the iteration complexity for the case of B0 = LI is O  min   κ log 1 ϵ , (d + C0)κ + log 1 ϵ , log 1 ϵ log 1 2 + q 1 4 + 1 dκ+C0κ log 1 ϵ     . Similarly, in this case of B0 = µI, the second result in Corollary 4.2 establishes a global complexity of O d log κ + κ log 1 ϵ , where the first term represents the number of iterations before the linear convergence rate in (19) begins, and the second term arises from the linear rate itself. Additionally, following the same argument, the second result in Corollary 5.3 indicates a complexity of O(C0d log κ + C0κ + log 1 ϵ ). Here, the first term accounts for the wait time until the convergence rate takes effect, and the second term is associated with the condition number-independent linear rate. For the superlinear convergence rate, when B0 = µI, to reach the accuracy of ϵ, we need the number of iterations t to be at least t ≥ log 1 ϵ log 1 2 + q 1 4 + 1 Ωµ log 1 ϵ , where Ωµ = C0d log κ+C0κ. The proof is the same as the proof for the case of B0 = LI. Therefore, the iteration complexity for the case of B0 = µI is O  min   d log κ + κ log 1 ϵ , C0(d log κ + κ) + log 1 ϵ , log 1 ϵ log 1 2 + q 1 4 + 1 C0(d log κ+κ) log 1 ϵ     . J Log Bisection Algorithm for Weak Wolfe Conditions 27 Algorithm 1 Log Bisection Algorithm for Weak Wolfe Conditions Require: Initial step size η(0) = 1, η(0) min = 0, η(0) max = +∞ 1: for i = 0, 1, 2, . . . do 2: if f(xt + η(i)dt) > f(xt) + αη(i)∇f(xt)⊤dt then 3: Set η(i+1) max = η(i) and η(i+1) min = η(i) min 4: if η(i) min = 0 then 5: η(i+1) = ( 1 2)2i+1−1 6: else 7: η(i+1) = q η(i+1) max η(i+1) min 8: end if 9: else if ∇f(xt + η(i)dt)⊤dt < β∇f(xt)⊤dt then 10: Set η(i+1) max = η(i) max and η(i+1) min = η(i) 11: if η(i) max = +∞then 12: η(i+1) = 2 2i+1−1 13: else 14: η(i+1) = q η(i+1) max η(i+1) min 15: end if 16: else 17: Return η(i) 18: end if 19: end for K Results and Discussion on the Bisection Scheme for Line Search in Section 7 K.1 Proof of Lemma K.1 First, we present major results concerning the complexity of the bisection method, which specifies a range of values that meet the conditions in (5) and (6). Lemma K.1. Suppose that Assumptions 2.1, 2.2 and 2.3 hold. Recall the definition of ρt in (21) and Ct in (9). At iteration t, there is unique ηr > 0 such that the sufficient decrease condition (5) is equity for ηr, i.e., f(xt + ηrdt) = f(xt) + αηr∇f(xt)⊤dt. (75) Then, ηt satisfies the sufficient decrease condition (5) if and only if ηt ≤ηr. We also have that 2(1 −α) 1 + Ct ρt ≤ηr ≤2(1 −α)(1 + Ct)ρt. (76) Similarly, there is also unique ηl > 0 such that the curvature condition (6) is equity for ηl, i.e., ∇f(xt + ηldt)⊤dt = β∇f(xt)⊤dt. (77) Then, ηt satisfies the curvature condition (6) if and only if ηt ≥ηl. Moreover, we have that ηr ηl ≥1 + β −α (1 −β)(1 + 2Ct) > 1. (78) Proof. Notice that Assumption 2.1 indicates that the objective function f(x) is strongly convex. Consider function h1(η) = f(xt + ηdt) −αη∇f(xt)⊤dt. We observe that this function h1(η) is strongly convex and h1(0) = f(xt), h′ 1(0) < 0. Hence, there is unique ηr > 0 such that h1(ηr) = f(xt) and ηt ≤ηr if and only if f(xt + ηtdt) ≤f(xt) + αηt∇f(xt)⊤dt. Denote that ¯xt+1 = xt+ηrdt. We know that f(¯xt+1)−f(xt) = αηrg⊤ t dt. Since f(¯xt+1)−f(xt) = ηrg⊤ t dt + 1 2η2 rd⊤ t ∇2f(xt + τ(¯xt+1 −xt))dt for τ ∈(0, 1), we have that ηrg⊤ t dt + 1 2η2 rd⊤ t ∇2f(xt + τ(¯xt+1 −xt))dt = αηrg⊤ t dt, 28 ηr = 2(1 −α) −g⊤ t dt d⊤ t ∇2f(xt + τ(¯xt+1 −xt))dt . which leads to ηr = 2(1 −α) −g⊤ t dt d⊤ t ∇2f(xt + τ(¯xt+1 −xt))dt ≤2(1 −α)(1 + Ct) −g⊤ t dt d⊤ t ∇2f(x∗)dt = 2(1 −α)(1 + Ct)−g⊤ t dt ∥˜dt∥2 = 2(1 −α)(1 + Ct)ρt. ηr = 2(1 −α) −g⊤ t dt d⊤ t ∇2f(xt + τ(¯xt+1 −xt))dt ≥2(1 −α) 1 + Ct −g⊤ t dt d⊤ t ∇2f(x∗)dt = 2(1 −α) 1 + Ct ρt. where we use the (32) from Lemma A.1 and the fact that f(¯xt+1) = f(xt) + αηrg⊤ t dt ≤f(xt). Hence, we prove the results in (76). Similarly, consider function h2(η) = ∇f(xt + ηdt)⊤dt. We observe that this function h2(η) is strictly increasing function for η ≥0 and h2(0) = ∇f(xt)⊤dt < β∇f(xt)⊤dt, h2(ηexact) = ∇f(xt + ηexactdt)⊤dt = 0 > β∇f(xt)⊤dt where ηexact := arg minη>0 f(xt + ηdt) is the exact line search step size satisfying ∇f(xt + ηexactdt)⊤dt = 0. Hence, there is unique ηl ∈(0, ηexact) such that h2(ηl) = β∇f(xt)⊤dt and ηt ≥ηl if and only if ∇f(xt + ηtdt)⊤dt ≥β∇f(xt)⊤dt. Notice that f(xt + ηrdt) = f(xt) + αηr∇f(xt)⊤dt. Using mean value theorem, we know there exists ¯η ∈(0, ηr) such that f(xt + ηrdt) = f(xt) + ηr∇f(xt + ¯ηdt)⊤dt. The above two equities indicates that ∇f(xt + ¯ηdt)⊤dt = α∇f(xt)⊤dt. Recall that ∇f(xt + ηldt)⊤dt = β∇f(xt)⊤dt. Combing the above two equities, we obtain that (∇f(xt + ¯ηdt) −∇f(xt + ηldt))⊤dt = −∇f(xt)⊤dt(β −α). Using mean value theorem again, we know there exists ˜η ∈(ηl, ¯η) such that (∇f(xt + ¯ηdt) −∇f(xt + ηldt))⊤dt = (¯η −ηl)d⊤ t ∇2f(xt + ˜ηdt)dt. Leveraging the above two equities, we obtain that ¯η −ηl = (β −α) −∇f(xt)⊤dt d⊤ t ∇2f(xt + ˜ηdt)dt . Notice that ¯η ≤ηr, we have that ηr −ηl ≥¯η −ηl = (β −α) −∇f(xt)⊤dt d⊤ t ∇2f(xt + ˜ηdt)dt . (79) Recall the definition of ηl in (77), we have that (∇f(xt + ηldt) −∇f(xt))⊤dt = −(1 −β)∇f(xt)⊤dt. Notice that there exists ˆη ∈(0, ηl), such that (∇f(xt + ηldt) −∇f(xt))⊤dt = ηld⊤ t ∇2f(xt + ˆηdt)dt. Combing the above two equities, we obtain that ηl = −(1 −β)∇f(xt)⊤dt d⊤ t ∇2f(xt + ˆηdt)dt . (80) 29 Leveraging (79) and (80), we have that ηr ηl = 1 + ηr −ηl ηl ≥1 + (β −α)d⊤ t ∇2f(xt + ˆηdt)dt (1 −β)d⊤ t ∇2f(xt + ˜ηdt)dt . Recall that ¯xt+1 = xt + ηrdt and notice that ˆη ≤ηr, ˜η ≤ηr. We have that xt + ˆηdt = xt + ˆτ(¯xt+1 −xt) and xt + ˜ηdt = xt + ˜τ(¯xt+1 −xt) with ˆτ = ˆη ηr ∈(0, 1) and ˜τ = ˜η ηr ∈(0, 1). Since f(¯xt+1) = f(xt + ηrdt) = f(xt) + αηr∇f(xt)⊤dt ≤f(xt), applying (36) in Lemma A.1, we prove the conclusion that ηr ηl ≥1 + (β −α)d⊤ t ∇2f(xt + ˆηdt)dt (1 −β)d⊤ t ∇2f(xt + ˜ηdt)dt = 1 + (β −α)d⊤ t ∇2f(xt + ˆτ(¯xt+1 −xt))dt (1 −β)d⊤ t ∇2f(xt + ˜τ(¯xt+1 −xt))dt ≥1 + β −α (1 −β)(1 + 2Ct). K.2 Bound on the Number of Inner Loops Proposition K.2. Suppose that Assumptions 2.1, 2.2 and 2.3 hold. Consider the BFGS method with inexact line search defined in (5) and (6) and we choose the step size ηt according to Algorithm 1. At iteration t, denote λt as the number of loops in Algorithm 1 to terminate and return the ηt satisfying the Wolfe conditions (5) and (6). Then λt is finite and upper bounded by λt ≤2 + log2 1 + (1 −β)(1 + 2Ct) β −α + 2 log2 1 + log2 2(1 −α)(1 + Ct) + max{log2 ρt, log2 1 ρt } . (81) Proof. At the first iteration, if η(0) = 1 satisfies the weak Wolfe conditions (5) and (6), the algorithm terminates and returns the unit step size ηt = 1. In this case, we have that λt = 1. Suppose that at the first iteration, η(0) = 1 doesn’t satisfy the sufficient decrease condition (5) but satisfies the curvature condition (6), we have that η(1) max = +∞, η(1) min = 1 and η(1) = 2. Assume that in the Algorithm 1, η(i) max is never set to a finite value and the algorithm never returns. This means that the condition in line 2 is never satisfied, and as a result, we keep repeating steps in line 12. Thus, η(i) = 22i−1 and since the condition in line 2 is never satisfied, we always have that f(xt + η(i)dt) ≤f(xt) + αη(i)∇f(xt)⊤dt. Notice that limi→∞η(i) →+∞and ∇f(xt)⊤dt < 0. We obtain that limi→∞f(xt + η(i)dt) →−∞, which is a contradiction since f is strongly convex. Hence, at some point, either the algorithm finds an admissible step size and returns, or η(i) max must become finite. Suppose that this happens at iteration K1 ≥1 of the loop in Algorithm 1. Then, we know that η(K1) = 22K1−1. In the first case that the algorithm finds an admissible step size and returns ηK1, ηK1 satisfies the Armijo-Wolfe conditions and therefore ηK1 ≤ηr. Using the upper bound result in (76) from Lemma K.1, we obtain that η(K1) = 22K1−1 ≤ηr ≤2(1 −α)(1 + Ct)ρt, which leads to λt = K1 ≤log2 1 + log2 2(1 −α)(1 + Ct)ρt . (82) In the second case that η(i) max becomes finite but the algorithm does not terminate, we have that η(K1−1) satisfies the sufficient condition (5) and η(K1−1) ≤ηr. Similarly, this implies that K1 ≤1 + log2 1 + log2 2(1 −α)(1 + Ct)ρt . (83) Then, we further go through the log bisection process. Notice that for any iteration i > K1, the sequence η(i) max is finite and non-increasing and the sequence η(i) min ≥1 and non-decreasing. The log bisection process indicates that log2 η(i+1) max η(i+1) min = 1 2 log2 η(i) max η(i) min , ∀i > K1. (84) 30 The Algorithm 1 implies that for any i > K1, we have that f(xt + η(i) maxdt) > f(xt) + αη(i) max∇f(xt)⊤dt, ∇f(xt + η(i) mindt)⊤dt < β∇f(xt)⊤dt. Hence, we know that for any i > K1, η(i) max ≥ηr and η(i) min ≤ηl where ηr, ηl are defined in (75), (77) from Lemma K.1. Therefore, using result (78) from Lemma K.1, we have that for any j ≥1, log2 η(K1+j) max η(K1+j) min ≥log2 ηr ηl > 0. (85) Notice that (84) implies that log2 η(K1+j) max η(K1+j) min = 1 2j−1 log2 η(K1+1) max η(K1+1) min , (86) which leads to 0 = limj→+∞ 1 2j−1 log2 η(K1+1) max η(K1+1) min = limj→+∞log2 η(K1+j) max η(K1+j) min ≥log2 ηr ηl > 0. This is a contradiction. Hence, Algorithm 1 must terminate after finite number of loops. Now suppose that Algorithm 1 terminates after K1 + Γ1 iterations, (85) and (86) indicate that when Γ1 ≥1, we have 1 2Γ1−1 log2 η(K1+1) max η(K1+1) min = log2 η(K1+Γ1) max η(K1+Γ1) min ≥log2 ηr ηl > log2 1 + β −α (1 −β)(1 + 2Ct) (87) where the last inequality holds since (78) in Lemma K.1. Notice that η(K1+1) max = 22K1−1 and ηK1+1 min = 22K1−1−1. Hence, we obtain that log2 η(K1+1) max η(K1+1) min = 2K1−1 ≤1 + log2 2(1 −α)(1 + Ct)ρt . (88) Combing (87), (88) and using log x ≥1 −1 x, we have that Γ1 ≤1 + log2 1 + log2 2(1 −α)(1 + Ct)ρt −log2 log2 1 + β −α (1 −β)(1 + 2Ct) ≤1 + log2 1 + log2 2(1 −α)(1 + Ct)ρt −log2 log 1 + β −α (1 −β)(1 + 2Ct) ≤1 + log2 1 + log2 2(1 −α)(1 + Ct)ρt −log2 1 − 1 1 + β−α (1−β)(1+2Ct) = 1 + log2 1 + log2 2(1 −α)(1 + Ct)ρt + log2 1 + (1 −β)(1 + 2Ct) β −α . (89) Leveraging (83) and (89), we prove that λt = K1 + Γ1 ≤2 + 2 log2 1 + log2 2(1 −α)(1 + Ct)ρt + log2 1 + (1 −β)(1 + 2Ct) β −α . (90) Similarly, suppose that at the first iteration, η(0) = 1 satisfies the sufficient decrease condition (5) but doesn’t satisfy the curvature condition (6), we have that η(1) max = 1, η(1) min = 0 and η(1) = 1 2. Assume that in the Algorithm 1, η(i) min is never set to a positive value and the algorithm never returns. This means that the condition in line 2 is always satisfied, and as a result, we keep repeating steps in line 5. Thus, η(i) = ( 1 2)2i−1 and since the condition in line 2 is always satisfied, we have that f(xt + η(i)dt) > f(xt) + αη(i)∇f(xt)⊤dt. Therefore, we know that η(i) ≥ηr where ηr > 0 is defined in (75) from Lemma K.1. Notice that η(i) ≥ηr > 0 for any i and limi→∞η(i) = 0, this leads to a contradiction. Hence, at some point either the algorithm returns a step size satisfying the weak Wolfe conditions or η(i) min must become positive. Suppose that this happens at iteration K2 ≥1 of the loop in Algorithm 1. Then, we know that η(K2) = ( 1 2)2K2−1. 31 In the first case that the algorithm finds an admissible step size and returns ηK2, ηK2 satisfies the Armijo-Wolfe conditions and therefore ηK2 ≤ηr. Using the upper bound result in (76) from Lemma K.1, we obtain that η(K2) = 22K2−1 ≤ηr ≤2(1 −α)(1 + Ct)ρt, which leads to λt = K2 ≤log2 1 + log2 2(1 −α)(1 + Ct)ρt . (91) In the second case that η(i) min becomes positive but the algorithm does not terminate, we have that η(K2−1) doesn’t satisfy the sufficient condition (5) and η(K2−1) ≥ηr. Using the lower bound result in (76) from Lemma K.1, we obtain that η(K2−1) = ( 1 2)2K2−1−1 ≥ηr ≥2(1−α) 1+Ct ρt, which leads to K2 ≤1 + log2 1 + log2 1 + Ct 2(1 −α)ρt . (92) Then, we further go through the log bisection process. Using the same techniques, we can assume that Algorithm 1 terminates after K2 + Γ2 iterations, where Γ2 ≥1 satisfies that 1 2Γ2−1 log2 η(K2+1) max η(K2+1) min = log2 η(K2+Γ2) max η(K2+Γ2) min ≥log2 ηr ηl > log2 1 + β −α (1 −β)(1 + 2Ct) (93) where the last inequality holds since (78) in Lemma K.1. Notice that η(K2+1) max = ( 1 2)2K2−1−1 and ηK2+1 min = ( 1 2)2K2−1. Hence, we obtain that log2 η(K2+1) max η(K2+1) min = 2K2−1 ≤1 + log2 1 + Ct 2(1 −α)ρt . (94) Combing (93), (94) and using log x ≥1 −1 x, we have that Γ2 ≤1 + log2 1 + log2 1 + Ct 2(1 −α)ρt −log2 log2 1 + β −α (1 −β)(1 + 2Ct) ≤1 + log2 1 + log2 1 + Ct 2(1 −α)ρt −log2 log 1 + β −α (1 −β)(1 + 2Ct) ≤1 + log2 1 + log2 1 + Ct 2(1 −α)ρt −log2 1 − 1 1 + β−α (1−β)(1+2Ct) = 1 + log2 1 + log2 1 + Ct 2(1 −α)ρt + log2 1 + (1 −β)(1 + 2Ct) β −α . (95) Leveraging (92) and (95), we prove that λt = K2 + Γ2 ≤2 + 2 log2 1 + log2 1 + Ct 2(1 −α)ρt + log2 1 + (1 −β)(1 + 2Ct) β −α . (96) Notice that α < 1 2 and thus 1 2(1−α) < 2(1 −α), combining (82), (90), (91) and (96), we prove the final conclusion λt ≤2 + log2 1 + (1 −β)(1 + 2Ct) β −α + 2 log2 1 + log2 2(1 −α)(1 + Ct) + max{log2 ρt, log2 1 ρt } . K.3 Proof of Theorem 7.1 Using result from Proposition K.2, we have that Λt = 1 t t−1 X i=0 λi ≤2 + 1 t t−1 X i=0 log2 1 + (1 −β)(1 + 2Ci) β −α + 2 t t−1 X i=0 log2 1 + log2 2(1 −α)(1 + Ci) + max{log2 ρi, log2 1 ρi } . (97) 32 Using Jensen’s inequality, we have that 1 t t−1 X i=0 log2 1 + (1 −β)(1 + 2Ci) β −α ≤log2 1 + 1 −β β −α + 2(1 −β) β −α Pt−1 i=0 Ci t . (98) 1 t t−1 X i=0 log2 1 + log2 2(1 −α)(1 + Ci) + max{log2 ρi, log2 1 ρi } ≤log2 1 + log2 2(1 −α) + 1 t t−1 X i=0 log2(1 + Ci) + 1 t t−1 X i=0 max{log2 ρi, log2 1 ρi } ≤log2 1 + log2 2(1 −α) + log2 1 + Pt−1 i=0 Ci t ) + 1 t t−1 X i=0 max{log2 ρi, log2 1 ρi } . (99) We also have that 1 t t−1 X i=0 max{log2 ρi, log2 1 ρi } = 1 t t−1 X i=0,ρi≥1 log2 ρi + 1 t t−1 X i=0,0≤ρi<1 log2 1 ρi = 1 t t−1 X i=0,ρi≥2 log2 ρi + 1 t t−1 X i=0,1≤ρi<2 log2 ρi + 1 t t−1 X i=0, 1 2 <ρi<1 log2 1 ρi + 1 t t−1 X i=0,ρi≤1 2 log2 1 ρi ≤2 + 1 t t−1 X i=0,ρi≥2 log2 ρi + 1 t t−1 X i=0,ρi≤1 2 log2 1 ρi , (100) where the inequality is due to log2 ρi ≤1 for ρi < 2 and log2 1 ρi ≤1 for ρi > 1 2. Using the definition of ω and (b) in Lemma G.1, we obtain that 1 t t−1 X i=0,ρi≥2 log2 ρi = log2 e t t−1 X i=0,ρi≥2 log ρi = log2 e t t−1 X i=0,ρi≥2 (ρi −1 −ω(ρi −1)) ≤log2 e t t−1 X i=0,ρi≥2 ( 2ρi ρi −1ω(ρi −1) −ω(ρi −1)) = log2 e t t−1 X i=0,ρi≥2 ρi + 1 ρi −1ω(ρi −1) ≤3 log2 e t t−1 X i=0,ρi≥2 ω(ρi −1). (101) Similarly, using (c) in Lemma G.1, we obtain that 1 t t−1 X i=0,ρi≤1 2 log2 1 ρi = log2 e t t−1 X i=0,ρi≤1 2 log 1 ρi = log2 e t t−1 X i=0,ρi≤1 2 (ω(ρi −1) + 1 −ρi) ≤log2 e t t−1 X i=0,ρi≤1 2 (ω(ρi −1) + 1 + ρi 1 −ρi ω(ρi −1)) = log2 e t t−1 X i=0,ρi≤1 2 2 1 −ρi ω(ρi −1) ≤4 log2 e t t−1 X i=0,ρi≤1 2 ω(ρi −1). (102) Combining (100), (101) and (102), we prove that 1 t t−1 X i=0 max{log2 ρi, log2 1 ρi } ≤2 + 1 t t−1 X i=0,ρi≥2 log2 ρi + 1 t t−1 X i=0,ρi≤1 2 log2 1 ρi ≤2 + 4 log2 e t t−1 X i=0 ω(ρi −1) ≤2 + 6 t Ψ( ˜B0) + 2 t−1 X i=0 Ci . (103) 33 where we use the fact that ω(ρi −1) ≥0 for any i ≥0 and the last inequality is due to (46) in Proposition G.2. Leveraging (97), (98), (99) and (103), we have that Λt ≤2 + log2 1 + 1 −β β −α + 2(1 −β) β −α Pt−1 i=0 Ci t + 2 log2 3 + log2 2(1 −α) + log2 1 + Pt−1 i=0 Ci t ) + 6 t Ψ( ˜B0) + 2 t−1 X i=0 Ci ≤2 + log2 1 + 1 −β β −α + 2(1 −β) β −α Pt−1 i=0 Ci t + 2 log2 log2 16(1 −α) + log2 1 + Pt−1 i=0 Ci t ) + 6Ψ( ˜B0) + 12 Pt−1 i=0 Ci t . We prove the final conclusion using (44) from the proof of Theorem 5.2 in Appendix F.2, i.e., t−1 X i=0 Ci ≤C0Ψ( ¯B0) + 3C0κ α(1 −β). K.4 Corollaries of Theorem 7.1 for B0 = LI and B0 = µI Corollary K.3 (B0 = LI). Suppose that Assumptions 2.1, 2.2 and 2.3 hold. Let {xt}t≥0 be the iterates generated by the BFGS method, where the step size satisfies the Armijo-Wolfe conditions in (5) and (6). For any initial point x0 ∈Rd and the initial Hessian approximation matrix B0 = LI, the average complexity of line search Algorithm 1 Tk is upper bounded by Λt ≤2 + log2 1 + 1 −β β −α + 2(1 −β) β −α 3C0κ α(1 −β)t + 2 log2 log2 16(1 −α) + log2 1 + 3C0κ α(1 −β)t + 6dκ + 36C0κ α(1−β) t . Moreover, when t ≥6dκ + 36 α(1−β)C0κ, we have that Λt ≤2 + log2 1 + 3(1 −β) β −α + 2 log2(5 + log2 2(1 −α)). (104) Proof. Since B0 = LI, we have ¯B0 = 1 LB0 = I and ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 = L∇2f(x∗)−1. Using results in the proof of Corollary 5.3, we have Ψ( ¯B0) = 0, Ψ( ˜B0) ≤dκ. Combining these two results with the result in Theorem 7.1, we prove the conclusion. Corollary K.4 (B0 = µI). Let {xt}t≥0 be the iterates generated by the BFGS method with inexact line search (5), (6) and suppose that Assumptions 2.1, 2.2 and 2.3 hold. For any initial point x0 ∈Rd and the initial Hessian approximation matrix B0 = µI, the average complexity of line search Algorithm 1 Tk is upper bounded by Λt ≤2 + log2 1 + 1 −β β −α + 2(1 −β) β −α C0d log κ + 3C0κ α(1−β) t + 2 log2 log2 16(1 −α) + log2 1 + C0d log κ + 3C0κ α(1−β) t + 6(1 + 2C0)d log κ + 36C0κ α(1−β) t . Moreover, when t ≥6(1 + 2C0)d log κ + 36C0κ α(1−β), we have that Λt ≤2 + log2 1 + 3(1 −β) β −α + 2 log2(5 + log2 2(1 −α)). (105) 34 Proof. Since B0 = µI, we have ¯B0 = 1 κB0 = I and ˜B0 = ∇2f(x∗)−1 2 B0∇2f(x∗)−1 2 = µ∇2f(x∗)−1. Using results in the proof of Corollary 4.2, we have Ψ( ¯B0) ≤d log κ, Ψ( ˜B0) ≤d log κ. Combining these two results with (26) in Theorem 6.4, we prove the conclusion. 35 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,419	MeshXL: Neural Coordinate Field for Generative 3D Foundation Models Sijin Chen1,2,∗, Xin Chen2,†, Anqi Pang2, Xianfang Zeng2, Wei Cheng2, Yijun Fu2, Fukun Yin1,2, Zhibin Wang2, Jingyi Yu3, Gang Yu2, Bin Fu2, Tao Chen1,‡ https://github.com/OpenMeshLab/MeshXL 1Fudan University 2Tencent PCG 3ShanghaiTech University † project lead ‡ corresponding author Figure 1: MeshXL can auto-regressively generate high-quality 3D meshes. We validate that Neural Coordinate Field (NeurCF), an explicit coordinate representation with implicit neural embeddings, is a simple-yet-effective sequence representation for large-scale mesh modelling. Abstract The polygon mesh representation of 3D data exhibits great flexibility, fast rendering speed, and storage efficiency, which is widely preferred in various applications. However, given its unstructured graph representation, the direct generation of high-fidelity 3D meshes is challenging. Fortunately, with a pre-defined ordering strategy, 3D meshes can be represented as sequences, and the generation process can be seamlessly treated as an auto-regressive problem. In this paper, we validate Neural Coordinate Field (NeurCF), an explicit coordinate representation with implicit neural embeddings, is a simple-yet-effective representation for large-scale sequential mesh modeling. After that, we present MeshXL, a family of generative pre-trained auto-regressive models that addresses 3D mesh generation with modern large language model approaches. Extensive experiments show that MeshXL is able to generate high-quality 3D meshes, and can also serve as foundation models for various down-stream applications. ∗Research done when Sijin Chen was a research intern at Tencent PCG. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1 Introduction The generation of high-quality 3D assets [61, 77, 29] is essential for various applications in video games, virtual reality, and robotics. Among existing 3D representations [51, 38, 57, 61], 3D mesh represents 3D data as graphs, which possesses the flexibility and accuracy representing sharp edges as well as both flat and curved surfaces. However, the direct generation of high-quality 3D meshes is challenging, given 1) the unstructured graph representation and 2) the demand for estimating accurate spatial locations and connectivity within vertices. To generate 3D meshes, many works adopt an indirect way by first producing data in other 3D representations, such as point clouds [97, 49, 54], SDF [88, 94], and multi-view images [46, 82, 30]. After that, they adopt re-meshing methods [37] to post-process the generated geometries. There are also attempts towards the direct generation of 3D polynomial meshes. PolyGen [53] adopts two separate decoder-only transformers for vertices generation and vertices connectivity prediction. MeshGPT [65] first builds a mesh VQVAE to first turn meshes into tokens, and then learns to generate the token sequences with a GPT model [59]. Meanwhile, PolyDiff [2] directly adopts discrete denoising diffusion [4] on the discretized mesh coordinates. Though these methods have achieved initial success in creating 3D assets, they suffer from certain limitations. To preserve sufficient high-frequency information, point clouds and voxels requires dense samplings on the object surfaces, which inevitably leads to great redundancy while representing flat surfaces. The reconstruction-based methods [82, 30, 67], however, rely heavily on the accuracy of the multi-vew generation pipelines [46]. Additionally, the VQVAE-based 3D generation methods [88, 65] require sophisticated multi-stage training, which less favors learning from large scale data. To tackle the above challenges and explore the potential of scaling up 3D generative pre-training, we introduce a simple-yet-effective way of 3D mesh representation, the Neural Coordinate Field (NeurCF). NeurCF represents the explicit 3D coordinates with implicit neural embeddings. We show that with a pre-defined ordering strategy, a 3D mesh can be represented by a one-and-only coordinate sequence, which further helps us formulate 3D mesh generation as an auto-regressive problem. After that, we present MeshXL, a family of generative pre-trained transformers [93, 59], for the direct generation of high-fidelity 3D meshes. Without resorting to intermediate 3D representations, NeurCF facilitates an end-to-end learning pipeline for the direct pre-training on large-scale 3D mesh data. By organizing high-quality 3D assets from ShapeNet [9], 3D-FUTURE [22], Objaverse [17], and Objaverse-XL [16], we achieve a collection of over 2.5 million 3D meshes to support large-scale generative pre-training. Extensive experiments demonstrate that the NeurCF representation facilitates MeshXL to generate higher-quality 3D meshes. By training on the collection of large-scale 3D mesh data, MeshXL can achieve better performance with larger numbers of parameters (Fig. 3 and Tab. 5), and surpass prior arts on multiple categories in the ShapeNet dataset [9] (Tab. 3). In summary, our contributions can be summarized as follows: • We validate that Neural Coordinate Field is a simple-and-effective representation of 3D mesh, which is also friendly to large-scale auto-regressive pre-training. • We present a family of MeshXLs that can be treated as strong base models for image-conditioned or text-conditioned 3D mesh generation tasks. • We show that MeshXL surpasses state-of-the-art 3D mesh generation methods, and can produce delicate 3D meshes compatible with existing texturing methods. 2 Related Work First, we present a concise review of existing 3D representations. Subsequently, we discuss related works on 3D generation and recent efforts in developing 3D foundation models. 3D Representations. Researchers have long sought for accurate and efficient methods to represent 3D data. Point Cloud [54, 57, 58, 89] captures the spatial positions of discrete points in the Euclidean space, which is preferred by various 3D sensors [15, 87, 66, 3, 7]. Mesh [53, 2, 65, 12] represents the 3D structure with graphs. By connecting the vertices with edges, mesh can also be interpreted into a set of polygons in the 3D space. Similar to point clouds, 3D Gaussians [38, 68] also record the discrete Euclidean distribution in 3D space. However, each point is represented by a 3D Gaussian 2 distribution function parameterized by its covariance matrix, color, and opacity. Given their fast convergence and rendering speed, 3D gaussians are often utilized for 3D reconstruction. Neural Radiance Field (NeRF) [51, 5] constructs a learnable volumetric function f using neural networks trained on multi-view images. Due to its derivability and flexibility, NeRF is also favored for 3D generative models [46, 99, 76, 56]. Additionally, there are other 3D representations such as multi-view images [74, 90, 100], voxel fields [61, 13, 45], and signed distance fields [94], among others [64, 88, 63]. In this paper, we consider the Neural Coordinate Field (NeurCF), an explicit spatial representation with implicit neural embeddings, and investigate its potential for scalable 3D asset generation. 3D Generation. With the exploration of various 3D representations and the collection of large-scale 3D datasets [17, 9, 16], researchers have also put much effort exploring the generation of high-fidelity 3D assets [42, 39]. The Generative Adversarial Network (GAN) [25, 80, 1, 33] produces synthetic 3D data with a generator G, and train a discriminator network D to distinguish the generated and real data. Additionally, the potential of diffusion models [54, 28, 62] in the direct generation of 3D data is also widely explored [97, 2, 54, 50, 47]. The key idea behind diffusion is to transform the desired data distribution into a simpler distribution (e.g. gaussian) and learn a desnoising model for the reverse process. Besides, researchers have also explored the potential of diffusion models in generating multi-view images [46, 16, 82, 43], and reconstruct them into 3D structures. In this paper, we mainly explore the auto-regressive methods for 3D generation. AutoSDF [52] and MeshGPT [65] learn to generate discrete tokens and reconstruct them into 3D representations with a VQVAE model [72]. PolyGen [53] adopts two decoder-only transformers that predict the location and connectivity of vertices, sequentially. In this paper, we explore the potential of an explicit sequential modelling method for 3D meshes, and present a family of generative pre-trained transformers, MeshXL, for high-fidelity 3D mesh generation. 3D Foundation Models. The collection of large-scale high-quality 3D data [17, 16, 9, 81, 71, 21, 22] builds up the foundation for various 3D-related tasks [83, 27, 10, 41]. To explore the scaling effects in 3D learning, researchers have made great endeavors in building 3D foundation models for 3D understanding [96, 44, 98, 85, 86, 92, 100], reconstruction [30, 78, 67, 46, 16, 84, 73], and generation [61, 29, 65, 8]. With the introduction of large-scale 3D data in both variety and granularity [34, 41, 16], existing 3D foundation models are capable of generalizing to unseen concepts [100, 86, 44], generating high-fidelity 3D assets [88, 36, 65], responding to complex instructions [31, 10, 32, 41], and generating actions that interacts with the 3D environments [20, 79, 95]. In this paper, we present a fully end-to-end 3D mesh generation pipeline, explore the scaling effect for large-scale pre-training, and test whether our method can serve as a well-trained foundation model for various down-stream tasks. 3 Data Data Sources. We provide details on the 3D data collections we use to train and evaluate our models. The whole data collection is built upon four widely-acknowledged 3D mesh datasets, i.e. ShapeNet V2 [9], 3D-FUTURE [22], Objaverse [17], and Objaverse-XL [16]. • ShapeNet V2 [9] collects about 51k 3D CAD models for 55 categories. We split the data in 9:1 for training and validation by each category. • 3D-FUTURE [22] present about 10k high-quality 3D mesh data for indoor furniture. However, because of the delicate design, the objects contain many faces. Therefore, only a small proportion of the data can be used to train our MeshXL models. • Objaverse [17] is a large 3D data collection with more than 800k 3D objects for about 21k categories collected from Sketchfab. We split the data in 99:1 for training and validation, respectively. • Objaverse-XL [16] further expand Objaverse [17] into a dataset with more than 10M 3D objects with additional data collected from GitHub, Polycam, Thingiverse, and Smithsonian. We split the Github and Thingiverse part of the Objaverse-XL dataset into 99:1 for training and validation. Data collection and filtering. To organize existing datasets, we build up a filtering and pre-processing pipeline to ensure that the meshes meet our demand. We first collect meshes with fewer than 800 faces, and ensure that they have corresponding UV maps for rendering. After that, we render the 3D 3 (b) Auto-regressive Mesh Generation (a) Neural Coordinate Field Face Embedding: 𝑝𝑝= (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ℱ𝑝𝑝= (ℰ𝑥𝑥, ℰ𝑦𝑦, ℰ𝑧𝑧) Next Coordinate Prediction ℰmesh ℳ= ℰface 𝑓𝑓(1) , ⋯, ℰface 𝑓𝑓(𝑛𝑛) Coordinate Embeddings: ℰ Mesh Embedding: ℰface 𝑓𝑓𝑖𝑖 = ℱ𝑝𝑝1 (𝑖𝑖) , ⋯, ℱ𝑝𝑝𝑘𝑘 (𝑖𝑖) Figure 2: Mesh Representation. We present the Neural Coordinate Field (NeurCF) to encode the discretized coordinates in the Euclidean space. Benefiting from NeurCF and a pre-defined ordering strategy, our proposed MeshXL can directly generate the unstructured 3D mesh auto-regressively. meshes, and discard those not center-aligned or occupying less than 10% of the rendered image. For those 3D objects with more than 800 but less than 20,000 faces, we use planar decimation to simplify the meshes. Finally, we achieve approximately 2.5 million 3D meshes (Tab. 1). Planar Decimation Pipeline. To ensure the quality of the decimated 3D meshes, we make sure either a lower Hausdorff distance δhausdorff [65] or a similar rendered views [11]. Collecting text-mesh pairs. We render the 3D meshes with 12 different views and use CogVLM [75] to annotate 1) the front view and 2) the concatenated multi-view image. Then, we adopt the Mistral7B-Instruct model [35] with in-context examples to generate a fused mesh caption. Data Statistics. We present the data statistics of our large-scale 3D mesh collection in Tab. 1. After organizing and combing 3D assets from ShapeNet [9], 3D-FUTURE [22], Objaverse [17], and Objaverse-XL [16], we could achieve a total of 2.5 million 3D meshes. Table 1: Statistics for the Training Data and Validation Data. After combining four data sources, our proposed MeshXL models are trained on approximately 2.5 million 3D meshes. Dataset Pre-training Text-to-3D Train Val Train Val ShapeNet [9] 16,001 1,754 15,384 1,728 3D-Future [22] 1,603 Objaverse [17] 85,282 854 83,501 820 Objaverse-XL [16] 2,407,337 15,200 1,347,802 13,579 Total 2,510,223 17,808 1,446,678 16,127 4 Neural Coordinate Field Neural Coordinate Field (NeurCF) is an explicit representation with implicit neural embeddings. To be specific, for a Euclidean 3D coordinate system, we can partition the vertices coordinates into an N 3 grid. Then, each discretized coordinate p = (x, y, z) can be encoded with the coordinate embedding layer E, where F(p) = (E(x), E(y), E(z)). Therefore, a k-sided polynomial face f (i) can be encoded with Eface(f (i)) = (F(p(i) 1 ), · · · , F(p(i) k )). For simplicity, we share the learnable coordinate embeddings E among axes. Ordering. Due to the graph representation, the order of the mesh vertices and the order of the edges among them are permutation-invariant. A pre-defined ordering strategy is essential to facilitate the sequence modelling in MeshXL. We employ the same ordering strategy as PolyGen [53] and MeshGPT [65]. The mesh coordinates are first normalized into a unit cube based on the mesh’s longest axis, and discretized into unsigned integers. Within each face, the vertices are cyclically permuted based their coordinates (z-y-x order, from lower to higher), which helps to preserve the direction of normal vectors. Then, we order these faces based on the permuted coordinates (lower to high). To this end, we can represent each 3D mesh with a one-and-only coordinate sequence, aiding large-scale generative pre-training on a large collection of 3D mesh data. With the NeurCF 4 0 20 40 60 80 100 120 Processed Tokens (Billions) 1.1 1.2 1.3 1.4 1.5 1.6 Train PPL 125M 350M 1.3B 0 50 100 Processed Tokens (Billions) 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.350 ShapeNet Validation PPL 125M 350M 1.3B 0 50 100 Processed Tokens (Billions) 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.350 Objaverse Validation PPL 125M 350M 1.3B 0 50 100 Processed Tokens (Billions) 1.250 1.275 1.300 1.325 1.350 1.375 1.400 1.425 1.450 Objaverse-XL Validation PPL 125M 350M 1.3B Figure 3: Training and Validation Perplexity (PPL) for MeshXL Models. We train all the models from scratch on 150 billion tokens. We observe that the performance grows with model sizes. representation, an n-faced 3D k-sided polynomial mesh can be represented as the coordinate sequence M ∈Zn×k×3, and further be encoded into Emesh = (Eface(f (1)), · · · , Eface(f (n))). A Sequential Mesh Representation. One direct way to represent the 3D meshes is to directly reshape M into a vector with (n · k · 3) tokens. As a special case, an n-faced triangular mesh can be represented by a vector with 9n tokens. Meanwhile, our representation can also be expanded to hybrid polynomial mesh representations with the proper introduction of separate tokens. For example, we can generate triangles within “<tri> · · · </tri>” and quadrilaterals within “<quad> · · · </quad>” also in one sequence. To identify the start and end of a mesh sequence, we add a <bos> (“begin-of-sequence”) token before the mesh sequence and an <eos> (“end-of-sequence”) token after. Comparisons. Since we represent each coordinate with learnable embeddings, NeurCF is an end-toend trainable representation for unstructured 3D meshes. Comparing to the decoupled vertex and polygon representation in PolyGen [53], NeurCF only requires one coordinate sequence for each 3D mesh. Additionally, NeurCF is storage and computation efficient comparing to voxel fields (O(N 3)), since it can easily scale up the resolution with a complexity of O(N). 5 Method We first present the architecture and training objective for MeshXL models. Then, we show that MeshXL models can take an additional modality as the condition for controllable 3D assets generation. After this, we investigate the effects of scaling. Architecture. In Sec. 4, we present a simple-yet-effective way to represent a 3D mesh into a sequence. Therefore, the learning of 3D mesh generation can be formulated as an auto-regressive problem, which can be seaminglessly addressed by modern Large Language Model (LLM) approaches. In our paper, we adopt the decoder-only transformers using the OPT [93] codebase as our base models. To adapt the pre-trained OPT models to our next-coordinate prediction setting, we fine-tune the whole model with newly-initialized coordinate and position embeddings. Generative Pre-Training. We train MeshXL models using the standard next-token prediction loss. Given the trainable weights θ and an \|s\|-length sequence s, the generation loss is calculated as: LMeshXL (θ) = − \|s\| X i=1 log P s[i]\|s[1,··· ,i−1]; θ . (1) For each mesh sequence, we add a <bos> token before the mesh tokens, and an <eos> token after to identify the ending of a 3D mesh. During inference, we adopt the top-k and top-p sampling strategy to produce diverse outputs. X-to-Mesh Generation. Here we mainly consider generating 3D meshes from images and texts. We first turn the extra conditions into tokens with pre-trained encoders [18, 19]. To align the additional 5 text/image feature with the mesh coordinate field, we adopt the Q-Former architecture [40] to compress the encoded feature into a fixed-length of 32 learnable tokens as the prefix of the MeshXL model. The overall training objective for the conditional mesh generation is shown in Eq. (2): LX-to-mesh (θ) = − \|s\| X i=1 log P s[i]\|s[1,··· ,i−1]; X . (2) During inference, the model predicts the mesh tokens after the fixed-length prefix. Scaling Up. We present MeshXL in various sizes, including 125M, 350M, and 1.3B. The detailed hyperparameters for training different models can be found in Tab. 2. To better analyze the scaling effects, we train all models from scratch on 150 billion tokens. We provide both training curve and validation perplexity for different models in Fig. 3. One can see that as the number of parameters grows, the model achieves a lower validation perplexity, indicating a higher probability to produce the validation data. Table 2: Hyperparameters for different MeshXL Base Models. We present three MeshXL models with 125M, 350M, and 1.3B parameters, respectively. Hyperparameters MeshXL(125M) MeshXL(350M) MeshXL(1.3B) # Layers 12 24 24 # Heads 12 16 32 dmodel 768 1,024 2,048 dFFN 3,072 4,096 8,192 Optimizer AdamW(β1=0.9, β2=0.999) Learning rate 1.0 × 10−4 1.0 × 10−4 1.0 × 10−4 LR scheduler Cosine Cosine Cosine Weight decay 0.1 0.1 0.1 Gradient Clip 1.0 1.0 1.0 Number of GPUs 8 16 32 # GPU hrs (A100) 1,944 6,000 23,232 6 Experiments We first briefly introduce the data, metrics, and implementation details in Sec. 6.1. Then, we provide evaluations and comparisons on the generated meshes (cf. Sec. 6.2) and ablations (cf. Sec. 6.3). We also provide visualization results in Sec. 6.4. 6.1 Data, Metrics, and Implementation Details Data. We pre-train the base model with 2.5 million 3D meshes collected from the combination of ShapeNet [9], 3D-FUTURE [22], Objaverse [17], and Objaverse-XL [16]. We use planar decimation on meshes with more than 800 faces following MeshGPT [65] and RobustLowPoly [11]. For generative mesh pre-training, we randomly rotate these meshes with degrees from (0◦, 90◦, 180◦, 270◦), and adopt random scaling along each axis within range [0.9, 1.1] for data augmentation. Metrics. We follow the standard evaluation protocols in MeshGPT [65] and PolyDiff [2] to measure the quality of the generated meshes with the following metrics. We use Coverage (COV) to quantify the diversity of the generated meshes, which is sensitive to mode dropping but cannot be used to assess the generation quality. Minimum Matching Distance (MMD) calculates the average distance between the reference set and their closest neighbors in the generated set, but is not sensitive to low-quality results. The 1-Nearest Neighbor Accuracy (1-NNA) quantifies the quality and diversity between the generation set and the reference set, whose optimal value is 50%. We also adopt the Jensen-Shannon Divergence (JSD) score. Among all the above metrics, we use Chamfer Distance to measure the similarity between two samples. We also render the generated meshes and adopt the Frechet Inception Distance (FID) and Kernel Inception Distance (KID) on the rendered images for feature-level evaluation. We multiply the MMD, JSD, and KID scores by 103. Implementation. We conduct all the experiments on a cluster consisting 128 A100 GPUs. We train our models under bfloat16 with the ZeRO-2 strategy [60] using the AdamW [48] optimizer with a 6 Table 3: Quantitative Comparisons with Prior Arts on ShapeNet [9]. We scale MMD, JSD, KID by 103. MeshXL can produce diverse and high-quality 3D meshes. Category Methods COV↑ MMD↓ 1-NNA JSD↓ FID↓ KID↓ Chair PolyGen [53] 7.79 16.00 99.16 228.80 63.49 43.73 GET3D [23] 11.70 15.92 99.75 155.25 67.84 42.10 MeshGPT [65] 42.00 4.75 69.50 55.16 39.52 8.97 MeshXL (125M) 50.80 3.11 56.55 9.69 28.15 1.48 MeshXL (350M) 50.80 3.17 55.80 9.66 28.29 1.39 MeshXL (1.3B) 51.60 3.23 55.80 9.48 9.12 1.84 Table PolyGen [53] 44.00 3.36 67.20 25.06 54.08 14.96 GET3D [23] 16.80 10.39 91.90 226.97 67.65 34.62 MeshGPT [65] 34.30 6.51 75.05 92.88 53.75 7.75 MeshXL (125M) 51.21 2.96 57.96 12.82 42.55 0.92 MeshXL (350M) 49.70 3.07 56.10 13.64 43.43 1.27 MeshXL (1.3B) 52.12 2.92 56.80 14.93 22.29 2.03 Bench PolyGen [53] 31.15 4.01 83.23 55.25 70.53 12.1 MeshGPT [65] 34.92 2.22 68.65 57.32 52.47 6.49 MeshXL (125M) 54.37 1.65 43.75 16.43 35.31 0.82 MeshXL (350M) 53.37 1.65 42.96 15.41 36.35 0.96 MeshXL (1.3B) 56.55 1.62 39.78 15.51 35.50 1.60 Lamp PolyGen [53] 35.04 7.87 75.49 96.57 65.15 12.78 MeshGPT [65] 41.59 4.92 61.59 61.82 47.19 5.19 MeshXL (125M) 55.86 5.06 48.24 43.41 34.61 0.84 MeshXL (350M) 53.52 4.18 49.41 34.87 25.94 1.92 MeshXL (1.3B) 51.95 4.89 47.27 41.89 31.66 0.99 learning rate decaying from 10−4 to 10−6 and a weight decay of 0.1. The detailed hyperparameters for different models can be found in Tab. 2. To train our base models, we load the weights from the pre-trained OPT models [93] and initialize the word embeddings and positional embeddings from scratch. Without further specification, we generate 3D meshes with the top-k and top-p sampling strategy with k = 50 and p = 0.95. 6.2 Evaluations and Comparisons We provide quantitative as well as qualitative comparisons on both unconditional and conditional 3D mesh generation on public benchmarks. Unconditional Generation. We evaluate MeshXL as well as other baseline methods using the ShapeNet [9] data in Tab. 3. We split the data by 9:1 for training and validation by each category. For evaluation, we fine-tune our pre-trained base model and sample 1,000 meshes for each category. Among the listed methods, we reproduce the MeshGPT [65] with a GPT2-medium model (355M) [59]. With a similar number of parameters, Mesh-XL (350M) out-performs MeshGPT by a large margin, showing a higher COV score, a lower MMD score, and a closer 1-NNA score to 50%. This indicates that MeshXL can produce diverse and high-quality 3D meshes. Table 4: User Study. Compared to baseline methods, the meshes generated by MeshXL are better aligned with human preference in terms of both geometry and designs. Methods Quality↑ Artistic↑ Triangulation↑ PolyGen [53] 2.53 2.72 3.15 GET3D [23] 3.15 2.46 3.15 MeshXL 3.96 3.45 3.72 Reals 4.08 3.33 3.75 User Study. To evaluate how well the generated 3D meshes align with human preference, we perform user studies on the chair category in Tab. 4 with several baseline methods [53, 23]. We recruit and instruct the participants to score each mesh from 0 to 5 (higher is better) based on its 1) quality: the smoothness of object surfaces and completeness of the mesh, 2) artistic: how much do you believe this object is designed and created by artists, and 3) triangulation: how well do the connectivity among vertices aligns with the models created by professional designing software [14]. As a baseline evaluation, we also ask the participants to score the ground truth 3D geometries sampled from the 7 Ground Truth Completed Mesh Input Figure 4: Evaluation of Partial Mesh Completion. Given some partial observation of the 3D mesh (white), MeshXL is able to produce diverse object completion results (blue). ShapeNet data. We have collected a total of 434 valid responses. The results show that the 3D meshes created by MeshXL are consistently preferred by human in all dimensions. 6.3 Ablation Studies Necessity of Mesh VQVAE. Comparing to MeshGPT [65], MeshXL is an end-to-end trainable model that produces 3D meshes with next-coordinate prediction. We show in Tab. 3 that, MeshXL out-performs MeshGPT with similar numbers of parameters. We also show that MeshXL can produce high quality 3D meshes with both sharp edges and smooth surfaces in Fig. 7. Furthermore, MeshXL can save the effort training a vector quantized mesh tokenizer [65, 72], which further facilitates generative pre-training on large scale datasets. Effectiveness of Model Sizes. To analyze whether pre-training a larger model benefits 3D mesh generation, we evaluate MeshXL base models with different sizes on the Objaverse [17] dataset in Tab. 5. We observe that as the model size grows, the generated samples exhibits a larger COV, smaller JSD score, and a closer 1-NNA to 50%, which indicates an improved diversity and quality. Table 5: Effectiveness of Model Sizes on Objaverse. As the model size grows, MeshXL achieves a closer 1-NNA to 50%, a larger COV and a smaller JSD, indicating better diversity and quality. Method COV↑ MMD↓ 1-NNA JSD↓ FID↓ KID ↓ MeshXL (125M) 39.76 5.21 67.34 26.03 17.32 4.48 MeshXL (350M) 40.79 5.20 65.68 23.71 15.14 3.33 MeshXL (1.3B) 42.86 4.16 61.56 20.99 12.49 2.94 Shape Completion. To analysis whether our method is capable of producing diverse outputs, we adopt MeshXL (1.3B) model to predict the whole object given some partial observations. In practice, we use 50% of the object mesh as input, and ask the model to predict the rest 50% in Fig. 4. One can see that Mesh-XL is able to produce diverse and reasonable outputs given the partial observation of the 3D mesh. X-to-Mesh Generation. To adopt the MeshXL base models to the X-to-mesh generation setting, we adopt the Q-Former [40] to encode the additional conditions as prefixes. We showcases several conditional generation results in Fig. 5. We show that MeshXL can generate high-quality 3D meshes given the corresponding image or text as the additional inputs. 8 Image Generated Ground Truth Text Generated Ground Truth “A basic chair with four leges and an open back.” “4 legs, solid seat and backing.” “A basic looking square wooden table.” Figure 5: Evaluation of X-to-mesh generation. We show that MeshXL can generate high-quality 3D meshes given the corresponding image or text as the additional inputs. Texturing. We adopt Paint3D [91], a coarse-to-fine texture generation pipeline, to generate textures for the 3D meshes produced by MeshXL in Fig. 6. We show that 3D meshes produced by MeshXL can easily fit in existing texturing methods to produce high-quality 3D assets. Generated Mesh Textured Mesh UV Map Generated Mesh Textured Mesh UV Map Figure 6: Texture Generation for the Generated 3D Meshes. We adopt Paint3D [91] to generate textures for 3D meshes produced by MeshXL. 6.4 Visualizations We provide qualitative comparisons on the generated meshes in Fig. 7. MeshXL is able to produce high-quality 3D meshes with both sharp edges and smooth surfaces. We also visualize the normal vectors to compare the smoothness of object surfaces. The results show that 3D meshes generated by GET3D [23] have rough surfaces with tens of thousands of triangles, while MeshXL depicts the 3D shape with much smoother surfaces and less triangles. 9 GET3D MeshXL PolyGen MeshGPT Figure 7: Qualitative comparisons. We visualize the generated meshes and normal vectors. MeshXL is able to produce high-quality 3D meshes with both sharp edges and smooth surfaces. 7 Discussions Difference with PolyGen [53]. PolyGen treats 3D mesh data as a vertex sequence and a face sequence. PolyGen first generates the ordered vertices with the vertex transformer, then predicts the connectivity among vertices with the face transformer. Comparing to PolyGen, our proposed MeshXL adopts a more straightforward approach that models the 3D mesh as a one-and-only coordinate sequence, which further supports the direct and end-to-end pre-training on a large collection of 3D data. Difference with MeshGPT [65]. MeshGPT consists of a mesh VQVAE [72] and a decoder-only transformer [59]. MeshGPT first learns a mesh VQVAE to quantize the 3D meshes into discrete tokens. After that, MeshGPT trains a decoder-only transformer to generate the discrete tokens for 3D mesh reconstruction. In comparison, our proposed MeshXL is an end-to-end method that learns the neural representation of coordinates and outputs 3D meshes directly. Extensibility. Our method, MeshXL, is built upon the concept of auto-regressive methods. Therefore, our method is not restricted to the decoder-only transformers [59, 93, 69, 70], and can also be extended to other causal language models (i.e. Mamba [26], RWKV [55], and xLSTM [6]). 8 Limitations, Future Work, and Conclusions Limitations and Future Work. The main drawback of MeshXLs is the inference time. During sampling, MeshXL will generate 7,200 tokens for an 800-faced 3D mesh, which takes a relatively long time because of the auto-regressive process. As for future works, recent endeavors on the RNN-related methods [6, 55, 26] and multiple tokens prediction for LLMs [24] might open up great opportunities in saving the inference cost. Conclusion. We validate that NeurCF, an explicit coordinate representation with implicit neural embeddings, is a simple-and-effective representation of 3D meshes. By modelling the 3D mesh generation as an auto-regressive problem, we seek help from modern LLM approaches and present a family of generative pre-trained models, MeshXL, for high-fidelity 3D mesh generation. We show that MeshXL performs better given larger-scale training data and increased parameters. Extensive results show our proposed MeshXL can not only generate high-quality 3D meshes, but also exhibits great potential serving as base models for conditional 3D assets generation. Acknowledgement This work is supported by National Natural Science Foundation of China (No. 62071127), National Key Research and Development Program of China (No. 2022ZD0160101), Shanghai Natural Science Foundation (No. 23ZR1402900), Shanghai Municipal Science and Technology Major Project (No.2021SHZDZX0103). 10 References [1] Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3d point clouds. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2639–2650, 2023. 16 A Appendix A.1 More Visualization Results Unconditional Results on ShapeNet. We visualize unconditional 3D mesh generation results for chair, table, lamp and bench in Fig. 8. One can see that MeshXL is able to produce diverse and high-quality 3D meshes. Figure 8: Gallery results. Additional generation results for chair, table, lamp, and bench. Unconditional Generation on Objaverse. We visualize 3D meshes randomly sampled from MeshXL base model in Fig. 9. After training on a large-scale collection of 3D mesh data, MeshXL is able to produce diverse and high-quality 3D meshes. 17 Figure 9: Gallery results. MeshXL is able to produce diverse 3D meshes with high quality. 18 A.2 Mesh Quality Assessment How well is the triangulation. We evaluate the aspect ratio, face area, and number of faces for a better evaluation in Tab. 6. Though the meshes generated by our MeshXL have a higher average aspect ratio, we manage to achieve a smaller variance with much less 3D faces. This indicates the stability of our generation ability and the efficiency of the direct mesh representation. Since we train our MeshXLs only on triangular meshes, long-thin triangles inevitably exist in our training data. By co-training our MeshXLs on triangular meshes, 3D quads, and even hybrid representations, we could reduce the existence of long thin triangles for better generation quality. Table 6: Mesh Quality Assessment. We evaluate the aspect ratio, face area and number of faces for the generated 3D meshes. Method Aspect Ratio Face Area Number of Faces mean std. mean std. mean std. GET3D [23] 6.27 116.03 0.000 0.000 27251.80 11535.135 MeshXL (125M) 10.47 16.88 0.031 0.096 327.34 174.53 MeshXL (350M) 10.25 16.09 0.032 0.099 342.24 193.97 MeshXL (1.3B) 10.23 15.91 0.034 0.102 320.36 195.43 A.3 Inference Time Analysis The inference cost of MeshXL is closely related to the numbers of generated faces and the model sizes. We perform inference cost analysis with a batch size of one using bfloat16 on a single RTX 3090. We carry out a an analysis (Tab. 7) on 1) the average inference time of generating a given number of triangles, and 2) the average inference time of generating 3D meshes. Table 7: Inference cost of MeshXL models. We carry out inference cost analysis on time duration and memory usage under bfloat16 with a single RTX 3090. num faces MeshXL (125M) MeshXL (350M) MeshXL (1.3B) time (s) GPU mem. (GB) time (s) GPU mem. (GB) time (s) GPU mem. (GB) 100 6.30 1.59 11.30 2..98 12.08 8.41 200 12.50 1.65 22.70 3.20 24.03 9.17 400 25.21 1.85 45.81 3.78 48.09 11.17 800 49.88 2.28 92.19 5.74 96.49 21.66 avg. 29.49 44.65 49.43 19 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The abstract and introduction of the paper clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,420	LexEval: A Comprehensive Chinese Legal Benchmark for Evaluating Large Language Models Haitao Li Department of Computer Science Tsinghua University Beijing, China 100000 liht22@mails.tsinghua.edu.cn You Chen Department of Computer Science Tsinghua University Beijing, China 100000 chenyou21@mails.tsinghua.edu.cn Qingyao Ai ∗ Quan Cheng Laboratory Department of Computer Science Tsinghua University Beijing, China 100000 aiqy@tsinghua.edu.cn Yueyue Wu Quan Cheng Laboratory Department of Computer Science Tsinghua University Beijing, China 100000 wuyueyue@mail.tsinghua.edu.cn Ruizhe Zhang Department of Computer Science Tsinghua University Beijing, China 100000 u@thusaac.com Yiqun Liu Quan Cheng Laboratory Department of Computer Science Tsinghua University Beijing, China 100000 yiqunliu@tsinghua.edu.cn Abstract Large language models (LLMs) have made significant progress in natural language processing tasks and demonstrate considerable potential in the legal domain. However, legal applications demand high standards of accuracy, reliability, and fairness. Applying existing LLMs to legal systems without careful evaluation of their potential and limitations could pose significant risks in legal practice. To this end, we introduce a standardized comprehensive Chinese legal benchmark LexEval. This benchmark is notable in the following three aspects: (1) Ability Modeling: We propose a new taxonomy of legal cognitive abilities to organize different tasks. (2) Scale: To our knowledge, LexEval is currently the largest Chinese legal evaluation dataset, comprising 23 tasks and 14,150 questions. (3) Data: we utilize formatted existing datasets, exam datasets and newly annotated datasets by legal experts to comprehensively evaluate the various capabilities of LLMs. LexEval not only focuses on the ability of LLMs to apply fundamental legal knowledge but also dedicates efforts to examining the ethical issues involved in their application. We evaluated 38 open-source and commercial LLMs and obtained some interesting findings. The experiments and findings offer valuable insights into the challenges and potential solutions for developing Chinese legal systems and LLM evaluation pipelines. The LexEval dataset and leaderboard are publicly available at https://github.com/CSHaitao/LexEval and will be continuously updated. ∗Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. 1 Introduction Recently, the rapid development of large language models (LLMs) has brought new opportunities to the research of general artificial intelligence. A series of models (e.g., ChatGPT), with their extensive knowledge and outstanding language processing ability, have demonstrated excellent performance in various language processing tasks such as text generation, machine translation, and dialogue systems [10, 4, 6, 45, 37]. Meanwhile, LLMs have profoundly impacted the work patterns of legal practitioners and the development of the legal field. Recent studies show that the GPT-4 has the ability to pass the U.S. Judicial Exam [23]. By interacting with large language models, lawyers and judges can analyze legal documents more efficiently, obtaining comprehensive and valuable information and judicial advice. This has led to a growing trend among legal practitioners to incorporate LLMs as a vital supportive instrument in legal proceedings[11, 25, 27]. Despite the considerable potential of large language models, there are still concerns about their application in the legal domain [38, 35, 26]. Firstly, unlike human decision-making, which is grounded in professional knowledge and logical reasoning, LLMs derive decisions from patterns and connections extracted from massive amounts of training data. Consequently, these models, predicated on probabilistic frameworks, often fall short of ensuring the reliability and explainability of their output [18]. Additionally, existing research has indicated that LLMs may produce misleading and factually incorrect content [34]. Substandard legal texts or flawed judicial guidance may mislead legal practitioners and increase their workload. Finally, the content generated by LLMs may reflect biases present in the training data, leading to unfair treatment of certain groups or specific events. This may undermine the effectiveness and fairness of judicial proceedings and judgments, bringing considerable systemic risks. The great potential and inherent risks of LLMs in the legal domain give rise to the urgent need for a standardized and comprehensive benchmark [41, 48]. Such a benchmark is essential to ensure that LLMs meet the high standards required for legal practice, minimizing the risks while maximizing their beneficial impact. Although numerous methods for evaluating the abilities of LLMs have been developed, most focus on assessing their generalist abilities on non-professional or semi-professional texts. These benchmarks provide limited guidance for highly specialized fields such as the legal domain[54, 22, 5]. For instance, the well-known Chinese language model evaluation framework, C-Eval [22], primarily uses test questions from high school and university courses. However, in judicial applications, tasks like case summarization, legal case retrieval, and judgment prediction require LLMs to consider precise legal knowledge and complex legal contexts. These tasks often involves highly specialized elements such as judicial interpretation and reasoning. To the best of our knowledge, existing general evaluation benchmarks are unable to reflect or capture the complexity of judicial cognition and decision-making. Furthermore, some researchers have utilized existing traditional natural language processing datasets to construct benchmarks, such as LawBench [17] and LaiW [13], to evaluate the performance of LLMs in the Chinese legal system. However, traditional datasets are typically designed to test specific capabilities from a computer-centric perspective, which does not always reflect the practical use of LLMs in legal applications. Moreover, these benchmarks often overlook aspects such as legal ethics, which are crucial for ensuring the safe application of LLMs in the legal domain. Also, the evaluation metrics for previous tasks vary significantly, complicating the standardization of model performance measurement. Simply integrating existing datasets cannot provide a standardized and comprehensive evaluation of LLMs’ capabilities in the legal domain. In light of these limitations, we present LexEval: a comprehensive Chinese legal benchmark for evaluating LLMs. LexEval focuses on practical legal applications, involving how legal professionals manage, contemplate, and resolve legal issues. Firstly, to systematically organize various evaluation tasks, we propose a Legal Cognitive Ability Taxonomy (LexAbility Taxonomy), which includes six aspects: Memorization, Understanding, Logic Inference, Discrimination, Generation, and Ethic. This taxonomy comprehensively analyzes various legal tasks and their intrinsic connections, constructing a systematic framework for evaluating LLMs. Then, based on the LexAbility Taxonomy, we collect 14,150 questions covering 23 legal tasks. To our knowledge, LexEval is the largest and most comprehensive Chinese legal benchmarking dataset for evaluating LLMs. Moreover, LexEval is constructed from existing legal datasets (reorganized into a unified format), exam datasets in reality, and manually curated datasets. It adopts standard evaluation methods and metrics, laying a solid groundwork for future expansion and the integration of diverse tasks. It’s important to acknowledge that despite its comprehensiveness, LexEval may not cover every practical application task within the 2 Ethic Generation Understanding Discrimination Logic Inference Memorization Memorize and recall legal information Legal Concept, Legal Rule, Legal Evolution Understanding the legal meaning and implications Legal Element Recognition, Legal Fact Verification, Reading Comprehension, Relation Extraction, Named-entity Recognition Perform logical reasoning with legal facts. Analyze and evaluate the value of legal information Create professional juridical documents and argumentative texts Make informed judgments about ethical issues in law Cause Prediction, Article Prediction, Penalty Prediction, Multi-hop Reasoning, Legal Calculation, Argument Mining Similar Case Identification, Document Proofreading Summary Generation, Judicial Analysis Generation, Legal Translation, Open-ended Question Answering Bias and Discrimination, Morality, Privacy Figure 1: Overview of the legal cognitive ability taxonomy. legal field. As a platform supporting further research, LexEval encourages individuals to contribute additional tasks to the taxonomy, collectively pushing the boundaries of what’s achievable in the field of legal language understanding and generation. We conduct a thorough evaluation of 38 popular LLMs, including General LLMs and Legal-specific LLMs. The experimental results show that the existing LLMs are ineffective and unreliable in addressing legal problems. We hope this benchmark can point out different directions for future work. 2 Related Work In recent years, large language models (LLMs) have drawn great attention in academia and industry for their excellent performance and wide applicability [36, 50, 28]. Models such as ChatGPT and ChatGLM achieve excellent performance across various tasks through mechanisms such as pretraining, supervised fine-tuning, and alignment with human or AI feedback [2, 8, 15, 16]. By learning from massive amounts of text data, LLMs can capture the subtle differences and complex patterns of language, demonstrating the great potential in understanding and generating human language. However, despite great success, they face significant challenges in the legal domain [32, 7, 14, 31]. In the legal domain, accuracy, reliability, and fairness are crucial, but LLMs often perform poorly in these aspects due to issues like hallucination [32] and inherent biases [52, 9, 29]. Hallucination refers to models generating information that is not based on facts, which can lead to misleading or entirely incorrect conclusions in legal documents and consultations. Additionally, due to biases in the training data, the model may inadvertently replicate and amplify these biases, affecting its fairness and accuracy in applications such as legal judgment prediction, case analysis, and contract review. To mitigate these issues, the community has proposed a series of evaluation criteria and benchmarks [19, 17, 13, 30]. For example, LegalBench [19] is dedicated to the collaborative evaluation of legal reasoning tasks in English LLMs, consisting of 162 tasks contributed by 40 contributors. Lawbench [17] and LaiW [13] have conducted evaluations on the Chinese legal system using existing traditional natural language processing datasets, contributing to the development of the community. However, these datasets all focus on the partial performance of LLMs and do not provide a comprehensive evaluation. In this paper, we devote to a more comprehensive evaluation of the performance of LLMs in the legal domain. Leveraging the proposed legal cognitive ability taxonomy, we constructed the largest legal benchmark in the Chinese community through various means. 3 LexEval 3.1 Design Principle The motivation behind LexEval is to help developers quickly understand the capabilities of LLMs within the legal domain across multiple dimensions, enabling them to focus on specific areas for enhancement. To this end, we advocate for considering the hierarchy and connections of abilities, rather than organizing evaluations based solely on difficulty or in a discrete manner. Nevertheless, 3 research on the hierarchical abilities of LLMs is still in the early stages, and to our knowledge, there isn’t a well-developed taxonomy describing the abilities of LLMs in legal applications[39]. Drawing inspiration from Bloom’s taxonomy [24] and real-world legal application scenarios, we propose a legal cognitive ability taxonomy (LexAbility Taxonomy) to guide the organization of tasks in LexEval. As depicted in Figure 1, the taxonomy categorizes the application of LLMs in the legal domain into six ability levels: Memorization, Understanding, Logic Inference, Discrimination, Generation, and Ethic. At the Memorization level, LLMs are tasked with memorizing and recalling legal information, including fundamental legal statutes, case law, basic legal principles, and specialized legal terminology, among other essential content. Moving to the Understanding level, LLMs must demonstrate an aptitude for comprehending the meaning and implications of legal information. They should possess the ability to interpret legal concepts, texts, and issues accurately. Logical Inference involves the capacity for legal reasoning and deductive logic. LLMs should be capable of deducing conclusions based on provided legal facts and rules, identifying and applying legal patterns and principles effectively. The Discrimination level necessitates LLMs to analyze and evaluate the significance of legal information according to specific criteria. At the Generation level, LLMs are expected to produce professional legal documents and argumentative texts within specific legal contexts. This includes drafting legal writings, contracts, and providing legal opinions. LLMs should generate precise, legally sound, and well-structured texts based on given conditions and requirements. Finally, the Ethics level requires LLMs to make judgments about ethical issues in the legal domain. Models should identify and analyze legal ethical issues, make ethical decisions, and weigh advantages and disadvantages. They must consider ethical principles of law, professional ethics, and social values in their decision-making processes. Each level contains several specific evaluation tasks corresponding to the respective abilities. Legal practitioners can employ this taxonomy to identify the cognitive levels attained by LLMs, thereby enhancing the planning of training objectives and downstream applications. It is important to note that this legal cognitive ability taxonomy does not imply a linear learning process. During training, the model can be designed to learn back and forth from different tasks at different levels. Different legal tasks may involve multiple levels at the same time, and evaluating model performance at one level also requires synthesis across multiple tasks. As these ability levels in LexEval are developed and refined, LLMs will become increasingly integrated into legal practice, enhancing the efficiency, accuracy, and ethical standards of legal work. This taxonomy not only provides a framework for assessing the current capabilities of LLMs but also guides future advancements in the field. Although the taxonomy is primarily designed for the Chinese legal system, we believe it can be extended to involve other legal tasks in other countries as well, as these ability levels are universal across different legal systems. 3.2 Data Collection and Processing Data Source: The data in LexEval comes from three sources. The first source comprises existing datasets and corpora, primarily including CAIL 2, JEC-QA [53], and LeCaRD [33]. As these resources are originally designed for non-LLM evaluation settings, we standardize the data format and adjust the prediction targets to align with the evaluation objectives of LLMs. The second source originates from the National Uniform Legal Profession Qualification Examination, which is a uniform national examination for assessing qualifications to practice the legal profession in China. We carefully select and adapt exam questions from previous years to suit our evaluation framework. The third task source is expert annotation, where we hire 18 experts in the legal field as annotators to craft precise and relevant evaluation questions. Detailed data sources and licenses can be found in Appendix C. Data Processing: We collect data in various formats, including PDF, JSON, LaTeX, and Word, among others. By using techniques such as OCR, we first convert PDF and Word documents into textual form. For those questions that are difficult to parse automatically, we process them manually. Subsequently, all questions are converted into structured data using JSON format. All questions (except for the Generation level task) are converted to multiple-choice format. This is because multiple-choice questions have clearly defined metrics (i.e., accuracy) and are also a simple and effective way to evaluate the capabilities of the LLMs, which have been widely used in various benchmarks [17, 13, 22]. Detailed construction processes for each task can be found in Appendix 2The data from CAIL can be found on the official website of the competitionhttp://cail.cipsc.org.cn/. 4 C.3. All questions have been verified by the authors in multiple rounds to ensure their accuracy and reasonableness. Data Annotation: For tasks lacking existing datasets, we hire professional annotators to create entirely new datasets. Our annotation team consists of 18 legal experts who have all passed the National Uniform Legal Profession Qualification Examination. The annotation experts are all from China, of whom 9 are men and 9 are women. Before the beginning of the annotation work, we signed a legally effective agreement with all annotation experts to protect their rights and interests. To ensure the quality of annotation, all annotators first go through several hours of interpretation to understand their respective tasks. After that, we provide several examples to help them understand the format of tasks. The annotator creates the questions and answers according to the appropriate rules and format. Our gold annotators, who hold a Ph.D. in law, cross-check and inspect all generated questions. Before formal annotation, each annotator creates 100 questions and answers corresponding to the task. Subsequently, only annotators who achieve a 90% approved rate through cross-checking and inspection are allowed to annotate formally. We remove questions that are too simple and try to ensure that the distribution of causes is as balanced as possible. For each approved question, we pay the legal expert 0.7 dollars. We have annotated a total of 6,250 questions, with a total payment of 4,375 dollars. Detailed annotation guidance for each task can be found in Appendix C.3 and Appendix F. Built upon the above processing, we finally select and construct 23 evaluation tasks in LexEval. For the existing datasets, we try our best to avoid using datasets that have already been extensively mined by existing LLMs (e.g. C-Eval) so that the risk of test data leakage could be minimized. To ensure the quality of LexEval, we also try to balance the distributions of legal documents from different causes, thereby avoiding bias or long-tail effects in the dataset. 3.3 Task Definition Based on the legal cognitive ability taxonomy, we construct a series of evaluation tasks. Table 1 shows the overview of tasks in LexEval. These tasks may simultaneously evaluate one or multiple ability levels, and we categorize them based on their primary ability level. Each task has at least 100 evaluation samples. Among these tasks, 11 tasks are derived from existing datasets, 2 tasks come from the National Uniform Legal Profession Qualification Examination, and 10 tasks are annotated by experts. Detailed task definition, construction process, and task statistics can be found in Appendix C.4. Based on these tasks, LexEval not only provides comprehensive coverage of legal knowledge and reasoning ability but also detects issues such as bias and discrimination in legal ethics, providing valuable insights for in-depth evaluation and analysis. 3.4 Legal and Ethical Considerations Due to the sensitivity of the legal domain, we conducted a thorough review for this benchmark. All open-source datasets we utilized are licensed. LexEval tasks are subject to different licenses. Appendix C.1 provides a summary of the licenses. The authors take full responsibility for any infringement and confirm the authorization of the dataset. Our evaluation task strictly avoids involving the speculation of sensitive information about individuals and the generation of insulting or sensitive statements. In addition, we have carefully screened and filtered the data sets in LexEval for any content that contains personally identifiable information, discriminatory content, explicit, violent, or offensive content. The data set has been ethically reviewed by legal experts. We strongly believe that our benchmarks have a very low risk of negative impact on safety, security, discrimination, surveillance, deception, harassment, human rights, bias, and fairness. Appendix B.2 discusses the potential social impacts. 4 Evaluation In this section, we present the experimental setup, evaluated models, and experimental results. 5 Table 1: Details of tasks within LexEval. Level ID Task Metrics Data Source Test Set Memorization 1-1 Legal Concept Accuracy JEC-QA [53] 500 1-2 Legal Rule Accuracy Expert Annotation 1000 1-3 Legal Evolution Accuracy Expert Annotation 300 Understanding 2-1 Legal Element Recognition Accuracy CAIL-2019 500 2-2 Legal Fact Verification Accuracy Expert Annotation 300 2-3 Reading Comprehension Accuracy CAIL-2021 100 2-4 Relation Extraction Accuracy CAIL-2022 500 2-5 Named-entity Recognition Accuracy CAIL-2021 500 Logic Inference 3-1 Cause Prediction Accuracy CAIL-2018 1000 3-2 Article Prediction Accuracy CAIL-2018 1000 3-3 Penalty Prediction Accuracy CAIL-2018 1000 3-4 Multi-hop Reasoning Accuracy Exams 500 3-5 Legal Calculation Accuracy Expert Annotation 400 3-6 Argument Mining Accuracy CAIL-2021 500 Discrimination 4-1 Similar Case Identification Accuracy LeCaRD [33]&CAIL-2019 500 4-2 Document Proofreading Accuracy Expert Annotation 300 Generation 5-1 Summary Generation Rouge-L CAIL-2020 1000 5-2 Judicial Analysis Generation Rouge-L Expert Annotation 1000 5-3 Legal Translation Rouge-L Expert Annotation 250 5-4 Open-ended Question Answering Rouge-L Exams 500 Ethic 6-1 Bias and Discrimination Accuracy Expert Annotation 1000 6-2 Morality Accuracy Expert Annotation 1000 6-3 Privacy Accuracy Expert Annotation 500 4.1 Setup We evaluate the LLMs in both zero-shot and few-shot settings. In the zero-shot setting, the inputs to LLMs are only instructions and queries. In the few-shot setting, we design three different examples for each task. These examples can be found on the GitHub website. When evaluating LLMs, we set the temperature to 0 to minimize the variance introduced by random sampling. For chat LLMs, we reserve the format of their dialog prompts. When the input length exceeds the maximum context length of LLMs, we truncate the input sequence from the middle since the front and end of the input may contain crucial information. The input prompts used during our evaluation can be found in the Appendix C.4. We standardize our evaluation metrics by using Accuracy to evaluate all multiple-choice questions and Rough-L to evaluate tasks at Generation level. The evaluation metrics for each task can be found in Table 1. We also discuss the limitations of the evaluation metrics in Appendix B.1. 4.2 Evaluated Models We evaluate a total of 38 popular models, categorized into two main groups: General LLMs and Legal-specific LLMs. There are 29 General LLMs, including GPT-4 [36], ChatGPT [4], LLaMA-2-7B [44], LLaMA-2-7BChat [44], LLaMA-2-13B-Chat [44], ChatGLM-6B [50], ChatGLM2-6B [50], ChatGLM3-6B [50], Baichuan-7B-base [49], Baichuan-13B-base [49], Baichuan-13B-Chat [49], Qwen-7B-chat [1], Qwen14B-Chat [1], MPT-7B [43], MPT-7B-Instruct [43], XVERSE-13B, InternLM-7B [42], InternLM7B-Chat [42], Chinese-LLaMA-2-7B [12], Chinese-LLaMA-2-13B [12], TigerBot-Base, ChineseAlpaca-2-7B [12], GoGPT2-7B, GoGPT2-13B, Ziya-LLaMA-13B [51], Vicuna-v1.3-7B, BELLELLAMA-2-13B [3], Alpaca-v1.0-7B, MoSS-Moon-sft [40]. The Legal-specific LLMs include 9 models, which are ChatLaw-13B [11], ChatLaw-33B [11], LexiLaw, Lawyer-LLaMA [21], Wisdom-Interrogatory, LaWGPT-7B-beta1.0, LaWGPT-7B-beta1.1, HanFei [20], Fuzi-Mingcha [46].. The specific description of evaluated models can be found in the Appendix D. 6 Table 2: Zero-shot performance(%) of various models at Memorization, Understanding, and Logic Inference level. Best preformance in each column is marked bold. Model Memorization(Acc.) Understanding(Acc.) Logic Inference(Acc.) 1-1 1-2 1-3 2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 3-5 3-6 GPT-4 34.0 35.4 14.0 79.8 51.0 94.0 78.0 96.2 80.3 68.3 53.7 33.2 66.0 57.8 Qwen-14B-Chat 28.0 38.6 11.4 93.4 45.3 90.0 85.6 91.8 80.2 91.0 27.9 31.6 44.7 50.4 Qwen-7B-Chat 22.8 38.9 8.4 79.8 43.3 87.0 67.2 92.0 79.2 83.9 53.2 24.2 36.3 45.0 ChatGPT 19.0 25.6 9.0 56.8 42.3 87.0 76.0 82.2 77.7 60.3 23.0 19.4 39.6 38.2 InternLM-7B-Chat 20.4 35.4 11.0 61.4 42.3 89.0 49.4 53.8 79.3 77.9 28.8 23.8 38.3 30.0 Baichuan-13B-Chat 14.6 33.9 10.0 54.2 35.0 72.0 62.2 75.4 77.0 58.0 41.8 20.2 33.5 21.0 ChatGLM3 19.2 28.9 7.7 41.0 34.3 80.0 62.8 81.4 73.4 61.2 19.4 21.4 25.6 37.0 Baichuan-13B-base 22.6 23.0 9.0 43.2 26.7 75.0 59.2 74.4 58.3 25.6 12.5 23.8 31.0 19.6 Fuzi-Mingcha 13.0 25.0 6.7 62.0 29.0 61.0 46.4 24.8 68.0 58.6 25.5 16.0 28.9 20.4 ChatLaw-33B 16.0 25.9 7.0 51.4 32.3 76.0 67.6 62.0 60.6 32.9 23.0 15.4 23.6 37.6 ChatGLM2 28.2 13.6 16.4 22.4 24.0 61.0 40.0 29.8 77.2 54.4 24.8 19.8 27.7 8.6 Chinese-Alpaca-2-7B 19.8 24.8 19.7 25.0 33.3 61.0 46.6 24.2 66.8 39.4 20.6 16.4 18.0 26.6 BELLE-LLAMA-2-Chat 15.0 25.7 7.0 31.4 27.3 77.0 61.6 46.2 64.1 47.3 8.2 19.8 33.2 24.4 XVERSE-13B 25.4 29.0 12.0 47.0 21.7 71.0 48.2 32.4 54.9 44.7 9.9 19.2 27.7 14.6 TigerBot-base 16.6 27.5 9.0 22.4 27.0 58.0 57.0 24.6 71.5 35.7 18.3 19.0 31.2 18.8 Table 3: Zero-shot performance(%) of various models at Discrimination, Generation, and Ethic level. Best preformance in each column is marked bold. Model Discrimination(Acc.) Generation(Rough-L) Ethic(Acc.) Average Rank 4-1 4-2 5-1 5-2 5-3 5-4 6-1 6-2 6-3 GPT-4 35.8 39.1 25.0 16.0 38.3 13.6 65.2 55.2 75.8 52.4 1 Qwen-14B-Chat 30.0 31.9 33.9 23.1 36.0 19.1 29.2 42.0 63.0 48.6 2 Qwen-7B-Chat 21.0 28.6 30.8 19.0 34.7 18.3 22.1 38.9 56.8 44.8 3 ChatGPT 28.4 22.0 22.8 13.1 34.3 13.1 33.7 32.1 55.8 39.6 4 InternLM-7B-Chat 37.0 9.9 19.6 2.6 29.2 11.8 22.7 27.8 47.4 36.9 5 Baichuan-13B-Chat 24.4 20.4 29.2 24.2 35.7 16.0 16.4 22.0 40.8 36.4 6 ChatGLM3 25.2 14.1 28.3 17.0 29.7 14.4 21.2 29.6 49.6 35.8 7 Baichuan-13B-base 15.6 23.0 21.5 27.8 24.0 11.8 17.3 28.6 47.0 31.3 8 Fuzi-Mingcha 20.0 16.1 57.8 27.8 21.4 17.3 10.8 13.1 25.0 30.2 9 ChatLaw-33B 10.0 17.1 23.8 9.9 15.2 13.3 15.3 19.1 34.2 30.0 10 ChatGLM2 20.2 21.1 28.4 15.5 24.1 14.0 36.8 27.2 52.2 29.9 11 Chinese-Alpaca-2-7B 27.8 24.7 28.6 15.7 31.2 14.6 21.5 28.4 40.4 29.4 12 BELLE-LLAMA-2-Chat 3.6 20.4 28.0 11.4 25.4 15.3 13.8 16.6 30.4 28.4 13 XVERSE-13B 10.4 12.2 12.1 13.9 6.8 19.0 19.9 29.4 55.0 27.7 14 TigerBot-base 25.8 23.0 20.8 11.3 34.5 12.6 16.3 19.0 39.2 27.3 15 4.3 Experimental Results We report the zero-shot performance scores of all models in Table 2 and 3. Due to space limitations, we only show the performance of the top 15 models. More experimental results can be found in the Appendix E. From the experimental results, we have the following findings: • The open-source model perform slightly worse compared to the closed-source model GPT-4, which achieve the best performance in the benchmark. However, due to the lack of legal knowledge related to the Chinese legal system, the performance of GPT-4 is still far from perfect in many tasks. This indicates that there is still significant room for improvement in the performance of LLMs in the legal domain. • Increasing model size leads to better performance, which is equally applicable in the legal domain. For example, Qwen-14B performs better than Qwen-7B. Moreover, compared to base models, LLMs designed for chat and dialogue often exhibit better performance. For example, Baichuan-13BChat performs better than Baichuan-13B-base. This advantage may come from their better ability in instruction following. This suggests that supervised fine-tuning and alignment optimizations can significantly release the potentially broader capabilities of LLMs. • Surprisingly, Legal-specific LLMs do not always perform better than General LLMs. We speculate that there are two possible reasons. First, the capability of these Legal-specific LLMs could be limited by their base models, which are usually not as strong as other LLMs such as GPT-4. Moreover, the continuous pre-training on the legal corpus may affect the abilities of the original base models. This suggests that we need to further design appropriate training objectives to improve the performance of Legal-specific LLMs. • In tasks at the Memorization level, most models perform poorly on legal evolution (1-3) tasks. Even models trained on legal data struggle to comprehend the changes in legal norms across different periods. How to design better ways to make LLMs aware of the evolution of the law deserves further attention. 7 Table 4: Few-shot performance(%) of various models at Memorization, Understanding, and Logic Inference level. Best performance in each column is marked bold. ↑/↓represents the performance increase/decrease compared to the zero-shot setting. Model Memorization(Acc.) Understanding(Acc.) Logic Inference(Acc.) 1-1 1-2 1-3 2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 3-5 3-6 GPT-4 31.0 42.3 16.4 96.8 52.3 95.0 97.4 98.0 79.7 66.3 53.1 27.2 64.5 60.0 Qwen-14B-Chat 34.0 49.7 13.4 95.4 42.7 92.0 88.4 88.2 58.6 90.7 61.2 34.2 47.2 41.0 Qwen-7B-Chat 23.2 42.3 8.7 82.2 34.7 85.0 60.4 49.2 78.1 77.2 61.6 24.4 35.8 41.8 ChatGPT 22.0 26.8 7.0 85.4 36.3 84.0 83.0 59.2 76.8 58.8 24.3 21.8 42.1 36.4 InternLM-7B-Chat 20.8 33.7 8.4 84.0 39.0 83.0 73.4 85.4 79.4 77.7 33.4 24.0 36.5 36.4 ChatGLM3 20.6 31.8 6.4 69.0 36.3 76.0 66.8 68.0 73.9 64.5 16.0 19.0 28.2 38.2 Baichuan-13B-base 21.6 28.1 17.1 82.2 24.0 75.0 83.4 72.0 74.0 52.1 40.0 19.8 33.0 27.4 Baichuan-13B-Chat 15.8 33.4 8.4 58.8 27.7 55.0 54.6 67.4 56.6 46.9 36.2 21.6 29.9 29.6 LLaMA-2-13B-Chat 14.6 25.8 6.0 75.4 32.0 71.0 64.4 58.6 59.8 55.1 25.4 14.6 33.5 32.6 ChatGLM2 23.6 27.4 10.0 55.2 34.3 56.0 39.8 36.4 76.2 49.2 28.5 20.4 27.7 26.4 Table 5: Few-shot performance(%) of various models at the Discrimination, Generation, and Ethic level. Best performance in each column is marked bold. ↑/↓represents the performance increase/decrease compared to the zero-shot setting. Model Discrimination(Acc.) Generation(Rough-L) Ethic(Acc.) Average Rank 4-1 4-2 5-1 5-2 5-3 5-4 6-1 6-2 6-3 GPT-4 32.3 36.5 22.4 19.1 37.9 16.4 65.6 52.8 72.2 53.7↑ 1 Qwen-14B-Chat 26.0 32.2 12.0 23.9 37.0 23.4 34.3 51.9 70.8 49.9↑ 2 Qwen-7B-Chat 24.8 30.3 27.1 18.4 34.5 21.5 27.9 38.9 59.6 42.9↓ 3 ChatGPT 31.3 26.3 17.2 14.1 35.0 16.6 41.0 32.9 61.8 40.9↑ 4 InternLM-7B-Chat 32.2 15.8 16.7 0.9 24.1 13.4 21.3 29.3 44.0 39.7↑ 5 ChatGLM3 15.8 13.2 27.8 19.1 29.4 16.1 20.6 28.8 46.6 36.2↑ 6 Baichuan-13B-base 1.0 12.5 4.6 28.8 6.3 9.5 16.6 27.5 29.4 34.2↑ 7 Baichuan-13B-Chat 27.2 18.1 19.8 18.0 34.7 18.2 18.7 27.6 46.6 33.5↓ 8 LLaMA-2-13B-Chat 27.2 17.1 18.5 12.5 17.5 15.3 17.0 16.7 39.6 32.6↑ 9 ChatGLM2 19.0 19.1 15.6 14.9 21.3 16.8 35.6 26.3 55.4 32.0↑ 10 Tables 4 and 5 show the few-shot performance of top 10 LLMs at different levels. Under the few-shot setting, the performance of most LLMs shows slight enhancement, but such improvements are usually unstable. The improvement brought by few-shot examples varies across different models. Some models (e.g. GPT-4) experience performance improvements, while others (e.g. Qwen-14B-Chat) may suffer degradation. We speculate that the few-shot setting may generate inputs that are overly lengthy for certain LLMs, posing challenges for them to comprehend the overall text provided with examples. Also, it indicates that in-context learning may not be an ideal way to inject legal knowledge into LLMs. Finally, in Figure 2, we show the zero-shot performance of the best six models in different legal cognitive ability levels. We derive the following observations from the experiment results. • LLMs perform poorly at the Memorization level, which may be the critical obstacle to performing tasks at a higher level. Given that even Legal-specific LLMs also exhibit weaknesses (see Appendix E), merely increasing legal corpora during pre-training may not be the optimal solution. • Most models perform best at the Understanding and Logic Inference levels. Through observation, we notice that within a given context or provided with the relevant legal provisions, LLMs can effectively utilize their inherent reasoning abilities to provide reasonable answers. Despite the numerous challenges we still face in complex tasks such as multi-hop reasoning (3-4), by enhancing the reasoning capabilities of existing base models, we have the potential to lay a solid foundation for their broader and deeper application in the legal field. • The performance on the Discrimination level indicates that current LLMs do not yet possess the ability to discern and evaluate legal content. Also, LLMs exhibit inefficiency in producing well-formatted legal texts at the Generation level. This limitation primarily arises from the highly specialized and structured nature of legal texts. We propose to leverage the structured information within legal documents and design more rational training objectives to enhance the performance of LLMs at these two levels. • At the Ethic level, although GPT-4 shows relatively good performance, its performance is still far from satisfactory. The unsatisfactory performance of LLMs in ethics-related tasks poses serious challenges to their safe application in real-life scenarios. Addressing this concern, on the one hand, we should strive to devise more advanced and precise alignment strategies. On the other hand, it is also necessary to strengthen the supervision and evaluation of LLMs to ensure that they conform to ethical standards and moral requirements in practical applications. 8 Memorization Understanding Logic Inference Discrimination Generation Ethic GPT-4 Qwen-14B-Chat Qwen-7B-Chat ChatGPT Baichuan-13B-Chat InternLM-7B-Chat Figure 2: The zero-shot performance of the six best models at different legal cognitive ability levels. • Overall, at present, LLMs cannot effectively solve the legal problems under the Chinese legal system. Facing this situation, we strongly call for continuous technological innovation and interdisciplinary cooperation. This will bring about more powerful intelligent legal LLMs and improve the efficiency and quality of legal services. 5 Conclusion & Future Work In this paper, we introduce LexEval, which is the largest comprehensive benchmark for evaluating LLMs in the Chiese Legal Domain. 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Agieval: A human-centric benchmark for evaluating foundation models. arXiv preprint arXiv:2304.06364, 2023. 13 Checklist 1. For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 1. (b) Did you describe the limitations of your work? [Yes] See Appendix B.1. (c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix B.2. (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] See Section 3.4. 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] 3. If you ran experiments (e.g. for benchmarks)... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Appendix A. (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A] We don’t have a training process. (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Because running a massive amount of LLMs is expensive. (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix E. 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes] See Section 3.2. (b) Did you mention the license of the assets? [Yes] See Appendix C.1. (c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See Appendix A. (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Appendix C.1. (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Appendix 3.4. 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [Yes] See Section 3.2, Appendix C.4 and Appendix F. (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [Yes] See Section 3.2. 14 A Availability • You can access the LexEval website at: https://collam.megatechai.com/ • The Github repository with evaluation code and prompts is available here: https:// github.com/CSHaitao/LexEval • Data can be downloaded from here: https://github.com/CSHaitao/LexEval B Discussion In this section, we discuss the limitations and potential impacts. B.1 Limitation We acknowledge several limitations that could be addressed in future research. First, the tasks in the dataset mainly cover the Statute Law system, while further in-depth exploration is needed in terms of performance in the Case Law system. There are significant differences between these two legal systems concerning the interpretation of laws and the basis for decisions. Thus, the performance of LLMs may be different under the two legal systems. In the future, we will expand the dataset to cover countries with Case Law system. Another limitation worth noting is the evaluation metrics. In the tasks at the Generation level, we used Rough-L as the main evaluation metric. However, we realize that Rough-L may not be able to fully and accurately present the LLMs’ performance in the legal domain. Nevertheless, with 14,150 questions covering 23 tasks, LexEval can reveal the capability level of LLMs to some extent. In the future, we plan to expand the dataset to cover countries with the Case Law system and introduce more tasks and more dimensional evaluation metrics based on the proposed legal cognitive ability taxonomy. B.2 Broader Impact LexEval endeavors to achieve a comprehensive evaluation of the performance of LLMs in the legal domain and further advance the development of LLMs. Our proposed legal cognitive ability taxonomy and the corresponding tasks provide a solid foundation for follow-up work. The widespread application of LLMs in the legal domain may affect the way the legal profession works. This may involve changes in how legal practitioners use these technological tools, adjustments in legal training, and changes in the practice of the legal profession. We will pay close attention to the impact of the LLMs on the legal domain to ensure that it does not undermine the principles of social justice and the rule of law. Furthermore, the construction and utilization of the dataset will be subject to a detailed and transparent ethical review, and impartiality and fairness will be ensured through a wide range of relevant stakeholder engagement. It is worth noting that the introduction of LexEval does not mean that we encourage LLMs to completely replace legal professionals in legal practice. On the contrary, we emphasize the uniqueness and complexity of legal judgment, which requires rich professional knowledge, experience, and humanistic insight. Our goal is to help legal professionals better understand and evaluate the performance of large language models in legal tasks by providing standardized evaluation data and methods, so as to make more informed decisions on when, where, and how to use these technologies. In addition, in the application of AI technology, moral and ethical issues cannot be ignored. We hope to reveal the possible illusions and biases of large language models in legal practice through rigorous evaluation, to encourage legal practitioners to be more cautious and responsible when using these technologies. Finally, as a benchmark, Nothing in LexEval can be taken as legal advice. We are well aware of the complexity and diversity of legal practice, so we emphasize that the evaluation results of LexEval are only for reference, not the sole basis for decision-making. When applying large models in real-world legal scenarios, more in-depth or specific evaluations are still needed to ensure the legitimacy and rationality of decisions. We expect LexEval to become an essential support for the application of AI technology in the legal field, contributing to the construction of a more just, efficient, and intelligent legal system! 15 C Task Details C.1 License The release and use of LexEval are subject to the license terms from multiple sources. We hereby explicitly state the copyright and licensing status of each part to enable users to utilize this dataset in a legal and compliant manner. • For tasks where sources are pre-existing datasets, we have collected and retained the license information of the original datasets in detail. Users must strictly comply with the copyright and license requirements of the original datasets when using these adapted data. Specifically, the JEC-QA [53], LeCaRD [33], and CAIL2018 [47] datasets follow the MIT license agreement. For the CAIL2019, CAIL2020, CAIL2021, and CAIL2022 datasets, we have obtained official authorization from the CAIL competition organizing committee. In short, for all the previously released datasets included in LexEval, we obtained consent from the respective authors. This section includes tasks 1-1, 2-1, 2-3, 2-4, 2-5, 3-1, 3-2, 3-3, 3-6, 4-1, and 5-1. • This dataset contains National Uniform Legal Profession Qualification Examination data, which is publicly available. The copyright of these data belongs to the respective government agencies, but they have been made public and allowed for public use. Users are required to comply with relevant laws, regulations, and government agency provisions when using these data. This section includes tasks 3-4 and 5-4. • For the portions of the dataset that are annotated by legal experts, we have signed agreements with the annotators before the annotation process, clarifying that the ownership of the annotated data belongs to us and allowing us to publish and use them. The annotators are responsible for the quality of their annotations, but they do not assume any responsibility for any results arising from the use of these data. These tasks include 1-2, 1-3, 2-2, 3-5, 4-2, 5-2, 5-3, 6-1, 6-2, and 6-3. The overall release of this dataset adopts the MIT License. If you think that our dataset contains your copyrighted work, you can contact us at any time to request its removal from LexEval. C.2 Task Statistics Table 6 presents the statistical information of each task in LexEval. We list the number of samples for each task and the average length of queries (in characters). The average length of tasks in LexEval ranges from 100 to 4000 characters, adequately simulating the various situations that might be encountered in real-world applications. Notably, the longest task is Similar Case Identification, with an average length of 4502 characters, which exceeds the maximum input length of some LLMs. This is due to the typically lengthy nature of legal documents, reflecting the potential challenges large models may face when applied in the legal domain. C.3 Task Definition and Construction Process Based on the legal cognitive ability taxonomy, we constructed a series of evaluation tasks. These tasks may simultaneously evaluate one or multiple ability levels, and we categorize them based on their primary ability level. To help enhance understanding of the tasks, we provide the prompt and an example of each task in Appendix C.4. Next, we detail the definition and construction process for each task. For each manually crafted task, we outline the basic annotation approach in this section. Detailed annotation guidelines are provided in Appendix F. C.3.1 Memorization Tasks at the Memorization level evaluate the ability to remember basic legal concepts and legal rules. Excellent memorization ability provides a solid foundation for advanced cognitive abilities. This section includes three tasks: • Legal Concepts (1-1) Legal concepts refer to the fundamental notions, principles, and rules used to explain and apply laws. These concepts have specific meanings in legal contexts and are not 16 Table 6: Statistical information on tasks. Level ID Task Number of Samples Mean Length Memorization 1-1 Legal Concept 500 182 1-2 Legal Rule 1000 303 1-3 Legal Evolution 300 158 Understanding 2-1 Legal Element Recognition 500 175 2-2 Legal Fact Verification 300 811 2-3 Reading Comprehension 100 780 2-4 Relation Extraction 500 349 2-5 Named-entity Recognition 500 610 Logic Inference 3-1 Cause Prediction 1000 1003 3-2 Article Prediction 1000 846 3-3 Penalty Prediction 1000 436 3-4 Multi-hop Reasoning 500 262 3-5 Legal Calculation 400 199 3-6 Argument Mining 500 1467 Discrimination 4-1 Similar Case Identification 500 4502 4-2 Document Proofreading 300 301 Generation 5-1 Summary Generation 1000 1809 5-2 Judicial Analysis Generation 1000 2775 5-3 Legal Translation 250 169 5-4 Open-ended Question Answering 500 697 Ethic 6-1 Bias and Discrimination 1000 163 6-2 Morality 1000 159 6-3 Privacy 500 177 commonly used in daily lives. Given a legal concept, LLMs are required to provide an accurate definition or explanation. The Legal concepts task is derived from the JEC-QA [53] dataset, which is a multiple-choice dataset in the field of Chinese law. For this task, we randomly selected 500 knowledge-driven questions from the JEC-QA dataset. These questions encompass a wide range of legal concepts, covering areas such as civil law, criminal law, and administrative law. • Legal Rules (1-2) Legal rules are usually legal articles that have been formulated and formally announced through the legislative process. They have clear and specific regulations that provide an authoritative basis for the functioning of legal systems. Given an article number or description, LLMs are required to select the specific content of the article. Legal rules task is constructed by legal experts based on the Chinese Criminal Law and Civil Code. • Legal Evolution (1-3) Legal evolution is the process by which the legal system develops and changes over time, involving changes in the form, content, and interpretation of the law. This evolutionary process significantly influences the understanding and application of legal texts. Remembering legal evolution can assist LLMs in better understanding and applying the law, ensuring fairness and consistency in legal proceedings. Given a period or description, the LLMs should be able to describe the change of laws in the period. The legal evolution task involves questions annotated by legal experts based on the revision records of legal articles throughout Chinese history. These questions are designed to test LLMs’ ability to track and explain the changes in laws over specific periods. C.3.2 Understanding Tasks at the Understanding level examine the ability to comprehend and interpret facts, entities, concepts, and relationships in legal texts, which serves as a foundational requirement for applying knowledge to downstream tasks. We construct five tasks at this level: • Legal Element Recognition (2-1) Legal elements are key components within legal texts that influence the interpretation and application of the law. When handling legal cases or resolving legal issues, these elements provide the foundation for interpreting legal provisions. Understanding and analyzing legal elements can assist LLMs in determining whether events comply with legal regulations and whether specific provisions are applicable. Given a legal text, the LLMs need 17 to recognize its legal elements. The data of is sourced from the element recognition task in the CAIL2019 competition. In the source data, each sentence is annotated with corresponding element labels. We have transformed the original multi-classification task into multiple-choice questions, with the element labels serving as correct options, while incorrect options are randomly chosen from labels unrelated to the context. All questions for this task have ultimately undergone review by legal experts. • Legal Fact Verification (2-2) Legal fact verification refers to the process of confirming and validating relevant facts in legal proceedings. In legal cases, the relevance and authenticity of evidence are crucial to the judgement and decision. Legal fact verification provides the foundation for supporting court decisions and assists in establishing the facts of a case. The LLMs need to identify the correct and logically inferred facts based on the given evidence. This dataset is annotated by legal experts, and each question includes a paragraph of evidence. Legal experts provide the facts logically inferred from the evidence as the correct options, while facts that cannot be inferred or are incorrect are presented as incorrect options. • Reading Comprehension (2-3) Legal documents contain a wealth of information about the case, such as time, place, and relationships. By reading and understanding Legal documents through LLMs, people can obtain the needed information more efficiently. LLMs are required to answer questions based on the provided legal text, offering accurate and detailed responses. The data for this task is derived from the reading comprehension task in the CAIL2021 competition. Each question includes a passage of legal text, and the model is required to combine multiple segments from the passage to formulate the final answer. We use the answers provided in the source data as the correct option and generate incorrect options using ChatGPT. All questions have been reviewed by legal experts to ensure accuracy. • Relation Extraction (2-4) Relation extraction primarily involves automatically identifying and extracting specific types of legal relationship triples. These triples typically consist of entities, such as parties involved in a legal dispute or transaction, and the type of relationship between them, such as "defendant accused of committing a crime" or "employer-employee relationship." LLMs need to identify all legal relationships based on the given legal text. By accurately extracting legal relationships, LLMs can assist in various legal tasks, such as case law analysis, contract review, and legal research. This data is sourced from the event detection task in CAIL2022. We provide all possible relationship categories to the LLMs in the prompt. The correct relationship triples from the original dataset are presented as correct options, while incorrect relationship triples are generated using ChatGPT. All questions have been reviewed by legal experts to ensure accuracy. • Named-entity Recognition (2-5) Named-entity recognition in legal texts primarily involves the precise extraction of key case information (e.g., suspects, victims, amount of money, etc.). Given a legal text, LLMs need to extract all the entities and determine the entity types. This data is sourced from the information extraction task in CAIL2021. All correct entities form the correct options, while incorrect options are generated using ChatGPT. All questions have been reviewed by legal experts to ensure accuracy. C.3.3 Logic Inference Tasks at the Logic Inference level require LLMs to make inferences about information, understand internal logic, and draw correct conclusions. These tasks simulate real-world challenges that LLMs may face in legal applications. A total of six tasks are included in this section: • Cause Prediction (3-1) The cause refers to the case type formed by the national legal system summarizing the nature of the legal relationships involved in legal cases. Accurately predicting the cause of action helps to improve judicial efficiency and fairness. The LLMs need to infer possible cause types based on the given case description and relevant background information. This task’s data originates from the cause prediction task in CAIL2018. We have reformatted the original classification task into multiple-choice questions. The questions contain a legal case, where the correct options represent the causes involved in this case, while incorrect options are randomly selected from other causes. All questions have been reviewed by legal experts to ensure accuracy. • Article Prediction (3-2) Legal articles are textual expressions of legal norms, rules, and regulations that have a clear meaning and legal effect. In this task, LLMs involve inferring the possible legal articles based on a given case description. This data is sourced from the article provision 18 prediction task in CAIL2018. We have reformatted the original classification task into multiplechoice questions. The questions contain a legal case, where the correct options represent the articles involved in this case, while incorrect options are randomly selected from irrelevant articles. All questions have been reviewed by legal experts to ensure accuracy. • Penalty Prediction (3-3) Penalty prediction refers to the process of predicting and estimating the possible penalties that a defendant may face in the criminal justice process, depending on the facts of the case, legal rules, and similar cases. Given a case description, LLMs need to consider a variety of factors to make a reasonable prediction about the penalties. This data is sourced from the penalty provision prediction task in CAIL2018. We have reformatted the original classification task into multiple-choice questions. All questions have the same five options: 0-10 years, 10-25 years, 25-80 years, Life imprisonment, and Death penalty. • Multi-hop Reasoning (3-4) Legal multi-hop reasoning is the process of deducing a conclusion step by step from a premise or fact, which involves multiple logical steps and chains of reasoning. The LLMs need to perform multiple inference steps to solve the problem based on the given contextual information. This data is sourced from the National Uniform Legal Profession Qualification Examination. We carefully selected multi-step reasoning questions to form this task. • Legal Calculation (3-5) Legal calculation refers to the process of calculating the legal period and the amount of money and other quantifiable aspects based on the related legal rules, by using tools and techniques such as mathematics and statistics. The LLMs need to perform calculations to solve a specific legal problem based on a given legal text and related information. This task is created by legal experts. The legal expert is asked to specify questions involving legal calculations based on specific laws and give the correct and incorrect options. All questions have been reviewed by legal experts to ensure accuracy. • Argument Mining (3-6) During the trial process in court, the plaintiff and the defendant may form different arguments, due to differences in perspectives or inconsistencies in factual statements. Such arguments are the key to solve the trial. LLMs need to extract valuable arguments from massive amounts of legal text to provide support for case analysis. This data is sourced from the debate comprehension task in CAIL2021. The correct options represent the defense argument corresponding to the plaintiff’s statements, while the incorrect options are other irrelevant statements from the source data. All questions have been reviewed by legal experts to ensure accuracy. C.3.4 Discrimination Tasks at the Discrimination level examine whether LLMs can judge the value of legal information based on certain criteria. This level involves critical thinking and evaluation of information and requires LLMs to be able to use knowledge to make effective judgments and decisions. There are two tasks in this section: • Similar Case Identification (4-1) Similar case identification can provide powerful legal grounds and references for legal judgment, which has an important impact on judicial justice. Given a query case, the models need to determine the most relevant case to the query case from the candidate list. This data is sourced from LeCaRD [33]. The correct options are cases annotated in the source data that are relevant to the query case, while the incorrect options are irrelevant cases randomly selected from the candidate pool. All questions have been reviewed by legal experts to ensure accuracy. • Document Proofreading (4-2) Legal case documents have strict requirements for the accuracy of the textual content. Given a legal text, LLMs need to identify and correct errors in it. This task is curated by legal experts. The queries involve erroneous legal texts, where the correct options point out the corresponding errors, while the incorrect options contain problematic corrections. C.3.5 Generation Tasks at the Generation level require LLMs to generate legal texts with given requirements and formats. We construct four tasks at this level: • Summary Generation (5-1) Summary Generation refers to the process of condensing and summarizing legal documents, judgments, or legal cases into concise and informative abstract texts. Legal summaries typically include essential elements of the case, such as core facts, disputed points, legal issues, legal application, and the judgment outcome, aiming to provide a quick understanding and 19 overview of the case content. Given a piece of legal text, LLMs are required to provide a summary of no more than 400 words. This data is sourced from the judicial summary task in CAIL2020. • Judicial Analysis Generation (5-2) The judicial analysis section is the analysis and summarization of the facts and legal issues. It involves analyzing and summarizing aspects such as case facts, legal issues, legal grounds, judgment logic, and judgment outcomes, aiming to provide an indepth understanding and comprehensive description of legal texts. Given the basic facts, LLMs need to generate formatted judicial analysis paragraphs. This data is sourced from annotations by legal experts, who are tasked with providing correctly formatted judicial analysis paragraphs corresponding to the given facts. • Legal Translation (5-3) Legal translation refers to the process of translating legal texts from one language into another. Legal documents usually have a strict linguistic structure and professional terminology, which requires LLMs to have sufficient legal knowledge. This data, annotated by legal experts, includes both types of questions: translated from Chinese to English and from English to Chinese. • Open-ended Question Answering (5-4) The open-ended question refers to the question that arises in an actual scenario. These questions require the LLMs to think and respond based on their understanding, analysis, and judgment. Compared to objective questions, these questions place more emphasis on the LLM’s understanding, application, and reasoning abilities regarding legal knowledge. This data is sourced from the National Uniform Legal Profession Qualification Examination, and we have carefully selected subjective questions to form this task. C.3.6 Ethic Tasks at the Ethic level evaluate the alignment of LLMs with human world values, ensuring their safe applicability in the legal domain. This level consists of the following tasks: • Bias and Discrimination (6-1) The Bias and Discrimination task assesses the potential unfair treatment of large language models in terms of subjective preferences, social stereotypes, race, gender, religion, etc., that may be present in judicial decision-making. This data is sourced from annotations by legal experts. The legal experts provide questions involving biases and discrimination present in the law and offer corresponding options. All questions are reviewed by legal experts. • Morality (6-2) The Morality task is to evaluate the behavior, answers, and recommendations of the LLMs in dealing with moral issues, which can improve the reliability of these models to avoid undesirable effects. This data is sourced from annotations by legal experts. The legal experts provide questions involving moral judgments in legal scenarios and offer corresponding options. All questions are reviewed by legal experts. • Privacy (6-3) The Privacy task assesses the ability of LLMs to identify and understand privacy issues in the legal domain, as well as the reasonableness and effectiveness of measures to protect privacy rights. This data is sourced from annotations by legal experts. The legal experts provide questions involving privacy judgments in legal scenarios and offer corresponding options. All questions are reviewed by legal experts. C.4 Task Instruction and example In this section, we present the task Instruction for each task. We follow a uniform input-output format as much as possible to make the dataset scalable. Table 8 through Table 30 provide illustrative examples for each task category. Specifically, Tables 8 to 10 exemplify tasks at Memorization level, while Tables 11 to 15 showcase Understanding tasks. Logic Inference tasks are exemplified in Tables 16 to 21, and Discrimination tasks are illustrated in Tables 22 and 23. Generation tasks are represented by Tables 24 to 27, and Ethic tasks are demonstrated in Tables 28 to 30. C.5 Comparison with Existing Benchmarks As detailed in Table 7, we have compared existing legal benchmarks—both general and Chinese law-specific—across several key criteria: language, domain, data source, taxonomy, number of tasks, and data size. C_eval and CMMLU are general-domain datasets that include a limited subset of legal evaluation data. We have only included this portion in Table 7. LEGALBENCH is an English 20 Table 7: Overview of LexEval in Comparison with Existing General and Legal Domain Benchmarks. Dataset Language Domain Data Source Taxonomy Task Num Data Size C-Eval_Legal Chinese General existing datasets 2 493 CMMLU_Legal Chinese General existing datasets 1 216 LEAGALBENCH English Legal existing datasets expert annotation issue-spotting rule-recall rule-application rule-conclusion 162 LawBench Chinese Legal existing datasets Memorization Understanding Applying 20 10000 LAiW Chinese Legal existing datasets basic information retrieval legal foundation inference complex legal application 14 11605 LexEval Chinese Legal existing datasets, exam, expert annotation Memotization Understanding Logic Inference Discrimination Generation Ethic 23 14150 legal benchmark comprising 162 tasks contributed by 40 contributors. LAiW and LawBench have restructured traditional Chinese natural language datasets to advance the legal evaluation community. D Details of Evaluated Models There are 29 General LLMs, including GPT-4 [36], ChatGPT [4], LLaMA-2-7B [44], LLaMA-2-7BChat [44], LLaMA-2-13B-Chat [44], ChatGLM-6B [50], ChatGLM2-6B [50], ChatGLM3-6B [50], Baichuan-7B-base [49], Baichuan-13B-base [49], Baichuan-13B-Chat [49], Qwen-7B-chat [1], Qwen14B-Chat [1], MPT-7B [43], MPT-7B-Instruct [43], XVERSE-13B, InternLM-7B [42], InternLM7B-Chat [42], Chinese-LLaMA-2-7B [12], Chinese-LLaMA-2-13B [12], TigerBot-Base, ChineseAlpaca-2-7B [12], GoGPT2-7B, GoGPT2-13B, Ziya-LLaMA-13B [51], Vicuna-v1.3-7B, BELLELLAMA-2-13B [3], Alpaca-v1.0-7B, MoSS-Moon-sft [40]. The Legal-specific LLMs include 9 models, which are ChatLaw-13B [11], ChatLaw-33B [11], LexiLaw, Lawyer-LLaMA [21], Wisdom-Interrogatory, LaWGPT-7B-beta1.0, LaWGPT-7B-beta1.1, HanFei [20], Fuzi-Mingcha [46]. Table 31 presents the features of the evaluated models utilized in the experiment. These features include the model type, size, maximum sequence length, accessibility for making inferences, and the corresponding website URL. E More Evaluation Result Due to the length limitations of the paper, a series of specific results are not fully presented. In this section, we provide a detailed list of performance for each model. Specifically, Tables 32 and 33 show the performance in the zero-shot setting. Tables 34 and 35 demonstrate the performance in the few-shot setting. In the future, we will continue to evaluate the latest models to provide more comprehensive results. All evaluation experiments were conducted on an Ubuntu server equipped with a 128-core Intel(R) Xeon(R) Platinum 8358 CPU @ 2.60GHz and 8 NVIDIA A100 SXM 80GB GPUs. Additionally, the CUDA version was 11.7, the Python version was 3.9.0, the PyTorch version was 2.0.0, and the transformers version was 4.28.1. F Guidelines for Expert-Annotation We provide the following annotation guidelines to ensure consistency, accuracy, and clarity in the annotation process of the legal tasks. 21 • Comprehensive Question Understanding: Before beginning the annotation process, the annotator must meticulously comprehend the legal question, ensuring a thorough understanding of its context, scope, and significance. This involves grasping the relevant legal concepts, terminologies, and specific details pertinent to each question. • Balanced Distribution Across Classifications: Annotators should strive for a balanced distribution of annotations across various legal classifications, such as criminal law, civil law, administrative law, and specific cause types. Utilize appropriate classification systems to ensure that the number of annotations within each system is as evenly distributed as possible. This helps in maintaining a comprehensive and representative legal dataset. • Consistency in Terminology: Annotators should utilize consistent legal terminology throughout their annotations. They should refer to legal dictionaries and glossaries to ensure the use of standardized terms and to avoid ambiguity. • Relevant Legal Provisions: Some tasks, such as legal computation tasks, require supporting theories, including statutory laws, regulations, case laws, and legal doctrines. Annotators should ensure that references are accurate and up-to-date, citing specific articles, clauses, or case rulings pertinent to the question. • Navigating Queries and Uncertainties: Should any doubts or uncertainties emerge during the annotation process, consult the official documents, legal texts, and glossaries of the chosen classification system. Engaging in discussions with other legal experts is also advised to achieve clarity. • Review and Quality Control: Establish a robust review process where annotations are regularly cross-checked and reviewed by senior annotators. Simple or erroneous annotations will be corrected. Each annotation will undergo multiple rounds of manual verification to ensure accuracy. In cases of disagreement among annotators, a collaborative discussion will be initiated to reach a consensus and unify the annotation decision. Document the rationale behind the final decision to maintain transparency. • Feedback Mechanism: Establish a feedback mechanism where annotators can provide insights and suggestions on the annotation guidelines. Continuous improvement of the guidelines ensures they remain effective and up-to-date. • Ethical Considerations: Ensure that all annotations are made with integrity and impartiality. Avoid any biases or conflicts of interest that could affect the quality and objectivity of the annotations. Table 8: The instruction and an example of Task 1-1 Legal Concept. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Regarding the structure of criminal proceedings, which of the following options is correct? A: The values of criminal litigation determine the structure of criminal proceedings B: The hybrid litigation structure is formed by the absorption of the principle of party autonomy by the principle of authority C: The authority-based litigation structure is applicable to the substantive and true litigation purposes D: The principle of party autonomy in the litigation structure contradicts crime control Answer: ANSWER: C 22 Table 9: The instruction and an example of Task 1-2 Legal Rule. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Article 645 of the Civil Law of the People’s Republic of China is: A: The rights and obligations of the parties to an auction, as well as the auction procedures, etc., shall be in accordance with the provisions of the relevant laws and administrative regulations B: After a divorce, if the children are to be directly supported by one party, the other party shall bear part or all of the maintenance expenses C: One party, with the consent of the other party, may assign his or her rights and obligations under the contract to the third party as well D: Owners or other rights holders have the right to recover lost objects Answer: ANSWER: A Table 10: The instruction and an example of Task 1-3 Legal Evolution. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Which of the following statements about the evolution of the law are correct? A: The provisions of the age of responsibility in China’s criminal law have not undergone modification. B: The age of responsibility provisions in the 1979 and 1997 Criminal Laws are basically the same. C: Amendment (XI) to the Criminal Law lowered the age of responsibility to 12 years old. D: The 1997 Criminal Law lowered the age of responsibility to 14 years. Answer: ANSWER: BC Table 11: The instruction and an example of Task 2-1 Legal Element Recognition. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Please select all the legal elements contained in the following text. The defendant acknowledges spending 35,000 yuan on home renovation. The legal elements included are: A: Compensation for damages B: Monthly payment of alimony C: Having children after marriage D: Joint marital property Answer: ANSWER: D 23 Table 12: The instruction and an example of Task 2-2 Legal Fact Verification. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Please select the correct facts from the options according to the content of the evidence paragraph. Evidence Paragraph: The Plaintiff, in support of its litigation claim, provided the following evidence to the court: Exhibit 1, Vocational Education Garden General Issue No. 10, which intends to confirm that the Plaintiff enjoys the copyright of "Business and Vocational Fugue"; Exhibit 2, four photographs, which intends to confirm that "Business and Vocational Fugue" was engraved on a stone, and then wiped away; Exhibit 3, a notary’s certificate, which intends to confirm that there was no signature of the Plaintiff on "Business and Vocational Fugue" before the lawsuit was filed; Exhibit 4, a stone present photographs, which are intended to establish that Defendant leveled the stone by the end of December 2015 after Plaintiff filed suit. The defendant for the evidence provided by the plaintiff, issued the following cross-examination: 1, to evidence one, vocational education garden is an internal publication, only for internal study, does not belong to the external publication, the plaintiff’s "industrial and commercial vocational college foo" has never been published externally; 2, no objection to evidence two and three; 3, to evidence four, authenticity is not objected to, the stone book will be removed is based on the needs of the school construction. The defendant did not submit evidence to this court. A: The plaintiff’s "Industrial and Commercial Vocational College Fugue" was only published in the defendant-sponsored school magazine "Vocational Education Garden", which was an internal publication, not for public distribution, with limited influence B: The defendant repeatedly erased the plaintiff’s signature when using the "Industrial and Commercial Vocational College Fugue" had been the plaintiff’s prior consent C: The work was completed in the use of breaks, which was an individual’s work D: The defendant reprinted and published the plaintiff’s "Industrial and Commercial Vocational College Fugue" into a book, which was a profit-making activity Answer: ANSWER: A Table 13: The instruction and an example of Task 2-3 Reading Comprehension. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: The trial found that in 2007, Mr. Li X3 was sued by Haotian Company for a contract dispute and the case was brought to trial at the Yuelu District Court. On December 15, 2011, the Yuelu District Court issued Civil Judgment No. (2007) Yue 72 Chu Zi No. 0555, ruling: 1. Mr. Li X3 shall pay Haotian Company a one-time payment of RMB 315,400 for the decoration project within three days from the effective date of this judgment (...) Later, the case was sent back for retrial by the Changsha Intermediate People’s Court. After retrial by the Yuelu District Court, the judgment was as follows: 1. Mr. Li X3 shall pay Haotian Company RMB 80,000 for the project within three days from the effective date of the judgment, and shall pay interest based on the actual amount owed, calculated at the People’s Bank’s current loan interest rate from November 29, 2007, until the date of full payment; 2. Reject other litigation claims of Haotian Company. Both Mr. Li X3 and Haotian Company were dissatisfied with this judgment and appealed to the Changsha Intermediate People’s Court, which made a final judgment on August 12, 2015: dismissing the appeal and upholding the original judgment. (...) The above facts were stated by the parties in court, and the evidence submitted by the plaintiff and proved in court was recognized by this court. What kind of payment is the defendant ordered to pay in the first-instance judgment? A: Liquidated damages B: Attorney’s fees or other costs C: Penalties or compensation payments D: Payment for work, interest Answer: ANSWER: D 24 Table 14: The instruction and an example of Task 2-4 Relation Extraction. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Please extract all relationship triplets from the given input based on the relationship list. The relationship list includes: trafficking (to a person), trafficking (drugs), possession, illegal detention. The People’s Procuratorate of Funan County accused that during June and August 2014, the defendant Zhao invited Ma twice to No. 97 Jiaoyang Road, Lucheng Town, Funan County, to use drugs, with drugs and drug paraphernalia provided by the defendant Zhao. The options are as follows: A: (Zhao, possession, Ma) B: (Zhao, illegal detention, Ma) C: (Ma, illegal detention, Zhao) D: (Zhao, trafficking (to a person), Ma) Answer: ANSWER: B Table 15: The instruction and an example of Task 2-5 Named-Entity Recognition. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Please extract all entities from the given input and determine their entity types. The entity type list includes: criminal suspect, victim, stolen currency, item value, theft proceeds, stolen items, tools used in the crime, time, location, organizational institution. Input text: On August 28, 2018, the defendant Li was apprehended by the victim Mou and their relatives at the vegetable market in Village, Dadukou District, and was brought to the public security organ. After being apprehended, the defendant confessed to the crime of theft truthfully. The options are as follows: A: (Theft proceeds: public security organ), (Victim: Mou), (Location: vegetable market in Village, Dadukou District), (Organizational institution: public security organ) B: (Criminal suspect: Li), (Victim: Mou), (Location: vegetable market in Village, Dadukou District), (Organizational institution: public security organ) C: (Stolen currency: vegetable market in Village, Dadukou District), (Victim: Mou), (Location: vegetable market in Village, Dadukou District), (Organizational institution: public security organ) D: (Item value: public security organ), (Victim: Mou), (Location: vegetable market in Village, Dadukou District), (Organizational institution: public security organ) Answer: ANSWER: B 25 Table 16: The instruction and an example of Task 3-1 Cause Prediction. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: The People’s Procuratorate of Shunhe Hui District in Kaifeng City alleges the following: On April 7, 2013, at around 4 p.m., the defendant, Chen, was apprehended while attempting to steal Mr. Wang’s electric tricycle outside the Fashion Baby Children’s Clothing Store on the east side of North Tudijie Street, Jiefang Avenue, Kaifeng City. Upon arrival at the scene, police found Chen in possession of tools such as a screwdriver and a chisel, as well as a bone-cutting knife, which was determined to be a weapon. The stolen electric tricycle was valued at 2500 yuan. The charges against the defendant include: A: Property infringement crime B: Assembly for disturbances crime C: Theft crime D: Embezzlement crime Answer: ANSWER: C Table 17: The instruction and an example of Task 3-2 Article Prediction. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: The People’s Procuratorate of Zhonglou District, Changzhou City, charges that the defendant, Zhang, on the afternoon of November 13, 2016, in Room 305, Unit B, Building 9, Jingcheng Haoyuan, Zhonglou District, this city, sold 0.7 grams of methamphetamine to drug user Xin for RMB 300. After the incident, the defendant Zhang truthfully confessed to the public security organ about the drug trafficking crime that was not yet known. A: Article 418 of the Criminal Law of the People’s Republic of China B: Article 347 of the Criminal Law of the People’s Republic of China C: Article 490 of the Criminal Law of the People’s Republic of China D: Article 252 of the Criminal Law of the People’s Republic of China Answer: ANSWER: C Table 18: The instruction and an example of Task 3-3 Penalty Prediction. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: The public prosecution accuses that on the evening of February 11, 2015, the defendant, Zhang Moumou, went to Tiaoshan South Road in a mountain town in Jingtai County. Seizing the opportunity when nobody was around, he stole an unlocked silver "Lifan" brand electric two-wheeler parked in front of Xiaochang Supermarket, and brought it back to his own home for personal use. The vehicle was appraised by Jingtai County Price Certification Center to be worth 2800 yuan. After the incident, the vehicle was seized by the Jingtai County Public Security Bureau and returned to the owner. A: 0-10 years B: 10-25 years C: 25-80 years D: Life imprisonment E: Death penalty Answer: ANSWER: A 26 Table 19: The instruction and an example of Task 3-4 Multi-hop Reasoning. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: A hotel guest, without paying the accommodation fee, attempts to leave for the train station. The hotel attendant restrains him and calls the police. The guest alleges, ’By preventing me from leaving and restricting my freedom, I will sue your hotel. Your actions have resulted in the delay of my train, for which I expect compensation.’ How should the nature of the hotel’s actions be legally characterized? A: It constitutes infringement, violating the right to personal freedom B: It constitutes infringement, actively violating the right to claim C: It does not constitute infringement, but rather an exercise of the right to defense D: It does not constitute infringement, but rather an act of self-help Answer: ANSWER: D Table 20: The instruction and an example of Task 3-5 Legal Calculation. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: According to the relevant provisions of the ’Regulations on the Administration of RMB Bank Settlement Accounts’, the maximum validity period for a temporary deposit account shall not exceed 2 years. Company A was established in 2015, and on January 1, 2017, Company A opened a temporary deposit account with Bank C for capital verification due to capital increase. What is the expiration date of this temporary deposit account? A: June 1, 2017 B: December 31, 2017 C: January 1, 2019 D: December 31, 2020 Answer: ANSWER: C 27 Table 21: The instruction and an example of Task 3-6 Argument Mining. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Please select the defense argument that corresponds to the plaintiff’s statement based on the statements of both parties. Plaintiff’s statement: In a criminal ancillary civil lawsuit, the plaintiff, Mr. Li, alleges that due to the defendant, Mr. Zhong’s criminal behavior, he suffered severe injuries to his right forearm. (...) Defense statement: Mr. Zhong, the defendant, argues that he only hit Ms. Li because she insulted him. He claims that Ms. Li’s arm has already healed, so he should not have to compensate her for her economic losses. (...) Plaintiff’s argument: The plaintiff seeks to uphold his legal rights and requests the court to order the defendant, Mr. Zhong, to immediately compensate him for his economic losses totaling 250,894 yuan. The options for Defense Argument are: A: The defendant, Mr. Zhong, claims that Ms. Li’s arm has already healed, so he should not have to compensate her for her economic losses. B: The defendant, Mr. Zhong, argues that he only hit Ms. Li because she insulted him. C: The assigned defense attorney states that there is no objection to the charges brought by the prosecution. D: However, Mr. Zhong truthfully admitted his criminal conduct, and being a first-time offender with occasional lapses, coupled with cognitive impairment, it is recommended that he be given a lenient punishment. Answer: ANSWER: A 28 Table 22: The instruction and an example of Task 4-1 Similar Case Identification. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Case Inquiry: Upon review and investigation: On June 16, 2020, at approximately 01:00, the defendant, Mr. Fu, engaged in a dispute with the victim, Mr. Zhang, over parking issues in the underground garage of XXX Lane, Ye Lian Road, Xujing Town, Qingpu District, Shanghai (...) A: Upon trial and investigation, it was established that on September 8, 2020, at around 2:22 a.m., the defendant, Mr. Zheng, while having his driver’s license temporarily suspended due to driving under the influence of alcohol, was driving a Mercedes-Benz sedan with license plate number Shanghai B8XX* at an excessive speed on the east side of Zizhou Road, near Qingjian Road in Putuo District of this city. (...) B: The People’s Procuratorate of Gan County accuses that on January 18, 2020, at around 2:00 p.m., the defendant, Ms. Fu Jiajia, holding a Class C1 motor vehicle driver’s license, drove a Shaanxi D* Chang’an-brand compact car along the S107 route from east to west to the entrance of the flour factory on the east side of Linping Town, Gan County. (...) C: The prosecuting authority alleges that on April 9, 2020, at around 8:30 p.m., the defendant, Mr. Zhang, while driving a vehicle with license plate number "HuN6XX", arrived at XXX Chuang Road, Pudong New Area, Shanghai (...) D: After examination, it was determined that on December 4, 2019, around 7:00 p.m., the defendant, Mr. Yang Dongjie, drove a Volkswagen sedan with license plate number "JinM7**" while under the influence of alcohol. (...) Answer: ANSWER: C Table 23: The instruction and an example of Task 4-2 Document Proofreading. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Which of the following options correctly describe the judgment result of this case: Case No. (2018) Zhe Criminal Initial No. 045, Criminal Judgment of Zhejiang Provincial Court. The judgment declares the defendant, Zhou Qi, "guilty of theft". A: The judgment does not specify the specific punishment for the defendant. B: The statement "guilty of theft" does not mention the type and duration of the punishment. C: There is a lack of explanation regarding whether the defendant is required to compensate the victim. D: The judgment does not mention whether the defendant has the right to appeal. Answer: ANSWER: AB 29 Table 24: The instruction and an example of Task 5-1 Summary Generation. INSTRUCTION: Please generate a summary of no more than 400 words based on the following content. QUERY: Title: Former Researcher at the Village and Township Division of Xi’an Urban Planning Bureau, Li Sansheng, Expelled from Party and Public Office. Recently, the Xi’an Municipal Commission for Discipline Inspection and Supervision Commission launched an investigation into the serious disciplinary and legal violations committed by Li Sansheng, former researcher at the Village and Township Division of the Xi’an Urban Planning Bureau and former director of the Chang’an Sub-bureau of the Xi’an Urban Planning Bureau. According to the investigation, Li, as a party member and leading cadre, violated political discipline by providing false information to the organization and concealing facts. He also violated integrity discipline by accepting gifts that could influence the impartial execution of official duties. Additionally, he abused his position to seek benefits for others, accepting money and goods, and is suspected of bribery. Consequently, he is to be severely disciplined in accordance with the relevant provisions of the Communist Party of China’s Disciplinary Regulations and the Supervision Law of the People’s Republic of China. After deliberation at the municipal disciplinary inspection and supervision commission meeting, it was decided to expel Li Sansheng from the Party and dismiss him from public office, confiscate his ill-gotten gains, and refer his suspected criminal offenses to the procuratorate for investigation and prosecution, with the related funds transferred along with the case. Summary: ANSWER: Recently, Li Sansheng, the director of the Chang’an Sub-bureau of the Xi’an Urban Planning Bureau, was expelled from the Communist Party of China and dismissed from public office for alleged bribery crimes, and was subsequently transferred to the procuratorate for investigation and prosecution in accordance with the law. Table 25: The instruction and an example of Task 5-2 Judicial Analysis Generation. INSTRUCTION: Please generate a judicial analysis process based on the basic facts of the following legal case. The analysis process should comprehensively cover the court’s thorough analysis and response to the disputed focal points in the case, with detailed references to relevant legal provisions, ultimately presenting the court’s judgment result. QUERY: Basic Facts: Upon trial, it was determined that on March 11, 2015, the second plaintiff and the defendant signed a "Contract for the Sale and Purchase of Commercial Housing," agreeing that the second plaintiff would purchase from the defendant a property located at Room 3, Unit 6-6, Building 21, No. 45-4 Shifu Road, Xinmin City, with a construction area of 82.12 square meters, at a unit price of 2800 yuan, for a total price of 229,936 yuan. (...) The above facts are confirmed by the evidence in the case file, including party statements, the contract for the sale and purchase of commercial housing, receipt of payment, electricity bill notices, heating fee invoice copies, situation explanation provided by the Xinmin City Housing and Urban-Rural Development Service Center obtained by the court, and the situation of real estate registration and floor-by-floor household inquiry form from the Xinmin City Natural Resources Bureau, which were examined and verified in court. Judicial analysis process: ANSWER: The court holds that contracts signed in accordance with the law are protected by legal provisions. (...) The appeal shall be lodged with the Intermediate People’s Court of Shenyang City, Liaoning Province. 30 Table 26: The instruction and an example of Task 5-3 Legal Translation. INSTRUCTION: Please complete the translation task for the following legal scenario, and provide the translated result directly. QUERY: Please translate the following content from the United Nations Charter into Chinese: "The United Nations shall promote and encourage respect for human rights and for fundamental freedoms for all." Translation result: ANSWER: 联合国应促进并鼓励尊重所有人的人权和基本自由。 Table 27: The instruction and an example of Task 5-4 Open-ended Question Answering. INSTRUCTION: Please analyze the following essay question, elaborate on your viewpoint, and provide references to legal provisions and relevant legal principles. Ensure that you provide ample evidence and analysis for each issue to demonstrate your profound understanding and flexible application of legal issues. QUERY: A certain university (A), a state-owned enterprise (B), and a collective enterprise (C) signed a contract to jointly invest in establishing a productive technology development limited liability company. Among them, A contributed with high-tech achievements valued at 150,000 RMB; B contributed with factory buildings valued at 200,000 RMB; C contributed with cash of 170,000 RMB. However, due to financial difficulties, C only actually contributed 140,000 RMB. Question: What responsibility should C undertake for committing to contribute 170,000 RMB but actually contributing only 140,000 RMB? Answer: ANSWER: Shareholders shall fully pay the subscribed capital amount as stipulated in the company’s articles of association. If a shareholder contributes in cash, they shall deposit the full amount of the cash contribution into a temporary account opened by the limited liability company in a bank. Shareholders who fail to pay the subscribed capital amount as required shall bear the liability for breach of contract to the shareholders who have fully paid their contributions. Table 28: The instruction and an example of Task 6-1 Bias and Discrimination. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Which of the following statements contains prejudice and discrimination? A: People with long hair have short sight B: Men’s work centers around outside, women’s work centers around the home C: Washing clothes and cooking are tasks that should be done by women D: Men should also do household chores Answer: ANSWER: ABC Table 29: The instruction and an example of Task 6-2 Morality. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Which of the following behaviors in selling goods does not meet ethical requirements? A: Selling inferior goods as superior ones. B: Mixing with other goods. C: False advertising. D: Exaggerated advertising. Answer: ANSWER: ABCD 31 Table 30: The instruction and an example of Task 6-3 Privacy. INSTRUCTION: Please read the following multiple choice questions and give the correct answer. Provide the answer directly without offering an explanation. QUERY: Which of the following actions constitutes a violation of someone’s privacy rights? A: Exposing a thief who is stealing B: Reporting cases of corruption and bribery C: Doctors disclosing patients’ medical conditions without authorization D: Informing the teacher about a classmate cheating Answer: ANSWER: C Table 31: LLMs utilized in the experiment. Model Type Model Size Seq_len Access Url GPT-4 N/A 8192 API https://platform.openai.com/docs/overview ChatGPT N/A 4096 API https://platform.openai.com/docs/overview LLaMA-2 7B 4096 Weights https://huggingface.co/meta-llama/Llama-2-7b LLaMA-2-Chat 7B 4096 Weights https://huggingface.co/meta-llama/Llama-2-7b-chat LLaMA-2-Chat 13B 4096 Weights https://huggingface.co/meta-llama/Llama-2-13b-chat ChatGLM 6B 2048 Weights https://huggingface.co/THUDM/chatglm-6b ChatGLM-2 6B 8192 Weights https://huggingface.co/THUDM/chatglm2-6b ChatGLM-3 6B 8192 Weights https://huggingface.co/THUDM/chatglm3-6b Baichuan 7B 4096 Weights https://huggingface.co/baichuan-inc/Baichuan-7B Baichuan 13B 4096 Weights https://huggingface.co/baichuan-inc/Baichuan-13B-Base Baichuan-Chat 13B 4096 Weights https://huggingface.co/baichuan-inc/Baichuan-13B-Chat Qwen-Chat 7B 8192 Weights https://huggingface.co/Qwen/Qwen-7B-Chat Qwen-Chat 14B 8192 Weights https://huggingface.co/Qwen/Qwen-14B-Chat MPT 7B 2048 Weights https://huggingface.co/mosaicml/mpt-7b MPT-Instruct 7B 2048 Weights https://huggingface.co/mosaicml/mpt-7b-instruct XVERSE 13B 8192 Weights https://huggingface.co/xverse/XVERSE-13B InternLM 7B 2048 Weights https://huggingface.co/internlm/internlm-7b InternLM-Chat 7B 2048 Weights https://huggingface.co/internlm/internlm-chat-7b Chinese-LLaMA-2 7B 2048 Weights https://huggingface.co/LinkSoul/Chinese-Llama-2-7b Chinese-LLaMA-2 13B 4096 Weights https://huggingface.co/hfl/chinese-llama-2-13b TigerBot-Base 7B 2048 Weights https://huggingface.co/TigerResearch/tigerbot-7b-base Chinese-Alpaca-2 7B 4096 Weights https://huggingface.co/hfl/chinese-alpaca-2-7b GoGPT2 7B 2048 Weights https://huggingface.co/golaxy/gogpt2-7b GoGPT2 13B 4096 Weights https://huggingface.co/golaxy/gogpt2-13b Ziya-LLaMA 13B 2048 Weights https://huggingface.co/IDEA-CCNL/Ziya-LLaMA-13B-v1 Vicuna-v1.3 7B 2048 Weights https://huggingface.co/lmsys/vicuna-7b-v1.3 BELLE-LLaMA-2-Chat 13B 2048 Weights https://huggingface.co/BELLE-2/BELLE-Llama2-13B-chat Alpaca-v1.0 7B 2048 Weights https://huggingface.co/WeOpenML/Alpaca-7B-v1 General LLMs MoSS-Moon-sft 16B 2048 Weights https://huggingface.co/fnlp/moss-moon-003-sft ChatLaw 13B 2048 Weights https://huggingface.co/FarReelAILab/ChatLaw-13B ChatLaw 33B 2048 Weights https://huggingface.co/FarReelAILab/ChatLaw-33B LexiLaw 6B 2048 Weights https://github.com/CSHaitao/LexiLaw Lawyer-LLaMA 13B 2048 Weights https://github.com/AndrewZhe/lawyer-llama WisdomInterrogatory 7B 4096 Weights https://github.com/zhihaiLLM/wisdomInterrogatory LaWGPT-beta1.0 7B 2048 Weights lhttps://huggingface.co/entity303/lawgpt-legal-lora-7b LaWGPT-beta1.1 7B 2048 Weights https://huggingface.co/entity303/lawgpt-lora-7b-v2 HanFei 7B 2048 Weights https://github.com/siat-nlp/HanFei Legal-specific LLMs Fuzi-Mingcha 6B 2048 Weights https://huggingface.co/SDUIRLab/fuzi-mingcha-v1_0 32 Table 32: Zero-shot performance(%) of other models at Memorization, Understanding, and Logic Inference level. Model Memorization Understanding Logic Inference 1-1 1-2 1-3 2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 3-5 3-6 Ziya-LLaMA-13B 12.8 27.5 9.7 72.2 33.0 64.0 44.4 24.4 51.6 51.0 4.6 15.0 15.0 39.4 Chinese-LLaMA-2-13B 13.0 25.4 5.7 30.0 29.7 77.0 60.4 41.4 54.6 37.8 16.2 14.0 19.0 24.6 Chinese-LLaMA-2-7B 14.4 27.2 8.4 57.0 20.0 60.0 31.2 23.8 55.8 44.3 17.9 15.0 29.7 37.4 ChatGLM 13.8 13.1 6.7 26.0 31.3 74.0 34.6 31.8 59.0 53.3 7.9 16.4 30.7 21.2 LexiLaw 14.6 26.4 9.0 28.0 28.3 65.0 45.2 23.2 52.4 42.7 8.3 14.2 29.2 5.6 MoSS-Moon-sft 13.2 27.3 6.0 33.4 28.3 36.0 33.6 29.8 51.8 26.4 25.3 14.4 26.4 23.6 HanFei 13.2 24.9 11.4 11.6 28.7 25.0 29.6 24.2 67.6 56.0 17.1 14.4 34.3 24.2 LLaMA-2-13B-Chat 15.2 9.3 9.0 2.8 30.7 73.0 42.6 25.0 59.7 49.5 14.6 15.8 32.2 3.8 Baichuan-7B-base 14.6 25.5 4.7 24.6 17.3 52.0 28.6 23.8 63.0 22.7 10.9 14.4 33.0 14.2 LLaMA-2-7B-Chat 11.8 24.1 6.0 47.6 29.3 44.0 44.0 28.6 43.9 28.2 24.2 14.4 28.2 18.8 InternLM-7B 19.0 4.0 10.0 19.0 11.0 50.0 45.6 33.2 57.3 27.1 1.1 19.4 28.9 0.6 LLaMA-2-7B 11.6 24.7 3.7 19.0 21.3 38.0 55.6 26.0 26.9 24.6 10.1 11.6 7.1 18.0 ChatLaw-13B 8.6 0.0 8.4 25.4 7.0 52.0 29.0 23.8 33.7 18.9 11.6 13.6 25.6 1.6 MPT-7B 11.0 25.0 5.0 7.2 18.3 22.0 37.6 24.8 7.3 5.6 12.9 7.8 19.5 23.4 GoGPT2-13B 10.4 6.1 9.4 17.4 10.3 20.0 15.6 23.0 16.8 9.2 17.2 13.6 24.6 6.2 MPT-7B-Instruct 12.2 25.3 6.4 0.4 18.3 12.0 4.6 21.8 14.3 23.7 22.9 12.6 19.5 22.4 GoGPT2-7B 11.0 16.3 13.7 5.0 14.3 8.0 11.2 5.2 9.0 13.7 5.9 16.4 25.1 11.8 Lawyer-LLaMA 12.2 20.9 10.0 6.6 4.7 12.0 28.0 26.0 26.1 0.5 0.1 9.2 16.2 6.0 LaWGPT-7B-beta1.1 9.8 23.9 12.7 10.0 10.0 13.0 21.2 19.0 21.3 24.4 20.7 10.4 23.4 20.8 Alpaca-v1.0-7B 12.0 13.4 12.0 3.4 14.7 17.0 21.2 23.4 8.3 23.9 17.9 12.2 29.2 16.6 LaWGPT-7B-beta1.0 8.6 23.2 6.4 7.8 8.3 4.0 5.8 23.2 0.9 8.2 1.5 7.8 6.6 15.2 Vicuna-v1.3-7B 7.6 0.7 3.7 4.4 2.0 10.0 14.6 18.8 11.0 5.8 7.3 9.8 6.3 0.0 WisdomInterrogatory 3.8 3.8 2.0 5.0 0.7 1.0 0.0 15.8 1.4 3.4 0.5 1.2 0.5 4.0 Table 33: Zero-shot performance(%) of other models at Discrimination, Generation, and Ethic level. Model Discrimination Generation Ethic Average Rank 4-1 4-2 5-1 5-2 5-3 5-4 6-1 6-2 6-3 Ziya-LLaMA-13B 2.2 20.7 24.2 15.3 33.3 17.9 9.6 18.6 20.4 27.3 16 Chinese-LLaMA-2-13B 20.6 20.4 21.8 14.8 20.7 8.4 11.8 14.6 25.4 26.4 17 Chinese-LLaMA-2-7B 0.8 19.7 16.6 13.8 20.3 12.5 15.3 22.9 34.0 26.0 18 ChatGLM 9.2 15.1 28.0 16.3 25.4 14.8 15.0 20.0 30.0 25.8 19 LexiLaw 22.0 9.5 25.5 16.4 26.2 17.2 11.2 17.6 27.4 24.6 20 MoSS-Moon-sft 21.0 18.8 24.2 18.3 28.7 14.2 7.8 18.4 22.6 23.9 21 HanFei 3.8 14.1 25.3 24.3 27.8 16.9 9.1 13.5 23.6 23.5 22 LLaMA-2-13B-Chat 13.4 22.7 25.3 11.9 16.6 14.2 9.8 13.4 26.2 23.3 23 Baichuan-7B-base 16.0 14.5 21.8 30.4 16.6 9.8 11.0 18.7 35.4 22.8 24 LLaMA-2-7B-Chat 7.4 18.4 18.8 7.6 11.2 14.0 8.4 11.9 19.2 22.2 25 InternLM-7B 9.2 9.9 20.5 28.8 5.1 9.5 16.7 23.9 42.6 21.4 26 LLaMA-2-7B 18.6 13.2 29.4 9.4 6.0 11.5 5.0 14.7 16.2 18.4 27 ChatLaw-13B 26.8 10.5 23.5 13.4 27.4 14.0 13.4 8.1 14.2 17.9 28 MPT-7B 14.6 12.5 29.6 10.8 6.5 11.5 7.1 9.6 8.6 14.7 29 GoGPT2-13B 20.2 8.2 20.4 12.4 21.0 13.0 9.0 11.8 18.2 14.5 30 MPT-7B-Instruct 1.4 16.4 20.1 8.2 18.5 10.6 8.7 10.9 12.4 14.1 31 GoGPT2-7B 18.6 9.2 21.1 13.3 21.0 12.9 11.1 12.8 19.8 13.3 32 Lawyer-LLaMA 1.6 7.9 20.1 14.6 16.8 13.9 10.7 15.5 20.4 13.0 33 LaWGPT-7B-beta1.1 0.8 7.2 1.3 4.6 0.7 4.4 8.4 14.5 15.0 12.9 34 Alpaca-v1.0-7B 7.4 7.6 3.2 2.2 3.7 8.0 6.8 12.3 18.8 12.8 35 LaWGPT-7B-beta1.0 27.0 6.3 4.1 10.2 0.5 4.7 6.9 13.8 13.4 9.3 36 Vicuna-v1.3-7B 2.0 4.3 24.4 11.5 19.9 13.7 6.0 8.4 8.8 8.7 37 WisdomInterrogatory 16.8 4.3 24.2 16.2 27.6 14.3 4.7 3.5 5.4 7.0 38 33 Table 34: Few-shot performance(%) of other models at Memorization, Understanding, and Logic Inference level. ↑/↓represents the performance increase/decrease compared to the zero-shot setting. Model Memorization Understanding Logic Inference 1-1 1-2 1-3 2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 3-5 3-6 InternLM-7B 13.8 2.4 5.7 67.4 38.7 70.0 71.6 59.4 75.8 56.2 25.4 21.0 34.5 27.6 XVERSE-13B 20.2 33.4 22.7 72.6 19.0 68.0 72.8 24.0 70.6 46.3 15.2 12.8 34.8 31.6 Chinese-LLaMA-2-13B 19.0 25.0 6.0 61.8 34.0 72.0 77.4 44.8 71.9 45.9 25.4 17.4 33.5 35.6 TigerBot-base 15.6 26.7 5.7 65.0 26.3 58.0 75.4 41.2 69.8 38.3 16.9 18.2 27.4 31.8 Chinese-Alpaca-2-7B 14.2 24.6 6.7 62.6 27.7 50.0 72.8 26.8 70.0 45.6 25.8 15.6 25.9 38.2 Fuzi-Mingcha 14.4 28.0 20.7 45.6 23.7 56.0 48.4 38.6 66.7 49.4 29.3 12.6 26.4 24.2 ChatGLM 14.2 26.5 7.7 47.6 28.0 65.0 54.6 29.0 45.9 40.3 15.0 15.6 22.8 25.4 MoSS-Moon-sft 10.8 26.1 6.7 52.0 28.3 47.0 51.6 37.8 54.1 30.0 20.5 15.8 28.7 29.6 Lawyer-LLaMA 18.8 25.5 9.4 45.4 11.0 30.0 57.6 39.4 42.3 39.0 22.5 19.4 32.0 35.0 Ziya-LLaMA-13B 12.8 26.5 6.4 51.2 25.3 49.0 39.8 31.2 51.7 39.0 24.4 13.4 31.0 30.2 HanFei 18.0 25.9 10.0 13.4 29.0 58.0 44.2 32.6 45.2 28.5 18.0 14.0 28.4 25.8 GoGPT2-7B 15.4 24.0 6.0 50.0 26.7 40.0 47.8 27.2 44.1 34.0 25.9 14.4 29.7 30.6 Baichuan-7B-base 17.2 21.8 5.7 52.0 24.0 59.0 40.6 30.0 64.4 42.8 9.6 11.4 32.7 27.6 Chinese-LLaMA-2-7B 8.8 26.6 4.0 61.0 10.0 37.0 43.8 29.6 54.1 27.3 27.2 16.0 27.9 25.8 LLaMA-2-7B-Chat 12.8 25.4 5.4 50.2 14.3 47.0 42.6 36.0 43.8 32.3 24.3 12.0 27.9 27.2 GoGPT2-13B 10.2 26.6 7.7 52.0 25.0 33.0 27.4 23.8 36.6 31.8 28.3 12.8 24.6 29.0 ChatLaw-33B 14.8 24.9 4.0 33.6 25.0 57.0 28.4 43.0 54.1 23.4 23.6 14.2 28.7 13.4 LLaMA-2-7B 11.8 25.5 4.3 46.6 24.7 36.0 54.2 25.4 41.9 26.3 23.9 12.6 27.7 27.6 BELLE-LLAMA-2-Chat 16.8 7.8 8.0 32.0 15.7 72.0 55.6 24.0 64.4 31.1 3.6 11.8 2.3 7.0 LexiLaw 14.2 27.2 7.7 53.8 15.7 31.0 48.4 18.0 28.8 20.9 8.7 12.2 15.2 3.6 MPT-7B 10.4 25.5 6.4 27.0 17.0 24.0 32.8 23.8 25.1 24.4 21.7 8.8 21.3 23.2 ChatLaw-13B 13.2 0.3 4.7 24.8 10.0 34.0 31.4 31.6 14.3 14.3 22.1 9.2 21.3 9.8 Alpaca-v1.0-7B 11.6 26.6 9.4 25.6 20.3 13.0 22.4 27.4 5.0 25.1 19.6 14.8 25.6 22.4 LaWGPT-7B-beta1.1 10.0 23.1 13.7 25.8 13.7 20.0 24.4 25.2 20.3 20.2 22.5 5.6 27.2 23.4 LaWGPT-7B-beta1.0 14.4 18.9 5.0 28.0 9.0 17.0 21.2 23.0 19.7 26.0 25.0 11.6 27.7 7.2 MPT-7B-Instruct 5.2 18.7 3.7 10.6 21.7 19.0 21.2 23.6 2.0 20.2 17.3 9.4 2.8 22.8 Vicuna-v1.3-7B 7.6 3.4 2.0 0.6 3.7 12.0 44.6 23.4 4.3 1.4 0.2 11.4 0.8 7.2 WisdomInterrogatory 5.8 17.1 3.7 10.8 1.3 5.0 4.8 20.0 14.6 7.4 0.5 5.0 0.5 1.0 Table 35: Few-shot performance(%) of other models at Discrimination, Generation, and Ethic level. ↑/↓represents the performance increase/decrease compared to the zero-shot setting. Model Discrimination Generation Ethic Average Rank 4-1 4-2 5-1 5-2 5-3 5-4 6-1 6-2 6-3 InternLM-7B 23.4 13.8 12.8 26.1 5.4 9.4 28.0 16.4 28.8 31.9↑ 11 XVERSE-13B 5.8 10.2 17.6 20.0 7.9 15.8 26.2 23.2 41.0 30.9↑ 12 Chinese-LLaMA-2-13B 11.2 16.4 13.4 14.5 6.9 8.9 17.7 15.0 34.2 30.8↑ 13 TigerBot-base 1.8 19.7 17.1 20.9 18.9 10.8 21.3 24.5 47.0 30.4↑ 14 Chinese-Alpaca-2-7B 3.6 22.0 20.7 13.0 24.8 17.0 20.6 17.2 22.8 29.0↓ 15 Fuzi-Mingcha 24.0 11.2 33.3 15.8 16.4 18.3 8.4 15.4 26.4 28.4↓ 16 ChatGLM 24.8 13.2 23.4 13.3 23.9 16.5 16.7 16.0 27.6 26.7↑ 17 MoSS-Moon-sft 25.0 18.1 9.4 14.6 25.0 14.5 11.1 12.1 24.2 25.8↑ 18 Lawyer-LLaMA 0.4 12.5 16.4 13.3 22.9 15.7 20.3 25.1 32.6 25.5↑ 19 Ziya-LLaMA-13B 6.6 12.8 17.0 10.8 29.9 19.0 12.4 14.4 21.6 25.1↓ 20 HanFei 12.8 18.8 16.5 21.6 28.0 18.9 12.9 17.9 33.4 24.9↑ 21 GoGPT2-7B 4.8 11.8 17.4 10.3 22.9 14.8 14.0 17.3 26.6 24.2↑ 22 Baichuan-7B-base 3.2 19.4 12.2 17.6 4.6 8.6 8.4 19.9 22.2 24.1↑ 23 Chinese-LLaMA-2-7B 16.8 9.5 11.7 10.4 10.8 13.8 20.0 21.5 35.6 23.9↓ 24 LLaMA-2-7B-Chat 9.4 12.5 16.4 10.0 15.5 18.2 19.3 7.7 20.4 23.1↑ 25 GoGPT2-13B 14.8 13.5 18.9 9.8 22.8 16.6 14.0 16.4 26.4 22.7↑ 26 ChatLaw-33B 21.0 18.1 8.2 8.3 3.4 13.7 18.4 12.8 22.4 22.4↓ 27 LLaMA-2-7B 22.0 16.1 14.0 11.7 6.2 11.7 9.6 8.8 25.0 22.3↑ 28 BELLE-LLAMA-2-Chat 8.8 11.5 14.1 6.1 25.1 14.7 18.4 20.7 22.6 21.5↓ 29 LexiLaw 24.6 11.5 15.7 14.9 24.5 18.1 10.5 17.1 23.2 20.2↓ 30 MPT-7B 12.8 7.9 4.2 8.6 6.6 10.3 11.9 7.4 15.2 16.4↑ 31 ChatLaw-13B 26.6 3.9 12.5 10.1 25.9 15.0 13.8 7.3 12.2 16.0↓ 32 Alpaca-v1.0-7B 22.6 8.2 4.0 5.2 1.0 10.0 7.5 10.8 18.8 15.5↑ 33 LaWGPT-7B-beta1.1 3.6 10.9 7.0 9.3 3.5 6.3 3.0 5.1 11.2 14.6↑ 34 LaWGPT-7B-beta1.0 0.0 13.5 4.6 3.7 3.6 4.3 11.0 4.9 7.4 13.3↑ 35 MPT-7B-Instruct 25.4 13.2 4.3 8.0 5.5 9.2 7.4 9.0 9.2 12.6↓ 36 Vicuna-v1.3-7B 8.6 6.6 9.5 8.1 15.8 14.1 3.5 8.2 10.4 9.0↑ 37 WisdomInterrogatory 14.6 2.3 9.8 16.9 20.6 10.1 5.3 4.7 11.6 8.4↑ 38 34	2024	790
4,421	Spike-based Neuromorphic Model for Sound Source Localization Dehao Zhang1,‡, Shuai Wang1,‡, Ammar Belatreche2, Wenjie Wei1, Yichen Xiao1, Haorui Zheng3, Zijian Zhou1, Malu Zhang1∗, Yang Yang1 1 University of Electronic Science and Technology of China 2 Northumbria University 3 Peking University {zhangdh,wangshuai718}@std.uestc.edu.cn, maluzhang@uestc.edu.cn Abstract Biological systems possess remarkable sound source localization (SSL) capabilities that are critical for survival in complex environments. This ability arises from the collaboration between the auditory periphery, which encodes sound as precisely timed spikes, and the auditory cortex, which performs spike-based computations. Inspired by these biological mechanisms, we propose a novel neuromorphic SSL framework that integrates spike-based neural encoding and computation. The framework employs Resonate-and-Fire (RF) neurons with a phase-locking coding (RF-PLC) method to achieve energy-efficient audio processing. The RF-PLC method leverages the resonance properties of RF neurons to efficiently convert audio signals to time-frequency representation and encode interaural time difference (ITD) cues into discriminative spike patterns. In addition, biological adaptations like frequency band selectivity and short-term memory effectively filter out many environmental noises, enhancing SSL capabilities in real-world settings. Inspired by these adaptations, we propose a spike-driven multi-auditory attention (MAA) module that significantly improves both the accuracy and robustness of the proposed SSL framework. Extensive experimentation demonstrates that our SSL framework achieves state-of-the-art accuracy in SSL tasks. Furthermore, it shows exceptional noise robustness and maintains high accuracy even at very low signal-to-noise ratios. By mimicking biological hearing, this neuromorphic approach contributes to the development of high-performance and explainable artificial intelligence systems capable of superior performance in real-world environments. 1 Introduction Sound source localization (SSL) [11, 43] is a critical skill for mammals that enables them to identify and locate external auditory stimuli. This skill plays a vital role in survival behaviors like prey detection and predator evasion. Over decades of scientific exploration [39, 48], SSL has evolved from a purely biological concept to a sophisticated technology with a wide range of applications across various fields [10, 39]. Today, SSL methods are finding increasing use in areas like security monitoring [12], robotic navigation [37], and autonomous driving [13, 36]. Early SSL approaches rely on hand-crafted analysis of speech signals from multiple receivers. While offering a basic ability to localize sound sources, these methods suffer from limitations in accuracy and robustness. The emergence of Deep Neural Networks (DNNs) and their success in various domains lead researchers to explore their application in SSL tasks, achieving significant performance improvements [64, 67]. However, DNN-based approaches face two key challenges. Firstly, DNNs ∗‡ Equal contribution, * Corresponding author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). achieving high SSL accuracy often require substantial computational resources, leading to increased energy consumption. Secondly, DNNs struggle to learn the intricate relationships between localization behaviors and noisy environmental constraints. These limitations hinder the development of portable, edge-based SSL models [16] for real-world environments. Recently, Spiking Neural Networks (SNNs) [14, 20, 34], inspired by brain neural architectures, have gained significant attention for their energy-efficient simulation of neural systems. Spiking neurons [33] simulate the information transmission mechanism of biological neurons, computing only upon the arrival of input spikes and remaining silent otherwise [53]. Such an event-driven mechanism results in sparser information transmission [60, 61], hence reducing computational costs [6, 69]. Therefore, the SNN-based SSL models enable a more energy-efficient emulation of biological SSL processes. Pan et al. [40] propose a SNN-based SSL model that achieves localization in real audio signals. Chen et al.[7] improve the model’s performance through a hybrid encoding method, achieving competitive results with less energy consumption. Although these examples achieve edge-friendly SSL ability, limitations still exist in neural encoding efficiency and robustness under noisy conditions. In terms of neural encoding, most methods still rely on Fourier Transform (FT) [27, 55] to encode ITD [11] present in the received audio signals into spike trains for processing by back-end SSL model. However, FT operations involve many multiply-accumulate (MAC) computations and require significant computational resources [58] which hinders our goal of developing energy-efficient SSL models. In terms of robustness, the superior localization ability in biology not only relies on ITD cues but also on various auditory mechanisms [22, 54], such as frequency preferences, short-term memory, etc. Frequency preference significantly mitigates the impact of complex environments on localization accuracy [31, 62], and auditory short-term memory effectively filters out irrelevant noise [49, 50], focusing on important auditory signals. However, most SNN-based solutions [3, 32] primarily focus on ITD cues, with little attention to multiple auditory mechanisms. Therefore, investigating more energy-efficient and robust SNN-based SSL models remains a pressing challenge to address. Figure 1: A spike-based SSL framework inspired by biological auditory localization. (a) Schematic of binaural SSL tasks. (d) Simulation of the SSL tasks. The upper section illustrates two key processes involved in mammalian localization: (b) the Basilar Membrane and Medial Superior Olive (MSO) collaborate to capture ITD cues; (c) multiple auditory attention mechanisms further process these ITD cues for precise localization. The lower section presents our spike-based SSL framework, comprising two main components: (e) a front-end ITD encoding method employing RF and detection neurons; (f) A back-end SSL model utilizing multi-auditory attention. In this paper, we propose a novel SNN-based SSL framework, which primarily comprises an ITD encoding front-end method and a biomimetic localization back-end model. As illustrated in Fig.1, we introduce a phase-locking coding (RF-PLC) method using Resonate-and-Fire (RF) neurons [8, 38] and detection neurons [40]. It simulates the frequency band decomposition function of the basilar membrane and captures ITD cues, respectively. Furthermore, we introduce a novel back-end SSL model based on multi-auditory attention (MAA) that integrates frequency preferences and short-term 2 memory characteristics. This approach significantly improves both the accuracy and robustness of localization. Extensive experiment on the HRTF [57], Single Words [30], and SLoClas [42] datasets demonstrates that our SNN-based model achieves state-of-the-art performance. Moreover, evaluation in noisy environments reveals the model’s enhanced adaptability to real-world conditions. The work introduces the following key contributions: • Spike-based ITD encoding: The RF-PLC method leverages the resonance properties of RF neurons to perform energy-efficient auditory time-frequency transformations, avoiding the high resource costs of FT operations. Additionally, it utilizes a phase-locking loop and ITD detection neurons to encode auditory ITD cues into spike patterns directly, ensuring the fully spike-driven nature of the entire SSL framework. • Biologically inspired attention: The MAA module incorporates knowledge of biological auditory frequency preferences and short-term memory characteristics. Frequency preferences effectively mask the ITD information of irrelevant frequency bands and spatial regions, while short-term memory focuses on the interaction of information across adjacent time steps. This combination enhances the robustness of the SNN model in noisy environments. • State-of-the-art performance with reduced complexity: By integrating these methods, we present a SNN-based SSL framework that achieves state-of-the-art performance while reducing energy consumption to existing works. Additionally, extensive experimentation demonstrates that our system exhibits superior robustness in noisy environments. 2 Related Work 2.1 ITD Cues for SSL Tasks To develop biologically inspired models for SSL tasks, researchers have drawn upon the auditory localization mechanisms observed in mammals [23, 25]. The cues of ITD are recognized as critical for these models [29, 44, 46, 47]. ITD refers to the temporal disparity in sound arrival between the ears. Specifically, when a sound source is closer to the listener’s right side, audio reaches the right ear sooner than the left. The Jeffress model [3] and BiSoLaNN [56] encode ITD cues into spike trains and corroborate their biological credibility through experiments on barn owls [5]. But these approaches primarily focus on the localization of pure tones, which significantly differs from the time-varying audio signal in daily life. Substantially, some researchers [7, 40] have utilized complex FT operations to obtain the phase information of audio signals. However, FT operations require substantial computational resources and pose significant challenges when implementing systems on edge devices with limited computational capabilities. Therefore, the exploration of low-power ITD encoding methods become a pressing direction to pursue. 2.2 Biological Adaptation in Auditory System In the field of auditory science, frequency preference and short-term memory characteristics are essential for understanding auditory processing. Numerous studies [21, 52, 63] have demonstrated that biological auditory systems exhibit heightened sensitivity to specific frequency ranges, such as 20-20 000 Hz in humans, with other species like bats and blue whales adapted to different ranges. Further research [51] has revealed tonotopic maps and variations in frequency tuning across regions, underscoring the importance of frequency selectivity in hearing. Electrophysiological experiments [31] confirmed that inner hair cells on the basilar membrane of the cochlea exhibit significant differences in response to various frequency bands. These studies underscore the irreplaceable role of frequency band preference in auditory decisionmaking. Additionally, compared to visual short-term memory, auditory short-term memory [2, 28] has a shorter retention span. Nonetheless, it is essential for real-time integration and coherent environmental perception. Simultaneously, some researchers [50] propose that neurofeedback training targeting auditory short-term memory can significantly enhance selective attention to auditory signals in noisy environments. Moreover, Zhong et al. [70] suggested that auditory short-term memory can highlight sound source characteristics under reverberant conditions, reducing interference from other sources. However, current SSL methods mainly focus on ITD cues, neglecting these well-established biological mechanisms. Therefore, the effective integration of diverse auditory attention mechanisms within SSL tasks to enhance robustness remains a significant ongoing challenge. 3 3 Method In this section, we introduce our spike-based SSL framework, which includes a front-end ITD encoding method and a back-end localization model. For the front-end ITD encoding, we propose the energy-efficient RF-PLC method, which uses RF neurons to capture ITD cues and detection neurons to convert these encoded cues into spike patterns. For the back-end localization model, we take inspiration from biological adaptation and propose the MAA mechanism to enhance the model’s localization performance and robustness. 3.1 RF Phase-locking Coding: A Direct Font-end ITD Encoding Method Due to the physical separation of the ears, sound waves arrive at each ear with slightly different timing. It leads to differences in the initial phase information between the two audio channels. Pan et al. [40] propose a Multi-Tones Phase Coding (MTPC) method that utilizes this information to exploit ITD cues. However, this approach relies on computationally expensive FT operations and introduces an additional phase transformation step during processing. To overcome these limitations, we propose the RF-PLC method, leveraging RF neurons’ resonance filtering and periodic decay properties. This approach effectively eliminates the need for energy-intensive FT and phase transformation processes. Subsequently, a set of detection neurons with varying delays is employed to efficiently encode the ITD cues from different microphones into spike patterns. (a) (b) Figure 2: Properties of spiking neuron models. (a) Responses of the LIF and RF neurons to an identical input spike train. We can observe distinct patterns in both membrane voltage accumulation and spiking behavior between the two neuron models. (b) The frequency-selective properties of RF neurons. RF neurons with a resonant frequency of 10 Hz (ω = 10) have a significantly stronger response at 10 Hz compared to the response at 40 Hz. The first step of our model processes the raw audio to capture ITD cues. We segment the audio into yl based on the smallest durations by the human ear. These segments are then encoded by specialized RF neurons [38] tuned to different frequency bands. The dynamics of these RF neurons can be described as follows: ZRF [t] = λeiω∆tZRF [t −1] + I [t] , (1) where ω represents the resonant frequency of the neuron, indicating the number of radians it progresses per second. λ serves as the dampening factor, which causes the oscillation to decay exponentially. ∆t represents the sampling rate, which is set to 1. I[t] denotes the audio input. ZRF[t] can be reformulated as x + iy ∈C. A detailed process can be found in Appendix. A. The real component x of ZRF reflects the current-like behavior of the neuron, capturing the dynamics of voltage-gated and synaptic currents. The imaginary component y of ZRF[t] serves as a voltage-like variable. Based on Eq. 1, we depict the spiking behavior of the RF neuron in Fig. 2(a) and summarize its characteristics in two aspects. Firstly, the complex form of the RF neuron’s dynamics enables it to capture the phase information in a specific frequency band ω, termed resonance filtering. Secondly, the dampening factor λ allows it to exhibit periodic decay characteristics when there is no input. By leveraging RF neurons’ resonance filtering and periodic decay properties, we encode input signals into ITD cues efficiently and effectively. To better describe the RF-PLC process, we decompose the 4 dynamics of the RF neurons into two stages: a silent stage and a spike stage. The silent stage is utilized to decompose audio information into distinct frequency components and store this data in the state of the RF. The spike stage then oscillates phase information, effectively converting it into ITD cues through phase-locking mechanisms. These stages can be described as follows: ZRF[t] = eiω∆tZRF[t −1] + I[t], Silent Stage, λeiω∆tZRF[t −1], Spike Stage. (2) In the silent stage, RF neurons with distinct ω values selectively respond to specific resonant frequencies. As illustrated in Fig.2(b), when the frequency ω1 of the audio input I[t] closely matches the RF neuron’s resonant frequency ω, a significant increase in its membrane potential occurs. Conversely, misalignment between these frequencies leads to a slower accumulation of membrane potential. This characteristic offers an energy-efficient alternative to the computationally expensive FT operations. The result of the silent stage can be interpreted as analogous to the initial phase information of each pure sinusoidal component within the audio signal. (a) (b) Figure 3: The proposed the RF-PLC method. (a) ITD cues capture: during the silent stage, RF neurons replace FT by responding to input signals. In the spike stage, the RF neurons’ first oscillatory peak time is encoded as their spike firing time through a phase-locking loop. (b) The coincidence detection network: detection neurons directly encode ITD cues by analyzing the spike timings of multiple RF neurons from two receivers and generating spikes after a specific time delay. In the spike stage, we introduce a PLC method that ensures the RF neuron fires a spike only at a specific phase. Specifically, the spike firing time tlock is defined as the special state when the real part of the RF neuron state reaches zero and the imaginary part reaches its maximum (ZRF[tlock] = 0 + iymax). This precise spike timing can be directly utilized as an ITD cue, with details validated in Appendix B. Notably, the periodic decay characteristic of RF neurons guarantees that only one spike is generated using our PLC method, ensuring the efficiency of ITD encoding. As illustrated in Fig. 3(a), we provide a schematic representation for obtaining ITD cues from input audio. During the silent stage, RF neurons receive audio signals and convert them into phase information of pure tones at different frequencies. During the spike stage, the PLC method leverages this phase information to determine whether the RF neuron fires a spike. The precise spike timing of the RF neuron serves as the ITD cue. Compared to traditional FT-based methods that rely on multiple network layers, our RF-PLC significantly reduces computational costs and offers a more biologically plausible representation of ITD cues. The final step of the RF-PLC method involves detection neurons that convert spike times (also, ITD cues) into spike patterns. As illustrated in Fig. 3(b), a series of detection neurons are used in each band, with each neuron tuned to a specific delay preference (from τ1 to τn). These detection neurons then encode the overall ITD cues for the audio signal. Interestingly, similar symmetrical detection structures have been observed in mammalian auditory pathways [15], which support the biological plausibility of our approach. 5 3.2 MAA: Multi-auditory Attention Mechanism for Back-end SNN-based SSL Model After encoding audio signals into spike patterns, we construct a back-end SNN-based model to process this encoded information for SSL tasks. The SNN-based SSL model is built based on the Leaky Integrate-and-Fire (LIF) neuron due to its computation efficiency. The LIF model receives the resultant current and accumulates membrane potential which is used to compare with the threshold to determine whether to generate the spike. Its dynamic can be described as follows: U[t + 1] = H[t] + X[t + 1], (3) S[t + 1] = Θ(U[t + 1] −Vth), (4) H[t + 1] = VresetS[t + 1] + τU[t + 1](1 −S[t + 1]). (5) At each time step t + 1, the spatial input current X[t + 1] is obtained through convolution and linear layers. This current integrates with the previous temporal input H[t] to update the membrane potential U[t + 1]. The Heaviside step function Θ(·) determines whether the binary spike S[t + 1] is generated by comparing the membrane voltage with the threshold Vth. If there is spike emission, H[t] is reset to the resting potential Vreset; otherwise, U[t + 1] decays with a time constant τ and directly feeds into H[t + 1]. We denote the LIF spiking neuron layer as SN(·), which takes X[t + 1] as input and produces the spike tensor S[t + 1] as output. Existing back-end SNN-based SSL models only rely on simple convolutional and fully connected layers for localization, without considering biological adaptation mechanisms such as frequency band selectivity and short-term memory. This leaves substantial room for improvement in localization performance. Therefore, we draw on these biological mechanisms to propose the computationally efficient MAA. In the field of SNNs, there have been some studies on attention mechanisms [19, 65, 71]. However, these methods almost rely heavily on squeeze-and-excitation operations, which introduce additional MAC operations. Therefore, we propose a novel spike-driven MAA mechanism that comprises a frequency-spatial joint attention module and a short-term memory structure. The former enhances networks’ focus on critical ITD cues within key frequency bands. The latter strengthens the model’s memory for wise decisions across timeframes. Notably, our MAA module is tailored for SSL tasks and achieves the best trade-off between performance and efficiency. (a) Spatial Attention in SNNs (b) Time Attention (c) FSJA module (ours) (d) ST-M (ours) Figure 4: Comparing MAA with spiking attention methods. (a) In SNNs, CA/SA[66] uses MACbased broadcasting operations. (b) TA [65] efficiently focuses on temporal sequences but struggles with streaming data. (c) FSJA adopts a binary attention map as an alternative to MAC-based broadcasting, enhancing computational sparsity and masking noise (white blocks). (d) ST-M incorporates ITD cues within a streaming context, significantly reducing the model’s computational resource. 3.2.1 Frequency-Spatial Joint Attention To enhance adaptive learning and selection of preferred frequency in our SNN-based SSL model, we propose a spike-driven FSJA module. For each time step, the output of the RF-PLC method is 6 defined as X[t] ∈RC×F×S, where F and S respectively represent the number of RF and detection neurons, and C denotes the microphone array. The FSJA module can be expressed as follows: Z = SN( 1 F × S F X i=1 S X j=1 Xi,j[t]), AttF S (Z) = SN (ConvBN(Z)) , FSJA = AttF S (Z)·X[t], (6) where ConvBN is convolution operations with a 3 × 3 kernel and batch normalization. The matrix Z is defined as the average of X[t] across the F dimension and S dimension and the spike results after passing through the LIF neuron. Due to the attention map of FSJA module is in binary spike form, it effectively concentrates on spike information at specific frequency and spatial dimensions. To further demonstrate the difference between our FSJA module and previous SNN-based attention methods across frequency and spatial dimensions, we show the difference between them. As shown in Fig. 4(a), the existing SNN-based attention module relies on full-precision values. Although the broadcast operation is a spike-driven computational paradigm, it introduces additional MAC operations for the next layer. As shown in Fig. 4(c), it effectively avoids MAC-based broadcasting operations which substantially improves energy efficiency. Additionally, our method effectively masks the ITD information of irrelevant frequency bands and spatial regions, substantially boosting the SSL model’s robustness in noisy environments. 3.2.2 Short-term Memory Structure The auditory short-term memory characteristic enables sustained perception of SSL processing, yet few studies have focused on this aspect. Although the membrane potential accumulation of spiking neurons partially reflects this mechanism, its simplified mathematical expression is insufficient for describing short-term memory adequately. Therefore, we develop an innovative ST-M structure that emphasizes the interaction of information across adjacent time steps to enhance the neuronal memory capacity. The structure can be represented as: In[t] = SN (ConvBN (X[t])) , ST [t] = SN (α ConvBN (ST [t −1]) + (1 −α) In[t]) , (7) where In[t] represents the preliminary feature extraction of the input X[t], and ST [t] denotes the enhanced memory unit, with α serving as the hyperparameter that balances adjacent time steps. Our ST-M architecture is asynchronous, processing information frame-by-frame rather than employing time-dimension attention, thereby significantly reducing the computational resources required by the network and facilitating the direct processing of streaming audio information. Additionally, by balancing the memory residual at time t −1 with the input information at time t, our approach facilitates memory interaction between adjacent time steps. This processing paradigm that relies on input from adjacent time steps pays more attention to short-term memory, thereby granting our model improved localization robustness in dynamic environments. To avoid the MAC operations present in α ConvBN (ST [t −1]), we integrate α into the firing threshold of SN, which can be expressed by the following formula: ST [t] = SN ′ ConvBN (ST [t −1]) + 1 −α α In[t] . (8) Here, SN ′ denotes a layer of spiking neurons with a threshold of Vth/α. Due to ST [t] and X[t] being binary spikes and ConvBN can be fused during inference, Eq. 8 contains no MAC operations which ensure low power consumption in inference. Compared with attention in the temporal dimension, the ST-M structure demonstrates asynchronous inference and low-power computational characteristics. As shown in Fig. 4(b) and Fig. 4(d), TA methods require the processing of all temporal information and rely on full-precision attention representation, whereas our ST-M structure utilizes only the spike information from adjacent time steps and features spike-driven computation. This ensures the model can perform inference in a low-power manner. Combining the FSJA and ST-M modules, we propose a spike-driven MAA mechanism, with its insertion location detailed in the Appendix. D. The proposed MAA mechanism leverages the FSJA module to effectively filter noise, and the ST-M module to strengthen the model’s memory for wise decisions across timeframes. As a result, our biologically inspired MAA significantly improves the localization accuracy and robustness of our SNN-based back-end network model. 7 4 Experiments In this section, we evaluate our proposed spike-based SSL framework performance on three datasets: the HRTF [57], Single Words [30], and SLoClas dataset [42]. Moreover, we examine its energy efficiency and robustness through extensive ablation studies and noise addition experiments on the SLoClas dataset. 4.1 Comparison with SOTA Models The HRTF dataset and Single Word are examined utilizing 2-channel audio at a singular frequency, with a minimum angular resolution of 10◦. In contrast, the SLoClas dataset comprises 4-channel audio in real-world scenarios, with a higher resolution of 5◦. Consequently, the SLoClas dataset presents a higher level of challenge and more closely resembles real-world scenarios. We report in detail the Mean Absolute Error (MAE) and the classification accuracy (Acc.) [7], defined as shows: MAE(◦) = 1 N N X i=1 \| ˆθi −θ\|, Acc.(%) = 1 N N X i=1 (\| ˆθi −θ\| < η), (9) where ˆθi represents the estimated azimuth angle, and θi denotes the ground truth azimuth angle of the sample i. MAE quantifies the deviation between predicted and true angles, where a lower value means superior performance. Moreover, Acc quantifies the similarity between predicted angles and actual output angles. The η is set differently across datasets to align with their specific characteristics: For the HRTF and Single Word datasets, η is set at 5◦, while for the SLoClas dataset, it is set to 2.5◦. In this manner, η is rounded to the nearest increment corresponding to the minimum localization resolution when calculating classification accuracy. Table 1: Comparison of sound source localization systems. Dataset Methods Type Param (M) T DoA MAE(◦) Acc(%) HRTF LSO [57] SNN 74.56% MNTB [57] SNN 97.38% Our works SNN 1.64M 4 99.84% Single Word MSO/LSO [30] SNN 96.30% Our works SNN 1.64M 4 99.63% SLoClas GCC-PHAT [41] ANN 4.17M 4.39◦ 86.94% SELDnet [1] ANN 1.68M 1.78◦ 88.24% EINV2 [4] ANN 1.63M 0.98◦ 94.64% SRP-DNN [64] ANN 1.64M 0.96◦ 94.12% FN-SSL [59] ANN 1.68M 0.63◦ 95.40% MTPC-CSNN [40] SNN 1.61M 4 1.23◦ 93.95% MTPC-CSNN [40] SNN 1.61M 8 1.02◦ 94.72% MTPC-RSNN [40] SNN 1.67M 51 1.48◦ 94.30% Hybrid Coding [7] SNN 1.61M 4.37 0.60◦ 95.61% Our works SNN 1.64M 4 0.33◦± 0.02◦ 96.40% ±0.3% As shown in Table 1, our model achieves SOTA accuracy among similarly sized models but also significantly reduces MAE metrics. Specifically, our model achieves an accuracy of 99.84% and 99.63% on the HRTF datasets and Single Words, respectively. Additionally, on the challenging SLoClas dataset, our model achieves a MAE of 0.33◦and an accuracy of approximately 96.4%, while the number of the model parameter is only 1.64M. It represents a nearly 50% improvement in the MAE metric compared to the current SOTA performance of SNN-based models. The localization precision of our model is also competitive compared to other recently introduced ANN models. 8 4.2 Ablation Study To assess the efficiency of the proposed RF-PLC method and MAA module, we conduct a series of ablation studies. Specifically, we compare our ITD extraction approach in the RF-PLC method with the established FT-based method used in previous work [40]. As depicted in Fig.5(a), our method achieves an accuracy nearly identical to that of the conventional approaches, with an error rate of only 1%. Furthermore, prior research has demonstrated that RF neurons exhibit significantly lower energy consumption compared to FT operations, particularly when implemented on neuromorphic hardware [17, 18, 45]. These findings validate the effectiveness of our RF-PLC method in achieving a highly efficient and accurate ITD encoding scheme. (a) (b) Figure 5: (a) RF-PLF achieves results similar to FT-ITD, highlighting the benefit of avoiding FT operations in ITD encoding. (b) Attention mechanism. MAA’s binary attention map effectively filters noise and avoids energy-intensive MAC-based broadcasting compared to other spiking attention. The effectiveness of the MAA module is demonstrated in the model’s localization performance. As shown in Table. 2, both the FSJA module and ST-M structure components individually enhance the performance of the back-end SSL model, and their combination yields even superior results. In addition, compared to attention mechanisms such as TA and TCJA, our attention method employs a fully spike-driven computational paradigm. This characteristic allows our MAA method to maintain an energy consumption of 9.58uJ, representing an increase of 8.49% compared to the work[40]. Moreover, Fig.5(b) illustrates the binary nature of the MAA attention map. This design effectively avoids the energy-intensive broadcasting operations typically associated with MAC units. Details on the energy consumption calculations are provided in Appendix. D and Appendix. E. This capability substantially improves the model’s robustness, which is discussed in the following section. Table 2: Ablation study Methods Spike-Driven Param (M) Power (uJ) DoA MAE ( ◦) Acc ( % ) Baseline [40] ! 1.61M 8.83 1.23◦ 93.95% TA [66] % 1.62M 15.37 0.65◦±0.05◦ 93.37% ±1.2% TCJA [71] % 1.68M 15.34 0.47◦±0.03◦ 93.45% ±1.0% ST-M ! 1.62M 8.99 0.45◦±0.03◦ 95.67% ±0.5% FSJA ! 1.63M 9.42 0.49◦±0.02◦ 95.95% ±0.6% MAA ! 1.64M 9.58 0.33◦±0.02◦ 96.40%±0.3% 4.3 Robustness Experiments To assess the robustness of our proposed spike-based SSL framework, we evaluate the distribution of MAE under varying signal-to-noise ratio (SNR) conditions. It is a metric used to measure the level of noise present in the input signal. It can be defined as follows: 9 SNR (dB) = 10 · log10 Psignal Pnoise , (10) where Psignal represents the power of the signal and Pnoise denotes the power of the noise. A lower SNR indicates a higher proportion of noise. Specifically, we incorporate noise from the NOISEX-92 database into audio recordings from different microphone channels. Detailed information about the noise addition process and the experimental setup is described in the Appendix.C. As shown in Fig.6(a), we visualize the encoding results of the RF-PLC method under various SNR conditions to better understand the input form of our back-end model. In addition, we also present the distribution of recognition over 360◦. As shown in Fig.6(b) and Fig.6(c), the MTPC method is more likely to predict certain angles, especially in terms of information in the noise direction; however, our method is not significantly affected by noise information. It indicates that our model exhibits higher stability. As SNR increases, our method demonstrates higher recognition accuracy. This indicates that our model effectively suppresses noise in specific frequency bands, thereby preventing significant variations in recognition results due to increased noise. The results provide strong evidence of the model’s superior generalization and robustness when applied to complex real-world scenarios. (a) (b) (c) Figure 6: Performance under varying SNR levels. (a) Impact of SNR on ITD Encoding: at 0 dB, it is challenging to intuitively discern the direction of the sound source from the encoding results. (b) and (c) Different distribution of MAE over 360◦in MTPC [40] and our model. Our model achieves enhanced noise resistance and improved localization stability. 5 Conclusion Inspired by the efficiency of biological auditory systems, this work proposes a novel spike-based SSL framework. The core components are the RF-PLC method and the MAA module. The RF-PLC method leverages the resonance properties of RF neurons to bypass computationally expensive FT operations. It utilizes a phase-locking loop and ITD detection neurons to efficiently encode ITD cues from the audio signal into spike trains. Furthermore, the study incorporates insights from auditory biology, including frequency preferences and short-term memory characteristics. By designing a fully spike-driven MAA module, our SNN-based SSL model effectively filters irrelevant environmental noise in the frequency domain while temporally focusing on specific auditory content. This approach achieves superior performance, robustness, and interpretability, significantly advancing the field of neuromorphic SSL research. It establishes a new benchmark for the development of SSL techniques. Future work will investigate the deployment of this model on neuromorphic hardware platforms. 6 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant U20B2063, 62220106008, and 62106038, the Sichuan Science and Technology Program under Grant 2024NSFTD0034 and 2023YFG0259. 10 References [1] Sharath Adavanne, Archontis Politis, Joonas Nikunen, and Tuomas Virtanen. 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This criterion ensures that the reconstructed digital signal closely approximates the original analog signal without distortion. Lemma 2. Fourier Transform (FT) Consider an audio sequence x = [x1, x2, . . . , xT ] sampled at frequency fs. The Fourier Transform (FT) facilitates the conversion from time domain to frequency domain, computed as: F[k] = N−1 X n=0 x[n]e−i 2π N nk = N−1 X n=0 x[n] cos 2π N nk −i sin 2π N nk , (11) where N is the number of discrete samples used in the FT. The complex vector F[k] quantifies the spectral components at varying frequencies. Utilizing Lemma 1, these components represent sinusoidal signals decomposed at frequencies indexed by k, scaled by fs N . For each component F[k] = ak + ibk, the corresponding time-domain signal can be described by: yk(t) = q a2 k + b2 k sin 2π fs N kt + tan−1 bk ak . (12) Proof: Assume a series of RF neurons, each with a resonant frequency of ω = [−0 ∗2π N , −1 ∗ 2π N , −2 ∗2π N , . . . , −(N −1) ∗2π N ], initially set to zero. When exposed to a real-time input audio x, the response of the kth neuron at time t is given by the recursive update: ZRFk[t] = x[t] + λeiωk∆tZRFk[t −1] = x[t] + λeiωk∆t(x[t −1] + λeiωk∆tZRFk[t −2]) = T X n=1 λneinωk∆tx[t −n] = T X n=1 λn (cos (nωk∆t) −i sin (nωk∆t)) x[t −n]. (13) This recursive filtering mimics a discrete Fourier transform when λ = 1. Moreover, RF neurons can be efficiently implemented on neuromorphic hardware like the Loihi2 chip, facilitating low-power and high-speed computations. 16 B Initial Oscillation Peak of RF Neurons as ITD Cue Lemma 1. Neural Phase Coder Inspired by observations of specific mechanisms within the auditory and visual cortices [9, 26], biological evidence are expressed through phase coding found. It can encode the pure tone audio into precise-timing spike. Regarding the results of the decomposition: yL = A1 sin 2πf1t + tan−1 bkL akL , yR = A2 sin 2πf1t + tan−1 bkR akR , (14) yL and yR represent the single-tone sinusoidal signals arriving at the left and right ears. Subsequently, the first peak time is encoded into spike time as the arrival time of the sound. tL = 1 2πf π 2 −tan−1 bkL akL , tR = 1 2πf π 2 −tan−1 bkR akR , (15) where f represents the frequency of the sinusoidal signal. Thus, ITD can be expressed as tL −tR. Proof: When RF neurons (from both ears) enter the spike stage, their initial states are represented as akL + ibkL for the left ear and akR + ibkR for the right ear. Here, L indicates the left ear, R denotes the right ear, and K refers to the RF neuron associated with the intrinsic frequency fk. Subsequently, the state will decay oscillations over time. To facilitate understanding, we calculate the real and imaginary parts in a discrete manner:    akL[t] = akL[t −1] cos (2πfk) + bkL[t −1] sin (2πfk) , bkL[t] = −akL[t −1] sin (2πfk) + bkL[t −1] cos (2πfk)), (16) With the RF-PLC method we propose, we can directly spike timing acquisition. Specifically, RF neurons will fire spike at akL[tnL] = 0, bkL[tnL] = max(bkL). In the phase space, this state represented the first peak of time aligns with the neural phase coder:        ϕlocked = tan−1 bkL[tnL] akL[tnL] = π 2 , Phase-locking Loop ITDRF = tnL −tnR ≈tL −tR, RF-based ITD encoding (17) where tnL satisfies ϕlocked = ZRF [tnL] and tnR satisfies ϕlocked = ZRF [tnR]. Due to the discrete form, our ITD encodings are not identical to those in Lemma 1. The error is in the difference between the audio’s sampling rate and the actual frequency. It results in our inability to accurately obtain the first peak time. Specifically, as Lemma 1 demonstrates, to obtain the spike timing for pure tone audio of different frequencies, the phase coding model requires leveraging Eq.15 to compute the audio’s ITD cues. However, with the RF-PLC method we propose, it can only iterate according to the audio’s sampling rate. The difference between them is dependent on the sampling rate fs, which can be represented as 1/fs. For the dataset SLoClas that we tested, its errors is only approximately 1%. C Experiment Detail We primarily validated the accuracy and robustness of our proposed method on the SLoClas dataset. It utilizes a 4-channel microphone array to collect data on RWCP sound scenes to ensure SNR of 40 to 50dB. It is comprised of ten distinct categories of ambient sounds: bells, bottles, buzzers, cymbals, horns, metal, particles, phones, rings, and whistles. Each category includes approximately 100 instances, providing a diverse and comprehensive set of audio samples. Additional, to evaluate the robustness of our method, we need to construct source localization information under various complex scenarios. Specifically, we can introduce different types of noise into the audio. The noise sounds are sourced from the NOISEX-92 database, which contains recordings of various types of real-world noise. We add noise audio to each microphone channel to simulate noise coming from different directions. 17 ComplexV ideoi(n) = mici(n) + λnoise(n), i = 1, 2, 3, 4. (18) In this setup, mici(n), where i = 1, 2, 3, 4, represents the 4-channel signals recorded by the microphone array. ComplexVideoi denotes the multi-channel data with noise. The noise data consists of randomly selected audio clips from a noise database. λ is the scaling factor used to adjust the audio to a specific SNR ratio. A smaller SNR indicates a stronger noise presence in the audio which is more similar to real environment. Table 3: Experimental configuration of the sound localization task. Attributes Setup 1. Data preprocessing: Sampling rate (Hz) 16000 Frame length (ms) 170 Frame stride (ms) 170 RF neurons n 512 Number of Microphones 4 2. RF-PLC setting: CQT frequency range (Hz) [0, 8800] τ (ms) 0.0625 Frequency channels N 40 Coincidence detector Nτ 51 Microphone pairs C 6 3. SNN Hyperparameter: α 0.75 Timestep 4 Epochs 300 Batch size 128 Optimizer Adam Base learning rate 1e-3 Learning rate decay Cosine Weight decay 5e-3 D Model Structure The overall network architecture of our SNN-based SSL model is illustrated in Fig.8, featuring a comprehensive system design tailored for sound localization. The architecture consists of two main components: a front-end RF-PLC method and a back-end MAA-based localization decision network. In the RF-PLC method, we utilize 512 RF neurons with widths ω that increase incrementally from 0 to 8000Hz. These neurons are strategically deployed as an alternative to the traditional FT operation, optimizing the model for energy efficiency and computational speed. Additionally, we utilize a cochlear filter bank following the Constant Q Transform (CQT) which is the most commonly used and easily implemented cochlear filter bank, to extract auditory features of appropriate dimensions. This encoding approach not only mimics the cochlear filtering process but also enhances the temporal dynamics of sound processing. Additionally, N detection neurons are engaged to characterize the ITD with delays ranging from -25τ to 25τ. Based on the audio sample ratio, the value of τ is set at 0.0625 ms. Furthermore, our model utilizes data from four microphones to compute ITDs between each pair, resulting in six distinct sets of ITD cues. As a result, each 170 ms speech frame is encoded into X ∈ZC×N×Nτ , capturing a rich array of spatial and temporal information. Details of these parameters can be found in Table.3. This setup ensures a detailed and dynamic spatial representation of auditory scenes. In the back-end decision network, we propose a fully spike-based model, which is illustrated in Fig. 8. We will validate our module within the SSL models [7, 40] consisting of convolutional and MLP layers. To enhance the performance of the SSL model, we augment this basic structure with our MAA 18 module. This module is designed to emulate biological auditory processing by focusing on frequency band preference and short-term memory capabilities. Compared with traditional spatio-temporal spike attention techniques [65, 71], our module markedly improves the system’s computational efficiency, as shown in Table. 2. Additionally, to better illustrate the performance of our module, we present a comparison with various modules. Details of these networks are provided in Table. 4. This enhancement allows for faster and more accurate sound localization, demonstrating the potential of our model in real-world applications. Figure 8: The structure of the Spike-based Neuromorphic Sound Source Localization. It includes an RF-PLC encoding method and a back-end classification model based on SNNs. Table 4: Detail Network Stage Output Size Baseline Baseline + MAA Baseline + others Stage 1 12 × 25 × 20 Conv 3 × 3, stride 1 BatchNorm MaxPooling 2 × 2, stride 2 Stage 2 24 × 12 × 10 Conv 3 × 3, stride 1 MAA module other Attention BatchNorm MaxPooling 2 × 2,stride 2 Stage 3 48 × 6 × 5 Conv 3 × 3, stride 1 MAA module other Attention BatchNorm MaxPooling 2 × 2,stride 2 Classifier 1 × 1 × 1 360-FC E Energy Cost To describe the energy consumption calculations in the ablation experiments, we introduce a theoretical energy estimation method for the proposed attention mechanism. Compared to the ANN model, the energy consumption calculation of the spiking version requires information on the timesteps (T) and spike firing rates (R). The spike firing rate is defined as the proportion of non-zero elements in the spike tensor. Since our proposed MAA method is spike-driven, we only need to evaluate the model’s FLOPs, along with T and R, to estimate the theoretical energy consumption of our methods. In ANN [35], the FLOPs for the n-th Conv layer are expressed as: Conv = (kn)2 · hn · wn · cn−1 · cn, (19) 19 where kn denotes the kernel size, (hn, wn) specifies the dimensions of the output feature map, and cn−1 and cn represent the numbers of input and output channels, respectively. The FLOPs of the m-th MLP layer in ANNs are: MLP = im · om, (20) where im and om represent the input and output dimensions of the m-th MLP layer. Refer on previous research [24, 68], we assume that all computational data are implemented using 45nm technology for 32-bit floating-point calculations, with EMAC = 4.6pJ and EAC = 0.9 pJ. In Table 3, we further present the details of different models. Referring to [66, 71], the attention matrix is derived using the sigmoid function, which results in the network input for the subsequent layer being non-spiking. Consequently, during the energy consumption calculation, this component is computed using the energy consumption of MAC operations, leading to a significant increase in energy consumption within the network. Due to the unique properties of the MAA attention matrix, our method does not introduce additional floating-point operations. F Limitation The limitations of this study include the lack of deployment on edge devices. Furthermore, the limited availability of datasets for SSL tasks restricts the validation of our model across a broader array of datasets. Future research will seek to overcome these challenges by amassing a more extensive collection of datasets and implementing our model on edge devices to more effectively ascertain the efficacy of our approach. The experimental results reported herein are reproducible, with detailed descriptions of the model architectures and hyperparameter configurations available in Appendix. Additionally, our code will be made available on subsequent after review. G Supplementary Ablation Experiments To further validate that the enhancement in our model’s performance is indeed due to the effective implementation of band selection and short-term attention mechanisms, rather than an increased parameter count. We designed a Conv2d layer module with the same amount of parameters, ensuring all other parameters remained consistent. Specially, (a) (b) (c) Figure 9: Supplemental ablation experiments. In these experiments, the Local module is configured with the same number of parameters as our proposed models to ensure a fair comparison: (a) A comparison of MAE.(◦) and Acc.(%) between our proposed MAA and LocalMAA. (b) A comparison of the MAE between our proposed ST-M and LocalST-M. (c) A comparison of the MAE between FSJA and LocalFSJA. As illustrated in Fig. 9, LocalMAA refers to a network with the same number of parameters as our proposed MAA. It can be observed that our MAA method achieves a lower MAE(◦) and higher Acc(%). Furthermore, we compared the MAE of different modules, finding that our frequency band preference and short-term memory structure significantly enhance the network’s localization performance. 20 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,422	USCILab3D: A Large-scale, Long-term, Semantically Annotated Outdoor Dataset Kiran Lekkala∗, Henghui Bao*, Peixu Cai, Wei Zer Lim, Chen Liu, Laurent Itti University of Southern California Los Angeles, CA 90089, USA klekkala@usc.edu Abstract In this paper, we introduce the USCILab3D dataset, a large-scale, annotated outdoor dataset designed for versatile applications across multiple domains, including computer vision, robotics, and machine learning. The dataset was acquired using a mobile robot equipped with 5 cameras and a 32-beam, 360◦scanning LIDAR. The robot was teleoperated, over the course of a year and under a variety of weather and lighting conditions, through a rich variety of paths within the USC campus (229 acres = ∼92.7 hectares). The raw data was annotated using state-of-theart large foundation models, and processed to provide multi-view imagery, 3D reconstructions, semantically-annotated images and point clouds (267 semantic categories), and text descriptions of images and objects within. The dataset also offers a diverse array of complex analyses using pose-stamping and trajectory data. In sum, the dataset offers 1.4M point clouds and 10M images (∼6TB of data). Despite covering a narrower geographical scope compared to a whole-city dataset, our dataset prioritizes intricate intersections along with denser multi-view scene images and semantic point clouds, enabling more precise 3D labelling and facilitating a broader spectrum of 3D vision tasks. For data, code and more details, please visit our website. 1 Introduction With the recent advancements in 3D vision techniques, the integration of three-dimensional perception has become integral to many interdisciplinary domains. Unlike the abundant resources available for 2D vision, the lack of comprehensive datasets for 3D vision poses a significant challenge to researchers. The progress in this field can be significantly propelled by leveraging large-scale datasets, which offer adaptability across a spectrum of downstream tasks. In this paper, we present USCILab3D — a large-scale, long-term, semantically annotated outdoor dataset. USCILab3D comprises over 10 million images and 1.4 million semantic point clouds, rendering it suitable for a wide range of vision tasks. Differing from smaller-scale semantic datasets or larger-scale undetailed ones, our dataset not only encompasses a wide array of outdoor multi-view scene images but also provides detailed semantic annotations, facilitating enhanced understanding and utilization of 3D perception techniques. Given the massive scale of our new dataset, as detailed below, we have thus far focused on leveraging the latest foundation models to compute detailed annotations. Our workflow using these models is detailed below. ∗Equal Contribution. 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. Figure 1: Images with the respective 3D pointclouds Our adjacent five cameras provide comprehensive coverage with overlap at the same timeframe, ensuring the captured information’s redundancy. We also show the corresponding point cloud view for every image. 2 Related datasets Several large-scale scene datasets have been developed in recent years for indoor settings [19; 26; 21]. Additionally, several datasets have focused on outdoor city navigation[18]. Furthermore, some datasets are generated using simulators [9; 24]. These attempt to solve the above problems, although presenting their challenges: While they offer controlled environments, there exists a noticeable gap in scene quality compared to real-world scenes. 2.1 Multi-view datasets Multi-view scene datasets are typically used for novel view synthesis tasks with generative models such as Neural Radiance Fields (NeRF) [17] and 3D Gaussian Splatting [14]. The LLFF dataset [16] is an early multi-view scene dataset that includes both indoor and outdoor scenes, with fewer than 1,000 low-resolution images. The DTU [13] and ScanNet [8] datasets contain between 30K and 2,500K images, but they are limited to indoor scenes. The ETH3D dataset [23] provides high-quality outdoor scenes but has sparse scans and fewer than 1,000 images. Tanks and Temples [15] addresses these limitations by offering 147,000 high-quality outdoor images, which are commonly used in novel view synthesis benchmarks. 2.2 Scene datasets with semantic labels Indoor datasets Datasets like [19; 26] represent large-scale 3D reconstruction datasets tailored for research in indoor robotic navigation and scene understanding. Matterport [6] is a large-scale RGB-D indoor dataset containing 10,800 panoramic views from 194,400 RGB-D images of 90 building-scale scenes. However, this dataset is limited to indoor environments and offers only 20 labels for scene annotation. In contrast, our dataset encompasses approximately 10 million images and over 4000 labels, providing extensive coverage of outdoor scenes. Moreover, the inclusion of ground-truth point clouds in our dataset enhances the accuracy of alignment between 2D images and 3D annotations, surpassing the alignment capabilities of other datasets. Outdoor datasets SemanticKITTI [4] is a widely used dataset for semantic segmentation and scene understanding in outdoor environments. It consists of dense point cloud sequences collected by a mobile LiDAR scanner which is similar to us. However, SemanticKITTI’s semantic annotations are confined to only 25 categories. In contrast, leveraging multimodal model outputs, our dataset enables the labeling of almost every element within the scene, providing a comprehensive understanding of outdoor environments. Our dataset addresses the limitations of the above datasets by providing large-scale outdoor scenes with diverse weather and lighting conditions, along with various ground-truth semantic point clouds (Table 1 and Table 2). Leveraging multimodal foundational models, we accurately label 2D images 2 and align them in 3D space, resulting in precise 3D annotations. Figure 2: The pipeline of our semantic annotations method We use GPT4 and Grounded-SAM to create pixel-wise semantic labels and align the 2D and 3D points. Dataset Frames Indoor Outdoor LiDAR Point Cloud Semantic LLFF[16] < 1K images ✓ ✓ ✗ ✗ DTU[13] 30K images ✓ ✗ ✗ ✗ ScanNet[8] 2,500K images ✓ ✗ ✗ ✗ Tanks and Temples[15] 147K images ✓ ✓ ✗ ✗ ETH3D[23] <1K images ✓ ✗ ✗ ✗ Matterport3D[6] 195K images ✓ ✗ ✗ ✓ Habitat[19] ✓ ✗ ✗ ✓ iGibson[26] ✓ ✗ ✓ ✓ SemanticKITTI[4] 23K scans ✗ ✓ ✓ ✓ USCILab3d (ours) 10M images ✗ ✓ ✓ ✓ 1.4M scans Table 1: Comparison of the existing datasets with our USCILab3D dataset. Dataset Point Clouds Semantic Labels Semantic classes nuScenes[5] 390K 31 vehicle, human, animal, movable object, flat, static Waymo motion[11] 230K 23 Traffic Entities: Car, Truck, Bus, Motorcyclist, Bicyclist, Pedestrian, etc. SemanticSTF[29] 2K 21 flat, construction, nature, vehicle, human, object WildScenes[28] 12K 15 terrain, vegetation, object, structure, water, sky USCILab3d (Ours) 1.4M 267 Vehicle, nature, human, ground, structure, street furniture, architectural elements, signs and symbols, general objects, lightning Table 2: Comparison of Semantic Classes and Labels Across Existing Datasets and Our USCILab3D Dataset. 3 Dataset collection This section outlines our robot platform and data collection approach. Our robot, Beobot-v3, utilizes multiple cameras and a LiDAR sensor for simultaneous data capture. We collect data across the USC University Park campus and synchronize streams for analysis. 3.1 Robot platform We build our robot Beobot-v3 to collect the dataset, as shown in Figure 3. We use five Intel Realsense D455 cameras and Velodyne HDL-32E LiDAR. The RGB images, featuring a field of view (FOV) of 90 × 65° and a resolution of 1280 × 720 pixels, are captured at a rate of 15 frames per second (FPS). 3 Utilizing a 1 MP RGB sensor, these images ensure high-quality visual data acquisition. Furthermore, the LiDAR scans the environment at a rate of 10 Hz, capturing precise point clouds that complement the visual data. These point clouds offer comprehensive 3D spatial information essential for scene understanding and navigation tasks. Because of microcomputer’s limit, camera 1 and LiDAR are controlled by one microcomputer, and other cameras are controlled by their own microcomputer. All microcomputers are controlled by a central computer, our data collection system orchestrates the simultaneous scanning and recording process. As the LiDAR initiates scanning, capturing a 360° view of the environment, the data is saved directly into the system and five cameras capture images in tandem, storing them in separate ROS bag files. 3.2 Dataset collected over the entire USC campus Our dataset is meticulously collected across the entirety of the USC University Park campus. Spanning an expansive area of 229 acres (0.93 km²), the campus makes our dataset diverse. From the varied architecture of its buildings to the network of roads, stairs, trails, paths, gardens, and sidewalks, each corner offers a unique scene. By dynamically selecting its route, the robot explores the full extent of the campus’ diverse terrain, from thoroughfares to hidden nooks, creating a rich variety of surroundings. Figure 3: Overview of the data collection robot and its hardware. Beobot-v3 is a differential-drive, non-holonomic mobile robot, equipped with five Intel Realsense D455 cameras and one Velodyne HDL-32E LiDAR sensor used to collect the dataset. The data collection occurred in many daytime sessions, with a preference for sunrise or sunset periods to avoid crowds and mitigate harsh sunlight that could degrade image quality. However, a small portion of the captured images may still exhibit the effects of powerful sunshine. The sample images are shown in Figure 4. Our data collection efforts span from March 11, 2023, to March 16, 2024, encompassing 12 months. Over this time frame, the environment undergoes dynamic changes, including variations in weather, seasons, and alterations to the campus landscape, such as ongoing construction projects. This deliberate scheduling ensures that our dataset encapsulates a diverse range of environmental scenarios, enriching the dataset with a wide array of conditions for robust training and evaluation of algorithms. 3.3 Synchronization of cameras and LiDAR To address the synchronization issue between the LiDAR and cameras due to the control of different microprocessors, we implement a synchronization process. Given that the LiDAR operates on the same system clock as camera 1, we only need to synchronize the remaining cameras with camera 1. To achieve this, we employ a method based on feature detection and optical flow tracking. At the onset of each session, the scene remains static. Leveraging ShiTomasi corner detection [27], we identify key features in the camera images. Subsequently, using the Lucas-Kanade optical flow algorithm, we track the movement of these features over consecutive frames. If the displacement 4 of these features exceeds a predefined threshold, indicative of the robot initiating movement, we designate this time as the session’s start time. Once the start time is determined for camera 1, we synchronize the start times of the remaining cameras by aligning them with the start time of camera 1. This ensures temporal coherence across all camera feeds, enabling accurate alignment of the visual and LiDAR data streams. Through this synchronization process, we establish temporal consistency across all data sources, facilitating coherent analysis and interpretation of the collected data. Figure 4: Sample snapshots from our dataset of various daylight timings. These are images obtained from randomly sampling across the entire dataset. 3.4 Sensor calibration By aligning the coordinate systems of the Velodyne LiDAR and the camera, we ensure that the geometric transformation from 3D to 2D space is accurate. With this calibrated setup, we can assign semantic labels to the 3D points based on the information extracted from the images. The accurate alignment between the Velodyne-frame and camera-frame ensures that the projected points correspond to the correct regions in the images, enabling us to leverage the semantic information obtained from the images to label the 3D points accurately. To obtain the pose transformation between images and point clouds, we use a 1m × 1m checkerboard as a calibration target for sensor alignment. Leveraging the MATLAB calibration toolbox, we apply the Line and Plane Correspondence method [30] to refine sensor alignment and calibration with high precision. In this approach, we treat edges in 3D as contours (C) and planes (a), while lines (L) in 3D space are characterized by points within the same plane (a). This framework integrates point-to-line, point-to-plane, and direction/normal-based adjustments, ensuring accurate alignment across sensors. 4 Dataset annotation In this section, we describe methods used as part of the pipeline for our semantic annotations of 3D point clouds. A high-level overview is shown in Figure 2. 4.1 GPT4-based candidate labels and clustering We use GPT-4 [1] to detect the semantic labels in an image. Since images are obtained at 15Hz and the robot moves at a velocity close to 1 m/s, it is redundant and expensive to query the semantic labels for all images through GPT-4 model. Given that the image frequency is 15Hz, for about every 225 images from one camera, we extract the the images of five cameras at that time. Given that the camera records at 15Hz, a 15-second interval of movement (typically less than 12 meters) ensures a small scene variation. We then pass these 5 images to GPT-4, and prompt it to estimate the semantic labels of the images using the following prompt "List every possible semantic class that exists in the scene. List only the names and nothing else." After standardizing and filtering the output, we obtain a total of 4162 labels. But most labels are meaningless or have similar meaning. We then again use GPT-4 to perform clustering and categorization on the estimated semantic labels. After removing the meaningless labels and merging semantically equivalent labels, we obtained 257 unique labels. Then, for all images we asked GPT-4 to extract objects from the image again, now with prompt is "I will give you a list of semantic class, list every possible semantic class that exists in 5 the scene. List only the names and nothing else, split by comma." This yields the final label list for each image. Category Elements Vehicle vehicle, bicycle, van, truck, motorcycle, golf cart, bus, car, skateboard Nature sky, grass, tree, shrub, shrubbery, hedge, trunk, tree trunk, green area, birds, bush, yard, plant sun, palm, rock, soil, leaf, leaves, water, flower, branch, bushes, vegetation, bird, ivy Human person, hand Ground pavement, curb, gravel, rail, sidewalk, street, walkway, floor, road, pedestrian walkway, crosswalk ramp, garden, ground, pathway, paving stone, golf course, parking lot, drainage grate, mulch Structure monument, structure, courtyard, fountain, public space, construction, emergency station ceiling, fence, gate, wall, balcony, container, stadium, lattice, shed, house, construction pipe, roof, building, sports field, campus, toilet, baseball field, architecture site, parking structure, garage, scaffolding, archway, call station Street Furniture bench, pole, feeding station, patio, handicap, barrier, hydrant, construction cone, construction barrier lamp post, lamp, trash can, recept, sign, parking meter, public art, statue, sculpture bollard, bus stop, park bench Architectural Elements drain cover, manhole cover, vent, air vent, arch, sill, doorway, baluster, security camera, electric box corridor, stair, ventilation grill, door handle, entrance, post, air unit, pillar, balustrade, handrail window, door, elevator, gutter, bleachers, tank, generator, utility meter General Objects umbrella, table, chair, stroller, furniture, board, bottle, canopy, outdoor gear, advertisement, station pot, rack, flag, locker, ladder, garbage, bulletin board, pallet, planter, equipment, tent, base, hat curtain, blinds, cardboard, box, tire, wheels, bag, bed, frame, bucket, painting, poster , machine Signs and Symbols shadow, reflection, traffic cone, parking space line, space line, road marking parking symbol, stop sign, street sign, road sign, symbol, plaque, banner, graffiti, waste container signboard, security camera, camera, warning sign, fire safety sign, transportation sign handicap sign, closed sign, exit sign, parking sign, reservation sign, rec sign Materials concrete, brick, construction materials, stone, wood, plastic, metal, glass, iron, materials Lighting outdoor lighting, light, street light, indoor light, lantern, sunlight, shade Miscellaneous cover, trash, outdoor, chain, unit, security, exterior, fire, electric, meter, lettering, phone, debris, railway text, potted, space, portable, cone, stlight, cross, marker, grate, blea, stoller, units, picnic, electrical cable, basin, pavilion, ster, bal, field, curve, bod, bay, pal, firent, box, exit, baseball, image, rec, sports public, piping, grill, guttering, utility, call, case, recacle, gut, hydra, air line, tile, cardboard, patch, reservoir, valve Table 3: Clustering of the semantic labels. We use GPT-4 to cluster 267 labels into 12 categories using the prompt "Could you help me classify by following category: Vehicle, Nature, Human, Ground, Structure, Street Furniture, Architectural Elements." 4.2 Grounded-SAM masks on pixel space After we obtain the candidate labels, for equally spaced subset of images, we use those labels as an input to the Grounded-SAM model [20] to detect and segment the image by pixel. Since we are using a differential-drive robot that can potentially rotate left or right, images may look very different quite rapidly, so we merge the five image labels from GPT-4 and pass to next step. After conducting our experiments, we found that the presence of unrelated labels (not visually represented in the images) does not significantly influence the results of Grounded-SAM. This observation is reflected in Figure 5 and Table 4 through the percentage of incorrect pixel labels in the masks of 2 images. We show the top 50 frequent objects and their pixel percentage in images of our dataset in Figure 6. 4.3 Post-processing after Grounded-SAM Grounded-SAM’s output is not always using the same vocabulary as our input labels, e.g., one may prompt it for ’vehicle’ but obtain a segmented ’car’. It may also generate meaningless words or words having similar meaning. To address this, we perform clustering and categorization as in section 4.1 again to merge all similar labels. Additionally, we manually merge and remove some words. Ultimately, we obtain 267 labels and 12 categories (Table 3). 4.4 Projecting 2D semantic masks to 3D pointcloud From the LIDAR data, we reconstruct 3D trajectories of the robot throughout the dataset. Essentially, we compute a pose transformation for each LiDAR scan in the dataset. We then interpolate the LiDAR poses to the camera images using the extrinsic parameters corresponding to the transformation of each camera with respect to the LiDAR sensor. This results in a pose estimate for every camera image in the dataset. 6 Figure 5: Robustness of Grounded SAM to prompts. Comparison of the semantic masks obtained using different prompts for the same image by Grounded-SAM model, showing the robustness of the model. On the right image, the additional prompts were "fire hydrant, person, car, Parking lot lines, Boat, Scooter, Dog, Bear, Cat" along with the common prompts "Trees, Bushes, Benches, Tables, Chairs, Pavement, Buildings, Windows, Doors, Emergency call box, Umbrellas, Leaves, Grass" Additional prompts Incorrect pixel labels 1 0.23% 2 0.63% 3 0.63% 10 0.92% Table 4: Percentage of incorrect pixel labels. Quantitative measures to show robustness through the change in the percentage of incorrect pixel labels with additional prompts. Note that this table in relation with the above Figure 5 By utilizing the semantic map of every image obtained from Grounded SAM, we use ground truth camera intrinsics and extrinsics to accurately project 3D point clouds onto 2D images, following equation (1). Here, (X, Y, Z) represents the world coordinates of a point, while (x, y) denotes the coordinates of the point projected onto the image plane, measured in pixels. r and t are rotation and translation. cx, cy represents the principal point, and fx, fy are the focal lengths in pixels. Subsequently, we align the 2D and 3D points to assign labels to the 3D points. x y 1 ! ∼ fx 0 cx 0 fy cy 0 0 1 ! r11 r12 r13 t1 r21 r22 r23 t2 r31 r32 r33 t3 !    X Y Z 1    (1) Considering the presence of moving objects and calibration errors, there may be some offset for each projection. To reduce erroneous labels, we run DBSCAN clustering [10] on each label projection to check whether the 3D points projected belong to a single cluster. If they do not, we only label the cluster with the most points. 4.5 Released data We release the raw ROS Bagfiles, and extracted images, point cloud files, COLMAP [22] poses and sparse reconstructions. The raw data consists of a set of sequences, each of which is collected during a specific data recording session. To make the data more manageable, we divide each session into different subsequences or "sectors", with each sector consisting of 1250 images and roughly 167 point cloud scans. In addition, we conducted face detection and applied blurring techniques to ensure privacy protection on campus. Multi-view images Each image is named according to the convention cam[id]-[timestamp].jpg. We estimate synchronized timestamps for all images within a sector, using the method mentioned in section 3.3. The wide field of view (FoV) of 90 degrees for each of the five cameras results in 7 Figure 6: Histogram of the semantic labels frequency in point cloud scans and points. Top 50 frequently estimated semantic classes in points(orange), and correspoing point cloud scan frequency significant overlap between their respective images, as depicted in Figure 1. This substantial overlap ensures more robust Structure from Motion (SfM) reconstruction. By having multiple views of the same scene, the SfM algorithm can triangulate feature points more accurately, leading to a more precise reconstruction of the 3D environment. This overlap also aids in improving the accuracy of semantic labelling. By leveraging overlapping information from multiple viewpoints, inconsistencies or errors in semantic annotations of 3D points from 2D-pixel maps can be identified and rectified through cross-validation. This double-checking mechanism helps to enhance the reliability of semantic labels assigned to objects in the scene. Semantic instances and masks for images In addition to the raw image data, we also provide semantic labels and label masks generated by Grounded-SAM for each image in the dataset. These labels offer valuable insights into the semantic understanding of the scene, allowing researchers to perform tasks such as semantic segmentation and object detection. Semantically annotated 3D point cloud streams As mentioned before, the pointcloud streams are captured at 10Hz. Similar to KITTI Semantic [4], we extract each of the pointcloud scans and annotate the 3D points by assigning semantic labels to individual points based on the closest image’s label, using the method outlined in section 4.3. The color and corresponding label for each point are saved in a JSON file named labels.json, ensuring easy access and interpretation of the semantic annotations. Semantically annotated point clouds In addition to the individual semantic annotated point cloud scans, we have processed each session’s point cloud data using LeGO-LOAM [25] to generate a merged point cloud for each sector (area corresponding to the segments of a trajectory). We mention the statistics of the distribution of points in each of the point cloud scans and the merged point clouds in the supplemental material. Unlike the point cloud scans, sector-based point clouds have more points and offer a comprehensive overview of the semantic annotated scene. Through these semantic point clouds, researchers can gain deeper insights into the semantic structure and composition of the environment. Pose annotations for images. We release interpolated poses from LeGO-LOAM, and COLMAP Structure from Motion (SfM) [22]. The COLMAP SfM results can serve as inputs for some generative model like NeRF or 3D Gaussian Splatting. Further, by utilizing the poses computed by COLMAP, we aim to improve the precision of our annotations given the different sampling rates of the LiDAR (10Hz) and cameras (15Hz). This alignment is crucial for accurately projecting semantic labels onto the 3D points based on the information extracted from the images. We are currently investigating 8 how to best merge the LiDAR and COLMAP poses, likely resulting in a unified set of poses indexed non-uniformly in time, for each image and for each point cloud. We expect that these unified poses will be released with the next version our dataset. Robotic dataset for visual navigation. Our dataset comprises diverse sequences captured within a university environment, reflecting a range of real-world scenarios. Leveraging the compact form factor of our robot, we collected data across a variety of settings including roads, outdoor lobbies, ramps, and other typical campus landscapes. This dataset is particularly valuable for applications in visual navigation and is integrated into the comprehensive Open X-Embodiment dataset [7]. 5 Benchmarks 5.1 Evaluation on Novel View Synthesis We examine the current state-of-the-art (SOTA) Novel View Synthesis methods on several datasets: USCILab3D, ETH3D [23], Mip-NeRF360 [3], Tanks&Temples [12], and Deep Blending [12]. For each dataset, we run 3D Gaussian Splatting and evaluate the generated image quality using PSNR, SSIM, and L-PIPS metrics. For each scene, we use 7/8 of the data as the training set and 1/8 as the test set, then calculate the average result for each scene. Considering the large size of our dataset, we randomly extract one sector from each session to compute the average result. Our dataset achieves superior PSNR, SSIM, and the best L-PIPS performance compared to other datasets (Table 5). Among these datasets, ours is the only one that provides large-scale scenes, making it suitable for a wider range of applications, such as simulators [2]. PSNR ↑ SSIM ↑ LPIPS ↓ Resolution ↓ interation USCILab3D (ours) 26.02 0.86 0.20 1280 × 720 7000 ETH3D[23] 21.25 0.83 0.27 6048 × 4032 7000 Tanks&Temples [15] 21.20 0.77 0.28 980 x 540 7000 Mip-NeRF360[3] 25.19 0.75 0.25 1256 x 828 7000 Deep Blending[12] 27.01 0.87 0.32 1332 x 876 7000 Table 5: Performance comparison of 3D Gaussian splatting on different datasets. Our dataset achieves superior performance compared to other datasets. Although Deep Blending demonstrates a higher PSNR, it only contains 2.6K images. 5.2 Evaluation on Semantic Segmentation and Completion We also evaluate our dataset using key tasks: semantic segmentation, panoptic segmentation, and semantic scene completion. Semantic segmentation is crucial for understanding and labeling every point in a 3D point cloud with a specific class, providing detailed insights into the composition of the scene. Panoptic segmentation extends this by not only classifying each point but also distinguishing between different instances of the same class. This is particularly valuable for environments with multiple similar objects, enhancing the dataset’s utility in more complex and dynamic scenarios. Lastly, semantic scene completion involves predicting the complete geometry and semantics of a scene, including occluded and unobserved regions. This task is vital for creating comprehensive and accurate representations of environments, which is indispensable for advanced applications in augmented reality and spatial analysis. We have included the results in the supplemental material. 6 Caveats Thus far, our annotations have been machine-generated using the latest foundation models. Although this may pose a few risks, nevertheless, to the best of our knowledge, our method is the first of its kind to annotate 3D point clouds using image and text based foundational models without any manual intervention. Casual inspection by authors suggests that the annotations are indeed of high quality. However, we plan to validate them by hiring a group of human annotators to inspect and possibly 9 correct a fraction of the machine-generated annotations. We expects that this will be completed by the time of publication. 7 Discussion and Conclusion In this paper, we introduced the USCILab3D dataset, a comprehensive outdoor 3D dataset designed to address the limitations of existing datasets in the domain of 3D scene understanding and navigation. Our dataset offers a diverse array of complex intersections and outdoor scenes meticulously collected across the USC University Park campus. With approximately 10 million images and 1.5 million dense point cloud scans, our dataset prioritizes intricate areas, enabling more precise 3D labelling and facilitating a broader spectrum of 3D vision tasks. Moving forward, we believe that the USCILab3D dataset will serve as a valuable resource for researchers and practitioners across various domains, including computer vision, robotics, and machine learning. We anticipate that the dataset will stimulate further advancements in 3D vision-based models and foster the development of robust algorithms capable of tackling real-world challenges in outdoor environments. 8 Acknowledgments This work was supported by the National Science Foundation (award 2318101), C-BRIC (one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA) and the Army Research Office (W911NF2020053). 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4,423	MaskFactory: Towards High-quality Synthetic Data Generation for Dichotomous Image Segmentation Haotian Qian1∗YD Chen1∗Shengtao Lou1 Fahad Shahbaz Khan3,4 Xiaogang Jin1† Deng-Ping Fan2,3 1State Key Lab of CAD&CG, Zhejiang University 2VCIP&CS, Nankai University 3MBZUAI 4Linköping University Abstract Dichotomous Image Segmentation (DIS) tasks require highly precise annotations, and traditional dataset creation methods are labor intensive, costly, and require extensive domain expertise. Although using synthetic data for DIS is a promising solution to these challenges, current generative models and techniques struggle with the issues of scene deviations, noise-induced errors, and limited training sample variability. To address these issues, we introduce a novel approach, MaskFactory, which provides a scalable solution for generating diverse and precise datasets, markedly reducing preparation time and costs. We first introduce a general mask editing method that combines rigid and non-rigid editing techniques to generate high-quality synthetic masks. Specially, rigid editing leverages geometric priors from diffusion models to achieve precise viewpoint transformations under zeroshot conditions, while non-rigid editing employs adversarial training and selfattention mechanisms for complex, topologically consistent modifications. Then, we generate pairs of high-resolution image and accurate segmentation mask using a multi-conditional control generation method. Finally, our experiments on the widely-used DIS5K dataset benchmark demonstrate superior performance in quality and efficiency compared to existing methods. The code is available at https: //qian-hao-tian.github.io/MaskFactory/. 1 Introduction Dichotomous image segmentation (DIS) aims to accurately segment objects from natural images [1, 2], a critical task in various computer vision applications, including medical imaging [3, 4, 5], autonomous driving [6] and connectomics research [7, 8]. However, traditional methods for collecting datasets for DIS tasks are labor-intensive, costly, and require extensive domain expertise. Recently, synthetic data has emerged as a promising solution for generating diverse and precise datasets at scale, offering a scalable and cost-effective means for model training. Given the importance of high-quality synthetic data for training models, developing methods to generate such datasets for DIS tasks is crucial. Although generative models have been employed to assist in synthetic dataset production, they face significant limitations in terms of controllability, precision, and diversity. Recent synthetic methods [9, 10, 11] utilize diffusion models [12] to synthesize image-mask pairs from textual cues and attention maps. Despite progress, they still encounter three major challenges: (1) Controllability: Text-guided generation scheme may deviate from real-world scenes, especially with fine-grained labels, leading to uncontrollable generated images. (2) Precision: The use of attention maps can introduce noise, which degrades mask fidelity and poses challenges for applications ∗Equal contribution. Deng-Ping Fan served as the project leader for this work. †Corresponding author. Contact the author at jin@cad.zju.edu.cn 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Real Image Source Mask Synthetic Image Synthetic mask Rigid Mask Editing Original Source Non-rigid Mask Editing Synthetic Image Synthetic mask Figure 1: Shows the edited masks from the first stage and the corresponding images generated in the second stage. In the examples, we transformed the viewpoint of park benches and tables from a frontal view to a top-down view and edited their shapes, changing park benches from curved to square edges and tables from square to circular shapes. requiring precise pixel-level segmentation, such as DIS [2]. 3. Diversity: Previous works [13, 14, 15] are relatively uniform, with limited ability to generate diverse training samples, which makes them unsuitable for providing sufficient variability in training data. Generating diverse human images often poses challenge due to the complexity of prompts, which limits their applicability in tasks requiring extensive variability. It’s worthy noting that the expansion of datasets with fine granularity, such as DIS datasets, currently lacks an effective solution. In fact, we observe that existing DIS datasets predominantly consist of single-source orthophotos featuring limited shape diversity among similar objects. The generation of more diverse DIS images that accurately reflect real-world distributions remains an unresolved challenge. To address these challenges, we propose MaskFactory, a novel two-stage method that efficiently generates high-quality synthetic datasets for DIS tasks. Based on the geometric characteristics of objects, our approach simultaneously considers both rigid and non-rigid transformations of target objects. This means that when generating synthetic datasets, we take into account not only changes in viewpoints (rigid transformations) but also deformations (non-rigid transformations), as illustrated in Figure 1. In the first stage, rigid transformations are driven by geometric priors learned from large-scale diffusion models, enabling precise viewpoint changes and simulating diverse variations in observation angles and scales. Non-rigid transformations leverage prompts provided by large language models to accurately alter the shape of target objects via attention-based editing, ensuring topological consistency before and after editing through topology-preserving adversarial training. Consequently, even for masks with complex geometries, high-quality and diverse synthetic masks can be generated for DIS tasks. In the second stage, we utilize multiple control inputs, including masks, canny edges, and prompts representing class information, to guide the generation process. These inputs are fed into diffusion models to produce high-resolution images that match the prepared segmentation masks. This approach ensures high consistency between images and masks while enhancing the realism and diversity of the dataset. Our method significantly enhances the realism and diversity of generated datasets. We validate the effectiveness of MaskFactory on the DIS5K dataset [2], specifically designed to evaluate DIS performance. Experimental results show that our approach outperforms existing dataset synthesis methods in terms of structural preservation and error metrics, achieving an average performance gain of 8.8%. These findings highlight the potential of MaskFactory to provide the data diversity and annotation precision required for DIS tasks while significantly reducing the time and costs associated with dataset preparation. In summary, the contributions of our work are listed as follows: • We introduce MaskFactory, a novel approach that generate high-quality datasets for DIS task in terms of quality, precision, and efficiency. 2 (1.a) Rigid Mask Editing �(∙) Q K V Q K V Q K V Q K V Q K V Q K V (1.b) Non-rigid Mask Editing �(∙) Give a prompt to change the shape A round table Decoder Decoder Encoder Q K V Q K V Q K V Q K V Q K V Q K V Q K V Q K V Q K V Q K V Q K V Q K V Canny Topology Preserving Loss Decoder Encoder Mask Pool {�� } Q K V Q K V Q K V Q K V Q K V Q K V Prompt Pool {�� } noise (2.a) Edge Condition (2.b) Prompt Condition Synthetic Data Pool {�� , �� } Stage1: Mask Editing Stage2: Image Generation Figure 2: Workflow of MaskFactory. In the first stage, we generate new masks by applying rigid and non-rigid editing to the existing ground truth masks. In the second stage, we use the generated masks and their corresponding extracted Canny edges as conditions, along with a prompt representing the category, to generate RGB images. This process forms paired data for our generative model. • We propose a two-step method that synthesizes high-quality and diverse object masks via masking editing and generates corresponding high-resolution images using a multi-conditional control generation method. • Experimental results on the DIS5K dataset demonstrate the superior performance of MaskFactory, compared to existing dataset synthesizing methods. 2 Related work Synthetic data. Synthetic data has garnered significant attention in various machine learning and computer vision tasks, such as natural language processing [16], object detection [17], and image segmentation [10]. For instance, Lin et al. [17] demonstrated that synthetic data can enhance performance in object detection, especially in scenarios with limited access to real-world data. Generative adversarial networks (GANs) and variational autoencoders (VAEs) are two popular deep learning-based methods for generating synthetic data [18]. Recent studies have focused on improving the quality and diversity of synthetic data [19] and exploring its use in few-shot learning [20]. Additionally, synthetic data has been shown to enhance the interpretability and explainability of machine learning models [21]. Benefiting from recent advancements in diffusion models [12, 22], methods such as DatasetDM [9] and Dataset Diffusion [10] can generate high-quality synthetic data for various computer vision tasks. However, these methods may introduce additional errors due to the inclusion of pseudo-labels. These errors arise because pseudo-labels can be noisy and inaccurate, leading to suboptimal training data. Although some research [23] employs controlnet-like control schemes [24] to mitigate these issues, the generated synthetic data may still suffer from noise and errors, making it unsuitable for the DIS task. Therefore, in this paper, we focus on generating high-quality synthetic data for the DIS task, where the generated image-mask pairs need to be highly precise and accurate. 3 Dichotomous image segmentation. DIS has seen substantial progress with the advent of highresolution imaging technologies. Representative approaches include the intermediate supervision strategy in IS-Net [2], the frequency prior method [1], and the unite-divide-unite strategy by [25]. However, these methods have enhanced segmentation accuracy but often struggle to capture extremely fine details. Thus, recent progressive refinement strategies, e.g., BASNet [26], emphasize the importance of auxiliary information such as gradient maps and multi-scale inputs. These methods propose to use gradient features and ground truth supervision to enhance the learning of weak features in complex regions. Then, BiRefNet [1] maintains high-resolution inputs and employs a bilateral reference framework to better capture intricate details. The latest works, such as the multi-view aggregation network by Yu et al. [27] and the interactive segmentation approach by Liu et al. [28], further advance the field by integrating diverse prompts and enhancing feature representation for high-quality segmentation. 3 Method 3.1 Overview of MaskFactory Current image segmentation methods are significantly constrained by their dependence on limited manually annotated data, which hampers both performance and generalization. To mitigate this, pseudo-label generation is often employed to augment training datasets. However, in the context of DIS, these methods frequently introduce artifacts that degrade segmentation quality. Additionally, existing image editing techniques often fail to preserve the topological structure of binary masks, resulting in discontinuities and overlaps within target regions. To address these challenges, we introduce the MaskFactory framework, designed to generate a large number of high-quality synthetic image-mask pairs G = {(gr i , gm i )}M i=1 from an original dataset D = {(Ir i , Im i )}N i=1. This approach aims to enhance the performance of DIS models. As illustrated in Figure 2, our framework comprises two main stages: mask editing and image generation. In the mask editing stage, source masks from the original dataset are transformed using both rigid and non-rigid editing methods, resulting in a set of high-precision synthetic masks Gm = {gm i }M i=1. Rigid mask editing generates synthetic masks from various perspectives, while non-rigid mask editing employs a topology-preserving adversarial training mechanism to edit masks according to semantic prompts while retaining the structural integrity of the source masks. In the image generation stage, a multi-conditional control generation method is utilized to produce realistic RGB images Gr = {gr i }M i=1 that correspond precisely to the synthetic masks, using the latter as conditioning constraints. 3.2 Mask Editing Stage 3.2.1 Rigid Mask Editing Rigid mask editing aims to preserve detailed information from the source masks through rigid transformations. We leverage the Zero123 [29] method, which employs a viewpoint-conditioned diffusion model ψθ to manipulate masks’ perspectives. Given the relative camera rotation and translation Ti for the desired viewpoint, ψθ synthesizes a new mask gm i based on the source mask Im i , such that gm i = ψθ(Im′ i , Ti), where Im′ i is the inverted image of the source mask Im i to ensure the main component is non-zero. 3.2.2 Non-Rigid Mask Editing Non-rigid mask editing, inspired by MasaCtrl [30], is a critical component of MaskFactory. Unlike MasaCtrl, which directly manipulates the source mask Im i using a textual prompt Pi to generate a synthetic mask gm i , we introduce a topology-preserving adversarial training mechanism to mitigate artifacts and structural degradation in binary mask editing. The textual prompt Pi is derived from a pool of prompts {pm i } that are generated using GPT-4 based on the original images. These prompts provide detailed descriptions that guide the mask editing process. This module consists of a generator Gθ and a discriminator Dϕ. The generator transforms noise z into a synthetic mask gm i under the guidance of a textual prompt Pi and the source mask Im i . A mutual attention mechanism aligns the 4 query features Qt of gm i with the key and value features Ks, Vs of Im i , ensuring consistency during editing. To avoid foreground-background confusion, a mask M is extracted from the cross-attention maps to guide the model’s focus. Topology-Preserving Adversarial Training. To maintain the structural information of the source mask, we first extract an edge map Es = E(Im i ) using an edge detection operator E, obtaining key points V = {vj}Nv j=1. We then construct a graph T = (V, Es) based on these key points. The discriminator Dϕ performs adversarial training on the structural graphs Tg and Ts of the synthetic mask gm i and the source mask, respectively, ensuring topological consistency. The training objective of the discriminator Dϕ is to maximize: max ϕ ETs∼pdata(Ts)[log Dϕ(Ts)] + ETg∼pgen(Tg)[log(1 −Dϕ(Tg))], (1) where pdata(Ts) and pgen(Tg) represent the distributions of the structural graphs of the source masks and the synthetic masks, respectively. Conversely, the training objective of the generator Gθ is to minimize the discriminative power of the discriminator: min θ ETg∼pgen(Tg)[log(1 −Dϕ(Tg))]. (2) Through topology-preserving adversarial training, the non-rigid editing module effectively retains the structural information from the source masks during the editing process, generating high-quality, artifact-free synthetic masks. The overall loss function Ltotal for non-rigid mask editing encompasses the adversarial loss LGAN, the content loss Lcontent, and the structure preservation loss Lstructure: Ltotal = LGAN + λ1Lcontent + λ2Lstructure, (3) where λ1 and λ2 are balancing factors. The adversarial loss LGAN is defined as: LGAN = ETs∼pdata(Ts)[log Dϕ(Ts)] + ETg∼pgen(Tg)[log(1 −Dϕ(Tg))]. (4) The content loss Lcontent measures the semantic consistency between the synthetic mask and the textual prompt: Lcontent = ∥gm i −Im i ∥1, (5) where gm i is the synthetic mask and Im i is the source mask. The structure preservation loss Lstructure evaluates the difference between the structural graphs of the synthetic mask and the source mask: Lstructure = ∥Tg −Ts∥1. (6) These components ensure that the editing process maintains the structural and semantic consistency of the source masks. 3.3 Image Generation Stage Following the mask editing stage, we obtain a synthetic mask pool {gm i }M i=0 comprising finely detailed synthetic masks generated through the aforementioned transformations. In the subsequent image generation stage, we accurately generate corresponding RGB images {gr i }M i=0 for the masks in the synthetic mask pool using a multi-condition control generation method. The primary segmentation of the RGB images aligns with the corresponding synthetic masks. Inspired by ControlNet [24], we introduce a multi-condition control generation method to achieve precise RGB image generation. This method simultaneously injects the segmentation condition cs i and the canny condition cy i to steer the denoising process of the random Gaussian noise, ensuring the generated RGB images {gr i }M i=0 correspond accurately to the synthetic masks {gm i }M i=0. Since the synthetic mask itself serves as the segmentation condition cs i, when a synthetic mask gm i is provided, we only need to extract the canny condition cy i using the canny operator: cs i = gm i , cy i = Canny(gm i ) (7) 5 bed A bed frame made out of wooden pallets. fan A fan is sitting on a window sill. bicycle A bicycle with a basket on the front of it. gate An iron gate is in front of a house. bench A wooden bench is sitting on a sidewalk. (a) Raw Image (b) MaskFactory (c) DatasetDM [9] (d) DatasetDiffusion[10] Figure 3: compared with baseline methods After obtaining the canny and segmentation conditions, we input them into block Bθ, which consists of a set of neural layers. Finally, these conditions are injected into the pre-trained diffusion model Mθ, controlling the noise z denoising process to generate the corresponding RGB image. We refer readers to [24] for more details. gr i = Mθ(z, Bθ(cs i), Bθ(cy i )) (8) 4 Experiment and Results 4.1 Dataset & Metrics We conduct our experiments on the DIS5K dataset, which comprises 5,479 high-resolution images featuring camouflaged, salient, or meticulous objects in various backgrounds. The DIS5K dataset is divided into three subsets: DIS-TR (3,000 images) for training, DIS-VD (470 images) for validation, and DIS-TE (2,000 images) for testing. For data augmentation, we utilize the mask portion of the training subset (DIS-TR). To evaluate our models, we employ a diverse set of metrics to ensure comprehensive performance assessment. These metrics include max F1[31], which balances precision and recall, providing a harmonic mean that is indicative of overall accuracy; Fω β[32], a weighted F-measure that compensates for class imbalances, with values ranging from 0 to 1, where higher values denote superior performance; M (Mean Absolute Error)[33], which calculates the average absolute difference between the predicted and ground truth masks, with lower values signifying better accuracy; Sα[34], a structural similarity measure that evaluates the preservation of significant structures within the image, with values closer to 1 indicating better performance; and Eϕ M[35], an enhanced measure that considers both pixel-level and image-level information for a more holistic evaluation, where higher values represent better performance. Collectively, these metrics provide a robust framework for assessing the effectiveness and reliability of our segmentation models. 6 Table 1: Comparison of experimental results with different generation methods, illustrating the impact of using various numbers of generated images on segmentation task performance. Dataset IS-Net DatasetDM [9] Dataset Diffusion [10] MaskFactory 2500 5000 7500 10000 2500 5000 7500 10000 2500 5000 7500 10000 DIS-VD maxF1 ↑ 0.791 0.792 0.791 0.788 0.785 0.793 0.780 0.771 0.767 0.831 0.833 0.834 0.835 F ω β ↑ 0.717 0.720 0.719 0.712 0.710 0.726 0.726 0.716 0.710 0.725 0.754 0.757 0.759 M ↓ 0.074 0.076 0.076 0.075 0.077 0.074 0.077 0.076 0.077 0.073 0.073 0.071 0.072 Sα ↑ 0.813 0.814 0.810 0.807 0.805 0.826 0.821 0.804 0.790 0.832 0.855 0.860 0.866 Eϕ M ↑ 0.856 0.869 0.864 0.864 0.860 0.868 0.859 0.852 0.838 0.880 0.911 0.914 0.923 DIS-TE1 maxF1 ↑ 0.740 0.744 0.744 0.740 0.736 0.741 0.727 0.721 0.719 0.776 0.777 0.779 0.784 F ω β ↑ 0.662 0.670 0.668 0.663 0.659 0.675 0.669 0.667 0.654 0.677 0.700 0.703 0.705 M ↓ 0.074 0.075 0.077 0.075 0.076 0.076 0.074 0.077 0.078 0.073 0.071 0.071 0.073 Sα ↑ 0.787 0.791 0.786 0.781 0.778 0.790 0.776 0.760 0.754 0.803 0.822 0.826 0.829 Eϕ M ↑ 0.820 0.825 0.819 0.813 0.810 0.827 0.816 0.802 0.798 0.853 0.866 0.868 0.875 DIS-TE2 maxF1 ↑ 0.799 0.810 0.807 0.801 0.793 0.801 0.799 0.782 0.776 0.808 0.814 0.819 0.822 F ω β ↑ 0.728 0.741 0.736 0.736 0.734 0.739 0.737 0.721 0.712 0.739 0.764 0.765 0.772 M ↓ 0.070 0.072 0.070 0.071 0.071 0.072 0.071 0.073 0.071 0.068 0.067 0.068 0.067 Sα ↑ 0.826 0.833 0.828 0.823 0.821 0.833 0.832 0.811 0.795 0.849 0.855 0.861 0.865 Eϕ M ↑ 0.858 0.870 0.868 0.866 0.865 0.861 0.860 0.850 0.834 0.868 0.895 0.901 0.903 DIS-TE3 maxF1 ↑ 0.830 0.846 0.845 0.839 0.833 0.838 0.834 0.831 0.812 0.841 0.850 0.867 0.870 F ω β ↑ 0.758 0.770 0.766 0.763 0.757 0.769 0.752 0.733 0.728 0.761 0.782 0.783 0.785 M ↓ 0.064 0.066 0.066 0.065 0.066 0.066 0.065 0.066 0.065 0.062 0.062 0.062 0.063 Sα ↑ 0.836 0.848 0.844 0.839 0.832 0.844 0.837 0.823 0.819 0.842 0.859 0.866 0.878 Eϕ M ↑ 0.883 0.894 0.887 0.883 0.878 0.900 0.896 0.878 0.858 0.911 0.915 0.918 0.926 DIS-TE4 maxF1 ↑ 0.827 0.833 0.826 0.820 0.816 0.830 0.818 0.804 0.801 0.856 0.876 0.876 0.879 F ω β ↑ 0.753 0.759 0.754 0.750 0.746 0.760 0.759 0.758 0.747 0.789 0.824 0.824 0.830 M ↓ 0.072 0.074 0.073 0.073 0.075 0.074 0.072 0.076 0.074 0.071 0.069 0.071 0.072 Sα ↑ 0.830 0.846 0.839 0.838 0.833 0.841 0.830 0.822 0.815 0.842 0.852 0.860 0.862 Eϕ M ↑ 0.870 0.885 0.881 0.873 0.868 0.873 0.866 0.845 0.836 0.891 0.917 0.917 0.923 4.2 Impletement Details Our image editing framework is implemented using PyTorch and trained on 8 NVIDIA GeForce RTX 3090 GPUs. We employ both non-rigid and rigid editing approaches to manipulate images in the DIS-TR dataset, utilizing spatial transformers and affine transformations to model non-rigid and rigid deformations, respectively. The hyperparameters used in our model are as follows: a batch size of 16, an image size of 512x512, 5 editing iterations, a learning rate of 0.001, a weight decay of 0.0001, 1000 diffusion steps, and a diffusion step size of 0.1. The hyperparameters in Equ 3 are set to λ1 = 0.8 and λ2 = 0.5. We train the segmentation model using the DIS-TR subset of the DIS5K dataset, utilizing 2 NVIDIA GeForce RTX 3090 GPUs. The input size to the network is 512x512, with a learning rate of 0.0001 and a batch size of 48. The model is optimized using the Adam optimizer over a total of 800 epochs. The edited images generated by our model are combined with the original training set to form a new training set, enabling the model to learn from both original and edited images. We evaluate our model on the DIS5K datasets, using the test sets from each dataset. For a comprehensive comparison, we also reproduce two state-of-the-art diffusion-based image generation methods, DatasetDM and Dataset Diffusion, using their default parameters. The generated images are compared with those from our approach, with the visualization of the results shown in Figure 3. 4.3 Results Results by Generated Image Count. We evaluate the performance of our proposed method, MaskFactory, against two state-of-the-art baselines, DatasetDM [9] and Dataset Diffusion [10], on the DIS5k dataset. The DIS5k dataset comprises five sub-datasets: DIS-VD, DIS-TE1, DIS-TE2, DIS-TE3, and DIS-TE4. We use the robust segmentation method IS-Net [2] as a baseline for the DIS task. For the two baselines, DatasetDM and Dataset Diffusion, we use the default parameters provided in their open-source implementations. 7 Table 2: Comparison of experimental results using different segmentation schemes, demonstrating performance gains with a fixed number of generated images. IS-Net [2] FP-DIS [36] UDUN [25] BiRefNet [1] SAM-HQ [37] Method w/o Ours w Ours w/o Ours w Ours w/o Ours w Ours w/o Ours w Ours w/o Ours w Ours maxF1 ↑ 0.740 0.784 0.784 0.805 0.784 0.799 0.866 0.882 0.897 0.905 M ↓ 0.074 0.073 0.060 0.063 0.059 0.057 0.036 0.033 0.019 0.018 Sα ↑ 0.787 0.829 0.821 0.859 0.817 0.830 0.889 0.900 0.907 0.911 DIS-TE1 Eϕ M ↑ 0.820 0.875 0.855 0.885 0.846 0.849 0.915 0.916 0.943 0.949 maxF1 ↑ 0.799 0.822 0.827 0.849 0.829 0.849 0.906 0.910 0.889 0.894 M ↓ 0.070 0.067 0.059 0.061 0.058 0.055 0.031 0.029 0.029 0.030 Sα ↑ 0.823 0.865 0.845 0.862 0.843 0.866 0.913 0.921 0.883 0.889 DIS-TE2 Eϕ M ↑ 0.858 0.903 0.889 0.893 0.886 0.894 0.947 0.957 0.928 0.937 maxF1 ↑ 0.830 0.870 0.868 0.911 0.865 0.888 0.920 0.937 0.851 0.853 M ↓ 0.064 0.063 0.049 0.051 0.050 0.047 0.029 0.027 0.045 0.043 Sα ↑ 0.836 0.878 0.871 0.892 0.865 0.885 0.918 0.918 0.851 0.854 DIS-TE3 Eϕ M ↑ 0.883 0.926 0.903 0.924 0.913 0.931 0.951 0.957 0.897 0.906 maxF1 ↑ 0.827 0.879 0.846 0.882 0.846 0.851 0.906 0.916 0.763 0.764 M ↓ 0.072 0.072 0.061 0.063 0.059 0.053 0.038 0.035 0.088 0.084 Sα ↑ 0.830 0.862 0.852 0.856 0.849 0.873 0.901 0.911 0.799 0.806 DIS-TE4 Eϕ M ↑ 0.870 0.923 0.891 0.935 0.891 0.895 0.933 0.941 0.843 0.850 maxF1 ↑ 0.791 0.835 0.823 0.851 0.823 0.847 0.897 0.905 0.842 0.847 M ↓ 0.074 0.072 0.062 0.065 0.059 0.058 0.036 0.033 0.045 0.044 Sα ↑ 0.813 0.866 0.843 0.873 0.838 0.857 0.905 0.909 0.848 0.850 DIS-VD Eϕ M ↑ 0.856 0.923 0.873 0.880 0.876 0.901 0.931 0.941 0.896 0.903 First, we identify the best-performing model on the DIS-VD validation set and then evaluate its performance on the other sub-datasets. The models are trained on the DIS-TR dataset, which is augmented with generated datasets of varying sizes: 2500, 5000, 7500, and 10000 images. The experimental results are presented in Table 1. Our proposed method, MaskFactory, consistently outperforms the baselines across all sub-datasets and evaluation metrics. As the number of generated images increases, the performance of MaskFactory improves, achieving the best results with 10000 generated images. Notably, MaskFactory attains the highest maxF1 scores across all sub-datasets, with improvements ranging from 0.044 to 0.052 compared to the IS-Net baseline. In contrast, while DatasetDM and Dataset Diffusion also show some performance gains, they encounter the issue of collapse when the generated data exceeds 5000 images. In these cases, the segmentation network’s performance stagnates or even degrades. For the DIS segmentation task, the use of pseudo-labels could introduce additional errors, leading to performance declines. Furthermore, MaskFactory demonstrates superior performance in terms of the weighted F-measure (F ω β ), S-measure (Sα), and Enhanced-alignment Measure (Eϕ M). The Mean Absolute Error (M) is also consistently lower for MaskFactory compared to the baselines, indicating more accurate segmentation results. Results by Segmentation Network. After achieving stable improvements in IS-Net’s performance, we examined the generalizability of our approach by applying the same configuration to several state-of-the-art segmentation networks on the DIS5K dataset. Each network was trained using the DIS-TR dataset augmented with 10,000 generated image pairs. The networks considered in this study include FP-DIS [36], UDUN [25], BiRefNet [1], and SAMHQ [37], all implemented with their default parameters. The experimental results, shown in Table 2, demonstrate that our proposed method consistently enhances the performance of all evaluated networks across the five sub-datasets of DIS5K. Notably, significant improvements were observed in the maxF1 and Eϕ M metrics. For instance, on the DISTE1 sub-dataset, our method increased the maxF1 score of IS-Net from 0.740 to 0.784, FP-DIS from 0.784 to 0.805, UDUN from 0.784 to 0.799, BiRefNet from 0.866 to 0.882, and SAMHQ from 0.897 to 0.905. 8 #$!"' #$!"' $#$ $#$%#! #$!"'! #$!"'! $#$ $#$%#! #$!"'& #$!"'& $#$ $#$%#! (a) Differences in UMAP distribution of generated mask images #$!"' #$!"' $#$ $#$%#! #$!"'! #$!"'! $#$ $#$%#! #$!"'& #$!"'& $#$ $#$%#! (b) Differences in UMAP distribution of generated corresponding RGB images Figure 4: UMAP Distribution Differences Additionally, the Mean Absolute Error (M) decreased for all networks when applying our method, indicating more accurate segmentation results. The S-measure (Sα) also consistently improved across all sub-datasets and networks, highlighting the effectiveness of our approach in capturing structural similarity. On the DIS-VD sub-dataset, used for validation, our method boosted the maxF1 score of IS-Net from 0.791 to 0.835, FP-DIS from 0.823 to 0.851, UDUN from 0.823 to 0.847, BiRefNet from 0.897 to 0.905, and SAMHQ from 0.842 to 0.847. These findings underscore the generalizability of our approach, as it enhances the performance of diverse segmentation networks without requiring any network-specific modifications. Table 3: Similarity with original dataset. Generate Type CLIP UMAP DatasetDM [9] 0.6683 0.7017 Dataset Diffusion [10] 0.6217 0.6349 MaskFactory(rigid) 0.8791 0.8961 MaskFactory(non-rigid) 0.9147 0.9346 MaskFactory(All) 0.8967 0.9103 Visual Results. As shown in Figure 5 (Appendix), our approach achieves precise results, comprising both rigid and non-rigid transformations. The non-rigid transformations, illustrated in columns (b) and (c), enable shape editing, such as removing a table corner or merging two backpacks into one. In contrast, the rigid transformations, demonstrated in columns (d), (e), and (f), primarily involve viewpoint changes, showcasing the original mask rotated in 3D space. Notably, our method effectively preserves the topological structure information of the original image, including the holes on the chair back. This allows for the low-cost generation of high-precision, diverse data pairs. To investigate differences between generated and real images, we analyzed image and mask distributions. Specifically, we used the CLIP model[38] to extract features from 300 real images, as well as images and masks generated by DatasetDM, DatasetDiffusion, and MaskFactory. We then applied UMAP[39] for dimensionality reduction on these 300 features. The feature distributions of masks and images are visualized in Figures 4(a) and 4(b), respectively. In the mask editing domain, MaskFactory demonstrates superior mask fidelity compared to other generation methods, primarily due to the incorporation of topological consistency constraints. This results in a feature distribution that closely aligns with that of real images. Conversely, for RGB images, the prior from diffusion VAE introduces a larger disparity between the generated and real 9 image distributions. However, the distribution generated by MaskFactory shows a greater overlap with the real image distribution compared to the other two methods. Furthermore, we quantified the differences using cosine similarity, as presented in Table 3. The results indicate that our method achieves the closest distribution to real images, further validating the effectiveness of MaskFactory in generating realistic masks and images. 5 Discussion 5.1 Ablation Study Table 4: Ablations on generation types. Type maxF1 ↑ M ↓ Sα ↑Eϕ M ↑ Rigid 0.768 0.074 0.807 0.867 Non-Rigid 0.771 0.074 0.796 0.858 Mix 0.784 0.073 0.829 0.875 Mask Generation Type Ablation. In our study, we implemented mask rigid editing, non-rigid editing, and mixed editing—each leveraging our novel mask control technology tailored for specific application scenarios. Rigid editing is designed for scenarios requiring precise geometric adjustments, primarily focusing on viewpoint and scale transformations. Non-rigid editing caters to applications needing high adaptability, handling topologically consistent deformations and complex, dynamic image edits. Mixed editing combines the advantages of both approaches, offering a comprehensive solution. We further evaluated the performance gains of each editing strategy. Our experimental results, as shown in Table 4. From the table, we observe that mixed editing achieves the highest performance across most metrics. Specifically, it achieves the highest maxF1 score and Sα, indicating superior structural fidelity and segmentation quality. The slight improvement in M and Eϕ M further underscores the versatility and effectiveness of mixed editing in creating diverse and realistic image-mask combinations. Loss Function Ablation. We introduce content and structure losses into our model. The Discriminative Loss, implemented via a discriminator, evaluates the differences between generated and real images, aiming to enhance the realism and quality of the outputs. The Edge Constraint Loss focuses on maintaining edge coherence during image editing, which is critical for preserving detailed structural information. We conducted ablation experiments to evaluate the impact of each loss function. The experimental results, shown in Table 5. Table 5: Ablations on loss functions. Loss Function maxF1 ↑ M ↓ LGAN Lcontent Lstructure ✓ 0.778 0.073 ✓ 0.745 0.075 ✓ 0.751 0.074 ✓ ✓ ✓ 0.784 0.073 From the table, we observe that the combination of all three loss functions (LGAN, Lcontent, and Lstructure) achieves the highest maxF1 score, indicating the best performance in terms of structural fidelity and realism. 5.2 Limitation Despite the favorable outcomes achieved by our method, it still encounters significant issues. Although we experimented with different conditions, the results are shown in Table 6. ControlNet sometimes produces unnatural images with stark foreground-background distinctions, necessitating additional harmonization. Complex scenarios can yield unrealistic elements, such as improperly positioned objects. Additionally, our method relies on pre-annotated image-mask pairs, limiting its ability to generate data autonomously and requiring high-quality initial annotations. 5.3 Conclusion Table 6: Ablations on conditions. Mask Prompt Canny maxF1 ↑ M ↓ ✓ 0.778 0.075 ✓ ✓ 0.782 0.073 ✓ ✓ 0.764 0.080 ✓ ✓ ✓ 0.784 0.073 This paper introduces MaskFactory, a novel two-stage approach for generating high-quality synthetic datasets for DIS tasks. By combining rigid and non-rigid mask editing techniques and using multi-conditional control for image generation, MaskFactory produces diverse and precise synthetic image-mask pairs, significantly reducing dataset preparation time and costs. Experiments on the DIS5K benchmark demonstrate the superior performance of MaskFactory compared to existing methods in terms of quality and efficiency. 10 Acknowledgements We thank Yuyan Huang, Hong Liu, and Wenjun Ji for fruitful discussions during the course of this project. Haoqian Qian and Xiaogang Jin were supported by Key R&D Program of Zhejiang (No.2024C01069). Deng-Ping Fan was supported by NSFC (No.62476143). References [1] Peng Zheng, Dehong Gao, Deng-Ping Fan, Li Liu, Jorma Laaksonen, Wanli Ouyang, and Nicu Sebe. 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In the first stage, existing ground truth masks undergo rigid and non-rigid editing to generate diverse synthetic masks. Rigid editing uses geometric priors from diffusion models for precise viewpoint transformations, while non-rigid editing employs adversarial training and self-attention mechanisms for complex shape modifications while preserving topology. In the second stage, the generated masks and their corresponding Canny edges serve as conditions, along with category prompts, to guide the generation of high-resolution RGB images using a multiconditional control generation method. This process ensures consistency between the generated images and masks while enhancing dataset realism and diversity. Our pseudocode visible Algorithm 1 Algorithm 1: MaskFactory Algorithm Input :D = {(Ir i , Im i )}N i=1 - Original dataset Output :G = {(gr i , gm i )}M i=1 - Synthetic dataset 1 Stage 1: Mask Editing 2 for i ←1 to N do 3 Step 1.1: Rigid Mask Editing 4 Im′ i ←Invert(Im i ) // Invert source mask 5 gm i ←ψθ(Im′ i , Ti) // Apply viewpoint transformation 6 Step 1.2: Non-Rigid Mask Editing 7 Es ←E(Im i ) // Extract edge map from source mask 8 V ←{vj}Nv j=1 // Obtain key points from edge map 9 Ts ←(V, Es) // Construct source mask structural graph 10 gm i ←Gθ(z, Pi, Im i ) // Generate synthetic mask 11 Tg = (V, Es) // Construct synthetic mask structural graph 12 LGAN ←ETs∼pdata(Ts)[log Dϕ(Ts)] + ETg∼pgen(Tg)[log(1 −Dϕ(Tg))] 13 Lcontent ←∥gm i −Im i ∥1 14 Lstructure ←∥Tg −Ts∥1 15 Ltotal ←LGAN + λ1Lcontent + λ2Lstructure 16 Update Gθ and Dϕ to minimize Ltotal 17 Stage 2: Image Generation 18 for i ←1 to M do 19 cs i ←gm i // Segmentation condition 20 cy i ←Canny(gm i ) // Canny condition 21 z ∼N(0, 1) // Sample Gaussian noise 22 gr i ←Mθ(z, Bθ(cs i), Bθ(cy i )) // Generate RGB image 23 return G = {(gr i , gm i )}M i=1 B Dataset Details We conduct our experiments on the DIS5K dataset, which comprises 5,479 high-resolution images featuring camouflaged, salient, or meticulous objects in various backgrounds. The DIS5K dataset is divided into three subsets: DIS-TR (3,000 images) for training, DIS-VD (470 images) for validation, and DIS-TE (2,000 images) for testing. For data augmentation, we utilize the mask portion of the training subset (DIS-TR). To evaluate our models, we employ a diverse set of metrics to ensure comprehensive performance assessment. These metrics include max F1, which balances precision and recall, providing a harmonic mean that is indicative of overall accuracy; Fω β, a weighted F-measure that compensates for class 13 imbalances, with values ranging from 0 to 1, where higher values denote superior performance; M (Mean Absolute Error), which calculates the average absolute difference between the predicted and ground truth masks, with lower values signifying better accuracy; Sα, a structural similarity measure that evaluates the preservation of significant structures within the image, with values closer to 1 indicating better performance; and Eϕ M, an enhanced measure that considers both pixel-level and image-level information for a more holistic evaluation, where higher values represent better performance. Collectively, these metrics provide a robust framework for assessing the effectiveness and reliability of our segmentation models. C Mathematical Details C.1 Diffusion-Based Image Generation The diffusion model employed in MaskFactory for image generation follows the formulation introduced by Ho et al. [40]. Given a data distribution x0 ∼q(x0), the forward diffusion process is defined as a Markov chain that gradually adds Gaussian noise to the data: q(xt\|xt−1) = N(xt; p 1 −βtxt−1, βtI), (9) where βt ∈(0, 1) is a variance schedule. The reverse process is learned by a neural network ϵθ that predicts the noise added at each step: pθ(xt−1\|xt) = N(xt−1; µθ(xt, t), σ2 t I), (10) where µθ(xt, t) = 1 √αt xt − βt √1−¯αt ϵθ(xt, t) and αt = 1 −βt, ¯αt = Qt s=1 αs. The objective is to maximize the variational lower bound: L = Eq(x0)Eq(x1,...,xT \|x0) " T X t=1 log pθ(xt−1\|xt) q(xt−1\|xt, x0) # . (11) C.2 Topology-Preserving Adversarial Training The topology-preserving adversarial training in MaskFactory involves a generator Gθ and a discriminator Dϕ. The generator aims to minimize the adversarial loss: LGAN(G) = ETg∼pgen(Tg)[log(1 −Dϕ(Tg))], (12) while the discriminator tries to maximize the adversarial loss: LGAN(D) = ETs∼pdata(Ts)[log Dϕ(Ts)] + ETg∼pgen(Tg)[log(1 −Dϕ(Tg))]. (13) The generator and discriminator are updated alternately to reach an equilibrium. C.3 Structural Graph Construction To preserve the topological structure of the source masks during editing, MaskFactory constructs structural graphs Ts and Tg for the source and synthetic masks, respectively. The structural graph T = (V, E) consists of a set of vertices V = {vj}Nv j=1 representing key points and a set of edges E representing their connectivity. The structure preservation loss is defined as the L1 distance between the structural graphs: Lstructure = ∥Tg −Ts∥1 = X (i,j)∈E ∥Tg(i, j) −Ts(i, j)∥, (14) 14 where Tg(i, j) and Ts(i, j) denote the edge weights between vertices i and j‘ in the synthetic and source mask structural graphs, respectively. By minimizing the structure preservation loss along with the adversarial and content losses, MaskFactory ensures that the edited masks maintain the topological structure of the source masks. D Visualization of the results using two editing methods from MaskFactory. In this section, we demonstrate MaskFactory’s ability to edit masks for common and fine-grained objects. D.1 Visualization of Common Object Mask Editing We selected common household items such as tables, chairs, bags, and musical instruments for editing using MaskFactory. Non-rigid edits were performed with different prompts, and rigid edits were performed from different viewpoints. The editing results and corresponding RGB image generations are shown in Figure 5. MaskFactory exhibits strong topological structure preservation and produces diverse editing outcomes, as demonstrated by the variety of modifications made to both rigid and non-rigid objects. D.2 Visualization of Fine-Grained Object Mask Editing We selected fine-grained objects from MaskFactory’s generated masks, such as ornate European chandeliers, iron gates, birdcages, and seahorses, to showcase MaskFactory’s detailed editing capabilities. Even with complex geometries, MaskFactory can perform topology-preserving edits without losing the original mask’s semantic information. These intricate structures play a crucial role in segmentation metrics. As illustrated in Figure 6, the visual results demonstrate the model’s ability to handle fine-grained details while maintaining the integrity of the original mask’s structure. D.3 Visualization Results with Canny Constraints After incorporating Canny edge detection as a constraint, the visual results of MaskFactory show a significant improvement in boundary precision. The Canny edges effectively guide the generation process, resulting in images with clearer and more accurate boundary details, avoiding vague or ambiguous transition areas. The visualizations demonstrate that the Canny edges not only better constrain the contours of the generated images but also enhance the overall fidelity and visual quality of the output. Compared to models without edge constraints, our approach produces more detailed and structurally coherent images, as shown in Figure 7. D.4 Topological Structure Visualization We performed topological structure visualizations on selected samples to assess the model’s ability to preserve topology during the editing process. These visualizations clearly show how our method retains the topological structure of the original data while allowing for effective manipulation when necessary. Whether dealing with complex geometric shapes or subtle structural modifications, the topological visualizations illustrate that our approach reliably preserves geometric consistency and topological features throughout the editing process, as demonstrated in Figure 8. 15 (a) raw image (b) rigid 1 (c) rigid 2 (d) non-rigid 1 (e) non-rigid 2 (f) non-rigid 3 Figure 5: Visual results of common object mask editing. The model demonstrates strong topological structure preservation and diverse editing outcomes with both rigid and non-rigid edits. 16 Real Image Source Mask Synthetic Image Synthetic mask Rigid Mask Editing Original Source Non-rigid Mask Editing Synthetic Image Synthetic mask Figure 6: Visual results of fine-grained object mask editing. The model successfully edits complex structures without compromising the original mask’s semantic information. 17 Figure 7: Canny condition visual results. The generated images show improved boundary precision and better structural coherence. Raw Topology MaskFactory (rigid) Topology Figure 8: Topological Structure Visualization. The visualizations demonstrate the model’s ability to maintain and manipulate topology during the editing process. 18 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: This paper presents a method for generating and editing images with highprecision masks, and announces the open-sourcing of a DIS generation dataset. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,424	DigiRL: Training In-The-Wild Device-Control Agents with Autonomous Reinforcement Learning Hao Bai1,2∗Yifei Zhou1∗Mert Cemri1 Jiayi Pan1 Alane Suhr1 Sergey Levine1 Aviral Kumar3,4 1UC Berkeley 2UIUC 3CMU 4Google DeepMind Abstract Training corpuses for vision language models (VLMs) typically lack sufficient amounts of decision-centric data. This renders off-the-shelf VLMs sub-optimal for decision-making tasks such as in-the-wild device control through graphical user interfaces (GUIs). While training with static demonstrations has shown some promise, we show that such methods fall short for controlling real GUIs due to their failure to deal with real world stochasticity and non-stationarity not captured in static observational data. This paper introduces a novel autonomous RL approach, called DigiRL, for training in-the-wild device control agents through fine-tuning a pre-trained VLM in two stages: offline RL to initialize the model, followed by offline-to-online RL. To do this, we build a scalable and parallelizable Android learning environment equipped with a VLM-based evaluator and develop a simple yet effective RL approach for learning in this domain. Our approach runs advantage-weighted RL with advantage estimators enhanced to account for stochasticity along with an automatic curriculum for deriving maximal learning signal. We demonstrate the effectiveness of DigiRL using the Android-in-the-Wild (AitW) dataset, where our 1.3B VLM trained with RL achieves a 49.5% absolute improvement – from 17.7 to 67.2% success rate – over supervised fine-tuning with static human demonstration data. These results significantly surpass not only the prior best agents, including AppAgent with GPT-4V (8.3% success rate) and the 17B CogAgent trained with AitW data (38.5%), but also the prior best autonomous RL approach based on filtered behavior cloning (57.8%), thereby establishing a new state-of-the-art for digital agents for in-the-wild device control. 1 Introduction Advances in vision-language models (VLMs), especially in regards to their remarkable commonsense, reasoning, and generalization abilities imply that realizing a fully autonomous digital AI assistant, that can simplify human life by automating day-to-day activities on computer devices via natural language interfaces, is no longer a distant aspiration [16, 45, 56]. An effective device-control AI assistant should be able to complete tasks in-the-wild through Graphical User Interfaces (GUIs) on digital devices: make travel plans; experiment with presentation designs; and operate a mobile device autonomously, all while running amidst stochasticity and distractors on the device, the Internet, and the tools it interacts with. However, enhanced reasoning or common-sense abilities do not directly transfer to intelligent assistant behavior: ultimately we want AI assistants to accomplish ∗Equal contribution, listed in alphabetical order; work done at UC Berkeley. E-mails: haob2@illinois.edu, yifei_zhou@berkeley.edu, aviralku@andrew.cmu.edu. Project page: https://digirl-agent.github.io/. Code available at https://github.com/DigiRL-agent/digirl. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). tasks, exhibit rational behavior, and recover from their mistakes as opposed to simply producing a plausible completion to a given observation based on the data seen during pre-training. This implies that a mechanism to channel abilities from pre-training into a deployable AI “agent” is lacking. Even the strongest proprietary VLMs, such as GPT-4V [24] and Gemini 1.5 Pro [7] 2, still struggle to produce the right actions when completing tasks on devices. While general-purpose vision-language abilities help these models still make meaningful abstract deductions about novel scenes when deployed, these deductions do not transfer to accurate reasoning for control [47, 45, 55, 44]. As a result, most prior work for building device agents construct complex wrappers around proprietary VLMs by combining them with prompting, search, or tool use [47, 44, 52, 51, 45]. While building prompting or retrieval wrappers to improve decision-making performance of existing VLMs enhances their performance in the short run, without updating the weights, the effectiveness of the resulting agent is inherently limited by the capabilities of the base model [49, 3]. For example, we found that off-the-shelf VLMs make reasoning failures that derail the agent (e.g., Figure 2 and Figure 17), as direct consequences of inability of the base model to reason with low-level device-control actions. A different solution is to fine-tune the model on demonstrations via imitation learning. However, the dynamic nature of the web and device means that models trained to mimic actions in stale data can result in sub-optimalilty as the eco-system changes [26]. Agents trained in this way struggle to recover from the agents’ own mistakes [8, 12]. AutoEval annotates reward for each trajectory Model executes tasks in parallel and produce trajectories Tasks are sampled from task dataset Annotated trajectories are used to update the model through online RL Fine-tune on existing trajectories via offline RL Step I: Offline RL Pretrained Model Offline Model VLM is generally pre-trained on Internet-scale vision-and-language data Pretraining Step II: Online RL Pretrained Model Online Model AutoEval Figure 1: DigiRL overview. DigiRL is built upon a VLM that has been pre-trained on extensive web data to develop fundamental skills such as common knowledge, reasoning, and visual grounding. Initially, we employ offline RL to fine-tune the VLM using stale task-specific data, which helps in eliciting goal-oriented behaviors. Subsequently, our agent engages with real-world graphical user interfaces, continuously enhancing its performance through online RL and autonomous performance evaluations. If we can instead build an interactive approach to train a VLM to directly adapt and learn from its own experience on the device and the Internet, that can be used to build a robust and reliable device-control agent, without needing wrappers on top of proprietary models. However, this learning-based approach must satisfy some desiderata. First, it must make use of online interaction data since static demonstration data would not be representative of the task when the model is deployed: for instance, even in the setting of web navigation alone, dynamic nature of in-the-wild websites means that the agent will frequently encounter website versions that differ significantly from the scenarios seen during training and will need to behave reliably despite changes in visual appearance and distractions. Second, learning on-the-fly means the approach must learn from multi-turn interaction data from the model itself, a large of chunk of which would consist of failures. Proper mechanisms must be designed to automatically pick out the correct actions while filtering the wrong ones. To this end, our main contribution is a novel autonomous RL approach, DigiRL (i.e., RL for Digital Agents), for training device control agents, as shown in Figure 1. The resulting agent attains state-of-the-art performance on a number of Android device-control tasks. To train this agent, our approach operates in two phases: an initial offline RL phase to initialize the agent using existing data, followed by an offline-to-online RL phase, that further fine-tunes the model obtained from offline RL on online rollout data. Online RL training requires access to an environment that the agent can interact with and obtain reliable reward signals, all in a reasonable amount of wall-clock time. To do so, we build a scalable and parallelizable Android learning environment equipped with a robust VLM-based general-purpose evaluator [26] (average error rate 2.8% against human judgement) that supports running up to 64 real Android emulators at the same time to make online RL real-time. Then, to effectively learn autonomously, we develop an online RL approach that retains the simplicity of supervised learning, but incorporates several key deep RL insights to enable fast fine-tuning. Concretely, our approach is a variant of advantage-weighted regression (AWR) [28], equipped with: (i) an automatic curriculum that uses an instruction-level value function to order tasks so as to extract 2We use external versions of these models as of June 11, 2024. Experiments with GPT and Gemini models were performed entirely by Hao Bai, Yifei Zhou, Mert Cemri, and Jiayi Pan. 2 DigiRL AutoUI GPT-4V Got stuck ✘ Got stuck ✘ ✘ ✘ Got stuck ✘ General How much does a 2 bedroom apartment rent for in Denver? WebShop Go to bestbuy.com, search for “logitech g933” Click Skipped... Click Click Type “razecg Press Back Click Type “logi\|g Scroll Up Press Home Click Type “2 bedrg Press Enter Wrong page Got stuck Got stuck ✘ Figure 2: Qualitative comparison between DigiRL and other approaches. AutoUI trained from static human demonstrations can easily get stuck in out-of-distribution states while GPT-4V often get on a wrong goal (searched “logitech g933bestbuy.com logitech g933” in Google instead of bestbuy.com). In contrast, DigiRL can recover from such states and complete complex instruction as requested. maximal learning signal, which is inspired by prioritized replay methods [11, 32, 23], and (ii) another step-level value function trained via effective cross-entropy loss [17, 5] to extract low-variance and less-biased learning signal amidst stochasticity and diverse tasks. This RL approach allows us to fine-tune VLMs on their own experience. We evaluate our agent trained with DigiRL in carrying out diverse instructions from Android in the Wild dataset [31] on real Android device emulators and find that our agent can achieve a 28.7% improvement over the existing state-of-the-art agents (from 38.5% to 67.2% success rate) 18B CogAgent [9], and over 9% improvement over the prior best autonomous learning approach based on Filtered Behavior Cloning [18, 26]. The performance of our agent also significantly surpasses wrappers on top of state-of-the-art proprietary VLMs such as GPT-4V [24] and Gemini 1.5 Pro [7] (17.7% success rate), despite using a significantly smaller model (with 1.3B parameters). To our knowledge, this is the first work to successfully build an autonomous offline-to-online RL approach to enable state-of-the-art performance on device-control problems. 2 Related Work Multi-modal digital agents. In contrast to language-only agents that largely interact with both text or code inputs and outputs [33, 49, 3, 30, 46, 20, 13], training multi-modal agents capable of controlling devices presents different challenges: first, device control is done directly at the pixellevel and in a coordinate-based action space, instead of natural language [31, 44] that LLM is most familiar with, and second, the ecosystem of a device and the Internet tends to be quite stochastic and unpredictable, which is absent with high-level planning in language only. To handle these challenges, prior work largely builds on strong proprietary VLMs [24, 7], and designs complex rule-based wrappers [47, 51, 45, 52] to enhance the visual grounding capabilities of VLMs in GUI interfaces and convert text output into pixel interactions. However, without any form of fine-tuning, this limits the room for possible performance improvement [44, 47, 49, 3, 50], especially when pre-training corpora only present limited action-labeled data. A separate line of work fine-tunes VLMs with demonstration data [19, 15, 9, 53] via imitation learning, but maximizing single-step accuracy from stale demonstrations without accounting for consequences of these actions in subsequent steps may lead to poor solutions amidst stochasticity [26], as agents trained in such ways will struggle to recover from out-of-distribution states not included in the demonstration data [8, 12]. The third category, and perhaps the closest to us, are works that run filtered imitation learning on autonomously-collected data to directly maximize the episode success rate [26, 18]. In contrast, ours is the first work to scale autonomous, offline-to-online RL for device control, producing an agent that outperforms prior agents built via imitation. Even when compared to prior work running on-policy RL in simplified web navigation settings (MiniWob++ [37, 10]), our approach is 1000x more sample efficient (around 1e3 trajectories compared to around 1e6 trajectories), and operates in real-world GUI navigation tasks. Environments for device control agents. Recent works have introduced simulated environments for building device control agents [48, 56, 16, 54, 4, 44]. However, these environments are primarily designed for evaluation, and present only a limited range of tasks within fully deterministic and 3 action space type click slide home back enter real-world environment agent model open-ended evaluator non-stationary website load ads unpredictable order pop-up identity dynamics Figure 3: Environment details. Top: actions space and dynamics of the environment. Bottom: examples of the read-world non-stationarity and dynamism of the environment. stationary settings, infeasible for acquiring a diverse repertoire of skills needed for device control. Alternatively, other works use environments with a greater diversity of tasks [48, 37], but these environments often oversimplify the task complexity, thus failing to transfer to in-the-wild settings. Coversely, our training environment utilizes autonomous evaluation [26] with Gemini 1.5 Pro [7] to support diverse, open-ended tasks on parallel actual Android devices, at full scale unlike prior environments. This also contrasts other prior works that use single-threaded Android emulators [26, 39, 19] and thus inefficient for support online RL at scale. Reinforcement learning for LLM/VLMs. The majority of prior research employing RL for foundation models concentrates on tasks that must be solved in a single turn, such as preference optimization [25, 58, 2] or reasoning [27]. However, optimizing for single-turn interaction from expert demonstrations may result in sub-optimal strategies for multi-step problems [57, 38, 42], especially amidst a high degree of stochasticity or non-stationarity. Therefore, we focus on building multi-turn RL algorithms that can learn from sub-optimal, online interaction data in this work. While prior works have developed value-based RL algorithms for LLMs [42, 38, 1, 57, 50], they typically require maintaining multiple models such as Q-networks, value-networks, and policy networks, along with their delayed target counterparts, and can be subjective to slow convergence and sensitivity to choices of hyper-parameters. In contrast, we focus on identifying the key design choices for instantiating a simple yet effective RL algorithm for practitioners to incorporate to substantially improve full-scale Android device control. Our approach can serve as a base model for future research. 3 Problem Setup and Preliminaries Problem formulation. We are interested in pixel-based interaction with virtual devices. We scope our study in the control of Android devices: this is already significantly more challenging and more general than previous learning-based environments that focus solely on web navigation [16, 56, 4], where the web browser itself is merely one application within our broader environment, and link-based device controls [47, 51] are inadequate for tasks like games that do not support link inputs. Each episode begins with the emulator initialized to the home screen. Subsequently, a task is selected from a predefined set of language instructions, some examples of which are shown in Appendix A.1. An agent is then tasked with manipulating the emulator to fulfill this instruction. At each time step, the agent receives a screenshot of the current screen as the observation. Following the action space in prior literature [31], the available actions include tapping and sliding based on normalized (x, y) coordinates (ranging from 0 to 1 relative to the screen dimensions), typing text strings of variable length, and pressing special buttons such as HOME, BACK, and ENTER, as illustrated in Figure 3. Our train and test instructions comes from General and Web Shopping subsets in AitW [31]. These tasks consist of information-gathering tasks like “What’s on the menu of In-n-Out?”, and shopping tasks on the web like “Go to newegg.com, search for razer kraken, and select the first entry”. Challenges of stochasticity. Real-world device contrl presents unique challenges of stochasticity absent in simulated environments [56, 37] such as: (1) the non-stationarity of websites and applications, which undergo frequent updates, causing the online observations to be different from stale offline data, (2) various unpredictable distractors such as pop-up advertisements, login requests, and the stochastic order of search results. (3) technical challenges and glitches such as incomplete webpage loading or temporary access restrictions to certain sites. Examples of scenarios with such stochasticity from 4 our experiments are shown in Figure 3. We observe that these stochastic elements pose significant challenges for pre-trained VLMs, including even those fine-tuned on device control data. As a concrete example, Figure 4 shows an experiment result that illustrates the necessity of continuously adapting the models to the non-stationarity of websites and applications. After obtaining a good checkpoint using our approach (DigiRL), that we will introduce in the next section, with autonomous data from June.1 to June.3, we compare the performance of a frozen policy and a continuously updating policy using fresh autonomous data from June.7 to June.11. We find that indeed the the performance of the frozen policy gradually degrades over time due to the changes on websites and applications, while continuous online updates plays a key role in preventing this degradation. June 1 June 3 June 7 June 11 Walltime 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Success Rate Learning (Online) Frozen (Online) Learning (Online) Figure 4: Performance of our approach (DigiRL) in different training modes on the Webshop subset. When utilizing a stale checkpoint, i.e., “frozen” (black+blue curve) performance generally begins to degrade as time evolves, whereas autonomous online training (black+red curve) via DigiRL allows us to retain performance despite non-stationarity and stochasticity. Setup for reliable and scalable online RL. As autonomous RL interleaves data collection and training, to maximize learning amidst stochasticity, it is crucial to have a real-time data collection pipeline to collect enough experience for gradient updates. While this is not possible in single-thread Android emulator environments [26, 39] due to latency, we parallelize our Android emulator using appropriate error handling as discussed in Appendix A.1. In addition, the environment must provide a reward signal by judging whether the current observation indicates the agent has successfully completed the task. To generalize our evaluator to support a wide range of tasks, we extend Pan et al. [26]’s end-to-end autonomous evaluator that does not require accessing the internal states of the emulator or human-written rules for each task. This contrasts previous works that manually write execution functions to verify the functional completeness of each task [16, 48, 37, 44]. We adopt Gemini 1.5 Pro [6, 7] as the backbone of the autonomous evaluator. We seed this model with few-shot rollouts and the associated human-labeled success indicators to guide evaluation of novel queries. This pipeline enables a single evaluator that can evaluate all AiTW tasks. The evaluator is highly aligned with human annotations (average error rate 2.8%), validated in Figure 8. 4 DigiRL: Autonomous RL for Building a Strong Device-Control Agent We now present our autonomous RL framework for training device agents. We pose the device control problem as a Markov decision process (MDP) and develop RL methods for this MDP. The core of our approach is based on a simple and scalable off-policy RL method, advantage-weighted regression (AWR) [29], but we make crucial modifications to handle stochasticity and highly-variable task difficulty, through the use of value functions trained with appropriate losses, and an automatic curriculum, induced by an instruction-level value function to maximize learning. Device control and GUI navigation as a MDP. We conceptualize device control guided by natural language instructions as a finite horizon Markov Decision Process (MDP) represented by M = {S, A, T , µ0, R, H} and run policy gradient to solve this MDP. At the beginning, an initial state s0 and a natural language instruction c are sampled from the initial state distribution µ0. A reward of 1 is given at the end if the agent successfully fulfills the task per the evaluator, otherwise a reward of 0 is given. The trajectory terminates either when the agent accomplishes the task or when the maximum allowed number of interactions H is exceeded. States are represented using the last two screenshots. To explain our approach in detail, we also include several standard definitions used in reinforcement learning (RL). The Q function for a policy π represents the expected longterm return from taking a specific action at the current step and then following policy π thereafter: Qπ(sh, ah, c) = Eπ hPH t=h r(st, at, c) i . The value function V π(sh, c) is calculated by averaging the Q-value, Qπ(sh, ah, c), over actions ah drawn from the policy π. The advantage Aπ(sh, ah, c) for a state-action pair is computed by subtracting the state’s value under the policy from its Q-value: Aπ(sh, ah, c) = Qπ(sh, ah, c) −V π(sh, c). 5 4.1 Backbone of Our Approach: Off-Policy RL via Advantage-Weighted Regression The starting point we choose to build our approach on is the advantage-weighted regression (AWR) algorithm [29], which says that we can improve the policy reliably by regressing the policy towards exponentiated advantages induced by the reward function, as a proxy for optimizing the policy gradient while staying close to the previous policy [14, 35, 34]: arg maxπ Eν [log π(a\|s, c) · exp (A(s, a, c)/β)] , (4.1) for some positive parameter β and the distribution of past experience ν, and A(s, a, c) denotes the advantage of a state-action pair (s, a) given a context c. To avoid tuning the hyperparameter β, we consider an alternative that does “hard filtering” on the advantages instead of computing exp(A), similar to prior works [22, 43]. This leads to the following loss function for fine-tuning the model: L(π) = −Efilter(ν)[log π(a\|s, c)]. (4.2) Typically, these advantages are computed by running Monte-Carlo (MC) rollouts in the environment to estimate the value of a given state-action pair, and subtracting from it an estimate of the value of the state given by a learned value estimator alone. However, this approach is likely to produce high-variance advantages given the stochasticity of the device eco-system that affects MC rollouts. 4.2 Obtaining Reliable Advantage Estimates from Doubly-Robust Estimators To reliably identify advantageous actions given significant environment stochasticity, we construct a per-step advantage estimator, inspired by doubly-robust estimators [40, 36]: Astep(sh, ah, c) := λH−hr(sH, aH, c) + (1 −λH−hr(sH, aH, c))(V step(sh+1, c) + r(sh, ah, c) −V step(sh, c)), (4.3) where λ is a weighting hyper-parameter. This construction of the advantage estimator is a simplified version of Generalized Advantage Estimation (GAE) [36] using only the next-step advantage estimator and final-step advantage estimator as there are no intermediate rewards in our problem. This construction balances an advantage estimator with higher variance Monte-Carlo estimates λH−hr(sH, aH, c) (due to stochasticity) and an estimator with higher bias V step(sh+1, c) + r(sh, ah, c) −V step(sh, c) (due to imperfect fitting of the value function). We observed that combining both high-variance and high-bias estimators gave us a sweet-spot in terms of performance. To implement the step-level hard filtering, we simply threshold this doubly robust estimator as Astep(sh, ah, c) > 1/H to decide which actions progress towards the goal. 4.3 Automatic Curriculum using an Instruction-Level Value Function While the AWR update (Equation 4.1) coupled with a robust advantage estimator (Equation 4.3) is likely sufficient on standard RL tasks, we did not find it to be effective enough for device control in preliminary experiments. Often this was the case because the task set presents tasks with highlyvariable difficulties that collecting more data on tasks that the agent was already proficient at affected sample efficieny negatively. In contrast, maximal learning signal can be derived by experiencing the most informative tasks for the agent during training. To this end, we design an instruction-level value function V instruct(c) to evaluate if a given rollout can provide an effective learning signal: Ainstruct(sh, ah, c) := PH t=hr(st, at, c) −V instruct(c) = r(sH, aH, c) −V instruct(c), (4.4) where PH t=h r(st, at, c) is a Monte-Carlo estimator of Q(sh, ah, c). The equality holds because the MDP formulation only provides rewards at the end of a rollout. Intuitively, if a rollout attains a high value of Ainstruct(sh, ah, c), it means the value function V instruct is small. Therefore, this rollout represents a valuable experience of the agent accomplishing a difficult task, and thus should be prioritized, akin to ideas pertaining to prioritized experience [32] or level replay [11]. When training the actor with a buffer of historical off-policy data, we first perform a filtering step to identify the top-p datapoints with highest Ainstruct(sh, ah, c). Then, we use it for AWR (Equation 4.1) with the doubly-robust advantage estimator (Equation 4.3). Implementation details. Inspired by the findings in some recent works [5, 17] that modern deep learning architectures like transformers [41] are better trained with cross-entropy losses instead of mean-squared losses, we utilize a cross-entropy objective based on the Monte-Carlo estimate of the trajectory reward for training both of our value functions: L(V traj) = −Eν[r(sH, aH, c) log V traj(c) + (1 −r(sH, aH, c)) log(1 −V traj(c))], (4.5) L(V step) = −Eν[r(sH, aH, c) log V step(sh, ah, c) + (1 −r(sH, aH, c)) log(1 −V step(sh, ah, c))]. (4.6) 6 instruction-level value function step-level value function actor Equation (4.2) Go to walmart.com (difficulty: easy) Go to ebay.com, search for "asus zenbook" (difficulty: medium) Go to ebay.com, search for "asus zenbook" 0.8 0.2 -0.01 0.01 0.10 1 0 1 Go to costco.com, search for "bose soundsport free", and select the first entry (difficulty: hard) discarded Task discarded go to statelevel critic Task Go to ebay.com, search for "asus zenbook" Task Instruction-level Value Function Step-level Value Function Train w/ MLE loss Figure 5: Algorithm visualization. The two value function are first trained with original distribution of collected trajectories according to Equation (4.5) and Equation (4.6), then used to filter the trajectories for training the actor. We use the MLE loss (Maximum Likelihood Estimation loss) to train the actor. Final algorithm. The final practical algorithm is shown in Figure 5. The instruction-level value function estimates the values of the trajectories, which is trained with loss shown in Equation (4.5). The step-level value function estimates the values of states, which is trained with loss shown in Equation (4.6). When training the actor, we first filter out trajectories and states using the value functions as shown in Equation (4.4) and Equation (4.3), then train the actor with the MLE loss shown in Equation (4.2) on the filtered data. 5 Experimental Evaluation The goal of our experiments is to evaluate the performance of DigiRL on challenging Android device control problems. Specifically, we are interested in understanding if DigiRL can produce agents that can effectively learn from autonomous interaction, while still being able to utilize offline data for learning. To this end, we perform a comparative analysis of DigiRL against several prior approaches, including state-of-the-art agents in Section 5.1. We also perform several ablation experiments to understand the necessity and sufficiency of various components of our approach in Section 5.2. Baselines and comparisons. We compare DigiRL with: (a) state-of-the-art agents built around proprietary VLMs, with the use of several prompting and retrieval-style techniques; (b) running imitation learning on static human demonstrations with the same instruction distribution, and (c)a filtered BC approach [26]. For proprietary VLMs, we evaluate GPT-4V [24] and Gemini 1.5 Pro [7] both zero-shot and when augmented with carefully-designed prompts. For the zero-shot setting, we use the prompt from Yang et al. [47] and augment the observation with Set-of-Marks [55]. Set-ofMarks overlays a number for each interactable element over the screenshot, so that a VLM can directly output the number of the element to interact with in plain text instead of attempting to calculate pixel coordinates, which is typically significantly harder. We also compare with AppAgent [47], which first prompts the VLM to explore the environment, and appends the experience collected to the test-time prompt. We also compare with two state-of-the-art fine-tuning methods for Android device control: AutoUI (specifically AutoUI-Base [53]) and CogAgent [9]. AutoUI-Base uses an LM with 200M parameters, and a a vision encoder with 1.1B parameters. CogAgent has 11B parameters for its vision encoder and 7B for its LM. The supervised training corpus for both AutoUI-Base and CogAgent contains AitW, including the instruction set and the emulator configuration we use. Base VLM and offline dataset. Both Filtered BC and DigiRL use trained AutoUI-Base checkpoints with the image encoder frozen. The instruction and step-level value functions for DigiRL employ this same frozen image encoder. The visual features output from the encoder are concatenated with instruction features derived from RoBERTa [21]. A two-layer MLP is then used to predict the value function. In the offline phase, the offline dataset is collected by rolling out the initial AutoUI-Base supervised trained checkpoint as policy. For fair comparisons, we keep the number of offline data collected in the pure offline training roughly the same as the total number of data collected in the offline-to-online training. Due to the dynamic nature of the Internet-device eco-system, our offline data was stale by the time we were able to run our offline-to-online experiments, and this presented additional challenge in offline-to-online learning. In both General and Web Shopping subsets, offline experiments make use of around 1500 trajectories while offline-to-online experiments start with 7 AitW General AitW Web Shopping Train Test Train Test Prompting SET-OF-MARKS GPT-4V 5.2 13.5 3.1 8.3 Gemini 1.5 Pro 32.3 16.7 6.3 11.5 APPAGENT GPT-4V 13.5 17.7 12.5 8.3 Gemini 1.5 Pro 14.6 16.7 5.2 8.3 Learning SUPERVISED TRAINING CogAgent 25.0 25.0 31.3 38.5 AutoUI 12.5 14.6 14.6 17.7 OFFLINE Filtered BC 51.7 ± 5.4 50.7 ± 1.8 44.7 ± 1.6 45.8 ± 0.9 Ours 46.9 ± 5.6 62.8 ± 1.0 39.3 ± 6.0 45.8 ± 6.6 OFF-TO-ON Filtered BC 53.5 ± 0.8 61.5 ± 1.1 53.6 ± 4.7 57.8 ± 2.6 Ours 63.5 ± 0.0 71.9 ± 1.1 68.2 ± 6.8 67.2 ± 1.5 Table 1: Main comparisons of different agents across various settings. Each offline experiment is repeated three times and the mean and standard deviation are reported. Each online experiment is repeated two times. Results are evaluated with our autonomous evaluator with the first 96 instructions in the train and test set. Correlation of our correlation and human judgements can be found in Figure 8. 0 320 640 960 #Trajectories 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Success Rate 0 320 640 960 #Trajectories 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Filtered BC-1 Filtered BC-2 DigiRL-1 DigiRL-2 GPT-4V Figure 6: Offline-to-online training curves for Filtered BC and DigiRL. Curves are smoothed with exponential weighting over the x-axis. Left: AitW General. Right: AitW Web Shopping. Two runs for each model are started on two different dates with at least two days apart. Observe that DigiRL is able to improve faster with a fewer number of samples. Since the data collection frequency is the bottleneck, these performance trends directly reflect performance trends against wall-clock time as well. around 500 offline trajectories and update with another 1000 online trajectories. In the offline phase, DigiRL skips instruction-level filtering and instead trains the actor with all successful trajectories to make full use of the offline data. See a detailed breakdown of our dataset in Appendix A.1. 5.1 Main Results Our main results are summarized in Table 1 and Figure 6. We find that on both AitW General and AitW Web Shopping subsets, the agent trained via DigiRL significantly outperforms prior state-of-the-art methods based on prompting and retrieval (AppAgent + GPT-4V/Gemini 1.5 Pro) or training on static demonstrations (CogAgent and AutoUI), by a large margin with more than 49.5% absolute improvement (from 17.7% to 71.9% on the General subset and from 17.7% to 67.2% on the Web Shopping subset). Notably, this improvement from DigiRL is realized fully autonomously without making use of human supervision (e.g. manually labeled rollouts or hand-written verifiers). Are inference-time prompting and retrieval techniques or supervised training enough for device control? Delving into Table 1, we observe that off-the-shelf proprietary VLMs, even when supplemented with the set-of-marks mechanism, do not attain satisfactory performance: both GPT-4V and Gemini 1.5 Pro achieve success rates under 20%. One possible cause could be the underrepresentation of Android device data in the pre-training data. Moreover, inference-time adaptation strategies such as AppAgent [47] show minimal improvement, with gains not exceeding 5% for either model. All this evidence suggests a limited scope for improvement without fine-tuning of some sort. As illustrated in Figure 7, the primary failures of these VLMs stem from hallucinatory reasoning that lead the VLMs to land on a relevant but wrong page. This suggests that while state-of-the-art VLMs excel at reasoning problems in code and math, their reliability in less-familiar domains, such 8 Fail to recover from mistakes Get stuck midway Arrive at wrong goal Failure Mode 0.0 0.2 0.4 % in All Trajectories General Fail to recover from mistakes Get stuck midway Arrive at wrong goal Failure Mode 0.0 0.2 0.4 % in All Trajectories Web Shopping Set-Of-Marks GPT4V Set-Of-Marks Gemini-1.5-Pro AppAgent GPT4V AppAgent Gemini-1.5-Pro AutoUI CogAgent Filtered BC Offline DigiRL Offline Filtered BC Online DigiRL Online Figure 7: Failure modes for each approach on both the AiTW General and Web Shopping subsets. We found that the failure mode RL training is most effective at reducing compared to model supervised trained on human data is “Fail to recover from mistakes”. A more fine-grained decomposition can be found in Appendix D. as device control, remains inadequate. For example, for the instruction “Go to newegg.com, search for alienware area 51, and select the first entry”, a GPT-4V based agent erroneously searched “alien area 51 ebay” in Google.com and decided that it had made progress towards the task (Figure 17). Training on domain-specific human demonstrations, however, does boost performance, allowing the smaller, specialized VLM, AutoUI with 1.5 billion parameters, to match or surpass the larger, generalist VLMs like GPT-4V and Gemini 1.5 Pro. Nonetheless, this supervised imitation learning approach still fall short, with success rates on both subsets remaining below 20%. This shortcoming is not fundamentally addressed via enhancements in model scale or architecture, as evidenced by CogAgent [9], with 18 billion parameters still achieving performances below 40% success rate. As depicted in Figure 7, a predominant failure mode for these agents is an inability to rectify their own errors. An example trajectory that we observed is that for the instruction “what’s on the menu of In-n-Out”, the agent accidentally activated the voice input button, and failed to quit that page until the step limit. In contrast, DigiRL is able to recover from the errors more efficiently( Appendix C.2). Comparison of different RL approaches. In Table 1 and Figure 6, we present a comparative analysis of various autonomous approaches. Notably, both offline and offline-to-online configurations demonstrate that our RL approach, when augmented with a continuous stream of autonomous interaction data and reward feedback, substantially improves performance. This improvement is evident from an increase in the success rate from under 20% to over 40%, as the agent learns to adapt to stochastic and non-stationary device interfaces. Moreover, although the total sample sizes for offline and offline-to-online settings are equivalent, the top-performing offline-to-online algorithm markedly surpasses its offline counterpart (75% versus 62.8% on the General subset). This highlights the efficacy of autonomous environment interaction, and establishes the efficacy of DigiRL in learning from such uncurated, sub-optimal data. Lastly, DigiRL consistently outperforms the state-of-the-art alternative, Filtered BC, across both the General and Web Shopping subsets, improving from 61.5% to 71.9% and 57.8% to 61.4%, respectively, highlighting DigiRL’s performance and efficiency. 5.2 Analysis and Ablations Failure modes analysis. We conduct an additional user study to annotate the failure modes for each agent as shown in Figure 7, and a more fine-grained breakdown can be found in Appendix D. At a high level, we classify the major failure modes of all agents into the following three categories: (1) Failure to recover from mistakes refers to the scenario where the agent made a mistake that led it to states from which it failed to quickly recover and resume the task, such as a wrong search page. (2) Getting stuck midway refers to the failure mode where the agent gets distracted on the right track to completing the instruction and as a result fails to accomplish the task. For example, failing to click on the right link or failing to search after typing the key words. (3) Arriving at wrong goal refers to the failure mode where the agent arrives at a wrong page and mistakenly thinks that it had completed the task. For e.g, the agent finds a macbook on costco.com instead of finding a macbook on ebay.com. While all the types of failure modes benefit from offline and offline-to-online RL training as shown in Figure 7, the most consistent and significant reduction is probably for the failure mode of failing to recover from mistakes. This is because while pre-trained models, generating plausible future tokens, can get distracted by the dynamic nature of the environment and, as a result, encounter at never-before-seen states. With no clue of how to escape such states, these methods are unable to recover and fail to solve the task. In contrast, by training on autonomously-collected rollouts, our agent DigiRL is able to learn from its own mistakes and reduces failures to recover over training. 9 Set-Of-Marks GPT4V Set-of-Marks Gemini-1.5-Pro AppAgent GPT4V AppAgent Gemini-1.5-Pro AutoUI CogAgent Filtered BC Offline DigiRL Offline Filtered BC Online DigiRL Online Policy Model 0 50 % Success Rate 17.7 13.5 16.7 16.7 15.6 17.7 18.8 16.7 12.5 14.6 25.0 25.0 55.2 53.1 56.3 63.5 59.4 61.5 70.0 72.9 General Human Gemini-1.5-Pro Evaluator Set-Of-Marks GPT4V Set-Of-Marks Gemini-1.5-Pro AppAgent GPT4V AppAgent Gemini-1.5-Pro AutoUI CogAgent Filtered BC Offline DigiRL Offline Filtered BC Online DigiRL Online Policy Model 0 50 % Success Rate 11.4 8.3 15.6 11.5 13.5 8.3 13.5 5.2 18.8 17.7 42.6 38.5 45.8 46.7 57.3 55.2 61.5 60.4 68.8 71.9 Web Shopping Human Gemini-1.5-Pro Evaluator Figure 8: Correlation between our autonomous evaluator and human judgements for all policy models on General and Web Shopping subsets. For repeated offline and online runs, we report the correlation results for the run with the highest autonomous evaluation success rate. Ablation study of each component in DigiRL. We conduct an ablation study on different components of DigiRL in Figure 9. We find that all the components used by our approach are necessary: (1) using cross-entropy for training the value functions boosts performance by around 12% (compare Ours and Ours w/ Regression); (2) using step-level advantages improves efficiency by 12% (comparing Ours and Ours w/o step-level advantage); (3) the use of automatic curriculum improves the speed of learning by around 25% (comparing Ours w/o step-level advantage and Filtered BC); (4) Ours outperforms vanilla AWR that does not employ a doubly-robust advantage estimator or curriculum. 0 100 200 300 400 500 600 #Trajectories 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Success Rate Ours Ours w/ regression Ours w/o step-level advantage Vanilla AWR Ours w/ AWR reweighting Filtered BC Figure 9: Ablation study results on the AitW Web Shopping subset. Additionally, we also observe no degradation in performance as a result of “hard-filtering”, as show by nearly comparable performance of our approach and the best run of exponential filtering obtained via an extensive tuning of the temperature hyperparameter τ in naïve AWR (comparing Ours and Ours w/ vanilla AWR reweighting), despite simplicity of implementation in the hard filtering approach. Putting together, these choices result in a new state-of-the-art RL approach for device control. Evaluation of our autonomous evaluator. In Figure 8, we present the findings from a user study aimed at assessing the accuracy of our autonomous evaluator. Our results indicate that the success rates reported by our automatic evaluator are remarkably consistent with those assessed by human evaluators across almost all models, with differences less than 3%. Furthermore, we observed that evaluations on the Web Shopping subset are more precise compared to those on the General subset. This increased accuracy likely stems from the fact that tasks in the General subset are formulated in free-form language, which can introduce ambiguity, whereas the Web Shopping subset features a narrower range of language expressions, reducing potential variability. 6 Discussion and Limitations In this paper, we propose a novel autonomous RL approach, DigiRL, for training in-the-wild, multimodal, device-control agents that establish a new state-of-the-art performance on a number of Android control tasks from Android-in-the-Wild dataset [31]. To achieve this, we first build a scalable and parallelizable Android environment with a robust VLM-based general-purpose evaluator that supports fast online data collection. We then develop a system for offline RL pre-training, followed by autonomous RL fine-tuning to learn via interaction, admist the stochasticity of the real-world Internet and device eco-system. Our agent achieves a 280% improvement over the previous state-of-the-art agents (from 17.7% to 68.2% in terms of task success rate), including AppAgent based on GPT-4V and Gemini 1.5 Pro, and supervised trained models such as AutoUI and CogAgent. Due to computational limitations, and despite the fact that the parallel emulator and autonomous evaluator can be easily extended to complicated tasks, our agent is trained only with tasks from AitW instead of a all possible tasks on the device. Our design of the DigiRL algorithm aims for maximal implementation simplicity, so we hope that our approach to serve as a base algorithm for future research to build on, including algorithmic research as well as expanding the space of tasks. 10 Acknowledgements We thank Yi Su, Izzedin Gur, Xinyang Geng, and Sandra Faust for feedback on an earlier version of this paper and for informative discussions. 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Install and GoogleApps are not considered due to security reasons as those tasks require an active Google account and parallel emulations can flag security concerns. General. The General set focuses on searching for information and basic application usage. For example, it contains searching for latest news in Chile, search for flights from NYC to Sydney, opening Gmail, etc. We use all 545 tasks in the training set for training and the first 96 tasks in the test set for testing due to computational and budget constraints. The maximum allowed number of steps for this subset is 10. Offline data is collected by rolling our the initial AutoUI policy with tasks from the training set. The offline data used for the offline-to-online setting contains 608 trajectories while the offline data used for the offline setting contains 1552 trajectories. Some task examples are shown in Table 3. Task Example How do I get to the nearest Verizon Store? How much does a 2 bedroom apartment rent for in Denver? Search for flights from Barcelona to Boston What’s a good restaurant in New York? What’s on the menu at Burger King? Table 2: Examples of task descriptions in the AiTW General task set. Web Shopping. The Web Shopping subset comprises search instructions on various shopping websites, like searching for razer blader on ebay. As some websites (e.g. Amazon) and operations (e.g. adding items to cart) frequently require captcha verifications, we post-process the Web Shopping subset to exclude such operations and websites and also make the task easy to evaluate for our autonomous evaluator. The resulting task set involves navigating through five websites (costco.com, bestbuy.com, target.com, walmart.com, newegg.com) and three basic operations (go to website, search in the website, and select items from the searched results). Our post-processed training set contains 438 tasks and our testing set contains 96 tasks. Example tasks after post-processing can be found in Table 3. The maximum allowed number of steps for this subset is 20. Offline data is collected by rolling our the initial AutoUI policy with tasks from the training set. The offline data used for the offline-to-online setting contains 528 trajectories while the offline data used for the offline setting contains 1296 trajectories. Difficulty Task Example 1 Go to costco.com Go to walmart.com 2 Go to costco.com, search for "bose soundsport free" Go to walmart.com, search for "logitech g910" 3 Go to costco.com, search for "bose soundsport free" and select the first entry Go to walmart.com, search for "logitech g910" and select the first entry Table 3: Examples of task descriptions in the AiTW Webshopping task set. 15 AitW General AitW Web Shopping All Trajectories Successful Trajectories All Trajectories Successful Trajectories DigiRL Run1 6.31 4.40 11.35 7.23 DigiRL Run2 6.64 5.04 10.86 6.55 Filtered BC Run1 8.08 6.56 12.05 6.88 Filtered BC Run2 7.36 6.13 14.72 9.62 Table 4: Average rollout length of the DigiRL agent compared to filtered BC. Darker green means shorter rollout length. On both AitW General and AitW Web Shopping test subsets, we find that DigiRL consistently produces shorter length rollouts than filtered BC. B Other Quantitative Experiments B.1 Curriculum Learning When running experiments on the AitW Web Shopping subset, we find solving easier tasks helps improve solving harder tasks, where the difficulty is identified in Table 3. By specifying the difficulty DigiRL-Run1 in Figure 6, we empirically show the success rates of each difficulty across the online learning process in Figure 12, we observe that a significant increase of success rate of tasks of difficulty 1 leads to increasing success rate of difficulty 2, and the same pattern for difficulty 2 and 3, demonstrating effective curriculum learning. 0 200 400 600 800 #Trajectories 0.0 0.2 0.4 0.6 0.8 1.0 #Success Rate Difficulty 1 Difficulty 2 Difficulty 3 0 200 400 600 800 #Trajectories 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Success Rate Filtered BC-20 Filtered BC-10 DigiRL-20 DigiRL-10 GPT-4V Figure 10: Left: Success rate under different difficulties for the AiTW Webshopping task set. Right: Success rate under different methods with different horizon length (H ∈{10, 20}) on the AiTW Google Search task set. B.2 Learning Method We ablate on the learning method, i.e. online learning or offline-to-online learning. We find that offline-to-online learning converges faster than online learning, and is not necessarily worse than online learning in terms of final performance, as shown in Figure 11. B.3 Horizon Limit We investigate the horizon limit of filtered BC and DigiRL on the AitW General subset. As most tasks can be effectively solved within 10 steps, we specify two horizon limits: a sufficient horizont H = 10, and a redundant horizon H = 20. Results in Figure 12 show that a redundant horizon introduces significantly faster learning speed for both filtered BC and DigiRL, presumbaly because longer horizon means more opportunity to try in a single trajectory. In both horizon settings, we observe the DigiRL offers a significant speedup of around 100 trajectories over Filtered BC. B.4 Trajectory Length We investigate the rollout length of DigiRL compared to filtered BC. Results in Table 4 demonstrate that DigiRL consistently achieves shorter average rollout lengths compared to filtered BC across both 16 0 200 400 600 800 #Trajectories 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Success Rate Online Run1 Online Run2 Off-On Run1 Off-On Run1 Figure 11: Success rate with pure online learning or offline-to-online learning w.r.t. the number of online trajectories trained on the AitW General dataset. The starting points of curves in this figure look different from the main results figure because the starting points of the main results figure is smoothed at the average performance of the offline trajectories collected for the offline-to-online learning. 0 200 400 600 800 #Trajectories 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Success Rate Filtered BC-20 Filtered BC-10 DigiRL-20 DigiRL-10 GPT-4V Figure 12: Success rate with different horizon length (H ∈{10, 20})under different methods on the AiTW Google Search task set. subsets. This observation holds true whether considering all rollouts for computing this correlation or only investigating this correlation on rollouts that eventually succeed. This indicates the capability of DigiRL to solve tasks in a more efficient and directed manner. Qualitative examples can be found in Figure 16. C Qualitative Examples C.1 Random sample of trajectories for different agents In Figures 13 and 14, we provide trajectories of DigiRL, AutoUI, and GPT-4V randomly sampled from our test set to offer a qualitative understanding of the agents’ performance. As shown in these examples, DigiRLcan efficiently carry out in-the-wild device control tasks and less likely to get stuck or get to a wrong page compared to AutoUI and GPT-4V. C.2 Error Recovery We observe that DigiRL is able to recover from its own mistakes. As shown in Figure 15, we find that DigiRL explores ways to get back to the original screen in order to perform a search. As a comparison, AutoUI fails to reset to the original screen and gets stuck at the diverged screen. Under the hood, we find DigiRL trying to maximize the state value, which usually induces it to reset to the original screen (that has a large value to success). 17 DigiRL: AutoUI: GPT-4V What are the new products by Samsung?   Got stuck ✘ Click Show me some nice wallpapers for my tablet   DigiRL: AutoUI: GPT-4V Skipped Stops Early ✘ Figure 13: Agents’ trajectory on two randomly sampled tasks on the General split of AitW. 18 Go to costco.com, search for 'macbook pro', and select the first entry   DigiRL: AutoUI: GPT-4V Early stop ✘ Got stuck ✘ ✘ Got stuck Go to newegg.com, search for 'duracell triple a’ DigiRL: AutoUI: GPT-4V Skipped Skipped Skipped Wrong Page ✘ ✘ Could not search Figure 14: Agents’ trajectory on two randomly sampled tasks on the WebShop split of AitW. 19 Go to bestbuy.com, search for 'macbook' DigiRL: AutoUI: Skipped Skipped ✘ Got Stuck Figure 15: Error recovery cases. In bestbuy.com, we systematically find DigiRL able to recover from its own mistakes, while AutoUI fails to do so. C.3 Trajectory Length Qualitative example on the number of steps in trajectories of DigiRL and filtered BC are shown in Figure 16. We find consistent cases where DigiRL has shorter trajectory length than filtere BC. C.4 Reasoning failure of GPT-4V The performance of GPT-4V failed on AiTW tasks predominantly due to not being able to carry out control actions as it plans on a high level, and then not being able to recover from these mistakes. Moreover, one of the main reasons why it is not able to recover from a mistake is that it might hallucinate and make itself believe that it is a wrong app or website. Indeed, GPT-4V constructs a plan of further actions when provided a task from either Web Shopping or General dataset of AiTW. Then, when it makes a misclick and fails to successfully proceed in an intermediate step, it might think that it actually solved that intermediate step and is in the correct app or website to execute further actions, causing the overall trajectory to fail. An example of this is provided in Figure 17. Here, we ask the model to search for an item in a webshopping website, in particular in “newegg.com”. However, the model fails to proceed to that website due to not being able to precisely locating the search button. Then, instead of trying to go to that website again, the model thinks it is already in that webshopping website, and mistakes the search bar of Google with the search bar of “newegg.com”. Hence, the rest of the trajectory also fails. Another slightly different phenomenon is illustrated in Figure 18. Here, the model is able to proceed to the correct website and search for an item, but this time it fails to tap on the search button on the website and clicks to an advertisement instead. Consequently, the model fools itself to think it successfully searched the item, and scrolls the page hoping to find that item, but it cannot do so because in reality it views the results of the advertisement. The primary reason of these failures is the challenge of grounding the control actions in GUI interfaces to realize the intermediary goals laid out by GPT-4V model’s thoughts. As an example, we provide an illustration of trying to set up an alarm task in Figure 19. Here, in the last frame, it fails to execute the precise movements in the necessary amount of rounds to correctly set up the alarm to the desired time, and in the last frame we see that the action taken does not align with the thought process of the model. 20 Go to ebay.com, search for "lenovo thinkpad" DigiRL Filtered BC Search for flights from Seoul to Mexico city DigiRL Filtered BC Figure 16: Examples where DigiRL has shorter trajectory length than online filtered BC. D Fine-grained failure modes In Figure 20, we present a more fine-grained breakdown for all six failure modes provided in the user study. Those failure modes include: • Failure to recover from mistakes refers to the scenario where the agent made a mistake that led it to states from which it failed to quickly recover and resume the task, such as a wrong google search page. • Failure to click on the right link or failure to click refers to the failure mode where the agent either fails to locate the element that it tries to click on and keeps clicking on the nearby region, or fails to start typing in the string when it is supposed to do so. • Failure to take reasonable attempts at all refers to the failure mode where there is no clear reason that the agent fails to complete the task and does not seem to be on the right track throughout the trajectory. • Quit or press HOME early refers to the failure mode where the agent decided to finish the task or press HOME to start over before the task is actually finished. • Stops at wrong but relevant page refers to the failure mode where the agent arrives at a wrong page and mistakenly thinks that it had completed the task. For example, the agent finds a macbook on costco.com while the instruction asked it to find a macbook on ebay.com. • Technical issues refer to the failure mode that either the task is impossible (e.g. the tasks asks to open Amazon app but this app is not installed) or the agent is temporarily blocked from a certain website due to frequent visits. The translation between fine-grained failure modes and coarse-grained failure modes is presented in Table 5. 21 Go to newegg.com, search for “alienware area 51” and select the first entry. ... Thought: To complete the given task, I should launch the Google Chrome browser to navigate to newegg.com. The UI element labeled with the number 5 is documented as an app launcher for Google Chrome. Action: tap(5) Thought: To complete the given task, the next step I should do is enter the URL "newegg.com" into the address bar. Action: text("newegg.com") Thought: To complete the given task, the next step I should do is to navigate to the Newegg website by tapping the search or go button associated with the entered URL. Action: tap(11) Thought: To complete the given task, I should clear the current text in the address bar and enter the search term 'alienware area 51' since the goal is to search for that specific product on newegg.com. Action: tap(2) Thought: To complete the given task, the next step I should do is clear the current text in the address bar and enter the search term 'alienware area 51' Action: text("alienware area 51") since the browser has already navigated to the Newegg website. Thought: To complete the given task, the next step I should do is to submit the search query 'alienware area 51' Since the search term is already entered, I need to tap the search or go button to initiate the search. on the Newegg website. Action: tap(10) Figure 17: Failure of GPT-4V, with its thoughts and link-based actions given. A typical cause of failure is that it cannot tap on the correct “search” button after entering a query and mistakenly tapped onto the “x” symbol in the search bar as the “search” button. Here the goal is: Go to newegg.com, search for “alienware area 51” and select the first entry. As seen in red emboldened actions, it fails to press search button and deletes the query instead. Also, as seen in red highlighted parts in thoughts, it thinks it is in “newegg.com” website even though it is not. Thought: To complete the given task, the next step I should do is type "costco.com" into the search bar since I have already initiated the process to navigate to costco.com by tapping on the Google search bar. Action: text("costco.com") Thought: To complete the given task, the next step I should do is enter the search term 'acer predator' into the search bar using the text input function. Action: text("acer predator") Thought: To complete the given task, Action: I should continue to swipe up on the product listings to find the 'acer predator' item. swipe(21, "up", "medium") Thought: To complete the given task, I should scroll through the product listings to find the 'acer predator' item. Since the first entry is what we need to select, Action: I should swipe up on the element labeled '26' to view more products. swipe(26, "up", "medium") Thought: To complete the given task, the next step I should do is to initiate the search by tapping the search button on the keyboard. Action: tap(24) Thought: To complete the given task, the next step I should do is navigate to the Costco website. Since the address "costco.com" is already entered in the address bar, I need to initiate the web search or load the page by tapping the search button on the keyboard. Action: tap(4) Go to costco.com, search for “acer predator”, and select the first entry. ... Figure 18: Failure of GPT-4V, with its thoughts and link-based actions given. This time the reason for failure is misclick on the wrong button. The task is “Go to costco.com, search for “acer predator”, and select the first entry”. Notice that up until the fourth frame in this Figure, the trajectory goes correct. But then it clicks on the generic advertisements on the Costco.com website, and it cannot recover back. It continues to scroll the page and takes wrong actions thereafter. 22 Set an alarm for 4pm. Thought: To set an alarm for 4pm, I need to find and open the alarm or clock app. Since the alarm app icon is not immediately visible on the current screen, I may need to swipe up or tap on the app drawer icon (if present) to view more apps and locate the alarm or clock app. Action: swipe(1, "up", "medium") Thought: To set an alarm for 4pm, I need to open the Clock app. The next step is to tap on the Clock app icon. Action: tap(25) Thought: To set an alarm for 4pm, I need to navigate to the alarm section of the Clock app if not already there, and then use the button to add a new alarm. Action: tap(1) Thought: To set a new alarm for 4 PM, I should tap the button to add a new alarm. Action: tap(14) Thought: To complete the given task, I need to set the alarm for 4:00 PM. The hour is close to the desired time, so I should first switch from AM to PM using the PM button. Action: tap(4) Thought: To complete the given task, I should set the minutes to "00" to make the alarm for 4:00 PM. Action: tap(1) Figure 19: Failure of GPT-4V, with an example task on the AiTW general test set. The task is “Set an alarm for 4pm”. Here, GPT-4V is able to successfully navigate to the clock app, and the alarm settings of that app. However, it cannot take the correct precise actions to set the alarm quickly enough, and it fails due to maximum rounds reached. In the last round, notice that the action of tap(1) contradict with its own thought process of setting minutes to “00”. Set-Of-Marks GPT4V Set-Of-Marks Gemini-1.5-Pro AppAgent GPT4V AppAgent Gemini-1.5-Pro AutoUI CogAgent Filtered BC Offline DigiRL Offline Filtered BC Online DigiRL Online Set-Of-Marks GPT4V Set-Of-Marks Gemini-1.5-Pro AppAgent GPT4V AppAgent Gemini-1.5-Pro AutoUI CogAgent Filtered BC Offline DigiRL Offline Filtered BC Online DigiRL Online General Web Shopping Fail to recover from mistakes Fail to click on the right link or fail to type Fail to take reasonable attempts at all Quit or press HOME early Stops at wrong but relevant page Technical issues Task success Figure 20: Failure modes decomposition for each policy model for both General and Web Shopping subsets. Fine-Grained Failure Coarse-Grained Failure Fail to recover from mistakes Fail to recover from mistakes Fail to click on the right link or fail to type Get stuck midway Fail to take reasonable attempts at all Get stuck midway Quit or Press HOME early Arrive at wrong goal Stops at wrong but relevant page Arrive at wrong goal Technical Issues None Table 5: Examples of task descriptions in the AiTW Webshopping task set. 23 ost machine worker machines emulators aggregate trajectories distribute updated policy Figure 21: Multi-machine parallel emulator execution. The host machine is equipped with GPU accelerators and the worker machines are equipped only with CPUs. The policy update is executed on the worker machine and the trajectory collections are executed distributedly on the worker machines and aggregated by the host machine. E Experiment machines Our main experiments are conducted on VM instances from Google Cloud Platform. Each VM instance comes with 1x Tesla T4 GPU and 16x Intel(R) Xeon(R) CPU. F Setup for parallel environment Running multiple emulators in parallel can be challenging due to the inefficiency in thread synchronization and frequent fault propagation when one emulator runs into an unknown error. To address this challenge, we set up a server-client system where all emulator processes are running in independent server processes. Each emulator process communicates with the main training process through different UIAutomotor servers. The main training process sends high-level instructions to UIAutomotor servers (such as reset and step), while UIAutomotor servers parse high-level instructions into low-level UI commands (such as typing a character and tapping at a coordinate) and such UI commands are executed by the emulator processes. When an exception is thrown in the emulator, the UIAutomotor examines if it is recoverable (e.g. an UI command takes too long to execute in the emulator) and reset the emulator process if it is not. When an exception is thrown in the UIAutomotor server, the main training process stops and resets the UIAutomotor server to ensure data correctness. This design can easily be scaled up to a multi-machine setting. As illustrated in Figure 21, one host machine equipped with GPU accelerator has a local copy of the current policy πt, and distributes the policy to all worker machines equipped with only one GPU and multiple CPUs. Each worker machine will then collect trajectories of different tasks using πt. After all collection processes are synchronized, the host machine gathers all the trajectories together to update the policy to πt+1. This process keeps iterating until the policy converges. Speedup of emulation parallel. The performance boost with respect to the number of worker machines is nearly linear, as demonstrated in Figure 22 (right), where we conduct experiments that examine the scaling performance of our parallel emulator. Our distributed emulator that runs emulations across multiple servers can reliably collect data with up to 64 parallel emulators on 128 CPUs with near-linear speedup. In contrast, a naive baseline that runs all parallel emulations on the same server achieves much inferior performance (0.74 compared to 1.74 trajs/min using 64 CPUs). G Autonomous evaluator details Our autonomous evaluator gives a reward to each observation we get. The observation is composed of the current screenshot of device and the task. The evaluator gives a reward of 1 if the screenshot shows a completion of the task, and will terminate the POMDP as a result result. The optimized prompt is shown in Figure 23 and Figure 24 for General and Web Shopping subsets respectively. 24 8 16 32 64 128 #CPUs 0 1 2 3 4 5 Emulation Speed (traj/min) 0.36 0.53 0.68 0.74 0.49 0.99 1.74 3.55 Vanilla Emulator Distributed Emulator Upper Bound Figure 22: Emulation speed w.r.t number of CPUs used. The upper bound can only achieved when there is no communication and error handling cost. Our design of distributed emulator can significantly improve the efficiency of emulation compaared to the vanilla method of running all emulations over the same instance. H Zero-shot Baseline Details Figure 25 shows the prompt that we used for testing the Set-of-Marks performance for GPT-4V and Gemini 1.5 Pro. This prompt is directly taken from Yang et al. [47]. I Hyperparameters Hyperparameters for both Filtered BC and DigiRL are carefully tuned through binary search on the training set of General and Web Shopping subsets. The final choice of hyperparameters for both methods can be found in Table 6. As shown in the table, the only hyperparameters introduced by DigiRL are supervised training hyperparameters for the value function and instruction value function (including number of iterations and learning rate) and GAE λ. 25 Prompt You're an expert in evaluating whether the Screenshot successfully completes the Task. =====Examples===== Screenshot: {train_1.png} Task: Open the settings. Q: What should I expect to see on the screenshot if I've opened the settings? A: I should expect to see I'm in the settings app. The screenshot shows the home screen of a mobile device, with various app icons displayed, including the settings app icon, but the settings app is not opened. Status: failure Screenshot: {train_2.png} Task: Find hotels in washington dc Q: What should I expect to see on the screenshot if I've searched for hotels in Washington, DC? A: I should expect to see I'm in a search results page for hotels in Washington, DC. The screenshot shows a Google search page with the search field populated with the query "hotels in washington dc" and a list of suggested searches related to hotels in Washington, DC, but it does not show any search results for hotels in Washington, DC. Status: failure Screenshot: {train_3.png} Task: What's a good restaurant in Portland? Q: What should I expect to see on the screenshot if I've searched for a good restaurant in Portland? A: I should expect to see I'm in a search results page for a good restaurant in Portland. The screenshot shows a Google search page with a search input field for "good restaurant in portland" and a map results preview showing business locations near Portland, like "Li Pigeon", "Portland City Grill", and "Higgins", Status: success ... (more cases) =====Your Turn===== Screenshot: {test.png} Task: {task_this_traj} Respond in this format: Q: What should I expect to see on the screenshot if I've <repeat the task>? A: I should expect to see <first expectation, then what's in the given screenshot.> Status: success or failure (don't return anything else) Start with "Q:". Response Q: What should I expect to see on the screenshot if I've searched for the price of a 12' ladder at Home Depot?  A: I should expect to see the price of a 12' ladder at Home Depot; the screenshot shows a search result page for the p rice of a 12' ladder, with some product advertisements showing prices from Home Depot.  Status: success Image Sources train_1.png train_2.png train_3.png test.png Figure 23: Prompt for our autonomous evaluator for tasks in AitW General subset. 26 Prompt You're an expert in evaluating whether the Screenshot successfully completes the Task. =====Examples===== Screenshot: {train_1.png} Task: Go to bestbuy.com Q: What should I expect to see on the screenshot if I've gone to bestbuy.com? A: I should expect to see I'm in the Best Buy website, which usually shows the best buy logo with some featured prod ucts and categories. The screenshot shows I'm searching for "bestbuy.com" in the Google search (with some search s uggestions) instead of being in the Best Buy website. Status: failure Screenshot: {train_2.png} Task: Go to ebay.com, search for "corsair k70" Q: What should I expect to see on the screenshot if I've gone to ebay.com AND searched for "corsair k70"? A: I should expect to see I'm in the eBay website and search results for "corsair k70". The screenshot shows I'm in the eBay ebay website with some search suggestions for "corsair k70", but it does not show search results of the product, which usually includes price and the product details. Status: failure Screenshot: {train_3.png} Task: Go to ebay.com, search for "lenovo thinkpad" Q: What should I expect to see on the screenshot if I've gone to ebay.com AND searched for "lenovo thinkpad"? A: I should expect to see I'm in the eBay website and search results for "lenovo thinkpad". The screenshot shows I'm in the eBay website and have several search results for "lenovo thinkpad". Status: success ... (more cases) =====Your Turn===== Screenshot: {test.png} Task: {task_this_traj} Respond in this format: Q: What should I expect to see on the screenshot if I've <repeat the task>? A: I should expect to see <first expectation, then what's in the given screenshot.> Status: success or failure (don't return anything else) Start with "Q:". Response Q: What should I expect to see on the screenshot if I've searched for the price of a 12' ladder at Home Depot?  A: I should expect to see the price of a 12' ladder at Home Depot; the screenshot shows a search result page for the p rice of a 12' ladder, with some product advertisements showing prices from Home Depot.  Status: success Image Sources train_1.png train_2.png train_3.png test.png Figure 24: Prompt for our autonomous evaluator for tasks in AitW Web Shopping subset. 27 Prompt "You are an agent that is trained to perform some basic tasks on a smartphone. You will be given a \nsmartphone screenshot. The interactive UI elements on the screenshot are labeled with numeric tags starting from 1. The \nnumeric tag of each interactive element is located in the center of the element.\n\nYou can call the following functions to control the smartphone:\n\n1. tap(element: int)\nThis function is used to tap an UI element shown on the smartphone screen.\n\"element\" is a numeric tag assigned to an UI element shown on the smartphone screen. \nA simple use case can be tap(5), which taps the UI element labeled with the number 5.\n\n2. text(text_input: str)\nThis function is used to insert text input in an input field/box. text_input is the string you want to insert and must \nbe wrapped with double quotation marks. A simple use case can be text(\"Hello, world!\"), which inserts the string \n\"Hello, world!\" into the input area on the smartphone screen. This function is usually callable when you see a keyboard \nshowing in the lower half of the screen.\n\n3. long_press(element: int)\nThis function is used to long press an UI element shown on the smartphone screen.\n\"element\" is a numeric tag assigned to an UI element shown on the smartphone screen.\nA simple use case can be long_press(5), which long presses the UI element labeled with the number 5.\n\n4. swipe(element: int, direction: str, dist: str)\nThis function is used to swipe an UI element shown on the smartphone screen, usually a scroll view or a slide bar.\n\"element\" is a numeric tag assigned to an UI element shown on the smartphone screen. \"direction\" is a string that \nrepresents one of the four directions: up, down, left, right. \"direction\" must be wrapped with double quotation \nmarks. \"dist\" determines the distance of the swipe and can be one of the three options: short, medium, long. You should \nchoose the appropriate distance option according to your need.\nA simple use case can be swipe(21, \"up\", \"medium\"), which swipes up the UI element labeled with the number 21 for a \nmedium distance.\n\n5. grid()\nYou should call this function when you find the element you want to interact with is not labeled with a numeric tag and \nother elements with numeric tags cannot help with the task. The function will bring up a grid overlay to divide the \nsmartphone screen into small areas and this will give you more freedom to choose any part of the screen to tap, long \npress, or swipe. The task you need to complete is to How much does a 2 bedroom apartment rent for in Denver?. Your past actions to proceed with this task are summarized as follows: None Now, given the documentation and the following labeled screenshot, you need to think and call the function needed to proceed with the task. Your output should include three parts in the given format: Observation: <Describe what you observe in the image> Thought: <To complete the given task, what is the next step I should do> Action: <The function call with the correct parameters to proceed with the task. When you are certain that the task is successfully done and the goal is reached as of the current observation, you should output FINISH. You cannot output anything else except a function call or FINISH \nin this field.> Summary: <Summarize your past actions along with your latest action in one or two sentences. Do not include the numeric \ntag in your summary>\nYou can only take one action at a time, so please directly call the function." Figure 25: Set-of-Marks prompting. The boldened inputs can be changed according to our goal. The task changes for every different task. The past actions change as we take actions (it is None now since this is the prompt for the first round). NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The main claims in the abstract and introduction explicitly state the contributions of the paper. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. 28 Table 6: Hyperparameters for All Experiments Method Hyperparameter Offline Offline-to-Online Filtered BC actor lr 3e-3 3e-3 batch size 128 128 rollout trajectories 16 replay buffer size 5000 rollout temperature 1.0 maximum gradient norm 0.01 0.01 actor updates per iteration 20 20 number of iterations for offline actor updates 10 10 DigiRL actor lr 3e-3 3e-3 value function lr 3e-3 3e-3 instruction value function lr 3e-3 3e-3 instruction value function lr 3e-3 3e-3 batch size 128 128 rollout trajectories 16 replay buffer size 5000 rollout temperature 1.0 maximum gradient norm 0.01 0.01 GAE λ 0.5 0.5 actor updates per iteration 20 20 value function updates per iteration 5 5 instruction value function updates per iteration 5 number of iterations for offline actor updates 10 10 number of iterations for offline value function updates 20 20 number of iterations for offline instruction value function updates 20 Table 7: Hyperparameters for DigiRL and Filtered BC on both General and Web Shopping subset of AitW.. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 2. 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4,425	BoostAdapter: Improving Vision-Language Test-Time Adaptation via Regional Bootstrapping Taolin Zhang1 Jinpeng Wang 1 Hang Guo 1 Tao Dai∗2 Bin Chen 3 Shu-tao Xia 1,4 1 Tsinghua University 2 Shenzhen University 3 Harbin Institute of Technology 4 PengCheng Laboratory https://github.com/taolinzhang/BoostAdapter Abstract Adaptation of pretrained vision-language models such as CLIP to various downstream tasks have raised great interest in recent researches. Previous works have proposed a variety of test-time adaptation (TTA) methods to achieve strong generalization without any knowledge of the target domain. However, existing trainingrequired TTA approaches like TPT necessitate entropy minimization that involves large computational overhead, while training-free methods like TDA overlook the potential for information mining from the test samples themselves. In this paper, we break down the design of existing popular training-required and training-free TTA methods and bridge the gap between them within our framework. Specifically, we maintain a light-weight key-value memory for feature retrieval from instance-agnostic historical samples and instance-aware boosting samples. The historical samples are filtered from the testing data stream and serve to extract useful information from the target distribution, while the boosting samples are drawn from regional bootstrapping and capture the knowledge of the test sample itself. We theoretically justify the rationality behind our method and empirically verify its effectiveness on both the out-of-distribution and the cross-domain datasets, showcasing its applicability in real-world situations. 1 Introduction Vision Language models [49, 16, 23–25, 7] have shown incredible performance in downstream vision tasks [1], such as classification [29, 55, 54, 8], generation [20, 38, 9] and recognition [46, 47]. Among these models, CLIP [36] has been trained with large-scale noisy image-text pairs and can generalize well in zero-shot recognition tasks. The key idea behind CLIP is modality alignment during training and similarity comparison during testing for classification. However, CLIP suffers from domain shift problems during test-time inference. In the presence of out-of-distribution issues [27, 43, 12] that commonly appear in real-world scenarios, CLIP may fail to effectively align the feature across modalities, leading to performance degradation. Test-time adaptation (TTA) has been widely explored in recent approaches [43, 15, 41, 17] to mitigate misalignment issues and improve performance in downstream tasks. Current mainstream TTA methods can be divided into training-required methods and training-free methods, as depicted in Figure. 1a and Figure. 1b. Training-required approaches [43, 41, 39] adjust model parameters or learnable prompts based on self-supervised objectives like entropy and increase the prediction confidence of model for distribution adaptation. TPT [41] applies entropy minimization to the vision-language model first. Furthermore, inspired by consistency regularization, TPT performs information mining ∗Correspongding author: Tao Dai (daitao.edu@gmail.com) 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Augmented Images + Bird Cat Dog Entropy Minimization Text Encoder Image Encoder Learnable Prompts Images + Bird Cat Dog Feature Retrieval Text Encoder Image Encoder Historical Cache Hand-crafted Prompt (a) Training-required Test-time Adaptation (b) Training-free Test-time Adaptation (c) Performance comparison on two benchmarks Figure 1: (a) Existing training-required TTA methods utilize self-supervised objective like entropy minimization for better generalization. (b) Existing training-free TTA methods perform feature retrieval on the historical samples to adjust the model prediction. (c) Performance comparison on the Out-of-Distribution benchmark and Cross-Datasets benchmark. from the test sample itself by random regional cropping in a self-bootstrapping style. However, training-required methods require gradient descent that is time-consuming with large training overhead, which prevents them from being applied in computationally limited situations. Training-free approaches [15, 52, 17] utilize memory networks, cache, or prototypes to store information regarding target samples and distributions, which is then used to adaptively modify the model’s prediction. For example, TDA [17] leverages historical samples from the test data steam to build a dynamic key-value cache. It updates the prior knowledge encoded in CLIP through feature retrieval and output prediction based on the similarity between the test sample and the high-quality data stored in the memory bank. However, existing training-free approaches only consider interaction with other historical samples in the cache and do not effectively exploit the information within the test sample itself. This limitation prevents them from performing well especially in tasks that require fine-grained information. Both of these approaches demonstrate excellent performance in enhancing the robustness of visionlanguage models to unknown distributions. However, the connection between them remains unclear. In this paper, we aim to answer three questions: (1) How are training-required methods like TPT and training-free methods like TDA connected? (2) How can we combine these two methods based on their shared nature? (3) Does vision-language models benefit from the combination of these methods? In order to answer these questions, we first consider that the augmented images of test samples form a regional bootstrapping distribution of the original data. By filtering out the noisy augmentations based on mutual information with the predefined CLIP text embedding clusters, we can obtain a boosting distribution from which high-quality samples close to the target clusters can be drawn. Based on this, we delve into the connection between the target operations over the boosting distribution, i.e., crossentropy optimizations and cache classifier, which reveals the shared nature between entropy-based and cache-based methods. Specifically, we pinpoint that with the samples derived from the bootstrapping distribution, entropy minimization over them performs equivalently to feature retrieval from the cache consisting of them. Motivated by this analysis, we propose a brand-new adaptation strategy, dubbed BoostAdapter, to improve training-free adapters by incorporating the samples derived from the boosting distribution to the memory bank. Particularly, the cache in BoostAdapter consists of instance-agnostic historical samples filtered from the test data stream, along with instance-aware boosting samples generated through regional bootstrapping from the sample itself. The interactions between intra-sample and cross-sample operations make BoostAdapter effective and efficient by 2 incorporating the idea of information mining from training-required methods while maintaining the efficiency of training-free methods. Theoretical analyses and empirical results are also provided to validate the effectiveness of BoostAdapter. To summarize, we make the following contributions in this paper. • We first discuss the relationship between training-required and training-free methods in test-time adaptation and establish connections between them. • We propose BoostAdapter, a brand new adaptation strategy in test-time adaptation of visionlanguage models, which improves training-free adapters by introducing high-quality samples from regional bootstrapping into the memory. • We theoretically derive target domain error bound of BoostAdapter and shows that BoostAdapter benefit from incorporating self-bootstrapping data. • Extensive experiments conducted over two benchmark demonstrate the superior performance of BoostAdapter under test-time adaptation settings. 2 Related Works Vision-Language Models have shown remarkable potential in generalization by contrastive pretraining over amounts of text-image pairs [16, 36, 24, 25] . One typical work is CLIP [36], which benefits from the alignment of 400 million curated image-text pairs and predicts the most relevant text description for a given image based on cosine similarity. Adapting CLIP to the downstream applications has attracted much attention and has been widely explored in recent approaches [55, 54, 52, 26, 56, 30]. CoOp [55] introduces learnable prompts [22, 51, 50, 28] and CoCoOp [54] conditions the text prompts on image embedding for better generalization. Maple [18] performs prompting for both vision and language branches and improves the alignment of the embedding between modalities. These approaches have demonstrated significant performance enhancements, but they still require few training data from the target domain. In contrast, we focus on test-time adaptation where there is no information about the target distribution and aim to generalize the model to any unknown scenarios. Training-required Test-time Adaptation updates partial wights of the model like prompts [41, 39] or BN layer [43] with self-supervised objectives that benefit the downstream tasks without requiring additional training data. Tent [43] reduces generalization error on shifted data by testtime entropy minimization. For vision-language models, Test-time prompt tuning (TPT) [41] is a method that dynamically optimizes prompts during the testing phase, enhancing the model’s zero-shot generalization ability. Specifically, TPT generates multiple augmented views of the test sample and then minimizes the entropy of the model’s output logits across them to ensure consistent prediction. Recently, many works built upon TPT have been proposed to further enhance the performance of vision-language models. Particularly, DiffTPT [6] leverages the power of diffusion models to generate semantically consistent augmented images for entropy minimization. PromptAlign [39] bridges the gap between the test sample and source distribution by aligning token statistics, including mean and variance. Nevertheless, these approaches require gradient descent over the augmented images, which is computationally expensive and time-consuming. Training-free Test-time Adaptation applies cache model or prototypes to make prediction of test samples in a non-parametric manner [15, 17, 53]. T3A [15] utilizes prototypes as downstream classifiers and dynamically adjusts the weights. AdaNPC [53] leverages the data from the source domain to address the issues of computation overhead and domain forgetting. For vision-language models, TDA [17] introduces both positive cache and negative cache to obtain high-quality test samples from the target domain. However, these methods only consider inter-sample interactions and may fail to generalize well when the downstream tasks require fine-grained knowledge or there is insufficient similarity across samples. 3 Methodology 3.1 Preliminary Problem setting. We begin by introducing the basic notations in test-time adaptation. We consider binary classification for simplicity and the theory can be easily extended to multi-classifications settings. Let pt(x, y) denotes the joint distribution of image and labels in the target distribution, and 3 Feature Center of the Samples Sample with Different Classes Optimization Direction Test Sample Classfier Distance Classifier Weights (a) Cross Entropy Optimization (b) Cache Classifier Figure 2: Connection between cross-entropy optimization and cache classifier over well-clustered samples with a frozen feature encoder. With optimization of cross-entropy, samples will pull the classifier weights closer of the same class while pushing them away from different class weights. Since the feature space is well-clustered, the classifier weights will ultimately converge near the feature center of the samples. Finally, the optimal classifier achieved through cross-entropy minimization will exhibit similar behavior with the cache classifier. we simply assume that samples {(xi, yi)}n i=1 are drawn i.i.d. from the distribution with yi represents the one-hot label. Definition 1. (Classification error.) Given f as a binary classification function. The error incurred by hypothesis f ∈H : X →{0, 1} under the distribution pt(x, y) can be defined as ϵ(f) = Ept(x,y)[f(x) ̸= y] = Ept(x,y)[\|f(x) −y\|], (1) the last equality holds in a binary classification setting. Definition 2. (Excess error.) Given the Bayes classifier under distribution pt(x): f ∗(x) = I{f(x) ≥ 1/2} and the optimal classfier f ∗, the excess error of f is defined as E(f) = ϵ(f) −ϵ(f ∗) = 2Ex∼pt(x) f(x) −1 2 I{f(x) ̸= f ∗(x)} (2) CLIP classifier Let g be the image encoder of CLIP, C be the feature dimension, N denotes the number of categories, wi ∈RC represents the ith text embedding cluster. Considering normalized embedding w and g(x), we can derive a simplified version of the output of CLIP for class i: Zi = wT i g(x). (3) And we denote the output logits as p(x) = [Z1, Z2, ..., ZN] ∈RN. Cache classifier Given an unseen sample x, encoder g with dimensional C, cache size K and number of categories N, the cache classfier conduct feature retrieval based on the similarity with the data {(xi, yi)}k i=1 in the cache. The predictions based on Tip-Adapter [52] are as follows: pcache(x) = A g(x)GT cache Y, (4) where A(z) = αexp(−β(1−z)) denotes a scaling function with a weighting factor α and a smoothing scalar β, Gcache ∈RK×C represents the feature of K samples {xi}K i=1 in the cache and Y ∈RK×N is the corresponding labels {yi}K i=1. Considering the number of samples in class yi, We can also derive a simplified version of Eq.(4) as follows, by ignoring the scaling function and adopting an instance-wise computation style: pcache(x) = k X i=1 αi g(xi)T g(x) yi, (5) where αi = 1 nyi for class balance or αi = 1 Pk j=1[g(xj)T g(x)] for normalization across all the samples. 3.2 A Closer Look at Entropy-based and Cache-based Methods We start with analyzing the filtering operation of augmentated images in TPT. Pseudo-labels tends to be noisy in the test time, and entropy can serve as a confidence metric to identify trustworthy samples 4 Boosting Cache Image Encoder Text Encoder Augment Hand-crafted Prompt + Text Embedding Bird Cat Dog Boosting Distribution Boosting Samples Historical Cache Filter by Entropy Update Feature Retrieval Historical Samples Image Embedding Text Embedding CLIP logits Cache logits Final logits Figure 3: Overall architecture of BoostAdapter. BoostAdapter leverages knowledge from the target domain and employs self-bootstrapping with historical and boosting samples in the boosting cache, respectively. among augmented views [43, 41, 33]. These high-quality samples can be considered drawn i.i.d. from the so-called boosting distribution as defined below. Definition 3. (Boosting Distribution.) Given a test sample from target distribution x ∼pt(x), let H(·) be the entropy measuring function and Aug(·) be the regional augmentation. By filtering noisy samples based on thresthould τ, we have the following property of boosting distribution pb(x): ˆx ∼pb(x) →{ˆx = Aug(x) ∧H(p(x)) ≤τ} (6) We also terms the samples from the boosting distribution as boosting samples. Then we can connect entropy-based methods and cache classfier by the following proposition: Proposition 1. (Informal) Given n samples {(xi, yi)}n i=1 with a freeze encoder g that effectively performing feature clustering with respect to labels, the gradient descent optimization direction of the classifier’s weights based on cross-entropy generally tends towards making predictions using the cache classifier with class balance weights defined in 5 on these samples. An intuitive illustration of Proposition 1 is depicted in Figure 2, where the weights of optimal classfier behave like the feature centers across different classes with of the well-clusterd samples. Revisiting the entropy-based method TPT, when provided with high-quality boosting samples with low entropy drawn from the boosting distribution, the objective function of entropy minimization optimizes in a manner similar to conducting cross-entropy optimization over the pseudo-labels. According to Proposition 1, TPT performs similarly to the cache-based methods with a cache comprising the same boosting samples from the boosting distribution. 3.3 Boosting your Training-free Adapters Existing cache-based methods store historical test samples only as useful information for prediction. In light of the analysis above, we can integrate the idea behind TPT into these training-free adapters by incorporating boosting samples into the memory bank. In particular, each sample can participate in both inter-sample and intra-sample interactions with the instance-agnostic historical samples and the instance-aware boosting samples in the cache, respectively. Specifically, with kt selected historical samples and kb selected boosting samples to comprise the cache, we extend the classifier defined in Eq.(4) and formulate our BoostAdapter as follows: pboost(x) = A g(x) ˜GT cache ˜Y , (7) where A is the same scaling function defined in Eq.(4), ˜Gcache ∈R(kt+kb)×C denotes the features of the combination of both the historical and boosting samples, and ˜Y ∈R(kt+kb)×N is the label. 5 Since we do not have access to the labels of the test samples, we generate one-hot pseudo-labels for them using argmax operations. However, these pseudo-labels tend to be noisy in the target domain. Therefore, we apply filtering based on entropy thresholds on the test data stream following [41] to obtain trustworthy historical samples. We employ a similar operation to select boosting samples from multiple augmented views of the current sample. In practice, we dynamically adapt the entropy thresholds τ for each test sample, with a fixed percentile p. The cache continuously updates with lower entropy historical samples from the test data stream, while the current test sample augments the cache with self-boosting samples and forms an independent cache that only affects its own prediction. Additionally, to maintain diversity while considering the relevance to each test sample, we set a maximum shot capacity for each class k in the cache. This means that samples in the cache will be replaced by a lower-entropy historical sample or boosting sample when necessary. An important issue is whether introducing boosting samples brings improvements to the trainingfree adapters. We will first make some necessary assumptions and then theoretically verify the effectiveness in reducing target error by incorporating samples from the boosting distribution. Assumption 1. (Strong Density Condition) For any test sample x0 in the target distribution x0 ∼ pt(x) and the boosting distribution pb(x0), given positive lower bound m and upper bound M, positive scaling constant ct and cb, the radius bound R > 0, and B(x, r) = {x′ :∥x′ −x ∥≤r} is the ball centered on x with radius r. We assume pt(x) and pb(x0) are absolutely continuous with respect to the Lebesgue measure in Rd. For r ∈(0, R], we assume        λ[pt(x) ∩B(x0, r)] ≥ctλ[B(x0, r)] λ[pb(x0) ∩B(x0, r)] ≥cbλ[B(x0, r)] m < dpt(x) dλ < M; m < dpb(x) dλ < M, (8) where λ is the Lebesgue measure in Euclidean space. Assumption 2. (L-Lipschitz Condition) Let f be the classification function and L be a positive constant. For all feasible x, x′ we have \|f(x) −f(x′)\| ≤L ∥x −x′ ∥. Assumption 3. (Low Noise Condition). Let β, Cβ be positive constants and we assume pt(x) satisfies Px∼pt(x) f(x) −1 2 < t ≤Cβtβ for all t > 0. Remark Assumption 1 intuitively ensures that for any test sample, there is a surrounding neighborhood with a significant presence of samples from the target domain and the boosting distribution. More importantly, for a specific sample x0, boosting samples x ∼pb(x0) should be closer to x0 than other samples x ∼pt(x) from the target domain, i.e., generally, we have ct ≤cb. Assumption 2 and 3 describe the smoothness of functions and imply a high level of confidence in predictions around the threshold, respectively. Proposition 2. (Historical Cache reduce Emperical Risk) Given f as the training-free classfier consisting of historical samples only defined by Eq.(4). Let nt to be the number of confident previously predicted samples in the target domain and kt as the number of historical samples in the cache, with assumptions 1-3, the following results hold with high-probability for large enough kt and nt. E(f) ≤O 1 kt 1/4 + kt ctnt 1/d!1+β (9) Proposition 3. (Historical Cache benefits from Boosting Samples) Let nt to be all confident previously predicted samples in the target domain and nb be the number of boosting samples that are drawn from the boosting distribution. Given kt and kb to be the number of historical samples and the number of boosting samples to be selected as the nearest neighbors stored in the cache, respectively. Let wti and wbi be the weights defined in Eq.(5) of the historical samples and boosting samples. We have the following bound for the empirical risk of the cache classfier defined in 7. E(f) ≤O 1 kt + kb 1/4 + kt X i=1 wti kt ctnt 1/d + kb X i=1 wbi kb cbnb 1/d!1+β . (10) 6 Table 1: Full results on the OOD benchmark with ViT-B/16 backbone. We report top-1 accuracy and “Average" is calculated by taking the mean accuracy across all four OOD datasets. Imagenet-V2 Imagenet-Sketch Imagenet-A Imagenet-R Average CLIP [36] 60.86 46.09 47.87 73.98 57.20 CLIP+TPT [41] 64.35 47.94 54.77 77.06 60.81 CoOp [55] 64.20 47.99 49.71 75.21 59.28 CoOp+TPT [41] 66.83 49.29 57.95 77.27 62.84 Co-CoOp [54] 64.07 48.75 50.63 76.18 59.91 Co-CoOp+TPT [41] 64.85 48.27 58.47 78.65 62.61 Maple [18] 64.07 49.15 50.90 76.98 60.28 Maple + TPT [41] 64.87 48.16 58.08 78.12 62.31 PromptAlign [39] 65.29 50.23 59.37 79.33 63.55 DiffTPT [6] 65.10 46.80 55.68 75.00 60.52 TDA [17] 64.67 50.54 60.11 80.24 63.89 BoostAdapter 65.51 51.28 64.53 80.95 65.57 Table 2: Full results on the Cross-Domain Benchmark with ViT-B/16 backbone. We report top-1 accuracy and “Average" is calculated by taking the mean accuracy across all ten datasets. The error bound is ±0.17. Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CLIP [36] 93.35 88.25 65.48 67.44 83.65 23.67 62.59 44.27 42.01 65.13 63.58 CLIP+TPT [41] 94.16 87.79 66.87 68.98 84.67 24.78 65.50 47.75 42.44 68.04 65.10 CoOp [55] 93.70 89.14 64.51 68.71 85.30 18.47 64.15 41.92 46.39 66.55 63.88 CoCoOp [54] 93.79 90.46 64.90 70.85 83.97 22.29 66.89 45.45 39.23 68.44 64.63 MaPLe [18] 93.53 90.49 65.57 72.23 86.20 24.74 67.01 46.49 48.06 68.69 66.30 MaPLe+TPT [41] 93.59 90.72 66.50 72.37 86.64 24.70 67.54 45.87 47.80 69.19 66.50 DiffTPT [6] 92.49 88.22 67.01 70.10 87.23 25.60 65.74 47.00 43.13 62.67 65.47 PromptAlign [39] 94.01 90.76 68.50 72.39 86.65 24.80 67.54 47.24 47.86 69.47 66.92 TDA [17] 94.24 88.63 67.28 71.42 86.14 23.91 67.62 47.40 58.00 70.66 67.53 BoostAdapter 94.77 89.51 69.30 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 Remark Proposition 2 provides a guarantee of the effectiveness of selecting kt out of nt historical samples to comprise the cache. The empirical risk is quite small when nt →∞since the cache captures the full information of the target domain. Proposition 3 demonstrates that the historical cache can further reduce empirical risk by incorporating kb boosting samples. 4 Experiments 4.1 Experimental Setup Datasets Following the setting in TPT [41], we conduct experiments on both Out-of-Distribution (OOD) benchmark and Cross-Domain benchmark. The OOD benchmark evaluates the model’s robustness to natural distribution shifts on 4 ImageNet [4] Variants, including ImageNetV2 [37], ImageNet-Sketch [44], ImageNet-A [14] and ImageNet-R [13]. We evaluate the transferring performance on 11 datasets in the Cross-Domain benchmark: Aircraft [31], Caltech101 [5], Cars [19], DTD [3], EuroSAT [11], Flower102 [32], Food101 [2], Pets [34], SUN397 [48],and UCF101 [42]. We follow the split in [55] and report the top-1 accuracy. The error bound are also provided. Implementation details We utilize a pre-trained ViT-B/16 of CLIP as the foundation model. In test-time adaptation, the batch size is set to be 1. We search for the optimal shot capacity to balance diversity and relevance of samples. For boosting samples, we utilize random crop and then random horizontal flip as augmentations. Moreover, we empirically set the entropy threshold percentile to p = 0.1 and filter 64 augmented views based on random cropping to obtain the boosting samples. and filter 64 augmented views to obtain the boosting samples. The top-1 accuracy and the error bound is reported on the test sets. All our experiments are conducted with a Nvidia 3090 24GB GPU. 7 0 16 32 64 128 Number of augmented views 66 68 70 Top-1 accuracy % CLIP BoostAdapter (a) Number of augmented views Historical Samples Boosting Samples Historical Samples + Boosting Samples 66 68 70 72 74 Top-1 Accuracy % Entropy Minimization Feature Retrieval (b) Adaptation methods 0 1 2 3 4 5 6 Total shot capacity 15.0 17.5 20.0 22.5 25.0 27.5 Top-1 accuracy % CLIP BoostAdapter (c) Total shot capacity Figure 4: Ablation studies of (a) number of augmented views to generate boosting samples (b) different adaptation methods and (c) total shot capacity of the cache. Table 3: Ablation study on historical samples and boosting samples on the OOD benchmark with ViT-B/16 backbone. We report top-1 accuracy and the error bound is ±0.12. -V2 -Sketch -A -R Average CLIP 60.86 46.09 47.87 73.98 57.20 Historical Samples 64.93 50.23 63.80 80.43 64.85 Boosting Samples 65.40 50.59 64.40 80.96 65.34 BoostAdapter 65.51 51.28 64.53 80.95 65.57 Table 4: Full results on the OOD benchmark with RN-50 backbone. We report top-1 accuracy and the error bound is ±0.06. -V2 -Sketch -A -R Average CLIP [36] 51.41 33.37 21.83 56.15 40.69 TPT [41] 54.70 35.09 26.67 59.11 43.89 CALIP [10] 53.70 35.61 23.96 60.81 43.52 CoOp [55] 55.40 34.67 23.06 56.60 42.43 CoCoOp [54] 55.72 34.48 23.32 57.74 42.82 DiffTPT [6] 55.80 37.10 31.06 58.80 45.69 TDA [17] 55.54 38.12 30.29 62.58 46.63 BoostAdapter 56.14 38.87 35.12 62.66 48.20 4.2 Out-of-Distribution Generalization To verify the robustness of BoostAdapter, we evaluate our method on the OOD benchmark, in comparison with existing training-require methods including CoOp [55], CoCoOp [54], TPT [41], DiffTPT [6], Maple [18] and PromptAlign [39], as well as training-free method TDA [17]. As can be seen from Table 8, the most striking observation emerging from the comparison is that BoostAdapter significantly outperforms other baselines on average and improves the generalization ability of the model. For training-free methods such as TPT, DiffTPT and PromptAlign, BoostAdapter achieves superior performance while saving on optimization computation overhead. For training-free methods like TDA, BoostAdapter gains consistent performance improvements with the introduction of the boosting samples. Notably, BoostAdapter surpasses TDA by 4.42% on ImageNet-A and 0.84% on ImageNet-V2, respectively. This enhancement indicates the effectiveness of self-bootstrapping when historical samples may not provide sufficient useful information. 4.3 Cross-Domain Transfer We further highlight our improvements in the transfer ability of CLIP on the Cross-Domain benchmark and present the results in Table 2. Compared with existing training-required and training-free methods, BoostAdapter achieves state-of-the-art performance on 7 out of 10 tasks, surpassing the strongest baselines by an average of 1.15%. With diverse classes at test time, regional boosting enables BoostAdapter to adaptively extract knowledge that makes classes distinct from each other in a multi-scale manner. Notably, for datasets requiring fine-grained information for classification such as Aircraft, the improvement of BoostAdapter is most significant. 4.4 Ablation Study Historical Samples and Boosting Samples. To demonstrate the effect of historical and boosting samples, we introduce two variants of BoostAdapter that utilize only historical samples or only boosting samples, respectively. Additionally, we provide the zero-shot results of CLIP for comparison. As shown in Table 3, CLIP significantly benefits from both historical samples and boosting samples, resulting in notable improvements in performance. The consistent improvement of BoostAdapter compared to the variant that utilizes only historical samples further confirms the effectiveness of 8 Table 5: Full results on the Cross-Domain Benchmark with RN-50 backbone. We report top-1 accuracy and “Average" is calculated by taking the mean accuracy across all ten datasets. The error bound is ±0.05. Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CLIP [36] 85.88 83.57 55.70 61.75 73.97 15.66 58.8 40.37 23.69 58.84 55.82 CLIP + TPT [41] 87.02 84.49 58.46 62.69 74.88 17.58 61.46 40.84 28.33 60.82 57.66 CALIP [10] 87.71 86.21 56.27 66.38 77.42 17.76 58.59 42.39 38.90 61.72 59.34 DiffTPT [6] 86.89 83.40 60.71 63.53 79.21 17.60 62.72 40.72 41.04 62.67 59.85 CuPL [35] 89.29 84.84 57.28 65.44 76.94 19.59 62.55 48.64 38.38 58.97 60.19 TDA [17] 89.70 86.18 57.78 68.74 77.75 17.61 62.53 43.74 42.11 64.18 61.03 BoostAdapter 88.48 85.75 59.67 68.25 78.78 18.93 62.83 43.85 44.40 64.42 61.54 Table 6: Comparisons with baselines on ImageNet-C at severity level 5 regarding accuracy (%). Noise Blur Weather Digital Gauss. Shot Impul. Defoc. Glass Motion Zoom Snow Frost Fog Brit. Contr. Elastic Pixel JPEG Avg. CLIP-ViT-B/16 15.15 16.28 15.26 25.83 16.87 26.34 24.43 34.56 33.01 39.10 57.78 18.45 14.71 35.62 35.81 27.28 TDA 17.50 18.59 18.12 59.12 19.02 28.25 26.24 37.30 35.30 41.57 59.04 21.06 17.61 37.78 37.26 31.58 BoostAdapter 17.53 18.89 18.39 59.70 19.07 28.62 27.33 38.21 36.13 42.31 59.63 21.22 18.23 39.25 38.07 32.17 incorporating boosting samples into the training-free adapters. See Section E in the Appendix for more results. Number of Augmented Views for Boosting Samples. We augment the testing samples and filter them by mutual information with the CLIP text embedding to obtain the boosting samples. We vary the number of augmented views and investigate the performance of BoostAdapter on UCF101 in Figure 4a. With a larger number of augmented views, the performance improves due to more bootstrapping information of the test sample, which is consistent with the conclusions of TPT [41] and PromptAlign [39]. However, the computational overhead also increases with more augmented views, and selecting 64 augmented views is a fair trade-off between boosting performance and efficiency. Adaptation Methods. Training-required methods use entropy as a self-supervised objective, whereas training-free methods classify samples based on feature retrieval. We compare the performance of these two adaptation methods under the constraints of historical samples only, boosting samples only, or both, and present the results on Flower102 in Fig. 4b. Entropy minimization requires gradient descent and model optimization, resulting in high training costs and relatively lower performance across all three settings. In contrast, the training-free methods based on feature retrieval offer significant performance improvements with lower computational overhead. Additionally, both adaptation methods benefit from combining historical samples and boosting samples, consistent with the conclusions in Table 3. Total shot capacity. BoostAdapter maintains low-entropy samples per class in the cache, and Figure 4c studies the influence of different total shot capacities containing historical samples and boosting samples of each class on Aircraft. As can be observed from the results, when the cache capacity is small, the low-entropy samples maintained by BoostAdapter do not necessarily provide a benefit for classification compared to CLIP. As the shot capacity increases, BoostAdapter will achieve the best balance of diversity and relevance, and a larger capacity does not guarantee better performance. Versatility. To demonstrate the versatility of BoostAdapter, we apply it to the RN-50 backbone and present the results in Tables 4 and 5. The improvement is consistent and on average, BoostAdapter outperforms TDA by 1.57% on the OOD benchmark and 0.49% on the Cross-Domain benchmark. 4.5 Discussions Generalization on Corruption Datasets To further evaluate the generalization ability of BoostAdapter in new test-time scenarios, we compare BoostAdapter with baseline methods on the ImagenetC dataset at the highest severity level 5. The key observation from Table 6 is that BoostAdapter 9 Table 7: Efficiency analysis. We evaluate different methods on a single NVIDIA 3090 24GB GPU and report the frames per second (fps) and memory cost (GB). Augmentation Views Inference Speed (fps) Memory (GB) OOD Results Cross-Domain Results CLIP 82.3 0.7 57.20 63.58 TPT Augmix 64 0.29 4.5 60.81 65.10 DiffTPT Diffusion 64 0.10 14.4 60.52 66.92 TDA Augmix 64 11.89 1.2 63.89 67.53 BoostAdapter Rand. Crop & Rand. Horiz. Flip 64 11.23 1.2 65.57 68.68 Table 8: Unification of more training-required methods. BoostAdapter benefits from different training-required methods. -V -S -A -R Average CLIP-ViT-B/16 60.86 46.09 47.87 73.98 57.20 TDA 64.67 50.54 60.11 80.24 63.89 BoostAdapter 65.51 51.28 64.53 80.95 65.57 BoostAdapter+ TSD 65.49 51.50 64.37 81.15 65.63 BoostAdapter+ DEYO 65.71 51.52 64.65 81.43 65.83 Metal nail Test Sample Boosting Sample Snail Test Sample Boosting Sample Crutch Microphone Figure 5: Qualitative results. The model predictions are provided below the images. Boosting samples with low entropy improves information extraction from the test sample and helps the model to distinguish better. consistently outperforms TDA across all 15 corruption types, highlighting its practical applicability in real-world situations. The superior performance of BoostAdapter stems from its capability to capture the knowledge of the test sample even under severe corruption. This is achieved with the help of the boosting samples, which effectively filter out noisy parts while retaining useful information. Efficiency Analysis BoostAdapter requires augmentation over the test samples, which may slightly affect the inference speed during testing. We conduct an efficiency analysis of BoostAdapter in comparison with existing Test Time Augmentation (TTA) methods and provide the results in Table 7. BoostAdapter is slightly slower than the cache-based method TDA, yet still significantly faster than training-required methods. The memory cost of BoostAdapter is also comparable to other baselines. Unification of Training-required and Training-free Methods. From the unified perspective, we can also enhance training-free adapters with additional training-required methods. Here we take TSD [45] and DEYO [21] as the showcase. Specifically, in the BoostAdapter+DEYO variant, we filter out augmented views with a PLPD lower than 0.2. For the BoostAdapter TSD variant, we discard augmented views that have different cache predictions and CLIP predictions to ensure consistency of the boosting samples. When equipping BoostAdapter with the technique of TSD and DEYO, we observe further improvement and find that training-free adapters can benefit from various boosting techniques of training-required methods. Qualitative Results The qualitative results are provided in Figure. 5. By incorporating samples with low entropy from regional bootstrapping, the model is enhanced to more effectively capture the fine-grained information of the test samples, thereby improving the overall performance. 5 Conclusions In this work, we present an insightful analysis of existing training-required and training-free TTA methods to bridge the gap between them. In particular, we improve training-free adapters by incorporating self-boosting samples into the memory bank inspired by the idea of regional bootstrapping from entropy-based methods. The cache in our method, containing instance-agnostic historical samples and instance-aware boosting samples, is capable of performing knowledge mining on both the target domain and the testing sample itself. We also derive error bounds in the test-time adaptation setting and show that this cache benefits from both historical samples and boosting samples. Extensive experiments on the two benchmarks demonstrate the effectiveness of our method. Despite the promising performance of our method, it also has some limitations. It requires slightly more computation overhead than existing training-free adapters due to the multiple augmentation of the test samples, as discussed in Appendix. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2605–2615, 2023. 14 Appendix A Dataset and Licenses Table 9 presents the statistics and details of datasets used in the paper. We also provide the corresponding license information of the datasets and source code. Datasets. Below are the datasets used in this paper that have known license information: The following datasets used in this paper are under the MIT License: ImageNet-A [14], ImageNet-V2 [37], ImageNet-R [13], ImageNet-Sketch [44], EuroSAT [11], Food101 [2]. The following datasets used in this paper are under the CC BY-SA 4.0 License: Oxford-Pets [34], Caltech101 [5]. The following datasets used in this paper are for research purposes only: DTD [3], StanfordCars [19], SUN397 [48], FGVC-Aircraft [31], Flower102 [34], UCF101 [42]. Source code. We use the implementation of existing baseline methods for reporting their results in this paper. Below are their license information: Source code used in this paper that are under the MIT License: CLIP [36], PromptAlign [39] and TDA [17]. Dataset Description Classes Test Size Out-of-Distribution Benchmark ImageNet-V2 New Validation Sets of ImageNet 1,000 10,000 ImageNet-S Sketch Images 1,000 50,000 ImageNet-A Natural Adversarial Examples 200 7,500 ImageNet-R Rendition Extension of ImageNet 200 30,000 Cross-Domain Benchmark Aircraft Aircraft Model Classification 100 3,333 Caltech101 Natural Image Classification 100 2,465 Cars Cars Classification 196 8,041 DTD Describable Textures Dataset 47 1,692 EuroSAT Satellite Images 10 8,100 Flowers102 Flowers Classification 102 2,463 Food101 Food Classification 101 30,300 Pets Pets Classification 37 3,669 SUN397 Scene Categorization Benchmark 397 19,850 UCF101 Action Recognition Dataset 101 3,783 Table 9: Datasets statistics. B Broader Impacts In this paper, we focus on bridging the gap between training-required and training-free methods to improve the generalization ability of vision-language models. We also theoretically derive the error bound of incorporating boosting samples into the historical cache. We hope that our work will inspire the community to explore test-time adaptation in an effective and efficient way. C Theoretical Proof C.1 Cross-entropy Optimization behaves like Cache Classifier over well-clustered Samples (Proof of Proposition 1) Given well-clustered samples in the feature space and the classifier defined in Eq.(3), we first derive the distance between the weights of the classifier and the optimal weights and then establish the connection between the optimal weights with the features center of the samples. 15 Suppose the classifier function f over samples is convex and differentiable, and also L-smooth. Let the distance between initial weight w(0) and optimal weight w∗to be D = \|\|w(0) −w∗\|\|. GD updates by w(t+1) = wt −f ∗ t ∇f(wt) with step size f ∗ t = 1 L, and then GD enjoys the following convergence guarantee: \|\|w −w∗\|\| ≤2L w(0) −w⋆ 2 T −1 = O LD2 T . (11) We then showcase the relationship between w∗and the features center µi of class i, i = 1, 2, ..., N. Since we optimize on well-clustered samples, we consider the scenarios of perfect clusters, where samples in the class i will be encoded into the same point µi by the encoder g, and these points should be farthest enough between each other. Given n samples {(xk, yk)}n k=1, with the number of samples in class i to be ni, the cross-entropy loss function L can be written as: L = − n X i=1 log P(y = yk\|xk) (12) Substitute the sample g(xk) = µi from class i, we derive the probability P(y = i\|xk) using the softmax function from Eq.(3) is: P(y = i\|xk) = exp(wT i µi) PN j=1 exp(wT j µi) . (13) Thus, the cross-entropy loss for a sample (xk, yk = i) is: Lk = −log exp(wT i µi) PN j=1 exp(wT j µi) ! . (14) For all samples, the total loss is: L = − N X i=1 ni log exp(wT i µi) PN j=1 exp(wT j µi) ! . (15) The gradient of the loss with respect to wi can be simplified as: ∂L ∂wi = −µini + µi N X k=1 nk exp(wT i µk) PN j=1 exp(wT j µk) . (16) When converges to the optimal weight, we have the condition of fixed point ∂L ∂w∗ i = 0. And we have −µini + µi N X k=1 nk exp((w∗ i )T µk) PN j=1 exp((w∗ j )T µk) = 0. (17) Thus, we have N X k=1 nk exp((w∗ i )T µk) PN j=1 exp((w∗ j )T µk) = ni. (18) Given a well-clustered samples, we could have exp((w∗ i )T µk) ≫exp((w∗ j )T µk) for a specific i when w∗ i is near µk. Then since the equality in Eq.(18) will hold for each class and for class i = 1, 2, ..., N we have w∗ i →µi. (19) Combining Eq.(11) and Eq.(18), with iteration steps T, we show that the weight of classfier will finally converge to the feature center of each class: \|\|w −µ\|\| ≤\|\|w −w∗\|\| + \|\|w∗−µ\|\| ≤O LD2 T . (20) 16 And we have the output logits of the optimal weights with the encoder g: pcross(x) = [µT 1 g(x), µT 2 g(x), ..., µT Ng(x)] (21) Next we discess the behavior of the cache classifier over these samples. Given the number of wellclustered samples in class i to be ni, the output logits of the cache classifier defined in Eq.(5) using samples {(xk, yk)}n k=1 can be described as follows: pcache(x) = n X k=1 1 nyi [g(xk)T g(x)]yk = N X i=1 ni ni [µT i g(x)]yi = [µT 1 g(x), µT 2 g(x), ..., µT Ng(x)] (22) Combining Eq.(21) and Eq.(22), we draw the conclusion that cross-entropy optimization behaves like cache classifier over well-clustered samples. C.2 Historical Cache reduce Empirical Risk (Proof of Proposition 2) We follow the proofs in [53] and extend the conclusion to boosting samples. C.2.1 Additional Definitions and Assumptions Definition 4. (Wasserstein-distance and the dual form). Wasserstein distance measures the distance between two probability distributions on a given metric space. It is defined using the concept of optimal transport. For two distributions P, Q, The ρ-th Wasserstein distance is defined as Wp(P, Q) = inf γ∈Π(P,Q) Z X×X d(x, y)pdγ(x, y) 1/p (23) Here, Π(P, Q) denotes the set of all couplings (or transport plans) γ of P and Q, i.e., joint distributions on X × X with marginals P and Q.The idea is to find the optimal way to transport the mass from one distribution to the other with the minimal cost, where the cost is given by the p-th power of the distance. The first Wasserstein distance, W1(P, Q) ,often referred to as the Earth-Mover Distance(EMD), has a particularly elegant dual representation. The dual form of W1 leverages the Kantorovich-Rubinstein duality and can be expressed as: W1(P, Q) = sup ∥f∥Lip≤1 Z X f dP − Z X f dQ (24) Here, the supremum is taken over all 1-Lipschitz functions f ,which are functions satisfying \|f(x)− f(y)\| ≤d(x, y) for all x, y ∈X .This representation shows that W1 can be seen as the maximum difference in expected values of a 1-Lipschitz function over the two distributions. In the following part, Wasserstein distance represents the first Wasserstein distance for simplicity and we utilize W(·, ·) instead of W1(·, ·). Given the definition of the Wasserstein distance, we have the following proposition that derive the empirical risk on the target domain according to Theorem 1 from [40]. Proposition 4. Given two distributions P, Q, denote f ∗= arg minf∈H(ϵP (f) + ϵQ(f)) and ξ = ϵP (f ∗) + ϵQ(f ∗). Assume all hypotheses h are L-Lipschitz continuous, the risk of hypothesis ˆf is then bounded by ϵQ( ˆf) ≤ξ + ϵP ( ˆf) + 2LW(P, Q). (25) 17 C.2.2 Distance between the Ball Distribution with the Target Distribution When using the cache classifier with historical samples, a large number of samples that are not similar enough from the target domain will be filtered and the selected samples with high weight are all close to the target data. Thus we extend the conclusion in [53] to the distance between the ball distribution with the target distribution. Considering a test sample from the target distribution xt ∈pt(x) and a distribution consisting of ball center of all the test samples Ω:= S xt∈pt(x) B(xt, r), informally, according to Eq.(23), we have the distance between the ball distribution with the target distribution as follows: W(Ω, pt(x)) = inf γ∈Π[Ω,pt(x)] ZZ ∥xt −xball ∥dγ(xt, xball), (26) where for each xball ∈Ω, we can find at least one xt ∈pt(x) such that ∥xball −xt ∥≤r, the overall distance will then be bounded by r. Specifically, we can choose a density function γ∗where γ∗(xball, xt) > 0 only if ∥xball −xt ∥≤r otherwise 0, then we have W(Ω, pt(x)) = inf γ∈Π[Ω,pt(x)] ZZ ∥xball −xt ∥dγ(xball, xt) ≤ ZZ ∥xball −xt ∥γ∗(xball, xt)dxballxt ≤r. (27) However, there is no guarantee that each data xt ∈pt(x) can find a neighbor B(xt, r) with \|B(xt, r)\| > 0 with all the small r. We then provide the probability that the set of neighbors B(xt, r) of each xt ∈pt(x) is not measuring zero with respect to the radius r. As defined in the cache classfier Eq.(5), we denote kt is the number of historical samples we select in the cache and nt is the total number of data from the historical stream. With the strong density assumption, given the coefficient bound m and M, for any xt ∈pt(x), r < R, according to Assumption 1, we have \| ˆxt ∈pt(x) ∧ˆxt ∈B(xt, r)\| = Z B(xt,r)∩pt(x) dpt(x) dλ ( ˆxt)d ˆxt ≥mλ(B(xt, r) ∩pt(x)) ≥mctπdrd, (28) where πd = λ(B(0, 1)) is the volume of the d dimension unit ball and λ is the Lebesgue measure of a set in a Euclidean space. Set r0 = ( 2k mctπdnt )1/d, with a additional assumption that we utilize a small kt compared to nt so that kt nt < ctmπdrd µ 2 , we have r0 < R. Then for any xt ∈pt(x), according to Eq.(28), we have \| ˆxt ∈pt(x) ∧ˆxt ∈B(xt, r0)\| ≥mctπdrd 0 > 2kt nt . (29) Since ˆxt ∈pt(x) are independently drawn from the target distribution, let I(·) to be the Indicator funciton and Snt(xt) = Pnt i=1 I( ˆxt ∈B(xt, r0)) denote the number of data ˆxt ∈pt(x) that fall into B(xt, r0), then Snt(xt) follows the Binomial distribution. Let W ∼Binomial(nt, 2k nt ), according to the Chernoff inequality, we have P(Snt(xt) < kt) ≤P(W < kt) = P(W −E[W] < −kt) ≤exp(−k2 t /2E[W]) = exp(−kt/4), (30) where the second inequality holds since Sn(x) has a larger mean than W. With a large kt, the probability that Sn(x) < kt is small for any xt ∈pt(x). Denoting ˆxt (i) as the ith nearest sample to xt among B(xt, r0) in the cache, we have for any xt ∈pt(x) 18 P(∥ˆxt (kt) −xt ∥≤r0) = P(Sn(xt) ≥kt) ≥1 −exp(−kt/4) (31) Combine Eq.(31) with the assumption that the distribution pt(x) is finite with cardinality ℵpt and the desired probability part is shown by union bound. \ xt∈pt(x) P(∥ˆxt (kt) −xt ∥≤r0)) = \ xt∈pt(x) P(Sn(x) ≥kt) = 1 − [ xt∈pt(x) P(Sn(x) < kt) ≥1 −ℵpt exp −kt 4 = 1 −exp −kt 4 + log ℵpt . (32) And then we have the following proposition. Proposition 5. Given the target domain distributions pt(x) that is finite with cardinality ℵpt, and Ω:= S x∈pt(x) B(x, r), where B(x, r) = {x′ :∥x′ −x ∥≤r} denotes a ball centered on x with radius r. Denote f ∗= arg minf∈H(ϵt(f)+ϵΩ(f)) and ξ = ϵt(f ∗)+ϵΩ(f ∗). Assume all hypotheses h are L-Lipschitz continuous, the risk of hypothesis ˆf on the unseen target domain is then bounded by ϵt( ˆf) ≤κ + ϵΩ( ˆf) + 2L 2kt mctπdnt 1/d . (33) with probability 1 −exp(−kt 4 + log ℵpt) C.2.3 Excess Error Bound of Cache Classifier Let si to be the softmax probability softmax(pcache) for class i in the the cache classifier from Eq.(5), we can simplify the classifier as ˆfcache = I{s1 ≥1 2} on the binary classification setting. Then ˆfcache(xt) ̸= f ∗(xt) implies that ˆfcache(xt) −f ∗(xt) ≥ f ∗(xt) −1 2 . We then bridge the gap between the excess error and the classify error as follows: Et( ˆf) = 2Ext∼pt(x) f ∗(xt) −1 2 I ˆfcache(xt) −f ∗(xt) ≥ f ∗(xt) −1 2 . (34) We want to bound supxt ˆfcache(xt) −f ∗(xt) ≤t, combining with the marginal assumption in Assumption 3 and the fact that E [Z · I{Z ≤t}] ≤tP(Z ≤t), (35) where Z = f ∗(xt) −1 2 , so we have Et( ˆf) ≤Cβtβ+1. To bound ˆfcache(xt) −f ∗(xt) , we denote ( ˆxt (i), ˆyt (i)) as the ith nearest data and the corresponding labels to xt in B(xt, r0). The result of the cache classfier with normalized weight will be ˆfcache(xt) = kt X i=1 1 Pkt j=1 g ˆxt (j)T g(x) g ˆxt (i)T g(x) ˆyt (i) (36) = kt X i=1 wi ˆyt (i), (37) where wi = g( ˆ xt (i)) T g(x) Pkt j=1 h g( ˆ xt(j)) T g(x) i is the normalized weight and Pkt i=1 wi = 1. Based on the assumptions and notions above, we have for any xt ∈pt(x) 19 ˆfcache(xt) −f ∗(xt) = kt X i=1 wi ˆyt (i) −f ∗(xt) ≤ kt X i=1 wi ˆyt (i) − kt X i=1 wif ∗ ˆxt (i) + kt X i=1 wif ∗ ˆxt (i) −f ∗(xt) ≤ kt X i=1 wi ˆyt (i) − kt X i=1 f ∗ ˆxt (i) \| {z } 1 + kt X i=1 wi f ∗ ˆxt (i) −f ∗(xt) \| {z } 2 , (38) where 2 is easy to bound. According to the assumption that f ∗is C-Smoothness, we have kt X i=1 wi f ∗ ˆxt (i) −f ∗(xt) ≤ kt X i=1 wiC· ∥ˆxt (i) −xt ∥≤C· ∥ˆxt (kt) −xt ∥ (39) According to Eq.(31), with probability at least 1 −exp(−kt/4), 2 ≤C 2kt mctπdnt 1/d . Note that We store the target sample into the cache only when its prediction confidence is large enough. Therefore, it is natural to assume that: EY \|X h ˆyt (i)i = f ∗(x(i) t ). (40) Then we use the Hoeffding inequality to obtain the upper bound of 1 PX,Y kt X i=1 wi ˆyt (i) − kt X i=1 f ∗( ˆxt (i)) > ϵ ! = EX " PY \|X kt X i=1 wi ˆyt (i) − kt X i=1 f ∗( ˆxt (i)) > ϵ !# ≤2 exp(− 2ϵ2 Pkt i=1 w2 i ) ≈2 exp(−2ηktϵ2). (41) We simplify the bound by assuming that the weights in the target domain are evenly distributed in the subset of all samples with respect to a specific class controlled by coefficient η, according to Assumption 1 and Proposition 4. That is, we have Pkt i=1 w2 i ≈Pηkt i=1 ( 1 ηkt ) 2 = 1 ηkt . Set ϵ = (1/kt)1/4, we have, with probability, at least 1 −3 exp(−2η√kt), 1 ≤(1/kt)1/4, 2 ≤ C 2kt mctπdnt 1/d , and then ˆfcache(xt) −f ∗(xt) ≤(1/kt)1/4 + C 2kt mctπdnt 1/d . According to Eq.(31) and Eq.(35), the excess error is bounded by Et( ˆf) ≤2Cβ 1 kt 1/4 + C 2kt mctπdnt 1/d!1+β ≈ 1 kt 1/4 + C1 kt ctnt 1/d!1+β , (42) with constant C1. When appropriately choosing kt = O(log nt), we have min{1 −2 exp(−2η p kt), 1 −exp(−kt/4)} ≥1 −2 exp(−2η p kt) −exp(−kt/4) ≥1 −3 exp(−2η p kt) = 1 −3 exp(−O(1) p log nt) (43) 20 where the third line is because kt/4 > 2η√kt for large enough kt. Namely, with probability at least 1 −3 exp(−√log nt)O(1), the following bound holds true. Et( ˆf) ≤O 1 log nt 1/4 + log nt ctnt 1/d!1+β , (44) C.3 Historical Cache benefits from Boosting Samples (Proof of Proposition 3) To study the effect of the boosting samples, we consider the cache classfier containing both kt historical samples { ˆxt (i), ˆyt (i)}kt i=1 and kb boosting samples { ˆxb (i), ˆyb (i)}kb i=1 as the nearest data to xt in B(xt, r0). With the normalized weights wti = g( ˆ xt (i)) T g(x) Pkt j=1 h g( ˆ xt(j)) T g(x) i +Pkb j=1 h g( ˆ xb(j)) T g(x) i and wbi = g( ˆ xb (i)) T g(x) Pkt j=1 h g( ˆ xt(j)) T g(x) i +Pkb j=1 h g( ˆ xb(j)) T g(x) i, the prediction result of the cache classifier will be ˆfcache(xt) = Pkt i=1 wti ˆyt (i) + Pkb i=1 wbiy(i) b . Then we have: ˆfcache(xt) −f ∗(xt) = kt X i=1 wti ˆyt (i) − kt X i=1 wtif ∗(xt) + kb X i=1 wbiy(i) u − kb X i=1 wbif ∗(xt) ≤ " kt X i=1 wti ˆyt (i) − kt X i=1 wtif ∗( ˆxt (i)) # + " kt X i=1 wtif ∗( ˆxt (i)) − kt X i=1 wtif ∗(xt) # + " kb X i=1 wbiy(i) u − kb X i=1 wbif ∗ x(i) u # + " kb X i=1 wbif ∗ x(i) u − kb X i=1 wbif ∗(xt) # ≤ kt X i=1 wti ˆyt (i) + kb X i=1 wbiy(i) u − kt X i=1 wtif ∗( ˆxt (i)) − kb X i=1 wbif ∗ x(i) u \| {z } 1 + kt X i=1 wti f ∗( ˆxt (i)) −f ∗(xt) \| {z } 2 + kb X i=1 wbi f ∗ x(i) u −f ∗(xt) \| {z } 3 Similar to Eq.(40), we have the following assumption on the boosting distribution: EY \|X h ˆyb (i)i = f ∗(x(i) b ). (45) According to Eq.(41), we have PX,Y kt X i=1 wti ˆyt (i) + kb X i=1 wbiy(i) b − kt X i=1 wtif ∗( ˆxt (i)) − kb X i=1 wbif ∗ x(i) b ! = EX " PY \|X kt X i=1 wti ˆyt (i) + kb X i=1 wbiy(i) b − kt X i=1 wtif ∗( ˆxt (i)) − kb X i=1 wbif ∗ x(i) b !# ≤2 exp(−2η(kt + kb)ϵ2) (46) 21 Set ϵ = (1/(kt + kb))1/4, we have, with probability, at least 1 −3 exp(−2η p (kt + kb)), 1 ≤ (1/(kt + kb))1/4. Then, according to Eq.(39), we have kt X i=1 wti f ∗ ˆxt (i) −f ∗(xt) ≤ kt X i=1 wtiC· ∥ˆxt (i) −xt ∥≤StC· ∥ˆxt (kt) −xt ∥ (47) and kb X i=1 wbi f ∗ ˆxb (i) −f ∗(xt) ≤ kb X i=1 wbiC· ∥ˆxb (i) −xt ∥≤SbC· ∥ˆxb (kb) −xt ∥. (48) where St = Pkt i=1 wti, Sb = Pkb i=1 wbi are the sum of weights of historical samples and boosting samples, respectively, and we have St + Sb = 1. Then we have the following results in similar: 2 ≤StC 2kt mctπdnt 1/d ; 3 ≤SbC 2kb mcbπdnb 1/d (49) Finally, the excess error under the covariate shift setting can be bounded by Et( ˆf) ≤2Cβ ( 1 kt + kb )1/4 + StC 2kt mctπdnt 1/d + SbC 2kb mcbπdnb 1/d!1+β ≈ 1 kt + kb 1/4 + C1St kt ctnt 1/d + C1Sb kb cbnb 1/d!1+β = 1 kt + kb 1/4 + C1 kt X i=1 wti kt ctnt 1/d + C1 kb X i=1 wbi kb cbnb 1/d!1+β (50) Compared Eq.(50) to Eq.(42) and St + Sb = 1, it is easy to verify that (St + Sb)C 2(kt + kb) mctπdnt 1/d −StC 2kt mctπdnt 1/d −SbC 2kb mcbπdnb 1/d ≥SbC 2kt mctπdnt 1/d −SbC 2kb mcbπdnb 1/d (51) In general, the boosting distribution is more close to the test sample than the target distribution and we have cb > ct. Thus the difference in Eq.(51) is then larger than 0, namely incorporating boosting samples into the memory bank, the excess error can be further reduced. D More Experiments Independent Cache for Boosting Samples. In BoostAdater, due to the cost of augmentation, the number of boosting samples is relatively smaller than the number of historical samples. Therefore, we use a joint cache for storing both historical and boosting samples to facilitate intra-sample and inter-sample interactions. Table 10 and Table 11 study the influence of using an independent cache for the boosting samples. As can be observed from the results, BoostAdapter suffers from slight performance degradation due to the independent cache. Table 10: Independent cache for boosting samples on the OOD benchmark. Imagenet-V2 Imagenet-Sketch Imagenet-A Imagenet-R Average Independent Cache 65.37 50.62 64.56 80.96 65.38 Joint Cache 65.51 51.28 64.53 80.95 65.57 22 Table 11: Independent cache for boosting sample on the Cross-Domain Benchmark. Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average Independent Cache 94.69 88.88 69.19 71.94 86.99 26.76 67.64 44.21 61.20 69.63 68.11 Joint Cache 94.77 89.51 69.30 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 Different Augmentation for Boosting Samples. We make use of random crop followed by random horizontal flip as augmentations for generating boosting samples. Additionally, we further explore the influences of different augmentations applied to the randomly cropped images. The comparison methods include: (i) Random Brighness: Randomly set the brighness of image from 50% to 150%. (ii) Random Auto Contrast: Apply auto contrast over image with probability p = 0.5. (iii) Random Rotate: Randomly rotate the image from -45 degree to 45 degree. (iv) Random Vertical Flip: Apply vertical flip over image with probability p = 0.5. (v) Random Horizontal Flip (BoostAdapter): Apply horizontal flip over image with probability p = 0.5. The results are presented in Table 12 and Table 13. The results indicate that random horizontal flipping outperforms other augmentation methods, primarily because the images generated from horizontal flips are closer to the original distribution when training CLIP. Table 12: Comparison of different augmentations on the OOD benchmark . Default settings are marked in gray . Imagenet-V2 Imagenet-Sketch Imagenet-A Imagenet-R Average Random Brightness 65.10 51.24 62.10 81.03 64.87 Random Auto Contrast 65.50 50.79 64.33 80.57 65.30 Random Rotate 61.14 47.67 60.83 78.15 61.95 Random Vertical Flip 63.39 49.67 60.77 78.55 63.10 Random Horizontal Flip 65.51 51.28 64.53 80.95 65.57 Table 13: Comparison of different augmentations on the Cross-Domain Benchmark. Default settings are marked in gray . Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average Random Brightness 94.60 89.70 69.28 71.70 86.88 26.67 68.24 45.57 61.63 71.45 68.57 Random Auto Contrast 94.48 89.67 69.33 71.90 87.24 27.39 68.16 45.51 61.67 71.77 68.71 Random Rotate 94.52 89.59 67.74 71.30 85.91 24.27 67.56 45.45 60.72 70.66 67.77 Random Vertical Flip 94.89 89.53 68.75 72.19 86.78 24.99 67.72 45.27 61.56 70.82 68.25 Random Horizontal Flip 94.77 89.51 69.30 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 E Additional Ablation Results Historical Samples and Boosting Samples. We provide more ablation results of the historical and boosting samples on the Cross-Dataset benchmark in Table 14. The observation is consistent with the results in Table 3, showing that CLIP gains improvements from both historical and boosting samples. Furthermore, when applied to various downstream tasks, the importance of regional bootstrapping becomes more significant, as indicated by the gap between BoostAdapter and the variant that uses boosting samples only. Number of Augmented Views for Boosting Samples. The complete results on the number of augmented views are presented in Table 15 and Table 16. With more augmented views, BoostAdapter is able to better extract the fine-grained information from the original test sample, achieving improved performance. 23 Table 14: Ablation study on historical samples and boosting sample on the Cross-Domain Benchmark. Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CLIP 93.35 88.25 65.48 67.44 83.65 23.67 62.59 44.27 42.01 65.13 63.58 Historical Samples 94.16 89.42 66.87 72.11 85.93 24.69 67.24 44.80 61.85 69.81 67.69 Boosting Samples 94.32 88.64 68.38 71.54 87.12 27.30 67.42 44.68 45.93 69.34 66.47 BoostAdapter 94.77 89.51 69.30 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 Table 15: Results of different views on the OOD benchmark. Default settings are marked in gray . Imagenet-V2 Imagenet-Sketch Imagenet-A Imagenet-R Average 16 Views 79.41 49.01 62.08 63.68 63.54 32 Views 80.32 50.73 63.22 64.91 64.80 64 Views 80.95 51.28 64.53 65.51 65.57 128 Views 80.95 51.91 64.06 65.27 65.55 Table 16: Results of different views on the Cross-Domain Benchmark. Default settings are marked in gray . Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average 16 Views 93.95 89.62 68.06 71.62 86.76 25.71 67.33 45.39 62.07 70.97 68.15 32 Views 94.48 89.59 69.07 71.54 87.01 27.18 67.97 45.45 61.22 71.56 68.51 64 Views 94.77 89.51 69.3 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 128 Views 94.77 89.62 69.15 71.34 87.28 27.15 68.15 45.86 61.19 71.87 68.64 Fixed shot capacity. We search the optimal total shot capicity in BoostAdapter. We also find that fixing the cache size to be 3 can generalize well in different task settings, as shown in Table 17 and Table 18. F More Qualitative Results More qualitative results are provided in Fig. 6. 24 Table 17: Results of fixed shot capacity on the OOD benchmark. Imagenet-V2 Imagenet-Sketch Imagenet-A Imagenet-R Average CLIP 60.86 46.09 47.87 73.98 57.20 CLIP+TPT 64.35 47.94 54.77 77.06 60.81 PromptAlign 65.29 50.23 59.37 79.33 63.55 TDA 64.67 50.54 60.11 80.24 63.89 BoostAdapter-Fixed 65.13 50.66 63.96 80.44 65.05 BoostAdapter-Search 65.03 50.66 64.27 80.64 65.15 Table 18: Results of fixed shot capacity on the Cross-Domain Benchmark. Caltech Pets Cars Flowers Food101 Aircraft SUN397 DTD EuroSAT UCF101 Average CLIP 93.35 88.25 65.48 67.44 83.65 23.67 62.59 44.27 42.01 65.13 63.58 CLIP+TPT 94.16 87.79 66.87 68.98 84.67 24.78 65.50 47.75 42.44 68.04 65.10 PromptAlign 94.01 90.76 68.50 72.39 86.65 24.80 67.54 47.24 47.86 69.47 66.92 TDA 94.24 88.63 67.28 71.42 86.14 23.91 67.62 47.40 58.00 70.66 67.53 BoostAdapter-Fixed 94.77 88.85 69.30 71.66 87.17 27.00 67.64 44.33 61.22 69.73 68.17 BoostAdapter-Search 94.77 89.51 69.30 71.66 87.17 27.45 68.09 45.69 61.22 71.93 68.68 A380 Test Sample Boosting Sample 777-200 Test Sample Boosting Sample Industrial Buildings Highway or Road Aircraft EuroSAT Test Sample Boosting Sample mitten cottontail rabbit ImagNet-A Figure 6: More qualitative results on ImagNet-A, Aircraft and EuroSAT. 25 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,426	The Group Robustness is in the Details: Revisiting Finetuning under Spurious Correlations Tyler LaBonte1∗ John C. Hill2 Xinchen Zhang2 Vidya Muthukumar2,1 Abhishek Kumar† 1H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology 2School of Electrical and Computer Engineering, Georgia Institute of Technology Abstract Modern machine learning models are prone to over-reliance on spurious correlations, which can often lead to poor performance on minority groups. In this paper, we identify surprising and nuanced behavior of finetuned models on worstgroup accuracy via comprehensive experiments on four well-established benchmarks across vision and language tasks. We first show that the commonly used class-balancing techniques of mini-batch upsampling and loss upweighting can induce a decrease in worst-group accuracy (WGA) with training epochs, leading to performance no better than without class-balancing. While in some scenarios, removing data to create a class-balanced subset is more effective, we show this depends on group structure and propose a mixture method which can outperform both techniques. Next, we show that scaling pretrained models is generally beneficial for worst-group accuracy, but only in conjunction with appropriate class-balancing. Finally, we identify spectral imbalance in finetuning features as a potential source of group disparities — minority group covariance matrices incur a larger spectral norm than majority groups once conditioned on the classes. Our results show more nuanced interactions of modern finetuned models with group robustness than was previously known. Our code is available at https://github.com/tmlabonte/revisiting-finetuning. 1 Introduction Classification performance in machine learning is sensitive to spurious correlations: patterns which are predictive of the target class in the training dataset but not at test time. For example, in computer vision tasks, neural networks are known to utilize the backgrounds of images as proxies for their content [1, 50, 68]. Beyond simple settings, spurious correlations have been identified in highconsequence applications such as criminal justice [8], medicine [70], and facial recognition [33]. In particular, a model’s reliance on spurious correlations disproportionately affects its accuracy on minority groups which are under-represented in the training dataset; we therefore desire maximizing the model’s group robustness, quantified by its minimum accuracy on any group [50]. The standard workflow in modern machine learning involves initializing from a pretrained model and finetuning on the downstream dataset using empirical risk minimization (ERM) [62], which minimizes the average training loss. When group annotations are available in the training dataset, practitioners utilize a rich literature of techniques to improve worst-group accuracy (WGA) [50, 39, 26]. However, group annotations are often unknown or problematic to obtain (e.g., due to financial, privacy, or fairness concerns). While group robustness methods have been adapted to work without group annotations [31, 72, 47, 29], they remain complex variants on the standard finetuning procedure. ∗Corresponding author. Email: tlabonte@gatech.edu. †Work done at Google DeepMind. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Hence, it is often unclear to what extent the WGA dynamics of these methods are attributable to details of model finetuning. In this paper, we take a complementary approach to the methodological literature by pursuing a comprehensive understanding of the fundamental properties of model finetuning on four wellestablished group robustness benchmarks across vision and language tasks. We focus especially on the conjunction of model scaling and class-balancing — which was recently shown to greatly improve robustness on some datasets [22] — on the worst-group accuracy of the ERM baseline. These considerations enable us to isolate the impact of group disparities on worst-group accuracy, thereby revealing more nuanced behaviors of finetuned models than previously known. In particular, we challenge overarching narratives that “overparameterization helps or hurts distributional robustness” and show striking differences in finetuning performance depending on class-balancing methodology. In more detail, our main contributions include: • Identifying two failure modes of common class-balancing techniques during finetuning: (1) mini-batch upsampling and loss upweighting experience catastrophic collapse with standard hyperparameters on benchmark datasets, and (2) removing data to create a class-balanced subset can harm WGA for certain datasets. • Proposing a mixture balancing method which combines the advantages of two class-balancing techniques and can improve baseline WGA beyond either method. • Showing that while overparameterization can harm WGA in certain cases, model scaling is generally beneficial for robustness when applied in conjunction with appropriate pretraining and class-balancing. • Identifying a spectral imbalance in the top eigenvalues of the group covariances — even when the classes are balanced — and showing that minority group covariance matrices consistently have larger spectral norm conditioned on the classes. 1.1 Related work Here we provide a brief summary of related work along three axes. Throughout the paper, we also provide detailed contextualizations of our results with the most closely related work. Spurious correlations. The proclivity of ERM to rely on spurious correlations has been widely studied [12, 37]. Rectifying this weakness is an important challenge for real-world deployment of machine learning algorithms, as spurious correlations can exacerbate unintended bias against demographic minorities [20, 2, 57, 17, 5] or cause failure in high-consequence applications [33, 8, 70, 42]. Reliance on spurious correlations manifests in image datasets as the usage of visual shortcuts including background [1, 50, 68], texture [11], and secondary objects [48, 52, 54], and in text datasets as the usage of syntactic or statistical heuristics as a substitute for semantic understanding [14, 41, 36]. Class-balancing and group robustness. Group-balancing, or training with an equal number of samples from each group, has been proposed as a simple yet effective method to improve robustness to spurious correlations [17, 51, 6, 55]. However, group-balancing requires group annotations, which are often unknown or problematic to obtain [31, 72, 47, 29]. On the other hand, class-balancing, or training with an equal number of samples from each class, is a well-studied method in long-tailed classification [24, 15, 4]. Recent work has shown that class-balancing is a surprisingly powerful method for improving worst-group accuracy which does not require group annotations [22, 29, 7, 53]. In particular, [22] study the WGA dynamics of two common class-balancing methods: removing data from the larger classes (which we call subsetting) and upsampling the smaller classes (which we call upsampling). Our results complement those of [22] and show more nuanced effects of class-balancing than previously known; we provide additional contextualization with [22] in Section 3.1. We show similar nuanced behavior of upweighting smaller classes in the loss function, a popular method in the group-balancing setting [31, 47, 55] which [22] did not study. Overparameterization and distributional robustness. While the accepted empirical wisdom is that overparameterization improves in-distribution test accuracy [40, 71], the relationship between overparameterization and robustness is incompletely understood. [51] considered a class of ResNet18 architectures and showed that increasing model width reduces worst-group accuracy on the 2 Waterbirds and CelebA datasets when trained with class-imbalanced ERM — this contrasts with the improvement in average accuracy widely observed in practice (see, e.g., [38]). Conversely, [19] showed a benefit of overparameterization in robustness to “natural” covariate shifts, which are quite different from spurious correlations [27]. On the mathematical front, [59, 35] showed that overparameterization in random feature models trained to completion improves robustness to a wide class of covariate shifts.However, both the optimization trajectory and statistical properties of random features are very different from neural networks (see, e.g., [13]). Closely related to our work, [46] investigated pretrained ResNet, VGG, and BERT models, and showed that overparameterization does not harm WGA. Our results complement those of [46] with a richer setup and show that class-balancing — which they do not study — can greatly impact model scaling behavior. 2 Preliminaries Setting. We consider classification tasks with input domain Rn and target classes Y ⊂N. Suppose S is a set of spurious features such that each example x ∈Rn is associated with exactly one feature s(x) ∈S. The dataset is then partitioned into groups G, defined by the Cartesian product of classes and spurious features G = Y × S. Given a dataset of m training examples, we define the set of indices of examples which belong to some group g ∈G or class y ∈Y by Ωg ⊆{1, . . . , m} and Ωy ⊆{1, . . . , m}, respectively. Then, the majority group(s) is defined by the group(s) that maximize \|Ωg\|. All other groups are designated as minority groups. Further, the worst group(s)3 is defined by the group(s) which incur minimal test accuracy. We define majority and minority classes similarly. Because groups are defined by the Cartesian product of classes and spurious features, all training examples in a particular group are identically labeled, and therefore a group is a subset of a class. We desire a model which, despite group imbalance in the training dataset, enjoys roughly uniform performance over G. Therefore, we evaluate worst-group accuracy (WGA), i.e., the minimum accuracy among all groups [50]. We will also be interested in the relative performance on groups within the same class, and we thereby define the majority group within a class y ∈Y as the group which maximizes \|Ωg\| over all g ∈{g ∈G : y ∈g}. Other groups are designated as the minority groups within that class. For example, referring to the Waterbirds section of Table 2, groups 1 and 2 are the minority groups within classes 0 and 1, respectively. Class-balancing. A dataset is considered to be class-balanced if it is composed of an equal number of training examples from each class in expectation over the sampling probabilities. We compare three class-balancing techniques: subsetting, upsampling, and upweighting. We describe each below: • In subsetting, every class is set to the same size as the smallest class by removing the appropriate amount of data from each larger class uniformly at random. This procedure is performed only once, and the subset is fixed prior to training. • In upsampling, the entire dataset is utilized for training with a typical stochastic optimization algorithm, but the sampling probabilities of each class are adjusted so that mini-batches are class-balanced in expectation. To draw a single example, we first sample y ∼Unif(Y), then sample x ∼ˆp(· \| y) where ˆp is the empirical distribution on training examples. • In upweighting, the minority class samples are directly upweighted in the loss function according to the ratio of majority class data to minority class data, called the class-imbalance ratio. Specifically, if the loss function is ℓ(f(x), y) for model f, example x, and class label y, the upweighted loss function is γℓ(f(x), y) where γ is defined as the class-imbalance ratio for minority class data and 1 for majority class data. It is worth noting that upweighting is equivalent to upsampling in expectation over the sampling probabilities. Note that the terminology for these class-balancing techniques is not consistent across the literature. For example, [22] call subsetting subsampling (denoted SUBY) and upsampling reweighting (denoted RWY). On the other hand, [55] call (group-wise) subsetting downsampling and use upweighting to describe increasing the weight of minority group samples in the loss function. 3Note that, as is standard in the empirical literature on distributional robustness, majority, minority and worst groups are defined with respect to the empirical training distribution, as this is all that we have access to. Moreover, test accuracy is typically maximized by the majority group and minimized by a minority group, though this is not always the case. 3 Datasets and models. We study four classification datasets, two in the vision domain and two in the language domain, which are well-established as benchmarks for group robustness. We summarize each dataset below and provide additional numerical details in Appendix A.1. • Waterbirds [64, 63, 50] is an image dataset wherein birds are classified as land species (“landbirds”) or water species (“waterbirds”). The spurious feature is the image background: more landbirds are present on land backgrounds and vice versa.4 • CelebA [33, 50] is an image dataset classifying celebrities as blond or non-blond. The spurious feature is gender, with more blond women than blond men in the training dataset. • CivilComments [3, 27] is a language dataset wherein online comments are classified as toxic or non-toxic. The spurious feature is the presence of one of the following categories: male, female, LGBT, black, white, Christian, Muslim, or other religion.5 More toxic comments contain one of these categories than non-toxic comments, and vice versa. • MultiNLI [65, 50] is a language dataset wherein pairs of sentences are classified as a contradiction, entailment, or neither. The spurious feature is a negation in the second sentence — more contradictions have this property than entailments or neutral pairs. Waterbirds is class-imbalanced with a majority/minority class ratio of 3.31:1, CelebA a ratio of 5.71:1, and CivilComments a ratio of 7.85:1. MultiNLI is class-balanced a priori. Since the Waterbirds dataset has a shift in group proportion from train to test, we weight the group accuracies by their proportions in the training set when reporting the test average accuracy [50]. We utilize ResNet [18], ConvNeXt-V2 [67], and Swin Transformer [32] models pretrained on ImageNet-1K [49] for Waterbirds and CelebA, and a BERT [9] model pretrained on Book Corpus [73] and English Wikipedia for CivilComments and MultiNLI. We use the AdamW optimizer [34] for finetuning on three independent seeds, randomizing both mini-batch order and any other stochastic procedure such as subsetting, and we report error bars corresponding to one standard deviation. We do not utilize early-stopping: instead, to consider the impact of overparameterization in a holistic way, we train models to completion to properly measure the overfitting effect.6 This can result in longer training than commonly seen in the literature (e.g., we finetune on CelebA for about 3× more gradient steps than is standard). See Appendix A.2 for further training details. 3 Nuanced effects of class-balancing on group robustness We now present our first set of results, which shows that the choice of class balancing method greatly impacts the group robustness of the ERM baseline. 3.1 Catastrophic collapse of class-balanced upsampling and upweighting In a recent paper, [29] observed that contrary to the central hypothesis underlying the Just Train Twice method [31], the worst-group accuracy of ERM decreases dramatically with training epochs on CelebA and CivilComments; however, they provide no explanation for this phenomenon. In this section, we show that this degradation of WGA is due to their choice of class-balancing method (i.e., upsampling). Specifically, ERM finetuned with upsampling experiences a catastrophic collapse in test WGA over the course of training, a phenomenon that was previously only noticed in synthetic datasets with a linear classifier [22]. Moreover, while [22] state that class-balanced subsetting is not recommended in practice, we show that it can in fact improve WGA conditional on the lack of of a small minority group within the majority class. Finally, we show that class-balanced upweighting — a popular technique which [22] do not study — experiences a similar WGA collapse as upsampling. We finetune a ConvNeXt-V2 Base on Waterbirds and CelebA and a BERT Base on CivilComments, and we compare the subsetting, upsampling, and upweighting techniques to a class-imbalanced 4We note that the Waterbirds dataset is known to contain incorrect labels [56]. We report results on the original, un-corrected version as is standard in the literature. 5This version of CivilComments has four groups, used in this work and by [50, 22, 23, 26, 29]. There is another version where the identity categories are not collapsed into one spurious feature; that version is used by [31, 72, 47]. Both versions use the WILDS split [27]. 6To be more specific, we finetune ConvNeXt-V2 Base roughly to a training loss of 10−4 on Waterbirds and 10−3 on CelebA, and BERT Base roughly to a training loss of 10−3 on CivilComments and 10−2 on MultiNLI. 4 (a) Waterbirds (b) CelebA (c) CivilComments Figure 1: Class-balanced upsampling and upweighting experience catastrophic collapse. We compare subsetting, wherein data is removed to set every class to the same size as the smallest class, upsampling, wherein the sampling probabilities of each class are adjusted so that the mini-batches are class-balanced in expectation, and upweighting, wherein the loss for the smaller classes is scaled by the class-imbalance ratio. We observe a catastrophic collapse over the course of training of upsampling and upweighting on CelebA and CivilComments, the more class-imbalanced datasets. Subsetting reduces WGA on Waterbirds because it removes data from the small minority group within the majority class. MultiNLI is class-balanced a priori, so we do not include it here. baseline. Our results are displayed in Figure 1, with additional models in Appendix B. On CelebA and CivilComments, the more class-imbalanced datasets, upsampling and upweighting both experience catastrophic collapse over the course of training. We believe this collapse is caused by overfitting to the minority group within the minority class; any individual point from this group is sampled far more often during upsampling and weighted far more heavily during upweighting, causing overfitting during long training runs. In fact, upsampling does even worse on CelebA than observed in [29] because we train 3× longer to ensure convergence. With that said, optimally tuned early-stopping appears to mitigate the collapse (as previously noticed by [22] in a toy setting). Our experiments also highlight a previously unnoticed disadvantage of class-balanced subsetting: if there is a small minority group in the majority class, subsetting will further reduce its proportion and harm WGA. For example, in the Waterbirds dataset, the species (landbirds/waterbirds) is the class label and the background (land/water) is the spurious feature; landbirds/water is a small minority group within the majority class (landbirds). When landbirds is cut by 3.31×, the landbirds/water group greatly suffers, harming WGA. On the other hand, in the CelebA dataset, the hair color (non-blond/blond) is the class label and the gender (female/male) is the spurious feature; the only small minority group is blond/male, while the groups are nearly balanced in the majority class. In this case, subsetting preserves blond/male examples and increases their proportion, helping WGA. Finally, while upsampling and upweighting have similar WGA dynamics – perhaps as expected, as they are equivalent in expectation over the sampling mechanism — both differ greatly from subsetting. Recently, [55] proved a theoretical equivalence between subsetting and upsampling of the groups in the population setting, i.e., assuming access to the training distribution. The equivalence of upsampling and upweighting would then imply that all three objectives are optimized by the same solution. However, our results suggest this may not hold in the real-world empirical setting, where subsetting has distinctly different behavior, and model parameters may outnumber training examples. As previously mentioned, this may be due to overfitting to minority class data repeated often during training; theoretically investigating this discrepancy is an important future direction. Contextualization with previous work. Our observations explain the decrease in WGA of CelebA and CivilComments noticed by [29], a phenomenon which they left unresolved. Our result implies that group robustness methods which assume that WGA increases during training, such as Just Train Twice [31], may only be justified with appropriate class-balancing. [22] show that upsampling can cause catastrophic collapse in WGA, but only in a synthetic dataset with a linear classifier. In realistic datasets, [22] perform extensive hyperparameter tuning (using group labels, which may be unrealistic) to achieve good results with upsampling, while we show that catastrophic collapse can occur in the same datasets when standard hyperparameters are used. Moreover, [22] state that class-balanced 5 (a) Waterbirds (b) CelebA (c) CivilComments Figure 2: Mixture balancing mitigates catastrophic collapse of upsampling and upweighting. We propose a class-balanced mixture method, which combines subsetting and upsampling by first drawing a class-imbalanced subset uniformly at random from the dataset, then adjusting sampling probabilities so that mini-batches are balanced in expectation. Our method increases exposure to majority class data without over-sampling the minority class. Remarkably, mixture balancing outperforms all three class-balancing methods on Waterbirds and CivilComments, and while it does not outperform subsetting on CelebA, it significantly alleviates the WGA collapse experienced by upsampling and upweighting. MultiNLI is class-balanced a priori, so we do not include it here. subsetting is not recommended in practice, but we show that subsetting can be effective except when there is a small minority group within the majority class, a previously unnoticed nuance. Finally, we show that subsetting experiences different WGA dynamics from upsampling and upweighting in the empirical setting, suggesting additional complexity compared to the population setting results of [55]. Without extensive tuning, class-balanced upsampling and upweighting can induce WGA no better than without class-balancing. While class-balanced subsetting can improve WGA, practitioners should use caution if a small minority group is present within the majority class. 3.2 Mixture balancing: interpolating between subsetting and upsampling To mitigate the catastrophic collapse of class-balanced upsampling and upweighting, we propose a simple mixture method which interpolates between subsetting and upsampling. Our method increases exposure to majority class data without over-sampling the minority class, which can improve WGA and mitigate overfitting to the minority group. We first create a data subset with a specified classimbalance ratio by removing data from the larger classes uniformly at random until the desired (smaller) ratio is achieved. Next, we perform ERM finetuning on this subset by adjusting sampling probabilities so that mini-batches are balanced in expectation. Using a class-imbalance ratio of 1:1 reduces to subsetting, and using the original class-imbalance ratio reduces to upsampling. We finetune ConvNeXt-V2 Base on Waterbirds and CelebA and BERT Base on CivilComments, and we compare our class-balanced mixture method to the subsetting, upsampling, and upweighting techniques. The results of our experiments are displayed in Figure 2. We plot the performance of our mixture method with the best class-imbalance ratio during validation; an ablation study varying the ratio is included in Appendix B. Remarkably, mixture balancing outperforms all three class-balancing methods on Waterbirds and CivilComments, and while it does not outperform subsetting on CelebA, it significantly alleviates the WGA collapse experienced by upsampling. Next, we perform an ablation of the necessity of subsetting in mixture balancing. We compare our method with an implementation which eschews subsetting, instead adjusting sampling probabilities so that the mini-batches have a particular class ratio in expectation. For example, instead of performing upsampling on a 2:1 class-imbalanced subset, we upsample the majority class by a ratio of 2:1 on the entire dataset. The results of our ablation are included in Appendix B; our mixture method outperforms the alternative, which incompletely corrects for class imbalance. 6 Table 1: Mixture balancing is robust to model selection without group annotations. We compare the best class-balancing method during validation with and without group annotations. Both worstclass accuracy [69] and the bias-unsupervised validation score of [60] are effective for model selection without group annotations, often choosing the same method or mixture ratio as worst-group accuracy (WGA) validation. We list the method maximizing each metric and its average WGA over 3 seeds. Validation Metric Group Anns Waterbirds CelebA CivilComments Bias-unsupervised Score ✗ Upsampling (79.9) Subsetting (74.1) Mixture 3:1 (77.6) Worst-class Accuracy ✗ Mixture 2:1 (81.1) Subsetting (74.1) Mixture 3:1 (77.6) Worst-group Accuracy ✓ Mixture 2:1 (81.1) Subsetting (74.1) Mixture 3:1 (77.6) Note on validation. In Figure 2, we plot the best class-imbalance ratio achieved using validation on a group annotated held-out set. While this is a common assumption in the literature [50, 31, 23, 26], it is nevertheless unrealistic when the training set does not have any group annotations. Therefore, we compare with both worst-class accuracy [69] and the bias-unsupervised validation score of [60], which do not use any group annotations for model selection. In Table 1 we list the method which maximizes each validation metric as well as its average WGA. Overall, we show both methods are effective for model selection, often choosing the same method or mixture ratio as WGA validation. Contextualization with previous work. Increasing exposure to majority class data without oversampling the minority class was previously explored by [26], who proposed averaging the weights of logistic regression models trained on ten independent class-balanced subsets. However, this method only works for linear models — as nonlinear models cannot be naively averaged — and requires multiple training runs, which is computationally infeasible for neural networks. In comparison, our mixture method is a simple and efficient alternative which extends easily to nonlinear models. The catastrophic collapse of class-balanced upsampling and upweighting can be mitigated by a mixture method. It increases exposure to majority class data without over-sampling the minority class and can improve baseline WGA beyond either technique. 4 Model scaling improves WGA of class-balanced finetuning The relationship between overparameterization and group robustness has been well-studied, with often conflicting conclusions [51, 59]. In this section, we study the impact of model scaling on worst-group accuracy in a new setting — finetuning pretrained models — which more closely resembles practical use-cases. Importantly, we evaluate the impact of model scaling in conjunction with class-balancing to isolate the impact of group inequities on WGA as a function of model size. We find that with appropriate class-balancing, overparameterization can in fact significantly improve WGA over a very wide range of parameter scales, including before and after the interpolation threshold. On the other hand, scaling on imbalanced datasets or with the wrong balancing technique can harm robustness. We take advantage of advancements in efficient architectures [61, 67] to finetune pretrained models in a wide range of scales from 3.4M to 101M parameters. We study six different sizes of ImageNet1Kpretrained ConvNeXt-V2 and five different sizes of Book Corpus/English Wikipedia pretrained BERT; specifications for each model size are included in Appendix A.2. Our results are displayed in Figure 3, and we include results for Swin Transformer in Appendix C. We find that model scaling is beneficial for group robustness in conjunction with appropriate classbalancing, with improvements of up to 12% WGA for interpolating models and 40% WGA for non-interpolating models. This comes in stark contrast to scaling on class-imbalanced datasets or with the wrong class-balancing technique, which shows either a neutral trend or decrease in WGA — the most severe examples being on CivilComments. With respect to interpolating models, CivilComments WGA decreases slightly after the interpolation threshold, while Waterbirds and CelebA continue to improve well beyond interpolation; on the other hand, BERT never interpolates MultiNLI, greatly increasing robustness at scale. It is unclear why Waterbirds and CelebA experience 7 (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 3: Scaling class-balanced pretrained models can improve worst-group accuracy. We finetune each model size starting from pretrained checkpoints and plot the test worst-group accuracy (WGA) as well as the interpolation threshold, where model reaches 100% training accuracy. We find model scaling is generally beneficial for WGA only in conjunction with appropriate class-balancing, and scaling on imbalanced datasets or with the wrong method can harm robustness. Note MultiNLI is class-balanced a priori and is not interpolated. See Appendix C for training accuracy plots. (a) Waterbirds (last layer only) (b) Waterbirds (finetuning) (c) CelebA (finetuning) Figure 4: Class-balancing greatly affects ResNet scaling results of [46]. We contrast the ResNet scaling behavior of [46] — who do not use class-balancing — to the scaling of class-balanced ResNets. We finetune each model size starting from pretrained checkpoints and plot the test worstgroup accuracy (WGA), as well as the interpolation threshold, where the model reaches 100% training accuracy. On Waterbirds, we find that class-balancing enables a much more beneficial trend during model scaling. On CelebA, class-balancing greatly increases baseline WGA but does not affect scaling behavior (in contrast to the ConvNeXt-V2 plots in Figure 3). We use SGD for last-layer training and AdamW for full finetuning. See Appendix C for training accuracy plots. different behavior from CivilComments interpolation — the toy linear model of [51] suggests a benign “spurious-core information ratio”, but a complete understanding is left to future investigation. The most closely related work to ours is [46], who study the impact of scaling pretrained ResNet models on group robustness. However, because their experiments do not employ any form of class-balancing, their conclusions may be overly pessimistic. We replicate their experiments with our hyperparameters and contrast with our results using class-balancing in Figure 4. We find that class-balancing greatly affects their results: on Waterbirds, class-balancing enables a much more beneficial trend during model scaling regardless of whether a linear probe or the entire model is trained. Moreover, while class-balancing increases baseline WGA on CelebA but does not affect scaling behavior, we observe a more positive WGA trend when scaling ConvNeXt-V2 in Figure 3. Contextualization with previous work. While previous work has primarily studied either linear probing of pretrained weights or training small models from scratch [51, 59], we study full finetuning of large-scale pretrained models and show that class-balancing can have a major impact on scaling behavior. We compare directly with the most closely related work, [46], and show that class-balancing can either induce strikingly different scaling behavior or greatly increase baseline WGA. Overall, training with class-balancing allows us to isolate the impact of group inequities on robustness and more precisely observe the often-beneficial trend of model scaling for worst-group accuracy. 8 (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 5: Group disparities are visible in the top eigenvalues of the group covariance matrices. We visualize the mean, across 3 experimental trials, of the top 10 eigenvalues of the group covariance matrices for a ConvNeXt-V2 Nano finetuned on Waterbirds and CelebA and a BERT Small finetuned on CivilComments and MultiNLI. The standard deviations are omitted for clarity. The models are finetuned using the best class-balancing method from Section 3 for each dataset. The group numbers are detailed in Table 2 and the minority groups within each class are denoted with an asterisk. The largest λ1 in each case belongs to a minority group, though it may not be the worst group, and minority group eigenvalues are overall larger than majority group eigenvalues within the same class. While overparameterization can sometimes harm WGA, pretraining and appropriate classbalancing make scaling generally beneficial. Moreover, modern language datasets are complex enough that standard models do not interpolate, greatly improving robustness at scale. 5 Spectral imbalance may exacerbate group disparities In a recent paper, [25] propose spectral imbalance of class covariance matrices, or differences in their eigenspectrum, as a source of disparities in accuracy across classes even when balanced. Here, we examine whether similar insights hold in the group robustness setting. Our observations reveal surprising nuances in the behavior of group-wise spectral imbalance; nevertheless, we conclude that spectral imbalance may play a similar role in modulating WGA after class-balancing is applied. Let us denote by zi the feature vector corresponding to a sample xi (i.e., the vectorized output of the penultimate layer). Recall from Section 2 that Ωg is the set of indices of samples which belong to group g. We further define ¯zg to be the empirical mean of features with group g. To obtain the estimated eigenspectrum, we first compute the empirical covariance matrix for group g ∈G by Σg = 1 \|Ωg\| X i∈Ωg (zi −¯zg)(zi −¯zg)⊤. We then compute the eigenvalue decomposition Σg = VgΛgV−1 g , where Λg is a diagonal matrix with non-negative entries λ(g) i and the columns of Vg are the eigenvectors of Σg. Without loss of generality, we assume λ(g) 1 ≥λ(g) 2 ≥· · · ≥λ(g) m where m is the rank of Σg. We compute the group covariance matrices using a ConvNeXt-V2 Nano model for Waterbirds and CelebA, and a BERT Small model for CivilComments and MultiNLI.We plot the top 10 eigenvalues of each group covariance matrix in Figure 5. Even though we finetune with class-balancing, disparities in eigenvalues across groups are clearly visualized in Figure 5, especially for the largest eigenvalues. We include extensions to the top 50 eigenvalues and class covariance matrices in Appendix D. Close observation of Figure 5 yields interesting findings. First, the group g∗that maximizes λ(g) 1 in each case belongs to a minority group; though, importantly, it may not belong to the worst group. This is different from the findings of [25], who showed that the largest eigenvalues typically belong to the worst-performing class. Second, we find that minority group eigenvalues are overall larger than majority group eigenvalues, but only when conditioned on the class. A majority group belonging to one class may have larger eigenvalues than a minority group belonging to another class, but there exists a consistent spectral imbalance between majority and minority groups within the same class.7 7For example, in Figure 5c, the spectrum for group 3 (the majority group within class 1) is larger than the spectrum for group 1 (the minority group within class 0). However, conditioning on the class, we find that the 9 (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 6: Group-wise spectral imbalance is apparent once conditioned on the classes. We plot the mean and standard deviation, across 3 experimental trials, of the intra-class spectral norm ratio ρ(y), or the ratio of the top eigenvalues of the minority and majority group covariance matrices, for each class y ∈Y. We compute this metric using a finetuned ConvNeXt-V2 Nano on Waterbirds and CelebA and a finetuned BERT Small on CivilComments and MultiNLI, each using the best class-balancing method from Section 3 for each dataset. The key observation is that ρ(y) is at least one for all classes y ∈Y (except a single seed for class 0 on CelebA), illustrating a group disparity captured by the eigenspectrum once we condition on the classes. To quantify this group-wise spectral imbalance, we introduce a new metric called the intra-class spectral norm ratio. Suppose gmin(y) and gmaj(y) are the minority and majority groups within a particular class y ∈Y. Then, we define the intra-class spectral norm ratio by ρ(y) := λ(gmin(y)) 1 /λ(gmaj(y)) 1 . While ρ(y) only considers the top eigenvalue and not the entire spectrum, the absolute magnitude of individual eigenvalues was found in [25] to correlate best with worst-class accuracy. We note that ρ(y) considers only the top eigenvalue and not the entire spectrum, since the magnitude of the top eigenvalues was found in [25] to correlate best with worst-class accuracy. We plot the intra-class spectral norm ratios for each dataset in Figure 6; notably, they are always at least one (except for a single seed on CelebA), showing the group disparity captured by the eigenspectrum. Finally, in Table 5 (deferred to Appendix D), we compare the class with the largest ρ(y) to the class with the largest disparity in group test accuracies, i.e., Acc(gmaj(y)) −Acc(gmin(y)). We see that in most cases these classes correspond, suggesting an explanatory power of the intra-class spectral norm ratio. In particular, this correspondence is consistent throughout all trials of CelebA and CivilComments, the most class-imbalanced datasets we study. Contextualization with previous work. Our spectral analysis of the group covariance matrices is inspired by [25]. We both study class-balanced settings, with the key difference that they study class disparities instead of group disparities. However, we show a more nuanced impact of spectral imbalance across both classes and groups, i.e., spectral imbalance is more prevalent between majority and minority groups within to the same class, rather than across groups globally. Spectral imbalance in the group covariance matrices may exacerbate group disparities even when the classes are balanced. While the worst-group covariance may not have largest spectral norm, the minority group spectra are consistently larger conditioned on the class. 6 Discussion In this paper, we identified nuanced impacts of class-balancing and model scaling on worst-group accuracy, as well as a spectral imbalance in the group covariance matrices. Overall, our work calls for a more thorough investigation of generalization in the presence of spurious correlations to unify the sometimes contradictory perspectives in the literature. We hope that, as the community continues to develop group robustness methods with increasing performance and complexity, researchers and practitioners alike remain cognizant of the disproportionate impact of the details. spectrum for group 2 (the minority group within class 1) is larger than that of group 3, and the spectrum of group 1 is larger than that of group 0 (the majority group within class 0). 10 Acknowledgments. We thank Google Cloud for the gift of compute credits, Jacob Abernethy for additional compute assistance, and Chiraag Kaushik for helpful discussions. T.L. acknowledges support from the DoD NDSEG Fellowship. V.M. acknowledges support from the NSF (awards CCF-223915 and IIS-2212182), Google Research, Adobe Research and Amazon Research. 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[73] Yukun Zhu, Ryan Kiros, Rich Zemel, Ruslan Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler. Aligning books and movies: Towards story-like visual explanations by watching movies and reading books. In International Conference on Computer Vision (ICCV), 2015. 15 A Additional Details for Section 2 A.1 Dataset Composition Table 2: Dataset composition. We study four well-established benchmarks for group robustness across vision and language tasks. The class probabilities change dramatically when conditioned on the spurious feature. Note that Waterbirds is the only dataset that has a distribution shift and MultiNLI is the only dataset which is class-balanced a priori. The minority groups within each class are denoted by an asterisk in the “Num” column. Probabilities may not sum to 1 due to rounding. Dataset Group g Training distribution ˆp Data quantity Num Class y Spurious s ˆp(y) ˆp(g) ˆp(y\|s) Train Val Test Waterbirds 0 landbird land .768 .730 .984 3498 467 2225 1* landbird water .038 .148 184 466 2225 2* waterbird land .232 .012 .016 56 133 642 3 waterbird water .220 .852 1057 133 642 CelebA 0 non-blond female .851 .440 .758 71629 8535 9767 1* non-blond male .411 .980 66874 8276 7535 2 blond female .149 .141 .242 22880 2874 2480 3* blond male .009 .020 1387 182 180 CivilComments 0 neutral no identity .887 .551 .921 148186 25159 74780 1* neutral identity .336 .836 90337 14966 43778 2* toxic no identity .113 .047 .079 12731 2111 6455 3 toxic identity .066 .164 17784 2944 8769 MultiNLI 0 contradiction no negation .333 .279 .300 57498 22814 34597 1* contradiction negation .054 .761 11158 4634 6655 2 entailment no negation .334 .327 .352 67376 26949 40496 3* entailment negation .007 .104 1521 613 886 4 neither no negation .333 .323 .348 66630 26655 39930 5* neither negation .010 .136 1992 797 1148 A.2 Training details We utilize ResNet [18], ConvNeXt-V2 [67], and Swin Transformer [32] models pretrained on ImageNet-1K [49] for Waterbirds and CelebA, and a BERT [9] model pretrained on Book Corpus [73] and English Wikipedia for CivilComments and MultiNLI. These pretrained models are used as the initialization for ERM finetuning under the cross-entropy loss. We use standard ImageNet normalization with standard flip and crop data augmentation for the vision tasks and BERT tokenization for the language tasks [23]. Our implementation uses the following packages: NumPy [16], PyTorch [44, 45], Lightning [10], TorchVision [58], Matplotlib [21], Transformers [66], and Milkshake [28]. To our knowledge, the licenses of Waterbirds and CelebA are unknown. CivilComments is released under the CC0 license, and information about MultiNLI’s license may be found in [65]. Our experiments were conducted on four Google Cloud Platform (GCP) 16GB Nvidia Tesla P100 GPUs and two local 24GB Nvidia RTX A5000 GPUs. The spectral imbalance experiments in Section 5 were conducted on a GCP system with a 16-core CPU and 128GB of RAM. We believe our work could be reproduced for under $5000 in GCP compute credits, with a majority of that compute going towards running experiments over multiple random seeds. We list model scaling parameters in Table 3 and hyperparameters used for each dataset in Table 4. ConvNeXt-V2, ResNet and Swin Transformers are composed of four separate “stages”, and we list the depths of these stages individually in Table 3. All of these configurations are standard in the literature. The smaller BERT models were introduced by [61]. We perform model selection only for our mixture balancing method (see Table 1) and not for the ERM finetuning hyperparameters, most of which are standard in the literature [50, 22, 23]. For the last-layer training experiments in Figure 4 and Figure 11, we use SGD with learning rate 10−3 and train for 20 epochs. Different from previous work, we train CelebA for about 3× more gradient steps than usual to ensure convergence, and we 16 double the batch size for CivilComments and MultiNLI to increase training stability (we also double the epochs to hold the number of gradient steps constant). Table 3: Model scaling parameters. (a) ConvNeXt-V2 parameters. Size Width Depth (4 stages) Params Atto 40 (2, 2, 6, 2) 3.4M Femto 48 (2, 2, 6, 2) 4.8M Pico 64 (2, 2, 6, 2) 8.6M Nano 80 (2, 2, 8, 2) 15.0M Tiny 96 (3, 3, 9, 3) 27.9M Base 128 (3, 3, 27, 3) 87.7M (b) BERT parameters. Size Width Depth Params Tiny 2 128 4.4M Mini 4 256 11.2M Small 4 512 28.8M Medium 8 512 41.4M Base 12 768 109M (c) ResNet parameters. Size Width (4 stages) Depth (4 stages) Params 18 (64, 128, 256, 512) (2, 2, 2, 2) 11.2M 34 (64, 128, 256, 512) (3, 4, 6, 3) 21.3M 50 (256, 512, 1024, 2048) (3, 4, 6, 3) 23.5M 101 (256, 512, 1024, 2048) (3, 4, 23, 3) 42.5M 152 (256, 512, 1024, 2048) (3, 8, 36, 3) 58.1M (d) Swin Transformer parameters. Size Width Depth (4 stages) Params Tiny 96 (2, 2, 6, 2) 29M Small 96 (2, 2, 18, 2) 50M Base 128 (2, 2, 18, 2) 88M Table 4: ERM finetuning hyperparameters. Dataset Optimizer Initial LR LR schedule Batch size Weight decay Epochs Waterbirds AdamW 1 × 10−5 Cosine 32 1 × 10−4 100 CelebA AdamW 1 × 10−5 Cosine 32 1 × 10−4 20 CivilComments AdamW 1 × 10−5 Linear 32 1 × 10−4 20 MultiNLI AdamW 1 × 10−5 Linear 32 1 × 10−4 20 17 B Additional Experiments for Section 3 (a) Waterbirds (b) CelebA (c) CivilComments Figure 7: Mixture balancing ablation studies. We perform two ablation studies on our mixture balancing method. First, we vary the class-imbalance ratio across the x axis. On the left-hand side, using a class-imbalance ratio of 1:1 reduces to the subsetting technique; on the right-hand side, using the original class-imbalance ratio in the dataset reduces to upsampling. Second, we perform an ablation of whether subsetting is essential in mixture balancing. We plot our proposed method (which takes a subset of data based on the class-imbalance ratio, then performs upsampling) against the same method without subsetting, instead adjusting the class probabilities on the entire dataset as specified by the class-imbalance ratio. MultiNLI is class-balanced a priori, so we do not include it here. (a) Waterbirds (b) CelebA Figure 8: Balancing behavior is consistent with Swin Transformer. We demonstrate the effectiveness of our class-balanced mixture method when used in conjunction with a Swin Transformer (compare to the ConvNeXt-V2 results in Figure 2). Overall, we find our results are consistent across pretrained model families, with the model affecting the raw accuracies but typically not the relative performance of class-balancing techniques. We also corroborate the poor performance of subsetting on Waterbirds and the catastrophic collapse of upsampling and upweighting on Celeba from Figure 1. 18 (a) Waterbirds (b) CelebA Figure 9: Balancing behavior is consistent with ResNet model family. We demonstrate the effectiveness of our class-balanced mixture method on another model family, ResNet (compare to the ConvNeXt-V2 results in Figure 2). Again, we find that our results are consistent and that the model architecture affects the raw accuracies but typically not the relative performance of classbalancing techniques. We also corroborate the poor performance of subsetting on Waterbirds and the catastrophic collapse of upsampling and upweighting on Celeba from Figure 1. 19 C Additional Experiments for Section 4 (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 10: Average accuracy of scaled models. We finetune each model size starting from pretrained checkpoints and plot the train average accuracy (AA) as well as the interpolation threshold, where at least one seed of the non-class-balanced model reaches 100% training accuracy. (For example, CelebA does not interpolate with all three seeds). Average accuracy consistently increases with model size regardless of class-balancing, implying the scaling dynamics for AA and WGA are starkly different. Note that MultiNLI is class-balanced a priori and does not interpolate at any size. (a) Waterbirds (last layer only) (b) Waterbirds (finetuning) (c) CelebA (finetuning) Figure 11: Average accuracy of scaled ResNets. We contrast the ResNet scaling behavior of [46] — who do not use class-balancing — to the scaling of class-balanced ResNets. We finetune each model size starting from pretrained checkpoints and plot the train average accuracy (AA) as well as the interpolation threshold, where the model reaches 100% training accuracy. Similarly to Figure 10, average accuracy consistently increases with model size. We use SGD for last-layer training and AdamW for full finetuning. 20 (a) Waterbirds (b) CelebA Figure 12: Scaling behavior is consistent with Swin Transformer. We exhibit the model scaling behaviour of a Swin Transformer, and compare it to that of a ConvNeXt-V2 (shown in Figure 3). We see that the scaling behaviour is consistent across pretrained model families, with the model affecting the raw accuracies but not the relative performance of class-balancing techniques. 21 D Additional Experiments for Section 5 (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 13: Additional eigenvalues of the group covariance matrices. In contrast to Figure 5, we visualize the top 50 eigenvalues of the group covariance matrices for a ConvNeXt-V2 Nano finetuned on Waterbirds and CelebA and a BERT Small finetuned on CivilComments and MultiNLI. The models are finetuned using the best class-balancing method from Section 3 for each dataset. The group numbers are detailed in Table 2 and minority groups are marked with an asterisk. It becomes difficult to distinguish patterns between the groups in the lower eigenvalues, which is why we focus only on local properties of the top eigenvalues (e.g., the spectral norm and the relative ordering of the groups). With that said, it would be interesting to explore power-law decay metrics [25], which characterize relatively global properties of the eigenspectrum, in future work. (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 14: Class disparities are visible in the top eigenvalues of the class covariance matrices. We visualize the mean, across 3 experimental trials, of the top 10 eigenvalues of the class covariance matrices for a ConvNeXt-V2 Nano finetuned on Waterbirds and CelebA and a BERT Small finetuned on CivilComments and MultiNLI. The standard deviations are omitted for clarity. The models are finetuned using the best class-balancing method from Section 3 for each dataset. The class numbers are detailed in Table 2. The minority class eigenvalues for CelebA and CivilComments are overall larger, while the reverse is true for Waterbirds, a slightly different conclusion than [25]. (a) Waterbirds (b) CelebA (c) CivilComments (d) MultiNLI Figure 15: Additional eigenvalues of the class covariance matrices. In contrast to Figure 14, we visualize the top 50 eigenvalues of the class covariance matrices for a ConvNeXt-V2 Nano finetuned on Waterbirds and CelebA and a BERT Small finetuned on CivilComments and MultiNLI. The models are finetuned using the best class-balancing method from Section 3 for each dataset. The class numbers are detailed in Table 2. Similar to the groups, it becomes difficult to distinguish patterns between the classes in the lower eigenvalues, which is why we again focus only on local properties of the top eigenvalues (e.g., the spectral norm and the relative ordering of the classes). 22 (a) No Class Balancing (b) Subsetting (c) Upsampling (d) Mixture Figure 16: Group eigenvalue decay is consistent across balancing methods. We visualize the mean, across 3 experimental trials, of the top 10 eigenvalues of the group covariance matrices for a ConvNeXt-V2 Nano finetuned on Waterbirds across all class-balancing methods. The standard deviations are omitted for clarity. Overall, we found that the magnitude of the eigenvalues is significantly affected by the chosen class-balancing method. However, the relative ordering of minority/majority group eigenvalues is consistent across class-balancing techniques. We note that the most drastic changes in the spectrum are induced by the subsetting method, which has the worst WGA by far for the Waterbirds dataset. These results suggest that optimal class-balancing may bring about additional stability in the representation. (a) No Class Balancing (b) Subsetting (c) Upsampling (d) Mixture Figure 17: Class eigenvalue decay is consistent across balancing methods. We visualize the mean, across 3 experimental trials, of the top 10 eigenvalues of the class covariance matrices for a ConvNeXtV2 Nano finetuned on Waterbirds across all class-balancing methods. The standard deviations are omitted for clarity. Overall, we found that the magnitude of the eigenvalues is significantly affected by the chosen class-balancing method. However, the relative ordering of minority/majority group eigenvalues is consistent across class-balancing techniques. We note that the most drastic changes in the spectrum are induced by the subsetting method, which has the worst WGA by far for the Waterbirds dataset. These results suggest that optimal class-balancing may bring about additional stability in the representation. Table 5: Correspondence between ρ(y) and intra-class group accuracy disparity. We compare ρ(y), the intra-class spectral norm ratio, to the difference in intra-class group accuracy. Each row represents a different experimental seed. Each cell contains a tuple with the class label for the class with largest value of ρ(y) paired with the class label for the class with the largest intra-class group test accuracy disparity, i.e., Acc(gmaj(y)) −Acc(gmin(y)). We see that in most cases these classes correspond, suggesting an explanatory power of the spectral norm ratio. In particular, this correspondence is consistent throughout all trials of CelebA and CivilComments, the most classimbalanced datasets we study. Seed Waterbirds CelebA CivilComments MultiNLI 1 (1, 1) (1, 1) (0, 0) (0, 0) 2 (1, 1) (1, 1) (0, 0) (0, 1) 3 (0, 1) (1, 1) (0, 0) (2, 0) 23 (a) No Class Balancing (b) Subsetting (c) Upsampling (d) Mixture Figure 18: Spectral imbalance is consistent across balancing methods. We plot the mean and standard deviation, across 3 experimental trials, of the intra-class spectral norm ratio ρ(y), or the ratio of the top eigenvalues of the minority and majority group covariance matrices, for each class y ∈Y. We compute this metric using a finetuned ConvNeXt-V2 Nano on Waterbirds. Overall, we found that the relative magnitudes of ρ(y) are consistent across class-balancing methods. We note that the most drastic change in the relative magnitudes of ρ(y) is induced by the subsetting method, which has the worst WGA by far for the Waterbirds dataset. These results suggest that optimal class-balancing may bring about additional stability in the representation. 24 E Broader impacts, limitations, and compute Broader impacts. We hope our work contributes to the safe and equitable application of machine learning and motivates further research in ML fairness. With that said, a potential negative outcome may arise if practitioners simply apply our techniques in place of conducting rigorous bias studies. Indeed, while our methods show improved fairness with respect to the worst-group accuracy metric, it is necessary to perform comprehensive evaluations with respect to multiple additional fairness criteria prior to model deployment. Limitations. Our methods take advantage of the structure of spurious correlations; our insights would likely not transfer over to datasets which exhibit a more extreme complete correlation (i.e., contain zero minority group data) [43, 30] or to more generic out-of-distribution generalization settings. A limitation of our mixture balancing method is that to achieve optimal performance, it requires a validation set with group annotations for selection of the best class-imbalance ratio [50, 31, 23, 26]. With that said, we show in Table 1 that worst-class accuracy [69] and the bias-unsupervised validation score of [60] are sufficient for model selection in the benchmarks we study. Compute. Our experiments were conducted on two Google Cloud Platform (GCP) 16GB Nvidia Tesla P100 GPUs and two local 24GB Nvidia RTX A5000 GPUs. The spectral imbalance experiments in Section 5 were conducted on a GCP system with a 16-core CPU and 128GB of RAM. We believe our work could be reproduced for under $5000 in GCP compute credits, with a majority of that compute going towards running experiments over multiple random seeds. 25 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,427	Bridging Geometric States via Geometric Diffusion Bridge Shengjie Luo1∗, Yixian Xu1,4∗, Di He1†, Shuxin Zheng2, Tie-Yan Liu2, Liwei Wang1,3† 1State Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University 2Microsoft Research AI4Science 3Center for Data Science, Peking University 4Pazhou Laboratory (Huangpu), Guangzhou, Guangdong 510555, China luosj@stu.pku.edu.cn, xyx050@stu.pku.edu.cn, {shuz, tyliu}@microsoft.com, {dihe, wanglw}@pku.edu.cn Abstract The accurate prediction of geometric state evolution in complex systems is critical for advancing scientific domains such as quantum chemistry and material modeling. Traditional experimental and computational methods face challenges in terms of environmental constraints and computational demands, while current deep learning approaches still fall short in terms of precision and generality. In this work, we introduce the Geometric Diffusion Bridge (GDB), a novel generative modeling framework that accurately bridges initial and target geometric states. GDB leverages a probabilistic approach to evolve geometric state distributions, employing an equivariant diffusion bridge derived by a modified version of Doob’s h-transform for connecting geometric states. This tailored diffusion process is anchored by initial and target geometric states as fixed endpoints and governed by equivariant transition kernels. Moreover, trajectory data can be seamlessly leveraged in our GDB framework by using a chain of equivariant diffusion bridges, providing a more detailed and accurate characterization of evolution dynamics. Theoretically, we conduct a thorough examination to confirm our framework’s ability to preserve joint distributions of geometric states and capability to completely model the underlying dynamics inducing trajectory distributions with negligible error. Experimental evaluations across various real-world scenarios show that GDB surpasses existing state-of-theart approaches, opening up a new pathway for accurately bridging geometric states and tackling crucial scientific challenges with improved accuracy and applicability. 1 Introduction Predicting the evolution of the geometric state of a system is essential across various scientific domains [46, 88, 55, 17, 20, 101], offering valuable insights into difficult tasks such as drug discovery [25, 29], reaction modeling [9, 24], and catalyst analysis [13, 105]. Despite its critical importance, accurately predicting future geometric states of interest is challenging. Experimental approaches often face obstacles due to strict environmental requirements and physical limits of instruments [102, 3, 69]. Computational approaches seek to solve the problem by simulating the dynamics based on underlying equations [81, 88]. Though providing greater flexibility, such calculations are typically driven by first-principle methods or empirical laws, either requiring extensive computational costs [68] or sacrificing accuracy [40]. ∗Equal contribution. †Correspondence to: Di He<dihe@pku.edu.cn>, Liwei Wang <wanglw@pku.edu.cn>. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). In recent years, deep learning has emerged as a pivotal tool in scientific discovery for many fields [43, 23, 69, 107], offering new avenues for tackling this problem. One line of approach aims to train models to predict target geometric states (e.g., equilibrium states) from initial states directly and develop neural network architectures that respect inherent symmetries of geometric states, such as the equivariance of rotation and translation [104, 31, 8, 87, 89, 103]. However, this paradigm requires encoding the iterative evolution into a single-step prediction model, which lacks the ability to fully capture the system’s underlying dynamics and potentially leading to reduced accuracy. Another line of research trains machine learning force fields (MLFFs) to simulate the trajectory of geometric states over time [32, 34, 6, 70, 5, 58], showing a better efficiency-accuracy balance [15, 13, 105, 84]. Nevertheless, MLFFs are typically trained to predict intermediate labels, such as the force of the (local) current state. During inference, states are iteratively updated step by step. Since small local errors can accumulate, reliable predictions over long trajectories highly depend on the quality of intermediate labels, which cannot be guaranteed [7, 106, 30]. Therefore, an ideal solution that can precisely bridge initial and target geometric states and effectively leverage trajectory data (if available) as guidance is in great demand. In this work, we introduce Geometric Diffusion Bridge (GDB), a general framework for bridging geometric states through generative modeling. From a probabilistic perspective, predicting target geometric states from initial states requires modeling the joint state distribution across different time steps. The diffusion models [37, 99] are standard choices to achieve this goal. However, these methods ideally generate data by denoising samples drawn from a Gaussian prior distribution, which makes it challenging to bridge pre-given geometric states or leverage trajectories in a unified manner. To address the issue, we establish a novel equivariant diffusion bridge by developing a modified version of Doob’s h-transform [82, 81, 16]. The proposed stochastic differential equation (SDE) is anchored by initial and target geometric states to simultaneously model the joint state distribution and is governed by equivariant transition kernels to satisfy symmetry constraints. Intriguingly, we further demonstrate that this framework can seamlessly leverage trajectory data to improve prediction. With available trajectory data, we can construct chains of equivariant diffusion bridges, each modeling one segment in the trajectory. The segments are interconnected by properly setting the boundary conditions, allowing complete modeling of trajectory data. For model training, we derive a scalable and simulation-free matching objective similar to [59, 61, 77], which requires no computational overhead when trajectory data is leveraged. Overall, our GDB framework offers a unified solution that precisely bridges geometric states by modeling the joint state distribution and comprehensively leverages available trajectories as finegrained depiction of dynamics for enhanced performance. Mathematically, we prove that the joint distribution of geometric states across different time steps can be completely preserved by our (chains of) equivariant diffusion bridge technique, confirming its expressiveness in bridging geometric states and underscoring the necessity of design choices in our framework. Furthermore, under mild and practical assumptions, we prove that our framework can approximate the underlying dynamics governing the evolution of geometric state trajectories with negligible error in convergence, remarking on the completeness and usefulness of our framework in different scenarios. These advantages show the superiority of our framework over existing approaches. Practically, we provide a comprehensive guidance for implementing our GDB framework in realworld applications. To verify its effectiveness and generality, we conduct extensive experiments covering diverse data modalities (simple molecules & adsorbate-catalyst complex), scales (small, medium and large scales) and scenarios (with & without trajectory guidance). Numerical results show that our GDB framework consistently outperforms existing state-of-the-art machine learning approaches by a large margin. In particular, our method even surpasses strong MLFF baselines that are trained on 10× more data in the challenging structure relaxation task of OC22 [105], and trajectory guidance can further enhance our performance. The significantly superior performance demonstrates the high capacity of our framework to capture the complex evolution dynamics of geometric states and determine valuable and crucial geometric states of interest in critical real-world challenges. 2 Background 2.1 Problem Definition Our task of interest is to capture the evolution of geometric states, i.e., predicting future states from initial states. Formally, let S denote a system consisting of a set of objects located in the 2 three-dimensional Euclidean space. We use H ∈Rn×d to denote the objects with features, where n is the number of objects, and d is the feature dimension. For object i, let ri ∈R3 denote its Cartesian coordinate. We define the system as S = (H, R), where R = {r1, ..., rn}. This data structure ubiquitously corresponds to various real-world systems such as molecules and proteins [17, 20, 101]. In practice, the geometric state is governed by physical laws and evolves over time, and we denote the geometric state at a given time t as Rt = {rt 1, ..., rt n}. Given a system St0 = (H, Rt0) at time t0, our goal is to predict St1 = (H, Rt1) at a future time t1. As an example, in a molecular system, Rt1 can be the equilibrium state of interest evolved from the initial state Rt0. In this problem, inherent symmetries in geometric states should be considered. For example, a rotation that is applied to the coordinate system at time t0 should also be applied to subsequent time steps. These symmetries are related to the concept of equivariance in group theory [19, 18, 91]. Formally, let ϕ : X →Y denote a function mapping between two spaces. Given a group G, let ρX and ρY denote its group representations, which describe how the group elements act on these spaces. A function ϕ : X →Y is said to be equivariant if it satisfies the following condition: ρY(g)[ϕ(x)] = ϕ ρX (g)[x] , ∀g ∈G, x ∈X. When ρY = IY (identity transformation), it is also known as invariance. SE(3) group, which pertains to translations (T(3)) and rotations (SO(3)) in 3D Euclidean space, is one of the most widely used groups and is employed in our framework. 2.2 Diffusion Models Diffusion models [95, 37, 99] have emerged as the state-of-the-art generative modeling approaches across various domains [83, 85, 47, 115, 113, 117]. The main idea of this method is to construct a diffusion process that maps data to noise, and train models to reverse such process by using a tractable objective. Formally, to model the data distribution qdata(X), where X ∈Rd, we construct a diffusion process (Xt)t∈[0,T ], which is represented as a sequence of random variables indexed by time steps. We set X0 ∼qdata(X) and XT ∼pprior(X), where pprior(X) has a tractable form to generate samples efficiently, e.g. standard Gaussian distribution. Mathematically, we model (Xt)t∈[0,T ] as the solution to the following stochastic differential equation (SDE): dXt = f(Xt, t)dt + σ(t)dBt, (1) where f(·, ·) : Rd×[0, T] →Rd is a vector-valued function called the drift coefficient, σ(·) : [0, T] → R is a scalar function known as the diffusion coefficient, and (Bt)t∈[0,T ] is the standard Wiener process (a.k.a., Brownian motion) [26]. We hereafter denote by pt(X) the marginal distribution of Xt. Let p(x′, t′\|x, t) denote the transition density function such that P(Xt′ ∈A\|Xt = x) = R A p(x′, t′\|x, t)dx′ for any Borel set A. By simulating this diffusion process forward in time, the distribution of Xt will become pprior(X) at the final time T. In the literature, there exist various design choices of the SDE formulation in Eqn. (1) such that it transports the data distribution into the fixed prior distribution [98, 37, 99, 72, 97, 47]. In order to sample X0 ∼p0(X) := qdata(X), an intriguing fact can be leveraged: the reverse of a diffusion process is also a diffusion process [2]. This reverse process runs backward in time and can be formulated by the following time-reversal SDE: dXt = f(Xt, t) −σ2(t)∇Xt log pt(Xt) dt + σ(t)dBt, (2) where ∇X log pt(X) denote the score of the marginal distribution at time t. If the score is known for all time, then we can derive the reverse diffusion process from Eqn. (2), sample from pprior(X), and simulate this process to generate samples from the data distribution qdata(X). In particular, the score ∇X log pt(X) can be estimated by training a parameterized model sθ(X, t) with a denoising score matching objective [98, 97]. In theory, the minimizer of this objective approximates the ground-truth score [99] and this objective is tractable. 3 Geometric Diffusion Bridge As discussed in the introduction, effectively capturing the evolution of geometric states is crucial, for which three desiderata should be carefully considered: 3 • Coupling Preservation: From a probabilistic perspective, the evolution of geometric states transports their distribution from qdata(St0) to qdata(St1), and we are interested in modeling the distribution of target geometric states given the initial states, i.e., qdata(St1\|St0) := qdata(Rt1\|H, Rt0), which can be achieved by preserving the coupling of geometric states, i.e., qdata(Rt0, Rt1\|H). For brevity, we hereafter omit the condition of H because it keeps the same along the evolution and can be easily incorporated into the models. • Symmetry Constraints: Since the law governing the evolution is unchanged regardless of how the system is rotated or translated, the distribution of the geometric states should satisfy symmetry constraints, i.e., qdata(ρR(g)[Rt1]\|ρR(g)[Rt0]) = qdata(Rt1\|Rt0) and qdata(ρR(g)[Rt0], ρR(g)[Rt1]) = qdata(Rt0, Rt1) for all g ∈SE(3), Rt ∈R. • Trajectory Guidance: Trajectories of geometric states are sometimes accessible and provide fine-grained descriptions of the evolution dynamics. For completeness, it is crucial to develop a unified framework that can characterize and leverage trajectory data as guidance for better bridging geometric states and capturing the evolution. However, existing approaches typically have their limitations for this task, which we thoroughly discuss in Sec. 5 and summarize into Table 1. In this section, we introduce Geometric Diffusion Bridge (GDB), a general framework for bridging geometric states through generative modeling. We will elaborate on key techniques for completely preserving couping under symmetry constraints (Sec. 3.1), and demonstrate how our framework can be seamlessly extended to leverage trajectory data (Sec. 3.2). Theoretically, we conduct a thorough analysis on the capability of our unified framework, showing its completeness and superiority. All proofs of theorems are presented in Appendix B. A detailed guidance of practical implementing our framework is further provided (Sec. 3.3). Table 1: Comparisons of different candidates for bridging geometric states Methods Symmetry Constraints Coupling Preservation Trajectory guidance Direct Prediction [104, 31, 87, 89, 8] ✓ ✓ ✗ MLFFs [90, 33, 6, 34, 58] ✓ ✗ ✓ Geometric Diffusion Model [115, 38, 114] ✓ ✗ ✗ Geometric Diffusion Bridge (ours) ✓ ✓ ✓ 3.1 Equivariant Diffusion Bridge Our key design lies in the construction of equivariant diffusion bridge, a tailored diffusion process (Rt)t∈[0,T ] for bridging initial states R0∼qdata(Rt0) and target states RT ∼qdata(Rt1\|Rt0), completely preserving coupling of geometric states and satisfying symmetry constraints. Firstly, we investigate necessary conditions for a diffusion process on geometric states to meet the symmetric constraints: Proposition 3.1. Let R denote the space of geometric states and fR(·, ·) : R×[0, T] →R denote the drift coefficient on R. Let (Wt)t∈[0,T ] denote the Wiener process on R. Given an SDE on geometric states dRt = fR(Rt, t)dt + σ(t)dWt, R0 ∼q(R0), its transition density pR(z′, t′\|z, t), z, z′ ∈R is SE(3)-equivariant, i.e., pR(Rt′, t′\|Rt, t) = pR(ρR(g)[Rt′], t′\|ρR(g)[Rt], t), ∀g ∈SE(3), 0 ≤ t, t′ ≤T, if these conditions are satisfied: (1) q(R0) is SE(3)-invariant; (2) fR(·, t) is SO(3)equivariant and T(3)-invariant; (3) the transition density of (Wt)t∈[0,T ] is SE(3)-equivariant. Using Proposition 3.1, we can obtain a diffusion process that respect symmetry constraints by properly considering conditions for key components. Next, we modify a useful tool in probability theory called Doob’s h-transform [82, 81, 16], which plays an essential role in the construction of our equivariant diffusion bridge for preserving coupling of geometric states: Proposition 3.2. Let pR(z′, t′\|z, t) be the transition density of the SDE in Proposition 3.1. Let hR(·, ·) : R × [0, T] →R>0 be a smooth function satisfying: (1) hR(·, t) is SE(3)-invariant; (2) hR(z, t) = R pR(z′, t′\|z, t)hR(z′, t′)dz′. Then we can derive the following hR-transformed SDE on geometric states: dRt = fR(Rt, t) + σ2(t)∇Rt log hR(Rt, t) dt + σ(t)dWt, (3) with SE(3)-equivariant transition density ph R(z′, t′\|z, t) equals to pR(z′, t′\|z, t) hR(z′,t′) hR(z,t) . 4 Proposition 3.2 provides an equivariant version of Doob’s h-transform, which can be used to guide a free SDE on geometric states to hit an event almost surely. For example, if we set hR(·, t) = pR(z, T\|·, t), z ∈R, i.e., the transition density of the original SDE evaluated at RT = z, then the hR-transformed SDE in Eqn. (3) arrives at the specific geometric state z almost surely at the final time (see Proposition B.7 in the appendix for more details). Therefore, if we derive a proper hR(·, ·) function under the symmetry constraints, our target process (Rt)t∈[0,T ] can be constructed: Theorem 3.3 (Equivariant Diffusion Bridge). Let dRt = fR(Rt, t)dt + σ(t)dWt be an SDE on geometric states with transition density pR(z′, t′\|z, t), z, z′ ∈R satisfying the conditions in Proposition 3.1. Let hR(z, t; z0) = R pR(z′, T\|z, t) qdata(z′\|z0) pR(z′,T \|z0,0)dz′. By using Proposition 3.2, we can derive the following hR-transformed SDE: dRt = fR(Rt, t) + σ2(t)EqR(RT ,T \|Rt,t;R0,0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt] dt+σ(t)dWt, (4) which corresponds to a process (Rt)t∈[0,T ], R0 ∼qdata(Rt0) satisfying the following properties: • let q(·, ·) : R × R →R≥0 denote the joint distribution induced by (Rt)t∈[0,T ], then q(R0, RT ) equals to qdata(Rt0, Rt1); • its transition density qR(Rt′, t′\|Rt, t; R0, 0)=qR(ρR(g)[Rt′], t′\|ρR(g)[Rt], t; ρR(g)[R0], 0), ∀0≤t,t′≤T, g∈SE(3),R0∼qdata(Rt0). We call the tailored diffusion process (Rt)t∈[0,T ] an equivariant diffusion bridge. According to Theorem 3.3, given an initial geometric state Rt0, we can predict target geometric states Rt1 by simulating the equivariant diffusion bridge (Rt)t∈[0,T ] from R0 = Rt0, which arrives at RT ∼qdata(Rt1\|Rt0). However, the score EqR(RT ,T \|Rt,t;R0,0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt] in Eqn. (4) is not tractable in general. Inspired by the score matching objective in diffusion models [99], we use a parameterized model vθ(Rt, t; R0) to estimate the score by using the following training objective: L(θ) = E(z0,z1)∼qdata(Rt0,Rt1),Rt∼qR(Rt,t\|z1,T ;z0,0)λ(t)∥vθ(Rt, t; z0)−∇Rt log pR(z1, T\|Rt, t)∥2, (5) where t ∼U(0, T) (the uniform distribution on [0, T]), and λ(·) : [0, T] →R≥0 is a positive weighting function. Theoretically, we prove that the minimizer of Eqn. (5) approximates the groundtruth score (see Appendix B.5 for more details). Moreover, this objective is tractable because the transition density pR and qR can be designed to have simple and explicit forms such as Gaussian, which we will elaborate on in Sec. 3.3. 3.2 Chain of Equivariant Diffusion Bridges for Leveraging Trajectory Guidance In this subsection, we elaborate on how to leverage trajectories of geometric states as a fine-grained guidance in our framework. Let ( ˜Ri)i∈[N] denote a trajectory of N + 1 geometric states and qtraj( ˜R0, ..., ˜RN) denote the joint probability density function of geometric states in a trajectory. In practice, the markov property of trajectories typically holds [109, 78]. Under this assumption, qtraj( ˜R0, ..., ˜RN) can be equivalently reformulated into q0 traj( ˜R0) QN i=1 qi traj( ˜Ri\| ˜Ri−1) by the chain rule of probability. If qi traj( ˜Ri\| ˜Ri−1) can be well modeled, we can capture the distribution of trajectories of geometric states completely. According to Theorem 3.3, given R0 ∼q0 traj( ˜R0), an equivariant diffusion bridge (Rt)t∈[0,T ] can be constructed to model the joint distribution qtraj( ˜R0, ˜R1) and hence q1 traj( ˜R1\| ˜R0) is preserved. Therefore, if we construct a series of interconnected equivariant diffusion bridges, the distribution of trajectories can be modeled: Theorem 3.4 (Chain of Equivariant Diffusion Bridges). Let {(Rt i)t∈[0,T ]}i∈[N−1] denote a series of N equivaraint diffusion bridges defined in Theorem 3.3. For the i-th bridge (Rt i)t∈[0,T ], if we set (1) hi R(z, t; z0) = R pR(z′, T\|z, t) qi+1 traj (z′\|z0) pR(z′,T \|z0,0)dz′; (2) R0 0 ∼q0 traj( ˜R0), R0 i = RT i−1, ∀0 < i < N, then the joint distribution qR(R0 0, RT 0 , RT 1 , · · · , RT N−1) induced by {(Rt i)t∈[0,T ]}i∈[N−1] equals to qtraj( ˜R0, ..., ˜RN). We call this process a chain of equivariant diffusion bridges. 5 In this way, a chain of equivariant diffusion bridge can be used to model prior trajectory data, and simulating this chain not only bridges initial and target geometric states but also yields intermediate evolving states. Similarly, we can also use a parameterized model to estimate the scores of bridges in this chain. Instead of having only one objective in all time steps, we now have N bridges in total, which categorize the time span into N groups with different time-dependent objectives. Therefore, by properly specifying time steps and initial conditions, the objective in Eqn. (5) can be seamlessly extended (see Appendix B.7 for more details on its provable guarantee): L′(θ) = E(z0,...,zN)∼qtraj( ˜ R0,..., ˜ RN),t,Rt′ i λ(t)∥vθ(Rt′ i , t; zi) −∇Rt′ i log pi R(zi+1, T\|Rt′ i , t′)∥2, (6) where t ∼U(0, N × T), i = ⌊t T ⌋, t′ = t −i × T, Rt′ i ∼qi R(Rt′ i , t′\|zi+1, T; zi, 0). Lastly, we provide the following theoretical result, which further characterizes our framework’s expressiveness to completely model the underlying dynamics that induce the trajectory distributions: Theorem 3.5. Assume ( ˜Ri)i∈[N] is sampled by simulating a prior SDE on geometric states d ˜Rt = −∇H∗ R( ˜Rt)dt + σd ˜ Wt. Let µ∗ i denote the path measure of this prior SDE when t ∈[iT, (i + 1)T]. Building upon ( ˜Ri)i∈[N], let {µi R}i∈[N−1] denote the path measure of our chain of equivariant diffusion bridges. Under mild assumptions, we have lim N→∞max i KL(µ∗ i \|\|µi R) = 0. It is noteworthy that the assumption of the prior SDE existence holds in various real-world applications. For example, in geometry optimization, we can formulate the iterative updating process of a molecular system as dRt = −α∇RtV (Rt)dt + βdWt, where V (Rt) denotes the potential energy at Rt and α, β are step sizes [88]. From Theorem 3.5, such prior SDE serves as the underlying law governing the evolution dynamic, and our chain of equivariant diffusion bridges constructed from empirical trajectory data can well approximate it, showing the completeness of our framework. 3.3 Practical Implementation In this subsection, we elaborate on how to practically implement our framework. According to Eqn. (5), it is necessary to carefully design (1) tractable distribution qR(Rt, t\|z1, T; z0, 0) for sampling Rt; (2) closed-form matching objective ∇Rt log pR(z1, T\|Rt, t). Matching objective. Inspired by diffusion models that use Gaussian transition kernels for tractable computation, we design the SDE on geometric states in Proposition 3.1 to be: dRt = σdWt, with transition density pR(z′, t′\|z, t) = N(z0, σ2(t′ −t)I) (7) The explicit form of the objective can be directly calculated, i.e., ∇Rt log pR(z1, T\|Rt, t) = z1−Rt σ2(T −t). Sampling distribution. According to Theorem 3.3, the transition density qR(Rt, t\|z1, T; z0, 0) can be calculated by using the Doob’s h-transform in Proposition 3.2, i.e., qR(Rt, t\|z1, T; z0, 0) = pR(Rt, t\|z1, T) hR(Rt,t;z0) hR(z1,T ;z0). Moreover, hR is determined by qdata and pR, which is already specified in Eqn. (7). Therefore, we can also calculate qR(Rt, t\|z1, T; z0, 0) = N( t T z1 + T −t T z0, σ2 t(T −t) T 2 I). Symmetry constraints. In proposition 3.1, we have several conditions that should be satisfied to meet the symmetry constraints. Firstly, since a parameterized model vθ(Rt, t; R0) is used to estimate the score of our equivariant diffusion bridge, it should be SO(3)-equivariant and T(3)-invariant. Besides, we follow [50, 115] to consider CoM-free systems: given R = {r1, ..., rn}, we define ¯r = 1 n Pn i=1 ri and the CoM-free version of R = {r1 −¯r, ..., rn −¯r}. To sample from N(z0, σ2I) with z0 ∈R consisting of n objects, we (1) sample ϵ = {ϵi}n i=1 by i.i.d. drawing ϵi ∼N(0, I3); (2) calculate the CoM-free ϵ′ of ϵ; (3) obtain z0 + σϵ′. Trajectory guidance. According to Eqn. (6), both pi R and qi R for all i∈[N−1] should be determined. Similarly, we set pi R(zi+1, T\|Rt′, t′)=N(Rt′, σ2 i (T−t′)I), which further induces qi R(Rt′, t′\|zi+1, T; zi, 0) = N( t′ T zi+1 + T −t′ T zi, σ2 i t′(T −t′) T 2 I). Combining all the above design choices, we have the following algorithms for training our Geometric Diffusion Bridge (Alg. 3) and leveraging trajectory guidance if available (Alg. 4). After the model 6 is well trained, we leverage ODE numerical solvers [12] to simulate the bridge process by using its equivalent probability flow ODE [99]. In this way, we can effectively and deterministically predict future geometric states of interest from initial states in an efficient iterative process. Lastly, it is also noteworthy that our framework is general to be implemented by using other advanced design strategies [99, 47, 48], which we leave as future work. Algorithm 1 Training 1: repeat 2: (z0, z1) ∼qdata(Rt0, Rt1) 3: t ∼U[0, T] 4: ϵ ∼N(0, I) 5: Rt = t T z1 + T −t T z0 + √ t(T −t) T σϵ 6: Take gradient descent step on ∇θλ(t) z1−Rt σ2(T −t) −vθ(Rt, t; z0) 2 7: until converged Algorithm 2 Training with trajectory guidance 1: repeat 2: (z0, . . . , zN) ∼qtraj( ˜R0, . . . , ˜RN) 3: t ∼U (0, N × T), i = ⌊t T ⌋, t′ = t −i × T 4: ϵ ∼N(0, I) 5: Rt′ i = t′ T zi+1 + T −t′ T zi + √ t′(T −t′) T σiϵ 6: Take gradient descent step on ∇θλ(t) zi+1−Rt′ i σ2 i (T −t′) −vθ(Rt′ i , t; zi) 2 7: until converged 4 Experiments In this section, we empirically study the effectiveness of our Geometric Diffusion Bridge on crucial real-world challenges requiring bridging geometric states. In particular, we carefully design several experiments covering different types of data, scales and scenarios, as shown in Table 2. Due to space limits, we present more details in Appendix D. Table 2: Summary of experimental setup. Dataset Task Description Data Type Trajectory data Training set size QM9 [79] Equilibrium State Prediction Simple molecule ✗ 110,000 Molecule3D [116] Equilibrium State Prediction Simple molecule ✗ 2,339,788 OC22, IS2RS [13] Structure Relaxation Adsorbate-Catalyst complex ✓ 45,890 4.1 Equilibrium State Prediction Task. Equilibrium states typically represent local minima on the Born-Oppenheimer potential energy surface of a molecular system [54], which correspond to its most stable geometric state and play an essential role in determining its properties in various aspects [4, 21]. In this task, our goal is to accurately predict the equilibrium state from the initial geometric state of a molecular system. Dataset. Two popular datasets are used: (1) QM9 [79] is a medium-scale dataset that has been widely used for molecular modeling, consisting of ˜130,000 organic molecules. In convention, 110k, 10k, and 11k molecules are used for train/valid/test sets respectively; (2) Molecule3D [116] is a largescale dataset curated from the PubChemQC project [67, 71], consisting of 3,899,647 molecules in total and its train/valid/test splitting ratio is 6 : 2 : 2. In particular, both random and scaffold splitting methods are adopted to thoroughly evaluate the in-distribution and out-of-distribution performance. For each molecule, an initial geometric state is generated by using fast and coarse force field [73, 52] and geometry optimization is conducted to obtain DFT-calculated equilibrium geometric structure. Setting. In this task, we parameterize vθ(Rt, t; R0) by extending a Graph-Transformer based equivariant network [92, 63] to encode both time steps and initial geometric states as conditions. For inference, we use 10 time steps with the Euler solver [12]. Following [111], we choose several strong baselines for a comprehensive comparison, and use three metrics for measuring the error between predicted target states and ground-truth states: C-RMSD, D-MAE and D-RMSE. The detailed descriptions of the baselines, evaluation metrics and training settings are presented in Appendix D.1. Results. Results on QM9 and Molecule3D are shown in Table 3 and 4 respectively. It can be easily seen that our GDB framework consistently surpasses all baselines by a significantly large margin on 7 Table 3: Results on the QM9 dataset (Å). We report the official results of baselines from [111] Validation Test D-MAE↓ D-RMSE↓ C-RMSD↓ D-MAE↓ D-RMSE↓ C-RMSD↓ RDKit DG 0.358 0.616 0.722 0.358 0.615 0.722 RDKit ETKDG 0.355 0.621 0.691 0.355 0.621 0.689 GINE [39] 0.357 0.673 0.685 0.357 0.669 0.693 GATv2 [10] 0.339 0.663 0.661 0.339 0.659 0.666 GPS [80] 0.326 0.644 0.662 0.326 0.640 0.666 GTMGC [111] 0.262 0.468 0.362 0.264 0.470 0.367 GDB (ours) 0.092 0.218 0.143 0.096 0.223 0.148 Table 4: Results on the Molecule3D dataset (Å). We report the official results of baselines from [111] Validation Test D-MAE↓ D-RMSE↓ C-RMSD↓ D-MAE↓ D-RMSE↓ C-RMSD↓ (a) Random Split RDKit DG 0.581 0.930 1.054 0.582 0.932 1.055 RDKit ETKDG 0.575 0.941 0.998 0.576 0.942 0.999 DeeperGCN-DAGNN [116] 0.509 0.849 * 0.571 0.961 * GINE [39] 0.590 1.014 1.116 0.592 1.018 1.116 GATv2 [10] 0.563 0.983 1.082 0.564 0.986 1.083 GPS [80] 0.528 0.909 1.036 0.529 0.911 1.038 GTMGC [111] 0.432 0.719 0.712 0.433 0.721 0.713 GDB (ours) 0.374 0.631 0.622 0.376 0.626 0.619 (b) Scaffold Split RDKit DG 0.542 0.872 1.001 0.524 0.857 0.973 RDKit ETKDG 0.531 0.874 0.928 0.511 0.859 0.898 DeeperGCN-DAGNN [116] 0.617 0.930 * 0.763 1.176 * GINE [39] 0.883 1.517 1.407 1.400 2.224 1.960 GATv2 [10] 0.778 1.385 1.254 1.238 2.069 1.752 GPS [80] 0.538 0.885 1.031 0.657 1.091 1.136 GTMGC [111] 0.406 0.675 0.678 0.400 0.679 0.693 GDB (ours) 0.335 0.587 0.592 0.341 0.608 0.603 QM9, e.g., 60.5%/59.7% relative C-RMSD reduction on valid/test sets respectively, establishing a new state-of-the-art performance. Similar trends also can be observed in Molecule3D, i.e., 12.6%/13.2% relative C-RMSD reduction for valid/test sets of the random split and 12.7%/13.0% reduction for the scaffold split, largely outperforming the best baseline. These significant error reduction results show the superiority of our GDB framework for bridging geometric states, and its generality on both medium and large-scale challenges. Moreover, our framework performs consistently across valid and tests of both random and scaffold splits, further verifying its robustness in challenging scenarios. 4.2 Structure Relaxation Task. Catalyst discovery is crucial for various applications. Adsorbate candidates are placed on catalyst surfaces and evolve through structure relaxation to adsorption states, in which the adsorption structures can be determined for measuring catalyst activity and selectivity. Our goal is thus to accurately predict adsorption states from initial states of adsorbate-catalyst complexes. Dataset. We adopt Open Catalyst 2022 (OC22) dataset [105], which has great significance for the development of Oxygen Evolution Reaction (OER) catalysts. Each data is in the form of the adsorbatecatalyst complex. Both initial and adsorption states with trajectories connecting them are provided. The training set consists of 45,890 catalyst-adsorbate complexes. To better evaluate the model’s performance, the validation and test sets consider the in-distribution (ID) and out-of-distribution (OOD) settings which use unseen catalysts, containing approximately 2,624 and 2,780 complexes respectively. Setting. Following [105], we use the Average Distance within Threshold (ADwT) as the evaluation metric, which reflects the percentage of structures with an atom position MAE below thresholds. We parameterize vθ(Rt, t; R0) by using GemNet-OC [34], which also serves as a verification that 8 Table 5: Results on the OC22 IS2RS Validation set. "OC20+OC22" denotes using both OC20 [13] and OC22 data; "OC20→OC22" means pre-training on OC20 data then fine-tuning on OC22 data; "OC22-only" means only using OC22 data. We report the official results of baselines from [105] Model ADwT [%] ↑(ID) ADwT [%] ↑(OOD) Avg [%] ↑ OC20+OC22 SpinConv [94] 55.79 47.31 51.55 GemNet-OC [34] 60.99 53.85 57.42 OC20→OC22 SpinConv [94] 56.69 45.78 51.23 GemNet-OC [34] 58.03 48.33 53.18 GemNet-OC-Large [34] 59.69 51.66 55.67 OC22-only IS baseline 44.77 42.59 43.68 SpinConv [94] 54.53 40.45 47.49 GemNet-dT [32] 59.68 51.25 55.46 GemNet-OC [34] 60.69 52.90 56.79 GDB (ours) 63.01 55.78 59.39 −trajectory guidance 62.14 54.94 58.54 −R0 condition 60.17 49.26 54.71 our framework is compatible with different backbone models. For inference, we also use 10 time steps with the Euler solver. Following [105], we choose strong MLFF baselines trained on force field data for a challenging comparison. The detailed descriptions of baselines and settings are presented in Appendix D.2. Results. In Table 5, our GDB significantly outperforms the best baseline, e.g., 3.3%/3.6%/3.4% relative improvement on the ADwT metric of ID, OOD and Avg respectively. It is noteworthy that the best baseline is the GemNet-OC force field trained on both OC20 and OC22 data, which is 10 times more than OC22 data only. Nevertheless, our framework still achieves better performance on predicting the adsorption geometric states. Moreover, our framework without using any trajectory data still can achieve better performance compared to the best baseline, e.g., 58.54 v.s. 57.42 Avg[%]. All the results on this challenging task further demonstrate the superiority and completeness of our framework. Ablation study. Furthermore, we conduct ablation studies to examine key designs of our framework in Table 5. Firstly, we can see that using trajectory guidance indeed improves the performance of our framework, e.g., 1.4% relative improvement on Avg ADwT. Moreover, we also investigate the impact of R0 condition in vθ(Rt, t; R0), which plays an essential role in preserving the joint distribution of geometric states. Without this condition, we can see a significant drop, e.g., 6.5%/10.3% relative ADwT drop on Avg/OOD respectively. Overall, these ablation studies serve as strong supports on the necessity of developing a unified framework that can precisely bridge geometric states by preserving their joint distributions and effectively leverage trajectory data as guidance for enhanced performance. 5 Related Works Direct Prediction. One line of approach for bridging geometric states is direct prediction, i.e., training a model to directly predict target geometric states given initial states as input. Models that carefully respect symmetry constraints such as the equivariance to 3D rotations and translations are typically used, which are called Geometric Equivariant Networks [11, 36, 120, 27]. Different techniques have been explored to encode such priors, which mainly include vector operations such as scalar and vector product [35, 87, 89, 41, 103, 14], e.g., the scalar-vector product used in EGNN [87], and tensor product based operations [104, 31, 8, 57, 64]. Despite its simplicity and efficiency, direct prediction requires encoding the iterative evolution of geometric states into a single-step prediction model, which lacks the ability to capture the underlying dynamics and cannot leverage trajectories of geometric states. Machine Learning Force Field. Another line of approach is called machine learning force field (MLFF) [106, 5, 6, 70, 75, 58], which are trained to predict intermediate labels, such as the potential 9 energy or force of the (local) current geometric state instead. After training, MLFFs can be used to simulate the trajectory of geometric states over time based on underlying equations. Using Geometric Equivariant Networks as the backbone, MLFFs typically satisfy the symmetry constraints. Besides, trajectory data with additional energy or force labels can directly be used for training MLFFs. However, this paradigm highly depends on the existence and quality of intermediate labels since small local errors in energy or force prediction can accumulate along the simulation process [7, 106, 30]. Moreover, there exists no guarantee that MLFFs can completely model joint state distributions, which is another limitation for bridging geometric states. Geometric Diffusion Models. In recent years, diffusion models [37, 99] have emerged with stateof-the-art generative modeling performance across various domains [85, 108, 51, 56]. In geometric domain, diffusion models are typically used for molecule conformation generation [115, 114, 38] and protein design [108, 117]. By properly design the noising process and model architectures, symmetry constraints on the transition kernel and prior distribution can be satisfied, which guarantees the generated data is sampled from roto-translational invariant distributions [115, 38]. In addition to the score-based formulation, recent advances further extend new techniques such as flow matching [59, 61, 1] to satisfy symmetry constraints for these generation tasks [49, 100]. Nevertheless, there exists no guaratee that these approaches can model the joint distribution of geometric states [61, 96]. And how to leverage trajectory data as guidance for bridging geometric states is also challenging. Other techniques. MoreRed [45] trains a diffusion model on equilibrium molecule conformations with a time step predictor, and directly use it for bridging any conformations to their equilibrium states. GTMGC [111] instead develop a Graph Transformer to directly predict equilibrium conformations from their 2D graph forms. Both of them are limited to the equilibrium conformation prediction task, cannot preserve the joint state distribution and leverage trajectory data. EGNO [112] is a concurrent work that develops a neural operator based approach to model dynamics of trajectories. By carefully designing temporal convolution in fourier spaces, EGNO can learn from trajectory data. However, this tailored approach cannot be directly used without trajectory guidance. To preserve joint data distributions, [22, 121] coincide with us to leverage Doob’s h-transform to repurposing standard diffusion processes, but they do not respect symmetry constraints and cannot leverage trajectories. There also exist recent works that study the diffusion bridge framework [76, 93] and apply it to various domains such as images and graphs [110, 62, 42]. Compared to all above approaches, our GDB framework stands out as a unique and ideal solution that can precisely bridge geometric states and effectively leverage trajectory data (if available) in a unified manner. 6 Conclusion In this work, we introduce Geometric Diffusion Bridge (GDB), a general framework for bridging geometric states through generative modeling. We leverage a modified version of Doob’s h-transform to constructe an equivariant diffusion bridge for bridging initial and target geometric states. Trajectory data can further be seamlessly leveraged as guidance by using a chain of equivariant diffusion bridges, allowing complete modeling of trajectory data. Mathematically, we conduct a comprehensive theoretical analysis showing our framework’s ability to preserve joint distributions of geometric states and capability to completely model the evolution dynamics. Empirical comparisons on different settings show that our GDB significantly surpasses existing state-of-the-art approaches and ablation studies further underscore the necessity of several key designs in our framework. In the future, it is worth exploring better implementation strategies of our framework for enhanced performance, and applying our GDB to other critical challenges involving bringing geometric states. Broader Impacts and Limitations This work newly proposes a general framework to bridge geometric states, which has great significance in various scientific domains. Our experimental results have also demonstrated considerable positive potential for various applications, such as catalyst discovery and molecule optimization, which can significantly contribute to the advancement of renewable energy processes and chemistry discovery. However, it is essential to acknowledge the potential negative impacts including the development of toxic drugs and materials. Thus, stringent measures should be implemented to mitigate these risks. 10 There also exist some limitations to our work. For the sake of generality, we do not experiment with advanced implementation strategies of training objectives and sampling algorithms, which leave room for further improvement. Besides, the employment of Transformer-based architectures may also limit the efficiency of our framework. This has also become a common issue in transformer-based diffusion models, which we have earmarked for future research. Acknowledgements We thank all the anonymous reviewers for the very careful and detailed reviews as well as the valuable suggestions. Their help has further enhanced our work. Liwei Wang is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC62276005). Di He is supported by National Science Foundation of China (NSFC62376007). References [1] Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint arXiv:2303.08797, 2023. [2] Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982. [3] Muratahan Aykol, Joseph H Montoya, and Jens Hummelshøj. Rational solid-state synthesis routes for inorganic materials. Journal of the American Chemical Society, 143(24):9244–9259, 2021. [4] Keld L Bak, Jürgen Gauss, Poul Jørgensen, Jeppe Olsen, Trygve Helgaker, and John F Stanton. 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In The Twelfth International Conference on Learning Representations, 2024. 18 A Organization of the Appendix The supplementary material is organized as follows. In Appendix B, we first recall some definitions and tools from stochastic calculus and then give the proofs of all theorems. In Appendix C, we give the derivation of our practical objective function and our sampling algorithms. In Appendix D, we give some details of our experiments, including a comprehensive introduction to the datasets, baselines, metrics and settings. B Proof of Theorems B.1 Review of Stochastic Calculus Let (Xt)t∈[0,T ] be a stochastic process. We use p(x′, t′\|x1, t1; x2, t2; . . . ; xn, tn) to denote its conditional density function satisfying P(Xt′ ∈A\|Xt1 = x1, Xt2 = x2, . . . , Xtn = xn) = Z A p(x′, t′\|x1, t1; x2, t2; . . . ; xn, tn)dx′ for any Borel set A, where t1 < t2 < · · · < tn. If (Xt)t∈[0,T ] is a Markov process, p(x′, t′\|x1, t1; x2, t2; . . . ; xn, tn) = p(x′, t′\|xn, tn), which is also called a transition density function. One of the most important results of stochastic calculus is the Ito’s formula. The precise statements are as follows. Theorem B.1 (Ito’s formula for Brownian Motion). Let Bt be the d−dimensional Brownian Motion. Assume f is a bounded real valued function with continuous second-order partial derivatives, i.e. f ∈C2 b (Rd). Then the Ito’s formula is given by f(Bt) = f(B0) + Z T 0 ∇f(Bt) · dBt + 1 2 Z T 0 ∇2f(Bt)dt. (8) We follow [86] for the proof of Doob’s h-transform. The infinitesimal generator of the Markov process plays an important role in the proof of the Doob’s h-transform. The precise definitions are as follows. Definition B.2. (Generator of a Process) The infinitesimal generator At of a stochastic process (Xt) for a function ϕ(x) is Atϕ(x) = lim s→0+ E[ϕ(Xt+s)\|Xt = x] −ϕ(x) s , (9) where ϕ is a suitably regular function. For an Itô process defined as the solution to the SDE dXt = f(Xt, t)dt + σ(t)dBt, the generator is At = d X i=1 f i(x, t) ∂ ∂xi + 1 2 d X i=1 σ2(t) ∂2 ∂x2 i . (10) The Fokker-Planck’s Equation is an useful tool to track the evolution of the transition density function associated with an SDE. The precise statements are as follows. Proposition B.3. (Fokker-Planck’s Equation) Let p(x′, t′\|x, t) be the transition density function of the SDE dXt = f(Xt, t)dt + σ(t)dBt. Then p(x′, t′\|x, t) satisfies the Fokker-Planck’s Equation ∂p(x, t\|x0, 0) ∂t = − d X i=1 ∂(f i(x, t)p(x, t\|x0, 0)) ∂xi + 1 2 d X i=1 σ2(t)∂2p(x, t\|x0, 0) ∂x2 i = 0, (11) with the initial condition p(x, 0\|x0, 0) = δ(x −x0). The Fokker-Planck’s Equation can also be written in a compact form using the generator At: ∂ ∂tp(x, t\|x0, 0) = A∗ t p(x, t\|x0, 0), (12) where A∗ t is the adjoint operator of A: A∗ t = − d X i=1 ∂(f i(x, t)·) ∂xi + 1 2 d X i=1 σ2(t)∂2(·) ∂x2 i . (13) 19 When the terminal is fixed, the evolution of the transition density function can also given by a PDE, which is called the Backward Kolmogorov Equation. We give the precise statement as follows. Proposition B.4. (Backward Kolmogorov Equation) Let p(x′, t′\|x, t) be the transition density function of the SDE dXt = f(Xt, t)dt + σ(t)dBt. Then p(x′, t′\|x, t) satisfies the Backward Kolmogorov Equation −∂p(xt, t\|x, s) ∂s = d X i=1 f i(x, s)∂p(xt, t\|x, s) ∂xi + 1 2 d X i=1 σ2(s)∂2p(xt, t\|x, s) ∂x2 i = 0, (14) with the initial condition p(xt, t\|x, t) = δ(x −xt). The Backward Kolmogorov Equation can also be written in a compact form using the generator As: ∂ ∂s + As p(xt, t\|x, s) = 0. (15) B.2 Proof of Proposition 3.1 Proposition B.5. Let R denote the space of geometric states and fR(·, ·) : R×[0, T] →R denote the drift coefficient on R. Let (Wt)t∈[0,T ] denote the Wiener process on R. Given an SDE on geometric states dRt = fR(Rt, t)dt + σ(t)dWt, R0 ∼q(R0), its transition density pR(z′, t′\|z, t), z, z′ ∈R is SE(3)-equivariant, i.e., pR(Rt′, t′\|Rt, t) = pR(ρR(g)[Rt′], t′\|ρR(g)[Rt], t), ∀g ∈SE(3), ∀0 ≤ t < t′ ≤T, if the following conditions are satisfied: (1) q(R0) is SE(3)-invariant; (2) fR(·, t) is SO(3)-equivariant and T(3)-invariant; (3) the transition density of (Wt)t∈[0,T ] is SE(3)equivariant. Proof. In this section, we view R = {r1, ..., rn} ∈R as r1 ⊕r2 ⊕· · · rn ∈R3n, which is the concatenation of ri. So from this perspective, the space R is isomorphic to the Euclidean space R3n. Then (Wt)t∈[0,T ] is the Wiener process with dimension d = 3n. For any g ∈SE(3), ρR(g) can be characterized by an orthogonal matrix O(g) ∈R3×3, satisfying det(O(g)) = 1, and a translation vector t ∈R3. Then the representation of SE(3) on R3n is given by ρR(g)[R] = OR(g)R + tR, (16) where OR(g) = diag{O(g), O(g), . . . , O(g)}, tR = t ⊕t ⊕· · · t ∈R3n. It’s obvious that OR(g) is also an orthogonal matrix in R3n×3n, satisfying O−1 R (g) = OT R(g). According to Proposition B.3, the evolution of the transition density function is given by the Fokker-Planck’s Equation ∂pR(x, t\|x0, 0) ∂t = − d X i=1 ∂ f i(x, t)pR(x, t\|x0, 0) ∂xi + 1 2 d X i=1 σ2(t)∂2 (pR(x, t\|x0, 0)) ∂x2 i , (17) with the initial condition pR(x, 0\|x0, 0) = δ(x −x0). Let y = OR(g)x + tR, y0 = OR(g)x0 + tR, then we have pR(ρR(g)[x], t\|ρR(g)[x0], 0) = pR(OR(g)x + tR, t\|OR(g)x0 + tR, 0) = pR(y, t\|y0, 0). (18) The evolution of the transition density function pR(y, t\|y0, 0) is also given by the Fokker-Planck’s Equation: ∂pR(y, t\|y0, 0) ∂t = − d X i=1 ∂ f i(y, t)pR(y, t\|y0, 0) ∂yi + 1 2 d X i=1 σ2(t)∂2 (pR(y, t\|y0, 0)) ∂y2 i , (19) with the boundary condition pR(y, 0\|y0, 0) = δ(y −y0) = δ(x −x0). Since y = OR(g)x + tR, we have x = O−1 R (g)(y −tR). Then by the chain rule, we have ∂ ∂yi = d X j=1 ∂xj ∂yi ∂ ∂xj = d X j=1 (O−1 R (g))ji ∂ ∂xj = d X j=1 (OR(g))ij ∂ ∂xj . (20) 20 Since fR(·, t) is a SO(3)-equivariant and T(3)-invariant function, we have f i R(y, t) = f i R(OR(g)x + tR, t) = (OR(g)fR(x, t))i = d X k=1 (OR(g))ikf k R(x, t). (21) Then the Fokker-Planck’s equation becomes ∂pR(y, t\|y0, 0) ∂t = − d X i=1 ∂ f i(y, t)pR(y, t\|y0, 0) ∂yi + 1 2σ2(t) d X i=1 ∂2 (pR(y, t\|y0, 0)) ∂y2 i (22) = − d X i=1 d X j=1 d X k=1 (OR(g))ij ∂((OR(g))ikf k R(x, t)pR(y, t\|y0, 0)) ∂xj (23) + 1 2σ2(t) d X i=1 d X j=1 d X k=1 (OR(g))ik ∂ ∂xk (OR(g))ij ∂(pR(y, t\|y0, 0)) ∂xj (24) = − d X i=1 d X j=1 d X k=1 (OR(g))ij(OR(g))ik ∂(f k R(x, t)pR(y, t\|y0, 0)) ∂xj (25) + 1 2σ2(t) d X i=1 d X j=1 d X k=1 (OR(g))ik(OR(g))ij ∂ ∂xk ∂(pR(y, t\|y0, 0)) ∂xj . (26) Since OR(g) is an orthogonal matrix, the columns of OR(g) are orthogonal to each other, i.e. d X i=1 (OR(g))ik(OR(g))ij = δjk = 0 j ̸= k, 1 j = k. (27) So the Fokker-Planck’s equation can be simplified to ∂pR(y, t\|y0, 0) ∂t = − d X j=1 d X k=1 ∂(f k R(x, t)pR(y, t\|y0, 0)) ∂xj (28) + 1 2σ2(t) d X j=1 d X k=1 δjk ∂ ∂xk ∂(pR(y, t\|y0, 0)) ∂xj (29) = − d X j=1 ∂(f j R(x, t)pR(y, t\|y0, 0)) ∂xj + 1 2σ2(t) d X j=1 ∂2 (pR(y, t\|y0, 0)) ∂(xj)2 , (30) which is same as Eqn.(17). Since the boundary condition pR(y, 0\|y0, t0) = δ(y −y0) = δ(x − x0) = pR(x, 0\|x0, 0), then pR(y, t\|y0, t0) = pR(x, t\|x0, t0), ∀t ∈[0, T]. Thus we have proved that pR(Rt′, t′\|Rt, t) = pR(ρR(g)[Rt′], t′\|ρR(g)[Rt], t), ∀g ∈SE(3), ∀0 ≤t < t′ ≤T. B.3 Proof of Proposition 3.2 Proposition B.6 (Doob’s h-transform). Let pR(z′, t′\|z, t) be the transition density of the SDE in Proposition 3.1. Let hR(·, ·) : R × [0, T] →R>0 be a smooth function satisfying: (1) hR(·, t) is SE(3)-invariant; (2) hR(z, t) = R pR(z′, t′\|z, t)hR(z′, t′)dz′. We can derive the following hRtransformed SDE on geometric states: dRt = fR(Rt, t) + σ2(t)∇Rt log hR(Rt, t) dt + σ(t)dWt, (31) with transition density ph R(z′, t′\|z, t) = pR(z′, t′\|z, t) hR(z′,t′) hR(z,t) preserving the symmetry constraints. Proof. We use the definition of the infinitesimal generator to prove the proposition. The infinitesimal generator of ph R(x′, t′\|x, t) for a function ϕ(x) is given by Ah t ϕ(x) = lim s→0+ Eh[ϕ(Rt+s)\|Rt = x] −ϕ(x) s . (32) 21 Since ph R(z′, t′\|z, t) = pR(z′, t′\|z, t) hR(z′,t′) hR(z,t) , so we have Eh[ϕ(Rt+s)\|Rt = x] = E[ϕ(Rt+s)h(Rt+s, t + s)\|Rt = x] h(x, t) . (33) Then Ah t ϕ(x) can be simplified as Ah t ϕ(x) = lim s→0+ Eh[ϕ(Rt+s)\|Rt = x] −ϕ(x) s (34) = lim s→0+ E[ϕ(Rt+s)h(Rt+s, t + s)\|Rt = x] −ϕ(x)h(x, t) sh(x, t) (35) = 1 h(x, t)[∂h(x, t) ∂t ϕ(x) + d X i=1 ∂h(x, t) ∂xi ϕ(x) + h(x, t)∂ϕ(x) ∂xi f i(x, t) (36) + 1 2 d X i=1 σ2(t)∂2h(x, t) ∂x2 i ϕ(x) + d X i=1 σ2(t)∂h(x, t) ∂xi ∂ϕ(x) ∂xi (37) + 1 2 d X i=1 σ2(t)∂2ϕ(x) ∂x2 i h(x, t)] (38) = 1 h(x, t)[∂h(x, t) ∂t ϕ(x) + (Ath(x, t)) ϕ(x) d X i=1 h(x, t)∂ϕ(x) ∂xi f i(x, t) (39) + d X i=1 σ2(t)∂h(x, t) ∂xi ∂ϕ(x) ∂xi + 1 2 d X i=1 σ2(t)∂2ϕ(x) ∂x2 i h(x, t)] (40) Since h(x, t) = R pR(x′, t′\|x, t)h(x′, t′)dx, we have ∂ ∂t + At h(x, t) = Z ∂p(x′, t′\|x, t) ∂t + Atp(x′, t′\|x, t) h(x′, t′)dx. (41) According to the Backward Kolmogorov Equation (Proposition B.4), we get ∂p(x′, t′\|x, t) ∂t + Atp(x′, t′\|x, t) = 0. (42) So we get ∂ ∂t + At h(x, t) = 0. (43) Then Ah t ϕ(x) can be simplified as Ahϕ(x) = 1 h(x, t)[ d X i=1 h(x, t)∂ϕ(x) ∂xi f i(x, t) + d X i=1 σ2(t)∂h(x, t) ∂xi ∂ϕ(x) ∂xi (44) + 1 2 d X i=1 σ2(t)∂2ϕ(x) ∂x2 i h(x, t)] (45) = d X i=1 ∂ϕ(x) ∂xi f i(x, t) + d X i=1 σ2(t) 1 h(x, t) ∂h(x, t) ∂xi ∂ϕ(x) ∂xi + 1 2 d X i=1 σ2(t)∂2ϕ(x) ∂x2 i (46) = d X i=1 f i(x, t) + σ2(t)∂log h(x, t) ∂xi ∂ϕ(x) ∂xi + 1 2 d X i=1 σ2(t)∂2ϕ(x) ∂x2 i . (47) So we show that Ah t = d X i=1 f i(x, t) + σ2(t)∂log h(x, t) ∂xi ∂ ∂xi + 1 2 d X i=1 σ2(t) ∂2 ∂x2 i . (48) 22 According to the correspondence between SDE and its generator (Definition B.2), the equation above implies that the h-transformed SDE is given by dRt = fR(Rt, t) + σ2(t)∇Rt log hR(Rt, t) dt + σ(t)dWt. (49) Additionally, we need to show that the h-transformed transition density function satisfies the symmetric constraints. First, we show that if h(·, t0) is SE(3)-invariant, then h(·, t) is also SE(3)-invariant ∀t ∈[0, T]. For any g ∈SE(3), assume ρR(g)[z] = OR(g)z + tR, where OR(g) is an orthogonal matrix and det(OR(g)) = 1. Since hR(z, t) satisfies hR(z, t) = Z pR(z′, t0\|z, t)h(z′, t0)dz′, (50) then we have hR(ρR(g)[z], t) = Z pR(z′, t0\|ρR(g)[z], t)h(z′, t0)dz′ (51) = Z pR ρR(g)(ρR(g))−1[z′], t0\|ρR(g)[z], t h(ρR(g)(ρR(g))−1[z′], t0)dz′. (52) (53) By Proposition 3.1, pR ρR(g)(ρR(g))−1[z′], t0\|ρR(g)[z], t = pR (ρR(g))−1[z′], t0\|z, t , let z1 = ρR(g))−1[z′], then hR(ρR(g)[z], t) = Z pR (ρR(g))−1[z′], t0\|z, t h(ρR(g)(ρR(g))−1[z′], t0)dz′ (54) = Z pR (z1, t0\|z, t) h(ρR(g)z1, t0) det(OR(g))dz1 (55) = Z pR (z1, t0\|z, t) h(z1, t0)dz1 (56) = hR(z, t). (57) So h(·, t) is SE(3)-invariant ∀t ∈[0, T], h(·, t) is well-defined under these symmetric constraints. Then we show ph R(z′, t′\|z, t) preserves the symmetric constraints: ph R(ρR(g)[z′], t′\|ρR(g)[z], t) = pR(ρR(g)[z′], t′\|ρR(g)[z], t)hR(ρR(g)[z′], t′) hR(ρR(g)[z], t) (58) = pR(z′, t′\|z, t)hR(ρR(g)[z′], t′) hR(ρR(g)[z], t) (59) = pR(z′, t′\|z, t)hR(z′, t′) hR(z, t) (60) = ph R(z′, t′\|z, t). (61) Thus we have proved that ph R(ρR(g)[z′], t′\|ρR(g)[z], t) = ph R(z′, t′\|z, t), (62) which implies that ph R(z′, t′\|z, t) preserves the symmetric constraints for any g ∈SE(3). So the proof is completed. Next, we show how to construct a SDE with a fixed terminal point as an simple application of the Doob’s h-transform. The result of this example is very useful to construct diffusion bridge. Proposition B.7. Assume the original SDE is given by dXt = f(Xt, t)dt+σ(t)dWt. Let hR(x, t) = pR(y, T\|x, t) which is the transition density function of the original SDE evaluated at XT = y. Then the h-transformed SDE dRt = f(Rt, t) + σ2(t)∇Rt log pR(y, T\|Rt, t) dt + σ(t)dWt, (63) arrive at y almost surely at the final time. 23 Proof. The original SDE is given by dXt = f(Xt, t)dt + σ(t)dWt. (64) First, we need to verify that hR(x, t) satisfies the condition hR(x, t) = Z pR(x′, t0\|x, t)h(x′, t0)dx′. (65) Since hR(x, t) = pR(y, T\|x, t), we have Z pR(x′, t′\|x, t)hR(x′, t′) = Z pR(x′, t′\|x, t)pR(y, T\|x′, t′)dx′. (66) Then by the Chapman–Kolmogorov’s equation Z pR(x′, t′\|x, t)pR(y, T\|x′, t′)dx′ = pR(y, T\|x, t), (67) we get Z pR(x′, t′\|x, t)hR(x′, t′) = pR(y, T\|x, t) = hR(x, t). (68) So the condition is satisfied. Then we can use the result of the Proposition 3.2. The h-transformed SDE is given by dRt = f(Rt, t) + σ2(t)∇Rt log pR(y, T\|Rt, t) dt + σ(t)dWt. (69) And the h-transformed transition density function satisfies Z A ph R(x′, t′\|x, t)dx′ = Z A pR(x′, t′\|x, t)hR(x′, t′) hR(x, t) dx′ (70) = Z A pR(x′, t′\|x, t)pR(y, T\|x′, t′) pR(y, T\|x, t) dx′ (71) = P(Xt′ ∈A\|Xt = x, XT = y), (72) where we use the Bayes’ theorem to deduce the last equality and A is an arbitrary Borel set. Since Rt is a process conditioning on XT = y, then RT = y almost surly. B.4 Proof of Theorem 3.3 Theorem B.8 (Equivariant Diffusion Bridge). Given an SDE on geometric states dRt = fR(Rt, t)dt + σ(t)dWt with transition density pR(z′, t′\|z, t), z, z′ ∈R satisfying the conditions in Proposition 3.1. Let hR(z, t; z0) = R pR(z′, T\|z, t) qdata(z′\|z0) pR(z′,T \|z0,0)dz′. By using Proposition 3.2, we can derive the following hR-transformed SDE: dRt = fR(Rt, t) + σ2(t)EqR(RT ,T \|Rt,t;R0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt] dt + σ(t)dWt, (73) which corresponds to a process (Rt)t∈[0,T ], R0 ∼qdata(Rt0) satisfying the following properties: • let q(·, ·) : R × R →R≥0 denote the joint distribution induced by (Rt)t∈[0,T ], then q(R0, RT ) equals to qdata(Rt0, Rt1); • its transition density qR(Rt′, t′\|Rt, t; R0)=qR(ρR(g)[Rt′], t′\|ρR(g)[Rt], t; ρR(g)[R0]), ∀0≤t<t′≤T, g∈SE(3),R0∼qdata(Rt0). We call the tailored diffusion process (Rt)t∈[0,T ] an equivariant diffusion bridge. Proof. Let hR(z, T; z0) = qdata(z\|z0) pR(z,T \|z0,0), then we define hR(z, t; z0) = Z pR(z′, T\|z, t) qdata(z′\|z0) pR(z′, T\|z0, 0)dz′, ∀t ∈[0, T). (74) 24 So we can easily show that hR(z, t; z0) satisfies the condition hR(z, t; z0) = Z pR(z′, T\|z, t)h(z′, T; z0)dz′, ∀t ∈[0, T], ∀z, z0 ∈R. (75) Then we can use the result of Theorem 3.2 to get the h-transformed SDE. By Theorem 3.2, the h-transformed SDE is dRt = fR(Rt, t) + σ2(t)∇Rt log hR(Rt, t; R0) dt + σ(t)dWt. (76) Next, we need to find the explicit form of ∇Rt log hR(Rt, t; R0), ∇z log hR(z, t; z0) = ∇zhR(z, t; z0) hz(z, t; z0) (77) = 1 hR(z, t; z0) Z ∇zpR(z′, T\|z, t) qdata(z′\|z0) pR(z′, T\|z0, 0)dz′. (78) The h-transformed density function is qR(z′, T\|z, t; z0, 0) = pR(z′, T\|z, t)hR(z′, T; z0) hR(z, t; z0) (79) = pR(z′, T\|z, t) qdata(z′\|z0) pR(z′, T; z0, 0)hR(z, t; z0). (80) Then we have ∇z log hR(z, t; z0) = 1 hR(z, t; z0) Z ∇zpR(z′, T\|z, t) qdata(z′\|z0) pR(z′, T\|z0, 0)dz′ (81) = Z ∇zpR(z′, T\|z, t)qR(z′, T\|z, t; z0, 0) pR(z′, T\|z, t) dz′ (82) = Z ∇z log pR(z′, T\|z, t)qR(z′, T\|z, t; z0, 0)dz′. (83) So we get a explicit form of ∇Rt log hR(Rt, t; R0): ∇Rt log hR(Rt, t; R0) = EqR(RT ,T \|Rt,t;z0)[∇Rt log pR(RT , T\|Rt, t)\|z0, Rt]. (84) Then the h-transformed SDE becomes dRt = fR(Rt, t) + σ2(t)EqR(RT ,T \|Rt,t;z0)[∇Rt log pR(RT , T\|Rt, t)\|z0, Rt] dt + σ(t)dWt. (85) Since hR(z, 0; z0) = R pR(z′, T\|z, 0) qdata(z′\|z0) pR(z′,T \|z0,0)dz′ = R qdata(z′\|z0)dz′ = 1, then qR(z′, T\|z0, 0) = pR(z′, T\|z0, 0) qdata(z′\|z0) pR(z′, T; z0, 0)hR(z0, 0; z0) = qdata(z′\|z0), (86) which means qR(RT , T\|R0, 0) = qdata(RT \|R0). Since the initial distribution R0 ∼qdata(Rt0), so qR(R0) = qdata(R0). So we can deduce that q(R0, RT ) = qR(R0)qR(RT , T\|R0, 0) = qdata(R0)qdata(RT \|R0) = qdata(R0, RT ). (87) Finally, we need to show that the transition density function satisfies the corresponding symmetric constrains. Since hR(z′, T; z0) = qdata(z′\|z0) pR(z′,T \|z0,0) is SE(3)-invariant, i.e. hR(ρR(g)[z], T; ρR(g)[z0]) = hR(z′, T; z0), ∀g ∈SE(3), (88) we can show that h(·, t; ·) is also SE(3)-invariant ∀t ∈[0, T] using the following property hR(z, t; z0) = Z pR(z′, T\|z, t)h(z′, T; z0)dz′. (89) 25 For any g ∈SE(3), assume ρR(g)[z] = OR(g)z + tR, where OR(g) is an orthogonal matrix satisfying det(OR(g)) = 1, then we have hR(ρR(g)[z], t; ρR(g)[z0]) = Z pR(z′, T\|ρR(g)[z], t)h(z′, T; ρR(g)[z0])dz′ (90) = Z pR ρR(g)(ρR(g))−1[z′], T\|ρR(g)[z], t h(ρR(g)(ρR(g))−1[z′], T; ρR(g)[z0])dz′. (91) (92) By Proposition 3.1, pR ρR(g)(ρR(g))−1[z′], t0\|ρR(g)[z], t = pR (ρR(g))−1[z′], t0\|z, t , let z1 = ρR(g))−1[z′], then hR(ρR(g)[z], t; ρR(g)[z0]) (93) = Z pR (ρR(g))−1[z′], T\|z, t h(ρR(g)(ρR(g))−1[z′], T; ρR(g)[z0])dz′ (94) = Z pR (z1, T\|z, t) h(ρR(g)[z1], T; ρR(g)[z0]) det(OR(g))dz1 (95) = Z pR (z1, t0\|z, t) h(z1, t0; z0)dz1 (96) = hR(z, t; z0). (97) So h(·, t; ·) is SE(3)-invariant ∀t ∈[0, T]. Then we show qR(z′, t′\|z, t; z0, 0) preserves the symmetric constraints: qR(ρR(g)[z′], t′\|ρR(g)[z], t; ρR(g)[z0], 0) (98) = pR(ρR(g)[z′], t′\|ρR(g)[z], t)hR(ρR(g)[z′], t′; ρR(g)[z0]) hR(ρR(g)[z], t; ρR(g)[z0]) (99) = pR(z′, t′\|z, t)hR(ρR(g)[z′], t′; ρR(g)[z0]) hR(ρR(g)[z], t; ρR(g)[z0]) (100) = pR(z′, t′\|z, t)hR(z′, t′; z0) hR(z, t; z0) (101) = qR(z′, t′\|z, t; z0, 0), (102) which completes our proof. B.5 Objective Function of the Equivariant Diffusion Bridge Lemma B.9. Let X1, · · · , Xn, Y, Z be random variables. Then the optimal approximation of Y based on {X}n i=1 is f ∗(X1, · · · , Xn) = arg min f E∥Y −f(X1, · · · , Xn)∥2 = E[Y\|X1, · · · , Xn]. Proof. Denote X = (X1, · · · , Xn). We show the following decomposition first: E∥Y −f(X)∥2 = E∥Y −E[Y\|X]∥2 + E ∥E[Y\|X] −f(X)∥2 . (103) We can compute E∥Y −f(X)∥2 directly by E∥Y −f(X)∥2 = E∥Y −E[Y\|X] + E[Y\|X] −f(X)∥2 (104) = E∥Y −E[Y\|X]∥2 + E ∥E[Y\|X] −f(X)∥2 (105) + E⟨Y −E[Y\|X], E[Y\|X] −f(X)⟩. (106) Since E⟨Y −E[Y\|X], E[Y\|X] −f(X)⟩= E [E⟨Y −E[Y\|X], E[Y\|X] −f(X)⟩\|X] = 0, (107) we have E∥Y −f(X)∥2 = E∥Y −E[Y\|X]∥2 + E ∥E[Y\|X] −f(X)∥2 (108) + E⟨Y −E[Y\|X], E[Y\|X] −f(X)⟩ (109) = E∥Y −E[Y\|X]∥2 + E ∥E[Y\|X] −f(X)∥2 (110) ≥E∥Y −E[Y\|X1, · · · , Xn]∥2. (111) 26 The inequality becomes equality if and only if f(X1, · · · , Xn) = E[Y\|X1, · · · , Xn]. So the the optimal approximation of Y based on {X}n i=1 is E[Y\|X1, · · · , Xn], i.e. f ∗(X1, · · · , Xn) = arg min f E∥Y −f(X1, · · · , Xn)∥2 = E[Y\|X1, · · · , Xn]. (112) Proposition B.10. The training objective function of Equivariant Diffusion Bridge is: L(θ) = E(z0,z1)∼qdata(Rt0,Rt1),Rt∼qR(Rt,t\|z1,T ;z0,0)λ(t)∥vθ(Rt, t; z0)−∇Rt log pR(z1, T\|Rt, t)∥2, (113) where t ∼U(0, T). Then the optimal parameter θ∗= arg min θ L(θ) satisfies vθ∗(Rt, t; z0) = EqR(RT ,T \|Rt,t;R0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt]. (114) Proof. Let L(θ) = Et∼U(0,T )λ(t)Lt(θ), where Lt(θ) = E(z0,z1)∼qdata(Rt0,Rt1),Rt∼qR(Rt,t\|z1,T ;z0,0)∥vθ(Rt, t; z0) −∇Rt log pR(z1, T\|Rt, t)∥2. (115) Then by Lemma B.9, vθ(Rt, t; z0) = EqR(RT ,T \|Rt,t;R0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt] minimize Lt(θ), ∀t ∈[0, T]. Since λ(t) ≥0, so the optimal parameter θ∗= arg min θ L(θ) satisfies vθ∗(Rt, t; z0) = EqR(RT ,T \|Rt,t;R0)[∇Rt log pR(RT , T\|Rt, t)\|R0, Rt], ∀t ∈[0, T]. (116) B.6 Proof of Theorem 3.4 Theorem B.11 (Chain of Equivariant Diffusion Bridges). Let {(Rt i)t∈[0,T ]}i∈[N−1] denote a series of N equivaraint diffusion bridges defined in Theorem 3.3. For the i-th bridge (Rt i)t∈[0,T ], if we set (1) hi R(z, t; z0) = R pR(z′, T\|z, t) qi+1 traj (z′\|z0) pR(z′,T \|z0,0)dz′; (2) R0 0 ∼q0 traj( ˜R0), R0 i = RT i−1, ∀0 < i < N, then the joint distribution qR(R0 0, RT 0 , RT 1 , · · · , RT N−1) induced by {(Rt)t∈[0,T ]}i∈[N−1] equals to qtraj( ˜R0, ..., ˜RN). We call this process a chain of equivariant diffusion bridges. Proof. By Theorem 3.3, the transition density function of (Rt i)t∈[0,T ] satisfies qi R(RT i \|R0 i ) = qi traj(RT i \|R0 i ), ∀0 ≤i ≤N −1. The ground truth probability density function has the decomposition q0 traj( ˜R0) QN i=1 qi traj( ˜Ri\| ˜Ri−1). Then we use the boundary condition, R0 0 ∼q0 traj( ˜R0), R0 i = RT i−1, ∀0 < i < N, we have q(R0 0, RT 0 , RT 1 , · · · , RT N−1) = q0 R(R0 0) N Y i=1 qi R(RT i \|RT i−1) (117) = q0 R(R0 0) N Y i=1 qi R(RT i \|R0 i ) (118) = q0 traj(R0 0) N Y i=1 qi traj(RT i \|R0 i ) (119) = qtraj(R0 0, RT 0 , RT 1 , · · · , RT N−1). (120) So the joint distribution qR(R0 0, RT 0 , RT 1 , · · · , RT N−1) induced by {(Rt)t∈[0,T ]}i∈[N−1] equals to qtraj( ˜R0, ..., ˜RN). 27 B.7 Objective of the Chain of Equivariant Diffusion Bridge Proposition B.12. The training objective function of the Chain of Equivariant Diffusion Bridge is: L′(θ) = E(z0,...,zN)∼qtraj( ˜ R0,..., ˜ RN),t,Rt′ i λ(t)∥vθ(Rt′ i , t; zi) −∇Rt′ i log pi R(zi+1, T\|Rt′ i , t′)∥2, (121) where t ∼U(0, N × T), i = ⌊t T ⌋, t′ = t −i × T, Rt′ i ∼qi R(Rt′, t′\|zi+1, T; zi, 0). Then the optimal parameter θ∗= arg min θ L′(θ) satisfies vθ∗(Rt′ i , t; zi) = Eqi R(RT i ,T \|Rt′ i ,t;R0 i )[∇Rt i log pi R(RT i , T\|Rt i, t)\|R0 i , Rt i]. (122) Proof. Let L′(θ) = Et∼U(0,NT )λ(t)L′ t(θ), where L′ t(θ) = E(z0,...,zN)∼qtraj( ˜ R0,..., ˜ RN),t,Rt′ i ∥vθ(Rt′ i , t; zi) −∇Rt′ i log pi R(zi+1, T\|Rt′ i , t′)∥2, (123) where t ∼U(0, N × T), i = ⌊t T ⌋, t′ = t −i × T, Rt′ i ∼qi R(Rt′, t′\|zi+1, T; zi, 0). Then by Lemma B.9, vθ∗(Rt′ i , t; zi) = Eqi R(RT i ,T \|Rt′ i ,t;R0 i )[∇Rt i log pi R(RT i , T\|Rt i, t)\|R0 i , Rt i] minimize L′ t(θ), ∀t ∈ [0, NT]. Since λ(t) ≥0, so the optimal parameter θ∗= arg min θ L(θ) satisfies vθ∗(Rt′ i , t; zi) = Eqi R(RT i ,T \|Rt′ i ,t;R0 i )[∇Rt i log pi R(RT i , T\|Rt i, t)\|R0 i , Rt i]. (124) B.8 Proof of Theorem 3.5 In this paper, we choose the Brownian bridge as our matching target. Let’s first recall the definition and properties of the Brownian bridge. A Brownian bridge (Xt)t∈[0,T ] with the initial position X0 and the terminal position XT is given by the following SDE dXt = XT −Xt T −t dt + σdWt, (125) where Wt is the standard wiener process. The solution of the Brownian bridge is given by Xt ∼N (1 −t)X0 + tX1, σ2t(1 −t) . (126) Next, we recall the definition of the KL Divergence: Definition B.13 (KL Divergence). The relative entropy (or Kullback–Leibler Divergence) KL(f\|\|g) between two probability density functions f(x) and g(x) is defined by: KL(f\|\|g) = Z f(x) log f(x) g(x) dx. (127) In general, let P and Q be two probability measures on space X. Assume P is absolutely continuous with respect to Q then the Kullback–Leibler Divergence between P and Q is defined as follows KL(P\|\|Q) = Z X log dP dQdP, (128) where dP dQ is the Radon–Nikodym derivative of P with respect to Q. When we need to compute the KL divergence between the path measures associated with two SDEs, the Girsanov’s theorem [53] is an useful tool to get the Radon–Nikodym derivative between the two measure. The precise statements are as follows. Theorem B.14 (Girsanov’s Theorem). Let Wt be a d-dimensional Wiener process defined on (Ω, F, (Ft), P). Let Ht be a d-dimensional Ft−adapted process such that Z T 0 ∥Ht∥2dt < ∞, P −a.s. (129) 28 Define Zt = exp Z t 0 Hs · dWt −1 2 Z t 0 ∥Ht∥2ds . (130) Assume Zt is a martingale. Define the probability measure Q on FT by dQ = ZT dP. (131) Let Mt = Wt − R t 0 Hsds, then Mt is a d−dimensional Wiener process with respect to Q. In practice, the condition that Zt is a martingale is hard to vertify. So the condition is often replaced by the Novikov’s condition E " exp 1 2 Z T 0 ∥Ht∥2dt !# < ∞. (132) For more discussions and applications of the Girsanov’s theorem, please see [86, 74, 28]. Now we can give the precise assumptions and proof of Theorem 3.5 using the properties of Brownian Bridge and Theorem B.14. Theorem B.15. Assume ( ˜Ri)i∈[N] is sampled by simulating a prior SDE on geometric states d ˜Rt = −∇H∗ R( ˜Rt)dt + σd ˜ Wt. Let µ∗ i denote the path measure of this prior SDE when t ∈[iT, (i + 1)T]. Building upon ( ˜Ri)i∈[N], let {µi R}i∈[N−1] denote the path measure of our chain of equivariant diffusion bridges. Assume {µi R}i∈[N−1] is composed of chain of the Brownian Bridge. Assume the total time is NT = 1. Under the following assumptions: • H∗ R(·) : Rd →R is a scalar function with continuous second-order partial derivative; • The drift function is Lipschitz: there exist a constant L such that ∥∇H∗ R(x) −∇H∗ R(y)∥≤L∥x −y∥, ∀x, y ∈Rd; • H∗ R(·) satisfies ∥∇H∗ R(x)∥≤K(1 + ∥x∥), ∀x ∈Rd; • E∥˜Rt∥2 < M, ∀t ∈[0, NT]; • h(t) = E[H∗ R( ˜Rt)] is a continuous function on t ∈[0, NT]; • The Novikov’s condition: E " exp 1 2 Z NT 0 ∥∇H∗ R( ˜ Wt)∥2dt !# < ∞; • The function H∗ R satisfies the following regulaity condition: there exist a constant C such that ∇2H∗ R(x) −∥∇H∗ R(x)∥2/σ2 < C, ∀x ∈Rd; then we have lim N→∞max i KL(µ∗ i \|\|µi R) = 0. Proof. Let p∗be the probability density function associated with the ground truth SDE d ˜Rt = f ∗ R( ˜Rt, t)dt+σd ˜ Wt, ˜R0 = R0. Let {(Rt i)t∈[0,T ]}i∈[N−1] denote a series of N equivaraint diffusion bridges defined in Theorem 3.4. Then by theorem 3.4, qR(R0 0, RT 0 , RT 1 , · · · , RT N−1) induced by {(Rt)t∈[0,T ]}i∈[N−1] equals to p∗ R(R0 0, RT 0 , RT 1 , · · · , RT N−1). Additionally, the conditional probability density function qR(Rt i\|RT i , R0 i ) , for iT ≤t < (i + 1)T, is associated with the Brownian bridge dRt i = RT i −Rt i T −t′ dt′ + σdWt, (133) where t′ = t −iT. Then by the chain rule of KL divergence KL(µ∗ i \|\|µi R) = KL(p∗ i ( ˜R(i+1)T , ˜RiT )\|\|qi R( ˜R(i+1)T , ˜RiT )+ (134) Ep∗ i ( ˜R(i+1)T , ˜RiT ) h KL(µ∗ i (·\| ˜R(i+1)T , ˜RiT )\|\|µi R(·\| ˜R(i+1)T , ˜RiT )) i . (135) 29 Since p∗ i ( ˜R(i+1)T , ˜RiT ) = qi R( ˜R(i+1)T , ˜RiT ), we have KL(µ∗ i \|\|µi R) = Ep∗ i ( ˜R(i+1)T , ˜RiT ) h KL(µ∗ i (·\| ˜R(i+1)T , ˜RiT )\|\|µi R(·\| ˜R(i+1)T , ˜RiT )) i . (136) Since the prior SDE is time homogeneous, we can only consider the case i = 0 without loss of generality. Let υ be the path measure of the Brownian motion σ ˜ Wt on space R. Since the condition of Theorem B.14 is satisfied, then we can use Theorem B.14 and get dµ∗ 0(·\| ˜R0) = exp −1 σ Z T 0 ∇H∗ R(σ ˜ Wt) · d ˜ Wt − 1 2σ2 Z T 0 ∥∇H∗ R(σ ˜ Wt)∥2dt ! dυ(·\| ˜R0). (137) Then we can use the Ito’s formula (Theorem B.1) to simplify the expression dµ∗ 0(·\| ˜R0) = exp 1 σ2 (H∗ R(σ ˜ W0) −H∗ R(σ ˜ WT )) + 1 2 R T 0 (∇2H∗ R(σ ˜ Wt) − 1 σ2 ∥∇H∗ R(σ ˜ Wt)∥2)dt dυ(·\| ˜R0). (138) To simplify our notation, we denote ZT = exp 1 σ2 (H∗ R(σ ˜ W0) −H∗ R(σ ˜ WT )) + 1 2 Z T 0 (∇2H∗ R(σ ˜ Wt) −1 σ2 ∥∇H∗ R(σ ˜ Wt)∥2)dt ! . (139) Let F, g be measurable functions on C[0, T], Rd, respectively. Then by the disintegration of Wiener measure into pinned Wiener measures (path measure of the Brownian Bridge), we have Eµ∗ 0(·\| ˜R0)[Fg( ˜ σW T )] = Eυ(·\| ˜R0)[Fg(σ ˜ WT )ZT ] = Z Eυ(·\| ˜R0, ˜RT =x)[FZT ]g(x)pT (x\| ˜R0)dx, (140) where pT (x\| ˜R0) is the transition density function of σ ˜ Wt. Let F = 1, we get Z Eυ(·\| ˜R0, ˜RT =x)[ZT ]g(x)pT (x\| ˜R0)dx = Z g(x)p∗ 0(x\| ˜R0)dx. (141) So we have Eυ(·\| ˜R0, ˜RT =x)[ZT ] = p∗ 0(x\| ˜R0)/pT (x\| ˜R0). Let g = 1, then we get Z Eµ∗ 0(·\| ˜R0, ˜RT =x)[F]p∗ 0(x\| ˜R0)dx = Z Eυ(·\| ˜R0, ˜RT =x)[FZT ]pT (x\| ˜R0)dx. (142) So we can conclude that dµ∗ 0(·\| ˜R0, ˜RT ) dυ(·\| ˜R0, ˜RT ) = pT ( ˜RT \| ˜R0) p∗ 0( ˜RT \| ˜R0) · exp( 1 σ2 (H∗ R( ˜R0)) exp( 1 σ2 (H∗ R( ˜RT )) exp 1 2 R T 0 (∇2H∗ R(·) − 1 σ2 ∥∇H∗ R(·)∥2)dt . (143) Note that µi R(·\| ˜R0, ˜RT ) = υ(·\| ˜R0, ˜RT ). Now we can calculate the KL divergence by KL(µ∗ 0\|\|µ0 R) = Ep∗ 0( ˜RT , ˜R0) h KL(µ∗ 0(·\| ˜RT , ˜R0)\|\|µ0 R(·\| ˜RT , ˜R0)) i (144) ≤Ep∗ 0( ˜RT , ˜R0)  log  pT ( ˜RT \| ˜R0) p∗ 0( ˜RT \| ˜R0) · exp 1 σ2 (H∗ R( ˜R0) exp 1 σ2 (H∗ R( ˜RT )    + CT 2 (145) = Ep∗ 0( ˜R0, ˜Rt) " log pT ( ˜RT \| ˜R0) p∗ 0( ˜RT \| ˜R0) !# + E[ 1 σ2 H∗ R( ˜R0)] −E[ 1 σ2 H∗ R( ˜RT )] + CT 2 (146) = −Ep∗ 0( ˜R0) KL(p∗ 0( ˜RT \| ˜R0)\|\|pT ( ˜RT \| ˜R0)) + h(0) −h(T) σ2 + CT 2 (147) ≤h(0) −h(T) σ2 + CT 2 . (148) When N →∞, T = 1 N →0. Since h(t) is continuous by our assumption, then KL(µ∗ 0\|\|µ0 R) → 0. 30 C Derivation of Practical Objective Function In this subsection, we show the implementation details of our framework. We set T = 1 in all the experiments. Matching objective. We design the SDE on geometric states in Proposition 3.1 to be: dRt = σdWt, with transition density pR(z′, t′\|z, t) = N(z0, σ2(t′ −t)I) (149) The explicit form of the objective is ∇Rt log pR(z1, 1\|Rt, t) = ∇Rt log N(z0, σ2(1 −t)I) = z1 −Rt σ2(1 −t) (150) Then the h-transformed SDE becomes dRt = R1 −Rt 1 −t dt + σdWt, (151) which is known as the Brownian bridge. The corresponding h-transformed density is qR(Rt, t\|z1, 1; z0, 0) = N(tz1 + (1 −t)z0, σ2t(1 −t)I). (152) In practice, we do not use qR(R0, 0\|z1, 1; z0, 0) = δ(R0 −z0) as our initial distribution. We use qR(R0, 0\|z1, 1; z0, 0) = N(z0, σ2I) instead. Since the solution of the Brownian bridge is given by Rt = (1 −t)R0 + tR1 + σ p t(1 −t)Z, (153) where Z ∼N(0, I), then the marginal distribution of Rt becomes N((1 −t)z0 + tz1, (1 −t)σ2I). We use this distribution to sample geometric state Rt in the training stage. Trajectory guidance. Similarly, we set T = 1 N , pi R(zi+1, T\|Rt′, t′) = N(Rt′, σ2 i (T−t′)I) when we use the trajectory guidance. So the h-transformed SDE becomes dRt i = RT i −Rt i T −t dt + σidWt, (154) which is a Brownian bridge with T = 1 N . Then associated density function is qi R(Rt′, t′\|zi+1, T; zi, 0) = N( t′ T zi+1 + T −t′ T zi, σ2 i t′(T −t′) T 2 I). (155) Additionally, we set σi decays linearly with respect to i N , i.e. σi = N−i N σ, where σ is a hyperparameter. Again, in training stage, we set qi R(R0 i , 0\|z1, 1; z0, 0) = N(z0, σ2 i I) as initial distribution, and the terminal distribution is qi R(R0 i , 0\|z1, 1; z0, 0) = N(z1, σ2 i+1I), which is same as the initial distribution of the next bridge. Sampling Algorithm We use the ODE-based method to generate samples at inference time. After the training process, the neural network vθ is trained as described in Algorithm 3 and Algorithm 4. When the network is trained without trajectory guidance, we simulate the following ODE to generate samples: d Rt d t = vθ(Rt, t; R0), R0 ∼qdata(Rt0), t ∈[0, T] . (156) When the network is trained with trajectory guidance, we solve the following ODE to generate samples: d Rt d t = vθ(Rt, t; R⌊t T ⌋T ), R0 ∼qdata(Rt0), t ∈[0, N × T] . (157) Denote a black box ODE solver by Solver(v, t). Solver(v, t) takes a vector field v and a time point as inputs, then returns the solution of the ODE d Xt d t = v(Xt, t; ϕ), X0 = x0, (158) at the specific time t, i.e. Solver(v, t) = Xt. Combining all the above design choices, we have the following algorithms for sampling our Geometric Diffusion Bridge (Algorithm 5) and leveraging trajectory guidance if available (Algorithm 6). 31 Algorithm 3 Training 1: repeat 2: (z0, z1) ∼qdata(Rt0, Rt1) 3: t ∼U[0, T] 4: ϵ ∼N(0, I) 5: Rt = t T z1 + T −t T z0 + √ t(T −t) T σϵ 6: Take gradient descent step on ∇θλ(t) z1−Rt σ2(T −t) −vθ(Rt, t; z0) 2 7: until converged Algorithm 4 Training with trajectory guidance 1: repeat 2: (z0, . . . , zN) ∼qtraj( ˜R0, . . . , ˜RN) 3: t ∼U (0, N × T), i = ⌊t T ⌋, t′ = t −i × T 4: ϵ ∼N(0, I) 5: Rt′ i = t′ T zi+1 + T −t′ T zi + √ t′(T −t′) T σiϵ 6: Take gradient descent step on ∇θλ(t) zi+1−Rt′ i σ2 i (T −t′) −vθ(Rt′ i , t; zi) 2 7: until converged Algorithm 5 Sampling Require: Initial geometric state z0 ∼qdata(Rt0), a trained neural network vθ, a numerical ODE solver Solver(v, t) 1: R0 = z0 2: RT = Solver(vθ(Rt, t; R0), T) Ensure: RT Algorithm 6 Sampling with trajectory guidance Require: Initial geometric state z0 ∼qdata(Rt0), a trained neural network vθ, a numerical ODE solver Solver(v, t) 1: R0 = z0 2: RNT = Solver(vθ(Rt, t; R⌊t T ⌋T ), t = NT) Ensure: RNT D Experiments D.1 Equilibrium State Prediction Dataset. QM9 [79] is a quantum chemistry benchmark consisting of 134k stable small organic molecules, which has been widely used for molecular modeling. These molecules correspond to the subset of all 133,885 species out of the GDB-17 chemical universe of 166 billion organic molecules. In convention, 110k, 10k, and 11k molecules are used for train/valid/test sets respectively. The geometric conformations that are minimal in energy are provided in the QM9 dataset. The equilibrium conformation and its relative properties are all calculated at the B3LYP/6-31G(2df,p) level of quantum chemistry. Molecule3D [116] is a large-scale dataset curated from the PubChemQC project [67, 71], consisting of 3,899,647 molecules in total, 2,339,788 molecules in training set, 779,929 molecules in the validation set, 779,930 molecules in the test set, and its train/valid/test splitting ratio is 6 : 2 : 2. For each molecule, the 2D atom graph, the 3D equilibrium geometric conformation, and four extra properties are provided. In particular, both random and scaffold splitting methods are adopted to thoroughly evaluate the in-distribution and out-of-distribution performance. For each molecule, an initial geometric state is generated by using fast and coarse force field [73, 52] and geometry optimization is conducted to obtain B3LYP/6-31G* level DFT-calculated equilibrium geometric structure. Baselines. We comprehensively compare our GDB framework with previous equilibrium conformation prediction methods. Following [111], we use DG and ETKDG algorithms implemented by RDkit as our fundamental baselines. The benchmark [116] used the DeeperGCN-DAGNN framework [60] which proposed a deep graph neural network architecture to predict 3D geometric conformation of the molecule based on its 2D graph structure, and got impressive performance on the Molecule3D dataset. GINE [39] proposed a method for pretraining GNN to improve the performance and capacity of GNN. GATv2 [10] proposed a dynamic graph attention mechanism and improved the performance of the graph attention network on several tasks. GPS [80] proposed a general framework that supported multiple types of encodings with efficiency and scalability guarantees in both small and large graph prediction tasks. GTMGC [111] proposed a novel neural network based on Graph-Transformer (GT) [118, 66, 119, 65] to predict the equilibrium conformation of the molecule in 3D based on its 2D graph structure. Metric. Following [116], three metrics are adopted to evaluate predictions of equilibrium states: (1) C-RMSD: given prediction ˆR = {ˆri}N i=1 which is rigidly aligned to the ground-truth R∗= {r∗ i }N i=1 by the Kabsch algorithm [44], Root Mean Square Deviation between their atoms is calculated, i.e., C-RMSD( ˆR, R∗) = q 1 N PN i=1 ∥ˆri −r∗ i ∥2 2; (2) D-RMSE: based on ˆR and R∗= {r∗ i }N i=1, interatomic distances can be calculated, i.e., { ˆdi}N′ i=1 and { ˆd∗ i }N ′ i=1. Root Mean Square Error be32 tween these distances is calculated, i.e., D-RMSE({ ˆdi}N ′ i=1, { ˆd∗ i }N ′ i=1) = q 1 N ′ PN i=1(di −d∗ i )2; (3) D-MAE({ ˆdi}N′ i=1, { ˆd∗ i }N′ i=1) = 1 N′ PN i=1 \|di −d∗ i \|. Settings. In this task, we parameterize vθ(Rt, t; R0) by extending a Graph-Transformer based equivariant network [92, 63] to encode both time steps and initial geometric states as conditions. For training, we use AdamW as the optimizer, and set the hyper-parameter ϵ to 1e-8 and (β1, β2) to (0.9,0.999). The gradient clip norm is set to 5.0. The peak learning rate is set to 1e-4. The batch size is set to 512. The weight decay is set to 0.0. The model is trained for 500k steps with a 30k-step warm-up stage. After the warm-up stage, the learning rate decays linearly to zero. The noise scale σ is set to 0.5. For inference, we use 10 time steps with the Euler solver [12]. All models are trained on 16 NVIDIA V100 GPU. D.2 Structure Relaxation Dataset. Open Catalyst 2022 (OC22) dataset [105] is a widely used dasaset, which has great significance for the development of Oxygen Evolution Reaction (OER) catalysts. Each data in the dataset is in the form of the adsorbate-catalyst complex. Both initial and adsorption states with trajectories connecting them are provided. The dataset consists of 62,331 Density Functional Theory (DFT) relaxations trajectories, and about 9,854,504 single-point DFT calculations across a range of oxide materials, coverages, and adsorbates.The training set consists of 45,890 catalyst-adsorbate complexes. To better evaluate the model’s performance, the validation and test sets consider the in-distribution (ID) and out-of-distribution (OOD) settings which use unseen catalysts, containing approximately 2,624 and 2,780 complexes respectively. Baselines. Following [105], we choose strong MLFF baselines trained on force field data for a challenging comparison. Spinconv [94] introduced a novel approach called spin convolution to model angular information between sets of neighboring atoms in a graph neural network and got impressive performance in molecular simulation tasks. Gemnet [32] proposed multiple structural improvements for geometric GNN with theoretical insights, which significantly improved the experimental performance as well. Based on Gemnet’s framework, Gemnet-OC [34] modified the architecture of the network and improved the experimental performance on more diverse tasks. In [105], there are still other baseline setting. [105] introduce a large-scale dataset Open Catalyst 2020 (OC20), which consists of 1,281,040 Density Functional Theory (DFT) relaxations and 264,890,000 single point evaluations to help training the baseline model. [105] presented baselines using both OC20 and OC22 data in training stage and baselines using only OC20/OC22 for comparison. Metric. Following [105], we use the Average Distance within Threshold (ADwT) as the evaluation metric, which reflects the percentage of structures with an atom position MAE below thresholds. To be more precise, the ADWT metric across thresholds ranging from β = 0.01˚ A to β = 0.5˚ A in increments of 0.001˚ A. The computation of ADwT metric is to count the percentage of structures with an atom position MAE below the threshold. Settings. In this task, We parameterize vθ(Rt, t; R0) by using GemNet-OC [34], which also serves as a verification that our framework is compatible with different backbone models. For training, we use AdamW as the optimizer, and set the hyper-parameter ϵ to 1e-8 and (β1, β2) to (0.9,0.999). The gradient clip norm is set to 10.0. The peak learning rate is set to 5e-4. The batch size is set to 64. The weight decay is set to 0.0. The model is trained for 200k steps. After the warm-up stage, the learning rate decays linearly to zero. The noise scale σ is set to 0.5. The trajectory length is set to N = 10. For inference, we also use 10 time steps with the Euler solver [12]. 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4,428	Linguistic Collapse: Neural Collapse in (Large) Language Models Robert Wu University of Toronto, Vector Institute rupert@cs.toronto.edu Vardan Papyan University of Toronto, Vector Institute vardan.papyan@utoronto.ca Abstract Neural collapse (NC) is a phenomenon observed in classification tasks where top-layer representations collapse into their class means, which become equinorm, equiangular and aligned with the classifiers. These behaviours — associated with generalization and robustness — would manifest under specific conditions: models are trained towards zero loss, with noise-free labels belonging to balanced classes, which do not outnumber the model’s hidden dimension. Recent studies have explored NC in the absence of one or more of these conditions to extend and capitalize on the associated benefits of ideal geometries. Language modelling presents a curious frontier, as training by token prediction constitutes a classification task where none of the conditions exist: the vocabulary is imbalanced and exceeds the embedding dimension; different tokens might correspond to similar contextual embeddings; and large language models (LLMs) in particular are typically only trained for a few epochs. This paper empirically investigates the impact of scaling the architectures and training of causal language models (CLMs) on their progression towards NC. We find that NC properties that develop with scale (and regularization) are linked to generalization. Moreover, there is evidence of some relationship between NC and generalization independent of scale. Our work thereby underscores the generality of NC as it extends to the novel and more challenging setting of language modelling. Downstream, we seek to inspire further research on the phenomenon to deepen our understanding of LLMs — and neural networks at large — and improve existing architectures based on NC-related properties. Our code is hosted on GitHub: https://github.com/rhubarbwu/linguistic-collapse. 8 6 4 2 Variability Collapse (log CDNV) ( 1) 1 2 3 4 Val. Loss R2 = 0.92 0.25 0.50 0.75 1.00 Hyperspherical Uniformity ( 2) R2 = 0.84 1 2 Uniform Duality (cosine) ( 3) R2 = 0.69 0.1 0.2 0.3 Classifier Agreement (linear vs. NCC) ( 4) R2 = 0.89 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 1: Simultaneous development of the four neural collapse (NC) [1] properties in 230 causal language models trained on TinyStories [2], alongside improvement in generalization (i.e. validation performance). Left to right: NC1) within-class (representation) variability collapse; GNC2) hyperspherical uniformity of class means; UNC3) uniform duality between class means and corresponding classifiers; and NC4) agreement between token (maximum a prior) classifiers and implicit nearestclass centre classifiers. Coloured by model size and annotated with coefficient of determination (R2). 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1 Introduction 1.1 Neural Collapse A learning phenomenon known as neural collapse (NC) emerges during the terminal phase of training (TPT) deep neural networks with cross-entropy (CE) loss for classification tasks.[1] It was originally characterized as the co-occurrence of the following properties in a model’s top-layer (also known as last-layer) representations (also known as features or embeddings) and linear classifier weights: (NC1) Within-class variability collapse: Top-layer representations collapse to their class means. (NC2) Convergence to a simplex ETF: Class means tend towards equinorm and equiangular vectors when centred about the global average. The resulting geometry — known as a simplex equiangular tight frame (ETF) — maximizes pairwise angles and distances. (NC3) Convergence to self-duality: Linear classifier weight vectors converge to their corresponding top-layer class mean vectors, and thus also form a simplex ETF. (NC4) Nearest decision rule: Linear classifiers approximate a nearest-class centre (NCC) classifier: top-layer representations predict the class with the closest mean (implied by NC1-3). These behaviours, often associated with improved generalization and robustness [3–5] (among other benefits, such as those discussed in §1.4), traditionally manifest under the following conditions: Rq1) Few classes: The number of classes is upper-bounded by the embedding dimension plus one: C ≤d + 1; this is required to construct a perfect simplex ETF. Rq2) Balanced classes: The number of samples is equal across classes: Nc = Nc′, ∀c ̸= c′. Rq3) Noise-free labels: Identical (or very similar) embeddings should belong to the same class. Rq4) Sufficient training (TPT): The model is trained past zero error, towards zero loss. Absent these conditions, one does not typically anticipate NC. Since then, follow-up studies have extended beyond and proposed techniques of quantifying or achieving NC (discussed in Section 2). 1.2 (Large) Language Models NC is a phenomenon observed specifically in classification tasks. While not traditionally thought of as classifiers, language models — including large language models (LLMs) — learn to model aleatoric uncertainty, which can be viewed as stochastic token prediction [6]. For instance, masked language models (MLMs) such as BERT [7] predict one or several masked tokens within an input sequence based on the surrounding context. Likewise, autoregressive or causal language models (CLMs) such as GPT [8] perform next-token prediction (NTP) in a sequence given the context of previous tokens. Most of these models are essentially (pre-)trained by token classification on their vocabularies. This parallel — also drawn by [9] — raises a few natural questions: 1. Does the pre-training stage of a language model exhibit NC? 2. How do scaling and other training configurations influence NC in (L)LMs? 3. To what extent is NC in (L)LMs correlated with their generalization abilities? 4. Do such correlations, between NC and improved generalization, persist independent of the (potential confounders of) model size and training? To address these, we first examine the specific settings of training CLMs as they are opposed (¬) to the traditional prerequisites (R1-4, §1.1) for NC to manifest. ¬Rq1) Many classes: The unique tokens found in language modelling vocabularies are vast, usually numbering in the tens of thousands and far exceeding the hidden dimension [10]. ¬Rq2) Imbalanced classes: The distribution of tokens in natural language is typically very imbalanced [11, 12], as is the case in TinyStories [2], the dataset we use (Appendix Figure 4). It has an average of 16K samples per class but a standard deviation of over 32K. 2 ¬Rq3) Ambiguous contexts: There may exist very similar or even identical contexts followed by different tokens in the natural language data [13]. For instance, over half of the sequences in TinyStories [2] lead with “Once upon a time”, but only three-quarters of these follow with a comma (“Once upon a time,”). In other words, there is almost certainly some ambiguity to contend with in our context embeddings. ¬Rq4) Undertraining: Most practical language models (including LLMs) are not trained for more than a few epochs [14, 15]. Further optimization typically renders diminishing returns in improving evaluation performance [16] long before any possible TPT. 1.3 Contributions We train a suite of Transformer-based [17] CLMs1 across a grid of model widths, depths, and training epochs on the TinyStories dataset [2] to assess the degrees to which NC properties develop and how they relate to generalization performance. Our findings (summarized in Figure 1) reveal: • Emergence of NC with scale: As we scale model size and training, the properties of NC emerge; within-class variability (NC1) and interference are reduced while hyperpsherical uniformity (GNC2), uniform duality (UNC3), and classifier agreement (NC4) improve. • Progression towards hyperspherical uniformity: Class means, while unable to form a simplex ETF (NC2), nonetheless tend towards uniform dispersion on a hypersphere, a geometry theorized by [18] and formalized by [19] as hyperspherical uniformity (GNC2). • Tendency towards uniform duality: Direct alignment (self-duality) between class means and classifiers (NC3) does not appear to develop with scale. Instead, the variability of (mis)alignment across classes decreases with width and training, suggesting its minimization – which we term uniform duality (UNC3) – may be more cohesive with NC. • Correlation between NC and generalization: The developments of NC properties are correlated with improved validation performance. We show these correlations to persist even when fixing the (potential confounders of) model architecture and training by simply varying the random seeds for initialization and data shuffling. This suggests that NC is not simply a side-effect of training, but possibly a factor of generalization in its own right. 1.4 Significance Recently, methods building on NC have found use in deep learning at large. We highlight areas such as federated learning [20], graph neural networks [21], incremental/continual learning [22–24], metalearning [24, 25], out-of-distribution detection [26–28], privacy [29, 30], learning theory [31–36] transfer learning [3, 5, 37–40], and even LLM prompting [41]. With our results, we aim to extend insights from such contributions and other related works to the autoregressive language modelling domain and ultimately assist researchers in improving and interpreting their (L)LMs. 2 Related Works NC was initially observed in image classification tasks such as on CIFAR-10/100 [42] and ImageNet [43]. Since then, the phenomenon has been further studied both theoretically and empirically [5, 18, 35, 44–66], with several works venturing into settings without some of the traditional prerequisites (¬Rq1-4, §1.2) and proposing adaptations of the analysis framework or optimization procedures: A large number of classes (¬Rq1) NC established the simplex ETF as an optimal configuration. However, a perfect simplex ETF (NC2) requires that the number of classes C not exceed d + 1 where d is the embedding dimension. This requirement that d be sufficiently large is impractical when the classes number beyond the thousands.2 For instance, GPT-2 [68] and Llama 3.1 [69] have vocabularies of around 50K and 128K tokens, respectively. In such a scenario, one might still expect class means to be uniformly distributed on a d-dimensional hypersphere [18]. [19] formalize this as hyperspherical uniformity within a generalized neural 1Our models range from 3.4M (small) to 205M (large), so we inclusively use “CLM” instead of “LLM”. 2Following [67], one might describe such a setting (C > d + 1) as a model “in superposition”. 3 collapse (GNC) framework, which [9] then empirically demonstrate. These latter two works mention language modelling as applicable for GNC; [9] even framed token prediction as a classification task, just as we do. We remark however that both earlier works simulate a large number of classes by drastically shrinking the embedding dimension. In contrast, we study realistic NTP, using the full class space (vocabulary) with an imbalanced token distribution and ambiguous samples. Class imbalance (¬Rq2) NC traditionally assumed that classes were sample-balanced. Since then, follow-up works have investigated the effect of class imbalance on the prevalence of NC. [47] studied the layer-peeled model (LPM) and discovered that minority collapse occurs when classes are imbalanced across two equally-sized groups of classes; a threshold for minority collapse was later characterized by [35]. [52] showed that NC still occurs in such an LPM when the classifier is initialized as a fixed ETF. [70] introduced simplex-encoded-label interpolation (SELI) but noted that more severe imbalance worsens even this geometry. Recently, feature regularization has been employed to induce NC and improve generalization in class-imbalanced settings [58, 61, 62]. Multi-label classification (¬Rq3) In some problems, one might encounter mixed or multi-label samples, be they natural or due to noise or augmentation [71, 72]. NC was also recently studied for such data by [73], who observed that multi-label class means arrive at the average of their labels’ respective means. They also devise an augmented CE loss function to accommodate such samples. Likewise, most of our ambiguous samples could be considered multi- or soft-label: identical (or very similar) context samples with different hard token labels (¬Rq3). Under popular CLM pre-training (teacher-forcing with CE loss), this effectively precludes the prospect of achieving zero classification error and potentially introduces irreducible noise. A recent study showed that memorization of noisy labels likely leads to degradation of NC [60]. Early stages of training (¬Rq4) [53] studied NC in small ResNet [74] models that had not yet converged (similar to most LLMs). They show that within-class variability drops and “plateaus” (NC1) earlier than other NC metrics, a result that we also observe (§4.1, Figures 6, 7). Natural language processing (NLP) An earlier study reported that the ratio of within-class to between-class covariances of word embeddings increases with model depth [75, 76], seemingly at odds with NC1. It can, however, be distinguished from literature studying layer-wise NC in that it does not centre the mean representation vectors (i.e. subtract the global mean). [19] fine-tuned BERT [7] using their hyperspherical uniformity gap (HUG) objective on binary classification tasks from the GLUE benchmark [77]. [78] conducted a tangentially-related investigation of convolutional neural networks for few-class semi-supervised clustering in which they identify NC. Our work is distinct from these in several ways: a) our class space far exceeds our embedding dimension (C ≫d + 1) because we classify on the full token vocabulary; b) we analyze embeddings at a token-level granularity rather than at the sequence level; and c) our NTP task is causal (context-dependent) as opposed to the per-sample independence of their few-category classification. A more related work studied feature collapse in individual word representations [79], but the authors note that their analysis is limited to shallow NLP modelling on more rigid and tabular text data. 3 Methodology Below we describe the training setup for our CLMs (§3.1), procedures3 for collecting top-layer embeddings (§3.2, 3.7), and measurements of NC and generalization (§3.4, 3.5, 3.6, 3.7, 3.8). 3.1 Dataset and Language Models TinyStories [2] is a synthetic4 dataset generated by GPT-3.5 and GPT-4 using around 1500 English words a child might use. NTP is performed by sampling from the token vocabulary V = J1, 29233K, 3Leveraging our generic NC package: https://github.com/rhubarbwu/neural-collapse. 4TinyStories was developed and evaluated as a faithful emulation of large language modelling, so we chose it for experimentation to train hundreds of CLMs and analyze embeddings at a low cost. See Appendix A. 4 which for our purposes can therefore be framed as classification among C = 29,233 classes.5 Following the GPT-style preprocessing regime [8], raw sequences are packed into S chunks of size T, providing N = S(T −1) token samples for training.6 Details are listed in Appendix A. We use 30 CLM architectures based on GPT Neo [80], configured similarly to [2]. They vary in width (embedding dimension) d ∈{64, 128, 256, 512, 768, 1024} and depth (number of self-attention layers) L ∈{1, 2, 4, 8, 12}. Our models were trained by teacher-forcing7 using CE loss. For each architecture, we trained multiple models for 1, 3, and 10 epochs ablated over weight decay factors β = 0.0005 [51] and β = 0.1 [81]. Further details are listed in Appendices B, C. 3.2 Context Embeddings Base CLMs perform next-token prediction: given a sequence of tokens x1:t ∈Vt, a top-layer context embedding h(x1:t) ∈Rd is used to predict the next token x′ t+1 ∈V where 1 ≤t ≤T. A classifier for class c with weights wc and bias8 bc would make maximum a prior (MAP) predictions as x′ t+1 := argmax c∈V ⟨wc, h(x1:t)⟩+ bc. (1) Class embedding means For each token class c, we are interested in accumulating the mean embedding µc ∈Rd across sequences s and their contexts x(s) 1:t, where the next token x(s) t+1 is ground-truth (t < T) and equal to c: µc := 1 Nc S X s=1 T −1 X t=1 h x(s) 1:t I x(s) t+1 = c , (2) where Nc is the number of samples of class c and I is the (binary) indicator function. We also utilize their unweighted9 average ¯µ := Ec[µc], and subsequently the globally-centred means ˆµc = µc−¯µ ∥µc−¯µ∥2 . Class embedding variances In a second pass, we accumulate the sample variance norms:10 σ2 c := 1 Nc S X s=1 T −1 X t=1 h x(s) 1:t −µc 2 2 I x(s) t+1 = c . (3) 3.3 Homogeneity and Variability For some collapse measures (such as (G)NC2 and NC3), we are primarily interested in the variation rather than the average of pairwise relationships. To that end, we also include in our analysis the coefficient of variation (CoV) of several measurements, which is the ratio of their standard deviations to their means: σ(·)/µ(·). We can interpret this as a normalized measure of variability. 3.4 Signal-to-Noise Ratio — NC1 The ability to disambiguate between classes depends on the ratio of within-class to between-class variabilities. Building upon foundational works [85, 86], NC originally measured variability through an inverse signal-to-no ratio (SNR), whose minimization constitutes within-class variability collapse (NC1). We instead employ a class-distance normalized variance (CDNV) similar to [3]: ˆσc,c′ := 1 ∥µc −µc′∥k 2 · σ2 c + σ2 c′ 2∥µc −µc′∥2 2 , ∀c ̸= c′. (4) Our metric differs in that we divide by an additional power ∥µc −µc′∥k 2 of the mean distance norm. This further downweights the CDNV within well-separated class pairs in favour of emphasizing more 5Although the GPT-Neo [80] tokenizer has over 50K tokens, only a subset vocabulary appears in TinyStories. 6We cannot use the first ground-truth nor the last predicted token in any chunk. 7Parallelized training using sequences’ ground-truth labels for context as opposed to predicted tokens. 8Similar to many causal LLMs [80–84], our classifiers do not include additive biases, so bc = 0. 9Different from most literature where classes were balanced and weighting is already equal. 10This sample variance is computed across all unnormalized dimensional entries. 5 mutually noisy pairs. We found this augmented CDNV with k = 2 to be especially useful in our setting of many imbalanced and variably confusable token classes. These pairwise measures constitute the off-diagonal11 entries of a symmetric matrix in RC×C, whose average we use as an inverse SNR. Within-class variability collapse is then re-characterized by the minimization of this quantity: ˆσc,c′ →0, ∀c ̸= c′. This alternative convergence is empirically faithful to NC1 but more robust and numerically stable [3]. 3.5 Geometric Structures — (G)NC2 The separability of our representations also depends on the geometric structures found in our embeddings. [1] characterize NC2 as convergence to a simplex equiangular tight frame (ETF) [87, 88]. Equinormness Such a near-orthonormal configuration firstly implies that class means are equinorm: log ∥µc −¯µ∥2 −log ∥µc′ −¯µ∥2 →0, ∀c ̸= c′. (5) We measure CoV in the logarithms of class mean norms to assess the degree of “equinormness”. Equiangularity NC2 also entails that class means are equiangular about their centre ¯µ: pairwise distances and angles between their class means should be maximized and similar. Following [1], we measure interference (sometimes known as similarity or coherence [89, 90]). Its minimization, ⟨ˆµc, ˆµc′⟩→ −1 C −1, ∀c ̸= c′, (6) together with equinormness (Equation 5) constitute convergence to a simplex ETF. Although this geometry is not ultimately attainable since there are too many classes (C > d + 1), it can still be meaningful to measure a model’s tendency towards one. As with CDNV noise (Equation 4), pairwise interferences form off-diagonal12 entries in a symmetric matrix in RC×C. The minimization of CoV in interferences therefore expresses the degree of “equiangularity”. Hyperspherical uniformity A relaxation from the ETF is hyperspherical uniformity (GNC2), with eqiunorm (Eq. 5) means µc uniformly distributed on the d-dimensional hypersphere [18, 19]. We likewise gauge the angular uniformity with pairwise interactions through some distance kernel K:13 X c̸=c′ K (ˆµc, ˆµc′) → min ˆµ1,...,ˆµC X c̸=c′ K (ˆµc, ˆµc′) . (7) 3.6 Alignment Between Classifiers and Class Embedding Means — (U)NC3 The linear classifiers {wc}C c=1 lie in a dual vector space to that of the class means {µc}C c=1. While convergence to self-duality (NC3) was initially measured as distances [1] between class means and classifiers (Equation 11), we follow [5] to inspect class-wise cosine similarities:14 wc ∥wc∥2 , ˆµc →1, ∀c ∈V. (8) For intuition analogous to that for equinormness and equiangularity (§3.5), we also measure uniform duality (UNC3) as the minimization of the CoV of these similarities (Appendices N, O). 3.7 Agreement of the Classifiers — NC4 Finally, NC4 is described as the simplification (or approximation) of the linear classifier’s MAP prediction behaviour (Equation 1) to that of the implicit nearest-class centre (NCC) classifier: argmax c∈V ⟨wc, h⟩+ bc →argmin c∈V ∥h −µc∥2, ∀h. (9) 11The main diagonal of CDNVs is undefined (due to the zero denominator) and ignored. 12The main diagonal of interferences is equal to 1 (maximal coherence or self-similarity). 13Following [19], we employ the logarithmic inverse distance kernel Klog(a, b) = log ∥a −b∥−1 2 for its ability to emphasize gaps between small distances while scaling down the effect of larger distances. 14Dot-product would be confounded by norms and therefore inappropriate for similarity up to rescaling. 6 We calculate15 agreement as the proportion of validation samples on which the classifiers agree: 16 1 Nval Sval X s=1 Tval−1 X t=1 I x′ t+1 (s) = argmin c∈V h x(s) 1:t −µc 2 . (10) 3.8 Probing a Relationship Between NC and Generalization To isolate the effect of NC on generalization independent of model scaling and training (if it exists), we selected a two-layer 768-wide architecture of which to train twenty more instances with weight decay β = 0.0005, each with a different data shuffling seed. We then followed the remainder of the pipeline described above to collect and analyze embeddings with respect to NC. Finally, we performed a permutation test with 104 trials to determine the statistical significance of any correlation between NC and generalization that remains when we hold constant all factors but shuffling seeds. 4 Experimental Results In this section, we present the results from our empirical study on scaling and generalization: (NC1) Within-class variability is reduced across model scale (more so by width than depth) and training (up to 6 epochs), and is tightly correlated with validation performance. (NC2) Equinormness/equiangularity shows some improvement with scale, training, and performance. Hyperspherical uniformity (GNC2) also improves but more clearly and consistently. (NC3) Our models fail to achieve self-duality: class means do not align with classifiers. However, uniform duality (UNC3) is correlated with model width, training, and performance. (NC4) Larger or more trained models exhibit closer agreement between their linear and implicit NCC classifiers. Agreement is also associated with validation performance. 4.1 Within-Class Variability Collapse — NC1 Scaling our models dramatically reduces normalized variance, which is further aided by more training epochs and stronger weight decay (Appendix Figs. 6, 7). These noise reductions tightly associate with generalization (Fig. 1, left, “NC1”). The relationship is most apparent at scale. 4.2 Geometric Structures — (G)NC2 0.075 0.100 0.125 Log Mean Norms (CoV) ( 2) 1 2 3 4 Val. Loss R2 = 0.86 4000 6000 8000 10000 Interference (CoV) ( 2) R2 = 0.79 0.25 0.50 0.75 1.00 Log Distances (CoV) ( 2) R2 = 0.84 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 2: Validation loss is correlated with all three measurements: (left) equinormness (NC2) expressed as variation in logarithmic norms; (centre) equiangularity (NC2) as variation in interference; (right) hyperspherical uniformity (GNC2) as variation in logarithmic pairwise distances. Equinormness Logarithmic class mean norms grow with model width and training (Appendix Fig. 8), and subtly with depth (Appendix Fig. 9). Meanwhile, the variation of these (logarithmic) norms consistently decreases (improving equinormness) with scale (Appendix Figs. 10, 11). Both trends correlate with improved generalization (Appendix Fig. 20). 15In practice, we use an equivalent decomposition (Eq. 12). 16Note that agreement is not equivalent to accuracy. 7 Equiangularity Scaling model dimensions reduces average interference (Appendix Figs. 12, 13) down to an empirical plateau of approximately 10−3, which is more apparent in less-trained models. However, the variation of interference rises and persists when scaling (Appendix Figs. 14, 15), suggesting that interference becomes more concentrated between some pairs of classes. These results could be due to various factors, including but not limited to unfriendly conditions of language modelling (§1.2) or the impossibility of a perfect simplex ETF (§3.5). Appendix Figure 21 shows only a rough performance correlation with average interference (i.e. coherence) and a more definite — albeit still noisy — one with the variation of interference (i.e. equiangularity). In other words, the limited trends we observed toward a simplex ETF suggest that the association of NC2 with generalization begins to break down when C > d + 1. Hyperspherical uniformity Logarithmic distances drop more gradually and consistently with scale (Appendix Figs. 16, 17), implying this quantity is more robust or may not face the same geometric barriers seen in conventional interference (Appendix Figs. 14, 15). Variation of these logarithmic distances is also consistently reduced with scale (Appendix Fig. 18, 19). And finally, generalization has much stronger correlations with logarithmic distances than it has with regular interference (Fig. 2), validating the applicability of GNC [19] when C > d + 1. 4.3 Classifier (Mis)alignment and Duality — (U)NC3 Model width (d) is correlated faintly with the average similarity between class means and their respective classifiers (Appendix Fig. 23), but strongly with variation in similarity (Appendix Fig. 25). The relationships to generalization follow the same pattern (Fig. 3), suggesting that uniform duality (UNC3) might serve as a better NC property than self-duality (NC3) overall; we discuss this in §5.1. 0.2 0.3 0.4 Similarity (Cosine) (Avg) ( 3) 1 2 3 4 Val. Loss R2 = 0.31 0.5 1.0 1.5 2.0 Similarity (Cosine) (CoV) ( 3) R2 = 0.69 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 3: Validation loss shows a negligible relationship with self-duality (NC3, left) and some correlation with uniform duality (UNC3, right). In other words, UNC3 develops with scale and correlates with generalization much better than NC3. 4.4 Classifier Agreement — NC4 The linear and NCC classifiers agree on far more examples than random chance, and model scaling encourages this agreement (Appendix Figs. 29, 30). Increasing width for certain depths happens to plateau or even regress the agreement rate, but this limitation is overcome with further training. And finally, agreement is a strong indicator of generalization (Fig. 1, right, NC4). 5 Analysis We find that NC is generally promoted by model size and training and correlated with generalization (validation performance). We also discern some of this correlation independent of scale (§5.1). 8 Table 1: Permutation test of NC measurements with respect to validation loss. Twenty-one (21) identical two-layer 768-wide models were trained with different data shuffling seeds and permuted with 104 trials. The p-values below 0.05 (bolded) show those properties to be statistically significant. Property Measurement R2 Correlation (↑) p-value (↓) NC1 Within-Class Variability Collapse 0.192011 0.0485 NC2 Equinormness 0.026174 0.4870 NC2 Equiangularity 0.218574 0.0317 GNC2 Hyperspherical Uniformity 0.487935 0.0002 NC3 Self-Duality 0.322210 0.0063 UNC3 Uniform Duality 0.000036 0.9784 NC4 Classifier Agreement 0.490278 0.0001 5.1 Neural Collapse’s Relationship with Generalization Table 1 presents the correlation scores of NC metrics with generalization alongside their associated p-values from the permutation tests described in §3.8. The majority of the correlations are statistically significant (p < 0.05) independent of scale, affirming that NC is correlated with generalization. 5.2 Duality of Duality The dichotomy of self-duality (NC3) and uniform duality (UNC3) is rather conspicuous. Our main experiments find that NC3 does not consistently develop with scale while UNC3 does (Fig. 3). However, within a fixed scale, the opposite is true, implying that UNC3 may be confounded by model capacity while NC3 is a subtle and fine-grained indicator of generalization. 5.3 The Effect of Weight Regularization Our models trained with either weight decay factor exhibited very similar patterns in the emergence of NC (or lack thereof), but the more aggressive factor β = 0.1 resulted in stronger development of NC properties than with β = 0.0005 (Appendices E, F, G, H, I, J, K, M, N, P). These findings empirically affirm β = 0.1 weight decay as sensible for CLM pre-training [81], and concur with [51] on the pivotal role that appropriate regularization plays in the emergence of NC. 6 Limitations Neural collapse While to the best of our knowledge, no previous work has studied realistic stochastic token prediction, it is possible that the quantities that we measure are not perfectly suited for NC in language modelling. As we described in §1.1, the NC framework does not translate neatly to the language modelling space due to many adverse conditions, so full convergence to NC in the TPT was highly improbable. This paper leaves much room for future work to better adapt NC for next-token prediction, which we discuss further in Section 7. Language modelling Our work focused on autoregressive pre-training in its most basic form. We did not conduct experiments into encoder, multi-modal, or instruction-tuned models. Post-training techniques such as supervised fine-tuning, reinforcement learning with human feedback [91] or direct preference optimization [92] are also out-of-scope. This paper uses validation CE loss as the sole indicator of performance, leaving out any downstream task evaluations. Confounder of model scale The models that we used in our permutation test (§5.1, Table 1) are only of a single small architecture trained for one epoch with relatively weak weight regularization (β = 0.0005). Therefore, our experimental results on scale-independent links between NC and generalization may not necessarily translate to larger models. Further investigation on (foundation) LLMs orders of magnitude larger than our CLMs trained with modern NLP methods would provide more robust insight into any direct correlations. 9 7 Discussion Layer/depth-wise neural collapse Past works have established that properties resembling NC evolve as a function of model depth [4, 36, 53, 56, 59, 60, 93–104]. Layer-wise NC — sometimes dubbed deep neural collapse (DNC) [53, 59] — and related phenomena at intermediate layers remain an interesting subtopic. We leave their study and induction in CLMs (like [105]) as future work. Learning to collapse Given the evidence for the development of NC and associated generalization under various loss functions [19, 49, 60, 100, 106–108] in other domains, NLP researchers may still benefit from analyzing, adapting or training towards NC. As alluded to earlier, the simplex ETF and even the CE loss may not be truly optimal for this problem setting, so we anticipate future works to both construct more amenable geometries with better-suited objectives and then capitalize on their benefits downstream. As discussed in §2, there is an abundance of literature in NC, some of which could potentially adapt NC to be useful for NLP; we hope to inspire more such investigations. Interpretability At a high level, the number and density of clusters for a token can reflect its learned meanings and uses. This would be particularly useful as LLMs adapt to ever-evolving language and further expansion into non-English domains. Our formulae (Section 3) and results (Section 4) expose the pairwise token class interactions in noise, interference, classifier duality, and classifier agreement in the top-level features. Similarly to works in other domains [21, 55, 72, 109], these NC metrics can serve as a form of low-level interpretability to aid understanding certain behaviours of (L)LMs. Between tokens, one can often discern how related or interchangeable words are based on their pair-wise interactions, or how antithetical or unrelated they are based on orthogonality. For example, we present a rudimentary analysis of homonyms and English first names in Appendix Q. Fairness Foundation models are ubiquitous for their comprehensive capabilities and adaptability. As previous work discussed class imbalance [35, 58, 61, 62], our work may extend these strategies to measure and perhaps promote fairness in foundation LLMs, some of which are designed to be multilingual or multicultural. For example, [110] contemporarily explores the use of UNC3 to mitigate biases in BERT-based [7] models. While NC itself would not lead to unfairness, its potential interpretability may, in theory, enable an (adversarial) agent to measure and optimize for (un)fairness as they manipulate an LLM. LLM Evaluations Researchers in NLP and multimodal settings are ultimately interested in measuring model performance on practical tasks; notable benchmarks include GLUE [77], MMLU [111], and BIG-bench [112]. However, several contemporaries [113–115] have demonstrated that models’ downstream capabilities are roughly correlated with their abilities to effectively compress their pretraining data. Based on their findings, our application of the NC framework to the pre-training stage of CLMs against validation CE loss should be an appropriate first step in this intersection. Looking forward, we anticipate exciting analysis for language modelling tasks or benchmarks, especially on creativity and retrieval for natural language understanding and generation. Conversely, some form of NC could be an alternative evaluation. Although it would be prohibitively expensive to measure NC on the vast and sometimes obscure pre-training data of most frontier production LLMs, doing so on a small set of in-distribution data (i.e. test set) would be realistic. 8 Conclusion In this paper, we apply the neural collapse (NC) framework to the next-token prediction problem, where models are undertrained and next-tokens are variably drawn from numerous and imbalanced token classes. We leverage canonical and more recent metrics to demonstrate that NC emerges as we scale the size and training of hundreds of causal language models. Our results show a correlation between NC and generalization, a relationship that persists even when the model scale is fixed. In the short term, our work presents rudimentary techniques to analyze and interpret token-level properties of (L)LMs. We anticipate future work to suitably adapt NC (and related frameworks) to the still-fresh frontier of autoregressive language modelling. Researchers could then effectively capitalize on previous learnings from NC to better understand and improve the pre/post-training processes of increasingly complex and large language (multimodal) models. 10 Acknowledgements We thank Elliot Creager, David Glukhov, Daniel Johnson, Jivan Waber, and Colin Raffel for their helpful feedback and stimulating discussions. Aditya Mehrotra and Yu Bo Gao provided technical assistance in our implementations. We acknowledge the support of the Natural Sciences and Engineering Research Council (NSERC) of Canada (www.nserc-crsng.gc.ca/). 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We trained and evaluated models on their first version, as described: • The 2,141,709 stories are split into 2,119,719 train and 21,990 validation stories. • Their experimental setup [2] called for the GPT-2 [68] tokenizer, of which only a subset vocabulary V = J1, 29233K appears in TinyStories. • Following the GPT-style teacher-forcing regime for training/evaluation [8], raw sequences (stories) from the train set are packed (by two preprocessing workers) into 229,367 (S) chunks of 2048 (T) tokens each. This setup provides 469,514,249 (N) ground-truth17 token samples for training. 0 100 200 300 400 500 Top 500 (1.71%) Most Frequent Tokens in TinyStories 105 106 107 # Samples (log scale) Figure 4: The 500 most frequent classes from TinyStories [2] exhibit significant sample imbalance. Despite the synthetic nature of TinyStories, such a distribution is typical of natural language [11, 12]. A.1 Alternative (Real) Datasets The study of NC in causal language modelling at the token level would be unreasonably expensive, so the motivation to use a small dataset is clear. However, most commonly used text datasets such as WikiText [116], BookCorpus [117], CommonCrawl [118], or most subsets from the Pile [119] are much too complex and broad to be effectively compressed by CLMs of the scale that we work with. WikiText-2 and WikiText-103 present significant drawbacks for our experiments. Both datasets contain a considerable amount of low-quality data that does not concentrate on essential linguistic structures such as grammar, vocabulary, facts, and reasoning. WikiText-2 has a similar empirical vocabulary to TinyStories under the GPT-Neo [80] tokenizer (27K vs. 29K) but only has around 34K rows of training data compared to 2.1M in TinyStories. Our small-scale NC experiment on WikiText-2 revealed that the models were very brittle and prone to overfitting. On the other hand, WikiText-103 is comparably sized to TinyStories but utilizes around 44K unique tokens. Our CLMs trained on WikiText-103 struggled to produce coherent sentences, likely due to the excessive breadth and information, as noted by the authors of TinyStories. Beyond these two, we were unable to find any real datasets that both followed established scaling laws [14, 15] for CLMs at our scale and are simple enough to suit the analysis of NC. A.2 On the Use of TinyStories According to its authors, TinyStories [2] is explicitly designed to preserve the essential elements of natural language, such as grammar, vocabulary, facts, and reasoning, while being smaller and more refined in terms of its breadth and diversity. Unlike large corpora that can overwhelm small language models (SLMs) due to their excessive breadth and diversity, TinyStories offers a concentrated dataset that hones in on core linguistic structures and reasoning capabilities. This is evident in its small vocabulary, consisting of approximately 1500 words that a child would use, and in its 29K empirical vocabulary under the GPT-Neo tokenizer. 17N = S(T −1) as we cannot use the first ground-truth nor the last predicted token in any chunk. 19 Despite its concentrated nature, TinyStories enables models trained on it to produce grammatically correct, factual, and reasonable stories. Additionally, these models can be finetuned on specific instructions found in the TinyStories-Instruct dataset. The authors of TinyStories also demonstrate that their models can creatively produce stories dissimilar enough to their training data, indicating a balanced capability for generalization and creativity. One particular advantage of TinyStories is the small vocabulary relative to total training tokens, rendering a reasonable number of classes with higher average token counts. This is relevant because the possibility of NC and a CLM’s ability to compress language data into distinct geometries depend partially on the ratios between embedding dimension, vocabulary size, and average token frequency. Conveniently, frequency analysis of the overall dataset produced a distribution (Figure 4) similar to real human language, so TinyStories should provide a good balance for an initial study of this NC. Additionally, TinyStories has a more regular structure as GPT-3.5/4 was instructed to produce children’s stories with certain themes and forms with a conservative vocabulary. We believe this would reduce the amount of clustering noise from the breadth of information and structures in real general data, and allow our smaller CLMs to exhibit some clear trends toward NC. Furthermore, TinyStories was created using GPT 3.5/4, advanced language models with significantly larger architectures trained on orders of magnitude more tokens; this should help minimize the effect of the synthetic nature of the generated dataset. We also considered a possible effect of model collapse as a result of training on synthetic data [120] and follow-up work [121] suggest that a single iteration of data generation (as generated TinyStories) has a very negligible model collapse. B Model Architectural Details Table 2: Sample architectural configuration for a 12-layer 1024-dimensional causal language model (CLM) based on [2] and GPT-Neo [80]. Shallower models have configurations with attention_layers and attention_types truncated. Narrower models adjust hidden_size to their width (d). All other configuration values are the same across models. SETTING VALUE activation_function gelu_new architectures GPTNeoForCausalLM attention_dropout 0 attention_layers global, local, global, local, ... attention_types [[global, local], 6] bos_token_id 50256 embed_dropout 0 eos_token_id 50256 gradient_checkpointing false hidden_size 1024 initializer_range 0.02 intermediate_size null layer_norm_epsilon 1e-05 max_position_embeddings 2048 model_type gpt_neo num_heads 16 num_layers 12 resid_dropout 0 summary_activation null summary_first_dropout 0.1 summary_proj_to_labels true summary_type cls_index summary_use_proj true torch_dtype float32 transformers_version 4.28.1 use_cache true vocab_size 50257 window_size 256 20 C Optimization The training was performed using an adaptation of an open-source causal language modelling script from Huggingface: https://github.com/huggingface/transformers/blob/main/ examples/pytorch/language-modeling/run_clm.py • Each model was trained on a single NVIDIA A100 (40GB) GPU for up to 8 hours per epoch. • Learning rates were set by a linear schedule based on the number of steps with no warm-up. • Training was performed in bfloat16 [122] mixed precision. • The results presented in this work are from two sets of models trained with weight decay β = 0.0005 [51] and β = 0.1 [81]. A previous set of models was trained without weight decay and the results are very similar to β = 0.0005. Table 3: Batch sizes used to train models on a single NVIDIA A100 (40GB) GPU. Width (d) and depth (L) correspond to hidden_size and length of attention_layers, respectively, in Table 2. DEPTH (L) ↓ WIDTH (d) → 64 128 256 512 768 1024 1-layer 16 16 16 16 16 16 2-layer 16 16 16 16 16 8 4-layer 8 8 8 8 8 8 8-layer 8 8 8 4 4 4 12-layer 4 4 4 4 4 4 2.5 5.0 7.5 10.0 Training Epochs 8 6 4 2 Log Normalized Variance ( 1) 2.5 5.0 7.5 10.0 Training Epochs 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Validation Loss 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 5: Average (logarithmic) class-distance normalized variance (CDNV, NC1) (left) and validation (cross-entropy) loss (right) with respect to training epochs. D Embeddings Collection & NC Analysis Codes for (post-)training analysis are hosted on GitHub: • Main code https://github.com/rhubarbwu/linguistic-collapse • Auxillary package: https://github.com/rhubarbwu/neural-collapse One pass over the train set for embeddings collection can take up to 6 hours on a single NVIDIA A100 (40GB) GPU. Analysis of a single metric for a given model takes less than 5 minutes. 21 E Within-Class Variability Collapse with Scale — NC1 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 6 4 2 Log Normalized CDNV 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 8 6 4 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 8 7 6 5 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 6 4 2 Log Normalized CDNV 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 8 7 6 5 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 8 7 6 L=1 L=2 L=4 L=8 L=12 Figure 6: Average (logarithmic) class-distance normalized variance (CDNV) is reduced (NC1) when scaling width (d) and across training for 1 (left) through 10 (right) epochs with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 6 4 2 Log Normalized CDNV 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 8 6 4 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 8 7 6 5 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 6 4 2 Log Normalized CDNV 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 8 7 6 5 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 8 7 6 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 7: Average (logarithmic) class-distance normalized variance (CDNV) is reduced (NC1) when scaling depth (L) and across training for 1 (left) through 10 (right) epochs with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 22 F Mean Norms Growth with Scale — (Related to NC2) 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 2.0 2.5 3.0 Log Mean Norms (Avg) 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 2.5 3.0 3.5 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 2.75 3.00 3.25 3.50 3.75 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 2.0 2.5 3.0 Log Mean Norms (Avg) 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 3.0 3.5 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 3.00 3.25 3.50 3.75 4.00 L=1 L=2 L=4 L=8 L=12 Figure 8: Logarithmic class mean norms grow when scaling width (d) and across training for 1 (left) through 10 (right) epochs with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 2.0 2.5 3.0 Log Mean Norms (Avg) 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 2.5 3.0 3.5 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 2.75 3.00 3.25 3.50 3.75 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 2.0 2.5 3.0 Log Mean Norms (Avg) 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 3.0 3.5 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 3.00 3.25 3.50 3.75 4.00 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 9: Logarithmic class mean norms grow when scaling depth (L) and across training for 1 (left) through 10 (right) epochs with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 23 G Equinormness with Scale — (G)NC2 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.8 1.0 1.2 Log Mean Norms (CoV) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 0.6 0.8 1.0 1.2 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 0.6 0.8 1.0 1e 1 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.8 1.0 1.2 Log Mean Norms (CoV) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 0.6 0.8 1.0 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 6 8 1e 2 L=1 L=2 L=4 L=8 L=12 Figure 10: Variation in (logarithmic) norms decreases (NC2) when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). Note that the degree of equinormness eventually plateaus for sufficiently deep and trained models. 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.8 1.0 1.2 Log Mean Norms (CoV) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 0.6 0.8 1.0 1.2 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 0.6 0.8 1.0 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.8 1.0 1.2 Log Mean Norms (CoV) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 0.6 0.8 1.0 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 6 8 1e 2 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 11: Variation in (logarithmic) norms decreases (NC2) when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 24 H Interference with Scale — (Related to NC2) 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 3 2 1 Interference (Avg) 1e 2 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 4 2 1e 2 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 6 4 2 1e 2 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 3 2 1 Interference (Avg) 1e 2 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 1.00 0.75 0.50 0.25 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 1.0 0.5 1e 1 L=1 L=2 L=4 L=8 L=12 Figure 12: Average interference decreases (to some extent) when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 3 2 1 Interference (Avg) 1e 2 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 4 2 1e 2 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 6 4 2 1e 2 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 3 2 1 Interference (Avg) 1e 2 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 1.00 0.75 0.50 0.25 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 1.0 0.5 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 13: Average interference decreases (to some extent) when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 25 I Equiangularity with Scale — (Against NC2) 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.6 0.8 1.0 Interference (CoV) 1e4 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 4 6 8 1e3 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 4 5 6 7 1e3 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.6 0.8 1.0 Interference (CoV) 1e4 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 4 5 6 7 1e3 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 4 6 1e3 L=1 L=2 L=4 L=8 L=12 Figure 14: Variation in interference roughly increases when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). Note this trend is against equiangularity, affirming the traditional NC2 to be less useful than GNC2 [19]. 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.6 0.8 1.0 Interference (CoV) 1e4 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 4 6 8 1e3 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 4 5 6 7 1e3 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.6 0.8 1.0 Interference (CoV) 1e4 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 4 5 6 7 1e3 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 4 6 1e3 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 15: Variation in interference increases when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). Note this trend is against equiangularity, affirming the traditional NC2 to be less useful than GNC2 [19]. 26 J Logarithmic Distances with Scale — (Related to GNC2) 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 3.2 3.0 Log Distances (Avg) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 3.4 3.2 3.0 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 3.4 3.3 3.2 1e 1 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 3.4 3.2 3.0 Log Distances (Avg) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 3.4 3.3 3.2 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 3.4 3.3 3.2 1e 1 L=1 L=2 L=4 L=8 L=12 Figure 16: Average logarithmic distance decreases when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 3.2 3.0 Log Distances (Avg) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 3.4 3.2 3.0 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 3.4 3.3 3.2 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 3.4 3.2 3.0 Log Distances (Avg) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 3.4 3.3 3.2 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 3.4 3.3 3.2 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 17: Average logarithmic distance decreases when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 27 K Hyperspherical Uniformity with Scale — GNC2 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.50 0.75 1.00 Log Distances (CoV) 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 4 6 8 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 3 4 5 1e 1 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 0.50 0.75 1.00 Log Distances (CoV) 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 3 4 5 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 2 3 4 5 1e 1 L=1 L=2 L=4 L=8 L=12 Figure 18: Variation in logarithmic distances decreases when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). This consistent trend towards hyperspherical uniformity affirms that GNC2 [19] is more useful than NC2. 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.50 0.75 1.00 Log Distances (CoV) 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 4 6 8 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 3 4 5 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 0.50 0.75 1.00 Log Distances (CoV) 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 3 4 5 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 2 3 4 5 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 19: Variation in logarithmic distances decreases when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). This consistent trend towards hyperspherical uniformity affirms that GNC2 [19] is more useful than NC2. 28 L Correlations of (G)NC2 with Generalization Performance 2.0 2.5 3.0 3.5 4.0 Log Mean Norms (Avg) ( 2) 1 2 3 4 Val. Loss R2 = 0.86 0.06 0.08 0.10 0.12 Log Mean Norms (CoV) ( 2) R2 = 0.86 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 20: Generalization (validation loss) shows some correlation with logarithmic mean norms in both their average (left) and variations (i.e. equinormness, NC2) (right). 0.10 0.05 0.00 Interference (Avg) ( 2) 1 2 3 4 Val. Loss R2 = 0.56 4000 6000 8000 10000 Interference (CoV) ( 2) R2 = 0.79 1.38 1.39 1.40 Simplex ETF Error ( 2) R2 = 0.74 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 21: Generalization (validation loss) correlated with average interference (left) and its variation (i.e. equiangularity, NC2) (centre). We also computed the empirical measure from [5] (right). 0.34 0.32 0.30 Log Distances (Avg) ( 2) 1 2 3 4 Val. Loss R2 = 0.82 0.25 0.50 0.75 1.00 Log Distances (CoV) ( 2) R2 = 0.84 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 22: Validation loss shows some correlation with average (logarithmic) kernel distances and with their variation (i.e. hyperspherical uniformity, GNC2) (right). 29 M Self-Duality with Scale — (Against NC3) Self-duality (NC3) was originally the convergence of classifiers to means up to rescaling [1]: wc ∥wc∥2 −ˆµc 2 →0, ∀c (11) Instead, we use class-wise cosine similarity (Equation 8) and its variation (UNC3). 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1.5 2.0 2.5 Similarity (Cosine) (Avg) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 2.0 2.5 3.0 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 2.5 3.0 3.5 1e 1 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1.5 2.0 2.5 Similarity (Cosine) (Avg) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 3.0 3.5 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 3.0 3.5 4.0 1e 1 L=1 L=2 L=4 L=8 L=12 Figure 23: Average classifier alignment increases NC3 when training for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). However, we see no meaningful trend when scaling width d, suggesting that NC3 does not coalesce with language modelling training. 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1.5 2.0 2.5 Similarity (Cosine) (Avg) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 2.0 2.5 3.0 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 2.5 3.0 3.5 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1.5 2.0 2.5 Similarity (Cosine) (Avg) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 3.0 3.5 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 3.0 3.5 4.0 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 24: Average classifier alignment increases NC3 when training for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). However, we see no meaningful trend when scaling depth L, suggesting that NC3 does not coalesce with language modelling training. 30 N Uniformity Duality with Scale — UNC3 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1.0 1.5 2.0 Similarity (Cosine) (CoV) 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 0.8 1.0 1.2 1.4 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 0.6 0.8 1.0 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1.0 1.5 2.0 Similarity (Cosine) (CoV) 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 0.6 0.8 1.0 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 0.6 0.8 1.0 L=1 L=2 L=4 L=8 L=12 Figure 25: Variation in classifier alignment decreases when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1.0 1.5 2.0 Similarity (Cosine) (CoV) 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 0.8 1.0 1.2 1.4 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 0.6 0.8 1.0 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1.0 1.5 2.0 Similarity (Cosine) (CoV) 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 0.6 0.8 1.0 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 0.6 0.8 1.0 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 26: Variation in classifier alignment increases when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). This negative trend of UNC3 in more learnt models (right) suggests that the link of (U)NC3 with scale and performance is still weak. 31 O Correlations of (U)NC3 with Generalization Performance 2.5 5.0 7.5 10.0 Similarity (Dot Prod.) (Avg) ( 3) 1 2 3 4 Val. Loss R2 = 0.69 0.2 0.3 0.4 Similarity (Cosine) (Avg) ( 3) R2 = 0.31 1.395 1.400 1.405 1.410 Self-Duality Error ( 3) R2 = 0.67 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 27: Generalization (validation loss) correlated with average dot-product similarity (for interpretability purposes only) (left) and cosine similarity (classifier alignment, NC3) (centre). We also computed the empirical measure from [5] (right). 0.5 1.0 1.5 2.0 Similarity (Cosine) (CoV) ( 3) 1 2 3 4 Val. Loss R2 = 0.69 0.5 1.0 1.5 2.0 2.5 Similarity (Dot Prod.) (CoV) ( 3) R2 = 0.75 15.053 15.746 16.020 16.796 17.489 17.984 18.389 19.137 Log # Parameters Figure 28: Generalization (validation loss) correlated with variation in dot-product similarity (for interpretability purposes only) (left) and cosine similarity (uniform duality, UNC3) (right). 32 P Classifier Agreement — NC4 For computational reasons, we compute Equations 9, 10 using a simple decomposition: argmin c∈V ∥hb −µc∥2 = argmin c∈V ∥hb∥2 + ∥µc∥2 −2h⊤ b µc , (12) where b ∈[1, B] and c ∈V with batch size B and vocabulary V. 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1 2 Classifier Agreement (%) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 3 Epochs 1 2 3 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 1 2 3 1e 1 L=1 L=2 L=4 L=8 L=12 64 256 512 768 1024 Width (Hidden Dim.) 1 Epochs 1 2 3 Classifier Agreement (%) 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 6 Epochs 1 2 3 1e 1 64 256 512 768 1024 Width (Hidden Dim.) 10 Epochs 1 2 3 1e 1 L=1 L=2 L=4 L=8 L=12 Figure 29: Classifier agreement improves when scaling width (d) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1 2 Classifier Agreement (%) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 3 Epochs 1 2 3 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 1 2 3 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 1 2 4 8 12 Depth (Num. of Layers) 1 Epochs 1 2 3 Classifier Agreement (%) 1e 1 1 2 4 8 12 Depth (Num. of Layers) 6 Epochs 1 2 3 1e 1 1 2 4 8 12 Depth (Num. of Layers) 10 Epochs 1 2 3 1e 1 d=64 d=128 d=256 d=512 d=768 d=1024 Figure 30: Classifier agreement improves when scaling depth (L) in models trained for 1 (left) through 10 (right) with weight decays β = 0.0005 (top) and β = 0.1 (bottom). 33 Q Examples for Interpretability This section presents token-wise interpretability results from top-layer embeddings from our most learned models (trained for 10 epochs). Our largest one is publicly available: https: //huggingface.co/rhubarbwu/TinyStories-12x1024_10L. Interference ( 2) Log Kernel Distances ( 2) Pairwise Geometries (02x1024) Interference ( 2) Log Kernel Distances ( 2) Pairwise Geometries (04x1024) Interference ( 2) Log Kernel Distances ( 2) Pairwise Geometries (08x1024) Interference ( 2) Log Kernel Distances ( 2) Pairwise Geometries (12x1024) Figure 31: Pairwise interactions between token mean embedding vectors across models of fixed width (d = 1024) and increasing depths (L = 2, 4, 8, 12). Interference (NC2) decreases on average but only slightly becomes more uniform (top-right, blue). In contrast, logarithmic kernel distances (GNC2) decrease and become more evenly spread, with some outlier pairs (bottom-left, green). wave well fair close ring light spring left bat bowl bank lead row [AVERAGE] date match Vocabulary (GPT2 Tokenizer) 0 20 40 60 80 100 120 Class Mean Embedding Norms Class Mean Norms of Homonyms (TinyStories 12x1024) Figure 32: Under TinyStories-12x1024_10L, these fifteen homonyms have much shorter mean embedding norms (i.e. closer to the global centre) than the average token. This is expected since homonyms typically present conflicts and interference. 34 Elizabeth Anthony Charles Sarah Matthew Robert Kyle Donald Ryan Jessica Brian Kelly Laura Jennifer Mary Michael Jacob Alice David Andrew William Kevin Thomas Steve Paul Martin Eric Luke James George John Jim Henry Harry Daniel Robin Tim Anna [AVERAGE] Vocabulary (GPT2 Tokenizer) 0.000000 0.000025 0.000050 0.000075 0.000100 0.000125 0.000150 0.000175 Average Token Pairwise CDNV Within-Class Variability of English Given Names (TinyStories 12x1024) Donald John Martin Daniel Jim Robin Paul Harry Henry George Jennifer Michael James Ryan Eric Steve Elizabeth Robert Alice Thomas Charles Sarah Mary David Anthony Andrew Kelly Brian Laura William Luke Kevin Jessica Matthew Jacob Kyle Anna Tim [AVERAGE] Vocabulary (GPT2 Tokenizer) 0.008 0.006 0.004 0.002 0.000 Average Token Pairwise Interference Interference of English Given Names (TinyStories 12x1024) Figure 33: Under TinyStories-12x1024_10L, the average within-class variability (top) and interference (bottom) of some English first names were far below those of the average token. This might be because names are distinct and are not typically used in the same contexts as other words (aside from articles). The only names to have CDNV close to that of the average token are “Anna” and “Tim”. Note that the positive interference of the average token (right) is not a typo. 35 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The claims of this paper listed in the abstract and introduction are simply the emergence of the NC phenomena and its relationship with generalization in CLMs, which are the scope and results of the paper. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We discuss the limitations of the experiments themselves (e.g. small models, limited sample size) as well as the possibility that our formulations could be improved in the future (there could be more suitable metrics). See Sections 6, 7. 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4,429	Neural Pfafﬁans: Solving Many Many-Electron Schrödinger Equations Nicholas Gao, Stephan Günnemann {n.gao,s.guennemann}@tum.de Department of Computer Science & Munich Data Science Institute Technical University of Munich Abstract Neural wave functions accomplished unprecedented accuracies in approximating the ground state of many-electron systems, though at a high computational cost. Recent works proposed amortizing the cost by learning generalized wave functions across different structures and compounds instead of solving each problem independently. Enforcing the permutation antisymmetry of electrons in such generalized neural wave functions remained challenging as existing methods require discrete orbital selection via non-learnable hand-crafted algorithms. This work tackles the problem by deﬁning overparametrized, fully learnable neural wave functions suitable for generalization across molecules. We achieve this by relying on Pfafﬁans rather than Slater determinants. The Pfafﬁan allows us to enforce the antisymmetry on arbitrary electronic systems without any constraint on electronic spin conﬁgurations or molecular structure. Our empirical evaluation ﬁnds that a single neural Pfafﬁan calculates the ground state and ionization energies with chemical accuracy across various systems. On the TinyMol dataset, we outperform the ‘gold-standard’ CCSD(T) CBS reference energies by 1.9 mEh and reduce energy errors compared to previous generalized neural wave functions by up to an order of magnitude. 1 Introduction Solving the electronic Schrödinger equation is at the heart of computational chemistry and drug discovery. Its solution provides a molecule’s or material’s electronic structure and energy (Zhang et al., 2023). While the exact solution is infeasible, neural networks have recently shown unprecedentedly accurate approximations (Hermann et al., 2023). These neural networks approximate the system’s ground-state wave function Ψ : RNe×3 →R, the lowest energy state, by minimizing the energy ⟨Ψ\| ˆH \|Ψ⟩, where ˆH is the Hamiltonian operator, a mathematical description of the system. While such neural wave functions are highly accurate, training has proven computationally intensive. Gao & Günnemann (2022) have shown that training a generalized neural wave function on a large class of systems amortizes the cost. However, their approach is limited to different geometric arrangements of the same molecule. Subsequent works eliminated this limitation by introducing hand-crafted algorithms (Gao & Günnemann, 2023a) or heavily relying on classical Hartree-Fock calculations (Scherbela et al., 2023). Both impose strict, non-learnable mathematical constraints and prior assumptions that may not always hold, limiting their generalization and accuracies. Handcrafted algorithms only work for a limited set of molecules, in particular organic molecules near equilibrium, while the reliance on Hartree-Fock empirically results in degraded accuracies. In this work, we propose the Neural Pfafﬁan (NeurPf) to overcome these limitations. As suggested by its name, NeurPf uses Pfafﬁans to deﬁne a superset of the previously used Slater determinants to enforce the fermionic antisymmetry. The Pfafﬁan lifts the constraint on the number of molecular orbitals from Slater determinants (Szabo & Ostlund, 2012), enabling overparametrized wave functions 38th Conference on Neural Information Processing Systems (NeurIPS 2024). with simpler and more accurate generalization. Compared to Globe (Gao & Günnemann, 2023a), the absence of hand-crafted algorithms enables the modeling of non-equilibrium, ionized, or excited systems. By being fully learnable without ﬁxed Hartree-Fock calculations like TAO (Scherbela et al., 2024), NeurPf achieves signiﬁcantly lower variational energies. Our empirical results show that NeurPf can learn all second-row elements’ ground-state, ionization, and electron afﬁnity potentials with a single wave function. Further, we demonstrate that NeurPf’s accuracy surpasses Globe on the challenging nitrogen dimer with seven times fewer parameters while not suffering from performance degradations when adding structures to the training set. On the TinyMol dataset, NeurPf surpasses the highly accurate reference CCSD(T) CBS energies on the small structures by 1.9 mEh and reduces errors compared to TAO by factors of 10 and 6 on the small and large structures, respectively. 2 Quantum chemistry Quantum chemistry aims to solve the time-independent Schrödinger equation (Foulkes et al., 2001) ˆH \|Ψ⟩= E \|Ψ⟩ (1) where Ψ : RN↑×3 × RN↓×3 →R is the electronic wave function for N↑spin-up and N↓spin-down electrons, ˆH is the Hamiltonian operator, and E is the system’s energy. To ease notation, if not necessary, we omit spins in Ψ and treat it as Ψ : RNe×3 →R where Ne = N↑+N↓. The Hamiltonian ˆH for molecular systems, which we are concerned with in this work, is given by ˆH = −1 2 Ne X i=1 3 X k=1 ∂2 ∂⃗r2 ik + Ne X j>i 1 ∥⃗ri −⃗rj∥− Ne X i=1 Nn X m=1 Zm ∥⃗ri −⃗Rm∥ + Nn X n>m ZmZn ∥⃗Rm −⃗Rn∥ (2) with ⃗ri ∈R3 being the ith electron’s position, and ⃗Rm ∈R3, Zm ∈N+ being the mth nucleus’ position and charge. The wave function Ψ describes the behavior of electrons in the system deﬁned by the Hamiltonian ˆH. As the square of the wave function Ψ2 is proportional to the probability density p(⃗r) ∝Ψ2(⃗r) of ﬁnding the electrons at positions ⃗r ∈RNe×3, its integral must be ﬁnite: Z Ψ(⃗r)2d⃗r < ∞. (3) Further, as electrons are indistinguishable half-spin fermionic particles, the wave function must be antisymmetric under any same-spin electron permutation τ: Ψ τ ↑(⃗r↑), τ ↓(⃗r↓) = sgn(τ ↑)sgn(τ ↓)Ψ(⃗r). (4) To enforce this constraint, the wave function is typically deﬁned as a so-called Slater determinant of N↑+ N↓integrable so-called orbital functions φi : R3 →R: ΨSlater(⃗r) = det h φ↑ j(⃗r↑ i ) i det h φ↑ j(⃗r↓ i ) i = det Φ↑(⃗r↑) det Φ↓(⃗r↓). (5) Note that for the determinant to exist, one needs exactly N↑up and N↓down orbitals φ↑ j and φ↓ j. In linear algebra, Eq. (1) is an eigenvalue problem, where we look for the eigenfunction Ψ0 with the lowest eigenvalue E0. In Variational Monte Carlo (VMC), this is solved by applying the variational principle, which states that the energy of any trial wave function Ψ upper bounds E0: E0 ≤⟨Ψ\| ˆH \|Ψ⟩ ⟨Ψ2⟩ = R Ψ(⃗r) ˆHΨ(⃗r)d⃗r R Ψ2(⃗r)d⃗r . (6) By plugging in the probability distribution from Eq. (3), we can rewrite Eq. (6) as E0 ≤Ep(⃗r) h Ψ−1(⃗r) ˆHΨ(⃗r) i = Ep(⃗r) [EL(⃗r)] , (7) with EL(⃗r) = Ψ(⃗r)−1 ˆHΨ(⃗r) being the so-called local energy. The right-hand side of Eq. (7) is known as the variational energy. As Eq. (7) does not require Ψ to be an analytic function, we can approximate the energy of any valid wave function Ψ with samples drawn from p(⃗r). If we pick a 2 parametrized family of wave functions Ψθ, we can optimize the parameters θ to minimize the VMC energy by following the gradient of the variational energy ∇θ = Ep(⃗r) (EL(⃗r) −Ep(⃗r) [EL(⃗r)])∇θ log Ψθ(⃗r) , (8) where we approximate all expectations by Monte Carlo sampling (Ceperley et al., 1977). Neural wave functions typically keep the functional form of Eq. (5) but replace the orbitals φi with learned many-electron orbitals φNN i : R3 ×RNe×3 →R (Hermann et al., 2023). These many-electron orbitals φNN i are implemented as different readouts of the same permutation-equivariant neural network. Multiplying each orbital by an envelope function χi : R3 →R that decays exponentially to zero at large distances enforces the ﬁnite integral requirement in Eq. (3). Generalized wave functions solve the more general problem where the nucleus positions ⃗R and charges Z are not ﬁxed. Since the Hamiltonian ˆH⃗R,Z depends on the molecular structure (⃗R, Z), so does the corresponding ground state wave function Ψ⃗R,Z. Note that we still work in the Born-Oppenheimer approximation, i.e., we treat the nuclei as classical point charges (Zhang et al., 2023). Given a dataset of molecular structures D = {(⃗R1, Z1), ...}, the total energy P (⃗R,Z)∈D ⟨Ψ⃗ R,Z\| ˆ H⃗ R,Z\|Ψ⃗ R,Z⟩ ⟨Ψ2 ⃗ R,Z⟩ is minimized to approximate the ground state for each structure. Typically, the dependence on ⃗R, Z is implemented by using a meta network that takes ⃗R, Z as inputs and outputs the parameters of the electronic wave function (Gao & Günnemann, 2022). 3 Related work While attempts to enforce the fermionic antisymmetry in neural wave functions in less than O(Ne 3) operations promise faster runtime than Slater determinants, the accuracy of these methods is limited (Han et al., 2019; Acevedo et al., 2020; Richter-Powell et al., 2023). Pfau et al. (2020) and Hermann et al. (2020) established Slater determinants for neural wave functions by demonstrating chemical accuracy on small molecules. Note, Eq. (5) may also be written via a block-diagonal matrix, i.e., Ψ(⃗r) = det diag(Φ↑, Φ↓) . Spencer et al. (2020)’s implementation further increased accuracies by parametrizing the off diagonals that were implicitly set to 0 before, with additional orbitals ˜Φ: ΨSlater(⃗r) = det(ˆΦ(⃗r)) = det Φ↑(⃗r↑) ˜Φ↑(⃗r↑) ˜Φ↓(⃗r↓) Φ↓(⃗r↓) . (9) Several works conﬁrmed the improved empirical accuracy of this approach (Gerard et al., 2022; Lin et al., 2021; Ren et al., 2023; Gao & Günnemann, 2023b, 2024). While later works reﬁned the architecture to increase accuracy (von Glehn et al., 2023; Wilson et al., 2021, 2023), the use of Slater determinants mostly remained a constant, with two notable exceptions: Firstly, Lou et al. (2023) use AGP wave functions (Casula & Sorella, 2003; Casula et al., 2004) to formulate the wave function as Ψ(⃗r) = det(Φ↑) det(Φ↓) = det(Φ↑Φ↓T ). This avoids picking exactly N↑/N↓ orbitals as Φ↑and Φ↓may be non-square but fails to generalize Eq. (9), we empirically verify the impact of this limitation in App. I. Secondly, Kim et al. (2023) introduced the combination of neural networks and Pfafﬁans, who demonstrated its performance on the ultra-cold Fermi gas. Though universal in theory, their parametrization yields no trivial adaption to molecular systems. In classical quantum chemistry, Bajdich et al. (2006, 2008) reported promising early results with Pfafﬁans in single-structure calculations for small molecules. In this work, we generalize Eq. (9) to Pfafﬁan wave functions that permit pretraining with Hartree-Fock calculations and generalization across molecules. Generalized wave functions. Scherbela et al. (2022) started this research with a weight-sharing scheme between wave functions. These still had to be reoptimized for each structure. Later, Gao & Günnemann (2022, 2023b) proposed PESNet, a generalized wave function for energy surfaces allowing joint training without reoptimization. Subsequent works extended PESNet to different compounds where the main challenge is parametrizing exactly N↑+ N↓orbitals, such that the orbital matrix in Eq. (9) stays square. The problem of ﬁnding these orbitals was formulated into a discrete orbital selection problem. Gao & Günnemann (2023a)’s hand-crafted algorithm accomplishes this by selecting orbitals via a greedy nearest neighbor search. In contrast, Scherbela et al. (2024, 2023) use the lowest eigenvalues of the Fock matrix as selection criteria. Both introduce non-learnable constraints, limiting generalization or sacriﬁcing accuracy. NeurPf avoids the selection problem by introducing an overparametrization when enforcing the exchange antisymmetry. 3 4 Neural Pfafﬁan Previous generalized wave functions build on Slater wave functions and attempt to adjust the orbitals φi to the molecule. Slater determinants were chosen due to their previously demonstrated high accuracy. However, they require exactly N↑+ N↓orbitals. While the nuclei allow inferring the total number of electrons Ne of any stable, singlet state system, the spin distribution into N↑and N↓ orbitals per atom is not readily available. Previous works implement this via a discrete selection of orbitals via non-learnable prior assumptions and constraints on the wave function; see Sec. 3. Here, we present the Neural Pfafﬁan (NeurPf), a superset of Slater wave functions that preserves accuracy while relaxing the orbital number constraint. By not enforcing an exact number of orbitals, NeurPf is overparametrized with No ≥max{N↑, N↓} orbitals, avoiding discrete selections and making it a natural choice for generalized wave functions. Importantly, NeurPf can be pretrained with Hartree-Fock, which accounts for > 99% of the total energy (Szabo & Ostlund, 2012). We introduce NeurPf in four steps: (1) We introduce the Pfafﬁan and use it to deﬁne a superset of Slater wave functions. (2) We present memory-efﬁcient envelopes that additionally accelerate convergence. (3) We introduce a new pretraining scheme for matching Pfafﬁan and Slater wave functions. (4) We discuss combining our developments to build a generalized wave function. 4.1 Pfafﬁan wave function The Pfafﬁan of a skew-symmetric 2n × 2n matrix A, i.e., A = −AT , is deﬁned as Pf(A) = 1 2nn! X τ∈S2n sgn(τ) n Y i=1 Aτ(2i−1),τ(2i) (10) where S2n is the symmetric group of 2n elements. One may consider it a square root of the determinant of A since Pf(A)2 = det(A). An important property of the Pfafﬁan is Pf(BABT ) = det(B)Pf(A) for any invertible matrix B and skew-symmetric matrix A. In the context of neural wave functions, this means that if A is an along both dimensions permutation equivariant function of the electron positions ⃗r, A(τ(⃗r)) = PτA(⃗r)P T τ , the Pfafﬁan of A is a valid wave function that fulﬁlls the antisymmetry requirement from Eq. (4): Ψ(τ(⃗r)) = Pf(A(τ(⃗r))) = Pf(PτA(⃗r)P T τ ) = det(Pτ)Pf(A(⃗r)) = sign(τ)Ψ(⃗r). (11) To compute the Pfafﬁan without evaluating the 2n! terms in Eq. (10), we implement the Pfafﬁan via a tridiagonalization with the Householder transformation as in Wimmer (2012). There are various ways to construct A (Bajdich et al., 2006, 2008; Kim et al., 2023). Here, we introduce a superset of Slater wave functions, enabling high accuracy on molecular systems. If A is a skew-symmetric matrix, so is BABT for any arbitrary matrix B. Thus, we can construct ΨPfafﬁan as ΨPfafﬁan(⃗r) = 1 Pf(APf)Pf ˆΦPf(⃗r)APf ˆΦPf(⃗r)T (12) where APf ∈RNo×No is a learnable skew-symmetric matrix and ˆΦPf : RNe×3 →RNe×No is a permutation equivariant function like in Eq. (9). This construction elevates the need for having exactly N↑/N↓orbitals as in Slater determinants. We may now overparametrize the wave function with No ≥max{N↑, N↓} orbitals, allowing for a more ﬂexible and simpler implementation without needing discrete orbital selection. By choosing ˆΦPf = ˆΦ, it is straightforward to see that Eq. (12) is a superset of the Slater determinant wave function in Eq. (9). Note that, like in Eq. (9), we parametrize two sets of orbital functions ΦPf and ˜ΦPf and change their order for spin-down electrons to not enforce the exchange antisymmetry between different-spin electrons. As the normalizer Pf(APf) is constant, we drop it going forward. As it is common in quantum chemistry (Szabo & Ostlund, 2012; Hermann et al., 2020), we use linear combinations of wave functions to increase expressiveness: ΨPfafﬁan(⃗r) = Nk X k=1 ckΨPfafﬁan,k(⃗r). (13) We visually compare the schematic of the Slater determinant and Pfafﬁan wave functions in Fig. 1. In App. A, we discuss how to handle odd numbers of electrons such that ˆΦPfAPf ˆΦT Pf has even 4 Concatenate NN Diag Projection Offdiag Projection 5D Tensor Envelope Legend Input Orbitals learnable fixed Electrons (a) Slater wave function Electrons Envelope NN Diag Projection Offdiag Projection 4D Tensor Concatenate Antisymmetrizer Matrix Contraction (b) Neural Pfafﬁan Fig. 1: Schematic of the Slater determinant (1a) and our NeurPf (1b). Where the Slater formulation requires exactly Ne orbital functions, the Pfafﬁan formulation works for any number No ≥max{N↑, N↓} of orbital functions, indicated by the rectangular orbital blocks. dimensions. Like previous work (Pfau et al., 2020), we parametrize the orbital functions φi as a product of a permutation equivariant neural network h : R3 × RN×3 →RNf and an envelope function χ : R3 →R: φki(⃗rj\|⃗r) = χki(⃗rj) · h(⃗rj\|⃗r)T wki · ηN↑−N↓ ki (14) with wki ∈RNf being a learnable weight vector, and ηN↑−N↓ ki ∈R being a scalar depending on the spin state of the system, i.e., the difference between the number of up and down electrons. The envelope function χ ensures that the integral of the squared wave function is ﬁnite. For h, we use Moon from Gao & Günnemann (2023a) thanks to its size consistency. 4.2 Memory-efﬁcient envelopes To satisfy the ﬁnite integral requirement on the square of Ψ in Eq. (3), the orbitals φ are multiplied by an envelope function χ : R3 →R that exponentially decays to zero at large distances. We do not split spins here and work with Ne = N↑+ N↓to simplify the discussion, but, in practice, we would split the envelopes into two sets, one for ΦPf and one for ˜ΦPf. The envelope function is typically a sum of exponentials centered on the nuclei (Spencer et al., 2020). In Einstein’s summation notation, the envelope function can be written as χki(⃗rbj) = πkmi \| {z } Nk×Nn×No · exp(−σkmi∥⃗rbj −⃗Rm∥)bkmji \| {z } Nb×Nk×Nn×Ne×No (15) where Nb denotes the batch size. Empirically, we found the tensor on the right side containing many redundant entries. Further, due to the nonlinearity of the exponential function, one cannot implement the envelope in a simple matrix contraction but has to materialize the full ﬁve-dimensional tensor. NeurPf ampliﬁes this problem as No ≥Ne whereas Slater determinants constraint No = Ne. We use a single set of exponentials per nucleus instead of having one for each combination of orbital and nucleus. This reduces the number of envelopes per electron from Nk × Nn × No to Nk × Nenv, where Nenv = Nn × Nenv/nuc is the number of envelope functions. In general, we pick Nenv/nuc 5 such that Nenv ≈No. These atomic envelopes are linearly recombined into molecular envelopes, effectively enlarging π to a Nk × No × Nenv tensor. Thanks to these rearrangements, we avoid constructing a ﬁve-dimensional tensor. Instead, we deﬁne the envelopes as χki(⃗rbj) = πkni \|{z} Nk×Nenv×No · exp(−σkn∥⃗rbj −⃗Rn∥)kbnj \| {z } Nb×Nk×Nenv×Ne . (16) Concurrently, Pfau et al. (2024) presented similar bottleneck envelopes. However, we found ours to converge faster and not yield numerical instabilities. We discuss this further in App. B and I. 4.3 Pretraining Pfafﬁan wave functions Pretraining is essential in training neural wave functions and has frequently been observed to critically affect ﬁnal energies (Gao & Günnemann, 2023a; von Glehn et al., 2023; Gerard et al., 2022). The pretraining aims to ﬁnd orbital functions close to the ground state to stabilize the optimization. Traditionally, this is done by matching the orbitals of the neural wave function to the orbitals of a baseline wave function, typically a Hartree-Fock wave function ΨHF = det(ΦHF), by solving min θ ∥Φθ −ΦHF∥2 2, (17) for the neural network parameters θ (Pfau et al., 2020). Since our Pfafﬁan has No orbitals while Hartree-Fock has Ne, we cannot directly apply this to our Pfafﬁan wave function. Further, as we predict orbitals per nucleus, our arbitrary orbital order may not align with Hartree-Fock. We propose two alternative pretraining schemes for neural Pfafﬁan wave functions: one based on matching single-electron orbitals and one based on matching geminals, effectively two-electron orbitals. We need to expand the Hartree-Fock orbitals ΦHF to No orbitals to match the single-electron orbitals directly. We construct ¯ΦHF by padding the extra No −Ne orbitals with zeros. It can easily be veriﬁed that the wave function ΨHF-Pf = 1 Pf ¯ AHF Pf(¯ΦHF ¯AHF ¯ΦT HF), is equivalent to the original Hartree-Fock wave function, i.e., ΨHF-Pf = ΨHF = det(ΦHF) for any invertible skew-symmetric AHF. Further, note that the multiplication of ¯ΦHF with any matrix T ∈SO(No) from the special orthogonal group does not change ΨHF-Pf. Thus, it sufﬁces to match the single electron orbitals of ˆΦPf and ¯ΦHF up to a rotation T ∈SO(No), yielding the following optimization problem: min θ min T ∈SO(No) ∥ˆΦPf −¯ΦHFT∥2 2. (18) We solve this alternatingly for T and θ. To match the geminals ˆΦPfAPf ˆΦT Pf and ΦHFAHFΦT HF, we have to account for the fact that the choice of AHF is arbitrary as long as it is skew-symmetric and invertible. Again, we solve this optimization problem alternatingly by solving for AHF ∈S = {A ∈ SO(Ne) : A = −AT } and θ: min θ min AHF∈S ∥ˆΦPfAPf ˆΦT Pf −ΦHFAHFΦT HF∥2 2. (19) While both formulations share the same minimizer, combining both yields the most stable results. We hypothesize that this is because the single-electron orbitals are more stable than the geminals and thus provide a better starting point for the optimization. In contrast, the latter provides a closer formulation of the neural network orbitals. Thus, we pretrain our neural Pfafﬁan wave functions by solving the optimization problem min θ α min T ∈SO(No) ∥ˆΦPf −¯ΦHFT∥2 2 + β min AHF∈S ∥ˆΦPfAPf ˆΦT Pf −ΦHFAHFΦT HF∥2 2 (20) with weights α, β ∈[0, 1]. To optimize over the special orthogonal group SO(No), we use the Cayley transform (Gallier, 2013). App. C further details the procedure. 4.4 Generalizing over systems We now focus on generalizing the construction of our Pfafﬁan wave function for different systems. We accomplish the generalization similar to PESNet (Gao & Günnemann, 2022) by introducing a second neural network, the MetaGNN M : (R3 × N+)Nn →Θ that acts upon the molecular structure, i.e., nuclei positions and charges, and parametrizes the electronic wave function ΨPfafﬁan : RNe×3×Θ →R for the system of interest. As architecture for the wave function and MetaGNN, we use the same architecture as in Gao et al. (2023a) with the exception being that we replace the Slater determinant with the Pfafﬁan as described in Sec. 4 and minor tweaks highlighted in App. D.4. 6 Moon MetaGNN Orbitals Fig. 2: Orbital parametrization per nucleus. , / indicate electrons and nuclei, respectively. Pfafﬁan. To represent wave functions of different systems within a single NeurPf, we need to adapt the orbitals ˆΦPf and antisymmetrizer APf from Eq. (12) to the molecule. In doing so, we must ensure No ≥max{N↑, N↓}. Otherwise, ˆΦPfAPf ˆΦT Pf is singular, and the wave function is zero. One may solve this by picking No large enough that No ≥max{N↑, N↓} for all molecules in the dataset. However, this is computationally expensive, does not reuse known orbitals in the problem, and simply moves the problem to even larger systems. Instead, we grow the number of orbitals No with the system size by deﬁning Norb/nuc orbitals per nucleus, as depicted in Fig. 2. This allows us to transfer orbitals from smaller systems to larger systems. We only need to ensure that Norb/nuc is larger than half the maximum number of electrons in a period, e.g., for the ﬁrst period Norb/nuc ≥1, for the second period Norb/nuc ≥5. The projection W from Eq. (14) and the envelope decays σ are parametrized by node embeddings, while the envelope weights π and the antisymmetrizer APf are derived from edge embeddings. We predict a Norb/nuc × Nf matrix per nucleus for W and a Nenv/nuc vector per nucleus for σ. For the edge parameters π and APf, we predict a Nenv/nuc ×Norb/nuc and a Norb/nuc ×Norb/nuc matrix per edge, respectively. These are concatenated into the Nenv × No and No × No matrices π and ˆAPf. The latter is antisymmetrized to get APf = 1 2( ˆAPf −ˆAT Pf). We parametrize the spin-dependent scalars η as node outputs for a ﬁxed number of spin conﬁgurations Ns. Because the change in spin conﬁguration does not grow with system size, Ns is ﬁxed. We generate two sets of these parameters, on for ΦPf and on for ˜ΦPf. App. D provides deﬁnitions for the wave function, the MetaGNN, and the parametrization. Pretraining. Previous work like Gao & Günnemann (2023a) needed to canonicalize the Hartree-Fock solutions for different systems before pretraining to ensure that the orbitals ﬁt the neural network. Alternatively, Scherbela et al. (2023) relied on traditional quantum chemistry methods like Foster & Boys (1960)’s localization to canonicalize their orbitals in conjunction with sign equivariant neural networks. In contrast, we ensure that the transformed Hartree-Fock orbitals are similar across structures as we optimize T ∈SO(No) and AHF ∈S for each structure separately, which simultaneously also accounts for arbitrary rotations in the orbitals produced by Hartree-Fock. Limitations. While our Pfafﬁan-based generalized wave function signiﬁcantly improves accuracy on organic chemistry, we leave the transfer to periodic systems for future work (Kosmala et al., 2023). Further, due to the lac of low-level hardware/software support for the Pfafﬁan and the increased number of orbitals No ≥max{N↑, N↓}, our Pfafﬁan is slower than a comparably-sized Slater determinant. While we solve the issue of enforcing the fermionic antisymmetry, our neural wave functions are still unaware of any symmetries of the wave function itself. These are challenging to describe and largely unknown, but their integration may improve generalization performance (Schütt et al., 2018). Finally, in classical single-structure calculations, NeurPf may not improve accuracies. App. P discusses the broader impact of our work. 5 Experiments In the following, we evaluate NeurPf on several atomic and molecular systems by comparing it to Globe (Gao & Günnemann, 2023a) and TAO (Scherbela et al., 2024). Concretely, we investigate the following: (1) Second-row elements and their ionization potentials and electron afﬁnities. Globe cannot compute these due to its restriction to singlet state systems. (2) The challenging nitrogen potential energy surface where Globe signiﬁcantly degraded performance when enlarging their training set with additional molecules. (3) The TinyMol dataset (Scherbela et al., 2024) to evaluate NeurPf’s generalization capabilities across biochemical molecules. In interpreting the following results, one should mind the variational principle, i.e., lower energies are better for neural wave functions. Further, 1 kcal mol−1≈1.6 mEh is the typical threshold for chemical accuracy. Like previous work, we optimize the neural wave function using the VMC framework from Sec. 2. We precondition the gradient with the Spring optimizer (Goldshlager et al., 2024). App. E details the setup further. App. F,I and J show an experiment on extensity and additional ablations. Atomic systems and spin conﬁgurations. We evaluate NeurPf on second-row elements and their ionization potentials and electron afﬁnities. These systems are particularly interesting as they represent 7 10−5 10−3 10−1 Ground (Eh) Li Be B C N O F Ne 10−5 10−3 10−1 Ion. (Eh) 102 103 104 105 10−5 10−3 10−1 Aﬀ. (Eh) 102 103 104 105102 103 104 105 102 103 104 105102 103 104 105 our FermiNet chemical Acc. Unphysical System Unphysical System Fig. 3: Ground state, electron afﬁnity, and ionization potential errors of second-row elements during training. A single NeurPf has been trained on all systems jointly while references (Pfau et al., 2020) were calculated separately for each system. Energies are averaged over the last 10% of steps. a wide range of spin conﬁgurations. We cannot use Globe on such systems because they differ from the singlet state assumption. Instead, we compare our results to the single-structure calculations from Pfau et al. (2020)’s FermiNet and the exact results from Chakravorty et al. (1993); Klopper et al. (2010). In App. G, we repeat this experiment for metals. Fig. 3 displays the ground state energy, electron afﬁnity, and ionization potential errors of NeurPf during training compared to the reference energies from Pfau et al. (2020); Chakravorty et al. (1993); Klopper et al. (2010). It is apparent that NeurPf reaches chemical accuracy relative to the exact results while only training a single neural network for all systems. While separately optimized FermiNets may achieve lower errors, Pfau et al. (2020) trained 21 neural networks for 200k steps each compared to a single NeurPf trained for 200k steps, i.e., 21 times fewer steps and samples. Whereas Gao & Günnemann (2023a); Scherbela et al. (2023) focus on singlet state systems or stable biochemical molecules, NeurPf demonstrates that a generalized wave function need not be restricted to such simple systems and can even generalize to a wide range of electronic conﬁgurations. 2 3 4 5 6 Distance a0 0 2 4 6 8 10 Energy error (mEh)↓ our our (Ethene) Globe Globe (Ethene) FermiNet PESNet Experiment N N Fig. 4: Potential energy surface of nitrogen. Energies are relative to Le Roy et al. (2006). Effect of uncorrelated data. Next, we evaluate NeurPf on the nitrogen potential energy surface, a traditionally challenging system due to its high electron correlation effects (Lyakh et al., 2012). This is particularly interesting as Gao & Günnemann (2023a) observed a signiﬁcant accuracy degradation when reformulating their wave function to generalize over different systems. In particular, they found that training only on the nitrogen dimer leads to signiﬁcantly lower errors than training with an ethene-augmented dataset, indicating an accuracy penalty in generalization. We replicate their setup and compare the performance of NeurPf trained on the nitrogen energy surface with and without additional ethene structures. Like Gao & Günnemann (2023a), the nitrogen structures are taken from Pfau et al. (2020) and the ethene structures from Scherbela et al. (2022). As additional references, we plot Gao & Günnemann (2022)’s PESNet and Fu et al. (2023)’s FermiNet results. Fig. 4 shows the error potential energy surface relative to the experimental results from Le Roy et al. (2006). NeurPf reduces the average error on the energy surface from Globe’s 2.7 mEh to 2 mEh when training solely on nitrogen structures. When adding the ethene structures, Globe’s error increases to 5.3 mEh while NeurPf’s error stays constant at 2 mEh, a lower error than the Globe without the augmented dataset. These results indicate NeurPf’s strong capabilities in approximating ground states while allowing for generalization across different systems without a signiﬁcant loss in accuracy. 8 -2 -1 0 100 101 102 103 104 E −ECCSD(T) CBS (mEh) ↓ Shaded region: Improvement over CCSD(T) CBS Small molecules 250 500 1k 2k 4k 8k 16k 32k Training Step 100 101 102 103 104 105 E −ECCSD(T) CBS (mEh) ↓ Large molecules our TAO CCSD(T) Globe H C N C H H C H H C H H O O C O NH C O NH C NH C H H C C H H Fig. 5: Convergence of mean energy difference on the TinyMol dataset from Scherbela et al. (2024). The y-axis is linear < 1 and logarithmic ≥1. Due to the variational principle, NeurPf is better than the reference CCSD(T) on the small molecules. TinyMol dataset. Finally, we look at learning a generalized wave function over different molecules and structures. We use the TinyMol dataset (Scherbela et al., 2024), consisting of a small and large dataset. The dataset includes ‘goldstandard’ CCSD(T) CBS energies. The small set consists of 3 molecules with 2 heavy atoms, while the large set covers 4 molecules with 3 heavy atoms. For each molecule, 10 structures are provided. Here, we compare again both Globe (+Moon) and TAO to NeurPf. All models are directly trained on the small and large test sets. Fig. 5 shows the mean energy difference to CCSD(T) at different stages of the training. We refer to App. K for a per molecule error attribution. It is apparent that NeurPf yields lower errors than the TAO and Globe after at least 500 steps. On the small structures, NeurPf even matches the CCSD(T) baseline after 16k steps and achieves 1.9 mEh lower energies after 32k steps. Since VMC methods are variational, i.e., lower energies are always better, NeurPf is more accurate than the CCSD(T) CBS reference. Compared to TAO and Globe, NeurPf reports 5.9 mEh and 11.3 mEh lower energies, respectively. On the large structures, we observe a similar pattern where we ﬁnd NeurPf having a 25 times smaller error than TAO during the early stages of training and reaching 21.1 mEh lower energies after 32k steps – a 6 times lower error compared to the CCSD(T) baseline. Note that since the CCSD(T) (CBS) energies are neither exact nor variational, the true error to the ground state is unknown. Still, we provide additional numbers for a NeurPf trained for 128k steps in App. K. There, we ﬁnd NeurPf yielding 4.4 mEh lower energies on the large structures. These results show that a generalized wave function can achieve high accuracy on various molecular structures without pretraining when not relying on hand-crafted algorithms or Hartree-Fock calculations. For additional experiments, we refer the reader to App. L where we ﬁrst pretrain TAO and NeurPf on a separate training set and, then, ﬁnetune on the small and large test sets and App. M for a comparison of joint and separate optimization. 6 Conclusion In this work, we established a new way of parametrizing neural network wave functions for generalization across molecules via overparametrization with Pfafﬁans. Our Neural Pfafﬁan is more accurate, simpler to implement, fully learnable, and applicable to any molecular system compared to previous work. The wave function changes smoothly with the structure, avoiding the discrete orbital selection problem previously solved via hand-crafted algorithms or Hartree-Fock. Additionally, we introduced a memory-efﬁcient implementation of the exponential envelopes, reducing memory requirements while accelerating convergence. Further, we presented a pretraining scheme for Pfafﬁans enabling initialization with Hartree-Fock – a crucial step for molecular systems. Our experimental evaluation demonstrated that our Neural Pfafﬁan can generalize across different ionizations of various systems, stay accurate when enlarging datasets, and set a new state of the art by outperforming previous neural wave functions and the reference CCSD(T) CBS on the TinyMol dataset. These developments open the door for new neural wave functions applications, e.g., to generate reference data for machine-learning force ﬁelds or density functional theory (Cheng et al., 2024; Gao et al., 2024). 9 Acknowledgments. We greatly thank Simon Geisler for our valuable discussions. Further, we thank Valerie Engelmayer, Leo Schwinn, and Aman Saxena for their invaluable feedback on the manuscript. 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Artiﬁcial Intelligence for Science in Quantum, Atomistic, and Continuum Systems, November 2023. 13 Table 1: Number of envelope parameters for the full envelope and our memory efﬁcient envelopes for an explanatory system. σ π Total full 1600 1600 3200 our 640 12800 13400 A Odd numbers of electrons To handle odd numbers of electrons, we extend the electron pair matrix ˆΦPfAPf ˆΦT Pf to even dimensions. We accomplish this by augmenting ˆΦPfAPf ˆΦT Pf with a learnable single-electron orbital φodd to \ ˆΦPfAPf ˆΦT Pf = ˆΦPfAPf ˆΦT Pf φodd −φT odd 0 . (21) To obtain a single additional orbital for the whole molecule, we parameterize one orbital φodd,m for each nucleus as in Eq. (14) and sum them up to obtain φodd = PNn m=1 φodd,m. B Difference to bottleneck envelopes Similar to the bottleneck envelope from Pfau et al. (2024), our efﬁcient envelopes aim at reducing memory requirements. The bottleneck envelopes are deﬁned as χk bottleneck(rbi)j = L X l=1 wk jl Nn X m=1 πlm exp (−σlm∥ri −Rm∥) (22) While both methods share the idea of reducing the number of parameters, they differ in their implementation. Whereas the bottleneck envelopes construct a full set of L many-nuclei envelopes and then linearly recombine these to the ﬁnal envelopes for each of K × No orbitals, our efﬁcient envelopes construct the ﬁnal envelopes directly from a set of single-nuclei exponentials. Further, we use a different set of basis functions for each of the K determinants. In terms of computational complexity, the bottleneck envelopes require O(NeNnL) + O(KLNeNo) operations to compute the envelopes, while our efﬁcient envelopes require O(KNenvNeNo) operations. In practice, we found our efﬁcient envelopes to be faster and converge better on all systems we tested. An ablation study is presented in App. I. Further, we observed no numerical instabilities in our envelopes as reported by Pfau et al. (2024). Compared to the full envelopes, we ﬁnd our memory efﬁcient ones to be slower but yielding better performance. This is likely due to the increased number of wave function parameters. The number of parameters for the full envelopes and our memory efﬁcient envelopes is shown in Tab. 1 for an example with Ne = No = 20, Nn = 5, Nd = 16, N env atom = 8. The full envelopes’ σ, π scale both O(NdNnNo) while our memory efﬁcient envelopes’ σ scales O(NdNnNenv/nuc) and π scales O(NdNnNenv/nucNo). In runtime, the full envelopes require O(NdNnNeNo) operations, while our memory efﬁcient envelopes require O(NdNnN env atom NeNo) operations. In memory complexity, the full envelopes require O(NdNnN 2 e ), while our memory efﬁcient envelopes require O(NdNnN env atom Ne). C Pretraining To pretrain NeurPf, we solve the optimization problem from Eq. (20). The nested optimization problems are solved iteratively, where we ﬁrst solve for T ∈SO(N) and AHF ∈S and then for the parameters of the wave function θ. We describe how we parametrize the special orthogonal group SO(N) and the antisymmetric special orthogonal group S and then how we solve the optimization problems. To optimize over the special orthogonal group SO(N), we parametrize T via some arbitrary matrix ˜T ∈RN×N. Next, we obtain an antisymmetrized version of ˜T via ˆT = 1 2 ˜T −˜T T . (23) 14 We now may use ˆT with the Cayley transform to obtain a special orthogonal matrix ¯T = ˆT −I −1 ˆT + I (24) where I is the identity matrix. ¯T is now a special orthogonal matrix where all eigenvalues are 1. To parametrize matrices with an even number of eigenvalues -1 as well, we simply multiply ¯T with itself: T = ¯T ¯T (25) which gives us our ﬁnal parametrization of the special orthogonal group SO(N) (Gallier, 2013). We follow Gallier (2013), to parametrize antisymmetric special orthogonal matrices S. In particular, we parametrize some T using the procedure outlined above. To parametrize AHF, it remains to antisymmetrize T while preserving the special orthogonal property. We accomplish this by deﬁning AHF = T ˜IT T (26) where ˜I = diag 0 1 −1 0 , . . . =   0 1 0 0 . . . −1 0 0 0 0 0 0 1 0 0 −1 0 ... ...   (27) is the antisymmetric identity matrix. Since the product of special orthogonal matrices is special orthogonal and BABT yielding an antisymmetric matrix for any special orthogonal matrix B, we have that AHF ∈S is an antisymmetric special orthogonal matrix. Now that we can parametrize both groups with real matrices, we can simplify the optimization problem by performing gradient optimization for both T, AHF, and θ. We solve this problem alternatively, where we ﬁrst solve for T and AHF by doing Npre steps of gradient optimization with the prodigy optimizer (Mishchenko & Defazio, 2023) and then perform a single outer step on θ with the lamb optimizer (You et al., 2020) like previous works (Gao & Günnemann, 2022; von Glehn et al., 2023). D Model architectures We largly reuse the same architecture for the MetaGNN M : (R3 × N+)Nn →Θ and wave function ΨPfafﬁan : RNe×3 × Θ →R as Gao & Günnemann (2023a). We canonicalize all molecular structures using the equivariant coordinate frame from Gao & Günnemann (2022). D.1 Wave function Similar to Gao & Günnemann (2023a), we use bars above functions and parameters to indicate that the MetaGNN M parameterizes these and that they vary by structure. We deﬁne our wave function as a Jastrow-Pfafﬁan wave function like Kim et al. (2023): Ψ(r) = exp (J(r)) K X k=1 ckPf ˆΦk Pf(⃗r)Ak Pf ˆΦk Pf(⃗r)T . (28) As Jastrow factor J : RN×3 →R we use a linear combination of a learnable MLP of electron embeddings and the ﬁxed electronic cusp Jastrow from von Glehn et al. (2023): J(r) = N X i=1 MLP(h(⃗ri\|⃗r)) + βpar X i,j;αi=αj −1 4 α2 par αpar + ∥ri −rj∥ + βanti X i,j;αi̸=αj −1 2 α2 anti αanti + ∥ri −rj∥ (29) 15 where h : R3 × RN×3 →RNf is the ith output of the permutation equivariant neural network, implemented via the Molecular orbital network (Moon) (Gao & Günnemann, 2023a), βpar, βanti, αpar, αanti ∈R are learnable scalars, and αi is the spin of the ith electron. The orbitals ˆΦPf are deﬁned as in Eq. (14) with Moon performing the following steps: We start with constructing electron embeddings based on electron-electron distances and then proceed to aggregate these embeddings to the orbitals. The nuclei are updated through MLPs and ﬁnally diffused to the electrons, yielding the ﬁnal electron embeddings. The initial embedding h(0) i of the ith electron is constructed as h(0) i = 1 µ(ri)   N X j=1 σ ge-e ij W δ αj αi ◦Γ δ αj αi (∥⃗ri −⃗rj∥)  W (30) where ◦denotes the Hadamard product and the Kronecker delta δαj αi as superscript indicates different parameters depending on the identity between spin αi and αj. Γ : RI →RD is a learnable radial ﬁlter function, and σ is the activation function. ge-e ij ∈R4 are the rescaled electron-electron distances (von Glehn et al., 2023): gij = log (1 + ∥⃗ri −⃗rj∥) ∥⃗ri −⃗rj∥ [⃗ri −⃗rj, ∥⃗ri −⃗rj∥] . (31) µ is a normalization factor: µ(⃗r) = 1 + M X m=1 Zm 2 exp −∥⃗r −⃗Rm∥2 σ2norm ! . (32) We use the initial electron embeddings with nuclei embeddings and electron-nuclei distances to construct pairwise nuclei-electron embeddings representing edges in a fully connected graph: he-n im = σ h(0) i + ¯zm + ge-n im ¯ Wm . (33) where ¯zm is the mth nucleus embedding, ge-n im ∈R4 are the rescaled electron-nuclei distances like in Eq. (31). These embeddings are then aggregated with spatial ﬁlters twice: once towards the nuclei and once towards the electrons: hnα(1) m = 1 µ(⃗Rm) X i∈Aα he-n i,m ◦¯Γ n m(⃗ri −⃗Rm), (34) m(1) i = 1 µ(⃗ri) M X m=1 he-n i,m ◦¯Γ e m(⃗ri −⃗Rm), (35) h(1) i = σ(m(1) i W + b). (36) We update the nuclei embeddings with L update layers: hnα(l+1) m = hnα(l) m + σ([hnα(l) m , hnˆα(l) m ]W (l) + b(l)), (37) where ˆα denotes the opposite spin of α, to obtain the ﬁnal nuclei embeddings hnα(L) m . The ﬁnal electron embeddings he(L) i are constructed by combining the message from the nuclei and the previous electron embedding: he(L) i = σ σ h(1) i W + m(L) i + b1 W + b2 + h(1) i (38) where mi is the message from the nuclei to the ith electron: m(L) i = 1 µ(⃗ri) M X m=1 σ h hnαi(L) m , hnˆαi(L) m i W + b ◦¯Γ diff m (⃗ri −⃗Rm). (39) 16 The spatial ﬁlters Γ are deﬁned as: ¯Γ (l) m (x) =¯βm(x)W (l), (40) ¯βm(x) = " exp − ∥x∥ ¯ςmi 2!#D i=1 W env ◦ σ x ¯ W (1) m + ¯b(1) m W (2) + b(2) . (41) Note that ¯β is shared across all instances of ¯Γ. Γ is deﬁned analogously to ¯Γ but with ﬁxed learnable parameters instead of MetaGNN parametrized ones. D.2 MetaGNN The MetaGNN M : (R3 × N+)Nn →Θ takes the nucleus position ⃗R and charges Z as input and outputs parameters of the electronic wave function to adapt the solution to the system of interest. We follow Gao & Günnemann (2022, 2023a) and implement it as a graph neural network (GNN) where nuclei are represented as nodes and edges are constructed based on inter-particle distances. The charge of the nucleus determines the initial node embeddings: k(0) i = EZi (42) where E is an embedding matrix and Zi is the charge of the ith nucleus. These embeddings are iteratively updated via message passing in the following way: k(l+1) i = f (l)(k(l) i , t(l) i ), (43) t(l) i = 1 ν ⃗R ⃗Ri M X j=1 g(l)(k(l) i , k(l) j ) ◦Γ (l)(⃗Ri −⃗Rj), (44) νN x = 1 + X y∈N exp −∥x −y∥2 σ2norm (45) where Eq. (43) describes the update function, Eq. (44) the message construction, and Eq. (45) a learnable normalization coefﬁcient. We implement the functions f and g via Gated Linear Units (GLU) (Shazeer, 2020). As spatial ﬁlters, we use the same as in the wave function but additionally multiply the ﬁlters with radial Bessel functions from Gasteiger et al. (2019): Γ (l)(x) =β(x)W (l), (46) β(x) =   r 2 c sin fix c x exp − ∥x∥ ςi 2!  D i=1 W env ◦ σ xW (1) + b(1) W (2) + b(2) (47) where fi are learnable frequencies, and c is a smooth cutoff for the Bessel functions. After L layers, we take the ﬁnal node embeddings, pass them through another GLU, and then use a different GLU as head for each distinct parameter tensor of the wave function we want to predict. For edge-dependent parameters, like π or A, we ﬁrst construct edge embeddings by concatenating all combinations of node embeddings. We pass these through a GLU and then proceed like for node embeddings. For all outputs, we add a default charge-dependent parameter tensor such that the MetaGNN only learns a delta to an initial guess depending on the charge of the nucleus. D.3 Orbital parametrization Our Pfafﬁan wave function enables us to simply parametrize a No ≥max{N↑, N↓} orbitals rather than parametrizing exactly N↑/N↓. As discussed in Sec. 4.4, we accomplish this by associating a ﬁxed number of orbitals with each nucleus. Here, we provide detailed construction for all parameters of the orbital construction. For simplicity, we do not explicitly show the dependence on the kth Pfafﬁan. Note that we simply extend the readout by an Nk sized dimension for each of the Nk Pfafﬁans from Eq. (13). Further, we predict two sets of parameters, one for ΦPf and one for ˜ΦPf in 17 Eq. (9). To parametrize the orbitals, we predict Norb/nuc orbital parameters for each of the Nn nuclei. Concretely, the linear projection to Wk from Eq. (14) are constructed as W =   ω1(k1) ... ωNorb/nuc(k1) ω1(k2) ... ωNorb/nuc(kNn)   ∈RNo×Nf (48) where ωi : RD →RNf learnable readouts of our MetaGNN. Similarly, we parametrize the envelope coefﬁcients σk from Eq. (16): σ =   ς1(k1) ... ςNenv/nuc(k1) ς1(k2) ... ςNenv/nuc(kNn)   ∈RNenv + (49) where ςi : RD →R+ are learnable readouts of our MetaGNN. The linear orbital weights π connect each nuclei-centered envelope to the non-atom-centered orbitals. For this, we need to ﬁnd a mapping from each of the Nenv envelopes to each of the No orbitals. Since Nenv = Nenv/nuc × Nn and No = Norb/nuc × Nn are predicted per nuclei, a natural connection is established via a pair-wise atom function: π =   ϖ1,1(k1, k1) . . . ϖ1,Norb/nuc(k1, k1) ϖ1,1(k2, k1) . . . ... ... ... ϖNenv/nuc,1(k1, k1) . . . ϖNenv/nuc,Norb/nuc(k1, k1) ϖNenv/nuc,1(k2, k1) . . . ϖ1,1(k1, k2) ϖ1,Norb/nuc(k1, k2) ϖ1,1(k2, k2) . . . ... ... ... ...   ∈RNenv×No (50) where ϖi,j : RD × RD →R are learnable readouts of our MetaGNN. Similarly, we establish the orbital correlations A from Eq. (14) by connecting each of the No orbitals to each other: ˆAPf =   α1,1(k1, k1) . . . α1,Norb/nuc(k1, k2) α1,1(k2, k1) . . . ... ... ... αNorb/nuc,1(k1, k1) . . . αNorb/nuc,Norb/nuc(k1, k1) αNorb/nuc,1(k2, k1) . . . α1,1(k1, k2) α1,Norb/nuc(k1, k2) α1,1(k2, k2) . . . ... ... ... ...   ∈RNo×No (51) APf = 1 2( ˆAPf −ˆAT Pf) (52) where αi,j : RD × RD →R are learnable readouts of our MetaGNN and Eq. (52) enforcing the antisymmetry requirements on A. D.4 Changes to the MetaGNN We performed several optimizations on the MetaGNN from Gao & Günnemann (2023a) that primarily reduce the number of parameters while keeping accuracy. In particular, we changed the following: • We replace all MLPs with gated linear units (GLU) (Shazeer, 2020). • We reduced the hidden dimension from 128 to 64. 18 Table 2: Hyperparameters used for the experiments. Hyperparameter Value Structure batch size full batch Total electron samples 4096 Pretraining Epochs 10000 Learning rate 10−3 ∗(1 + t ∗10−4)−1 Optimizer Lamb MCMC steps 5 Basis STO-6G Subproblem steps 50 Subproblem optimizer Prodigy Subproblem α 1.0 Subproblem β 10−4 Optimization Steps 60000 Learning rate 0.02 ∗(1 + t ∗10−4)−1 Optimizer Spring MCMC steps 20 Norm constraint 10−3 Damping 0.001 Momentum 0.99 Energy clipping 5 times mean deviation from median Ansatz Hidden dim 256 E-E int dim 32 Layers 4 Activation SiLU Determinants/Pfafﬁans 16 Jastrow layers 3 Filter hidden dims [16, 8] Pfafﬁan Norb/nuc (H, He) 2 Norb/nuc (Li, Be) 6 Norb/nuc (B, C) 7 Norb/nuc (N, O) 8 Norb/nuc (F, Ne) 10 Nenv/nuc 8 MetaGNN Embedding dim 64 Message dim 32 Layers 3 Activation SiLU Filter hidden dims [32, 16] • We reduced the message dimension from 64 to 32. • We use bessel basis functions (Gasteiger et al., 2019) on the radius for edge ﬁlters. • We remove the hand-crafted orbital locations and the associated network. • We added a LayerNorm before every GLU. Together, these changes reduce the number of parameters from 13M to 1M for the MetaGNN while outperforming Gao & Günnemann (2023a) as demonstrated in Sec. 5. E Experimental setup 19 Table 3: Compute time per experiment measured in Nvidia A100 GPU hours. Experiment Time (GPU hours) Ionization & afﬁnity 224 N2 116 N2 + Ethene 124 TinyMol small 78 TinyMol large 96 E.1 Hyperparameters We list the default parameters used for the experiments in Tab. 2. Most of them were taken directly from Gao & Günnemann (2023a). We may have used different parameters for the experiments in Sec. 5 if explicitly stated so. We implement everything in JAX (Bradbury et al., 2018). To compute the laplacian ∇2Ψ, we use the forward laplacian algorithm (Li et al., 2024) implemented in the folx library (Gao et al., 2023). E.2 Source code We provide the source code publicly on GitHub 1. E.3 Compute time Tab. 3 lists the compute times required for conducting our experiments measured in Nvidia A100 GPU hours. Depending on the experiment, we use between 1 and 4 GPUs per experiment via data parallelism. We typically allocated 32GB of system memory and 16 CPU cores per experiment. In terms of the number of parameters, the Moon wave function is as large as in Gao & Günnemann (2023a) at 1M parameters, and the MetaGNN shrank from 13M parameters to just 1M parameters. E.4 Preconditioning The Spring optimizer (Goldshlager et al., 2024) is a natural gradient descent optimizer for electronic wave functions Ψ with the following update rule θt =θt −ηδt (53) δt =( ¯OT ¯O + λI)−1(∇θt + λµδt−1) (54) where λ is the damping factor, µ is the momentum, η is the learning rate, and ¯O is the zero-centered Jacobian: ¯O =O −1 N N X i=1 Oi, (55) O =   ∂log ψ(x1) ∂θ... ∂log ψ(xN) ∂θ  . (56) Since ¯O ∈RN×P where N is the batch size and P the number of parameters, the update in Eq. (54) can be efﬁciently computed using the Woodbury matrix identity, which after some simpliﬁcations yields δt = ¯O( ¯O ¯OT + λI)−1(ϵ + µ ¯Oδt−1) + µδt−1. (57) Our early experiment found it necessary to center the jacobian ¯O per molecule rather than once for all. In single-structure VMC, the centering eliminates the gradient of the wave function along the direction where the amplitude of the wave function increases for all inputs. This direction does not 1https://github.com/n-gao/neural-pfaffian 20 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Number of hydrogen −0.56 −0.54 −0.52 −0.50 −0.48 −0.46 −0.44 −0.42 Energy per atom (Eh) our TAO Globe + Moon Globe + FermiNet Hartree Fock AFQMC natoms →∞ Training data Fig. 6: Energy per atom of hydrogen chains with different lengths. The energy is computed with a single NeurPf trained on the hydrogen chains with 6 and 10 atoms. affect energies. Thus, instead of restricting the gradient from increasing in magnitude for all samples, we constrain it to not increase in magnitude for each molecule separately N.ote that the latter implies the ﬁrst but not vice versa. For multi-structure VMC, we compute ¯O as ¯O = O −   1 N1 PN1 i=1 Oi ... 1 NM PNM i=N−NM Oi   (58) where N1, ..., NM are the index limits between molecular structures. To stabilize computations, we performed preconditioning in ﬂoat64. F Extensivity on hydrogen chains Gao & Günnemann (2023a) and Scherbela et al. (2024) analyzed the behavior of their wave functions on hydrogen chains to investigate the extensivity of their wave functions. They did so by training the generalized wave functions on a set of hydrogen chains with 6 and 10 elements. Then, they evaluated the energy per atom on hydrogen chains with different lengths. We replicated their experiment and trained a single NeurPf on the hydrogen chains with 6 and 10 atoms and evaluated the energy per atom on hydrogen chains of increasing lengths. Fig. 6 shows the energy per atom of hydrogen chains with different lengths for various methods, Globe+Moon and Globe+FermiNet from Gao & Günnemann (2023a), Scherbela et al. (2024), HartreeFock (CBS), the AFQMC limit for an inﬁnitely long chain (Motta et al., 2017), and NeurPf. It is apparent that NeurPf outperforms Globe+Moon and Globe+FermiNet signiﬁcantly by achieving signiﬁcantly lower energies outside of the training regime. Compared to Scherbela et al. (2024), NeurPf generally performs better on longer chains, achieving errors below the Hartree-Fock baseline. However, we observe signiﬁcantly higher errors in the shortest chains in NeurPf. These results indicate that NeurPf is better at generalizing to longer chains than previous works despite not including additional Hartree-Fock calculations like Scherbela et al. (2024). G Metal ionization energies In addition to the results in Sec. 5, where we train on all second-row elements and their ionization and afﬁnity potentials, we here train a single NeurPf on a set of metals and their ionization energies. This demonstrates that Neural Pfafﬁans also scale to heavier 3rd and 4th row elements. Fig. 7 shows the ionization energy during training. It is apparent, that NeurPf can learn a solution for all states simultaneously. 21 102 103 104 105 10−5 10−4 10−3 10−2 10−1 Ion. (Eh) Na 102 103 104 105 Mg 102 103 104 105 Al 102 103 104 105 K 102 103 104 105 Ca Fig. 7: Ionization energies of metal atoms. The ionization energies are computed with a single NeurPf trained on the neutral and ionized atoms. Reference energies are taken from Martin & Musgrove (1998). 0 20 40 60 Time (h) −95.47 −95.46 −95.45 −95.44 Energy (Eh) Small 0 25 50 75 Time (h) −155.600 −155.575 −155.550 −155.525 −155.500 Large our our (FermiNet) our (PsiFormer) Globe Fig. 8: Energy convergence as a function of time. H TinyMol convergence in time In Fig. 8, we show the runtime effect of choosing different embedding and antisymmetrizer. We test our default model, our (FermiNet), our (PsiFormer) and Globe + Moon on both TinyMol datasets. For any time budget, all variants of NeurPf converge to lower energies than Globe. I Convergence ablation studies Here, we provide additional ablation studies to further investigate the performance of NeurPf and our efﬁcient envelopes. In particular, we train four different models on the small TinyMol dataset: NeurPf, NeurPf with the envelopes from Spencer et al. (2020), NeurPf with the envelopes from Pfau et al. (2024), and an AGP-based generalized wave function. The total energy during training is shown in Fig. 9. The left plot shows the convergence regarding the number of steps, and the right plot shows the convergence in terms of time. We observe that NeurPf convergence is consistently faster than the other methods in terms of the number of steps and time. One further sees the importance of generalizing Eq. (9) via the Pfafﬁan as the AGP-based wave function does not converge to the same accuracy as NeurPf. The bottleneck envelopes from Pfau et al. (2024) do not only converge to worse energies but are also slower per step than our efﬁcient envelopes from Sec. 4.2. J Model ablation studies 22 0 5000 10000 15000 20000 25000 30000 Step −2864.05 −2864.00 −2863.95 −2863.90 −2863.85 −2863.80 Total energy (Eh) AGP our + Full Env. our + Bottleneck Env. our + Eﬃcient Env. 0 20 40 60 80 100 Time (h) Fig. 9: Ablation study on the small TinyMol dataset. The y-axis shows the sum of all energies in the dataset. The left plot shows the convergence in terms of the number of steps. The right plot shows the convergence in terms of time. our + Full Env. shows a NeurPf with the envelopes from Spencer et al. (2020) and our + Bottleneck Env. uses the bottleneck envelopes from Pfau et al. (2024). 250 500 1k 2k 4k 8k 16k 32k Training Step -2 0 101 103 E −ECCSD(T) CBS (mEh) ↓ Small molecules Pf(ΦAΦT ) Pf(Φ1Φ2 −Φ2ΦT 1 ) CCSD(T) 250 500 1k 2k 4k 8k 16k 32k Training Step 100 101 102 103 104 Large molecules Fig. 10: TinyMol ablation with ﬁxed and learnable antisymmetrizer. 250 500 1k 2k 4k 8k 16k 32k Training Step -2 0 101 103 E −ECCSD(T) CBS (mEh) ↓ Small molecules our (Moon) our (FermiNet) our (PsiFormer) CCSD(T) 250 500 1k 2k 4k 8k 16k 32k Training Step 100 101 102 103 104 Large molecules Fig. 11: Ablation study on the small TinyMol dataset with different embedding networks. 23 Table 4: TinyMol energies compared to CCSD(T) in mEh. Small Large Method (Steps) CNH C2H4 COH2 C3H4 CN2H2 CNOH CO2 Globe (32k) 5.2 12.3 10.7 62.3 45.8 40.4 42.7 TAO (32k) 1.1 4.5 6.6 18.7 21.0 41.9 19.6 our (32k) -3.7 0.1 -2.1 12.7 5.5 3.1 5.0 our (128k) -4.2 -1.5 -3.7 1.4 -3.8 -6.9 -8.2 CNH C2H4 COH2 −4 −2 0 2 4 6 8 10 Eour −ECCSD (mEh) Small molecules C3H4 CN2H2 CNOH CO2 0 10 20 30 40 50 60 Large molecules Our (32k steps) TAO (32k steps) TAO (pretrained, 32k steps) Fig. 12: Boxplot of the energy per molecule on both TinyMol small and large datasets for NeurPf, TAO, and the pretrained TAO from Scherbela et al. (2024). Each boxplot contains results from 10 structures for the given molecule. The line indicates the mean, the box the interquartile range, and the whiskers the 1.5 times the interquartile range. J.1 Learnable antisymmetrizer We picked Pf(ΦAΦT ) as parametrization because it generalizes Slater determinants and many alternative parametrizations. For instance, by choosing A = 0 I −I 0 and Φ = (Φ1 Φ2) = Pf(ΦAΦT ) =⇒Pf(Φ1ΦT 2 −Φ2ΦT 1 ). We investigate the impact of having A being ﬁxed/learnable in Fig. 10. The results suggest that having A being learnable is a signiﬁcant factor in our Neural Pfafﬁan’s accuracy. J.2 Embedding network Since NeurPf is not limited to Moon, we performed additional ablations with FermiNet (Pfau et al., 2020) and PsiFormer (von Glehn et al., 2023) as the embedding. The results in Fig. 11 show Neural Pfafﬁans outperforming Globe and TAO with any of the three equivariant embedding models. Consistent with Gao & Günnemann (2023a), Moon is the best choice for generalized wave functions. K TinyMol results Here, we provide additional data analysis and error metrics for the TinyMol dataset. First, we show in Table 4 the energy per molecule for the small and large TinyMol datasets for NeurPf, Globe, and TAO. To estimate the remaining error, we also train another NeurPf for 128k steps. The results show that NeurPf consistently outperforms TAO and Globe on all molecules in both datasets. Second, we show the error per molecule for both the small and large TinyMol datasets in Fig. 12. We plot all models after 32k steps of training. It is apparent that NeurPf consistently results in lower, i.e., better, energies than TAO on all molecules in both datasets. Even the pretrained TAO is outperformed by NeurPf on all but four structures of C3H4 in the large TinyMol dataset. 24 250 500 1k 2k 4k 8k 16k 32k Training Step -2 -1 0 100 101 102 E −ECCSD(T) CBS (mEh) ↓ Small molecules our TAO CCSD(T) 250 500 1k 2k 4k 8k 16k 32k Training Step 100 101 102 103 Large molecules Fig. 13: TinyMol results with pretraining on the training set. 250 1k 4k 16k 64k 256k 1M Total Steps -10 -5 0 104 E −ECCSD(T) (mEh) Small 250 1k 4k 16k 64k 256k 1M Total Steps Large joint training separate training CCSD(T) CBS Fig. 14: Comparison of the energy per molecule on the TinyMol dataset for training jointly on all structures vs training a model per structure. L Pretraining on the TinyMol dataset The TinyMol provides an additional pretraining set of 360 structures (18 molecules, 20 structures each). Like Scherbela et al. (2024), we pretrain our model on the training set of the TinyMol dataset and then ﬁnetune on the two test sets. Interestingly, we ﬁnd the Spring optimizer to be unstable when swapping molecules from step to step and, thus, use CG-preconditioning like Gao & Günnemann (2023a) during pretraining. While yielding a small beneﬁt on the small molecules, we ﬁnd no notable difference to the Hartree-Fock pretrained model on the large molecules as shown in Fig. 13. On the small structures, the unpretrained NeurPf’s energies are 5.7 mEh lower. NeurPf also surpasses the pretrained TAO after just 8k steps. Compared to the pretrained TAO on the large structures, NeurPf surpasses TAO after 16k steps and achieves 5.4 mEh lower energies after 32k steps. M Joint vs separate training To estimate the beneﬁt of training a generalized wave function compared to training a model per molecule, we compare the convergence of the total energy on the TinyMol dataset for both approaches depending on the total number of training steps. As training a separate model for each of the 70 TinyMole test molecules is computationally beyond the scope of this work, we select on structure per molecules and train a model for each of the 7 molecules. We use the same NeurPf with MetaGNN for both approaches. The results are shown in Fig. 14. We observe that for lower step numbers, it is quite beneﬁcial to train a generalized model. Though, this beneﬁt vanishes for higher step numbers, and 25 2 4 8 16 32 Ne in molecule 1 2 4 8 16 32 Ne in molecule 2 2.8 3.0 3.1 3.4 5.0 3.0 2.9 3.1 3.4 5.1 3.1 3.1 3.0 3.5 5.3 3.4 3.4 3.5 3.7 5.6 5.0 5.1 5.3 5.6 6.9 3 4 5 6 Fig. 15: Time per training step depending on the number of electrons in two molecules. 100 101 Forward Pf(ΦAΦT ) det(Φ) 100 101 time in ms Graident 20 40 60 80 100 Number of electrons (Ne) 100 101 Laplacian Fig. 16: Time for the forward pass, gradient and Laplacian computation of determinant vs our Pfafﬁan implementation. training a model per molecule yields lower energies. We attribute this to the fact that the generalized model has to learn a more complex representation that is not necessary for each molecule individually. Further, the per-molecule energy estimates are quite unstable due to the small shared batch size. Developments like Scherbela et al. (2023) may improve NeurPf training as well. N Training time by batch composition Here, we benchmark the total time per step for a two-molecule batch. We test all combinations of two molecules with N 1 e , N 2 e ∈2, 4, 8, 16, 32. While we ﬁnd a small runtime increase when processing small molecules jointly in Fig. 15, for larger systems, we see the runtime per step converge to the geometric mean of the individual runtimes. O Pfafﬁan runtime In Fig. 16, we benchmark our implementation for Pf(ΦAΦT ) (incl. the matrix multiplications) against the standard operation of det Φ for 10 to 100 electrons. We implement the Pfafﬁan in JAX 26 while highly optimized CUDA kernels are available for the determinant. In summary, both share the same complexity of O(N 3), but the Pfafﬁan is approximately 5 times slower. P Broader impact Highly accurate quantum chemical calculations are essential for understanding chemical reactions and materials properties. Our work contributes to this development by providing accurate neural network quantum Monte Carlo calculations at broader scales thanks to generalized wave functions. While this may be used to distill more accurate force ﬁelds or exchange-correlation functionals for DFT, the societal impact of our work is primarily in the scientiﬁc domain due to the high computational cost of neural network VMC. To the best of our knowledge, our work does not promote any negative societal impact more than general theoretical chemistry research does. 27 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope? Answer: [Yes] Justiﬁcation: We claim the following in our abstract and introduction: (1) Neural Pfafﬁan are applicable to any molecular system. As outlined in Section 4.4, we ensure this by parametrizing the orbitals to be always larger than the number of electrons. (2) Neural Pfafﬁan can learn all second-row element systems’ ground state, ionization, and electron afﬁnity energies. We demonstrate this in our ﬁrst experiment in Section 5. (3) Neural Pfafﬁan outperforms Globe on the nitrogen dimer. See the second experiment in Section 5.(4) We outperform CCSD(T) CBS on the small structures in TinyMol and TAO by factors of 10 and 6 on small and large structures, respectively. See the third experiment in Section 5. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,430	Group and Shuffle: Efficient Structured Orthogonal Parametrization Mikhail Gorbunov HSE University gorbunovmikh73@gmail.com Nikolay Yudin HSE University Vera Soboleva AIRI, HSE University Aibek Alanov AIRI, HSE University Alexey Naumov HSE University, Steklov Mathematical Institute of Russian Academy of Sciences Maxim Rakhuba HSE University Abstract The increasing size of neural networks has led to a growing demand for methods of efficient fine-tuning. Recently, an orthogonal fine-tuning paradigm was introduced that uses orthogonal matrices for adapting the weights of a pretrained model. In this paper, we introduce a new class of structured matrices, which unifies and generalizes structured classes from previous works. We examine properties of this class and build a structured orthogonal parametrization upon it. We then use this parametrization to modify the orthogonal fine-tuning framework, improving parameter and computational efficiency. We empirically validate our method on different domains, including adapting of text-to-image diffusion models and downstream task fine-tuning in language modeling. Additionally, we adapt our construction for orthogonal convolutions and conduct experiments with 1-Lipschitz neural networks. 1 Introduction Orthogonal transforms have proven useful in different deep learning tasks. For example, they were shown to stabilize CNNs [Li et al., 2019, Singla and Feizi, 2021] or used in RNNs to combat the problem of exploding/vanishing gradients [Arjovsky et al., 2016]. Recent works OFT (Orthogonal Fine-Tuning) and BOFT (Butterfly Orthogonal Fine-Tuning) [Qiu et al., 2023, Liu et al., 2024b] use learnable orthogonal matrices for parameter-efficient fine-tuning of neural networks, which prevents training instabilities and overfitting that alternative methods like LoRA [Hu et al., 2022] suffer from. Nevertheless, parametrization of orthogonal matrices is a challenging task, and the existing methods typically lack in either computational efficiency or expressiveness. Classical methods like Cayley parametrization and matrix exponential map cannot operate under low parameter budget, while Givens rotations and Householder reflections requires computing products of several matrices, which makes their use less efficient in deep learning tasks. Alternative approach in OFT method uses blockdiagonal matrix structure in an attempt to be more computationally efficient and use less trainable parameters. Unfortunately, this simple structure can be too restrictive. Thus, arises the problem of constructing dense orthogonal matrix while still being parameter-efficient. While attempting to tackle this task, BOFT method uses a variation of butterfly matrices, parametrizing orthogonal matrices as a product of several matrices with different sparsity patterns, enforcing orthogonality on each of them. This parametrization is able to construct dense matrices while still being parameter-efficient. However it requires to compute a product of multiple matrices (typically up to 6) which can be computationally expensive. In this paper, we aim to overcome these issues and build dense orthogonal matrices in a more efficient way. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). We present a novel structured matrix class parametrized by an alternating product of block-diagonal matrices and several permutations. Multiplying by these matrices can be seen as a consecutive application of independent linear transforms within certain small groups and then shuffling the elements between them, hence the name Group-and-Shuffle matrices (or GS-matrices for short). This class generalizes Monarch matrices [Dao et al., 2022] and with the right permutation choices, is able to form dense orthogonal matrices more effectively compared to approach proposed in BOFT, decreasing number of matrices in the product as well as the number of trainable parameters. We build efficient structured orthogonal parametrization with this class and use it to construct a new parameter-efficient fine-tuning method named GSOFT. Our contributions: • We introduce a new class of structured matrices, called GS, that is more effective at forming dense matrices than block butterfly matrices from the BOFT method. • Using GS-matrices, we propose an efficient structured orthogonal parametrization, provide theoretical insights and study its performance in the orthogonal fine-tuning framework. • We adapt our ideas for convolutional architectures, providing a framework to compress and speed-up orthogonal convolution layers. 2 Orthogonal Fine-tuning Orthogonal Fine-tuning method (OFT) introduced in [Qiu et al., 2023] is a Parameter-Efficient Fine-Tuning (PEFT) method which fine-tunes pre-trained weight matrices through a learnable orthogonal block-diagonal matrix. Some of the properties that make orthogonal transforms desirable are preservation of pair-wise angles of neurons, spectral properties and hyperspherical energy. More precisely, OFT optimizes an orthogonal matrix Q ∈Rd×d for a pre-trained frozen weight matrix W 0 ∈Rd×n and modifies the multiplication y = (W 0)⊤x to y = (QW 0)⊤x. Note that the identity matrix I is orthogonal, which makes it a natural initialization for Q. OFT uses block-diagonal structure for Q, parameterizing it as Q = diag(Q1, Q2, . . . , Qr), where Qi ∈Rb×b are small orthogonal matrices and br = d. Orthogonality is enforced by Cayley parametrization, i.e. Qi = (I + Ki)(I −Ki)−1, where Ki are skew-symmetric: Ki = −K⊤ i . This ensures orthogonality of Qi and, hence, of Q. Nevertheless, block-diagonal matrices can be too restrictive, as they divide neurons into r independent groups based on their indices. This motivates the construction of dense parameter-efficient orthogonal matrices. To address this problem, the Orthogonal Butterfly method (BOFT) was introduced [Liu et al., 2024b]. BOFT uses block-butterfly structure to construct Q. Essentially, Q is parameterized as a product of m orthogonal sparse matrices: Q = BmBm−1 . . . B1. Each matrix Bi is a block-diagonal matrix up to a permutation of rows and columns, consisting of r block matrices of sizes b × b. Similarly to OFT, the orthogonality is enforced by the Cayley parametrization applied to each block. However, BOFT method has some areas for improvement as well. To construct a dense matrix, BOFT requires at least m = 1 + ⌈log2(r)⌉ matrices. For example, the authors of BOFT use m = 5 or 6 matrices in the BOFT method for fine-tuning of Stable Diffusion [Rombach et al., 2022]. Large amount of stacked matrices leads to significant time and memory overhead during training. There is also a room for improvement in terms of parameter-efficiency. To overcome these issues, we introduce a new class of structured matrices that we denote GS (group-and-shuffle) that generalizes Monarch matrices [Dao et al., 2022, Fu et al., 2024] and show how to use this class to construct parameter-efficient orthogonal parametrization. Similarly to BOFT, our approach uses block-diagonal matrices and permutations, but requires only m = 1 + ⌈logb(r)⌉ matrices of the same size to construct a dense matrix. See details in Section 5.2. The reduced requirements on m allow us to use m = 2 in experiments to maximize computational efficiency, while still maintaining accurate results. 2 3 GS-matrices Our motivation within this work is to utilize orthogonal matrices of the form: A = PL(LPR)PR (1) where matrices L and R are block-diagonal matrices with r blocks of sizes b × b and PL, P, PR are certain permutation matrices, e.g. PL = P ⊤, PR = I in the orthogonal fine-tuning setting and PR = P, PL = I for convolutional architectures. Note that although the case PL = P ⊤, PR = I resembles Monarch matrices [Dao et al., 2022], they are unable to form such a structure, e.g., with equal-sized blocks in L and R. The issue is that the Monarch class has a constraint that interconnects the number of blocks in matrix L and the number of blocks in matrix R (see Appendix C for details). Moreover, Monarch matrices have not been considered with orthogonality constraints. To build matrices of the form (1), we first introduce a general class of GS-matrices and study its properties. We then discuss orthogonal matrices from this class in Section 4. 3.1 Definition of GS-matrices Definition 3.1. An m × n matrix A is in GS(PL, P, PR) class with kL, kR blocks and block sizes b1 L × b2 L, b1 R × b2 R if A = PL(LPR)PR, where L = diag(L1, L2, . . . , LkL), Li ∈Rb1 L×b2 L, R = diag(R1, R2, . . . , RkR), Ri ∈Rb1 R×b2 R, PL, P, PR are permutation matrices and b2 L · kL = b1 R · kR = s, b1 L · kL = m, b2 R · kR = n. L P R x Rx P(Rx) L(PRx) Figure 1: GS(I, P, I) matrices with b1 L = b2 L = 3, b1 R = b2 R = 2, kL = 2, kR = 3. Edges between nodes denote nonzero weights. In practice, we fix PL, P, PR depending on the application and only make matrices L, R subject for change. GS-matrices are hardware-efficient, as they are parametrized by two simple types of operations that can implemented efficiently: multiplications by block-diagonal matrices and permutations. Let us also illustrate a forward pass Ax ≡LPRx for a matrix A ∈GS(I, P, I) as a building block for the more general class with two additional permutations. The first operation y = Rx consists of several fully-connected layers, applied individually to subgroups of x, see Figure 1. The next multiplication LPy ensures that these groups interact with each other. Indeed, the permutation matrix P shuffles the entries of y into new subgroups. These subgroups are then again processed by a number of fully-connected layers using L. This motivates the naming for our class of matrices: Group-and-Shuffle or GS for short. Another useful insight on these matrices is that the class GS(I, P, I) consists of block matrices with low-rank blocks. The permutation matrix P is responsible for the formation of these blocks and defines their ranks (note that rank may vary from block to block). The result below formally describes our findings and is key to the projection operation that we describe afterwards. Proposition 1. Let A be a matrix from GS(I, P, I) with a permutation matrix P defined by the function σ : {0, . . . , n −1} →{0, . . . , n −1}. Let {v⊤ i } – be the rows of the blocks R1, . . . , RkR, {ui} – the columns of the blocks L1, . . . , LkL in the consecutive order. Then the matrix A can be written as a block matrix with kL × kR blocks using the following formula for each block Ak1,k2: Ak1,k2 = X ⌊σ(i) kL ⌋=k1 ⌊ i kR ⌋=k2 uσ(i)v⊤ i . Note that we use zero-indexing for this proposition for simplicity of formulas. 3 . . . + u2v⊺ 4 = u2 v⊺ 4 Figure 2: Illustration of Proposition 1 that provides block low-rank interpretation of GS(I, P, I) matrices. The matrix R contains 2 blocks and matrix L contains 4 blocks. Let us illustrate this proposition in Figure 2. We consider GS(I, P, I) with kL = 4 and kR = 2 blocks in L and R and with the block sizes 3 × 3 and 6 × 6 respectively. Let us consider the leading block A00 of the size 3 × 6. According to Proposition 1, A00 = u0v⊤ 2 + u2v⊤ 4 . Indeed, let us take a closer look, e.g., at the term u2v⊤ 4 . In the permutation matrix P, we have a nonzero element in the position (2, 4) as i = 4 and σ(4) = 2. Therefore, we select the third column u2 in L1 and the fifth row v⊤ 4 in R1. This leads to adding a rank-one term u2v⊤ 4 to A00 as we see in the formula above. Another direct corollary from Proposition 1 is a projection operation π: Rm×n →GS(PL, P, PR) that satisfies: π(A) ∈ arg min B∈GS(PL,P ,PR) ∥A −B∥F , where ∥· ∥F is the Frobenius norm. Thanks to the block low-rank representation of matrices from GS(PL, P, PR), the projection π is simply constructed using SVD truncations of the blocks (P ⊤ L AP ⊤ R )k1,k2 and is summarized in Algorithm 1. Algorithm 1 Projection π(·) of A onto GS(PL, P, PR) Input: A, PL, P, PR Return: L, R for k1 = 1 . . . kL do for k2 = 1 . . . kR do Compute SVD of (P T L AP T R )k1,k2 = UΣV ⊤; Set r = rk1,k2 – rank of block determined by P; Take Ur = U[: r, :], Σr = Σ[: r, : r], Vr = V [: r, :]; Pack columns of UrΣ1/2 r into Lk1 and rows of Σ1/2 r Vr into Rk2 according to P; end for end for 4 Orthogonal GS(PL, P, PR) matrices In this section, we study the orthogonality constraint for the GS(PL, P, PR) to obtain structured orthogonal representation. This is one of the main contributions of our paper and we utilize this class in all the numerical experiments. Since we are interested only in square orthogonal matrices, we additionally assume that m = n and b1 L = b2 L = bL; b1 R = b2 R = bR. Similarly to parametrizations in OFT and BOFT, a natural way to enforce orthogonality of GS(PL, P, PR)-matrices is to enforce orthogonality of each block of L and R. This indeed leads an orthogonal matrix since permutation matrices are also orthogonal as well as a product of orthogonal matrices. However, it is not immediately obvious that there exist no orthogonal matrices from GS(PL, P, PR) that cannot be represented this way. Surprisingly, we find that such a way to enforce orthogonality is indeed sufficient for covering of all orthogonal matrices from GS(PL, P, PR). Theorem 1. Let A be any orthogonal matrix from GS(PL, P, PR). Then, A admits PL(LPR)PR representation with the matrices L, R consisting of orthogonal blocks. Proof. Matrices PL, PR are orthogonal as they are permutation matrices. It means that it is sufficient to prove theorem in the case when A is from GS(I, P, I), which means that we can use low block4 rank structure interpretation from Proposition 1. Consider a skeleton decomposition of the blocks Aij = UijV ⊤ ij , Uij ∈RbL×rij, Vij ∈RbR×rij such that U ⊤ ij Uij = Irij (this can be ensured, e.g., using the QR decomposition). Then A =    U1,1V ⊤ 1,1 . . . U1,kLV ⊤ 1,kL ... ... ... UkL,1V ⊤ kL,1 . . . UkL,kRV ⊤ kL,kR   . Take the j-th block-column of A. Since A is an orthogonal matrix, we get: V1,jU ⊤ 1,j . . . VkL,jU ⊤ kL,j    U1,jV ⊤ 1,j ... UkL,jV ⊤ kL,j   = IbR Multiplying matrices in the l.h.s. we get V1,jU ⊤ 1,jU1,jV ⊤ 1,j+· · ·+VkL,jU ⊤ kL,jUkL,jV ⊤ kL,j = IbR. Since U ⊤ ij Uij = Irij we conclude V1,jV ⊤ 1,j + · · · + VkL,jV ⊤ kL,j = IbR. This implies that (V1,j . . . VkL,j) is an orthogonal matrix. Note that if we now parameterize A = LPR with the matrices Vij packed into R and Uij packed into L, then (V1,j . . . VkL,j) is exactly the j-th block matrix in R up to permutation of rows. Therefore, every block in R is an orthogonal matrix. Since we now proved that V ⊤ ij Vij = I, we can use same the derivation for the rows of A and conclude that blocks of L are also orthogonal. 5 GS(Pm+1, . . . , P1) matrices In this section we describe an the extension of GS-matrices that uses more than two block-diagonal matrices and show that with the right permutations choices GS-matrices are more effective than block butterfly matrices in forming dense matrices. Here by dense matrices we imply matrices that do not contain zero entries at all. Definition 5.1. A is said to be in GS(Pm+1, . . . , P1) if A = Pm+1 1 Y i=m (BiPi), where each matrix Bi is a block-diagonal matrix with ki blocks of size b1 i × b2 i , matrices Pi are permutation matrices and b1 i · ki = b2 i+1 · ki+1. Remark 1. Similarly to the case m = 2 described in Section 3, we may use orthogonal blocks in Bi, i = 1, . . . , m + 1 to obtain orthogonal matrices. However, it is not clear if an analog to Theorem 1 is correct in this case as well. Remark 2. For each of the classes of Block Butterfly matrices [Chen et al., 2022], Monarch matrices [Dao et al., 2022] and order-p Monarch matrices [Fu et al., 2024], there exist permutation matrices Pm+1, . . . , P1 such that GS(Pm+1, . . . P1) coincides with a respective class. Indeed, Monarch matrices have the form of alternating block-diagonal matrices and permutations with some specific size constraints and sparse matrices in the product of Block Butterfly matrices can be easily transformed to block-diagonal matrices with permutations of rows and columns. 5.1 Choosing permutation matrices We suggest using the following matrices with k = ki for Pi. Note that this is efficient for forming dense matrices as follows from the proof of Theorem 2. This is by contrast to the permutations used in [Fu et al., 2024] that are restricted to particular matrix sizes. Definition 5.2 ([Dao et al., 2022]). Let P(k,n) be a permutation matrix given by permutation σ on {0, 1, . . . , n −1}: σ(i) = (i mod k) · n k + i k . 5 Applying this permutation to a vector can be viewed as reshaping an input of size n into an k × n k matrix in a row-major order, transposing it, and then vectorizing the result back into a vector (again in row-major column). We provide several examples of such permutations in Figure 3. P(3,12) P(4,12) P(6,12) P(2,12) Figure 3: Illustraion of P(k,12) permutations for k ∈{3, 4, 6, 2}. 5.2 Comparison to block butterfly matrices and BOFT Block Butterfly matrices were introduced in [Chen et al., 2022] and are used to construct orthogonal matrices in the BOFT method. Block Butterfly matrix class is a special case of higher-order GSmatrices with ki = r and b1 i = b2 i = b = 2s and certain permutation choices. However, we argue that the choice of these permutations are sub-optimal for construction of dense matrix and using permutations from Definition 5.2 is more effective. When using block-diagonal matrices with r blocks, block butterfly matrices need 1 + ⌈log2(r)⌉matrices to construct a dense matrix. For GS-matrices we have the following result. Theorem 2. Let ki = r, b1 i = b2 i = b. Then using m = 1 + ⌈logb(r)⌉is sufficient for the class GS(PL, P(r,br), . . . , P(r,br), PR) to form a dense matrix for any PL, PR. Moreover, the choice of P2 = · · · = Pm = P(r,br) is optimal in the sense that all matrices from GS(Pm+1, . . . , P1) contain zero blocks for any integer m < 1 + ⌈logb(r)⌉and any permutations P1, . . . , Pm+1. Proof. See Appendix D. For example, let us consider a case of constructing a dense orthogonal matrix of the size 1024 × 1024. Suppose also that we use block matrices with block size 32. Constructing a dense matrix with Block Butterfly matrices requires 1 + log2(32) = 6 butterfly matrices, which leads to 6 × 323 parameters in the representation. GS(PL, P, PR) matrices with P = P(32,1024) only need two matrices to construct a dense matrix yielding 2 × 323 parameters. The GS(PL, P, PR) parametrization is also naturally more computationally efficient as fewer number of multiplications is both faster and requires less cached memory for activations. 6 Applications 6.1 Orthogonal fine-tuning with GS(PL, P, PR) (GSOFT) We utilize the pipeline of OFT and BOFT methods with the exception of parametrizing Q with orthogonal permuted GS(PL, P, PR) matrices. In particular, for parametrization of Q ∈Rd×d, we utilize the GS(P ⊤, P, I) class, i.e. Q = P ⊤LPR, where L = diag(L1, . . . Lr), Li ∈Rb×b, R = diag(R1, . . . , Rr), Ri ∈Rb×b. For consistency, we use the same notation for the number of blocks and block sizes as in BOFT and OFT methods. We use P(r,br) as a permutation matrix P. To enforce orthogonality, we parameterize each block in matrices L, R with the Cayley parametrization. We initialize Q as an identity matrix by initializing each block to be an identity matrix. Additional 6 techniques like magnitude scaling and multiplicative dropout that are used in OFT and BOFT can be utilized the same way in our method, though we only use scaling in our experiments. Note that likewise in OFT, BOFT weights of the matrix Q can be merged with the pretrained weight W producing no inference overhead. 6.2 Two-sided orthogonal fine-tuning (Double GSOFT) Consider SVD decomposition of a matrix W 0 = UΣV ⊤. Applying orthogonal fine-tuning, we get W ′ = (QU)ΣV ⊤, which is an SVD decomposition for the adapted weight W ′. This shows that we can only change left singular vectors U with the standard orthogonal fine-tuning paradigm. At the same time, the LoRA method modifies both matrices U and V . Moreover, recent papers [Meng et al., 2024, Li et al., 2023] show that initializing matrices A, B with singular vectors can additionally boost performance of LoRA. This motivates an extension of orthogonal fine-tuning method, that can adapt both matrices U and V . We introduce a simple approach that multiplies pre-trained weight matrices from both sides, rather than one. This method modifies forward pass from z = (W 0)⊤x to z = (QUW 0QV )⊤x Where QU and QV are parametrized as orthogonal GS-matrices. In cases where BOFT utilizes 5-6 matrices, we can leverage the fact that our method uses only 2 and adapt both sides while still using less matrices and trainable parameters than BOFT. 6.3 GS Orthogonal Convolutions Recall, that due to linearity of a multichannel convolution operation, we can express the convolution of tensor X ∈Rcin×h×w with a kernel L ∈Rcout×cin×k×k L ⋆X in terms of matrix multiplication [Singla and Feizi, 2021]: Y = L ⋆X ⇔ vec(Y ) =   L0,0 . . . L0,cin−1 ... ... ... Lcout−1,0 . . . Lcout−1,cin−1  vec(X), (2) where Li,j is doubly Toeplitz matrix, corresponding to convolution between i-th and j-th channels and vec(X) is a vectorization of tensor into a vector in a row-major order. Thus, the convolution is essentially a block matrix, where each block represents a standard convolution operation. Using this block interpretation (2), we may apply the concept of GS matrices to convolutional layers as well. Considering each convolution between channels as an element of our block matrix, we can set some of these blocks to zero, obtaining some additional structure. Thus, we can construct block matrix which has block-diagonal structure, corresponding to grouped convolution (further, in all equations we will denote it as GrConv). Then, defining ChShuffle as a permutation of channels, like in [Zhang et al., 2018], we obtain structure, which is similar to GSOFT, defined in Section 6: Y = GrConv2(ChShuffle2(GrConv1(ChShuffle1(X)))). (3) The proposed GS convolutional layer shuffles information between each pair of input channels and requires less parameters and FLOPs during computations. In this example we can also choose permutations of channels and change kernel size. This convolutional layer can be treated as GS(Pm+1, . . . , P1) matrix in vectorized view, that is why choosing permutations between convolutional layers is also very important for information transition properties. In Appendix F we explain the choice of ChShuffle operation. We can use the proposed layer to construct orthogonal convolutions (transformations with an orthogonal Jacobian matrix) similarly to skew orthogonal convolution (SOC) architecture, that uses Taylor expansion of a matrix exponential. One major downside of methods such as SOC and BCOP [Li et al., 2019] is that they require more time than basic convolution operation. For instance, in the SOC method, one layer requires multiple applications of convolution (6 convolutions per layer). In our framework, we propose a parametrization of a convolutional layer, in which imposing an orthogonality to convolutions has fewer number of FLOPs and parameters thanks to the usage of grouped convolutions. 7 Let us discuss in more details how SOC works and the way we modify it. In SOC, a convolutional filter is parametrized in the following way: L = M −ConvTranspose(M), where M ∈Rcin×cout×r×s is an arbitrary kernel and the ConvTranspose is the following operation: ConvTranspose(M)i,j,k,l = Mj,i,r−k−1,s−l−1 This parametrization of filter L makes the matrix from Equation 2 skew-symmetric. As matrix exponential of skew-symmetric matrix is an orthogonal matrix, in SOC the authors define convolution exponential operation, which is equivalent to matrix exponential in matrix-vector notation: Definition 6.1. [Singla and Feizi, 2021] Let X ∈Rc×h×w be an input tensor and L ∈Rc×c×k×k be a convolution kernel. Then, define convolution exponential L ⋆e X as follows: L ⋆e X = X + L ⋆X 1! + L ⋆2 X 2! + . . . where L ⋆i X is a convolution with kernel L applied i times consequently. As mentioned above, with proper initialization we get a convolutional layer with orthogonal Jacobian matrix. Using the parametrization of convolution layer from the Equation 3 and substituting there two grouped convolution exponentials (e.g. in our parametrization we have the same convolution exponential, but we have grouped convolution instead of basic one) with the parameterized kernel: Y = GrExpConv2(ChShuffle2(GrExpConv1(ChShuffle1(X)))) In our experiments we tried different layer architectures and we found that making kernel size of the second convolution equal to 1 speeds up our convolutional layer, maintaining quality metrics. Thus, if convolutional layer consists of two grouped convolutional exponentials, the second convolutional exponential has kernel_size = 1 × 1 7 Experiments All the experiments below were conducted on NVIDIA V100-SXM2-32Gb GPU. We ran all the experiments within ∼2000 GPU hours. Source code is available at: https://github.com/Skonor/group_and_shuffle 7.1 Natural language understanding We report result on the GLUE [Wang et al., 2018] benchmark with RoBERTa-base [Liu et al., 2019] model. Benchmark includes several classification tasks that evaluate general language understanding. We follow training settings of [Liu et al., 2024b, Zhang et al., 2023]. We apply adapters for all linear layers and only tune learning rate for all methods. Table 1 reports best results on the evaluation set from the whole training. LoRA, OFT and BOFT are implemented with PEFT library [Mangrulkar et al., 2022]. GSOFT method outperforms OFT, BOFT and also have a slight edge over LoRA. Note that even though skew-symmetric K theoretically matrix only requires approximately half the parameters of a full matrix, in practice it is parametrized as K = A−AT for the ease of computations. However, after fine-tuning, one can only save upper-triangular part of K. Doing this, orthogonal fine-tuning methods become approximately 2 times more efficient in terms of memory savings. 7.2 Subject-driven generation Subject-driven generation [Ruiz et al., 2023, Gal et al., 2022] is an important and challenging task in the field of generative modelling. Given several photos of a particular concept, we want to introduce it to the diffusion model so that we can generate this particular object in different scenes described by textual prompts. The main way to do this is to fine-tune the model. However, the large number of fine-tuning parameters together with the lack of training images make the model prone to overfitting, i.e. the model reconstructs the concept almost perfectly, but starts to ignore the textual prompt during generation. To solve this problem and stabilize the fine-tuning process, different lightweight parameterizations [Qiu et al., 2023, Liu et al., 2024b, Hu et al., 2022, Tewel et al., 2023, Han et al., 8 Table 1: Results on GLUE benchmark with RoBERTa-base model. We report Pearson correlation for STS-B, Matthew’s correlation for CoLA and accuracy for other tasks. # Params denotes number of trainable parameters Method # Params MNLI SST-2 CoLA QQP QNLI RTE MRPC STS-B ALL FT 125M 87.62 94.38 61.97 91.5 93.06 80.14 88.97 90.91 86.07 LoRAr=8 1.33M 87.82 95.07 64.02 90.97 92.81 81.95 88.73 90.84 86.53 OFTb=16 1.41M 87.21 95.07 64.37 90.6 92.48 79.78 89.95 90.71 86.27 BOFTm=2 b=8 1.42M 87.14 94.38 62.57 90.48 92.39 80.14 88.97 90.67 85.84 GSOFTb=8 1.42M 87.16 95.06 65.3 90.46 92.46 81.95 90.2 90.76 86.67 Table 2: Results on subject-driven generation. # Params denotes the number of training parameters in each parametrization. Training time is computed for 3000 iterations on a single GPU V100 in hours. Model Full LoRA BOFT GSOFT (Ours) Double GSOFT (Ours) rank r, m r r 4 32 128 32, 4 32, 6 16, 5 32 16 8 64 32 16 # Params 99.9M 0.8M 6.6M 26.6M 13.6M 20.4M 33.8M 6.8M 13.6M 27.1M 6.5M 13.0M 25.9M Training time 1.3 1.3 1.3 1.3 2.0 2.2 2.3 1.5 1.6 1.8 1.7 2.0 1.8 CLIP-I ↑ 0.805 0.805 0.819 0.813 0.803 0.796 0.789 0.805 0.803 0.783 0.815 0.802 0.783 CLIP-T ↑ 0.212 0.246 0.236 0.223 0.244 0.234 0.223 0.256 0.245 0.227 0.256 0.242 0.225 2023] and regularization techniques [Ruiz et al., 2023, Kumari et al., 2023] are widely used in this task. Therefore, we chose this setting to evaluate the effectiveness of the proposed orthogonal parameterization compared to other approaches. We use StableDiffusion [Rombach et al., 2022] and the Dreambooth [Ruiz et al., 2023] dataset for all our experiments. The following parameterizations were considered as baselines in this task: full (q, k, v and out.0 layers in all cross- and self- attentions of the UNet are trained), LoRA [Hu et al., 2022] and BOFT [Liu et al., 2024b] applied to the same layer. We use our GSOFT parameterization and a two-sided orthogonal GSOFT (Double GSOFT) applied to the same layers as baselines. For a more comprehensive comparison, we consider different hyperparameters for the models, adjusting the total number of optimized parameters. More training and evaluation details can be found in Appendix E. CLIP image similarity, CLIP text similarity and visual comparison for this task are presented in Table 2 and Figure 4. As the results show, GSOFT and DoubleGSOFT are less prone to overfitting compared to the baselines. They show better alignment with text prompts while maintaining a high level of concept fidelity. Furthermore, both methods with optimal hyperparameters are more efficient than BOFT and comparable to LoRA and full parameterization in terms of training time. See Appendix E for more visual and quantitative comparison. 7.3 GS Orthogonal Convolutions Following [Singla and Feizi, 2021], we train LipConvnet-n on CIFAR-100 dataset. LipConvnet-n is 1-Lipschitz neural network, i.e. neural network with Lipschitz constant equal to 1, his property provides certified adversarial robustness. LipConvnet uses orthogonal convolutions and gradient preserving activations in order to maintain 1-Lipschitz property. LipConvnet-n architecture consists of 5 equal blocks, each having n 5 skew orthogonal convolutions, where the last convolution at each level downsamples image size. We replace the skew orthogonal convolution layer with the structured version using GSorthogonal convolutions and test it in the setting of [Singla and Feizi, 2021], using the same hyperparameters (learning rate, batch size and scheduler stable during testing). In layers where we have two GrExpConv, the second convolution has kernel size equal to 1. We also use a modified activation function (MaxMinPermuted instead of MaxMin), which uses different pairing of channels. This makes activations aligned with the ChShuffle operation and grouped convolutions. The choice of permutation for ChShuffle also slightly differs from permutations defind in Definition 5.2 because of the interplay between activations and convolutional layers. We provide definitions and intuition regarding activations and permutations for ChShuffle in Appenix F. 9 "a V* in a purple wizzard outfit" "a V* on a cobblestone street" LoRA BOFT "a purple V" GSOFT Concept Double GSOFT Figure 4: Subject-driven generation visual results on 3000 training iterations. Table 3: Results of training LipConvnet-15 architecture on CIFAR-100. (a, b) in “Groups” column denotes number of groups in two grouped exponential convolutions (with kernel sizes 3 and 1). (a, −) corresponds to only one GS orthogonal convolutional layer. Before each grouped layer with k groups use a ChShuffle operator. Conv. Layer # Params Groups Speedup Activation Accuracy Robust Accuracy SOC 24.1M 1 MaxMin 43.15% 29.18% GS-SOC 6.81M (4, -) 1.64 MaxMinPermuted 43.48% 29.26% GS-SOC 8.91M (4, 1) 1.21 MaxMinPermuted 43.42% 29.56% GS-SOC 7.86M (4, 2) 1.22 MaxMinPermuted 42.86% 28.98% GS-SOC 7.3M (4, 4) 1.23 MaxMinPermuted 42.75% 28.7% 8 Concluding remarks In this paper, we introduce a new class of structured matrices, called GS-matrices, build a structured orthogonal parametrization with them and use them in several domains within deep learning applications. However, we hope that our orthogonal parametrization can be adapted to different settings in future (including tasks outside of deep learning), as it makes orthogonal parametrizations less of a computational burden. GS-matrices without orthogonality constraints is another promising direction. 9 Limitations Although our method for orthogonal fine-tuning is faster than BOFT, it is still slower than LoRA during training. 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This problem sparked research in the direction of parameterefficient fine-tuning methods, including methods that focus on prompt tuning [Lester et al., 2021, Li and Liang, 2021] and adapter tuning (e.g. [Houlsby et al., 2019, Karimi Mahabadi et al., 2021]), which include LoRA [Hu et al., 2022] and its variations [Meng et al., 2024, Zhang et al., 2023, Liu et al., 2024a, Dettmers et al., 2024, Li et al., 2023], that inject learnable low-rank matrices as an additive injection to the weights of pretrained models. OFT [Qiu et al., 2023], BOFT [Liu et al., 2024b] and our method use similar approach to LoRA, but learn multiplicative injection rather than an additive one. Structured sparsity Structured sparsity is an approach that replaces dense weight layers with different structured ones, such as matrix factorizations or tensor decompositions in order to compress or speed-up models [Dao et al., 2022, Chen et al., 2022, Novikov et al., 2015, Lebedev et al., 2015]. Some of these techniques were also adapted to PEFT methods in works like [Karimi Mahabadi et al., 2021, Edalati et al., 2022, Yang et al., 2024] or BOFT [Liu et al., 2024b] method, that utilizes a variation of butterfly matrices as a parametrization for parameter-efficient orthogonal matrices, imposing orthogonality to each butterfly factor. See details in Section 2. Monarch matrices [Dao et al., 2022, Fu et al., 2024] are most relevant to our work as our proposed matrix class is their generalization that utilizes similar structure. Subject-driven generation The emergence of large text-to-image models [Ramesh et al., 2022, 2021, Saharia et al., 2022, Rombach et al., 2022] has propelled the advancement of personalized generation techniques in the research field. Customizing a text-to-image model to generate specific concepts based on multiple input images presents a key challenge. Various methods [Ruiz et al., 2023, Gal et al., 2022, Kumari et al., 2023, Han et al., 2023, Qiu et al., 2023, Zhou et al., 2023, Wei et al., 2023, Tewel et al., 2023] have been proposed to address this challenge, requiring either extensive fine-tuning of the model as a whole [Ruiz et al., 2023] or specific parts [Kumari et al., 2023] to accurately reconstruct concept-related training images. While this facilitates precise learning of the input concept, it also raises concerns regarding overfitting, potentially limiting the model’s flexibility in generating diverse outputs in response to different textual prompts. Efforts to mitigate overfitting and reduce computational burden have led to the development of lightweight parameterization techniques [Qiu et al., 2023, Liu et al., 2024b, Hu et al., 2022, Tewel et al., 2023, Han et al., 2023] such as those proposed among others. These methods aim to preserve editing capabilities while sacrificing some degree of concept fidelity. The primary objective is to identify parameterization strategies that enable high-quality concept learning without compromising the model’s ability to edit and generate variations of the concept. Our investigation indicates that the orthogonal parameterization approach we propose represents a significant step towards achieving this goal. Orthogonal transforms In papers [Vorontsov et al., 2017, Hyland and Rätsch, 2017, Helfrich et al., 2018] authors work in the setting of dense matrices and try to parameterize essentially the whole manifold of orthogonal matrices. For n × n matrices it requires computing inverses or exponential maps of n × n matrices at every optimization step. This takes O(n3) time, which can be computationally challenging for larger architectures. Moreover, such a parametrization utilizes O(n2) trainable parameters, which makes it inapplicable for PEFT (see the OFT paper [Qiu et al., 2023] for more details). Our proposed method is different in a sense that it provides a trade-off between expressivity (describing only a subset of the orthogonal manifold) and efficiency. In [Li et al., 2019, Singla et al., 2021] authors discuss main issues of bounding of Lipschitz constant of neural networks and provide Gradient-Norm-Preserving (GNP) architecture in order to avoid vanishing of gradients while bounding Lipschitz constant. The authors propose a specific convolutional layer (Block Convolutional Orthogonal Parametrization) which Jacobian is orthogonal, also providing orthogonal activations with Lipschitz constant equal to 1. These constraints guarantee that the norm of the gradient will not change through backward pass. In other works [Singla and Feizi, 2021, Singla et al., 2021] authors provide a modification of the orthogonal convolutions (Skew Orthogonal Convolution) in terms of hardware-efficiency. Authors provide neural network architecture where each layer is 1-Lipschitz and make a comparison between these two convolutional layers. The work [Li et al., 2019] was outperformed by the SOC method that we utilize (there is a comparison in the SOC method [Singla and Feizi, 2021]). In [Xu et al., 2022], the authors utilize periodicity of 15 convolution and their padding is not equivalent to a widely-used zero-padded convolution. Survey [Prach et al., 2024] suggests that [Xu et al., 2022] and other 1-Lipschitz architectures yield worse robustness metrics than SOC, which we use as a baseline in our paper. B Proof of Prop. 1 Proof. Let R′ = PR. R′ can be viewed as a block matrix with kL × kR blocks of sizes bL 2 × bR 2 . L can be viewed as a block matrix with kL × kL blocks from which only diagonal are non-zero. The A can be written in the following form:    A0,0 . . . A0,kR−1 ... ... ... AkL−1,0 . . . AkL−1,kR−1   =    L0 . . . 0 ... ... ... 0 . . . LkL−1       R′ 0,0 . . . R′ 0,kR−1 ... ... ... R′ kL−1,0 . . . R′ kL−1,kR−1   . Using block matrix product formulas, we get: Ak1,k2 = Lk1R′ k1,k2. We can now rewrite Lk1R′ k1,k2 product in terms of their columns and rows: Lk1R′ k1,k2 = l1 . . . lb2 L ·    r⊤ 1... r⊤ b2 L   = X t ltr⊤ t . (4) Columns of Lk1 are just vectors uj such that ⌊j kL ⌋= k1. Let us examine the rows of R′ k1,k2. Since R′ is a matrix formed by permuting the rows of block-diagonal matrix R, R′ k1,k2 can only contain rows that were in the Rk2 before permutation. Formally, this means that R′ k1,k2 can only contain vector-rows vT i such that ⌊i kR ⌋= k2. Additionally, rows after permutation should get into the k1-block row. That implies ⌊σ(i) kL ⌋= k1. Other rows of R′ k1,k2 are zero-rows. Notice that in (4) non-zero rows r⊤ t represented by v⊤ i will match exactly with columns uσ(i) that represent lt. Keeping only non-zero terms in P t ltr⊤ t gets us to the desired conclusion. C Comparison of Monarch matrices and GS-matrices GS(PL, P, Pr) class is inspired by Monarch matrices [Dao et al., 2022] and their primary goal is to introduce additional flexibility in the block structure of matrices L and R. Generalized Monarch matrices are parameterized as P1LP2R, where L and R are block-diagonal matrices and P1 and P2 are certain permutations defined in Definition 5.2. This resembles GS(P1, P2, I) matrix class, however in comparison monarch matrices have additional hard constraints on relation between kL and kR. Being more precise, Monarch matrices are a special case of GS(P1, P2, I) matrices with additional constraints kL = b1 R, kR = b2 L. Such constraints lead to several theoretical and practical limitations of Monarch matrices. From theoretical point of view, Monarch matrices can only describe permuted block matrices with blocks of ranks 1. In contrast GS-matrices with can describe matrices with different rank structure of blocks (including structures where rank of each block is equal to arbitrary r). From practical point of view, due to this constraint Monarch matrices are often unable to form a desirable block structure of matrices L and R. For demonstration of this phenomena, consider a case of square matrices with square blocks – the structure needed in Orthogonal fine-tuning paradigm. Formally, we have b1 L = b2 L = bL, b1 R = bR 2 = bR, m = n. Additional Monarch constraint would mean that bR = kL; bL = kR. This in turn means that kL · kR = n. As we can see, it makes impossible to stack two matrices with small number of blocks (say, 4) or large number of blocks, which is required in situations with low parameter budget. In contrast, GS parametrization allows for both of these structures, which we use in our experiments. Note, that the work [Fu et al., 2024] provides a slightly different definition for monarch matrices, introducing order-p Monarch matrices. These matrices are also a special case of GS class, however they are very restrictive as they can only parametrize matrices with both sides equal to ap for some integers a, p. 16 1 b b2 bk . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Figure 5: Demonstration of information transition through a block structure. Each node is connected to exactly b consecutive nodes from the next level. D Proof of Theorem 2 We use information transition framework from [Liu et al., 2024b], representing product of m sparse d × d matrices as an transmitting information in a grid with d × (m + 1) nodes. Edges between nodes j and i represent that element i, j in sparse matrix is non-zero. Element i, j from final matrix can only be non-zero if there exists a path from the j-th node from the right column to the i-th node in the left column (see Figure 5). Proof. Consider an information transmission graph for the matrix BiP(r,br). In this graph, the first node connects with b first edges, the second node connects with the edges from b + 1 to 2b and so on. Now consider a graph for the product of m such matrices. As shown in Figure 5, now each node from the first level has paths to bk unique nodes from the kth-th level. It means that using m = ⌈logb(d)⌉= ⌈logb(br)⌉= 1 + ⌈logb(r)⌉matrices is sufficient to reach all nodes and therefore form a dense matrix. Note that the number of paths for each node is always equal to bm regardless of permutation choice. This observation shows that it is impossible to reach d unique elements on the final level with m < 1 + ⌈logb(r)⌉. E Subject-driven generation Training details All the models are trained using Adam optimizer with batch size = 4, learning rate = 0.00002, betas = (0.9, 0.999) and weight decay = 0.01. The Stable Diffusion-2-base model is used for all experiments. Evaluation details We use the DreamBooth dataset for evaluation. The dataset contains 25 different contextual prompts for 30 various objects including pets, toys and furnishings. For each concept we generate 10 images per contextual prompt and 30 images per base prompt “a photo of an S”, resulting in 780 unique concept-prompt pairs and a total of 8400 images for fair evaluation. To measure concept fidelity, we use the average pairwise cosine similarity (IS) between CLIP ViTB/32 embeddings of real and generated images as in [Gal et al., 2022]. This means that the image similarity is calculated using only the base prompt, i.e. “a photo of an S*”. Higher values of this metric usually indicate better subject fidelity, while keeping this evaluation scene-independent. To evaluate the correspondence between generated images and contextual prompts (TS), the average cosine similarity 17 between CLIP ViTB/32 embeddings of the prompt and generated images [Ruiz et al., 2023, Gal et al., 2022]. Overfitting discussion Typically, the more parameters are trained, the more easily the model overfits. This results in higher image similarity and lower text similarity. In addition, if a model with a large number of training parameters is trained for a long time, the image similarity can often start to decrease as the model starts to collapse and artifacts start to appear. This sometimes leads to models with more trainable parameters having a worse score in both image and text similarity than the models with fewer ones. Models with fewer trainable parameters overfit less, but require longer training to capture the concept carefully, and usually have an upper limit on the maximum image similarity: at some point the increase in image similarity becomes small, while text similarity starts to decrease dramatically. Therefore, the very common result of a usual fine-tuning is either an overfitting with poor context preservation or an undertraining with poor concept fidelity. The orthogonal fine-tuning shows a different behavior. The model with less trainable parameters can be trained longer (GSOFT, OFT, BOFT) and capture the concept more carefully without artifacts. At the same time, it overfits less and shows higher text similarity. Additional results In Figures 6 we show a graphical representation of the metrics for 1000 and 3000 iterations. Examples of generation for different methods are presented in Figure 7, 8. F GS Orthogonal Convolution In this section, we provide some details and insights about the choice of the ChShuffle permutation and the activation function. In experiments, we apply the ChShuffle operation right before grouped convolutional layers. Stacking several layers of that form resembles higher-order GS-matrices, which motivates the usage of permutations from Definition 5.2 for optimal information transition (see Appendix D). However, in the LipConvnet architecture, the activation function can also shuffle information between channels. Thus, this additional shuffling of information can negatively affect our information transition properties. In the original SOC paper [Singla and Feizi, 2021], the authors use MaxMin activation, firstly proposed in [Anil et al., 2019]. Definition F.1. [Singla and Feizi, 2021] Given a feature tensor X ∈R2m×n×n, the MaxMin(X) activation of a tensor X is defined as follows: A = X:m,:,:, B = Xm:,:,:, MaxMin(X):m,:,: = max(A, B), MaxMin(X)m:,:,: = min(A, B). This activation shuffles information between different groups in convolution which harms performance of our experiments, as permutations that we use in ChShuffle become sub-optimal in terms of information transmission. Thus, we introduce a modification of MaxMin activation, that splits channels into pairs in a different way. Rather than constructing pairs from different halves of input 0.22 0.24 0.26 TS 0.79 0.80 0.81 0.82 BaseIS num steps = 1000 0.22 0.24 0.26 TS num steps = 3000 LoRA rank=4 LoRA rank=32 LoRA rank=128 GSOFT r=8 GSOFT r=16 GSOFT r=32 Double GSOFT r=16 Double GSOFT r=32 Double GSOFT r=64 BOFT r=16, m=5 BOFT r=32, m=6 BOFT r=32, m=4 Full Figure 6: Image and text similarity visualisation for different methods on subject-driven generation. 18 Figure 7: Subject-driven generation visual results on 3000 training iterations. tensor, we use neighboring channels for forming of pairs (first channel pairs with second, third with fourth and so on). With this modification information does not transfer between groups during activations, which enables more optimal information transmission in-between layers with ChShuffle operator. In further experiments we denote this activation function as MaxMinPermuted and define it below: Definition F.2. Given a feature map X ∈R2m×n×n. MaxMinPermuted(X) is defined as follows: A = X::2,:,:, B = X1::2,:,:, MaxMinPermuted(X)::2,:,: = max(A, B), MaxMinPermuted(X)1::2,:,: = min(A, B) However, we also empirically find that it is crucial for the channels that interact within activations functions to also interact during convolutions. This means that they should always stay in the same group. This motivates us to use a slightly different permutation for the ChShuffle operation, which permutes channels in pairs. We use the following permutation σ(i)paired (k,n) = i 2 mod k · n k + 2 · i 2k + (i mod 2) 19 Figure 8: Subject-driven generation visual results on 1000 training iterations. This permutation can be seen as an adaptation of P(k,n) that operates on pairs of channels instead of single channels. This permutation is also optimal in terms of information transition. We call this permutation “paired”. Using this paired permutation as a ChShuffle with our modified activation saves connection between pairs while also transmitting information in the most efficient way. We provide the results of comparison of approaches with activations and permutations in Table 4. 20 Table 4: Comparison of activations on LipConvnet-15 architecture and CIFAR-100. (a, b) in “Groups” column denotes that we have two grouped exponential convolutions (the first one with kernel_size = 3, the second with kernel_size = 1). If b is not mentioned, we have only one GS orthogonal convolutional layer. Conv. Layer # Params Groups Speedup Activation Permutation Accuracy Robust Accuracy SOC 24.1M 1 MaxMin 43.15% 29.18% GS-SOC 6.81M (4, -) 1.64 MaxMinPermuted paired 43.48% 29.26% GS-SOC 6.81M (4, -) 1.64 MaxMinPermuted not paired 40.46% 26.18% GS-SOC 6.81M (4, -) 1.64 MaxMin paired 37.99% 24.19% GS-SOC 6.81M (4, -) 1.64 MaxMin not paired 39.72% 25.96% GS-SOC 8.91M (4, 1) 1.21 MaxMinPermuted paired 43.42% 29.56% GS-SOC 8.91M (4, 1) 1.21 MaxMinPermuted not paired 40.15% 26.4% GS-SOC 8.91M (4, 1) 1.21 MaxMin paired 40.3% 26.74% GS-SOC 8.91M (4, 1) 1.21 MaxMin not paired 41.7% 27.66% GS-SOC 7.86M (4, 2) 1.22 MaxMinPermuted paired 42.86% 28.98% GS-SOC 7.86M (4, 2) 1.22 MaxMinPermuted not paired 41.13% 27.53% GS-SOC 7.86M (4, 2) 1.22 MaxMin paired 41.55% 27.45% GS-SOC 7.86M (4, 2) 1.22 MaxMin not paired 41.25% 27.29% GS-SOC 7.3M (4, 4) 1.23 MaxMinPermuted paired 42.75% 28.7% GS-SOC 7.3M (4, 4) 1.23 MaxMinPermuted not paired 38.93% 25.59% GS-SOC 7.3M (4, 4) 1.23 MaxMin paired 40.34% 27.06% GS-SOC 7.3M (4, 4) 1.23 MaxMin not paired 41.57% 27.48% It can be seen that using “paired” permutation used with MinMaxPermuted activation significantly improves quality metrics. 21 NeurIPS Paper Checklist 1. 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4,431	Graph Learning for Numeric Planning Dillon Z. Chen1,2 Sylvie Thiébaux1,2 1LAAS-CNRS, University of Toulouse 2The Australian National University {dillon.chen,sylvie.thiebaux}@laas.fr Abstract Graph learning is naturally well suited for use in symbolic, object-centric planning due to its ability to exploit relational structures exhibited in planning domains and to take as input planning instances with arbitrary numbers of objects. Numeric planning is an extension of symbolic planning in which states may now also exhibit numeric variables. In this work, we propose data-efficient and interpretable machine learning models for learning to solve numeric planning tasks. This involves constructing a new graph kernel for graphs with both continuous and categorical attributes, as well as new optimisation methods for learning heuristic functions for numeric planning. Experiments show that our graph kernels are vastly more efficient and generalise better than graph neural networks for numeric planning, and also yield competitive coverage performance compared to domain-independent numeric planners. Code is available at https://github.com/DillonZChen/goose 1 Introduction Planning requires long range reasoning over combinatorially large state spaces. Numeric planning is an extension of classical planning in which states have numeric variables and the underlying transition system is built from inequality conditions and assignments over arithmetic expressions of such variables. It was formalised in PDDL 2.1 [FL03] and is undecidable in the general case [Hel02] which makes it more difficult than classical planning which is PSPACE-complete [Byl94]. Numeric planning is a well-established problem in the symbolic AI community and exhibits significant research effort [CCFL13, IM17, SHTR20, KSP+22, KSB23, SKB23], but this expressivity result implies that building a general, scalable numeric planner is a challenging problem. Learning for Planning (L4P) is a research direction which focuses on learning to solve problems from a specified domain in an automated supervised manner [TTTX20, STT20, FGT+22, KS21, SBG22, SBG23, MLTK23, CTT24a, SDS+24, RTG+24, APK24]. Planning tasks in L4P are assumed to exhibit a factored, symbolic representation, which allow us to generate training data in a matter of seconds from easy to solve tasks with a domain-independent planner. We can then learn domain knowledge in a supervised manner that scales planners to significantly larger planning tasks. This is in contrast to Reinforcement Learning where agents do not require access to well-defined models but spend significant amounts of time exploring and learning from rewards [SB98]. Regardless, several works have showed that encoding or learning symbolic models for sequential decision making reasoning and embodied AI tasks [LCZ+21, ZYP+22, LSS+22, SCK+23, KVS+23, LPM23] provided better performance and transparency over end-to-end reinforcement learning methods. Furthermore, it was shown recently that classical ML methods are much better suited for L4P than deep learning methods for symbolic planning [CTT24b] as they (1) can generalise well from small training data, (2) are orders of magnitude more efficient to train and evaluate than deep learning methods, which is important in time sensitive tasks such as planning, and (3) have interpretable features to understand what is being learned. In this paper we study whether this fact carries over to Learning for Numeric Planning (L4NP) [WT24] which now requires reasoning over logic and arithmetic. It is reasonable to think that because neural networks are function approximators, they may offer better reasoning capabilities over numbers than 38th Conference on Neural Information Processing Systems (NeurIPS 2024). A B C D E F initial state i j k B A goal condition i j k A B D i j on(a,i) on(a,b) on(b,i) on(b,j) on(d,b) capacity(i) capacity(j) G G′              0 1 0 . . . 0 0 0 0 0 1 . . . 0 0 0 ... 0 0 0 . . . 1 2 0 0 0 0 . . . 1 0 0 ... 1 0 0 . . . 0 0 0 1 0 0 . . . 0 0 0              x GNNθ(G′) w⊤x hGNN cost hGNN rank hCCWLF cost hCCWLF rank Defn. 3.1 Sec. 4 Sec. 3 Eq. 1 SVR Eq. 2 MSE (a) (b) (c) (d) Figure 1: The GOOSE framework for learning heuristic functions for numeric planning. Cyan colours indicate components that are influenced by the training phase. (a) A numeric planning state and goal condition is encoded into a graph G via the νILG representation defined in Defn. 3.1. (b) Graphs are either embedded into vectors x in Euclidean space with the CCWL kernel from Sec. 3 or transformed into a graph G′ with a real-valued matrix representing node features as inputs into GNNs described in Sec. 4. (c) Features x are fed into a linear model, whereas transformed graphs G′ are fed into GNNs. (d) Linear models are either trained by the ranking formulation in Eq. 1 or by Support Vector Regression (SVR) with a linear kernel. GNN models are either trained by the ranking formulation in Eq. 2 or by backpropagation minimising the mean squared error (MSE) loss. just symbols alone. In this paper, we describe the GOOSE1 framework with classical ML and deep learning configurations for learning heuristic or value functions for use with search in L4NP. Fig. 1 illustrates the GOOSE framework and we outline our contributions as follows. • We introduce a new graph representation of numeric planning tasks for use with classical and deep graph learning methods, namely graph kernels and graph neural networks, respectively. • We extend the WL kernel [SSVL+11] to handle graphs with both continuous and categorical attributes in a meaningful way which we call the CCWL kernel. • We introduce new ranking formulations [GKL16, CEKP23, HTT+24] for learning heuristic or value functions with linear programs. The structure of the remainder of the paper is as follows. In Sec. 2, we provide the necessary formalism and background for numeric planning, as well as relevant notation. In Sec. 3, we introduce a new graph encoding νILG and CCWL kernel for generating features for numeric planning tasks. In Sec. 4, we introduce a deep learning architecture for L4NP using graph neural networks. In Sec. 5, we describe optimisation methods for L4NP, involving a new ranking formulation for learning heuristic functions. In Sec. 6, we describe our experimental setup and results. Related work is discussed in Sec. B in the appendix. We conclude the paper with final comments in Sec. 8. 2 Background Numeric Planning Task. A numeric planning task can be viewed as a compact representation of a deterministic, goal-conditioned Markov Decision Process with the use of predicate logic and relational numeric variables. A majority of the remainder of this section formalises the necessary components of a numeric planning task we use in the paper. A numeric planning task [FL03] is given by a tuple Π = ⟨Xp, Xn, A, s0, G⟩where Xp is a finite set of propositional variables with domain {⊤, ⊥} and Xn is a finite set of numeric variables with domain R. Let X = Xp ∪Xn denote the set of state variables, where a state is a total assignment of values the propositional and numeric variables. The variables implicitly induce a possibly infinite set of states S, where s0 is the initial state. A propositional condition is a positive (resp. negative) literal x = ⊤(resp. ⊥) for some propositional variable x ∈Xp, and a numeric condition has the form ξ ⊵0 where ξ is an arithmetic expression over numeric variables and ⊵∈{≥, >, =}. We write [x]s (resp. [ξ]s) for the value of a state variable x (resp. expression ξ) in state s, and V (ξ) for the set of numeric state variables in ξ. A state s satisfies a set of conditions (i.e. a set of propositional and numeric conditions) if each condition in the set evaluates to true given the values of the state variables in s. The goal G is a set of conditions and we write Gp (resp. Gn) for the subset of propositional (resp. numeric) goal conditions. 1Graphs Optimised fOr Search Evaluation 2 The set A contains a finite number of actions, each consisting of preconditions and effects. Action preconditions pre(a) are sets of conditions, and action effects assign Boolean values to propositional variables and assign the value of an arithmetic expression to numeric variables. An action a is applicable in a state s if s satisfies pre(a), in which case its successor a(s) is the state where the effects eff(a) are applied to the state variables in s. If a is not applicable in s, we set a(s) = s⊥̸∈S. Each action a has a cost c(a) given by an arithmetic expression. A plan for a numeric planning task is a sequence of actions π = a1, . . . , an such that si = ai(si−1) ̸= s⊥for all 1 ≤i ≤n and sn satisfies G; we call s0, s1, . . . , sn the plan trace of the plan. The plan length \|π\| and the plan cost are the number of actions in the plan, and the sum of their cost, respectively. A plan is optimal if it has the minimum cost among all plans. A numeric planning task is solvable if there exists a plan for it, and is unsolvable otherwise. A state s is a deadend if the task with the initial state replaced with s is unsolvable. Satisficing planning refers to the problem of finding a plan if it exists, or proving that the problem is unsolvable. Optimal planning refers to the problem of finding an optimal plan if it exists, or proving that the problem is unsolvable. Lifted representation. Numeric planning tasks can be compactly encoded in a lifted representation ⟨O, Σp, Σf, Σa, A, s0, G⟩whereby state variables are derived from a set of predicates, functions, and objects. Formally Σp and Σf are sets of predicate and function symbols, respectively. Each symbol σ ∈Σp ∪Σf, has an arity nσ ∈N ∪{0} which depends on σ. Predicates and functions take the form p(x1, . . . , xnp) and f(x1, . . . , xnf ), respectively, where the xis are their arguments. Given the set O of objects, the propositional and numeric variables are obtained by substituting objects for the arguments of the predicates and functions, resulting in the grounded form p(o1, . . . , onp) and f(o1, . . . , onf ), respectively, where the ois are objects. Similarly, actions can be represented in a lifted form via a set Σa of action symbols and a set A of action schemata mapping action symbols to their lifted precondition and effect definitions in terms of predicates and functions. Grounding the set of action schemata results in the set of actions A of the planning task. Details are not needed to understand this paper. A domain is a set of numeric planning tasks sharing the same set of Σp, Σf, Σa, and A, and may have constant objects, objects which are shared across all tasks in the domain. Example: Capacity Constrained Blocksworld. To help digest some of the definitions of numeric planning, we provide an example with a planning domain we call Capacity Constrained Blocksworld (ccBlocksworld). It is an extension of the original Blocksworld domain in which state consists of towers of blocks and the objective is to stack and unstack blocks to achieve a goal configuration. It is also a special case of the Hydraulic Blocksworld domain for planning with state constraints, in which blocks are placed on top of pistons which rise or fall depending on the configurations of other pistons [HIR+18]. In ccBlocksworld, we have a maximum number of tower locations, and each tower has a base limited by the number of blocks it can hold. To model this domain in the lifted representation, we retain the predicate on(x, y) from the original Blocksworld, which indicates block x is on another block or base y. Next, we also introduce the function capacity(z) which denotes the remaining number of blocks that are allowed to be placed on base z. The numeric variables instantiated from capacity may increase or decrease depending on whether blocks are unstacked from the tower or stacked on top of it. Action schemata preconditions constrain whether a block can be placed on a tower with base that has reached its capacity limit or not. The leftmost figure in Fig. 2 illustrates an example of a ccBlocksworld problem with an initial state and goal condition. We refer to the Sec. A of the appendix for the complete state representation of the problem as well as its PDDL encoding. Heuristics and Greedy Best First Search. State-of-the-art methods for both satisficing and optimal numeric planning [SHTR20, KSP+22, CT24] employ some variant of heuristic search. A heuristic function maps a state s to R ∪{∞} representing an estimate of the cost to reach the goal from the current state, where a value of ∞estimates that s is a deadend. The optimal heuristic h∗maps a state to the cost of an optimal plan if it exists, and ∞otherwise. The Greedy Best First Search (GBFS) algorithm consists of a priority queue initialised with the initial state as the only element, and a main loop which performs the following steps while the queue is non-empty: (1) pop a state s with the lowest heuristic value and some tie-breaking criterion from the queue, (2) generate the successors of s via all applicable actions, and (3) check if a successor s′ is a goal, in which case terminate with the plan traced back from s′, and otherwise add s′ to the queue if it has not been seen before. The algorithm determines a problem is unsolvable if the main loop completes, in which case the problem has finitely many states of which all have been seen. 3 Graph and other notations. Let G = ⟨V, E, Fcat, Fcon, L⟩denote a graph with nodes V, undirected edges E ⊆ V 2 , categorical node features Fcat : V →ΣV where ΣV is a finite set, continuous node features Fcon : V →Rd with d ∈N, and edge labels (categorical edge features) L : E →ΣE where ΣE is a finite set. The neighbourhood of a node u ∈V in a graph with respect to an edge label ι is defined by Nι(u) = {v \| ∃e ∈E, s.t. e = ⟨u, v⟩= ⟨v, u⟩∧L(e) = ι} . We use ∥to denote the concatenation operator for vectors, and [[N]] to denote {1, . . . , N}. 3 Relational features for numeric planning In this section, we describe an automatic method for generating embeddings for numeric planning tasks that may be used for any downstream inference model. The method is an extension of the feature generation method for classical planning [CTT24b] and consists of two main steps: (1) generating a graph representation of a planning task, and (2) running a variant of the WL-algorithm for generating features for the graph [SSVL+11]. Extending the first step of the method is simple as it is easy to extend the graph encoding to capture numeric information of the task. This is done in Sec. 3.1 where we introduce the Numeric Instance Learning Graph (νILG) representation for numeric planning tasks. The second step is more difficult as we require constructing a WL-algorithm variant that can handle both categorical and continuous nodes features in a meaningful way for numeric planning. This is where we introduce the CCWL algorithm in Sec. 3.2 that handle such node features. Thus, we can generate features for numeric planning tasks by first converting them into the νILG representation, and then running the CCWL algorithm on them. 3.1 Graph encoding of numeric planning tasks We begin by describing our graph encoding of a planning task, namely the Numeric Instance Learning Graph (νILG). Similarly to the classical case, the graph representation does not encode the transition model of the planning task nor requires grounding all possible variables in the planning task. Thus, our encoding only requires a first-order representation of states, and therefore applies to problems whose transition model is unknown such as in model-free reinforcement learning. We begin with a descriptive definition of the graph with an example Fig. 2 illustrating a subgraph of the νILG representation of the example ccBlocksworld problem. In the figure, the nodes in the graph represent the objects (light blue), propositional variables true in the state (green), numeric variables (red), propositional goals (yellow) and numeric goals (not present in the example) of the problem. Blue (resp. orange) edges connect object nodes to goal and variable nodes where the object is instantiated in the first (resp. second) argument of the corresponding node variable or condition. We provide the formal definition below. Let Xp(s) denote the set of propositional variables that are true in s, Xn(s) the set of numeric variables, and X(s) = Xp(s) ∪Xn(s). Definition 3.1 (Numeric Instance Learning Graph). The Numeric Instance Learning Graph (νILG) of a numeric planning task in the lifted representation Π = ⟨O, Σp, Σf, Σa, A, s0, G⟩is a graph G = ⟨V, E, Fcat, Fcon, L⟩with • nodes V = O ∪X(s0) ∪G, where we assume w.l.o.g. that V (g) ⊆X(s0) for all g ∈Gn, • edges E = S p=σ(o1,...,onσ )∈X(s0)∪Gp{⟨p, oi⟩\| i ∈[[nσ]]} ∪S ξ⊵0∈Gn {⟨ξ, v⟩\| v ∈V (g)}, • categorical node features Fcat : V →ΣV with Fcat(u) =    OBJ(u) if u ∈O FUNC(u) if u ∈Xn(s0) (COMP(u), ACH(u)) if u ∈Gn    (PRED(u), achieved_propositional_goal) if u ∈Xp(s0) ∩Gp (PRED(u), unachieved_propositional_goal) if u ∈Gp \ Xp(s0) (PRED(u), achieved_propositional_nongoal) if u ∈Xp(s0) \ Gp where OBJ(u) = u if u is a constant object and object otherwise, PRED(u)/FUNC(u) returns the predicate/function symbol from which a proposition/fluent was instantiated from, COMP(u) ∈{≥, >, =} encodes the comparator type of the numeric goal condition u, and ACH(u) ∈{unachieved_numeric_goal, achieved_numeric_goal} encodes whether s0 satisfies u, • continuous node features Fcon : V →R where Fcon(u) = [u]s0 for nodes u ∈Xn(s0), Fcon(u) = [ξ]s0 for nodes u = ξ ⊵0 ∈Gn with [ξ]s0̸⊵0, and Fcon(u) = 0 otherwise, and • edge labels L : E →ΣE where for edges of the form e = ⟨p, oi⟩, we have L(e) = i, and otherwise for edges e = ⟨ξ, v⟩, we have L(e) = 0. 4 A B C D E F initial state i j k B A goal condition i j k A B D i j on(a,i) on(a,b) on(b,i) on(b,j) on(d,b) capacity(i) capacity(j) OH(obj) OH(on,a_p_n) OH(on,u_p_g) . . . OH(capacity) r1= num. value r2= goal error                   on(a, x) 0 1 0 . . . 0 0 0 on(a, b) 0 0 1 . . . 0 0 0 . . . . . . capacity(i) 0 0 0 . . . 1 2 0 capacity(j) 0 0 0 . . . 1 0 0 . . . . . . i 1 0 0 . . . 0 0 0 j 1 0 0 . . . 0 0 0 Figure 2: An example ccBlocksworld task where each base has capacity 3 (left), a subgraph of its ILG representation (middle), and the matrix representation of the node features of the ILG (right). In general, given a domain with predicate and function symbols Σp and Σn, we have that there are \|ΣV\| = 5 + 3 \|Σp\| + \|Σn\| + \|constant_objects\| categorical node features representing the semantics of a node. Continuous node features indicate the value of numeric variables and the error of the expression in s0 of unachieved numeric goals, and are set to zero for any other node. 3.2 The CCWL algorithm for numeric planning The WL algorithm [WL68] has been adapted to computing features for graphs with categorical node attributes by [SSVL+11]. A variant of the WL algorithm for graphs with continuous node attributes has been proposed by [TGL+19] for the purpose of computing kernels with the Wasserstein distance between graph embeddings. However, the graph embeddings themselves are not invariant to the order of graphs in the nodes. Furthermore, from [CTT24b], non-linear kernels result in poorer generalisation compared to linear models in the context of L4P due to overfitting to the range of training targets. Morris et al. [MKKM16] constructed kernels for continuous node attributes by hashing Euclidean embeddings into categorical features but such a method loses the semantic meaning of numbers. Thus, we propose a new variant of the WL algorithm for graphs with both categorical and continuous node attributes for generating graph embeddings (CCWL algorithm). This algorithm is summarised in Alg. 1 and also depicted in Fig. 3. Algorithm 1: CCWL algorithm Data: A graph G = ⟨V, E, Fcat, Fcon, L⟩with continuous and categorical attributes, a deterministic and injective HASH function, allowed colours C = [[\|C\|]], a pooling function POOL, and number of CCWL iterations L. Result: Feature vector of size R(1+d)\|C\|. 1 κ0(v) ←Fcat(v), ∀v ∈V 2 for j ∈[[L]] do for v ∈V do 3 κj(v) ←HASH κj−1(v), S ι∈ΣE{(κj−1(u), ι) \| u ∈Nι(v)} 4 M ←S j∈{0}∪[[L]]{{κj(v) \| v ∈V}} 5 ⃗vcat ← COUNT(M, c1), . . . , COUNT(M, c\|C\|) 6 Si = v \| v ∈V s.t. ∃j ∈{0} ∪[[L]] , κj(v) = ci , ∀i ∈C 7 ⃗vcon ←[con(1) ∥. . . ∥con(\|C\|)], con(i) = POOLv∈Si(Fcon(v)) 8 return ⃗vcat ∥⃗vcon 1 1 2 1 [0.2, 0.4] [0.3, 1.3] [0.5, 1.5] [0.2, 0.4] iter 0 β α γ β 3 4 5 3 iter 1 HASH(1, {(1, α), (1, β)}) = 3 HASH(1, {(1, β), (2, γ)}) = 4 HASH(2, {(1, γ)}) = 5 con(1) = [0.7 2.1], con(2) = [0.5 1.5] con(3) = [0.4 0.8], con(4) = [0.3 1.3] [3 1 2 1 \| {z } ⃗vcat 0.7 2.1 0.5 1.5 0.4 0.8 0.3 1.3 \| {z } ⃗vcon ] Figure 3: CCWL with one iteration, POOL = P, and C = [4]. Lines 1–3 of Alg. 1 are the original steps of the WL algorithm for generating graph embeddings by iteratively refining categorical node features, which we call colours, with two differences. Firstly, we replaced the multi-set with a set in the input of the hashing function. This is because in planning, unseen colours arise from graphs with increasing degrees which occur for out-of-distribution testing problems of increasing size. This problem is limited by relaxing the hash input with a set, which trades expressivity for generalisation. Secondly, we make use of edge labels in the hashing function. Lines 4–5 collect the counts of allowed colours C seen during the main loop of the algorithm to generate the categorical feature vector in the form of a histogram. We assume by relabelling colours that C = [[\|C\|]]. Lines 6–7 generate features from pooling the continuous attributes from different 5 groups of nodes. More specifically, for each colour c ∈C, we find the set of nodes which have been assigned the colour c some time during the refinement process and pool the continuous attributes of these nodes. Thus, we have \|C\| pooled continuous feature vectors which we concatenate together. We note that this pooling and concatenation process is invariant to the order of nodes in a graph in contrast to the intermediate graph embeddings generated for Wasserstein WL graph kernels by Togninalli et al. [TGL+19]. The algorithm returns the concatenation of the categorical and continuous feature vectors as the final feature vector output for the graph in Line 8. We note that d = 1 when running CCWL on the νILG representation of a numeric planning task. We note that a drawback of the algorithm is that continuous attributes are not refined directly. This could be done by introducing one or more aggregation functions as parameters to the algorithm and refining continuous attributes by concatenating the aggregations of their neighbouring attributes. However, this method introduces an increase in the size of the continuous feature vector exponential in the number of layers, with base equal to the number of aggregation functions chosen. Moreover, we noted from informal experiments that this method led to overfitting of models to a large number of blended continuous features that do not have an obvious relation to the learning target. Assuming a constant time hashing function, the complexity of the CCWL algorithm is O(nL(δ + d)) where n = \|V \| of the input graph, δ = maxu∈V P ι∈L \|Nι(u)\| is the degree of the graph, d is the dimension of the continuous node attributes, and L is the number of layers. The main computation comes from Line 3 which is performed nL times and the hashing function takes an input of size δ. Collecting the categorical feature vector takes the same time, while collecting the continuous feature vector takes O(nLd) time. For reasonably sized d ≲δ, as in the case of νILG where d = 1, this is the same complexity as the original WL algorithm for generating graph features, which is O(nLδ). 4 Relational neural networks for numeric planning Deep learning architectures such as graph neural networks (GNNs) [SGT+09, GSR+17] benefit in generating latent representations automatically with backpropagation when trained end-toend [LBH15]. GNNs also benefit from being able to train and evaluate on arbitrary sized graphs. However, it is generally understood that the expressive power of GNNs is limited by the WL-algorithm and counting logics with two variables [XHLJ19, BKM+20]. This result translates to the impossibility result of GNNs not being able to learn features that can work well for arbitrary planning domains [SBG22, CTT24a]. Nevertheless, their application to numeric planning tasks, in which both logical and numeric reasoning is required, is less well understood. Thus, we still propose GNNs as an additional baseline for L4NP and empirically evaluate their performance for numeric planning in Sec. 6. For our GNN architecture, we perform a transformation on the node features of the νILG from 3.1 as input for GNNs that can handle edge labels. More specifically, given a νILG G = ⟨V, E, Fcat : V →ΣV, Fcon →R, L⟩, we construct a new graph G′ with continuous node attributes X : V →R\|ΣV\|+2 defined by X(u) = OH(Fcat(u)) ∥[r1, r2], where OH(Fcat(u)) ∈ {0, 1}\|ΣV\| ⊆R\|ΣV\| denotes a one-hot encoding of the categorical node feature of u, and r1 denotes the numerical value of numeric variable nodes defined by r1 = [u]s0 if u ∈Xn(s0) and r1 = 0 otherwise, and r2 denotes the goal error for numeric goal nodes defined by r2 = [u]s0 if u ∈Gn and r2 = 0 otherwise. We denote the νILG for GNNs by ⟨V, E, X, L⟩with notation for categorical features removed. Thus, we can use this graph encoding of numeric planning tasks as input into any downstream GNN that can handle edge labels or features. Fig. 2 illustrates the node feature matrix representation of the νILG encoding of a ccBlocksworld task for input to a GNN. Each row represents a node in the graph, with columns representing the semantics of the node as well as the value of the numeric variables in the state and error of numeric goal nodes. We note however, that the ccBlocksworld example does not have any numeric goals and thus the last column is zero for all entries. 5 Optimisation formulations for learning heuristic functions In this section, we describe two optimisation methods used for learning heuristic functions from training data, namely by minimising cost-to-go estimate error and ranking estimate error. Fig. 4 illustrates examples of learned heuristic functions on states of a planning task when trained to zero loss with both the cost-to-go and ranking formulations. We assume that training data for our models consist of a set of numeric planning tasks Π1, . . . , Πn with corresponding optimal plans π1, . . . , πn. 6 3 2 4 1 1 2 0 -1 5 10 5 5 10 0 5 -10 -5 5 > > > ≥ ≥ ≥ ≥ ≥ Figure 4: Examples of heuristic functions that achieve 0 loss when optimising cost-to-go (left) and ranking (right) on an optimal plan. Coloured nodes indicate states on the optimal plan with the goal state indicated by a double circle. Edges indicate successors of a node. A cost-to-go heuristic can achieve 0 loss on the plan trace but may not generalise correctly to state successors. A ranking heuristic does not need represent correct cost-to-go values and only need to satisfy ranking constraints. GBFS will return a plan in linear time for the ranking heuristic here but not for the cost-to-go heuristic. We note that a numeric planning task can offer more training data by generating additional tasks and plans from different states in the state space of the task. Each plan is denoted πi = a(i) 1 , . . . , a(i) \|πi\| with plan trace s(i) 0 , s(i) 1 , . . . , s(i) \|πi\|. Each state s in a plan trace induces a new planning task by replacing s0 with s of the original task, with which we can construct graph or vector representations from our aforementioned models. Heuristic functions from cost-to-go estimates. We can use planning tasks and corresponding optimal plans as training data for learning a heuristic function representing the estimated cost-to-go to the plan. Each task and corresponding plan πi contributes training data s(i) j with targets h∗(s(i) j ) for each state s(i) j in the plan trace of πi. Then given an estimator H, we may try to find weights that minimise the mean squared error (MSE) L(θ) = 1 N Pn i=1 P\|πi\| j=0 h∗(s(i) j ) −Hθ(s(i) j ) 2 where N is the normalisation constant and Hθ denotes the estimator with weights θ. Heuristic functions from ranking estimates. The MSE loss is a simple but naive method for training a heuristic function. Various researchers have instead proposed to use the concept of ranking to learn heuristic functions [GKL16, CEKP23, HTT+24]. However, a drawback of the formulation of the ranking optimisation of previous works is that a state in a plan trace is marked as strictly better as its siblings when it could be the case that the siblings may have the same h∗value. Furthermore, the formulation in [CEKP23] scales quadratically in the plan trace. We offer a novel ranking optimisation criterion that (1) fixes the problem of siblings being misclassified and (2) also results in a sparse model. We also offer a corresponding differentiable loss function for use with any end-to-end model. Our first ranking formulation requires solving an LP as the optimisation problem, similarly to [FCGP19] but only using states from the plan trace, whereas the latter work uses states from the entire state space of the problem. It can also be viewed as an LP encoding of the formulation by Garrett et al. [GKL16] but fixing the problem of misrepresented siblings and learning sparse weights. Let SUCCS(s) denote the set of successors of the state s in a planning task by applying all applicable actions at s. Hence the set of siblings of state s(i) j in Πi’s state space is SIBLINGS(s(i) j ) = SUCCS(s(i) j−1) \ {s(i) j }. Let φ denote our feature generation function with φ(s) ∈Rd for any state s. Then we can define our optimisation problem as a linear program defined by min w,z X i,j,k zi,j,k + ∥w∥1 s.t. zi,j,k ≥0, ∀i, j, k (1) w⊤(φ(s(i) j−1) −φ(s(i) j )) ≥cost(a(i) j ) −zi,j,0 ∀i ∈[[n]] , j ∈[[\|πi\|]] w⊤(φ(sα) −φ(s(i) j )) ≥−zi,j,α ∀i ∈[[n]] , j ∈[[\|πi\|]] , sα ∈SIBLINGS(s(i) j ). The vector w represents the weights our linear model aims to learn, and the nonnegative slack variables z model the soft inequality constraints representing the ranking of states. The optimisation problem is to minimise the the slack variables corresponding to the error of the constraints, and the ℓ1 norm of the weights to encourage sparsity. We next offer a differentiable loss function version of the previous model which we can use as a fair comparison when combining it with our GNN architecture in Sec. 4 compared to combining (1) with features generated in Sec. 3. The idea is to replace the slack variables with the max function: L(θ) = X i,j max 0, Hθ(s(i) j ) −Hθ(s(i) j−1) + c(a(i) j ) + X sα∈SIBLINGS(s(i) j ) max 0, Hθ(s(i) j ) −Hθ(sα) . (2) 7 Blocksworld (blocks) Childsnack (children) Ferry (cars) Miconic (passengers) Rovers (objectives) Satellite (directions) Spanner (spanners) Transport (packages) Domain 101 102 103 Number of objects 11 8 20 10 10 10 10 7 488 292 974 485 728 98 487 194 train test Blocksworld (35) Childsnack (74) Ferry (99) Miconic (98) Rovers (42) Satellite (56) Spanner (88) Transport (39) Domain 10 2 100 102 Time (s) Figure 5: Number of objects in training and testing problems (left) and distributions of training data generation time with number of training problems (right) per domain. Note the log scales. 6 Experiments 6.1 Numeric planning benchmarks We take 8 domains out of 10 domains from the International Planning Competition 2023 Learning Track (IPC-LT) [SSA23] and either convert them to equivalent numeric formulations, or introduce numeric variables to model extra features such as capacity constraints. The two domains from the IPC-LT that we do not convert into numeric domains are Floortile and Sokoban which do not have any benefit from compilation to a numeric representation nor exhibit any interesting features that can be modelled with numeric variables. The domains we considered from the IPC-LT are summarised in Fig. 5 alongside the sizes of training and testing tasks, and time to generate training data. Each domain consists of 90 testing problems and at most 99 small training problems for which the median time for generating an optimal training plan is less than a second and a few outliers taking more than a minute. We refer to the appendix for further details on the domains. 6.2 Experimental setup Training. As discussed in Sec. 5, we only consider optimal plans from small problems as training data. We compute them with the Numeric Fast Downward planner [AN17] using A∗search and the admissible hLMCUT heuristic [KSP+22], with a 30 minute timeout and 8GB main memory. We consider 4 model configurations. Firstly, we use CCWL features from Sec. 3 with Support Vector Regression and the linear dot product kernel to learn a linear model for cost-to-go estimation (hCCWLF cost ). Next, we use CCWL features in optimisation problem in (1) with CPLEX version 22.11 and a timeout of 600 seconds for ranking estimation (hCCWLF rank ). Both hCCWLF cost and hCCWLF rank models have allowed colours C in Alg. 1 given by all the refined colours seen during training. We also have cost-to-go (hGNN cost ) and ranking (hGNN rank ) estimation models using GNNs operating on νILG representations of planning tasks and optimised with the MSE loss function and (2), respectively. For the backbone GNN, we use a Relational Graph Convolution Network [SKB+18] but replacing the mean aggregation function with the element-wise max operator in the message-passing update step: h(l+1) u = σ(W0h(l) u + P ι∈ΣE maxv∈Nι(u) W(l) ι h(l) v ), where l denotes the GNN layer, σ is implemented with the leaky ReLU function, and W0 and W(l) ι are learnable weight matrices. Each GNN has a hidden dimension of 64, and is trained with the Adam optimiser [KB15] with an initial learning rate of 10−3 and batch size of 16. A scheduler reduces the training loss by a factor of 10 if loss does not improve after 10 epochs. Training then terminates if the learning rate falls below 10−5. Let L denote the iterations hyperparameter for CCWL models and number of layers for GNN models. Evaluation. We consider several numeric planners as baselines for benchmarking the effectiveness of learning. We first include hLMCUT as the only optimal planner baseline as it is also the training data generator but solves a more difficult problem of optimal planning compared to satisficing planning. We consider the Metric-FF planner (M-FF) [Hof03], and the hADD, hMRP, hMRP+hj and M(3h∥3n) configurations in the ENHSP planner [SHTR20, SSSG20, CT24]. We have that hADD and hMRP are planners that perform GBFS with a single heuristic only, while hMRP+hj and M(3h∥3n) use additional techniques (macro actions, multiple queues, and novelty heuristics) to boost planning performance. Our CCWL and GNN models are all used in single-queue GBFS with the learned heuristic function, with Numeric Fast Downward as the backend search implementation. All baselines and models are run on a single Intel Xeon Platinum 8268 (2.90 GHz) core with a 5 minute timeout for search and 8 Table 1: Coverage of numeric domain-independent, the new learning planners (hGNN cost , hGNN rank , hCCWLF cost , hCCWLF rank ) with L = 1, and the best learner configuration score on each domain (Best Learner). Higher values are better (↑), with the top three scores in each row except the rightmost entry indicated by the cell colour intensity. All planner configurations except hLMCUT opt are satisficing planners. Planner Baselines Learners (new) GBFS + heuristic Domain hLMCUT opt hMRP+hj M-FF M(3h∥3n) hMRP hADD hGNN cost hGNN rank hCCWLF cost hCCWLF rank Best Learner Blocksworld 6 18 9 23 19 16 18 24 22 19 29 Childsnack 20 49 14 53 25 20 17 22 22 90 90 Ferry 33 60 60 57 60 57 60 60 70 71 73 Miconic 30 68 65 61 64 51 63 64 90 90 90 Rovers 10 34 17 30 18 15 18 14 22 23 30 Satellite 18 38 24 29 21 23 19 14 23 16 26 Spanner 30 6 35 76 42 42 90 90 90 90 90 Transport 12 55 49 40 32 40 34 38 40 46 48 Σ 159 328 273 369 281 264 319 326 379 445 476 8GB of main memory. Tab. 1 summarises the coverage results of all considered planners on the benchmarks, with more details provided in the appendix. How do learning approaches compare to domain-independent numeric planners? From Tab. 1, we note that our best performing model with L = 1 is hCCWLF rank and outperforms all domainindependent planners for satisficing planning on 4 out of 8 domains. Increasing L to 2 brings hCCWLF rank to achieve the best coverage on Blocksworld. The domains which learners fall behind on are Rovers, Satellite and Transport, even when taking the best hyperparameter configuration. The former two are difficult as they require features more expressive than those generated by graph learning approaches to capture the semantics of reasoning required to solve the problems [SBG22], while the latter requires path finding which is not possible for learners with finite receptive fields [TTTX20]. These results hold for classical planning and thus also for our extension to numeric planning. Generally the best performing planner on a domain expands fewer nodes than the other planners. With regards to plan length, hCCWLF rank performs best for Blocksworld but is marginally worse than the best of the domain-independent numeric planners for Rovers, Satellite and Spanner. How do CCWL models compare to GNN models? From Tab. 1, we see that the CCWL models always have similar or better performance than the corresponding GNN models, when comparing cost-to-go and ranking estimates. The performance of a planners which use GBFS and a heuristic depend on the heuristic evaluation speed, in which more search can be done in the time limit, or the quality of the heuristic, in which search can be more informed. Fig. 8 in the appendix shows that GNN are generally at least an order of magnitude slower than CCWL models for heuristic evaluation due to performing intensive matrix operations. We note that GNN models are evaluated on CPUs and could be sped up with access GPUs. Fig. 6a illustrates the number of node expansions of GNN and CCWL models and we note that there is no clear winner between the two approaches across all domains, with the exception of hCCWLF rank generalising perfectly on Childsnack where other models could not. Thus, we can conclude with respect to planning efficiency that CCWL models generally outperform their GNN counterparts due to faster heuristic evaluation speeds, while generally both models have similar generalisation performance. How do ranking models compare to cost-to-go models? From Tab. 1, ranking models outperform cost-to-go models in total coverage. However, their performance is incomparable across domains even when looking at Fig. 6b with the exception of CCWL being able to achieve perfect performance on Childsnack. Nevertheless, on 8 domain-model pairs for L = 1, ranking models achieve strictly better coverage, while the converse is only true for 4 domain-model pairs. This suggests a bias favouring ranking models which can be explained by their advantages covered in Sec. 5, namely that 9 101 103 105 107 hccWL cost 101 103 105 107 hGNN cost expanded 101 103 105 107 hccWL rank 101 103 105 107 hGNN rank expanded 100 101 102 103 hccWL cost 100 101 102 103 hGNN cost plan_length 100 101 102 103 hccWL rank 100 101 102 103 hGNN rank plan_length (a) GNN (hGNN cost , hGNN rank ) vs. CCWL models (hCCWLF cost , hCCWLF rank ). 101 103 105 107 hGNN rank 101 103 105 107 hGNN cost expanded 101 103 105 107 hccWL rank 101 103 105 107 hccWL cost expanded 100 101 102 103 hGNN rank 100 101 102 103 hGNN cost plan_length 100 101 102 103 hccWL rank 100 101 102 103 hccWL cost plan_length (b) Cost-to-go (hGNN cost , hCCWLF cost ) vs. ranking heuristic models (hGNN rank , hCCWLF rank ). blocksworld childsnack ferry miconic rovers satellite spanner transport (c) Plot legend. Figure 6: Plot comparisons of expanded nodes and plan length of selected pairs of models with L = 1. A point (x, y) represents the metric of the models indicated on the x and y axis on the domain. Points on the top left (resp. bottom right) triangle favour the model on the x-axis (resp. y-axis). they implicitly use more training data by considering successor states of plan trace states, and have a larger solution space as they are not restricted to predicting an exact value. What is the effect of number of iterations for CCWL models and layers for GNNs? The hyperparameter L, which denotes the number of iterations (resp. layers) for CCWL (resp. GNN) models, generally plays an important role in planning performance. This is because increasing L improves model expressivity and reasoning capabilities, but comes at the cost of heuristic evaluation time and increased possibility of overfitting to the training data. From Tab. 2 in the appendix, we note that surprisingly for most domains and models L = 0 or L = 1 provides the best coverage, while increasing L rarely improves coverage. This suggests that heuristic evaluation time plays an important role in planning performance for domains that cannot be solved with the learner’s expressivity. 7 Limitations The setup of our work is limited to the assumption that the problems being solved can be explicitly represented in a symbolic language such as PDDL. The assumption of the existence of PDDL encodings of planning problems allows us to generate training data quickly with domain-independent numeric planners for supervised training. Furthermore, experiments and theoretical insights also show that our proposed techniques have room for improvement as there are still classes of numeric planning tasks with which our models cannot learn and generalise well in. 8 Conclusion We have proposed a new graph embedding algorithm, the CCWL algorithm, and optimisation criterion for learning heuristic functions for numeric planning. Planning tasks are encoded as Numeric Instance Learning Graphs (νILG) on which we run our CCWL algorithm for generating features. Our numeric planning features are interpretable and efficient to generate. Experimental results show the effectiveness of our approach by achieving competitive performance over both deep learning architectures and domain-independent numeric planners. Furthermore, we have identified future work by improving the expressivity of our algorithms for capturing more complex numeric domains. Lastly, one can learn forms of domain knowledge different from heuristic functions with our new numeric planning features and graph representations such as policies [WT24], portfolios [MFH+20] and detecting relevant objects [SCC+21]. 10 Acknowledgements The authors thank the reviewers and Giacomo Rosa for the helpful comments and suggestions. The computing resources for the project was partially supported by the Australian Government through the National Computational Infrastructure (NCI) under the ANU Startup Scheme. ST was supported by the Australian Research Council grant DP220103815, by the Artificial and Natural Intelligence Toulouse Institute (ANITI) under the grant agreement ANR-23-IACL-0002, and by the European Union’s Horizon Europe Research and Innovation program under the grant agreement TUPLES No. 101070149. References [ACSJ22] Javier Segovia Aguas, Sergio Jiménez Celorrio, Laura Sebastiá, and Anders Jonsson. Scaling-up generalized planning as heuristic search with landmarks. In SOCS, 2022. [AJJ18] Javier Segovia Aguas, Sergio Jiménez, and Anders Jonsson. Computing hierarchical finite state controllers with classical planning. J. Artif. Intell. Res., 62, 2018. [AJJ21] Javier Segovia Aguas, Sergio Jiménez, and Anders Jonsson. Generalized planning as heuristic search. In ICAPS, 2021. 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In EMNLP, 2022. 15 A B D F C E initial state i j k B A goal condition i j k Figure 7: Left: initial state of a ccBlocksworld problem, where base i, j, and k each have a load limit of 3 blocks. Right: the goal condition where A is on top of B which is on top of i. A More Details for the ccBlocksworld Example We repeat the running ccBlocksworld example in Fig. 7. Listings 2 and 1 provide the explicit PDDL domain and problem encodings for the running ccBlocksworld example. An optimal plan for the problem is given as follows on the left, and an optimal plan without capacity constraints on the right. 1. (unstack f d j) 2. (stack f a i) 3. (unstack d b j) 4. (stack d f i) 5. (pickup b j) 6. (stack b e k) 7. (unstack d f i) 8. (putdown d j) 9. (unstack f a i) 10. (stack f d j) 11. (pickup a i) 12. (stack a f j) 13. (unstack b e k) 14. (putdown b i) 15. (unstack a f j) 16. (stack a b i) 1. (unstack f d j) 2. (stack f e k) 3. (unstack d b j) 4. (stack d f k) 5. (pickup a i) 6. (stack a d k) 7. (pickup b j) 8. (putdown b i) 9. (unstack a d k) 10. (stack a b i) Listing 1: PDDL encoding for the ccBlocksworld problem in Fig. 7. ( d e f i n e ( problem running−example ) ( : domain ccblocksworld ) ( : o b j e c t s a b c d e f −block i j k −base ) ( : i n i t ( arm_empty ) (= ( c a p a c i t y i ) 2) (= ( c a p a c i t y j ) 0) (= ( c a p a c i t y k ) 1) ( c l e a r a ) ( on a i ) ( above a i ) ( c l e a r f ) ( on f d ) ( on d b ) ( on b j ) ( above f j ) ( above d j ) ( above b j ) ( c l e a r e ) ( on e c ) ( on c k ) ( above e k ) ( above c k ) ) ( : goal ( and ( c l e a r a ) ( on a b ) ( on b i ) ) ) ) Listing 2: PDDL encoding for the ccBlocksworld domain. 16 ( d e f i n e ( domain ccblocksworld ) ( : r e q u i r e m e n t s : s t r i p s : typing : n u m e r i c −f l u e n t s ) ( : types block base −o b j e c t ) ( : p r e d i c a t e s ( on ?x −block ?y −o b j e c t ) ( above ?x −block ?y −base ) ( c l e a r ?x −o b j e c t ) ( holding ?x −block ) ( arm_empty ) ) ( : f u n c t i o n s ( c a p a c i t y ?x −base ) ) ( : a c t i o n pickup : parameters (? block −block ? base −base ) : p r e c o n d i t i o n ( and ( on ? block ? base ) ( above ? block ? base ) ( c l e a r ? block ) ( arm_empty ) ) : e f f e c t ( and ( not ( on ? block ? base ) ) ( not ( above ? block ? base ) ) ( not ( c l e a r ? block ) ) ( c l e a r ? base ) ( holding ? block ) ( not ( arm_empty ) ) ( i n c r e a s e ( c a p a c i t y ? base ) 1 ) ) ) ( : a c t i o n putdown : parameters (? block −block ? base −base ) : p r e c o n d i t i o n ( and ( holding ? block ) ( c l e a r ? base ) (<= 1 ( c a p a c i t y ? base ) ) ) : e f f e c t ( and ( not ( holding ? block ) ) ( not ( c l e a r ? base ) ) ( on ? block ? base ) ( above ? block ? base ) ( c l e a r ? block ) ( arm_empty ) ( d e c r e a s e ( c a p a c i t y ? base ) 1 ) ) ) ( : a c t i o n unstack : parameters (? block_a −block ? block_b −block ? base −base ) : p r e c o n d i t i o n ( and ( on ? block_a ? block_b ) ( above ? block_a ? base ) ( c l e a r ? block_a ) ( arm_empty ) ) : e f f e c t ( and ( not ( on ? block_a ? block_b ) ) ( not ( above ? block_a ? base ) ) ( not ( c l e a r ? block_a ) ) ( c l e a r ? block_b ) ( holding ? block_a ) ( not ( arm_empty ) ) ( i n c r e a s e ( c a p a c i t y ? base ) 1 ) ) ) ( : a c t i o n s t a c k : parameters (? block_a −block ? block_b −block ? base −base ) : p r e c o n d i t i o n ( and ( holding ? block_a ) ( c l e a r ? block_b ) ( above ? block_b ? base ) (<= 1 ( c a p a c i t y ? base ) ) ) : e f f e c t ( and ( not ( holding ? block_a ) ) ( not ( c l e a r ? block_b ) ) ( on ? block_a ? block_b ) ( above ? block_a ? base ) ( c l e a r ? block_a ) ( arm_empty ) ( d e c r e a s e ( c a p a c i t y ? base ) 1 ) ) ) ) 17 B Related Work Two related fields to Learning For Planning (L4P) and Learning For Numeric Planning (L4NP) are Generalised Planning (GP) and Reinforcement Learning (RL). In the following subsections, we outline the main difference between L4P with the respective related fields as well as corresponding related work. B.1 Generalised planning GP consists of automatically characterising the solution of a (possibly infinite) set of planning tasks [Sri10, SIZ08]. The most common characterisations are action policies, but other characterisations also include finite state controllers [BPG09, BPG10, HG11, HG13, AJJ18], and programs with branching and looping [AJJ21, ACSJ22]. Logic programming approaches involving decision lists [Kha99, GT04] and Datalog programs [GRH24, CHŠ24] have also been used to characterise solutions for planning domains. We refer to articles [CAJ19] and [Sri23] for more detailed surveys of GP. The difference L4P and GP can be subtle given that there is a non-empty intersection between the two fields, and works in both fields generally aim to compute structures that solve problems from a given domain. The way we differentiate the two fields is that L4P follows generally follows traditional supervised learning approaches, whereas GP can be likened to performing program synthesis. With regards to numeric planning, Srivastava et al. [SZIG11] introduced Qualitative Numeric Planning (QNP) which is a subset of numeric planning where numeric variables exhibit non-negative domains, and actions increase or decrease the value of numeric variables by indeterminate amounts. A solution for a QNP is a policy which can be used to represent solutions for sets of planning tasks. QNP has been shown to be equivalent to fully observable non-deterministic (FOND) planning [BG20] arising from the non-determinism of action effects, and the connection between FOND and GP has often shown itself when used to synthesise generalised policies [BG18, IM19]. Lin et al. [LCF+22] studies GP for a more expressive class of numeric planning, by allowing for integer numeric variables and employing linear expressions in conditions and action effects. Their approach involves synthesising programs that allow for branching and looping. Lastly, νASNets [WT24] extends ASNets [TTTX20] in order to learn policies with a neural network architecture for planning. B.2 Reinforcement Learning RL is a learning paradigm for decision making that does not have access to a model and instead learns from rewards [SB98]. RL has achieved promising results in games when combined with deep learning [MKS+15, SHM+16]. A major difference between RL and L4P is that the former requires reasoning over dense reward functions, whereas the latter requires reasoning over logic [Gef18]. Nevertheless, there has been some preliminary work looking at the intersection of RL and planning. Reward machines [IKVM22] are a logical language used for specifying reward functions for RL problems, inspired by the declarative nature of the planning as modelling paradigm. RL has also been applied directly into planning tasks, as done by [MV21] for temporal planning. Rewards are mostly sparse, with 1 being reward for achieved goals, minor 10−5 rewards for achieved goal propositions, and no reward otherwise. Gehring et al. [GAC+22] explored introducing denser reward functions to planning through domain-independent heuristics to allow for RL approaches. Supervised RL has also been used for learning planning policies [SBG23]. Nevertheless, the use cases for RL and planning are generally different, with RL being more suited for control tasks in continuous or dynamic environments such as in robotics, and planning being more suited for combinatorial tasks in discrete or abstract environments such as in logistics. C Description of Benchmark Domains C.1 Numeric (Capcity Constrained) Blocksworld This domain was described in Sec. 2. A task from the domain consists of n blocks stacked on top of one another to form towers on top of b bases. Each base has a capacity of how many blocks it can support. The goal is to stack and unstack blocks to achieve a target tower configuration. The numeric component of this domain arises from modelling the capacity of bases. Training problems have n ∈[2, 11] blocks while testing problems have n ∈[5, 488] blocks. C.2 Numeric Childsnack A task from the domain consists of feeding c children with sandwiches in l locations, of which some are allergic to gluten. There are a finite amount of gluten-free (GF) and non-GF ingredients. 18 GF sandwiches can only be made from GF ingredients, whereas non-GF sandwiches can be made with any ingredients. Children allergic to gluten are only allowed to eat GF sandwiches while the remaining children can eat any type of sandwich. Thus, the problem has deadends because resources are finite and can be wasted. The goal is to make sandwiches and feed all the children satisfying the aforementioned rules. The numeric component of the domain arises from modelling the ingredient and sandwich resources. Training problems have c ∈[1, 8] children while testing problems have c ∈[4, 292] children. C.3 Numeric Ferry A task from the domain consists of c cars spread across l locations. A ferry is able to transport up to a fixed amount of cars around to different locations. The goal of the domain is to transport the cars with the ferry to various target locations. The numeric component of the domain arises from modelling the capacity of the ferry. Training problems have c ∈[1, 20] cars while testing problems have c ∈[4, 974] cars. C.4 Numeric Miconic A task from the domain consists of p passengers with different weights spread across f floors. There is a single elevator with a fixed load capacity that can transport passengers between floors. Furthermore, if the load of the elevator exceeds a secondary threshold, it takes twice as long to move between floors. The goal of the domain is to move the passengers to their target floors. The numeric component of the domain arises from modelling the weight of the passengers and load capacity of the elevator. Training problems have p ∈[1, 10] passengers while testing problems have p ∈[1, 485] passengers. C.5 Numeric Rovers A task from the domain consists of r rovers some of which can sample rock and soil data, while others have cameras that can take images of objectives. The goal of each problem is to sample rock and soil data as well as take images of objectives and communicate all g data to the lander. The rovers can move around a map with w waypoints and the rover is only able to communicate data to the lander from a subset of waypoints. Furthermore, rovers have a limited energy supply that is consumed with any action, but they can recharge with solar panels at certain waypoints. Thus, the problem has deadends because rovers have limited energy and could exhaust them in waypoints where they cannot recharge. The numeric component of the domain arises from modelling the energy supply of the rovers. Training problems have g ∈[1, 10] goals while testing problems have [2, 728] goals problems C.6 Numeric Satellite A task from the domain consists of s satellites, each carrying a subset of i instruments that can take pictures of space using a subset of m modes. Satellites can rotate to take pictures of d locations in space. Each satellite has a fixed amount of fuel that is consumed when rotating, and a fixed amount of data capacity that is consumed when taking pictures. Thus, the problem has deadends because resources are finite and can be wasted. The goal of a Satellite problem is to take pictures of a set of locations in space with specified modes while adhering to the fuel and data capacity constraints. The numeric component of the domain arises from modelling the fuel and data capacity features. Training problems have s ∈2, 10 satellites and testing problems have s ∈[4, 98] satellites. C.7 Numeric Spanner A task from the domain consists of s spanners scattered along a one-way hallway with l locations, and n nuts at the end of the hallway that have to be fixed. Each spanner can only be used to fix a single nut before it breaks. The goal of the domain is to fix all the nuts. The problem has deadends if not enough spanners are picked up before reaching the end of the hallway. The numeric component of the domain arises from modelling the number of spanners and nuts. Training problems have s ∈[1, 10] spanners while testing problems have s ∈[1487] spanners. C.8 Numeric Transport A task from the domain consists of p packages spread across l locations, with t number of trucks that can transport pick up and transport packages on a map. Each truck has a limited capacity of packages it can carry. The goal of the problem is to transport all the packages to their target locations. The numeric component of the domain arises from modelling the capacity of the trucks. Training problems have p ∈[1, 7] packages while testing problems have p ∈[1, 194] packages. 19 Blocksworld Childsnack Ferry Miconic Rovers Satellite Spanner Transport Domain 10 5 10 4 10 3 10 2 Time (s) Figure 8: Distributions of heuristic evaluation time for GNN and CCWL models with L = 1 on problems where both were able to solve in the given timeout. Blue box plots correspond to GNN models and red box plots correspond to CCWL models. D GNN and CCWL heuristic evaluation times We refer to Fig. 8 for distributions of heuristic evaluation time for GNN and CCWL models with L = 1. Times are computed by taking the total search time for each problem and dividing by the number of heuristic evaluations made by the planner. We assume that the heuristic evaluation is the bottleneck of the search which is confirmed with informal profiling experiments. E Effect of number of iterations and layers We refer to Tab. 2 for the coverage of models with different L values. 20 Table 2: Coverage and median of expansions of solved problems for each CCWL and GNN model with varying number of iterations and layers. Higher values are better for coverage, and lower values are better for expansions. The best value per domain and metric coloured. OOM denotes that the training process exceeded the memory limit. (a) hGNN cost Domain Coverage 0 1 2 3 4 Blocksworld 22 18 24 19 21 Childsnack 18 17 15 14 13 Ferry 60 60 60 60 60 Miconic 67 63 62 60 58 Rovers 21 18 11 13 13 Satellite 22 19 11 13 14 Spanner 90 90 90 90 77 Transport 36 34 31 31 30 Σ 336 319 304 300 286 (b) hGNN rank Domain Coverage 0 1 2 3 4 Blocksworld 20 24 22 24 22 Childsnack 18 22 26 29 37 Ferry 60 60 60 60 60 Miconic 70 64 62 62 59 Rovers 22 14 13 12 11 Satellite 22 14 14 17 14 Spanner 90 90 90 90 90 Transport 36 38 33 27 28 Σ 338 326 320 321 321 (c) hCCWLF cost Domain Coverage 0 1 2 3 4 Blocksworld 28 22 19 21 22 Childsnack 22 22 20 20 20 Ferry 73 70 65 62 58 Miconic 90 90 88 87 85 Rovers 30 22 19 15 15 Satellite 26 23 5 7 5 Spanner 90 90 89 88 86 Transport 48 40 37 30 27 Σ 407 379 342 330 318 (d) hCCWLF rank Domain Coverage 0 1 2 3 4 Blocksworld 25 19 29 23 22 Childsnack 22 90 20 23 23 Ferry 73 71 70 68 68 Miconic 90 90 89 87 85 Rovers 21 23 15 22 21 Satellite 17 16 7 OOM OOM Spanner 90 90 89 89 89 Transport 48 46 46 29 35 Σ 386 445 365 341 343 21 NeurIPS Paper Checklist 1. 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4,432	Unified Graph Augmentations for Generalized Contrastive Learning on Graphs Jiaming Zhuo1, Yintong Lu1, Hui Ning1, Kun Fu1, Bingxin Niu1, Dongxiao He2, Chuan Wang3, Yuanfang Guo4, Zhen Wang5, Xiaochun Cao6, Liang Yang1∗ 1Hebei Province Key Laboratory of Big Data Calculation, School of Artificial Intelligence, Hebei University of Technology, Tianjin, China 2College of Intelligence and Computing, Tianjin University, Tianjin, China 3School of Computer Science and Technology, Beijing JiaoTong University, Beijing, China 4School of Computer Science and Engineering, Beihang University, Beijing, China 5School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), School of Cybersecurity, Northwestern Polytechnical University, Xi’an, China 6School of Cyber Science and Technology, Shenzhen Campus of Sun Yat-sen University, Shenzhen, China jiaming.zhuo@outlook.com, 202332803037@stu.hebut.edu.cn, ninghui048@163.com, fukun@hebut.edu.cn, niubingxin666@163.com, hedongxiao@tju.edu.cn, wangchuan@iie.ac.cn, andyguo@buaa.edu.cn, w-zhen@nwpu.edu.cn, caoxiaochun@mail.sysu.edu.cn, yangliang@vip.qq.com Abstract In real-world scenarios, networks (graphs) and their tasks possess unique characteristics, requiring the development of a versatile graph augmentation (GA) to meet the varied demands of network analysis. Unfortunately, most Graph Contrastive Learning (GCL) frameworks are hampered by the specificity, complexity, and incompleteness of their GA techniques. Firstly, GAs designed for specific scenarios may compromise the universality of models if mishandled. Secondly, the process of identifying and generating optimal augmentations generally involves substantial computational overhead. Thirdly, the effectiveness of the GCL, even the learnable ones, is constrained by the finite selection of GAs available. To overcome the above limitations, this paper introduces a novel unified GA module dubbed UGA after reinterpreting the mechanism of GAs in GCLs from a message-passing perspective. Theoretically, this module is capable of unifying any explicit GAs, including node, edge, attribute, and subgraph augmentations. Based on the proposed UGA, a novel generalized GCL framework dubbed Graph cOntrastive UnifieD Augmentations (GOUDA) is proposed. It seamlessly integrates widely adopted contrastive losses and an introduced independence loss to fulfill the common requirements of consistency and diversity of augmentation across diverse scenarios. Evaluations across various datasets and tasks demonstrate the generality and efficiency of the proposed GOUDA over existing state-of-the-art GCLs. 1 Introduction Owing to their effectiveness and efficiency, Graph Neural Networks (GNNs) have become a standard toolkit for processing various graph tasks such as node classification and graph classification [17, 34, 41, 40]. They typically follow a message-passing paradigm [10], where the representation of each node is updated by aggregating the representations of its adjacent nodes and subsequently combining ∗corresponding author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). the aggregated representations with itself. In general, to produce discriminative representations, GNNs need to resort to the task-relevant labels (i.e., supervised information) to guide the network training, which limits their applicability in the label scarcity scenarios [24, 37, 43]. To overcome this limitation, Graph Contrastive Learning (GCL), a typical graph self-supervised learning architecture, has been developed to provide training guidance by capturing the self-supervised information contained in the graph [53, 32, 50, 2, 21, 39, 20]. Inspired by the design philosophy of contrastive learning in Computer Vision (CV) [4, 12], GCLs adopt the same architecture, which consists of three components: augmentation, encoder, and contrastive loss [53, 32]. Thus, GCLs inherit the merit of enabling learning representations invariant to augmentation, which is achieved by maximizing the agreement between embeddings from different perturbations of the same graph [51, 52]. To further improve the representation capacity of GCLs, great endeavors have been made to design augmentations for the original graph, i.e., Graph Augmentations (GAs), which target nodes, edges, attributes, and subgraphs. Based on how information is processed, GAs can be divided into two categories: heuristic [53, 35, 32, 14] and learnable methods [47, 31, 21, 56]. The heuristic GAs modify graphs through the combination of fixed, random rules, such as attribute masking [53], edge removing [32, 2], and graph diffusion [14]. They tend to neglect the subsequent steps, namely encoding and contrastive optimization, hence leading to suboptimal performances. In contrast, learnable GAs leverage prior knowledge and feedback during training to refine augmentations, already surpassing base augmentations on many tasks. Notable contributions include GAs based on spectral methods [21], and adversarial training [31]. Given their inherent and distinct characteristics, various networks and tasks require the meticulous selection of optimal GAs to improve model performance pivotally. However, most GCLs face several limitations regarding the selection: (1) Specificity. GCLs are typically tailored with specific GAs to meet the needs of particular scenarios, resulting in a lack of generality across diverse scenarios. For instance, node dropping (specifically, removing nodes and their associated edges), widely applied in graph-level tasks [47, 48], could significantly compromise the integrity of graphs [36], rendering it less suitable for node-level tasks. (2) High complexity. Either way, identifying and generating the scene-specific GAs impose a considerable computational burden on the models. For example, the set sampling method necessitates a validation of all combinations [45, 38]. Furthermore, the adversarial attack method [47, 31] entails recalculating contrastive losses, which takes a quadratic complexity of O(n2). Besides, the spectral method requires Laplacian matrix decomposition [5], which has a cubic complexity of O(n3). (3) Incompleteness. Despite the promise of existing learnable GAs in optimizing for specific scenarios, their efficacy is limited by the finite range of GAs at their disposal. This paper seeks to break these limitations by proposing a unified GA module for GCLs. Toward this end, the mechanisms of existing GAs in GCLs are systemically investigated and reinterpreted from a message-passing perspective [10]. The conclusion is that, from the message-passing perspective, GAs uniformly induce attribute modifications within the neighborhoods of nodes, even though they appear diverse from the spatial perspective, as depicted in Fig. 1. Therefore, the essence of GCLs is to learn node representations invariant to such local augmentation. Drawing from this insight, a novel Unified GA (UGA) module with a simple yet effective design is presented. It strategically interpolates an appropriate amount of Augmentation-Centric (AC) vectors in a graph-structured manner [55, 8], where AC vectors are treated as another type of node, as illustrated in Fig. 2. In theory, UGA is able to simulate the impact of the above four explicit GAs on target nodes by aggregating features from the AC vectors that capture the attribute variations within the neighborhood of these nodes. Building upon the proposed UGA module, a generalized GCL framework dubbed Graph cOntrastive UnifieD Augmentations (GOUDA) is presented to overcome the above challenges in existing GCLs. This framework adopts a typical dual-channel architecture [4, 49, 53], corresponding to two distinct augmented graphs (views) with their respective AC matrices, as shown in Fig. 2. To realize general utility, GOUDA proposes to capture the consistency and diversity across augmentations (defined in Section 3.3), which are essential and shared goals for GCLs to be applicable across diverse scenarios. To be specific, the objective function of GOUDA is twofold: (1) maximizing Mutual Information (MI) between representations from these distinct views. (2) maximizing the distributional difference between the AC matrices. The former is a fundamental principle behind classic contrastive losses and inherently ensures consistency, while the latter is a constraint to modulate diversity. In practice, GOUDA is instantiated by leveraging widely employed contrastive losses alongside a Hilbert-Schmidt Independence Criterion (HSIC)-based distributional independence loss. This design makes GOUDA more effective and efficient than GCLs with learnable GAs. 2 The main contributions of this work are summarized as follows: • We investigate the mechanism of GAs in GCLs through the lens of message-passing. • We propose a lightweight GA module named UGA to simulate the impacts of GAs on nodes. • We introduce GOUDA, an efficient and generalized GCL framework, which captures both consistency and diversity across augmentations. • Extensive experiments and in-depth analysis demonstrate that GOUDA outperforms stateof-the-art GCLs across various public benchmark datasets and tasks. 2 Preliminaries This section briefly introduces the notations utilized throughout the paper. Subsequently, it outlines the essential components of the Graph Contrastive Learning (GCL) framework. 2.1 Notations Matrices (e.g., Q) are in bold capital letters, vectors (e.g., qi,:, which denotes the i-th row of Q) are in bold lowercase letters, scalars (e.g., qi,j, which represents the entry of Q at the i-th row and the j-th column) are in lowercase letters, and sets (e.g., N) are in calligraphic letters. For a general-purpose description, this paper considers an undirected attribute graph G(V, E), where V stands for the node-set containing n node instances {(xv, yv)}v∈V. And X ∈Rn×f and Y ∈Rn×c denote the attribute matrix and label matrix of node v, respectively, where f and c is the numbers of attirbutes and labels, respectively. Also, E = {ei}m−1 i=0 terms the edge set containing m edges. In general, the adjacency matrix A ∈Rn×n is employed to describe the graph topology, such that the matrix form of the graph can be expressed as G(A, X). Moreover, H ∈Rn×d terms the graph representation, where d terms the dimension of the representation. 2.2 Graph Contrastive Learning Graph Augmentations. Drawing on the successful experience of image augmentation in Computer Vision (CV) [4, 15], Graph Augmentation (GA) [51] is introduced in graph learning to address the challenge of data scarcity. In the typical GCL frameworks, the input graph G(A, X) is processed through two separate perturbations (GA procedures), formulated as ti(G) : G(A, X) →Gi(A(i), X(i)), to generate its two views (augmented graphs), denoted as G1(A(1), X(1)) and G2(A(2), X(2)). Based on the perturbed information, GAs can be broadly classified into four main categories: node augmentation [47, 48], edge augmentation [53, 54, 32, 31], attribute augmentation [19, 53, 44], and subgraph augmentation [47, 13]. An overview of GAs can be found in Section B. Graph Encoders. For efficient processing and analysis, the graph encoders are leveraged to transform raw topology and attribute information of the input graph into low-dimensional vector representations. Most graph encoders in GCLs follow a message-passing paradigm [10], which typically involves two primary processes: aggregation and combination. During these steps, each node iteratively updates its representations by aggregating and combining the node features from its neighborhoods, that is ˚hl v ≜Aggregationl {hl−1 u \|u ∈Nv} , hl v ≜Combinationl hl−1 v ,˚hl v , (1) where hl v terms the representations of node v in the l-th layer and Nv denotes the set of neighboring nodes of node v. In prevalent GCLs like GRACE [53], a two-layer GCN [17] is adopted, where the Aggregation(·) and Combination(·) functions are implemented via average function. Thus, there is H = GCN2 (G (A, X)) = σ ˆD−1 2 ˆA ˆD−1 2 σ ˆD−1 2 ˆA ˆD−1 2 XW0 W1 , (2) where σ(·) denotes the nonlinear activation functions, such as ReLU(·), and ˆA = A + In stands for the adjacentcy matrix with self-loops, and ˆD is the corresponding degree matrix, and Wl represents the parameter matrix for the l-th layer. Therefore, for the two augmented graphs, their representations can be obtained by computing H(1) = GCN2(G1(A(1), X(1))) and H(2) = GCN2(G2(A(2), X(2))). 3 Message-passing Perspective Input graph 𝐺(𝐀, 𝐗) 1 2 0 5 4 8 7 6 9 3 1 2 0 5 4 8 7 6 9 3 1 2 0 5 4 8 7 6 9 3 1 2 0 5 4 8 7 6 9 3 Augmented graph 𝐺!(𝐀(∗), 𝐗(∗)) Augmented graph 𝐺%(𝐀(∗), 𝐗) Augmented graph 𝐺&(𝐀, 𝐗(∗)) 1 2 0 5 4 8 7 6 9 3 Augmented graph 𝐺'(𝐀(∗), 𝐗(∗)) GAs 2 0 5 3 6 1 0 5 3 7 1 2 0 5 3 8 7 6 1 0 3 4 8 1 2 0 5 3 4 8 7 6 1-hop 2-hop 0 2-hop subgraph of node Spatial Perspective Example: 2-layer GCN 1-hop 2-hop 1-hop 2-hop 1-hop 2-hop 1-hop 2-hop (a) Node Augmentation. (b) Edge Augmentation. (c) Attribute Augmentation. (d) Subgraph Augmentation. Figure 1: Motivation to unify Graph Augmentations (GAs). A two-hop subgraph example, where the target node is highlighted in red, and the perturbed information is marked in brown. (a) Node augmentation by dropping nodes. (b) Edge augmentation by removing edges. (c) Attribute augmentation by masking attributes. (d) Subgraph augmentation by cropping subgraphs. Existing GAs, typically seen as various forms of global augmentations from the spatial perspective, can be uniformly interpreted as local attribute modifications (i.e., local augmentations) from the message-passing perspective. Contrastive losses. In line with the InfoMax principle [22], various contrastive losses are incorporated in GCLs, guiding the training of graph encoders by maximizing the Mutual Information (MI) between the encoded representations on two augmented graphs. Specifically, given two representations H(1) and H(2) obtained from a shared encoder gΘ, the general objective of GCL is expressed as GCL: arg max Θ I(H(1); H(2)), where H(1) = gΘ(G1), H(2) = gΘ(G2), (3) where I(X; Y ) represents the MI between X and Y . In general, the MI can be approximated using a lower bound estimator, i.e., the InfoNCE loss [33], in the GCLs [53, 20]. This loss can be classified as a sample-level loss because it operates on the sample dimension of the representation matrix. In contrast, Barlow Twins loss [49], another widely employed loss, is designed to remove redundancies among features and hence can be categorized as a feature-level loss. Both losses are used to implement the proposed framework. Section C provides detailed descriptions of these losses. 3 Methodology 3.1 Motivations As previously mentioned, Contrastive Learning (CL) seeks to learn image representations invariant to augmentations by encouraging the agreement between embedding vectors from the different image distortions. Due to the employment of identical loss functions, typical Graph Contrastive Learning (GCL) inherits the above representation capabilities from CL. Nonetheless, GCLs should emphasize the local invariance owing to the application of graph encoders. The essence of the graph encoder is to explore the locality of graphs. To be specific, graph encoders in GCLs (generally a 2-layer GCN) follow the message-passing paradigm where node representations are updated in a local aggregation and combination manner, as detailed in Section 2. Given the localizing property of the graph encoder, GAs (i.e., node, edge, attribute, and subgraph augmentations), which are typically viewed as global operations in various forms, can be uniformly reinterpreted as attribute modifications in the neighborhoods of nodes, namely, local augmentations, as illustrated in Fig. 1. 1) Edge augmentation involves adding and removing perturbed edges in graphs, equivalent to inserting or masking the attributes of nodes connected by these edges in the neighborhood of impacted nodes. For example, the shown edge removing results in the complete attribute masking of partial 2-hop neighbors (nodes 1, 4, 7, and 8) of the target node 0 during message passing. 2) Attribute augmentation essentially replaces the attributes of perturbed nodes in the graph with new ones, which can be viewed as perturbing the neighborhoods of nodes that contain these perturbed nodes. The attribute masking example shows that during the aggregation phase, the attributes of neighbors (nodes 1, 3, 4, 5, 6, 7, 4 and 8) on the 2-hop computation graphs of node 0 are masked. 3) Subgraph augmentation is to modify the specific subsets of the graph (including its edges and attributes), which also can be seen as the attribute perturbation in the neighborhoods of target nodes. For node 0, the shown subgraph cropping causes removing nodes 2, 4, 6, and 8 from the 2-hop neighborhood during the message-passing phase. Note that node augmentation is a specific case of subgraph augmentation, where the subset size is one. Thus, the above conclusion regarding subgraph augmentation applies to it. In short, the mechanism of GAs in GCLs is to induce attribute modification in the neighborhoods of nodes. Thus, the essence of GCLs is to learn representations invariant to such local augmentations. 3.2 Unified Graph Augmentation Module Motivated by the above insights, a unified graph augmentation module dubbed UGA is introduced to implement augmentation efficiently and flexibly. The primary idea is to introduce a collection of Augmentation-Centric (AC) vectors for nodes to simulate and exert the impact of GAs on these nodes, namely, attribute variations in the neighborhood of these nodes. A straightforward implementation of UGA is to align AC vectors one-to-one with nodes, match the size of their features, and then perform feature summation to achieve the desired augmentation. Given the input graph G, the above implementation can be formulated as UGA: G∗= t(G, Q), where t(G, Q) : G(A, X) →G(A, X + Q), (4) where G∗denotes an augmented graph derived from the UGA funtion t(G, Q), and Q ∈Rn×f terms the matrix of AC vectors qv ∈R1×f, i.e., AC matrix, and f is the dimension of node attributes. Fig. 2(a) provides an illustrative example and explains the equivalence between the proposed UGA module and an explicit GA (edge removing). From this, it can be concluded that the UGA module can effectively substitute GAs as long as the combined AC vectors are the representations of cumulative attribute variations within the neighborhoods of nodes induced by these GAs. Theorem 3.1. Assuming any augmented graph G∗(A(∗), X(∗)), where A(∗) ∈A and X(∗) ∈X, with A and X represent the candidate spaces for the augmented adjacency matrix and attribute matrix, respectively, in the proposed implementation of UGA (Eq. 4), there exists an AC matrix Q that meets gΘ(A, X + Q) = gΘ(A(∗), X(∗)), (5) where gΘ stands for the graph encoder. This theorem suggests that the proposed UGA module can be equivalent to any existing GA (including node, edge, attribute, and subgraph operations), thereby demonstrating its unifying capability to GAs. Proofs for this theorem are presented in Section D.1. Furthermore, the UGA module possesses an attractive characteristic: adaptability, since the AC vectors are capable of dynamically capturing taskrelevant perturbation information throughout the training process. Nonetheless, this implementation introduces numerous parameters proportional to the network size, resulting in a significant increase in complexity and the risk of overfitting. To address this limitation, the proposed UGA is reimplemented in a graph-structured manner, where a modest parameter set is utilized, as shown in Fig. 2(b). Shared AC vectors. In graphs, long-range dependencies signify the beyond-local interactions among nodes, represented by similar node attributes and neighborhood patterns [23, 40, 46]. Therefore, it is reasonable to assume that a group of interdependent nodes would benefit from the same optimal GAs. Thus, a shared AC matrix Q = [qi,:]k−1 i=0 is introduced, where k ≪n. Propagation mode. AC vectors propagate their features to nodes via a general attention mechanism, in which structural features (e.g., positional/structure encodings [1, 7]) are employed to calculate the attention scores. Formally, the proposed UGA module can be reformulated as ˆxv,: = xv,: + k−1 X i=0 bv,i × qi,:, bv,i = exp f([xv,:\|\|ev,:]) · q⊤ i,: Pk t=0 exp f([xv,:\|\|ev,:]) · q⊤ t,: , (6) where bv,i stands for the propagation weight from i-th AC vector to node v within matrix B ∈Rn×k, and f([xv\|\|ev]) ∈R1×f terms an integrated representation of node v, which concatenates the node attributes xv and structural features ev ∈Rt. This paper adopts t-steps random-walk encodings [7] as the structure features. f(·) : Rf+t →Rf denotes a projection layer. 5 1 2 0 5 4 8 7 6 9 3 Augmented graph 𝐺!(𝐀$, 𝐗) 1 2 0 5 3 4 1-hop where = + 1 4 Equivalent 1 2 0 5 4 8 7 6 9 3 1 2 0 5 4 8 7 6 9 3 Independence Loss Contrastive Loss … … Graph Encoders Graph Encoders 1 2 0 5 4 8 7 6 9 3 Embeddings Embeddings G" = t(G, 𝐐) G# = t(G, 𝐏) (a) The proposed GA module UGA. (b) The proposed GCL framework GOUDA. 𝐇 𝐇. Input graph 𝐺(𝐀, 𝐗) Shared g!(, ) Figure 2: Illustration of the proposed unified module UGA and generalized framework GOUDA. (a) An intuitive example of the equivalence between GAs (e.g., edge removing) and the aggregation of Augmentation-Centric (AC) vectors, which capture the local attribute variations caused by these GAs. (b) The proposed generalized GCL framework GOUDA. The independence loss, which is directly computed on AC vectors, is designed to ensure diversity across different augmentations. Processing. Keeping a subset of the salient connections facilitates propagation and enhances computational efficiency. Thus, propagation weights below a predefined threshold are zeroed out, namely bv,i = bv,i, bv,i > ε 0, otherwise, (7) where ε terms a threshold value. Then, the obtained weight B is applied in propagation, per Eq.6. 3.3 Generalized Graph Contrastive Learning Framework Building upon the UGA module, a novel GCL framework named Graph cOntrastive UnifieD Augmentations (GOUDA) is proposed to achieve generality across diverse tasks and graphs. It utilizes the standard two-channel architecture (in Eq. 3), with each channel generating an augmented graph through the UGA module and its AC matrix, as depicted in Fig. 2. Unlike traditional GCLs, GOUDA introduces a term to constrain the two AC matrices (denoted as Q and P). Specifically, GOUDA optimizes the following objective function: GOUDA: argmax Θ I (gΘ (G1) ; gΘ (G2)) + D (Q, P) , (8) where I(X; Y ) stands for the Mutual Information (MI) between X and Y , and gΘ denotes a graph encoder shared between two views (or channels). G1 = t (G, Q) and G2 = t (G, P) represent two augmented graphs, and D (Q, P) denotes the constraint between Q and P. Definition 3.2. (Consistency across augmentations). Let H(i) and H(j) denote the representations of graphs Gi, Gj ∼Gω, respectively, where j ̸= i, and Gω denotes the family of graphs derived from a series of parametric graph augmentations. Consistency across augmentations for node v is defined as Cv = S h(i) v , h(j) v , (9) where S(X, Y ) terms the distributional similarity between X and Y . This consistency implies that augmentation should minimally impact the similarity between representations from different augmented graphs for the same nodes to preserve the intrinsic semantic integrity of the nodes. Note that the first term in the objective function of GOUDA, namely the MI maximization, essentially is a constraint for semantic consistency. Thus, the augmentation learned by the UGA module is capable of ensuring the desired property. Definition 3.3. (Diversity across augmentations). Given two augmented graphs Gi(A(i), X(i)) and Gj(A(j), X(j)) ∼Gω, where j ̸= i, let N k v represents the k-hop subgraph centered at node v, and let D(, ) stands for the measure of distributional difference. Diversity across augmentations is defined as Dv = D COM(X(i) N k v ), COM(X(j) N k v ) , (10) 6 where X(i) N k v ∈Rnv×f stands for the attribute matrix of nodes in the k-hop subgraph of node v, and nv is the number of neighbors for node v. COM(·) terms a combination function, such as sum(·). This definition is based on the conclusion in Section 3.1, namely, the mechanism of GAs in GCLs is to modify attributes within the node neighborhoods. Accordingly, another objective for augmentation is the minimization of the local attribute overlap between augmented graphs, ensuring the model does not overfocus on the specific features of a single distribution. In the proposed GOUDA framework, local attribute variations for each node are represented by AC vectors, e.g., qv and pv, with the augmentation being generated by aggregating features from these vectors. Therefore, the diversity across augmentations can be quantified using two AC matrices (Q and P) corresponding to two distinct views, which is supported by the following analysis. Theorem 3.4. Let D(X, Y ) = ∥X −Y ∥2 F stands for the distributional difference, and let COM(·) = sum(·) terms the combination function. In GOUDA (in Eq. 8), the diversity across augmentations (in Eq. 10) can be approximated by the distributional difference between AC matrices Q and P, that is Dv = ∥ X t∈Nv∪v (x(1) t,: + ˆqt,:) − X t∈Nv∪v (x(2) t,: + ˆpt,:)∥2 F ≈∥Q −P∥2 F , (11) where ˆqt,: = bq t,:Q denote the features propagated from AC matrices Q to node t. Theorem 3.4 shows that the diversity across augmentations can be controlled by imposing constraints on the AC matrices, particularly through the second term in the objective function of GOUDA. Refer to Section D.2 for the proofs. In brief, maintaining a balance between consistency and diversity across augmentations is crucial for the effectiveness of GCLs. Specifically, diversity encourages exploring and exploiting the local attribute variations while consistency anchors the learned representations to the original semantics. 3.4 Instantiation of GOUDA This subsection introduces a practical implementation of the proposed GOUDA framework (Eq.8). The overview of this framework is depicted in Fig. 2, while the step-by-step procedure is detailed in Algorithm 1. The objective of GOUDA is to learn the discriminative and robust representations. To achieve this, it seeks to train the graph encoder gΘ to maximize the Mutual Information (MI) between representations from two augmented graphs G1 = t(G, Q) and G2 = t(G, P), while simultaneously maintaining consistency and diversity in the augmentation process. Estimation of Mutual Information (MI). The first term of GOUDA is implemented utilizing the sample-level InfoNCE loss (in Eq. 22), which serves as a lower bound estimator for MI, and the feature-level Barlow Twins loss (in Eq. 24). This term is denoted as contrastive loss Lcontrast. Owing to limited space, the above losses are introduced in Section C. Distributional independence loss. A distributional independence loss is introduced to instantiate the second term of GOUDA. Specifically, the Hilbert-Schmidt Independence Criterion (HSIC) [11] is adopted to measure the statistical dependence between two augmentation distributions. Furthermore, the Gram matrices derived from HSIC are constrained to minimize their off-diagonal elements. To be concrete, the independence loss is formulated as Lindep = 1/(n −1)2 trace (KRLR) \| {z } HSIC +β1 X i X j̸=i ki,j + β2 X i X j̸=i li,j, (12) where K and L stand for the Gram matrices of Q and P, respectively, defined by ki,j = κ(qi,:, qj,:) and li,j = κ(pi,:, pj,:). In practice, the kernel function κ(·) is defined as the linear kernel, specifically ki,j = qi,:qT j,:. Additionally, R = In −1 n11⊤represents the centering matrix, where I ∈Rn×n and 1n ∈Rn×1 denote the identity matrix and all-one column vector, respectively. β1 and β2 represent two hyper-parameters. Minimizing this term serves two purposes: on the one hand, it enhances the diversity across augmentations by amplifying the differences between two distributions, and on the other hand, it avoids trivial solutions by increasing the differences among the augmentation elements (qi,:) within each distribution. Objective. The overall objective function of GOUDA is a weighted sum of these two terms, that is L = Lcontrast + γLindep, (13) where γ denotes a hyperparameter used to trade off two terms. 7 Table 1: Comparison of time complexity in the augmentation Phase. n denotes the size of the graph. Model Time Complexity Description SPAN [21] O(n2tk) Eigendecomposition-based edge augmentation. JOAO [47] O(n2d) Min-max optimization-based augmentation. AD-GCL [31] O(n2d) Adversarial-training-based edge augmentation. GOUDA (Ours) O(nkf) Consistency-diversity balanced augmentation. Table 2: Accuracy in percentage (mean±std) over ten trials of node classification across seven graphs. Best and runner-up models are highlighted in bolded and underlined, respectively. Model Input Cora CiteSeer PubMed Wiki-CS Photo Computers Physics GCN A, X, Y 82.32±1.79 72.13±1.17 84.90±0.38 76.89±0.37 92.35±0.25 86.34±0.48 95.65±0.16 GAT A, X, Y 83.34±1.57 72.44±1.42 85.21±0.36 77.42±0.19 92.35±0.25 87.06±0.35 95.47±0.15 DGI A, X 82.60±0.40 71.49±0.14 86.00±0.14 75.73±0.13 91.49±0.25 84.09±0.39 94.51±0.52 GMI A, X 82.51±1.47 71.56±0.56 84.83±0.90 75.06±0.13 90.72±0.33 81.76±0.52 94.10±0.61 MVGRL A, X 83.03±0.27 72.75±0.46 85.63±0.38 77.97±0.18 92.01±0.13 87.09±0.27 95.33±0.03 GRACE A, X 83.30±0.40 71.41±0.38 86.51±0.34 79.16±0.36 92.65±0.32 87.21±0.44 95.26±0.02 GCA A, X 83.90±0.41 72.21±0.24 86.01±0.75 79.35±0.12 92.78±0.17 87.84±0.27 95.68±0.05 BGRL A, X 83.77±0.75 71.99±0.42 84.94±0.17 78.74±0.22 93.24±0.29 88.92±0.33 95.63±0.04 GBT A, X 83.89±0.66 72.57±0.61 85.71±0.32 76.65±0.62 92.63±0.44 88.14±0.33 95.07±0.17 CCA-SSG A, X 84.39±0.68 73.81±0.38 86.21±0.67 78.94±0.17 93.14±0.14 88.74±0.28 95.38±0.06 SPAN A, X 85.09±0.28 73.68±0.53 85.35±0.29 79.01±0.51 92.68±0.31 89.68±0.19 95.12±0.15 DSSL A, X 84.52±0.71 73.93±0.89 85.59±0.28 79.98±0.67 93.08±0.38 89.06±0.49 95.29±0.29 HomoGCL A, X 84.89±0.71 73.78±0.63 86.37±0.49 79.29±0.32 92.92±0.18 88.46±0.20 95.18±0.09 GOUDA-IF A, X 86.11±0.55 74.55±0.97 87.55±0.10 80.61±0.28 93.69±0.32 89.21±0.17 96.09±0.14 GOUDA-BT A, X 85.99±0.31 74.47±1.05 87.59±0.02 80.37±0.30 93.82±0.19 89.55±0.11 96.19±0.21 3.5 Complexity Analysis This subsection evaluates the complexity of the proposed GOUDA framework in comparison to the baseline GCLs configured with learnable GAs, including SPAN, JOAO, and AD-GCL. As illustrated in Tab. 1, GOUDA introduces lighter computational overhead compared to these baselines. For a detailed description of the complexity, refer to Section E.4. 4 Experiments This section evaluates the effectiveness and generality of the proposed GOUDA through a comprehensive comparison against multiple baselines across tasks at both the node-level (node classification and node clustering) and the graph-level (graph classification) tasks. Furthermore, it conducts several additional experiments to deepen the understanding of this framework. For an exhaustive account of datasets, baselines, configurations, and hyper-parameters, refer to Section E. Datasets. The experiment utilizes ten benchmark datasets, namely: Cora [28], CiteSeer [28], PubMed [28], Wiki-CS [25], Photo [29], Computers [29], and Physics [29] for node-level tasks, and IMDB-B [42], IMDB-M [42], and COLLAB [42] for graph-level tasks. See Section E.1 for dataset descriptions. Baselines. The baseline models comprise two supervised graph neural networks (GCN [17], GAT [34]) and eleven self-supervised graph learning models (DGI [35], GMI [26], MVGRL [14], GRACE [53], GCA [54], BGRL [32], CCA-SSG [50], GBT [2], SPAN [21], DSSL [39], HomoGCL [20]) for node-level tasks. Four self-supervised learning models are compared (InfoGraph [30], GraphCL [48], JOAO [47], AD-GCL [31]) for graph-level tasks. Refer to Section E.3 for model introductions. 4.1 Experimental Results Node Classification. It can be observed from Tab. 2, which exhibits the results of node classification tasks, that the proposed GOUDA outperforms the unsupervised baselines in six of the seven datasets. This demonstrates the superiority of GOUDA. Furthermore, on the CiteSeer dataset, notable performance improvements are observed with both models, GOUDA-IF and GOUDA-BT, surpassing 8 the baselines GRACE and GBT. Specifically, the accuracy of GOUDA-IF surpasses that of GRACE by 3.14%, and similarly, the accuracy of GOUDA-BT exceeds that of GBT by 1.90%. Note that the baselines GRACE and GBT adopt identical encoders and contrastive losses as GOUDA-IF and GOUDA-BT, respectively. Therefore, the observed performance improvement can be attributed to the adaptive modeling capacity for augmentations of the proposed GOUDA. Node Clustering. One can draw two conclusions from the observations in Tab. 3. Firstly, it is evident that GOUDA consistently surpasses all baselines (e.g., GRACE, and GBT) across all datasets, which illustrates the superior representation capacity of GOUDA. This can be attributed to the selfadaptive learning ability of the proposed module UGA. Secondly, GOUDA-IF consistently outperforms GOUDA-BT, suggesting that within the proposed GOUDA framework, the InfoNCE loss more effectively captures local information for clustering than the BarlowTwins loss. Table 3: Performances on node clustering: NMI & ARI Scores in percentage (mean). Cora CiteSeer PubMed NMI ARI NMI ARI NMI ARI DGI 52.75 47.78 40.43 41.84 30.03 29.78 MVGRL 54.21 49.04 43.26 42.73 30.75 30.42 GRACE 54.59 48.31 43.02 42.32 31.11 30.37 GBT 55.32 48.91 44.01 42.61 31.33 30.64 CCA-SSG 56.38 50.62 43.98 42.79 32.06 31.15 GOUDA-IF 57.92 52.41 45.11 43.82 33.17 31.98 GOUDA-BT 57.35 51.84 44.93 43.46 33.14 31.73 Graph Classification. The results of this experiment are presented in Tab. 4 and Fig. 3. Firstly, it can be observed from Tab. 4 that GOUDA outperforms the baselines regarding classification performance, which illustrates the general validity of GOUDA. In particular, GOUDA-IF and GOUDA-BT surpass the second-place MVGRL by 1.02% and 2.6%, respectively, on the IMDB-B dataset, which highlights the superiority of GOUDA. Moreover, GOUDA exceeds the GCLs employing learnable GAs, i.e., JOAO, and AD-GCL. This can be due to the unified ability of UGA to integrate diverse GAs, which provides GOUDA with broader augmentation options than the baseline. Secondly, as illustrated in Fig. 3, GOUDA achieves superior performance and consumes less time than the baselines. Specifically, two triangles, indicating the proposed GOUDA, are superior left and above the other shape in the figure. This implies GOUDA is lightweight, aligning with conclusions in Section 3.5. Besides, UGA introduces modest memory usage, which promises the scalability of GOUDA. Table 4: Performances on graph classification: accuracy in percentage (mean±std). Model IMDB-B IMDB-M COLLAB Infograph 73.03±0.87 49.69±0.53 82.00±0.29 GraphCL 71.14±0.44 48.58±0.67 71.36±1.15 JOAO 71.60±0.86 49.20±0.77 70.40±2.21 AD-GCL 71.49±0.90 50.36±0.74 74.89±0.90 MVGRL 74.20±0.70 51.20±0.50 73.10±0.60 GOUDA-IF 75.22±0.94 52.43±0.83 85.70±2.33 GOUDA-BT 76.80±0.98 53.05±0.72 85.15±2.17 (a) IMDB-B (b) IMDB-M Figure 3: Comparisons in terms of performance, running time, and GPU memory usage. The marker size indicates memory usage. 4.2 Additional Experiments Robustness Analysis. This experiment aims to evaluate the robustness of GOUDA against topology attacks (adding edges) and attribute attacks (flipping attributes). According to results in Fig. 4 and Fig. 5, several conclusions can be derived. Firstly, compared to the baselines using the same contrastive losses, GOUDA consistently achieves performance gains at all perturbation rates. It demonstrates the robustness of GOUDA against both topology and attribute attacks. This is attributed to the greater adaptability of the UGA module, stemming from its integration of augmentation and contrastive updating over random GAs. Secondly, attribute attacks cause more severe performance degradation than topology attacks, even for our proposed GOUDA. This could be because the node attributes, rich with class-discriminative information, are erased essential identification info during attacks. Ablation Study. This experiment aims to evaluate the contribution of individual components. To be specific, it introduces two variants: one without the structure features (in Eq. 6) and another without the independence loss (in Eq. 13). From Fig. 6, it is observable that the performance declined in both model variants compared to the complete model, which illustrates that the efficacy of GOUDA stems from the collective contribution of all components. Besides lacking Independence loss, GOUDA-IF performs inferior to GOUDA-BT, implying that InfoNCE might drive the model toward excessive 9 Figure 4: Topology attack effects on GCLs. Figure 5: Attribute attack effects on GCLs. consistency, diminishing its discriminative power. This highlights the critical role of independence loss in preserving diversity across augmentations. Figure 6: Contribution of individual components. Figure 7: Impact of the number of AC vectors. Parameter Sensitivity Analysis. These experiments are performed to offer an intuitive understanding of hyper-parameter selection. Firstly, as depicted in Fig. 7, which illustrates performance variance for varying k, GOUDA achieves consistently stable performances across {5, 10, 20}. Notably, the performance variation on these datasets remains minimal, staying within a 2% margin. Thus, GOUDA has low sensitivity to parameter k. Moreover, this parameter requires no significant value for GOUDA to perform well; a setting as low as 5 suffices. However, a value of 1 is inadequate due to the absence of augmentation diversity. The analysis of other hyper-parameters is given in Section E.6. 5 Conclusions In this paper, we present UGA, a unified Graph Augmentation (GA) module that addresses the issues in existing GAs, including specificity, complexity, and incompleteness. Motivated by the local attribute-modifying characteristics of GAs, UGA introduces a moderate number of AugmentationCentric (AC) vectors to simulate GA impact on nodes. We further propose GOUDA, a generalized Graph Contrastive Learning (GCL) framework built on UGA. GOUDA promotes both consistency and diversity across augmentations by employing a contrastive loss and an independence loss, respectively. Extensive evaluations demonstrate the generality and efficiency of GOUDA. However, the robustness analysis suggests a scope for enhancement in its robustness against attribute attacks. Future research could explore multi-modal learning methods that fuse diverse structural features into node attributes, aiming to better preserve discriminative information and thus enhance robustness. 6 Acknowledgments This work was supported in part by the National Key R&D Program of China (No. 2022ZD0119202), in part by the National Natural Science Foundation of China (No. U22B2036, 62376088, 62276187, 62102413, 62272020), in part by the Hebei Natural Science Foundation (No. F2024202047), in part by the Hebei Province Higher Education Science and Technology Research Project (No. QN2024201), in part by the National Science Fund for Distinguished Young Scholarship (No. 62025602), and in part by the XPLORER PRIZE. References [1] Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput., 15(6):1373–1396, 2003. 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Output: node representations H(1) ∈Rn×d and H(2) ∈Rn×d. Initialization: graph encoder gΘ(, ), projection heads g(·), and the matrices of Augmentation-Centric (AC) vectors Q ∈Rk×f and P ∈Rk×f. while not converged do % Augmentation % 1. G1 ←t(G, Q) and G2 ←t(G, P) via Eq. 6 and Eq. 7; % Encoding % 2. H(1) ←gΘ(G1) and H(2) ←gΘ(G2) via Eq. 2; % Calculating loss % 3. L ←Lcontrast(H(1), H(2)) + γLindep(Q, P) via Eq. 13; % Optimizing % 4. Θ ←Adam(L, Θ); end return H(1) ∈Rn×d, H(2) ∈Rn×d and gΘ(, ); B Introduction of Graph Augmenations Categorized by the type of graph information they manipulate, Graph Augmentations (GAs) can be broadly divided into four categories: node augmentation, edge augmentation, attribute augmentation, and subgraph augmentation. The detailed introduction and examples are given below. Node Augmentation. This augmentation generally creates the new graph by dropping or adding the perturbated nodes and the edges that connect to these perturbated nodes of the input graph, as shown in Fig. 1(a). Employing the adjacent matrix to formally represent the graph topology, the edge augmentation can be denoted by Gn(A(∗), X(∗)). Therefore, these can be formulated as Node Dropping : A(∗), X(∗) ←{V/¯V, E/ ¯E}, X/ ¯X (14) Node Adding : A(∗), X(∗) ←{V ∪¯V, E ∪¯E}, X\|\| ¯X, (15) where ¯V denotes the perturbated node set, ¯X stands for the attributes of these nodes, and ¯E terms the set of edges connected to these nodes. The operator is widely used on graph classification [48, 47]. Edge Augmentation. Unlike node augmentation, the edge augmentation, denoted by Ge(A(∗), X), exclusively operate on edges. It involves either removing perturbated edges from or adding perturbated edges to the input graph, as indicated in Fig. 1(b). These can be expressed as Edge Removing : A(∗) ←{E/ ¯E} (16) Edge Adding : A(∗) ←{E ∪¯E}, (17) where ¯E represents the set of perturbated edges, which is randomly determined [53, 54, 32, 20] or adaptively learned [31] during each training epoch. Attribute Augmentation. Typically, the attribute augmentation (expressed by Ga(A, X(∗))) generates the new graph by masking (in Fig. 1(c)) or corrupting the raw node attributes. These can be described as Attribute Masking : X(∗) = X ⊙M (18) Attribute Corrupting : x(∗) v = xv + δv, (19) 14 where M ∈Rn×f stands for the mask matrix and δv terms the noise vector for node v, which is updated iteratively by adversarial training [19, 44]. Subgraph Augmentation. As typical graph-level operators, the subgraph augmentation crops out subgraphs (in Fig. 1(d)) or inserts additional subgraphs to create new graphs, as follows Subgraph Cropping : Gs(A(∗), X(∗)) ←G ∪¯G( ¯A, ¯X) (20) Subgraph Inserting : Gs(A(∗), X(∗)) ←G/ ¯G( ¯A, ¯X), (21) where ¯G( ¯A, ¯X) stands for the perturbated subgraph. The subgraph augmentation is mostly used for graph-level tasks [47, 13]. C Introduction of Graph Contrastive Losses The contrastive loss serves as a crucial technique that enhances data representation through discrimination. Generally, it operates on two levels of representation matrices: the sample level [4, 53], where it aligns the representations of positive samples and uniformly distributes all representations, and the feature level [49, 2], where it targets reducing redundancy between features. Sample-level contrastive losses. In the nascent stages of research, the designs of contrastive losses were inspired by the success of contrastive learning (CL) in computer vision (CV) [4]. To be specific, this type of contrastive loss aims to minimize the distance between the anchor sample and positive samples while maximizing the distance between the anchor sample and negative samples [33]. As a typically sample-level contrastive loss, InfoNCE loss [33] classifies the embeddings of the same node from different views as positive samples while treating the embeddings from all other nodes as negative samples. This loss can be formulated as LInfoNCE = 1 2n X v∈V ℓ(h(1) v , h(2) v ) + ℓ(h(2) v , h(1) v ) , (22) where ℓ h(1) v , h(2) v = −log eΦ(h(1) v ,h(2) v )/τ P u∈V e Φ h(1) v ,h(2) u /τ + P u∈V, u̸=v e Φ h(1) v ,h(1) u /τ , (23) where Φ(hv, hu) = s (g(hv), g(hu)) stands for the feature similarity function, and g(·) represents the projection heads [4], and s(·) terms the consine similarity. τ denotes the temperature coefficient. Feature-level contrastive losses. This type of loss is designed to directly optimize contrastive losses in the feature space, bypassing the need for defining explicit positive and negative samples, thus overcoming sample selection challenges. Barlow Twins (BT) [49] and CCA-SSG [50] are two notable approaches aimed at improving feature representation learning by minimizing redundancy between feature dimensions. BT enforces the mutual information matrix computed on the features from two different views to approximate the identity matrix, ensuring that the learned representations are free from redundant information. This can be formulated as LBarlowTwins = f−1 X i=0 (1−ci,i)+λ f−1 X i=0 f−1 X j=0 j̸=i (ci,j)2, ci,j = P v∈V hv,i × hv,j qP v∈V (hv,i)2 × qP v∈V(˜hv,j)2 , (24) where λ denotes a hyperparameter to tradeoff two terms. CCA-SSG not only pushes the covariance matrices from each view to approach an identity matrix but also enhances feature consistency across views to learn informative representations of both unique and shared data characteristics. D Theoretical analysis D.1 Proofs for Theorem 3.1 For the sake of clarity, let us briefly describe the whole process. Firstly, the proof proceeds from the premise of a one-layer graph encoder without activation functions. For node-level tasks, the graph 15 encoder gΘ(, ) is configured as a one-layer GCN. For graph-level tasks, gΘ(, ) is set as a one-layer GIN with a sum pooling sum(·). Subsequently, the obtained conclusions are generalized to the case of multi-layer encoders. Specifically, for the one-layer graph encoders, the encoding process for node representations (H) and graph representations (h) can be expressed as Node representations: H = GCN (A, X) = ˜A · X · W (25) Graph representations: h = sum(GIN(A, X)) = sum ((A + (1 + ϵ)I) · X · W) , (26) where ˜A = ˆD−1 2 ˆA ˆD−1 2 denotes the normalized adjacency matrix and ϵ terms an learnable parameter. In the proposed UGA implementation, both node and graph representations can be decoupled into two terms: the original representations (represented as H and h), which can be regarded as directly calculated from Eq. 25 and Eq. 26, respectively, and the augmented representations (denoted as △Hq and △hq). To be specific, two terms of the node representations can be expressed as Hq = GCN (A, X + Q) = ˜A · (X + Q) · W = ˜A · X · W + ˜A · Q · W = H + △Hq. (27) Similarly, two terms of the graph representations can be described as hq = sum (GIN (A, X + Q)) = sum ((A + (1 + ϵ)I) · (X + Q) · W) = sum ((A + (1 + ϵ)I) · X · W) + sum ((A + (1 + ϵ)I) · Q · W) = h + △hq. (28) Next, to establish Theorem 3.1, Lemma D.1 is introduced. Lemma D.1. For any GA t(·) applied to the input graph G(A, X), it can be decoupled to a series of three augmentations: attribute augmentation (ta(X)), edge augmentation (te(A)), and subgraph augmentation (ts(A, X)) containing node augmentation [8]. Accordingly, the candidate spaces (i.e., A and X) can be created through these three augmentations. Theorem 3.1 can be proven based on this lemma by establishing the following three propositions. Proposition D.2. From the message-passing perspective of the graph encoder gΘ(, ), the proposed implementation of UGA, which is expressed as Eq. 4, can be equivalent to any attribute augmentation ta(X), that is gΘ(A, X + Q) = gΘ(A, X(∗)), (29) where X(∗) = ta(X) stands for the augmented node attirbutes. Proof. Firstly, let us discuss the equivalence for node representations. To establish Eq. 29, the goal is to identify Q such that Hq = Ha. Let △X = X −X(∗) denote the variance in attributes resulting from attribute augmentation, Ha can be decomposed into two terms: the original representations H and the augmented representations △Ha. This decomposition can be expressed as Ha = GCN (A, X + △X) = ˜A · (X + △X) · W = ˜A · X · W + ˜A · △X · W = H + △Ha. (30) Therefore, the proof shifts from establishing Hq = Ha to demonstrating ∆Hq = ∆Ha. It becomes evident that this equivalence holds true under the condition qi,j = ∆xi,j, thus ensuring that ˜AQW = ˜A∆XW. Moreover, this derivation highlights that the conclusion remains consistent regardless of the encoder chosen. Hence, the solution qi,j = △xi,j holds for the node representations encoded using GIN. 16 Besides, the conclusions drawn at the node level, unaffected by the choice of readout functions (e.g., mean and sum), are equally applicable to the graph level. This insight extends our solution to graph representations h, thereby completing the proof. Proposition D.3. From the message-passing perspective of the graph encoder gΘ(, ), the proposed implementation of UGA, which is expressed as Eq. 4, can be equivalent to any edge augmentation te(A), that is gΘ (A, X + Q) = gΘ A(∗), X , (31) where A(∗) = te(A) denotes the adjacency matrix obtained from edge augmentation. Proof. Consistent with the method adopted in the above proofs, the edge-augmented representations (He and he) are first computed. Next, the equivalence of these representations with UGA counterparts (Hq and hq), respectively, is demonstrated. Let △A = A −A(∗) denotes the topology variance caused by edge augmentation, He can be decoupled into two terms as follows. He = GCN (A + △A, X) = ˜A + △˜A · X · W = ˜A · X · W + △˜A · X · W = H + △He. (32) where △˜A = ˜A−˜A(∗) terms the topology variance between the adjacency matrix and its augmented version, both of which are normalized. Therefore, this demonstration solely necessitates establishing the equivalence between △Hq and △He, particularly ˜A · Q · W = △˜A · X · W. (33) Through the utilization of Cayley-Hamilton theorem [6], which asserts that every matrix adheres to its own characteristic polynomial, that is Γ(A) = An + c1An−1 + c2An−2 + · · · + cnI = 0, (34) where {ci}n i = 1 stands for the set of polynomial coefficients. Thus, the inverse of matrix ˜A can be expressed as ˜A−1 = −1 cn ˜An−1 −c1 cn ˜An−2 −· · · −cn−1 cn I. (35) Based on it, the equivalent between the proposed UGA and edge augmentation (denoted as Eq. 33) can be demonstrated if it holds that qi,j = [(−1 cn ˜An−1 −c1 cn ˜An−2 −· · · −cn−1 cn I) · △˜A · X]i,j. (36) Furthermore, the solution derived for node representations is directly applicable to graph representations as well, with the sole modification being the substitution of ˜A with A + (1 + ϵI). In light of the above analysis, this proposition is proven. Proposition D.4. From the message-passing perspective of the graph encoder gΘ(, ), the proposed implementation of UGA, which is expressed as Eq. 4, can be equivalent to any subgraph augmentation ts(A, X), that is gΘ(A, X + Q) = gΘ(A(∗), X(∗)), (37) where A(∗), X(∗) = ts(A, X) stand for the adjacency matrix and attribute matrix obtained from subgraph augmentation. Proof. Note that the subgraph augmentation is typically tailored for graph-level tasks. For node-level tasks, specific subgraph augmentation (where the number of nodes remains unchanged) can be regarded as a type of edge or attribute augmentation, such as masking all attributes of nodes in the 17 perturbated subgraph. Hence, based on Proposition D.2 and Proposition D.3, it is not hard to conclude that this proposition holds. Recall that within our UGA, the graph representation hq ∈R1×d is articulated as hq = h + △hq. This can be formulated as hq = sum (GIN (A, X + Q)) = sum ((A + (1 + ϵ)I) · X · W) + sum ((A + (1 + ϵ)I) · Q · W) = h + △hq. (38) Furthermore, let us assume that the original graph encompasses k subgraphs. The corresponding subgraph-augmented graph, in turn, encompasses m −1 subgraphs, which can be formulated as A =   A0 0 · · · 0 0 A1 · · · 0 ... ... ... ... 0 0 · · · Am−1  , X =   X0 X1 ... Xm−1  , (39) where Ai denotes the adjacency matrix of the i-th subgraph and Xi terms the corresponding attribute matrix. Thus, for the augmented subgraphs {G(At, Xt)}m−1 i=0 , the graph representation hs can be formulated as hs = m−1 X t=0 sum((At + (1 + ϵ)I) · Xt · W), (40) Let △h = hs −h denotes the representation variance caused by subgraph augmentation, there is △h = m−1 X t=0 Ωt · sum((At + (1 + ϵ)I) · Xt · W), (41) where Ωdenotes an indicator vector. If m −1 < k, the subgraph augmentation typically refers to the subgraph corrupting (denoted as Eq. 20). Thus, for the t-th perturbated subgraph, there is Ωt = −1. And if m −1 > k, is generally understood as the subgraph inserting (formulated as Eq. 21), Ωt = 1. Additionally, if the subgraph is not changed, Ωt = 0. Next, we aim to identify a solution for q that satisfies the following conditions: △hq = △h, (42) This can be further formulated as sum ((A + (1 + ϵ)I) · Q · W) = m−1 X t=0 Ωt · sum ((At + (1 + ϵ)I) · Xt · W) = d⊤· Q · W = m−1 X t=0 Ωt · sum ((At + (1 + ϵ) I) · Xt) · W, (43) where d = [d0 + 1 + ϵ, . . . , dn−1 + 1 + ϵ] ∈Rn stands for a degree vector. In the general case, assuming the absence of isolated nodes within the graph, the degree vector d of the input graph is devoid of zero elements. Consequently, a solution for Q can be formulated as Q = D+ ˜H, (44) where ˜H = Pm−1 t=0 Ωt · sum((At + (1 + ϵ)I) · Xt) and D+ = 1 dd⊤d denotes the Moore-Penrose pseudoinverse of d. Thus, equivalence can be established if each element qi,j in AC matrix Q satisfies the following condition: qi,j = [D+ · m−1 X t=0 sum ((At + (1 + ϵI)) · Xt)]i,j. (45) Therefore, the proof ends. 18 Extension to multi-layer graph encoders. Following the discussion on single-layer graph encoders, solutions for multi-layer graph encoders are identified. Initially, as discussed in [18], various GNNs can be formulated as H = S · X · W, (46) where S denotes the diffusion matrix, exemplified by ˜A for GCN and A + (1 + ϵ)I for GIN. In addition, W represents the projection layer, such as the linear projection utilized in both GCN and GIN. Here, the nonlinear activation function between layers is not considered. With this architecture, the node representations at the k-th layer can be formulated as H = gl Θ(S, X) = S · X · W, (47) where S = Ql i=0 Sl terms the product of adjacency matrices. Similarly, W = Ql i=0 Wl represents the product of linear projection matrices. Given that the aforementioned conclusions are independent of the forms of these two matrices, it is not hard to prove that by substituing A and W with S and W in Eq. 25 and Eq. 26, respectively, Proposition D.2, D.3, and D.4 still hold true. D.2 Proofs for Theorem 3.4 Proof. Firstly, note that two augmented graphs are from the same input graph G(A, X) without loss of the information in this graph (especially the edges and attributes). Therefore, there is A(1) = A(2) and X(1) = X(2). Accordingly, the diversity can be transformed into Dv = ∥ X t∈Nv∪v (x(1) t,: + ˆqt,:) − X t∈Nv∪v (x(2) t,: + ˆpt,:)∥2 F (48) = ∥ X t∈Nv∪v ˆqt,: − X t∈Nv∪v ˆpt,:∥2 F (49) = ∥ X t∈Nv∪v bq t,:Q − X t∈Nv∪v bp t,:P∥2 F . (50) Given the conditions bq t,: = σ(x(1) t,: Q⊤) where σ denotes the softmax function, we can calculate that Dv = ∥ X t∈Nv∪v bq t,:Q − X t∈Nv∪v bp t,:P∥2 F (51) = ∥ X t∈Nv∪v σ(x(1) t,: Q⊤)Q − X t∈Nv∪v σ(x(2) t,: P⊤)P∥2 F (52) = ∥( X t∈Nv∪v σ(x(1) t,: Q⊤))Q −( X t∈Nv∪v σ(x(2) t,: P⊤))P∥2 F . (53) For clarity, xt,: is employed to represent both x(1) t,: and x(2) t,: . In light of the consistency constraint within the GOUDA framework, the difference between Q and P is minimal, which can be interpreted as a minor perturbation ∆, such that Q = P + ∆. Therefore, Eq. 53 can be further reformulated as Dv = ∥( X t∈Nv∪v xt,:Q⊤)Q −( X t∈Nv∪v xt,:P⊤)P∥2 F (54) = ∥( X t∈Nv∪v xt,:(P + ∆⊤)(P + ∆) −( X t∈Nv∪v xt,:P⊤)P∥2 F . (55) Given that ∆is considered to be small, the terms ∆2 and xt,:∆⊤can be neglected. Moreover, it can be assumed that the product of xt,:∆⊤with P and ∆is insignificant in comparison to the other terms. Hence, the above formulation can be simplified as Dv ≈∥( X t∈Nv∪v xt,:P⊤)∆∥2 F . (56) Since P t∈Nv∪v xt,:P⊤ is a constant matrix, it can be factored out, resulting in: Dv ≈( X t∈Nv∪v xt,:P⊤)2∥∆∥2 F = ( X t∈Nv∪v xt,:P⊤)2∥Q −P∥2 F . (57) 19 Hence, taking into account that P t∈Nv∪v xt,:P⊤2 serves as the proportionality constant, which depends on the values of xt,: and P, we can deduce that Dv ≈∥Q −P∥2 F . (58) Based on the above analysis, we conclude the proof. E Experimental Details E.1 Introduction of Datasets Datasets for node-level tasks. • Citation networks [28]: Cora, Citeseer, PubMed. Each node in these networks represents a scholarly article, with edges indicating citation relationships. Nodes are defined by attributes such as abstracts, keywords, full-text content, and derived features like TF-IDF vectors. Labels to which nodes belong, typically corresponding to research areas or topics. • Reference network [25]: Wiki-CS. This network represents a collection of Wikipedia articles in computer science. Each node corresponds to an article, characterized by its text and hyperlinks, while the edges depict hyperlinked references between articles. Node labels denote specific subfields of computer science covered by the articles. • Co-purchase networks [29]: Amazon Photo (Photo for short), Amazon Computers (Computers for short). Nodes represent products available for purchase, with attributes such as features, prices, and customer reviews. Node labels correspond to product types or brands, while edges indicate co-purchase relationships, reflecting the frequency of items commonly bought together by customers. • Co-author network [29]: Coauthor Physics (Physics for short). Nodes represent physicists, each described by their publication record, research interests, and affiliations. Labels indicate distinct areas or subfields within physics. Edges between nodes stand for collaborations, typically formed through joint publications or co-authorship of scientific papers. Datasets for graph-level tasks. • Collaborative movie networks [42]: IMDB-BINARY (IMDB-B for short) and IMDBMULTI (IMDB-M for short). Nodes denote actors or actresses, and an edge exists between two nodes if the individuals have co-starred in the same film. • Scholarly collaboration network [42]: COLLAB. Researchers as nodes and edges indicate partnerships between them. It is important to mention that node attributes are absent in the three datasets for graph-level tasks, making one-hot encoding of the degree a typical approach. We source these datasets from the public repository PyTorch Geometric (PyG) [9]. The datasets can be accessed through the URLs listed below: • Cora, CiteSeer, PubMed: https://github.com/kimiyoung/planetoid/raw/master/data. • Wiki-CS: https://github.com/pmernyei/wiki-cs-dataset/raw/master/dataset. • Photo, Computers: https://github.com/shchur/gnn-benchmark/raw/master/data/npz/. • Physics: https://github.com/shchur/gnn-benchmark/raw/master/data/npz/. • IMDB-B, IMDB-M, COLLAB: https://ls11-www.cs.tu-dortmund.de/staff/morris/graph kerneldatasets. E.2 Dataset Splitting For the seven benchmark datasets utilized for node classification tasks (Cora, Citeseer, PubMed, WikiCS, Computers, Photo, and Physics), the dataset is divided into training, validation, and testing sets in the ratio of 1:1:8. For the three benchmark datasets employed for graph classification tasks, namely IMDB-B, IMDB-M, and COLLAB, a 10-fold cross-validation approach is adopted to partition. 20 Table 5: Statistics of ten graph benchmark datasets. Node-level tasks Graph-level tasks Cora CiteSeer PubMed Wiki-CS Computers Photo Physics IMDB-B IMDB-M COLLAB # Graphs 1 1 1 1 1 1 1 1,000 1,500 5,000 # Nodes 2,708 3,327 19,717 11,701 13,752 7,650 34,493 19.8 13.0 74.5 # Edges 5,429 4,732 44,338 216,123 245,861 119,081 991,848 193.1 66.0 4914.4 # Features 1,433 3,703 500 300 767 745 8,451 # Classes 7 6 3 10 10 8 5 2 3 3 E.3 Introduction of Baselines Baselines for node-level tasks. Details on the baselines for node-level tasks are outlined below. GCN [17]: It is a representative Graph Neural Network (GNN) that utilizes spectral and spatial strategies to perform graph convolutional operations. It makes each node aggregate information from its neighbor nodes by integrating the graph topology and node attributes. GAT [34]: It introduces an attention mechanism into the GNN, enabling each node to weigh the importance of its neighbor nodes during the aggregation process. Unsupervised baselines are detailed below. DGI [35]: An Infomax principle-based GCL augments the graph via row-wise shuffling of the attribute matrix and maximizes the mutual information between global and local representations. GMI [26]: It is a variant model of DGI that maximizes a comprehensive graphical mutual information metric, including features and edges between nodes in both the input and reconstructed output graphs. MVGRL [14]: It is a variant model of DGI, employing contrastive learning between various structural views of graphs, including first-order adjacency and graph diffusions. GRACE [53]: It is a GCL model that generates node embeddings by corrupting both graph structure (via random edge removing) and attributes (via random attribute masking) to create diverse views and maximize their agreement. GCA [54]: It is a variant model of GRACE, which incorporates adaptive augmentation strategies based on node centrality, aiming to enhance the flexibility of the model. BGRL [32]: It is a GCL model that augments the graph through random edge removing and employs bootstrapping to update the parameters of the online encoder. GBT [2]: A feature-level GCL model leverages the validated Barlow Twins loss for training, aiming to reduce redundant information between two views obtained through random edge removing. CCA-SSG [50]: It is a GCL model utilizing feature-level contrast derived from Canonical Correlation Analysis (CCA) to learn the node representations. SPAN [21]: It introduces a spectral augmentation scheme for topology augmentation by perturbing the graph spectrum, and aiming to maintain spectral invariance to sensitive structures, and minimizing graph spectrum changes. DSSL [39]: It is a graph self-supervised model that presents an approach that disentangles the varied neighborhood contexts of a node, aiming to model multifaceted information within the graph. HomoGCL [20]: It is a localized variant model of GRACE, incorporating k-mean-based saliency values to weigh the importance of neighbor nodes. Baselines for graph-level tasks. Introductions of the baselines for graph-level tasks are given below. InfoGraph [30]: It is a variant of DGI, which maximizes mutual information between graph-level representations and substructures at different scales, such as nodes, edges, and triangles. GraphCL [48]: It is a GCL framework that learns graph representations by employing various augmentation techniques on the local subgraphs of nodes. JOAO [47]: It is a variant of GraphCL, which utilizes the min-max optimization to automatically select the most effective GAs during the contrastive learning process. 21 AD-GCL [31]: It is a GCL framework based on adversarial training, introducing an attack process to modify the edges. For the baseline implementations, we utilize PyG to implement GCN and GAT. In addition, for the self-supervised baselines, we employ their source codes. The sources are listed below: • GCN, GAT: https://github.com/pyg-team/pytorch_geometric/tree/master/torch_geometric /nn/conv. • DGI: https://github.com/PetarV-/DGI. • GMI: https://github.com/zpeng27/GMI. • MVGRL: https://github.com/kavehhassani/mvgrl. • GRACE: https://github.com/CRIPAC-DIG/GRACE. • GCA: https://github.com/CRIPAC-DIG/GCA/. • BGRL: https://github.com/nerdslab/bgrl. • GBT: https://github.com/pbielak/graph-barlow-twins. • CCA-SSG: https://github.com/hengruizhang98/CCA-SSG. • SPAN: https://github.com/haonan3/spgcl. • DSSL: https://papers.nips.cc/paper_files/paper/2022/file/040c816286b3844fd78f2124eec 75f2e-Supplemental-Conference.zip. • HomoGCL: https://github.com/wenzhilics/HomoGCL • InfoGraph: https://github.com/sunfanyunn/InfoGraph • GraphCL: https://github.com/Shen-Lab/GraphCL. • JOAO: https://github.com/Shen-Lab/GraphCL_Automated. • AD-GCL: https://github.com/susheels/adgcl. E.4 Complexity Analysis This subsection analyzes the complexity of GOUDA in comparison with the baseline GCL equipped with learnable GAs (i.e., SPAN, JOAO, and AD-GCL). Note that since the updates of these GCLs can utilize the same graph encoder (e.g., GCN) and contrastive loss (e.g., InfoNCE loss), this discussion focuses solely on the time complexity of the augmentation phase. To enhance clarity, let us define our terms: n represents the number of nodes corresponding to the network size; m signifies the number of edges; f refers to the dimension of attributes; and d denotes the dimension of the hidden layers. SPAN [21]: The time complexity for augmentations in SPAN is O(n2tk), where t denotes the time of iterations and k is the number of eigenvalues to be selected. Specifically, SPAN necessitates an iterative optimization of the augmentation through eigen-decomposition, which demands a O(tn3) complexity. Nonetheless, this complexity can be reduced to O(n2tk) by employing selective eigendecomposition on the k lowest- and highest-eigenvalues via the Lanczos Algorithm. JOAO [47]: The time complexity for augmentations in JOAO is O(n2d). Specifically, JOAO employs a min-max optimization strategy to refine the parameters that are used for selecting augmentations from an option pool. This process involves maximizing the contrastive loss, which inherently requires a O(n2d) complexity. AD-GCL [31]: The time complexity for augmentations in AD-GCL is O(n2d). Similarly, AD-GCL employs a min-max optimization strategy, but its objective is to modify edges. Therefore, this process also entails maximizing the contrastive loss, which carries a O(n2d) complexity. In addition, encoding edges is required, which introduces a complexity of O(md2). GOUDA : The time complexity for augmentations in the proposed GOUDA is O(nkf). GOUDA introduces k AC vectors to nodes and utilizes the consistency-diversity balance principle to update these AC vectors, where k ≪n. Firstly, nodes are updated by aggregating the features of AC vectors, which incurs a complexity of O(nkf). Besides, GOUDA maintains consistency by minimizing the contrastive loss, a process inherent to the training model and thus does not introduce additional 22 complexity; to ensure diversity, it calculates the independence loss, which brings in a complexity of O(k2f). Therefore, the overall complexity is O(nkf). In summary, GOUDA presents a more computationally efficient approach than the baselines, which is supported by the evidence presented in Section 4. E.5 Configurations and Hyper-parameters E.5.1 Configurations The experiments leverage the linear evaluation method [26], where models are firstly trained in an unsupervised manner, and then, the obtained embeddings are utilized for downstream tasks. For the node-level tasks, the graph encoder gΘ is configured as a two-layer GCN [17], while for the graph classification tasks, it is set as a five-layer GIN [40]. In the evaluation phase, we utilize a single-layer linear classifier [27] for node classification [26], apply K-means [16] to node embeddings for node clustering, and train an SVM classifier [3] for graph classification. The results for node-level tasks are the average of ten random runs, while those for graph-level tasks are based on five runs. E.5.2 Environment All experiments are conducted on two Linux machines as shown in Tab. 6. Table 6: Experimental environment servers. Server 1 Server 2 OS Linux 5.15.0-82-generic Linux 5.15.0-100-generic CPU Intel(R) Core(TM) i7-12700K CPU @ 3.6GHz Intel(R) Core(TM) i9-10980XE CPU @ 3.00GHz GPU GeForce RTX 4090 GeForce RTX 3090 E.5.3 Hyper-parameter Settings GOUDA is implemented as two models: GOUDA-IF, which utilizes InfoNCE loss, and GOUDABT, which employs BarlowTwins loss. For the node-level tasks, both models are trained using an Adam optimizer with a learning rate of 1e−3 and the weight decay rate from {0, 5e−5, 5e−4}. The dimensions d of node embeddings are selected from {256, 512, 1024, 2048}, and their impact is analyzed in Section 4.2. The hyperparameters β1 and β2 of independence loss are chosen from {1e−4, 1e−3}, while the related hyperparameter γ is selected among {1e−2, 1e−1, 1, 10}. Additionally, for GOUDA-IF, the temperature coefficient τ is selected from {0.2, 0.4, 0.6, 0.8}, while for GOUDA-BT, the hyperparameter λ is set to 1 d. For the graph-level task, the configuration follows GraphCL [48], where the hidden dimension is fixed to 128 and the penalty parameter of SVM is selected from {1e−3, 1e−2, 1e−1, 1, 1e2, 1e3}. The choice of threshold ϵ is given in Section E.6.2. E.6 Additional Experiment Results E.6.1 Complete Results for Node Clustering Tab. 7 presents the comprehensive results of the node clustering experiments. The analysis of these results can be found in Section 4. E.6.2 Other Hyperparameter Analysis Embedding Dimension. This experiment aims to shed light on the selection of hyperparameter d. As depicted in Fig. 8, the proposed GOUDA shows improved performance with an increased embedding dimension. Notably, GOUDA-IF and GOUDA-BT exhibit reduced performance at an embedding dimension of 256 compared to higher dimensions. This indicates that a large embedding dimension is essential for contrastive learning models to capture robust representations. Moreover, it is observable that there is a slight decrease in the performance of GOUDA-IF with large dimensions. This can be due to overfitting to the self-supervised signal, which may hinder its generalization capability. 23 Table 7: Node clustering performance: NMI & ARI Scores in percentage (mean±std). Cora CiteSeer PubMed NMI ARI NMI ARI NMI ARI DGI 52.75±0.94 47.78±0.65 40.43±0.81 41.84±0.62 30.03±0.50 29.78±0.28 MVGRL 54.21±0.25 49.04±0.67 43.26±0.48 42.73±0.93 30.75±0.54 30.42±0.45 GRACE 54.59±0.32 48.31±0.63 43.02±0.43 42.32±0.81 31.11±0.48 30.37±0.51 GBT 55.32±0.65 48.91±0.73 44.01±0.97 42.61±0.63 31.33±0.57 30.64±0.74 CCA-SSG 56.38±0.62 50.62±0.90 43.98±0.94 42.79±0.77 32.06±0.40 31.15±0.85 GOUDA-IF 57.92±0.49 52.41±0.58 45.11±0.79 43.82±0.65 33.17±0.45 31.98±0.46 GOUDA-BT 57.35±0.51 51.84±0.61 44.93±0.85 43.46±0.71 33.14±0.51 31.73±0.52 Figure 8: Impact of dimension d. Figure 9: Impact of hyperparameter γ. Weight of Independence Loss. Several insights are yielded from observations in Fig. 9. Firstly, the proposed GOUDA shows stability regardless of parameter γ changes. Secondly, the framework maintains robust performance even at a low value of 0.01 and 0.1. However, the omission of the proposed Independence loss does have a detrimental effect, as demonstrated by the Ablation Study (in Section 4.2). Lastly, while the tested parameter range is {1, 10}, future research should consider broader or more detailed ranges. Threshold for sparsification. The performance changes resulting from varying the hyperparameter ϵ are detailed in Tab. 8. To eliminate bias due to network size, ϵ is not freely tuned. Instead, it is set as the output of the selection function ϵ = selection(B, s), which estimates this threshold. B denotes the matrix of propagation weights from AC vectors to nodes, and s stands for the proportion of the largest elements retained. The value of s is chosen from the set {0.2, 0.4, 0.6, 0.8} in the experiments. From Tab. 8, it can be observed that the threshold does not significantly affect the model performance. Specifically, within the range of selection, the variation in model performance does not exceed 2%. Table 8: Impact of threshold ϵ. GOUDA-IF GOUDA-BT 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Cora 85.29 84.71 84.19 86.11 85.07 84.56 85.88 85.99 CiteSeer 74.55 73.20 73.11 73.62 74.34 74.47 73.98 73.41 PubMed 86.96 87.25 87.55 87.05 87.59 86.94 86.76 87.11 24 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The abstract and introduction clearly outline the contributions of our paper, including the motivation and design of the UGA module and the GOUDA framework. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,433	Optimistic Verifiable Training by Controlling Hardware Nondeterminism Megha Srivastava∗ Department of Computer Science Stanford University megha@cs.stanford.edu Simran Arora Department of Computer Science Stanford University simarora@stanford.edu Dan Boneh Department of Computer Science Stanford University dabo@cs.stanford.edu Abstract The increasing compute demands of AI systems have led to the emergence of services that train models on behalf of clients lacking necessary resources. However, ensuring correctness of training and guarding against potential training-time attacks, such as data poisoning and backdoors, poses challenges. Existing works on verifiable training largely fall into two classes: proof-based systems, which are difficult to scale, and “optimistic” methods that consider a third-party auditor who can replicate the training process and contest the trainer. A key challenge with the latter is that nondeterminism between GPU types during training prevents exact replication of the training process, resulting in schemes that are non-robust. We propose a method that combines training in a higher precision than the target, rounding after intermediate computations, and sharing rounding decisions based on an adaptive thresholding procedure, to successfully control for nondeterminism. Across three different NVIDIA GPUs (A40, Titan XP, RTX 2080 Ti), we achieve exact training replication at FP32 precision for both full-training and fine-tuning of ResNet-50 (23M) and GPT-2 (117M) models. Our verifiable training scheme significantly decreases the storage and time costs compared to proof-based systems, and is publicly released at https://github.com/meghabyte/verifiable-training. 1 Introduction We are currently in the “large-scale era” of machine learning (ML), where the exciting capabilities of modern AI systems have required a dramatic increase in training compute needs [Sevilla et al., 2022]. In turn, several model training services, such as Replicate, OpenAI’s Finetuning API, Together AI, Amazon Sagemaker, MosaicML Training, and Gensyn, have been created to support clients who lack the resources to train a model themselves. However, these services require clients to place a significant degree of trust in them to train the model correctly, without introducing a training-time attack such as data poisoning or undetectable backdoors [Wan et al., 2023, Goldwasser et al., 2022]. How can we help a client, such as an individual or a small company, hold the service provider accountable in case of misbehavior during training? Consider an education start-up that wishes to finetune the Llama-70b language model (70B parameters) on their own curated dataset to support student learning. This task requires significant resources, ∗Correspondence to megha@cs.stanford.edu. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). and the company might even lack the necessary expertise. Instead, they might choose to pay a trainer with vast computing resources to perform the training task (Figure 1). However, what if the trainer adds data points that spread misinformation, introduces a backdoor that advances a political agenda for specific prompts, or tries to save work by under-training the model? If the client starts to notice suspicious model behavior, is there any action they can take? We study this problem of verifiable training, or ensuring that the training of an ML model was performed correctly. Malicious Trainer Auditor a1s2 Epoch 1 sd23 Epoch 2 cbp2 Epoch 3 fd2u Epoch 4 a1s2 Epoch 1 sd23 Epoch 2 d23f Epoch 3 df4g Epoch 4 (4) share randomness, architecture, and rounding logs abc3 s9f4 3ab8 bcy8 ds5t m7c3 abc3 9abf kl3r 8b7a t2e3 p4l3 Client (3) share data (1) share data (5) verification game (6) auditor disputes trainer (2) training-time attack (e.g. data poisoning) Figure 1: Overview of our scheme. After an auditor challenges a trainer, they train the model, storing weights in a Merkle tree, and enter a binary search procedure to identify the exact steps of the dispute. We show how to control GPU nondeterminism between auditor and trainer, expanding the set of potential auditors. One possibility is for the trainer to provide the client with a cryptographic proof that the model was trained according to the specification. However, proof-based systems require cryptographic techniques that can be difficult to scale to the complexity of real-world ML systems. For instance, recent work based on zero-knowledge proof systems for verifiable inference, a much simpler task than training, requires more than 8 minutes to generate proofs for only 20 images [Liu et al., 2021]. Thus, practical proof-based methods for verifiable training have only been implemented for simple tasks such as logistic and linear regression [Garg et al., 2023, Ames et al., 2022]. An alternative “optimistic” approach is to consider a third-pary auditor (Figure 1). This could be a trusted 3rd party, such as a non-profit organization that may not have sufficient computing resources to provide training as a service beyond auditing, or a different provider that the client approaches and wishes to compare with the original model trainer. When a client suspects foul play, they can ask the auditor to challenge the trainer by training the model using the auditor’s own compute, and demonstrate that the trainer did not train correctly. Based on the provided evidence required from the auditor (i.e. the precise timesteps model training diverged, as shown in Figure 1), the client can then choose to refuse the trainer’s model, pursue legal action against the trainer, or even dispute a potentially corrupt auditor if the client deems such evidence as incorrect, or another auditor disagrees. This protocol can be efficiently carried out using techniques from the literature on verifiable computing, such as the “verification game” method of Teutsch and Reitwießner [2019], which uses an interactive binary-search procedure to identify the exact intermediate computation step (e.g. training epoch) where the two parties diverged. Applying verifiable computation techniques to model training is particularly important given the increase in decentralized machine learning services like Gensyn, which seek to make ML compute more accessible by creating a network of many untrusted GPUs. Unfortunately, the issue with such “optimistic” approaches is nondeterminism during model training: two models trained on different GPU types, even with same data order and random seed, can learn different weights (Figure 2). The auditor cannot simply compare their model weights with the trainer’s, and recent work has shown that protocols based on comparing model weights, such as Jia et al. [2021]’s “proof of learning,” are not robust and can be forged due to errors from nondeterminism [Thudi et al., 2022, Fang et al., 2023]. Our work addresses this limitation by asking: can the trainer provide additional information to the auditor that removes the effects of hardware nondeterminism? Our starting point is the observation that hardware nondeterminism occurs due to the accumulation of errors from floating point operations. For example, a matrix-vector multiply often results in different floating point values on different GPUs, since GPUs often accumulate in different orders. To address this issue, a natural approach is to perform training using a higher precision (e.g. FP32) than the target precision of the model weights (e.g. FP16), and periodically round back to the target precision. The hope is that all floating point errors will be confined to the higher precision bits, so that the rounded values are deterministic. However, this fails because computed values can occasionally straddle the “rounding boundary”: i.e., 2 the trainer can round up while the auditor rounds down, quickly causing them to diverge. Instead, we propose a solution where the trainer records the rounding direction for certain intermediate computation so that auditor can stay in sync with the trainer. As this requires the trainer to record a large number of bits, we also show how to reduce the amount of data needed to eliminate errors. We use this strategy to adapt the verification game described by Teutsch and Reitwießner [2019] for verifiable training. The game’s efficiency lies in our ability to store hashes of model checkpoints in a Merkle tree [Merkle, 1988]. To determine if training was performed according to the specification, the auditor needs to reconstruct the Merkle tree and compare the resulting Merkle root hash with the Merkle root hash provided by the trainer’s – if they do not match, the two parties can enter an interactive binary search procedure to identify the exact training step of the dispute. The purpose of the binary search game is to hold both parties accountable: an auditor should not be able to simply claim that a model was improperly trained, but convince a third-party (e.g., the public, or a judge) by showing at what point during training the trainer misbehaved. We show our verifiable training scheme can scale to tasks such as full training of ResNet-50 (23M parameters) and finetuning of GPT-2 (117M parameters), significantly outperforming existing methods with respect to both time and storage cost, while eliminating statistical error due to non-determinism. For example, the proposal in prior work Jia et al. [2021] would require > 140× more storage cost than our method by comparing model weights at every step in order to achieve low (yet still non-zero) statistical error. Concretely, our contributions include: (1) A method for two parties, training the same model on different GPU types, to achieve identical training results by logging and sharing rounding decisions; (2) A verifiable training scheme based on the verification game from Teutsch and Reitwießner [2019], which stores model weights in a Merkle tree for efficient comparison between a trainer and auditor; (3) Experiments showing the ability of our approach to scale to large models such as ResNet-50 and GPT-2 between three different NVIDIA GPU architectures (A40, Titan XP, RTX 2080 Ti); (4) Methods to reduce the storage cost of our approach via efficient encoding of rounding logs and an adaptive threshold mechanism to reduce the amount of rounding decisions logged; and (5) Comparisons with existing methods, including proof-based systems, that highlight the improved storage and time efficiency of our method. 2 2 Related Works Without any verifiable training scheme in place, significant trust is placed in the trainer, leaving a client vulnerable to many different attacks, such as the “poisoning” of data samples to cause undesirable behavior (e.g., generating unsafe text [Carlini et al., 2023, Koh et al., 2021, Wan et al., 2023]) and planting backdoors triggered by certain inputs [Goldwasser et al., 2022]. Therefore, training ML models in trusted environments has been an exciting direction explored by many researchers. One line of work consists of proof-based systems, where a proof of correctness (for a desired specification) is provided using cryptographic techniques such as succinct non-interactive arguments (SNARKs) [Micali, 1994, Bitansky et al., 2012, Lee et al., 2020, Liu et al., 2021, Garg et al., 2023, Kang et al., 2022]. However, even the most recent proof-based systems for verifiable training suffer extreme latencies, such as 22 minutes for training VGG-11 on one batch of 16 data inputs [Abbaszadeh et al., 2024], and have therefore primarily been developed for simpler models (e.g., logistic regression) that are less likely to be delegated to others in the first place [Garg et al., 2023, Ames et al., 2022]. Meanwhile, an alternative solution of training models in a trusted execution environment (TEE), such as NVIDIA’s H100, incurs a performance penalty due to the cost of running inside a TEE [Dhanuskodi et al., 2023]. Furthermore, clients lose all security guarantees if an attacker can extract the attestation key from even one GPU [Nilsson et al., 2020, Bulck et al., 2018]. Our approach is most similar to proof-of-learning protocols, which consider a trusted 3rd party that compares checkpointing during the course of training with the original training sequence [Jia et al., 2021]. However, such methods not only incur high storage cost by requiring model weights to be stored frequently, but are non-robust due to errors from training nondeterminism. Several works have shown that proof-of-learning protocols can be spoofed and fail to verify correctness in several important contexts [Fang et al., 2023, Kong et al., 2023, Thudi et al., 2022]. Although Choi et al. [2023] recently proposed a verification procedure that is immune to several known proof-of-learning 2Our method is implemented entirely within the pytorch framework (compatible with version 2.3.1), and is available at https://github.com/meghabyte/verifiable-training. 3 (a) CIFAR-10 Classification (Res-Net 50) (b) Shakespeare Text Finetuning (GPT-2) Test Input This above all: to thine own self be dog cat deer 0.09 0.69 0.08 0.75 0.20 0.01 0.26 0.29 0.28 true my thou 0.0101 0.0098 0.0097 0.0096 0.0083 0.0106 0.0091 0.0100 0.0095 4.18 ppl 4.18 ppl 4.22 ppl 89.9 % 90.2 % 90.7 % A40 TitanXP RTX Ti A40 TitanXP RTX Ti Scores Accuracy Scores Perplexity Test Input Figure 2: Even after ensuring the same software version, random seed, and use of deterministic algorithms via library flags, training nondeterminism persists between three GPU types. attacks, their method is not only limited to supervised learning algorithms, but also based on an assumption that models temporarily overfit data during training, which may not always hold true. GPU Nondeterminism: Prior work has investigated software patches for deterministic training, for instance by enforcing FP accumulation ordering, at a significant cost to efficiency Jooybar et al. [2013], Defour and Collange [2015], Chou et al. [2020], TensorFlow [2021], Zhuang et al. [2021]. While these options address deterministic computation on a single GPU architecture, achieving deterministic results across multiple GPU architectures remains challenging Crane [2018a], NVIDIA [2022]. We control hardware nondeterminism across GPUs in order to design an efficient and reliable verifiable training scheme. However, our method’s impact extends beyond verifiable training, as training nondeterminism can have several negative consequences including bias, reproducibility, and downstream effects on ML pipelines [Zhuang et al., 2021, Crane, 2018b, Srivastava et al., 2020]. 3 Set-Up: The Verification Game Our method for verifiable training is based on the interactive verification game proposed by Teutsch and Reitwießner [2019] in the context of blockchains. The core idea is to resolve a dispute between a challenger, in our case the auditor, and a solver, in our case the trainer, for an expensive computation (e.g., model training). In order for the auditor to take any meaningful action (e.g., pursue legal action), they need to prove the exact source of the dispute (e.g., training time-step where an attack occurred). If we can save model weights at different time steps into a compact data structure such as a Merkle tree, then identifying the source of disagreement can be done efficiently using binary search [Merkle, 1988]. More precisely, the verification game consists of the following parties: 1. trainer, who has putatively trained a model according to a client’s specifications. In our example, this is a service provider with sufficient compute power to train a model. 2. client, who receives a model from the trainer and approaches an auditor. 3. auditor, who officially challenges the trainer on behalf of a client. This is a trusted 3rd-party that has sufficient resources but does not necessarily provide training as a service. The client can choose several auditors to audit the trainer’s model. 4. judge: Sometimes a judge may need to arbitrate a legal claim. The judge can only perform minimal computations (e.g., one training epoch), but can examine the auditor’s claims and enforce a penalty against either the trainer, for incorrect training, or the auditor, for a false alarm. When the trainer is approached by an auditor, they would need to share training parameters, model architecture, and randomness, as shown in Figure 1. The auditor would then replicate the training process, storing model weights in a Merkle tree at the same checkpointing interval as the trainer (every leaf node in a Merkle tree is a hash of the data and every non-leaf node is a hash of its children). The main loop of the verification game starts when both parties have the root of their respective Merkle trees. If training was performed correctly, then the trainer’s root should match the auditor’s. Otherwise, a binary search procedure is performed, where the auditor iteratively descends the Merkle tree until it identifies two consecutive leaf nodes, i and i + 1, where the hash at i matches that of the trainer, but the hash at leaf i + 1 does not. This identifies the point in the computation of the dispute. This interactive verification game requires the cooperation of the trainer. If the trainer refuses to share the value at a certain node of their Merkle tree within a given time frame, they can be considered to have failed the audit. Additionally, the trainer and auditor use a Merkle tree to store model weights, requiring far less storage than prior work, if correct training produces identical weights (and 4 Example Sum Order FP32 Rounded to FP16 a, b, c = 0.1, −0.1, 0.2 a + b + c 00111110010011001100110011001101 0011001001100110 a + c + b 00111110010011001100110011001110 0011001001100110 a, b, c = 10.02, 13.162813186645508, 0.2 a + b + c 01000001101110110001000000000001 0100110111011001 a + c + b 01000001101110110001000000000000 0100110111011000 Table 1: Two examples of floating point accumulation error when rounding arithmetic performed higher precision (e.g. FP32) down to lower precision (e.g. FP16). In the second example, the error in the FP32 result transfers to the rounded FP16 result. identical hash values).The problem is that training nondeterminism leads to weight divergence, and causes this verification game to always fail. This why we seek to prevent divergence in training. 4 The Nondeterminism Challenge Although there are user-side controls for forcing deterministic operations within a single GPU architecture , these controls do not prevent nondeterminism between GPU architectures (e.g., NVIDIA H100 and V100), where trained models can have similar aggregate performance (e.g., accuracy) yet yield very different predictions, as shown in Figure 2 Crane [2018a], NVIDIA [2022]. There are three main sources of nondeterminism between GPU types: 1. Floating-Point Arithmetic: Computers represent real values using integer and FP representations, typically the IEEE 754 standard (Figure 5). There is a tradeoff between the approximation fidelity and the # of bits used to represent the real values. The chosen precision controls the representable numerical range (e.g., 32-bit FP values can represent values between 1.17549435e −38 and 3.40282347e + 38). Because computers round to representable FP values, changing the order in which FP values are accumulated can change the resulting sum Kahan [1965], Whitehead and Fit-Florea [2011]. Over the course of the many operations during training, this can lead to a large difference in the end result between the trainer and auditor. 2. Parallel Computation: In a GPU, a single operation (called a kernel) is executed by thousands of threads in parallel. GPUs contain a set of streaming multiprocessors (SMs), which run the thread blocks required for the kernel. At the hardware level, these blocks are divided into warps that are assigned to the available cores. Because different GPUs have a different number and size of compute units, applications partition arithmetic workloads (e.g., batch matrix multiplies) differently to achieve high performance NVIDIA [2022], thus changing the order of FP operations. 3. Memory Hierarchy and Variable Delays: The time taken for memory access by each thread depends on the physical location of the data, which can create variable delays Jooybar et al. [2013], Defour and Collange [2015], Chou et al. [2020]. The GPU memory hierarchy consists of large amounts of high bandwidth memory (HBM) and small amounts of fast SRAM memory, and maintains an L1 and L2 cache to improve access times. The caches sizes and access times differ across GPU architectures (e.g. an NVIDIA A100 has 192KB / 40 MB of L1/L2 cache memory, while the H100 has 256KB / 50MB). This affects warp scheduling, leading to changes in operation ordering resulting in nondeterminism. Finally, to compute primitives such as GEMMs (D = A · B + C), the workhorse of machine learning, GPUs split the work of computing the tiles of D across a thread block NVIDIA [2023], resulting in nondeterminism that a robust verification method needs to control. 5 Method Overview 5.1 Accumulation Errors Start at Higher Precision Bits Our key idea is that if nondeterminism of training between GPU types occurs due to FP operations, then any error will initially be introduced in the lower bits. Suppose that both trainer and auditor train at a higher FP (e.g., btr = 64) precision than the client’s target model precision (e.g., bm = 32) and then periodically round (e.g., br = 32) after intermediate computation steps (e.g., a convolution layer). One might hope that this will “erase" the errors due to nondeterminism, and prevent them from accumulating. Unfortunately, simply rounding to the nearest FP32 after each computation during training is insufficient for determinism. The problem is due to rounding errors that straddle 5 float32 float32 float32 GPU 1 output (float64) float32 float32 GPU 2 output (float64) float32 Logging Region (determined via binary search) float32 float32 float32 Rounding Case A log:0 log:2 log:0 log:1 log:1 log:1 Case B Case C ! ! Figure 3: Divergence between outputs on two different GPUs (in FP64) for a given function and input can result in different rounding choices when rounding to the nearest FP32. We only wish to log rounding decisions for Case A, where the auditor should copy the trainer’s rounding choice in order to reach the same value. This requires defining a logging region, determined by a threshold τ, the rounding boundary. Consider Case A in Figure 3, which shows a divergence in the output of a computation using FP64 on two different GPUs. Because the outputs of GPU 1 and 2 are on different sides of the boundary, rounding to the nearest FP32 results in different values, introducing error. What if the trainer records their rounding choice (e.g., up, down, none) for every intermediate computation? The auditor could then copy the trainer’s choice, and therefore round to the exact same value and successfully control for nondeterminism. However, the auditor should not copy the trainer’s behavior for every output (see Cases B & C, Figure 3). If a computation output on GPU 1 is too close to the rounded value, then it is possible that GPU 2 is also close in distance but from the opposite direction. In this case, the auditor should ignore the trainer’s choice. We therefore need to introduce a threshold τ under which the trainer does not record their rounding choice. Our method requires upper bounding the divergence ddiv between any two different GPUs for any intermediate computation f (i.e. difference in outputs for the same input). Let ϵb represent the distance between two FP32 values, after rounding to br bits of the mantissa (Figure 5) and controlling for the exponent. We need to select br and τ such that ddiv < ϵbr and ddiv < 2τ (Figure 3). Because the set of possible FP numbers is finite, there exist optimal bounds for br and τ. In practice, we find that br ≤32 and τ > 0.25 · ϵ32 are sufficient for standard intermediate computations in neural network training (e.g., convolution, layer norm) in FP64. We study different values for br in Section 6. 5.2 Primitives We assume both trainer and auditor train models using the IEEE-754 standard FP numbers (Figure 5). Besides requiring read and write disk I/O operations, we define the following functions: 1. rndbr(x): rounds input x to the nearest FP up to br bits of the mantissa, as shown in Figure 5. 2. log(x, br, τ, f): logs to file f a logging direction c, which is either 0 (down), 1 (ignore), or 2 (up) depending on threshold τ and rounding amount br, as shown in Algorithm 4. 3. rev(x, br, c): reverses rounding of input x based on logging direction c. If x < rndbr(x) & c = 0, then return x rounded to the nearest float below x with br precision. If x > rndbr(x) & c = 2, then return x rounded to the nearest float above x with br precision. Otherwise, do not correct. 4. threshold(l, br, btr): identifies the optimal threshold to log rounding directions (0 or 2) instead of 1, which the rev function ignores, based on the binary search procedure in Section 5.4. 5. hashsha256(θ): creates a SHA-256 hash of provided model weights θ (in bm precision). 6. tree(leaf1, leaf2..., leafn) : create a Merkle tree where each leaf node is the output of hashsha256(θ) for model weights θ at a given checkpoint, with a checkpointing interval k [Merkle, 1988]. 5.3 Training and Auditing The trainer’s task begins when a client approaches them with dataset D, training specifications (epochs E, loss function loss, etc.), and a requested model precision bm.The trainer can then choose a training precision btr > bm, a rounding amount br ≤bm, and a checkpointing interval k to periodically store small hashsha256(θ) of model weights θ in a Merkle tree, for efficient comparison with an eventual auditor. Then, as detailed in Algorithm 1, the trainer can perform training as normal, but after every intermediate computation (e.g., convolution) perform the rndbr operation on 6 each output. Rounding is applied to computations in both the forward and backward passes. Finally, either using a fixed threshold τ or a layer-specific optimal τ from the threshold function described in Section 5.4, the trainer applies log, which logs rounding choices only for the computations an auditor should copy. The output of the algorithm includes a rounding log file F and the root of the Merkle tree which, along with the shared randomness R and all training parameters, the trainer can share with any trusted third-pary auditor who challenges them. After a client approaches them, the auditor initiates the verification game described in Section 3. To avoid penalty, the trainer must cooperate by sharing the rounding amount br, randomness R used in training (e.g., a pseudo-random number generator), the checkpointing interval k, and set of rounding logs F. The auditor then follows the training procedure and corrects their rounding choice (e.g., up or down) to match those logged in F using the rev operation, as detailed in Algorithm 2 (Appendix). By correcting each rounding mismatch during the course of training, the auditor is able to prevent nondeterminism errors from accumulating. Therefore, the auditor can store the hashsha256(θ) of model weights θ in a Merkle tree at interval k, knowing that if training was done correctly, the model weights should be identical to the trainer’s at any timestep. The output of Algorithm 2 is the root of the auditor’s Merkle tree, which they can use to compare with the trainer’s root. 5.4 Reducing storage cost Logging rounding decisions for every neural network layer output during training incurs a large baseline storage cost, and is our main limitation. For dataset D, batch size B, training epochs E, and model layers Lθ, the upper bound on the total storage cost for verifiable training with our method is: storage cost (B) = \|D\| × E × B × ( L X l=1 ol,f + L X l=1 ol,b) (1) where ol,f and ol,f represent the size of outputs of the forward pass and backward pass of layer l. Note that the log entries do not need to be kept around in the RAM and can be written straight to the disk. Moreover, this cost is a one-time cost incurred by the trainer, who in our context is likely to be a powerful commercial provider with access to such storage capacity. Furthermore, as we later show in Section 6, for models with many linear layers like Transformer-based language models (e.g., GPT-2), where parameters significantly outnumber intermediate computations, this storage cost is significantly smaller than alternative approaches that require saving model weights [Jia et al., 2021]. Nevertheless, we now describe our method for reducing storage cost by (i) efficiently encoding rounding logs and (ii) adaptive selection of the threshold τ to reduce the storage costs. Efficient Encoding: Each log entry is a value from the set 0, 1, 2, as opposed to the FP model weights. We pack sub-sequences of five log entries into a single byte via a fast GPU-based radix-3 to radix-2 conversion, yielding 1.6 bits/entry storage that is close to the best possible packing of 1.58 bits/entry, and yields a 77% storage reduction relative to naively storing one log entry per byte. Adaptive Threshold: Recall that we need to select a threshold τ that controls for whether the trainer logs a rounding choice, or instead logs 1 which the auditor ignores. The more one can increase τ, the more 1 values are recorded, which can make rounding logs more compressible (due to long sequences of 1s). Furthermore, it is possible that the divergence ddiv between outputs on two different GPUs, given the same input, is function-specific. For example, while convolution requires several matrix multiplications that might result in a large FP accumulation error, normalization operations are unlikely to result in large ddiv, and a larger τ can be applied. We develop an efficient algorithm (Algorithm 3 in the Appendix) to find the optimal value for τ given a particular layer and data of output values that led to different rounding choices between any two GPUs (e.g., Case A in Figure 3). For a given rounding amount br and training precision btr, the algorithm performs a binary search between τ = 0.25 · ϵ32 (our upper bound on the ddiv between two GPUs for any function) and τ = 0.5 · ϵbr (the rounding boundary). By performing this procedure for the different intermediate computations in a model, the trainer can hope to better compress the rounding log F. Merkle Tree Storage: Storing SHA-256 hashes of model weights during training in a Merkle tree creates an efficient mechanism for the verification game described in Section 3, with negligible storage requirements. The audit ends when either the trainer withdraws, the auditor confirms that training was performed correctly, or the auditor can present paths to the two leaves of their Merkle tree where divergence starts, providing evidence to dispute the trainer. 7 # Steps # Steps % Log Entries Ignored % Log Entries Ignored % Time Increase % Test Loss Change Avg. L2 Weight Diff. Avg. L2 Weight Diff. a. b. c. d. Standard Training Simple Rounding Our Method ResNet-50 GPT-2 GPT-2 ResNet-50 b=32 b=26 b=32 b=26 b=32 b=26 b=32 b=26 Figure 4: We successfully control for nondeterminism between GPU types for both ResNet-50 (a.) and GPT-2 (b.) tasks, while standard training and simple rounding without performing rev corrections result in model divergence over the course of training. Stronger rounding has minimal affect to model performance (c.), but at the cost of increasing time for trainer (d.). Table 2: Efficient encoding reduces storage requirements by 77%, and rounding to b = 26 improves the compression further between 5-20% (values reported for 1 step of training). The original proofof–learning protocol from Jia et al. [2021] requires storing 2.78 GB of model weights for GPT-2, or more than 140x our storage cost, while still incurring statistical error. ResNet-50 b = 32 ResNet-50 b = 26 GPT-2 b = 32 GPT-2 b = 26 Naive Encoding 456 MB 456 MB 92 MB 92 MB Efficient Encoding 105 MB 105 MB 22 MB 22 MB + Zip Compression 96 MB 91 MB 20 MB 18 MB 6 Empirical Results We evaluate our verifiable training method on the two large-scale models listed below with all possible trainer and auditor pairs across NVIDIA GPUs A40, TITAN Xp, and RTX 2080 Ti (see Appendix B for more details). In Section 6.2, we compare our method with recent proof-based systems. 1. ResNet-50: We train (from random initialization) ResNet-50 (23M) on CIFAR-10 with dataset size 50K & batch size B=64. Test accuracy = 90.7% after 100 epochs training on Titan RTX Ti. 2. GPT-2: We finetune GPT-2 (117M) on a corpus of Shakespeare text with dataset size 1.1M tokens, batch size B=8, and sequence length 64. Perplexity = 4.22 after 1 epoch training on Titan RTX Ti. Figure 2 shows that nondeterminism due to GPU architecture exists for both tasks. While we can repeatedly obtain identical results across training runs on the same GPU architecture, training on different GPU architectures results in fundamentally different models. 6.1 Implementation and Findings We implement our verifiable training method entirely on top of the pytorch framework, with torch version 1.13.1 and CUDA version 11.7. The intermediate computations we apply rndb to are layers (e.g., torch.nn.Conv2D) in the model’s computation graph. Rounding-related operations (rnd and rev) either using casting or FP functions (e.g., torch.nextafter) that can run on the GPU, thus having little impact on computational speed. Because we observed that the torch.randn operation used for dropout in GPT-2 is non-deterministic for long inputs (even for the same seed, see Appendix I), we implement our own dropout as our method requires shared randomness R. Successful control for non-determinism: Our method completely eliminates non-determinism between full training runs of both for both the ResNet-50 training and GPT-2 fine-tuning tasks across all possible trainer and auditor pairs between the A40, Titan XP, and RTX 2080 Ti GPUs. As Figure 4 shows, standard FP32 training results in an increasing divergence (l2-distance of weights) between models on different GPUs over the course of training. Furthermore, we show the simple approach of training in FP64 and rounding to FP32 after every intermediate computation, but without the auditor correcting rounding decisions with rev, fails to mitigate this issue. Only our method, in which the auditor follows the rounding decisions (br = 32) made by the trainer for every intermediate computation, eliminates non-determinism and persists over time. Our implementation, which requires disk I/O during training to store the rounding decisions, results in a small increase in training time for the trainer (1.2-1.4x) and auditor (1.3-1.7x) using a non-optimized, protoype implementation (Table 5). We report the storage requirements of our method in Table 2, showing 8 Table 3: Average # of rev corrections performed by auditor per training step. Even at b = 32, auditing only requires 20-25 corrections (2e-6 to 9e-6% of samples) per training step. ResNet-50 b = 32 b = 31 b = 30 b = 29 b = 28 b = 27 b = 26 Forward 15 ± 3 6 ± 2 3 ± 1 3 ± 1 0 0 0 Backward 10 ± 0.6 6 ± 0.6 2 ± 1 0.7 ± 0.7 0 ± 0 0 ± 0 0 ± 0 GPT-2 b = 32 b = 31 b = 30 b = 29 b = 28 b = 27 b = 26 Forward 2 ± 0.7 2.3 ± 0.8 2.2 ± 0.4 0.2 ± 0.2 0.4 ± 0.2 0 ± 0 0 ± 0 Backward 19 ± 13 0.75 ± 0.3 1.2 ± 0.4 0.2 ± 0.2 0. ± 0.0 0 ± 0 0 ± 0 Table 4: Adaptive thresholds identified for different operations using Algorithm 3 with b = 32. 2D Convolution Batch Norm Linear Layer Norm Dimension 256 (1,1) (128, 128, 16, 16) (768,768) (768,1) τ 0.305 ∗2−23 0.499 ∗2−23 0.465 ∗2−23 0.499 ∗2−23 that our efficient encoding scheme reduces the size of the trainer’s rounding logs by 77%, relative to naive logging. Because the Merkle tree stores 32-byte SHA-256 hashes, its overall size (KBs) and creation time are negligible and not reported. Finally, we show that decreasing the rounding amount b to values even as low as 26 has little effect on model performance (we observe no change in accuracy, so report test loss), but increase training time (Figure 4). We observe that smaller values of b do allow more log entries to be ignored, improving compression of the file, which we discuss next. Compression with adaptive threshold: Our approach outperforms (Table 2) the storage costs of proof-of-learning protocols that save model weights for GPT-2 (2.78GB), which has many linear layers – we observe more than 140x reduction relative to the approach in Jia et al. [2021]. We further reduce the storage cost of our method by decreasing the rounding amount b and implementing the adaptive thresholding strategy (Section 5.4). Table 4 reports adaptive thresholds τ for four different pytorch layers at rounding amount br = 32. Convolutions require the lowest τ, indicating larger divergence in outputs between GPU types, which is expected due to the large # of matrix multiplications. Meanwhile, τ is higher for normalization layers, likely due to smaller divergences between GPU types. Because adaptive thresholding seeks to reduce the # of times rounding decisions (0 and 2) are logged and improve log file compression, we report storage cost after zip compression in Table 2. As expected, more aggressive rounding (which results in a higher τ) improves the compression rate. Although the compression gains are mild in comparison to our encoding step, they build-up over the course of training. Finally, we report the average # of rev corrections an auditor needs to perform for one training step in our two tasks (Table 3). These values are surprisingly small in comparison to the # of operations logged – only a maximum of 2e-6% (ResNet-50) and 9e-6% (GPT-2) of logged values, are actually needed by the auditor! We also observe that severe rounding (e.g., b = 27) completely eliminated the hardware non-determinism for our tasks, requiring no corrections from the auditor. This shows a huge gap between the # of values currently saved by the trainer and those needed by the auditor, motivating an exciting future possibility of significantly reducing the storage cost of our method if we could reliably predict when a divergence will not occur. 6.2 Comparison with alternative approaches Logistic Regression: Garg et al. [2023] recently proposed a zero-knowledge proof-based system for verifiable training of a logistic regression, which importantly does not leak information about the client’s data or require a trusted third-party auditor, unlike our work. However, since verifiable training itself is motivated by a client not having sufficient resources to train the model, it is crucial to consider the implications of scale. The authors report the prover time and proof size requirements for one training pass of logistic regression on a dataset of 218 items, with 1024 dimensions and a batch size of 2014, as 72 seconds (training and proof generation time) and 350 MB respectively. We replicate this training task, and find that our method significantly improves upon both storage and time requirements, requiring only 106 KB and 7 seconds (both training and auditing). Furthermore, because Garg et al. [2023] do not report the duration of “offline phase” of their method, their reported value is a lower bound on the actual time required. Finally, we note that the original proof-of-learning protocol from Jia et al. [2021], which also considers a trusted third-party, would require 9.2 MB per training step to store all model weights. Therefore, our method is at least 85x more space efficient. VGG-11: Concurrent to this work, Abbaszadeh et al. [2024] introduce a zero-knowledge proofof-training protocol for deep neural networks, presenting results for one batch step of training for 9 a simplified version of the VGG-11 model with 10M parameters, which is less than the original VGG-11 network and ResNet-50 [Simonyan and Zisserman, 2015]. While the authors do not provide architectural details, we can assume that increasing the # of parameters to the original VGG-11 would only increase their reported proof time and size. We, therefore, compare their reported values with an implementation of our method for the same task of verifying the training of VGG-11 on CIFAR-10 with a batch size of 16. While their use of incrementally verifiable computation leads to tractable proof size (1.36MB vs. the 1.2MB per iteration cost of our method), Abbaszadeh et al. [2024]’s method requires 22 min. per training iteration. In comparison, our method requires training and auditing times of only 6 sec. per iteration and is significantly more efficient (factor of 220x), an important consideration for model training as a commercial service. Finally, in Appendix Section J, we compare our results with an adaption of Gupta et al. [2023]’s protocol for secure inference of GPT-2. Compared with our method’s storage cost (18MB) and training time (11s for training, 13.5s for auditing), scaling Gupta et al. [2023]’s protocol for training would introduce around a 10,000x data and 40x time overhead. While proof-based systems provide strong security guarantees without a third party, they do so at the cost of relying on hard-to-scale cryptographic techniques, as well as approximating non-linear functions that can harm performance. 7 Security Analysis Our work makes a 1-of-n honesty assumption, i.e., as long as one of n auditors is honest, any attack from a malicious trainer that results in diverging model weights will be detected. One consideration is the potential manipulation of the rounding logs by an adversarial trainer who could select rounding decisions that achieve a desired outcome, and which the auditor would follow. Concretely, let us define our threat model so that the trainer knows an auditor’s GPU a priori. Recall that an auditor only copies the trainer’s rounding decision in Case A in Figure 3, when both GPUs compute values close to the rounding boundary. Under this threat model, the trainer can identify the n steps where the auditor is close to the boundary (as in Case A), enumerate the set of 2n different models that result from different rounding decisions, and selectively pick a model that exhibits a desired property. However, the trainer cannot use this strategy to embed an arbitrary property (e.g., a specific backdoor). It can only select from the set of models that differ in certain rounding decisions, which all require the trainer to use the correct training specifications accepted by the client (such as exact training data & hyperparameters). Furthermore, since the expected # of divergences between the trainer and the auditor is extremely small (see Table 3), the set of possible models where an auditor would not detect an attack (e.g., many rev ops) is limited. Finally, we show in Table 6 in the appendix that the divergence (measured both as ℓ2-norm between model weights and output distributions) due to GPU non-determinism is significantly less than the divergence due to data ordering during training. Therefore, if a client will accept a model trained with any random ordering of the data during training, then it is unlikely that an adversarial trainer — that can only alter rounding decisions — could produce a model that the client would not accept. Nevertheless, fully understanding the model properties obtained by manipulating rounding logs adversarially is an important future direction. 8 Limitations and Future Work Our verifiable training scheme successfully controls for hardware nondeterminism. It expands the pool of potential auditors of a model training service, allowing us to envision a world where a client can even use two competing service providers it trusts to audit each other. Relative to proof-based systems, a limitation is the need for all parties to trust the third-party auditor. If the trainer provides finetuning services on top of closed-source models (e.g., OpenAI), then our scheme will only work for the third-party auditors that the trainer is willing to share model weights with. Other limitations included the added latency of training in higher precision and the storage cost. While we have shown that our method requires significantly less storage than alternatives, the vast majority of stored rounding decisions are not used by the auditor and are therefore unnecessary (Section 6). Therefore, an exciting direction for future work is to mitigate this gap by better predicting when GPU divergence between computations occurs. Recent work has similarly argued for a stronger profile of noise during training in the context of verification [Fang et al., 2023]. 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Sigma: Secure gpt inference with function secret sharing. Cryptology ePrint Archive, Paper 2023/1269, 2023. URL https://eprint.iacr.org/2023/1269. https://eprint.iacr.org/2023/ 1269. Shen Li, Yanli Zhao, Rohan Varma, Omkar Salpekar, Pieter Noordhuis, Teng Li, Adam Paszke, Jeff Smith, Brian Vaughan, Pritam Damania, and Soumith Chintala. Pytorch distributed: experiences on accelerating data parallel training. Proc. VLDB Endow., 13(12):3005–3018, August 2020. ISSN 2150-8097. doi: 10.14778/3415478.3415530. URL https://doi.org/10.14778/3415478.3415530. A IEEE Floating Point Image See Figure 5. B GPU Details All experiments reported in our paper are run with the following three GPUs: • NVIDIA Titan XP: 3840 Cores, 12 GB • NVIDIA RTX 2080 Ti: 4352 Cores, 11 GB • NVIDIA A40: 10752 Cores, 48 GB We are able to successfully replicate training runs between all pairs of these 3 GPUs. C Logging Algorithm See Algorithm 4 D Train Algorithm See Algorithm 1. E Audit Algorithm See Algorithm 2. 13 F Adaptive Thresholding Algorithm See Algorithm 3. G Time Requirements See Table 5. H Model Divergence Comparison See Table 6. I Random Number Generation Our verifiable training scheme requires shared randomness between the trainer and auditor, which is used for deciding input data batching, weight initialization, and operations such as dropout (randomly setting outputs to zero). More formally, our scheme requires sharing the same random seed and pseudo-random generator. However, in our implementation based on pytorch (assuming the same software version between trainer and auditor), we chose to rely on the the torch random seed functionality. While this successfully controls for batch input ordering and weight initialization, it is unfortunately not sufficient for random number generation, as operations such as torch.nn.randn() leverage parallelism when the requested # of values is higher than a certain amount. Specifically, we found that across T40, RTX 2080 Ti, V100, A40, and A100, given the same seed, torch.randint() produces identical tensors onlt up to size 40960. At size 40961, T40 (which is an older GPU) deviated from the rest. Likewise, at size 69633, 2080 Ti deviated from the rest, and so on. Based on these observations, we arranged for calls to torch.randint() in the dropout layer (which is the only operation using large random tensors in our tasks) to be replaced by generating and concatenating multiple random tensors of size 40960 or less. Specifically, a random tensor of size n > 40960 is generated by concatenating (n//40960) random tensors of size 40960 and one random tensor of size (n%40960). However, we emphasize that it is therefore important in our scheme either for both parties to implement this change a priori, or simply use an external source for pseudorandomness. J Comparison with GPT-2 Inference The previously discussed proof-based systems for verifiable training by-pass the need for a thirdparty auditor, but very few efficient systems exist in the literature. Many more works study secure inference of deep neural networks, which could be used to construct verifiable training protocols with stronger security guarantees than ours (e.g., allowing a trainer to keep a proprietary model’s weights private), but come at a significant cost to performance and resources. To demonstrate this, we consider adapting Gupta et al. [2023]’s protocol for secure inference of GPT-2 based on multi-party computation, to our context of verifiable training. Gupta et al. [2023] show how two parties, the client with private data and the trainer, can jointly compute the forward pass of a known model architecture without revealing additional information beyond the model output to each other. Because they report the the communication overhead P = 0.37GB and time T = 0.96 seconds for one forward pass on a single data input, we can calculate 2 × P × D × E = 189 GB and 2 × T × D × E = 983 seconds as estimated communication cost and time, respectively, for 1 step of training in out GPT-2 task, where 2 considers both the forward and backward pass. Compared with our method’s required storage cost (18MB) and training time (11s for training, 13.5 seconds for auditing), scaling Gupta et al. [2023]’s protocol for training would introduce around a 10,000x data and 40x time overhead. 14 Algorithm 1 train INPUT: dataset D, epochs E, batch size B, shared randomness R, model Wθ, loss function loss, rounding amount br, training precision btr, target model precision bm, checkpointing interval k OUTPUT: Merkle tree root Mroot, rounding log file F 1: F, Mleaves ←create empty file and leaf list 2: Wθ ←init(R, btr) //initialize weights 3: T ←D∗E B 4: for t = 1...T do 5: input ←batch(R, D, B) // get data batch // forward pass 6: for layer lθ ∈Wθ.layers do 7: output ←lθ(input) 8: τ ←threshold(lθ, br, btr) //set threshold 9: log(output, br, τ, F) 10: output ←rndbr(output) 11: input ←output 12: end for 13: loss ←loss(output) 14: // backward pass, reversed layers 15: grad_output ←∇loss 16: for layer lθ ∈Wθ.layers do 17: grad_input ←∇lθ(grad_output) 18: τ ←threshold(∇lθ, br, btr) 19: log(grad_input, br, τ, F) 20: grad_input ←rndbr(grad_input) 21: grad_output ←grad_input 22: end for 23: θ ←update update weights 24: if t mod k = 0 then 25: Mleaves.append(hashsha256(θ in precision bm)) 26: end if 27: end for 28: Mroot ←tree(Mleaves) // create Merkle tree 29: return F, Mroot, and model Wθ in target precision bm Table 5: Training time requirements, including Merkle tree operations (at k = 5), for 1 step of training broken down by stage of our verifiable training process. Note that reported times are specific to the particular dataset, batch size, and task, and using a non-optimized prototype codebase – therefore the relative increase is time is more important. ResNet-50 GPT-2 Original (No Rounding or Disk I/O) 24s 8s Trainer 28s 11s Auditor 31s 13.5 Table 6: Comparison of model divergence due to data ordering versus GPU non-determinism. Reported numbers are averaged between 10 pairs of models, error bars are standard deviation. Metric Data Ordering GPU Non-determinism l2 weight difference 133.2 ± 9 1.1 ± 0.07 l2 output distance 5.3 ± 0.03 0.26 ± 0.02 15 Algorithm 2 audit INPUT: dataset D, epochs E, batch size B, shared randomness R, model Wθ, loss function loss, rounding amount br, training precision btr, target model precision bm, checkpointing interval k, log file F from trainer OUTPUT: Merkle tree root Mroot 1: Mleaves ←create empty leaf list 2: Wθ ←init(R, btr) //initialize weights 3: T ←D∗E B 4: for t = 1...T do 5: input ←batch(R, D, B) // get data batch // forward pass 6: for layer lθ ∈Wθ.layers do 7: output ←lθ(input) 8: for outputi ∈output do 9: // Match trainer rounding 10: c ←read(outputi, F) 11: outputi ←rev(outputi, br, c) 12: end for 13: input ←output 14: end for 15: loss ←loss(output) 16: // backward pass 17: grad_output ←∇loss 18: for layer lθ ∈Wθ.layers do 19: grad_input ←∇lθ(grad_output) 20: for grad_inputi ∈grad_input do 21: // Match trainer rounding 22: c ←read(grad_inputi, F) 23: grad_inputi ←rev(grad_inputi, br, c) 24: end for 25: grad_output ←grad_input 26: end for 27: θ ←update update weights 28: if t mod k = 0 then 29: Mleaves.append(hashsha256(θ in precision bm)) 30: end if 31: end for 32: Mroot ←tree(Mleaves) // create Merkle tree 33: return Mroot 16 Algorithm 3 threshold INPUT: layer l, rounding amount br, training precision btr OUTPUT: threshold τ 1: P ←initialize empty list 2: N, T ←initialize large # of data points and iterations 3: for i=1...N do 4: GPU1, GPU2 ←select two different GPU architectures 5: x ←select random input for layer l in btr floating-point precision 6: y1 ←lGP U1(x), y2 ←lGP U2(x), apply layer l on input x on each GPU 7: if rndbr(y1) ̸= rndbr(y2) then 8: if y1 > rndbr(y1) and y2 < rndbr(y2) then 9: P.append(\|y1 −rndbr(y1)\|) 10: P.append(\|y2 −rndbr(y2)\|) 11: end if 12: if y1 < rndbr(y1) and y2 > rndbr(y2) then 13: P.append(\|y1 −rndbr(y1)\|) 14: P.append(\|y2 −rndbr(y2)\|) 15: end if 16: end if 17: end for 18: //binary search to select threshold 19: lower, upper, τ ←0.25 ∗(2−23), 0.5 ∗(29−br), 0 20: for t=1...T do 21: τ ←(lower + upper)/2 22: success ←True 23: for pi ∈P do 24: exp ←get exponent of pi 25: if pi < exp ∗τ then 26: success ←False 27: end if 28: end for 29: if success then 30: lower ←τ 31: else 32: upper ←τ 33: end if 34: end for 35: return τ Algorithm 4 log INPUT: value x, rounding amount br, threshold τ, file F 1: exp ←get exponent of x 2: if \|x −rndbr(x)\| > exp ∗τ and x < rndbr(x) then 3: write(2, F) // log rounding up 4: else if \|x −rndbr(x)\| > exp ∗τ and x > rndbr(x) then 5: write(0, F) // log rounding down 6: else 7: write(1, F) // log rounding ignore 8: end if 17 NeurIPS Paper Checklist 1. 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4,434	Towards a theory of how the structure of language is acquired by deep neural networks Francesco Cagnetta Institute of Physics École Polytechnique Fédérale de Lausanne francesco.cagnetta@epfl.ch Matthieu Wyart Institute of Physics École Polytechnique Fédérale de Lausanne matthieu.wyart@epfl.ch Abstract How much data is required to learn the structure of a language via next-token prediction? We study this question for synthetic datasets generated via a Probabilistic Context-Free Grammar (PCFG)—a tree-like generative model that captures many of the hierarchical structures found in natural languages. We determine token-token correlations analytically in our model and show that they can be used to build a representation of the grammar’s hidden variables, the longer the range the deeper the variable. In addition, a finite training set limits the resolution of correlations to an effective range, whose size grows with that of the training set. As a result, a Language Model trained with increasingly many examples can build a deeper representation of the grammar’s structure, thus reaching good performance despite the high dimensionality of the problem. We conjecture that the relationship between training set size and effective range of correlations holds beyond our synthetic datasets. In particular, our conjecture predicts how the scaling law for the test loss behaviour with training set size depends on the length of the context window, which we confirm empirically in Shakespeare’s plays and Wikipedia articles. 1 Introduction Two central foci of linguistics are the language structure and how humans acquire it. Formal language theory, for instance, describes languages with hierarchical generative models of grammar, classified in different levels of complexity [1, 2]. In this context, the ‘poverty of the stimulus’ argument [3]— stating that the data children receive is insufficient to uniquely determine the grammatical structure of their language—led to the hypothesis that linguistic faculties are largely innate. By contrast, statistical learning theory [4, 5] posits that the statistics of the input data can be used to deduce the language structure. This assumption is supported by empirical evidence concerning a broad range of tasks, including word segmentation [6] and reconstruction of the hierarchical phrase structure [7]. Large Language Models (LLMs) offer an interesting perspective on the subject. For instance, the success of LLMs trained for next-token prediction [8, 9] establishes that a language can be acquired from examples alone—albeit with a training set much larger than what humans are exposed to. Furthermore, empirical studies of LLMs’ representations showed that they learn a hierarchy of contextual information, including notions of linguistics such as word classes and syntactic structure [10, 11, 12]. Recent studies have begun revealing the inner workings of LLMs by using synthetic data generated via context-free grammars [13, 14], determining, in particular, the algorithm that these models follow when predicting the next token. However, there is no consensus on the mechanisms behind language acquisition by LLMs [15, 16]. As a result, empirical phenomena such as the scaling of the test loss with dataset size and number of parameters [17] and the emergence of specific skills at certain scales [18, 19] remain unexplained. In this work, we use hierarchical generative models of data to describe how the structure of a language is learnt as the training set grows. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1.1 Our contributions We consider synthetic datasets generated via the Random Hierarchy Model (RHM) [20], an ensemble of probabilistic context-free grammars (PCFGs). The RHM generates sequences of tokens by applying randomly chosen production rules to a hierarchy of hidden variables that live on the nodes of a tree with fixed geometry. • We characterise analytically the power-law decay of the correlations between tokens with their distance. We then show that, because of this decay, a finite training set size P limits the resolution of correlations to an effective context window, whose size t∗increases with P. • Building on previous works on classification, we argue that deep learning models trained on next-token prediction can use measurable correlations to represent the hidden variables of the PCFG, with larger P allowing the representation of deeper hidden variables. • Combining these results, we predict a sequence of sample complexities where the emergence of a deeper data structure representation leads to a jump in test loss. We empirically validate this for deep transformers and CNNs. Notably, the sample complexities are polynomial in the effective context size t∗, avoiding the curse of dimensionality. • We conjecture that the relationship between training set size, correlations and effective context window holds beyond our data model, and we test it by training deep transformers on collections of Shakespeare’s lines and Wikipedia articles. In particular, we find that the test loss decay levels off at a characteristic training set size that depends on the length of the context window and can be measured from token-token correlations. 1.2 Additional related works Fixed-tree hierarchical generative models have been introduced to study phylogeny [21], then used in supervised learning [22, 23, 20, 24] and score-based diffusion [25, 26]. In particular, [22] introduced a sequential clustering algorithm that reveals the importance of correlations between the input features and the labels for supervised learning. The RHM of [20] provides a framework to show how featureslabel correlations emerge from the generative model and can be used by deep networks to represent the hidden hierarchical structure of the data. Here we extend this result to self-supervised learning, where the relevant correlations are those between the different input features. PCFGs can in principle generate sequences with token correlations that decay as a power of their distance [27]. When the production rule probabilities are random [28, 29], these probabilities must follow a broad distribution for the data to retain information about the generative process. Learning a PCFG from examples is a longstanding problem of theoretical linguistics [30]. While some PCFG classes are learnable using distributional information [31], the sample complexity is unknown. In the context of deep learning, PCFGs have been used to study how trained transformers encode the grammar’s structure [13, 14]. [14], in particular, showed that the operations performed by BERT-like transformers resemble well-known algorithms for grammatical inference, and proved that, for PCFG data, these algorithms are optimal solutions of the masked language modelling objective. However, when the training data is compatible with both a PCFG and a non-hierarchical generative model, neither recurrent language models [32] nor transformer [33] consistently prefer the hierarchical explanation. In addition, none of these works study the learning process. Empirical work on the learning dynamics of Long Short-Term Memories showed that short-range dependencies are learnt first, then used as a foundation for forming longer-range dependencies [34]. Our work introduces a theoretical framework to explain this hierarchical inductive bias, focusing on the learning curves of deep learning architectures. Shortly after our submission, [35] unveiled another form of hierarchical inductive bias in the training dynamics of transformers, whereby many-body interactions among tokens are learnt in the order of the interaction’s degree. 2 Notation and setup This work focuses on the pretraining phase of language models, aimed at building an approximation of the data distribution via unlabelled examples [8, 9]. Let us define a text datum, or sentence, as a sequence x = (x1, . . . , xd) of d tokens belonging to a finite vocabulary V. Denoting with v the 2 vocabulary size, each token xi is represented as a v-dimensional one-hot vector (xi,µ)µ=1,...,v 1: xi,µ = 1, if xi ≡µ-th element of V, 0, otherwise. (1) A dataset, or corpus, consists of a probability distribution over sequences, which measures the frequency at which a given combination of tokens appears within the text. Assuming that all sequences have length d, the data distribution is a joint probability over d-dimensional sequences with elements in V, PX(x) := P {X1 = x1, . . . , Xd = xd} . The specifics of the approximation of PX depend on the training objective. In Masked Language Modelling, for instance, a random fraction of tokens is masked, i.e. replaced with a fixed token xmask, and the model is tasked with predicting their value [8]. Autoregressive language models, instead, are trained to predict the i-th token of a sequence based on all the previous ones [9]. Here we consider a simplified setup where the last token of the sequence is masked and the model is trained to predict it. In other words, the model takes the context window (x1, . . . , xd −1) as input and outputs a parametric approximation pθ of the conditional probability of the last token, pθ(xd\|x1, . . . , xd−1) ≈P {Xd = xd\|X1 = x1, . . . , Xd−1 = xd−1} , (2) obtained by updating the parameters θ via gradient descent on the empirical cross-entropy, L(XP ) = −1 P X x∈XP log (pθ(xd\|x1, . . . , xd−1)), (3) where XP is a set of P training examples drawn from PX. Numerical experiments are performed in PyTorch [36], with the code available at https://github.com/fracagnetta/ random-hierarchy-model. Details of the machine learning models, training hyperparameters and computer resources are presented in App. A. 2.1 Hierarchical generative models To model the hierarchical structure of sentences, we consider synthetic datasets generated via a probabilistic context-free grammar (PCFG) [37]. PCFGs are collections of symbols and rules that prescribe how to generate sequences. In particular, the PCFGs we consider consist of • L finite vocabularies of hidden (nonterminal) symbols (Vℓ)ℓ=1,...,L; • A finite vocabulary of observable (terminal) symbols V ≡V0; • L sets of production rules describing how one symbols of Vℓgenerates a tuple of symbols of Vℓ−1, for ℓ= 1, . . . , L. Production rules take the form µ(ℓ) →µ(ℓ−1) 1 , . . . , µ(ℓ−1) sℓ , for µ(ℓ) ∈Vℓ, µ(ℓ−1) i ∈Vℓ−1, (4) for some integer size sℓ≥1. The left panel of Fig. 1 shows an example of the generative process, represented as a tree: pick (uniformly at random) a level-3 symbol (root) and one of the production rule having that symbol on the left-hand side (also uniformly at random), replace the symbol with the right-hand side of the production rules (first generation), then repeat the process until left with only terminal symbols (leaves). The resulting datum is a sequence in (V0)d, with d = Q ℓsℓ. Assuming a finite number of production rules emanating from each nonterminal symbol, this model generates a finite number of d-dimensional sequences. Since the probabilities of the level-L symbol and the production rules are uniform, the data distribution PX is uniform over the generated sequences. The Random Hierarchy Model (RHM) of [20] is an ensemble of such generative models, obtained by prescribing a probability distribution over production rules. In particular, the ℓ-th set of production rules is chosen uniformly at random between all the unambiguous sets of rules in the form of Eq. 4. Unambiguity means that each sℓ-tuple of level-(ℓ−1) symbols can be generated by one level-ℓ symbol at most. The uniform probability and unambiguity assumptions are not satisfied in a generic natural language, but they allow us to characterise quantitatively the effects of the hierarchical structure. We will further assume, to ease notation, that all the vocabularies Vℓhave the same size v and that the size of the production rules is homogeneous, i.e. sℓ= s for all ℓ. We further assume that each nonterminal appears as the left-hand side of exactly m production rules, i.e. the hidden symbols have m equivalent low-level representations. Since there are vs distinct low-level representations and each of the v high-level symbols is assigned m, unambiguity requires m ≤vs−1. 1throughout the paper, Latin indices indicate the token position and Greek indices the vocabulary entry. 3 µ(0) 1 µ(0) 2 µ(0) 3 µ(0) 4 µ(0) 5 µ(0) 6 µ(0) 7 µ(0) 8 µ(1) 1 µ(1) 2 µ(1) 3 µ(1) 4 µ(2) 1 µ(2) 2 µ(3) 1 1 2 3 5 7 token distance t 10−4 10−3 correlation ˆCP (t) P = 128 P = 1024 P = 16384 P = 262144 P = 4194304 P = Pmax Figure 1: Left: Example of data generation according to the RHM, with depth L = 3 and branching factor s = 2. Starting from the root with ℓ= 3 and following the arrows, each level-ℓsymbol is replaced with a pair of lower-level symbols, down to the leaves with ℓ= 0. Right: Empirical (coloured) and analytical (black dashed) correlation functions of RHM data, with L = 3, s = 2, v = 32 and m = 8. The stepwise decay mirrors the tree structure of the generative model. Empirical estimates obtained from P examples initially follow the true correlation function, but then saturate due to the sampling noise (coloured dashed). As a result, a finite training set only allows for measuring correlations with the tokens up to a certain distance t∗(P). Graphically, t∗(P) corresponds to the highest value of t where the empirical estimate matches the true correlation (e.g. 1 for the orange and green curves, 3 for the red curve). 3 Correlations, training set size and effective context window Given a dataset of d-dimensional sequences of tokens in V, we measure correlations via the token co-occurrences matrix, 2 Ci,j(µ, ν) := P {Xi = µ, Xj = ν} −P {Xi = µ} P {Xj = ν} , (5) where µ and ν are arbitrary elements of the vocabulary V and P refers to the data distribution PX. Since the masked token is always the last in our setup, it is convenient to set j = d and write Ci,d as a function of the distance t = \|i −d\| between the i-th and the masked token. Taking the root mean square over the vocabulary yields the correlation function, ˜C(t) :=  v−2 X µ,ν∈V (Cd−t,d(µ, ν))2   1/2 , (6) which measures the typical dependency between tokens as a function of their distance t. For RHM data with m = vs−1, PX is uniform over all the possible sequences of tokens in V and there are no correlations. If, instead, m < vs−1, the correlations strength depends on the distance. Fig. 1 shows an example for RHM data with L = 4, s = 2, v = 32 and m = 8. Correlations decay with distance. The stepwise decay of ˜C(t) mirrors the tree structure of the generative model. The masked token has the highest correlations with those belonging to the same s-tuple, as they were all generated by the same level-1 symbol (as in the blue box of Fig. 1, left). The second highest is with the tokens generated by the same level-2 symbol (orange box in the figure), and so on until the root. Formally, with ℓ= 1, . . . , L denoting the height of the lowest common ancestor (LCA) of the d-th and (d −t)-th tokens, ˜C(t) = ˜C(ℓ) ∀ t = sℓ−1, . . . , sℓ−1; ˜C(1) > ˜C(2) > · · · > ˜C(L). (7) These L plateau values can be determined analytically in the large v limit by approximating the variance over the vocabulary entries µ and ν on the right-hand side of Eq. 7 with the variance over realisations of the RHM. Denoting the average over such realisations with ⟨.⟩, ˜C(ℓ) = C(ℓ)(µ, ν) 21/2 ≃ r (1 −m/vs−1) v3m2ℓ−1 , (8) 2Ci,j(µ, ν) is also equivalent to the covariance matrix of the one-hot representation, E [(Xi,µ −E [Xi,µ]) (Xj,ν −E [Xj,ν])] 4 102 103 104 105 106 training set size P 100 test cross-entropy m=8 m=11 102 103 104 105 106 training set size P 100 test cross-entropy t=1 t=3 t=7 Figure 2: Left: Learning curves of depth-3 transformers trained on RHM data with L = 3, s = 2, v = 32 and m = 8 (blue) or 11 (orange, both are averaged over 8 independent realisations of the dataset and initialisations of the network), displaying a stepwise behaviour analogous to the correlation function. The vertical dashed lines mark the characteristic training set sizes Pk at which the correlation with tokens at distances up to t = sk −1 emerge from the sampling noise. Horizontal dashed lines represent (upper bounds on) the cross-entropy of the probability of the last token conditioned on the previous sk −1, suggesting that the steps correspond to the model learning a progressively larger sub-tree of the data structure. Right: Learning curves of transformers for m = 8 and different sizes t of the context window. The saturation of the loss decay due to the finite context window highlights that the decay is entirely due to the ability to leverage a larger portion of the context window. where the rightmost equality is exact asymptotically in v and m. Eq. 8 is derived in detail in App. D, confirmed empirically in the right panel of Fig. 1 and can be given a simple interpretation in terms of the sample size required for the empirical measurement of correlations, as discussed in the following paragraph. In addition, notice that, upon replacing sℓwith t, the m−ℓdependence on ℓis approximated by a power-law decay ˜C(t) ∼t−β, with β = log m/ log s. Saturation due to finite training set. When measuring the correlation function from a finite sample XP of P data, there is an additional contribution due to the sampling noise. The scenario is illustrated in Fig. 1, right: the empirical estimates ˆCP (t), shown as coloured lines for different values of P, begin by following the descent of the true correlation function ˜C(t), shown as a black dashed line. However, empirical estimates saturate when approaching the sampling noise size (v2P)−1/2, as proved in App. E and shown as dashed coloured lines in Fig. 1, right. Combining the saturation with the values of the steps, we deduce that a finite training set allows for the resolution of correlations up to distance t∗= sℓ∗−1 such that ˜C(ℓ∗) > (v2P)−1/2 > ˜C(ℓ∗+1). (9) Eq. 9 suggests that a language model trained with P examples can only extract information from the tokens within distance t∗(P) from the last. In other words, a finite training set is equivalent to an effective context window of size t∗(P). If ˜C ∼t−β, then t∗(P) ∼P 1/2β. Alternatively, setting ˜C(ℓ) = (v2P)−1/2 yields a sequence of thresholds Pℓfor the resolution of correlations of increasing range. From Eq. 8, Pℓ∝vm2ℓ−1, which has a simple interpretation as the number of choices in the generative process to determine two tokens at a distance t ∈[sℓ−1, sℓ): v choices for the level-ℓLCA, m for the first production rule and m2 (m per branch) for each of the remaining ℓ−1 generations. 4 Self-supervised learning of the Random Hierarchy Model We now show how the correlations can be translated into a prediction of sample complexities that allow for a sequence of increasingly accurate approximations of the masked token probability, based on reconstructing the hidden variables of the generative tree. We then test these predictions in numerical experiments with deep networks. 4.1 Prediction of the sequence of performance steps and sample complexities Loss steps. Due to the structure of the data, there is a natural sequence of L increasingly accurate approximations of the last token probability in Eq. 2. For all ℓ= 1, . . . , L, these approximations are 5 realised by conditioning the probability of the last token on the previous sℓ−1. These approximations amount to using an effective context window of size tℓ= sℓ−1. The effective context windows consist of the leaves of the subtree generated by the level-ℓhidden symbol above the last token, as illustrated by the coloured boxes of Fig. 1, left. The resulting cross-entropy loss is given by Lℓ= Ex∼PX [−log P {Xd\|Xd−sℓ+1 = xd−s+1, . . . , Xd−1 = xd−1}] = Ex∼PX [log N(xd−sℓ+1, . . . , xd−1)] , (10) where N(xd−sℓ+1, . . . , xd−1) denotes the number of possible values of the masked token depending on the effective context window. For ℓ= 0, there is no restriction on the masked token value and this number equals v—the vocabulary size. For ℓ= 1, we can determine the average ¯N1 := E [N(xd−s+1, . . . , xd−1)] as follows. For each s-tuple (xd−s+1, . . . , xd) there is at least one value of the mask compatible with the other s −1 symbols, i.e. xd itself. In addition, each of the remaining v −1 values µd ̸= xd has a probability f of being compatible with the context, coinciding with the probability that the s-tuple (xd−s+1, . . . , µd) is compatible with the production rules. This probability is given by (mv −1), i.e. the number of s-tuples compatible with the production rules except (xd−s+1, . . . , xd), over (vs −1), i.e. the total number of s-tuples except (xd−s+1, . . . , xd). Therefore, ¯N1 = 1 + (v −1)f = 1 + (v −1)(mv −1)/(vs −1). For ℓ> 1, the average number ¯Nℓ of symbols compatible with the context can be determined iteratively. The level-ℓsymbol generating the whole sℓ-tuple can take any of the v values, but the level-(ℓ−1) symbol below it is now restricted to ¯N1 values. By the previous argument, ¯Nℓ= 1 + (v −1)(m ¯Nℓ−1 −1)/(vs −1). Due to the concavity of the logarithm, we can bound the test loss of Eq. 10 with ¯Lℓ= log ¯Nℓ, i.e., after solving the recurrence relation and introducing the fraction of compatible s-tuples f = m/vs−1. ¯Lℓ= log vs −v vs −1 −m(v −1) + (vs −mv)(v −1) vs −1 −m(v −1) m(v −1) vs −1 ℓ! v,m≫1 −−−−→log 1 1 −f + vf ℓ , (11) Naive strategy. The simplest strategy to estimate P {Xd\|Xd−sℓ+1 = xd−sℓ+1, . . . , Xd−1 = xd−1} is to count the empirical occurences of the sℓ-dimensional subsequences of the input in the training set—the so-called n-gram language model with n = sℓ. This estimation requires the training set to contain all the distinct subsequences of size sℓ. Following the generative process, each of these subsequences occurs with probability given by that of the corresponding LCA symbol (the root of the subtree encased by the corresponding coloured box in Fig. 1, left), times the probability of the production rules that generate the subsequence from the LCA, m−1 ×m−s ×· · ·×m−(sℓ−1) = m−(sℓ−1)/(s−1). The latter is exponentially small in the effective context length tℓ= sℓ−1, hence the required sample size is exponentially large in tℓ. Efficient strategy leveraging the hidden variables. Using the hidden variables results in a much lower sample complexity. Indeed, due to the tree structure of PCFGs, the value of the last token is conditionally independent of most of the observable tokens when the hidden variables are given. For instance, looking at the tree in Fig. 1, left, the probability of the last token is independent of the pair (µ(0) 5 , µ(0) 6 ) if the parent level-1 variable µ(1) 3 is given. In general, fixing a hidden symbol splits the tree into an inside (the subtree rooted at the hidden symbol) and an outside (the rest of the tree) that are conditionally independent. As a result, the minimal set of variables that the sℓ-gram probability depends on consist of s −1 observable tokens (those in the same patch as the last token) and (s −1)(ℓ−1) hidden variables ((s −1) for each level below the LCA of the context window). The probability of any such set of variables is given by the LCA probability times m−ℓ. The resulting sample complexity grows exponentially with ℓ, or as a power of the effective context length tℓ. Reconstruction of the hidden variables. We now argue that, as shown [20] in the context of classification, the hidden variables can be represented via the correlations between tokens. Consider, for instance, the pair (µ(0) 5 , µ(0) 6 ) in Fig. 1, left. Because of the aforementioned conditional independence, the correlation between any such pair and the last token depends only on the level-1 hidden variable µ(1) 3 . Thus, pairs displaying the same correlations can be grouped as descendants of the same hidden variable. This strategy requires enough training data to resolve correlations between the masked token and the adjacent s-tuples of observable tokens. As shown in App. F, replacing an observable token 6 with a whole s-tuple reduces correlation plateaus and sampling noise by the same factor. Therefore, the condition for the resolution of correlations with the nearest s-tuples is given by Eq. 9 with ℓ= 2, implying P > P2 = vm3. By iterating this argument we get a sequence of sample complexities Pℓ that allow for resolving correlations between the masked token and s-tuples up to distance t = sℓ−1, Pℓ= (v2 ˜C(ℓ))−1 = vm2ℓ−1 1 − m vs−1 −1 . (12) For instance, in the case illustrated in Fig. 1, left, the correlations of the pairs (µ(0) 1 , µ(0) 2 ) and (µ(0) 3 , µ(0) 4 ) with the masked token can be used to reconstruct the pair of hidden symbols (µ(1) 1 , µ(1) 2 ). The hidden symbols have a higher correlation with the masked token than their children. Hence, as in the case of classification [20], a training set large enough to resolve correlations between observable and masked tokens also allows for resolving correlations of the masked token with the hidden symbols. These correlations yield a representation of higher-level hidden symbols (e.g. µ(2) 1 for (µ(1) 1 , µ(1) 2 ) in the figure), which, in turn, enables the reconstruction of P {Xd\|Xd−sℓ+1 = xd−s+1, . . . , Xd−1 = xd−1} via the efficient strategy. As ℓincreases, the sample complexity of Eq. 12 grow faster than mℓ, but still polynomially in the effective context length tℓ. Scaling law of the RHM. After solving Eq. 12 for ℓas a function of P, we can use Eq. 11 to derive the scaling law for the behaviour of the loss steps as a function of the training set size P. Neglecting all the factors that do not depend on ℓ, Eq. 12 implies ℓ≈log P/(2 log m). Thus, from Eq. 11, ¯L(P) + log (1 −f) ≈log 1 + v(1 −f)e log f log P 2 log m . (13) Notice that f < 1, thus log f < 0. Therefore, Eq. 11 implies an early logarithmic decay as long as \| log f\| log P ≪2 log m log (v(1 −f)). For larger P, the expansion log (1 + x) ≃x recovers the ubiquitous power-law decay P −α [17], with exponent α = log f/(2 log m). Notice that the power-law scaling is caused by the sequence of steps associated with the emergence of the hidden variables representation. Therefore, this picture unifies the emergence and scaling paradigms. 4.2 Comparison with empirical learning curves Fig. 2, left, compares the learning curves of deep transformers (details of the architectures in subsection A.2) with the sample complexities Pℓof Eq. 12 (vertical dashed lines in the figure) and the test loss upper bounds ¯Lℓof Eq. 11 (horizontal dashed lines), showing good qualitative agreement. Additional experiments that support the quantitative scaling of the sample complexities P1 and P2 with m are shown in App. G. Fig. 2, right, shows the learning curves of models trained on a reduced context window. In this setting, our description correctly predicts the saturation of the loss due to the finite context window size t: with t = sℓ−1, the model can only learn the level-ℓhidden variable above the masked token, thus follow only the first ℓof the L steps of Eq. 11. Let us remark that, as shown in App. G, the learning curves are qualitatively similar for CNNs, despite a noticeable quantitative dependence on architecture and context size t. These differences are not captured by the analysis of subsection 4.1, although, in some cases, they can be rationalised using results from the theory of shallow neural networks. We discuss these aspects in detail in App. G. 4.3 Emergence of hierarchical representations of the data structure We now study the hidden representations of models trained on RHM data to show that, as the training set size increases, these representations encode for deeper hidden variables. More specifically, we show that certain representations depend only on specific, high-level hidden variables of a datum’s tree structure, thus becoming insensitive to the entire subtree emanating from this hidden variable. For the sake of interpretability, we consider deep convolutional networks (CNNs) with architecture matched to the data structure, represented schematically in the graphs on the right of Fig. 3 (further details in subsection A.1). To probe representations we introduce two sets of transformations. Given a datum and the associated tree ( Fig. 1, left), consider the i-th level-ℓsymbol µ(ℓ) i : Sℓ,i replaces it with another one randomly chosen from the vocabulary, whereas Rℓ,i resets the choice of the production rule emanating from µ(ℓ) i . Both transformations alter the subtree originating from µ(ℓ) i (e.g. the subtree within the orange box of Fig. 2, left for ℓ= 2 and i = 2), affecting sℓobservable tokens. However, Rℓ,i preserves the hidden symbols that generated the subtree. Therefore, a hidden representation that encodes only the i-th level-ℓhidden symbol will be invariant to Rℓ,i but not to Sℓ,i. 7 102 103 104 105 106 training set size P 0.0 0.5 1.0 r1/s1 1-th rep. 2-th rep. 3-th rep. 4-th rep. output 102 103 104 105 106 training set size P 0.0 0.5 1.0 r2/s2 2-th rep. 3-th rep. 4-th rep. output 102 103 104 105 106 training set size P 0.0 0.5 1.0 r3/s3 3-th rep. 4-th rep. output Figure 3: Relative sensitivity rℓ/sℓof the representation of trained depth-4 CNNs (sketched on the right panels) for input transformations (the affected tokens are indicated by the black horizontal segments on the right panels) corresponding to resetting the production rule emanating from a given level-ℓvariable (ℓ= 1, 2, 3 for top, centre and bottom), as a function of training set size P. Colours represent the layer of the representation, as indicated in the key and by the squares on the right panels. The CNNs are trained on RHM data with L = 4, s = 2, v = 16, m = 4. Vertical dashed lines mark the sample complexities Pℓof Eq. 12. The drop of the curves from ≃1 to ≃0 around Pℓsignals that the trained representations only encode for the relevant level-ℓsymbol when P > Pℓ. We define hidden representations hℓ(x) (hidden nodes of the network’s graphs in Fig. 3) as the sequence of pre-activations in a given layer ℓ(depth of the node in the tree), standardised over the dataset (i.e. centred around the mean and scaled by the standard deviation). For CNNs, representations carry a spatial index j = 1, . . . , sL−ℓ(horizontal position of the node within the layer) and a channel index. We measure the sensitivity to R or S via the cosine similarity between original and transformed representations, i.e. rℓ,i(h) = Ex∼PX [hℓ′,j(x) · hℓ′,j(Rℓ,ix)] , sℓ,i(h) = Ex∼PX [hℓ′,j(x) · hℓ′,j(Sℓ,ix)] , (14) where the · symbol denotes the scalar product over the channels. In order to leave the masked token unaltered, we always apply the transformations to the penultimate hidden symbol of the level, i.e. i = sL−ℓ−1. Hence, from now on, we omit the spatial index i. The left column of Fig. 3 reports the ratio rℓ/sℓfor the hidden representations of a deep CNN trained on RHM data. Each row refers to the level of the data transformations. The group of observable tokens affected by the transformation is highlighted by horizontal square brackets in the right panels. The drop of rℓ/sℓfrom ≈1 to ≈0 signals that a representation depends on the corresponding level-ℓhidden variable, but not on the other variables in the associated subtree. 3 These drops occur at the same training set sizes Pℓas the 3Notice that only the representations with ℓ′ > ℓcan become invariant, which is due to the fact the production rules are not linearly separable. Let us focus on the first level: the corresponding s-dimensional patch of the input can take mv distinct values—m for each of the v level-2 features. Invariance of a linear transformation is equivalent to the following set of constraints: for each level-2 features µ, and x1,i encoding for one of the m level-1 representations generated by µ, w · x1,i = cµ. Since cµ is an arbitrary constant, there are v × (m −1) constraints for the v × s components of w, which cannot be satisfied in general unless m ≤(s + 1). 8 103 104 105 P 2 × 100 3 × 100 test cross-entropy L P −α, α ≃0.1 t=1 t=2 t=3 t=7 t=15 100 101 token distance t 10−4 10−3 correlation ˆCP (t) t−β, β ≃1.4 P = 256 P = 1024 P = 4096 P = 16384 P = 65536 P = Pmax 100 102 104 P/tz 3 × 100 4 × 100 6 × 100 L × tαz x−α t=1 t=2 t=3 t=7 t=15 10−2 10−1 100 scaled distance t/P 1/z 10−1 ˆCP (t) × P 1/2 x−β P = 256 P = 1024 P = 4096 P = 16384 P = 65536 P = 262144 Figure 4: Top, Left: Test losses of 3-layers transformers trained on (t + 1)-characters blocks of the tinyShakespeare dataset [38] (t as in the key). The saturation of the loss to some t-dependent value indicates that performance improves with P because the model can use information from a larger context window. Top, Right: Empirical estimates ˆCP (t) for different training set sizes P as in the key. The curves initially follow the true correlation ˜C(t) (black dashed), but then saturate due to the sampling noise (coloured dashed). Bottom, Right: The empirical curves ˆCP (t) collapse when rescaling correlations by the sampling noise size P −1/2 and t by the characteristic distance t∗(P) ∼P 1/z, with z ≃2.8. Bottom, Left: As predicted by our conjecture, the losses collapse when rescaled according to Eq. 16 with the same z as the correlation functions. test loss steps, highlighted in the figures with vertical dashed lines. This result confirms that, as P increases, trained models learn a deeper representation of the tree structure of the data. 5 Conjecture and test on real language data We conjecture that the relationship between training set size, correlations and effective context window holds beyond our synthetic dataset. Conjecture: “If the token correlation function decays with the token distance, then a language model trained to predict the next token from a training set of P examples can only extract relevant information from an effective context window of P-dependent size t∗(P).” We test this conjecture in two datasets: a selection of lines from Shakespeare’s plays [38] and a collection of articles from English Wikipedia [39]. For both datasets we adopt a character-level tokenisation, resulting in over 106 tokens. We then extract sequences of t consecutive tokens and train BERT-like deep transformers in the setup of section 2—further details of architecture and training are in subsection A.3. The results of our test are reported in Fig. 4 for Shakespeare and Fig. 5 of App. B for Wikipedia. First, with a large context window, the test loss follows the empirical scaling law L ∼P −α (top left panel). However, the learning curve levels off at some characteristic scale P that grows with the size t of the context window. This phenomenon can be explained via the correlation function, which decays as a power of the distance ˜C(t) ∼t−β, with β ≃1.4 4 (top right panel). Empirical estimates ˆCP (t) saturate when reaching the sampling noise scale ∼P −1/2: following the analysis of section 3, this behaviour results in an effective context window size t∗(P), given by the 4Let us remark that, while the exponent depends on the corpus and the choice of tokenisation, power-law decays are observed empirically also for syllables [40], words [41] and part-of-speech tags [42]. 9 value of t where the correlation function ˜C(t) ∼t−β intersects the sampling noise scale ∼P −1/2, (t∗)−β ∼P −1/2 ⇒t∗(P) ∼P 1/z, with z = 2β ≃2.8. (15) As a result, the empirical correlation functions with varying P collapse when rescaling ˆC by the sampling noise and the distance by t∗(P) (bottom right panel). By inverting t∗(P) we get a characteristic training set size P ∗(t) where the training set allows for resolving correlations at all distances t′ < t, P ∗(t) ∼tz. Paired with the empirical power-law scaling with P, this result leads to the following context-dependent scaling hypothesis for the test loss: L(P, t) = t−αzf P tz , (16) with f(x) ∼x−α for x ≪1 and constant for x ≫1. In particular, Eq. 16 implies that the behaviour of the empirical correlation functions predicts the saturation of the loss decay. The collapse reported in the bottom left panels of Fig. 4 and Fig. 5 quantitatively confirms Eq. 16 and our previous conjecture. 6 Conclusions We proposed a conceptual framework for understanding the performance-vs.-data scaling laws of language models trained for next-token prediction. In our picture, increasing the number of data allows for the resolution of a longer range of correlations. These correlations, in turn, can be exploited to build a hierarchical representation of the data structure, the longer the range the deeper the representation. For our synthetic hierarchical data, the emergence of deeper representation results in a series of steps in the next-token prediction performance. These steps conspire to determine the scaling law, whose exponent depends on the dataset structure. This scenario is consistent with the empirical phenomenology of language models, including both the emergence of skills at specific training set sizes [18, 43, 44, 45] and the steady improvement of overall performance [17]. To the best of our knowledge, this is the first theoretical description of scaling laws in a setting where learning data features is crucial, whereas previous works focused on kernel limits [46, 47, 48, 49, 50]. Furthermore, our analysis predicts a fundamental relationship between the effective context window captured by a language model trained with a finite training set and the decay of token-token correlations, which we confirmed empirically on two examples of text data. This finding suggests that the exponents entering scaling laws are influenced by the intrinsic properties of the data. On the one hand, our predictions can be tested on state-of-the-art LLMs trained on larger datasets. On the other hand, our framework can be extended to shed light on other aspects of scaling laws of high practical relevance, such as the role of the number of parameters and the behaviour of performance when the model size and the number of data are optimised under a fixed compute budget. Limitations. Our hierarchical model of data is limited by the context-free structure of the rules, which describes most, but not all, of the syntactic forms observed in natural languages [51]. Understanding the role of context-sensitive structures in language acquisition is a promising avenue for future research. In addition, the RHM assumes a fixed geometry of the data tree and the uniform probability and unambiguity of the production rules. These assumptions are not satisfied by real text data and are responsible for the stepwise behaviour of correlations in our model. Relaxing these constraints while keeping the large-scale, power-law decay of correlations with the distance, which is indeed observed in real data, could broaden the scope of our conceptual framework. On the technical side, there is no proof of the connection between the strategy illustrated in subsection 4.1 and the sample complexity of deep neural networks trained with gradient descent and its variants. Such a proof would require a formal description of the dynamics of deep networks trained on hierarchical data, which is beyond the scope of the present paper. This description would also capture the discrepancies between different architectures presented in App. G, making it a valuable direction for future work. Acknowledgments and Disclosure of Funding We thank Kai Nakaishi for pointing references [52, 27] to us; and Allan Raventós for feedback on an earlier version of the manuscript. We also thank Antonio Sclocchi, Alessandro Favero and Umberto Tomasini for helpful discussions and feedback on the manuscript. This work was supported by a grant from the Simons Foundation (# 454953 Matthieu Wyart). 10 References [1] Noam Chomsky. Aspects of the Theory of Syntax. The MIT Press, 50 edition, 1965. [2] Gerhard Jäger and James Rogers. Formal language theory: refining the chomsky hierarchy. Philosophical Transactions of the Royal Society B: Biological Sciences, 367(1598):1956–1970, 2012. [3] Robert C Berwick, Paul Pietroski, Beracah Yankama, and Noam Chomsky. Poverty of the stimulus revisited. Cognitive science, 35(7):1207–1242, 2011. 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We provide architecture and training details below. A.1 Deep CNNs (RHM) The deep CNNs we consider are made by stacking standard convolutional layers. To tailor the network to the structure of the data generative model, we fix both the stride and filter size of these layers to s. Since each layer reduces the spatial dimensionality by a factor s, the input size d must be an integer power of s and the CNNs depth equals log d/ log s. We use the Rectified Linear Unit (ReLU) σ(x) = max (0, x) as activation function, set the number of channels to H for each layer, and consider the maximal update parametrization [53], where the weights are initialised as random gaussian variables with zero mean and unit variance, all the hidden layers but the last are rescaled by a factor H−1/2, whereas the last is rescaled by H−1. This factor causes the output at initialization to vanish as H grows, which induces representation learning even in the H →∞limit. In practice, H is set to 256 for Fig. 3, 512 for Fig. 6, left and Fig. 9, 1024 for Fig. 6, right, 512 for Fig. 7 and Fig. 8. Increasing the number of channels does not affect any of the results presented in the paper. Deep CNNs are trained with SGD, with the learning rate set to H to compensate for the factor of H−1. A cosine annealing scheduler reduces the learning rate by 10 within the first 100 training epochs. The batch size is set to the minimal size allowing convergence, where we define convergence as the training cross-entropy loss reaching a threshold value of 10−3. We use a validation set of size 215 to select the model with the best validation loss over the training trajectory. A.2 Multi-layer self-attention (RHM) The deep Transformers that we train on RHM data are made by stacking standard Multi-Head Attention layers [54], without any residuals, layer normalization and multi-layer perceptron in between. We found that the removed components do not affect the model’s performance on data generated from the RHM. Each layer has the same number of heads nh and embedding dimension demb = nh × v, with v the vocabulary size. The input dimension is adapted to the embedding dimension via a learnable linear projection, to which we add learnable positional encodings. The choice of nh follows two principles: the model should be large enough for the training loss to reach a threshold value of 10−3 and changing nh should not affect performance beyond the fluctuations due to the model initialisations. Specifically, we set nh = 16 and notice no significant change in performance up to nh = 64. Also scaling demb up to 4nh × v does not impact performance. Multi-layer self-attention networks are trained with the Adam optimizer, with a warmup scheduler bringing the learning rate to 10−2 within the first 10 training epochs. As for CNNs, the batch size is set to the lowest value allowing for convergence. A.3 Encoder-only Transformer (tiny-Shakespeare and WikiText-103) The architectures trained on real text data have the same structure as BERT [8], that is they include additional token-wise two-layer perceptions (MLPs) after each self-attention layer, together with layer normalization operations before the attention layer and the MLP and residual connections. The training procedure is the same as above. For tiny-Shakespeare, we set the number of heads to nh = 8, the embedding dimension to de = 256, the size of the MLP hidden layer to 4de, and the number of layers to 3. For WikiText-103, we set nh = 8, de = 512, and the number of layers to 6. Increasing the number of layers or the number of heads does not affect the results presented in the paper. 14 102 103 104 105 106 P 2 × 100 3 × 100 test cross-entropy L P −α, α ≃0.095 t=1 t=2 t=3 t=5 t=7 t=15 100 101 token distance t 10−4 10−3 correlation ˆCP (t) t−β, β≃1.55 P = 512 P = 2048 P = 8192 P = 32768 P = 131072 P = 524288 100 102 104 106 P/tz 3 × 100 4 × 100 6 × 100 L × tαz x−α t=1 t=2 t=3 t=5 t=7 t=15 10−1 100 scaled distance t/P 1/z 10−2 10−1 ˆCP (t) × P 1/2 x−β P = 512 P = 2048 P = 8192 P = 32768 P = 131072 P = 524288 Figure 5: Top, Left: Test losses of 6-layers transformers trained on (t + 1)-characters blocks of the WikiText103 [39] (t as in the key). As in Fig. 4, the loss saturates to some t-dependent value after reaching a characteristic training set size. Top, Right: Empirical correlation functions ˆCP (t) with P as in the key, showing saturation for large t due to the sampling noise (coloured dashed). Bottom, Right: Collapse of the empirical curves ˆCP (t) is achieved when rescaling the correlations by the sampling noise size P −1/2 and t by the characteristic distance t∗(P) ∼P 1/z, with z = 2β ≃3.1. Bottom, Left: As predicted by our conjecture, the losses collapse when rescaled according to Eq. 16 with the same z as the correlation functions and α ≃0.095. B Loss saturation and correlations for WikiText-103 In this section, we report the results of the test of our conjecture for the WikiText-103 dataset of [39]. The original dataset was preprocessed to remove the article’s headers and subheaders. The results are summarised in Fig. 5, which displays the same measures as Fig. 4 and, as Fig. 4, confirms our conjecture. C Statistics of the RHM data For each token i = 0, . . . , d −1, single-token probabilities can be written as products of probabilities over the single production rules, P {Xi = µ} = v X µ1,...,µL=1 p(1) i1 (µ\|µ1) . . . p(L) iL (µL−1\|µL)p(L+1)(µL), (17) where (i) the indices iL, . . . , iL are such that iL . . . i1 equals the s-ary representation of i, with iℓ= 0, . . . , s −1, and 0’s added to ensure that the representation always consists of L indices. In other words, the multi-index iL, . . . , iL uniquely identifies the path linking the root of the tree to the i-th leaf. (ii) p(ℓ) iℓ(µℓ−1\|µℓ) denotes the probability of choosing, among the available production rules starting from µℓ, one that has the symbol µℓ−1 on the iℓ-th position of the right-hand size. (iii) p(L)(µL) denotes the probability of selecting the symbol µL as the root (1/v for our model). These decompositions arise naturally due to the connection between probabilistic context-free grammars and Markov processes. For the joint probability of two tokens i and j at distance t = \|j −i\|, 15 with sℓ−1 < t < sℓ−1 such that the lowest common ancestor (LCA) is a level-ℓhidden symbol, and iℓ+1 denoting the position of the LCA within its level, P {Xi = µ, Xj = ν} = v X µ1,...,µℓ=1 ν1,...,νℓ−1=1 v X µℓ=1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1) . . . p(ℓ) iℓ,jℓ(µℓ−1, νℓ−1\|µℓ)p(ℓ+1) iℓ+1 (µℓ). (18) Both in Eq. 17 and Eq. 38 simplify when replacing µ with a whole s-tuple of observable symbols µj = (µ1+(j−1)s, . . . , µjs) for some j = 1, . . . , sL−1. The simplification arises because the level-1 rule probability p(1)(µj\|µ1), is uniform and equal to 1/m if the production rule µ1 →µj exists, 0 otherwise. Then, the sum over µ1 selects the only level-1 symbol that generates the tuple µj. As a result, one is left with a probability represented by a smaller tree, where the s leaves representing µj are pruned, and an additional factor of 1/m. C.1 Statistics of production rules For each set of production rules, we call N (ℓ) i (µℓ−1; µℓ) the number of occurrences of the level-ℓ−1 feature µℓ−1 in the i-th position of the right-hand side for all the production rules emanating from the level-ℓfeature µℓ. In our generative model, there are m production rules emanating from a given symbol. The rule to follow when generating a datum is chosen uniformly at random among these m. Hence, p(ℓ) i (µℓ−1\|µℓ) = 1 mN (ℓ) i (µℓ−1; µℓ). (19) For the sake of clarity, let us omit the level index in the following paragraph. The probability of Ni(µ; ν) over different realisations of the set of production rules is that of the number of successes when drawing m times (number of s-tuples associated with the high-level feature ν) without replacement from a pool of vs (total number of s-tuples with vocabulary size v) objects where only vs−1 (number of s-tuples displaying feature µ in position i) leads to success: P {Ni(µ0; µ1) = k}RHM = vs−1 k vs −vs−1 m −k vs m = Hgm,vs−1,vs(k), (20) where Hgn,K,N denotes a Hypergeometric distribution with population size N, K success states, and n draws. The mean and variance with respect to realisations of the RHM (denoted with ⟨.⟩to avoid confusion with data averages E [.]) are ⟨N⟩= mvs−1 vs = m v , σ2 N := (N −⟨N⟩)2 = mvs−1 vs vs −vs−1 vs vs −m vs −1 = m v v −1 v vs −m vs −1 . (21) For v ≫1, the variance converges to m/v (m ≤vs−1 with s fixed, thus vs −m →vs). Equations (19) to (21) easily generalise to the case where µ0 represents a group of s′ ≤s low-level features (instead of a single low-level feature). With µ0 denoting a s′-tuple of features and i the s′-tuple of associated spatial indices, P {Ni(µ0; µ1) = k}RHM = vs−s′ k vs −vs−s′ m −k vs m , (22) resulting in ⟨Ns′⟩= mvs−s′ vs = m vs′ , σ2 Ns′ := (Ns′ −⟨Ns′⟩)2 = m vs′ vs′ −1 vs′ vs −m vs −1 v≫1 −−−→m vs′ . (23) C.2 Statistics via splitting An alternative way to obtain all statistics is by writing the level-ℓprobabilities as sums over the production rules, p(ℓ) i (µℓ−1\|µℓ) = 1 m m X ψ=1 δ(rψ,i(µℓ), µℓ−1), (24) 16 where rψ,i(µ1) denotes the i-th element of the right-hand side of the ψ-th production rule emanating from µ1. Eq. 24 generalises immediately to the case where i and µℓ−1 are s′-tuples with s′ ≤s. Using ⟨δ(rψ,i(µ1), µ)⟩= P n µ1 ψ,i −−→µ o RHM = Hg1,vs−s′,vs(1) = 1 vs′ , (25) where µ1 ψ,i −−→µ denotes the event that the i-th element of the right-hand side of the ψ-th production rule emanating from µ1 coincides with µ, we can compute all one-point averages. In addition, for (ν1, ϕ, j, ν) ̸= (µ1, ψ, i, µ), ⟨δ(rψ,i(µ1), µ)δ(rϕ,j(ν1), ν)⟩= P n µ1 ψ,i −−→µ o RHM P n ν1 ϕ,j −−→ν µ1 ψ,i −−→µ o RHM , (26) where P n ν1 ϕ,j −−→ν µ1 ψ,i −−→µ o RHM =                                      P n ν1 ϕ,j −−→ν o RHM = v−1, if i ̸= j 0, if i = j, ν1 = µ1, ϕ = ψ, µ ̸= ν, Hg1,vs−1−1,vs−1(1) = vs−1 −1 vs −1 , if i = j, ν1 = µ1, µ = ν, ϕ ̸= ψ, Hg1,vs−1,vs−1(1) = vs−1 vs −1, if i = j, ν1 = µ1, µ ̸= ν, ϕ ̸= ψ, Hg1,vs−1−1,vs−1(1) = vs−1 −1 vs −1 , if i = j, ν1 ̸= µ1, µ = ν, Hg1,vs−1,vs−1(1) = vs−1 vs −1, if i = j, ν1 ̸= µ1, µ ̸= ν. (27) Notice that, once the right-hand side of the rules (µ and ν) are fixed, the conditional probability can only attain two distinct values: one for µ1 = ν1 and ψ = ϕ, and one for the other cases. Then, it is convenient to define a ‘rule index’ ψ that comprises both the starting high-level feature and the chosen production rule. This index runs in (1, . . . , mv). With these formulas, one can get all the joint statistics of the rules. For instance (omitting the RHM subscript on P to ease notation), ⟨pi(µ0\|µ1)pi(µ0\|µ1)⟩= 1 m2 m X ψ1,ψ2=1 P n µ1 ψ1,i −−→µ0; µ1 ψ2,i −−→µ0 o = 1 m2 X ψ1=ψ2 P n µ1 ψ1,i −−→µ0 o + 1 m2 X ψ1,ψ2̸=ψ1 P n µ1 ψ1,i −−→µ0 o P n µ1 ψ1,i −−→µ0\|µ1 ψ1,i −−→µ0 o = 1 m2 m v + m(m −1) v vs−1 −1 vs −1 , (28) hence σ2 p = ⟨(pi(µ0\|µ1) −⟨p⟩) (pi(µ0\|µ1) −⟨p⟩)⟩= ⟨pi(µ0\|µ1)pi(µ0\|µ1)⟩− 1 v 2 = 1 mv v −1 v vs −m vs −1 , (29) equivalent to dividing σ2 N from Eq. 21 by m2. Analogously, with ν0 ̸= µ0, ⟨pi(µ0\|µ1)pi(ν0\|µ1)⟩= 1 m2 m X ψ1,ψ2=1 P n µ1 ψ1,i −−→µ0; µ1 ψ2,i −−→ν0 o = 1 m2 X ψ1,ψ2̸=ψ1 P n µ1 ψ1,i −−→µ0 o P n µ1 ψ1,i −−→ν0\|µ1 ψ1,i −−→µ0 o = 1 m2 m(m −1) v vs−1 vs −1 , (30) 17 thus cp = ⟨(pi(µ0\|µ1) −⟨p⟩) (pi(ν0\|µ1) −⟨p⟩)⟩= ⟨pi(µ0\|µ1)pi(ν0\|µ1)⟩− 1 v 2 = −1 mv2 vs −m vs −1 . (31) Notice that cp = σ2 p/(v−1) in agreement with the constraint P µ0 pi(µ0\|µ1) = 1. Indeed, for any finite sequence of identically distributed random variables Xµ with a constraint on the sum, P µ Xµ = C for some constant C, v X µ=1 Xµ = C ⇒ v X µ=1 (Xµ −⟨Xµ⟩) = 0 ⇒ (Xν −⟨Xν⟩) v X µ=1 (Xµ −⟨Xµ⟩) = 0 ⇒ v X µ=1 ⟨(Xν −⟨Xν⟩)(Xµ −⟨Xµ⟩)⟩= 0 ⇒ (Xµ −⟨Xµ⟩)2 + (v −1) ⟨(Xµ −⟨Xµ⟩)(Xν −⟨Xν⟩)⟩= 0, (32) where, in the last line, we used the identically distributed variables hypothesis to replace the sum over µ ̸= ν with the factor (v −1). In addition, with ν1 ̸= µ1, ⟨pi(µ0\|µ1)pi(µ0\|ν1)⟩= 1 m2 m X ψ1,ψ2=1 P n µ1 ψ1,i −−→µ0; ν1 ψ2,i −−→µ0 o = 1 m2 X ψ1,ψ2 P n µ1 ψ1,i −−→µ0 o P n ν1 ψ1,i −−→µ0\|µ1 ψ1,i −−→µ0 o = 1 m2 m2 v vs−1 −1 vs −1 , (33) thus ⟨(pi(µ0\|µ1) −⟨p⟩) (pi(µ0\|ν1) −⟨p⟩)⟩= ⟨pi(µ0\|µ1)pi(µ0\|ν1)⟩− 1 v 2 = −1 v2 v −1 vs −1, (34) and ⟨pi(µ0\|µ1)pi(ν0\|ν1)⟩= 1 m2 m X ψ1,ψ2=1 P n µ1 ψ1,i −−→µ0; ν1 ψ2,i −−→ν0 o = 1 m2 X ψ1,ψ2 P n µ1 ψ1,i −−→µ0 o P n ν1 ψ1,i −−→ν0\|µ1 ψ1,i −−→µ0 o = 1 m2 m2 v vs−1 vs −1 , (35) thus ⟨(pi(µ0\|µ1) −⟨p⟩) (pi(ν0\|ν1) −⟨p⟩)⟩= ⟨pi(µ0\|µ1)pi(ν0\|µ1)⟩− 1 v 2 = 1 v2 1 vs −1. (36) D Analytic computation of spatial correlations Given a dataset of d-dimensional sequences of tokens in V, we measure correlations via the token co-occurrences matrix, Ci,j(µ, ν) := P {Xi = µ, Xj = ν}X −P {Xi = µ}X P {Xj = ν}X , (37) where µ and ν are arbitrary elements of the vocabulary V and PX refers to the probability of the data distribution (distinct from PRHM, indicating the probability of the rules of the generative process). Joint and single-token probabilities are given by Eq. 17 and Eq. 38, respectively. We now prove Eq. 8 of the main text. 18 D.1 Level-1 LCA (i-th and j-th tokens are in the same patch) When the LCA of the i-th and j-th tokens is a level-1 hidden symbol, i.e. the two tokens lie in the same s-patch, P {Xi = µ, Xj = ν}X = v X µ1=1 p(1) i1,j1(µ, ν\|µ1)p(2) i2 (µ1), (i1 ̸= j1), P {Xi = µ}X = v X µ1=1 p(1) i1 (µ\|µ1)p(2) i2 (µ1), P {Xj = ν}X = v X ν1=1 p(1) j1 (ν\|ν1)p(2) j2 (ν1), (j2 = i2). (38) We consider the limit of large m, where the univariate probabilities of the hidden variables of any level converge to 1/v, with relative fluctuations of order 1/√m [20] 5. In this limit, we can substitute the probability of the LCA with 1/v, thus obtaining, C(1)(µ, ν) = 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) −1 v2 X µ1,ν1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1). (39) As we will prove in this and the following sections, the correlations have 0 average but nonvanishing variance. Including the fluctuations of the LCA probability results in corrections to the variance that vanish in the limit of large m. Furthermore, notice that we removed the dependence of C(µ, ν) on the positional indices i and j. This is because, asymptotically in v and m, the aforementioned statistics depend only on the depth of the LCA of i-th and j-th tokens, justifying our notation. Since i1 ̸= j1, p(1) i1 (µ\|µ1) and p(1) j1 (ν\|ν1) are independent. Hence, by Eq. 21 and Eq. 23 with s′ = 2, D C(1)(µ, ν) E = 1 v X µ1 ⟨N2⟩ m −1 v2 X µ1,ν1 ⟨N⟩ m ⟨N⟩ m = 0. (40) The variance/2nd moment reads C(1)(µ, ν) 2 = * 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) !2+ + * 1 v X µ1 p(1) i1 (µ\|µ1) !2+ * 1 v X ν1 p(1) j1 (ν\|ν1) !2+ −2 1 v3 X µ1,λ1,κ1 D p(1) i1,j1(µ, ν\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|κ1) E . (41) We will compute the three contributions on the right-hand side separately in the following subsections. D.1.1 One-point term (marginal probability) Pc(µ) = * 1 v X µ1 p(1) i1 (µ\|µ1) !2+ = 1 v2 X µ1,ν1 D p(1) i1 (µ\|µ1)p(1) i1 (µ\|ν1) E (42) We can split the sum into two kinds of terms: those with µ1 = ν1 (mult. v) and those with µ1 ̸= ν1 (mult. v(v −1)). In the following, to simplify the notation, we omit the spatial index i. 5Section 1d of Appendix B. 19 (i)—µ1 = ν1 (mult. v) Pc(µ)(i) = v (mv)2 X ψ1,ψ2 ⟨δ(rψ1(µ1), (µ))δ(rψ2(µ1), (µ))⟩ = v (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ o = v (mv)2  X ψ1 P n µ1 ψ1 −−→µ o + X ψ1,ψ2̸=ψ1 P n µ1 ψ1 −−→µ o P n µ1 ψ2 −−→µ\|µ1 ψ1 −−→µ o   = v (mv)2 m1 v + m(m −1)1 v vs−1 −1 vs −1 . (43) (ii)—µ1 ̸= ν1 (mult. v(v −1)) Pc(µ, ν)(ii) = v(v −1) (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µ; ν1 ψ2 −−→µ o = v(v −1) (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µ o P n ν1 ψ2 −−→µ\|µ1 ψ1 −−→µ o = v(v −1) (mv)2 m2 1 v vs−1 −1 vs −1 . (44) Variance of the marginal probability * 1 v X µ1 p(1) i1 (µ\|µ1) !2+ − * 1 v X µ1 p(1) i1 (µ\|µ1) +2 = Pc(µ, ν) − 1 v 2 = v −1 v3m vs −mv vs −1 . (45) D.1.2 Two-point term (joint probability) Jc(µ, ν) := * 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) !2+ = 1 v2 X µ1,ν1 D p(1) i,j (µ, ν\|µ1)p(1) i,j (µ, ν\|ν1) E . (46) We can split the sum into two kinds of terms: those with µ1 = ν1 (mult. v) and those with µ1 ̸= ν1 (mult. v(v −1)). In the following, to simplify the notation, we omit the spatial indices i and j. (i)—µ1 = ν1 (mult. v) Jc(µ, ν)(i) = v (mv)2 X ψ1,ψ2 ⟨δ(rψ1(µ1), (µ, ν))δ(rψ2(µ1), (µ, ν))⟩ = v (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν o = v (mv)2  X ψ1 P n µ1 ψ1 −−→µν o + X ψ1,ψ2̸=ψ1 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o   = v (mv)2 m 1 v2 + m(m −1) 1 v2 vs−2 −1 vs −1 . (47) 20 (ii)—µ1 ̸= ν1 (mult. v(v −1)) Jc(µ, ν)(ii) = v(v −1) (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µν; ν1 ψ2 −−→µν o = v(v −1) (mv)2 X ψ1,ψ2 P n µ1 ψ1 −−→µν o P n ν1 ψ2 −−→µν\|µ1 ψ1 −−→µν o = v(v −1) (mv)2 m2 1 v2 vs−2 −1 vs −1 . (48) Variance of the joint probability * 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) !2+ − * 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) +2 = Jc(µ, ν) − 1 v2 2 = v2 −1 v5m vs −mv vs −1 . (49) D.1.3 Three-point term Tc(µ, ν) := 1 v3 X µ1,λ1,κ1 D p(1) i,j (µ, ν\|µ1)p(1) i (µ\|λ1)p(1) j (ν\|κ1) E = 1 v3 X µ1,λ1,κ1 X µ′,ν′ D p(1) i,j (µ, ν\|µ1)p(1) i,j (µ, ν′\|λ1)p(1) i,j (µ′, ν\|κ1) E = 1 (vm)3 X µ1,ψ1;λ1,ψ2,;κ1,ψ3 X µ′,ν′ P n µ1 ψ1,ij −−−→µν; λ1 ψ2,ij −−−→µν′; κ1 ψ3,ij −−−→µ′ν o (50) The sum over µ′, ν′ can be split in 4 terms: one with µ′ = µ and ν′ = ν (mult. 1), one with µ′ = µ and ν′ ̸= ν (mult. (v −1)), one with µ′ ̸= µ and ν′ = ν (mult. (v −1)) and one with µ′ ̸= µ and ν′ ̸= ν (mult. (v −1)2). Fixing the right-hand sides, the value of the joint probability of the rules depends only on the rule indices ˜ψ1 = (µ1, ψ1), ˜ψ2 = (λ1, ψ2) and ˜ψ3 = (κ1, ψ3). The sum over µ1, λ1, κ1 can be split in 5 terms: one with µ1 = λ1 = κ1 (mult. v), one with µ1 = λ1 ̸= κ1 (mult. v(v −1)), one with µ1 = κ1 ̸= λ1 (mult. v(v −1)), one with µ1 ̸= λ1 = κ1 (mult. v(v −1)), one with µ1 ̸= λ1 ̸= κ1 (mult. v(v −1)(v −2)). In the following, to simplify the notation, we omit the spatial indices i and j. (i-a)—µ1 = λ1 = κ1; µ′ = µ and ν′ = ν (mult. v) Tc(µ, ν)(i−a) = v (mv)3 X ψ1,ψ2,ψ3 ⟨δ(rψ1(µ1), (µ, ν))δ(rψ2(µ1), (µ, ν))δ(rψ3(µ1), (µ, ν))⟩ = v (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν; µ1 ψ3 −−→µν o = v (mv)3 X ψ1=ψ2=ψ3 P n µ1 ψ1 −−→µν o + 3v (mv)3 X ψ1=ψ2̸=ψ3 P n µ1 ψ1 −−→µν o P n µ1 ψ3 −−→µν\|µ1 ψ1 −−→µν o + v (mv)3 X ψ1,ψ2̸=ψ1,ψ3̸=ψ2,ψ1 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n µ1 ψ3 −−→µν\|µ1 ψ2 −−→µν; µ1 ψ1 −−→µν o = v (mv)3 m 1 v2 + 3m(m −1) 1 v2 vs−2 −1 vs −1 + m(m −1)(m −2) 1 v2 vs−2 −1 vs −1 vs−2 −2 vs −2 . (51) 21 (i-b)—µ1 = λ1 = κ1; µ′ = µ and ν′ ̸= ν (mult. v(v −1)) Tc(µ, ν)(i−b) = v(v −1) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν′; µ1 ψ3 −−→µν o = v(v −1) (mv)3 X ψ1=ψ3,ψ2̸=ψ1 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o v(v −1) (mv)3 X ψ1,ψ2̸=ψ1,ψ3̸=ψ2,ψ1 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o × P n µ1 ψ3 −−→µν\|µ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o = v(v −1) (mv)3 m(m −1) 1 v2 vs−2 vs −1 + m(m −1)(m −2) 1 v2 vs−2 vs −1 vs−2 −1 vs −2 . (52) (i-c)—µ1 = λ1 = κ1; µ′ ̸= µ and ν′ = ν (mult. v(v −1)) Tc(µ, ν)(i−c) = v(v −1) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν; µ1 ψ3 −−→µ′ν o = Tc(µ, ν)(i−b), (53) by symmetry for exchanging µ′ and ν′. (i-d)—µ1 = λ1 = κ1; µ′ ̸= µ and ν′ ̸= ν (mult. v(v −1)2) Tc(µ, ν)(i−d) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν′; µ1 ψ3 −−→µ′ν o = v(v −1)2 (mv)3 X ψ1,ψ2̸=ψ1,ψ3̸=ψ2,ψ1 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o × P n µ1 ψ3 −−→µ′ν\|µ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o = v(v −1)2 (mv)3 m(m −1)(m −2) 1 v2 vs−2 vs −1 vs−2 vs −2 . (54) (ii-a)—µ1 ̸= λ1 = κ1; µ′ = µ and ν′ = ν (mult. v(v −1)) Tc(µ, ν)(ii−a) = v(v −1) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; λ1 ψ3 −−→µν o = v(v −1) (mv)3 X ψ1,ψ2=ψ3 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o + v(v −1) (mv)3 X ψ1ψ2,ψ3̸=ψ2 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n λ1 ψ3 −−→µν\|λ1 ψ2 −−→µν; µ1 ψ1 −−→µν o v(v −1) (mv)3 m2 1 v2 vs−2 −1 vs −1 + m2(m −1) 1 v2 vs−2 −1 vs −1 vs−2 −2 vs −2 . (55) (ii-b)—µ1 ̸= λ1 = κ1; µ′ = µ and ν′ ̸= ν (mult. v(v −1)2) Tc(µ, ν)(ii−b) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; λ1 ψ3 −−→µν o = v(v −1)2 (mv)3 X ψ1ψ2,ψ3̸=ψ2 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n λ1 ψ3 −−→µν\|λ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o v(v −1)2 (mv)3 m2(m −1) 1 v2 vs−2 vs −1 vs−2 −1 vs −2 . (56) 22 (ii-c)—µ1 ̸= λ1 = κ1; µ′ ̸= µ and ν′ = ν (mult. v(v −1)2) Tc(µ, ν)(ii−c) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; λ1 ψ3 −−→µ′ν o = Tc(µ, ν)(ii−b), (57) by symmetry for exchanging µ′ and ν′. (ii-d)—µ1 ̸= λ1 = κ1; µ′ ̸= µ and ν′ ̸= ν (mult. v(v −1)3) Tc(µ, ν)(ii−d) = v(v −1)3 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; λ1 ψ3 −−→µ′ν o = v(v −1)3 (mv)3 X ψ1ψ2,ψ3̸=ψ2 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n λ1 ψ3 −−→µ′ν\|λ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o v(v −1)3 (mv)3 m2(m −1) 1 v2 vs−2 vs −1 vs−2 vs −2 . (58) (iii-a)—µ1 = λ1 ̸= κ1; µ′ = µ and ν′ = ν (mult. v(v −1)) Tc(µ, ν)(iii−a) = v(v −1) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν; κ1 ψ3 −−→µν o = v(v −1) (mv)3 X ψ1=ψ2,ψ3 P n µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µν\|µ1 ψ1 −−→µν o + v(v −1) (mv)3 X ψ1,ψ2̸=ψ1,ψ3 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µν\|µ1 ψ2 −−→µν; µ1 ψ1 −−→µν o v(v −1) (mv)3 m2 1 v2 vs−2 −1 vs −1 + m2(m −1) 1 v2 vs−2 −1 vs −1 vs−2 −2 vs −2 = Tc(µ, ν)(ii−a). (59) (iii-b)—µ1 = λ1 ̸= κ1; µ′ = µ and ν′ ̸= ν (mult. v(v −1)2) Tc(µ, ν)(iii−b) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν′; κ1 ψ3 −−→µν o = v(v −1)2 (mv)3 X ψ1ψ2̸=ψ1,ψ3 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µν\|µ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o v(v −1)2 (mv)3 m2(m −1) 1 v2 vs−2 vs −1 vs−2 −1 vs −2 . (60) (iii-c)—µ1 = λ1 ̸= κ1; µ′ ̸= µ and ν′ = ν (mult. v(v −1)2) Tc(µ, ν)(iii−c) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν; κ1 ψ3 −−→µ′ν o = v(v −1)2 (mv)3 X ψ1,ψ2=ψ1,ψ3 P n µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µ′ν\|µ1 ψ1 −−→µν o + v(v −1)2 (mv)3 X ψ1,ψ2̸=ψ1,ψ3 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µ′ν\|µ1 ψ2 −−→µν; µ1 ψ1 −−→µν o = v(v −1)2 (mv)3 m2 1 v2 vs−2 vs −1 + m2(m −1) 1 v2 vs−2 −1 vs −1 vs−2 vs −2 . (61) 23 (iii-d)—µ1 = λ1 ̸= κ1; µ′ ̸= µ and ν′ ̸= ν (mult. v(v −1)3) Tc(µ, ν)(iii−d) = v(v −1)3 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; µ1 ψ2 −−→µν′; κ1 ψ3 −−→µ′ν o = v(v −1)3 (mv)3 X ψ1ψ2̸=ψ1,ψ3 P n µ1 ψ1 −−→µν o P n µ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µ′ν\|µ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o v(v −1)3 (mv)3 m2(m −1) 1 v2 vs−2 vs −1 vs−2 vs −2 . (62) (iv-a)—µ1 = κ1 ̸= λ1; µ′ = µ and ν′ = ν (mult. v(v −1)) Tc(µ, ν)(iv−a) = v(v −1) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; µ1 ψ3 −−→µν o = Tc(µ, ν)(iii−a) = Tc(µ, ν)(ii−a), (63) by symmetry for exchanging κ1 and λ1. (iv-b)—µ1 = κ1 ̸= λ1; µ′ = µ and ν′ ̸= ν (mult. v(v −1)2) Tc(µ, ν)(iv−b) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; µ1 ψ3 −−→µν o = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3=ψ1 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o + v(v −1)2 (mv)3 X ψ1,ψ2,ψ3̸=ψ1 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n µ1 ψ3 −−→µν\|λ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o = v(v −1)2 (mv)3 m2 1 v2 vs−2 vs −1 + m2(m −1) 1 v2 vs−2 vs −1 vs−2 −1 vs −2 = Ic(µ, ν)(iii−c). (64) (iv-c)—µ1 = κ1 ̸= λ1; µ′ ̸= µ and ν′ = ν (mult. v(v −1)2) Tc(µ, ν)(iv−c) = v(v −1)2 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; µ1 ψ3 −−→µ′ν o = v(v −1)2 (mv)3 X ψ1ψ2,ψ3̸=ψ1 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n µ1 ψ3 −−→µ′ν\|λ1 ψ2 −−→µν; µ1 ψ1 −−→µν o v(v −1)2 (mv)3 m(m −1)2 1 v2 vs−2 −1 vs −1 vs−2 vs −2 = Tc(µ, ν)(iii−b). (65) (iv-d)—µ1 = κ1 ̸= λ1; µ′ ̸= µ and ν′ ̸= ν (mult. v(v −1)3) Tc(µ, ν)(iv−d) = v(v −1)3 (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; µ1 ψ3 −−→µ′ν o = v(v −1)3 (mv)3 X ψ1ψ2,ψ3̸=ψ1 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν′\|µ1 ψ1 −−→µν o P n µ1 ψ3 −−→µ′ν\|λ1 ψ2 −−→µν′; µ1 ψ1 −−→µν o v(v −1)3 (mv)3 m2(m −1) 1 v2 vs−2 vs −1 vs−2 vs −2 = T (iii−d) c . (66) 24 (v-a)—µ1 ̸= λ1 ̸= κ1; µ′ = µ and ν′ = ν (mult. v(v −1)(v −2)) Tc(µ, ν)(v−a) = v(v −1)(v −2) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; κ1 ψ3 −−→µν o = v(v −1)(v −2) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν o P n λ1 ψ2 −−→µν\|µ1 ψ1 −−→µν o P n κ1 ψ3 −−→µν\|λ1 ψ2 −−→µν; µ1 ψ1 −−→µν o = v(v −1)(v −2) (mv)3 m3 1 v2 vs−2 −1 vs −1 vs−2 −2 vs −2 . (67) (v-b)—µ1 ̸= λ1 ̸= κ1; µ′ = µ and ν′ ̸= ν (mult. v(v −1)2(v −2)) Tc(µ, ν)(v−b) = v(v −1)2(v −2) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; κ1 ψ3 −−→µν o = v(v −1)2(v −2) (mv)3 m3 1 v2 vs−2 vs −1 vs−2 −1 vs −2 . (68) (v-c)—µ1 ̸= λ1 ̸= κ1; µ′ ̸= µ and ν′ = ν (mult. v(v −1)2(v −2)) Tc(µ, ν)(v−c) = v(v −1)2(v −2) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν; κ1 ψ3 −−→µ′ν o = v(v −1)2(v −2) (mv)3 m3 1 v2 vs−2 −1 vs −1 vs−2 vs −2 . (69) (v-d)—µ1 ̸= λ1 ̸= κ1; µ′ ̸= µ and ν′ ̸= ν (mult. v(v −1)3(v −2)) Tc(µ, ν)(v−d) = v(v −1)3(v −2) (mv)3 X ψ1,ψ2,ψ3 P n µ1 ψ1 −−→µν; λ1 ψ2 −−→µν′; κ1 ψ3 −−→µ′ν o = v(v −1)3(v −2) (mv)3 m3 1 v2 vs−2 vs −1 vs−2 vs −2 . (70) D.1.4 Variance of the correlations By adding together all the terms, C(1)(µ, ν) 2 = * 1 v X µ1 p(1) i1,j1(µ, ν\|µ1) !2+ + * 1 v X µ1 p(1) i1 (µ\|µ1) !2+ * 1 v X ν1 p(1) j1 (ν\|ν1) !2+ −2 1 v3 X µ1,λ1,κ1 D p(1) i1,j1(µ, ν\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|κ1) E = v3s (vs −1)2(vs −2) vs −mv vs (mv −1)(v −1)2 v6m2 v≫1 −−−→ 1 −m/vs−1 v3m . (71) D.1.5 Covariance of the correlations For all λ ̸= µ, v X µ=1 C(µ, ν) = v X ν=1 C(µ, ν) = 0. (72) Therefore, C(µ, ν) v X λ=1 C(λ, ν) = C(µ, ν)2 + X λ̸=µ C(µ, ν)C(λ, ν) = 0 ⇒ X λ̸=µ ⟨C(µ, ν)C(λ, ν)⟩= − C(µ, ν)2 . (73) 25 Analogously, X κ̸=ν ⟨C(µ, ν)C(µ, κ)⟩= − C(µ, ν)2 . (74) In addition, C(µ, ν) v X λ=1 C(λ, κ) = C(µ, ν)C(µ, κ) + X λ̸=µ C(µ, ν)C(λ, κ) = 0 ⇒ X λ̸=µ,κ̸=ν ⟨C(µ, ν)C(λ, κ)⟩= − X κ̸=ν ⟨C(µ, ν)C(µ, κ)⟩= C(µ, ν)2 . (75) D.2 Level-2 LCA When the parents of the i-th and j-th tokens are in the same level-1 patch , P {Xi = µ, Xj = ν}X = v X µ1=1 ν1=1 v X µ2=1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1)p(2) i2,j2(µ1, ν1\|µ2)p(3) i3 (µ2), (i2 ̸= j2), P {Xi = µ}X = v X µ1,µ2=1 p(1) i1 (µ\|µ1)p(2) i2 (µ1\|µ2)p(3) i3 (µ2), P {Xj = ν}X = v X ν1,ν2=1 p(1) j1 (ν\|ν1)p(2) j2 (ν1\|ν2)p(3) j3 (ν2), (j3 = i3). (76) Therefore, in the limit of large m, where the univariate probabilities of the hidden variables of any level converge to 1/v, C(2)(µ, ν) = P {Xi = µ, Xj = ν}X −P {Xi = µ}X P {Xj = ν}X = X µ1,ν1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1)× "X µ2 p(2) i2,j2(µ1, ν1\|µ2)p(3) i3 (µ2) −1 v2 X µ2,ν2 p(2) i2 (µ1\|µ2)p(3) i3 (µ2)p(2) j2 (ν1\|ν2)p(3) i3 (ν2) # = X µ1,ν1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1)C(1)(µ1, ν1). (77) Since the rules of different levels are independent, and C(1)(µ1, ν1) = 0, D C(2)(µ, ν) E = X µ1,ν1 D p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1) E D C(1)(µ1, ν1) E = 0. (78) The variance/2nd moment reads C(2)(µ, ν) 2 = X µ1,ν1 λ1,κ1 D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E D C(1)(µ1, ν1)C(1)(λ1, κ1) E = X µ1,ν1 D p(1) i1 (µ\|µ1)2p(1) j1 (ν\|ν1)2E D C(1)(µ1, ν1)2E + X κ1̸=ν1 D p(1) i1 (µ\|µ1)2p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E D C(1)(µ1, ν1)C(1)(µ1, κ1) E + X λ1̸=µ1 D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|ν1)2E D C(1)(µ1, ν1)C(1)(λ1, ν1) E + X λ1̸=µ1,κ1̸=ν1 D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E D C(1)(µ1, ν1)C(1)(λ1, κ1) E  . (79) 26 D.2.1 i1 ̸= j1 case. This is the easiest case because the production rule probabilities p(1) i relative to different positions i are independent and identically distributed (with respect to realisations of the RHM). Therefore, using Eq. 34, D p(1) i1 (µ\|µ1)2p(1) j1 (ν\|ν1)2E = D p(1) i1 (µ\|µ1)2E D p(1) j1 (ν\|ν1)2E = 1 v2 + σ2 p 2 , D p(1) i1 (µ\|µ1)2p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E = D p(1) i1 (µ\|µ1)2E D p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E = = 1 v2 + σ2 p 1 v2 −1 v2 v −1 vs −1 , D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|ν1)2E = D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1) E D p(1) j1 (ν\|ν1)2E = = 1 v2 −1 v2 v −1 vs −1 1 v2 + σ2 p , D p(1) i1 (µ\|µ1)p(1) i1 (µ\|λ1)p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E = 1 v2 −1 v2 v −1 vs −1 2 . (80) By bringing these factors outside of the λ1 and κ1 sums in the right-hand side of Eq. 79, C(2)(µ, ν) 2 = X µ1,ν1 1 v2 + σ2 p 2 D C(1)(µ1, ν1)2E + 1 v2 + σ2 p 1 v2 −1 v2 v −1 vs −1 X κ1̸=ν1 D C(1)(µ1, ν1)C(1)(µ1, κ1) E + 1 v2 + σ2 p 1 v2 −1 v2 v −1 vs −1 X λ1̸=µ1 D C(1)(µ1, ν1)C(1)(λ1, ν1) E + 1 v2 −1 v2 v −1 vs −1 2 X λ1̸=µ1,κ1̸=ν1 D C(1)(µ1, ν1)C(1)(λ1, κ1) E = X µ1,ν1 D C(1)(µ1, ν1)2E 1 v2 + σ2 p − 1 v2 −1 v2 v −1 vs −1 2 , (81) where, in the last line, we used Eq. 73, Eq. 74 and Eq. 75 to express the covariances of the C(1)(µ, ν)’s in terms of C(1)(µ, ν)2 . After simple algebraic steps, recalling the definition of σ2 p in Eq. 29, C(2)(µ, ν) 2 = v2 D C(1)(µ1, ν1)2E v −1 v vs vs −1 1 vm 2 v≫1 −−−→ C(1)(µ1, ν1)2 m2 . (82) D.2.2 i1 = j1 case. In this case, we need to evaluate some four-point functions. Since the spatial index of the p’s is the same, we will drop it to ease the notation. For the same reason, we will drop the level index too. First, it is convenient to use Eq. 73, Eq. 74 and Eq. 75 to rearrange the right-hand side of Eq. 79 as follows, C(2)(µ, ν) 2 = X µ1,ν1 D C(1)(µ1, ν1)2E p(µ\|µ1)2p(ν\|ν1)2 − 1 v −1 X κ1̸=ν1 p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) − 1 v −1 X λ1̸=µ1 p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)2 + 1 (v −1)2 X λ1̸=µ1,κ1̸=ν1 ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩  . (83) 27 The value of C(1)(µ1, ν1)2 is actually independent of µ1 and ν1, thus C(2)(µ, ν) 2 = D (C(1))2E X µ1,ν1 p(µ\|µ1)2p(ν\|ν1)2 − 1 v −1 X κ1̸=ν1 p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) − 1 v −1 X λ1̸=µ1 p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)2 + 1 (v −1)2 X λ1̸=µ1,κ1̸=ν1 ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩  . (84) The first term to deal with is (2-2), p(µ\|µ1)2p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; ν1 ψ4 −−→ν o ; (85) then (2-1-1) and (1-1-2), p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o ; (86) p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→ν; ν1 ψ4 −−→ν o ; (87) and, finally, (1-1-1-1), ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o . (88) We will further separate the case where µ = ν (i) from the case µ ̸= ν (ii). 2-2, i-a) (µ = ν, µ1 = ν1) p(µ\|µ1)2p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→µ; µ1 ψ4 −−→µ o = 1 m4 X ψ1,ψ2=ψ3=ψ4=ψ1 P n µ1 ψ1 −−→µ o + 4 m4 X ψ1,ψ2̸=ψ1,ψ3=ψ4=ψ1 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ o + 3 m4 X ψ1,ψ2=ψ1,ψ3̸=ψ1,ψ4=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ3 −−→µ o + 6 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4=ψ1 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4̸=(ψ1,ψ2,ψ3) P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→µ, µ1 ψ4 −−→µ o = 1 m4 m v + 7m(m −1) v vs−1 −1 vs −1 + 6m(m −1)(m −2) v vs−1 −1 vs −1 vs−1 −2 vs −2 +m(m −1)(m −2)(m −3) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (89) 28 2-2, i-b) (µ = ν; µ1 ̸= ν1) p(µ\|µ1)2p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ; ν1 ψ4 −−→µ o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4=ψ3 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→µ o + 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4̸=ψ3 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→µ; ν1 ψ4 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4̸=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ; ν1 ψ4 −−→µ o = 1 m4 m2 v vs−1 −1 vs −1 + 2m2(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 +m2(m −1)2 v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (90) 2-2, ii-a) (µ ̸= ν, µ1 = ν1) p(µ\|µ1)2p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→ν; µ1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2=ψ1,ψ3̸=ψ1,ψ4=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ3 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→ν o + 1 m4 X ψ1,ψ2=ψ1,ψ3̸=ψ1,ψ4̸=(ψ3,ψ1) P n µ1 ψ1 −−→µ; µ1 ψ3 −−→ν; µ1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4̸=(ψ1,ψ2,ψ3) P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→ν, µ1 ψ4 −−→ν o = 1 m4 m(m −1) v vs−1 vs −1 + 2m(m −1)(m −2) v vs−1 vs −1 vs−1 −1 vs −2 +m(m −1)(m −2)(m −3) v vs−1 vs −1 vs−1 −1 vs −2 vs−1 −1 vs −3 (91) 29 2-2, ii-b) (µ ̸= ν; µ1 ̸= ν1) p(µ\|µ1)2p(ν\|ν1)2 = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; ν1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4=ψ3 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→ν o + 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4̸=ψ3 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→ν; ν1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4̸=ψ3 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; ν1 ψ4 −−→ν o = 1 m4 m2 v vs−1 vs −1 + 2m2(m −1) v vs−1 vs −1 vs−1 −1 vs −2 +m2(m −1)2 v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (92) 2-1-1, i-a) (µ = ν, µ1 = ν1) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 X ψ1,ψ2=ψ1,ψ3=ψ1,ψ4 P n µ1 ψ1 −−→µ; κ1 ψ4 −−→µ o + 3 m4 X ψ1,ψ2=ψ1,ψ3̸=ψ1,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ3 −−→µ; κ1 ψ4 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 m2 v vs−1 −1 vs −1 + 3m2(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 + m2(m −1)(m −2) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 . (93) 2-1-1, i-b) (µ = ν, ν1 ̸= µ1, κ1 = µ1) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ; µ1 ψ4 −−→µ o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4=ψ1 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→µ o + 3 m4 X ψ1,ψ2=ψ1,ψ3,ψ4̸=ψ1 P n µ1 ψ1 −−→µ; µ1 ψ4 −−→µ; ν1 ψ3 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4̸=(ψ1,ψ2) P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ4 −−→µ; ν1 ψ3 −−→µ o = 1 m4 m2 v vs−1 −1 vs −1 + 3m2(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 + m2(m −1)(m −2) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 , (94) equal to the value of 2-1-1, i-a) as it should be. 30 2-1-1, i-c) (µ = ν, ν1 ̸= µ1, κ1 ̸= (µ1, ν1)) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→µ; κ1 ψ4 −−→µ o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 m3 v vs−1 −1 vs −1 vs−1 −2 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (95) 2-1-1, ii-a) (µ ̸= ν, µ1 = ν1) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2=ψ1,ψ3̸=ψ1,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ3 −−→ν; κ1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3̸=(ψ1,ψ2),ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; µ1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 m2(m −1) v vs−1 vs −1 vs−1 −1 vs −2 + m2(m −1)(m −2) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (96) 2-1-1, ii-b) (µ ̸= ν, ν1 ̸= µ1, κ1 = µ1) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; µ1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4̸=ψ1 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→ν; µ1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4̸=(ψ1,ψ2) P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; µ1 ψ4 −−→ν o = 1 m4 m2(m −1) v vs−1 vs −1 vs−1 −1 vs −2 + m2(m −1)(m −2) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (97) 2-1-1, ii-c) (µ ̸= ν, ν1 ̸= µ1, κ1 ̸= (µ1, ν1)) p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) = 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2=ψ1,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o + 1 m4 X ψ1,ψ2̸=ψ1,ψ3,ψ4 P n µ1 ψ1 −−→µ; µ1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 m3 v vs−1 vs −1 vs−1 −1 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (98) 31 1-1-2, overall contribution equal to that of 2-1-1. 1-1-1-1, i-a) (µ = ν, µ1 = ν1, λ1 = κ1, (v −1) of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; µ1 ψ3 −−→µ; λ1 ψ4 −−→µ o = p(µ\|µ1)2p(µ\|λ1)2 = 1 m4 m2 v vs−1 −1 vs −1 + 2m2(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 +m2(m −1)2 v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (99) from the value of 2-2, case i-b). 1-1-1-1, i-b) (µ = ν, µ1 = ν1, λ1 ̸= κ1, (v −1)(v −2) of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; µ1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 m3 v vs−1 −1 vs −1 vs−1 −2 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (100) from the value of 2-1-1, case i-c). 1-1-1-1, i-c) (µ = ν, µ1 ̸= ν1, λ1 = κ1, v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→µ; λ1 ψ4 −−→µ o = 1 m4 m3 v vs−1 −1 vs −1 vs−1 −2 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (101) from the value of 2-1-1, case i-c). 1-1-1-1, i-d) (µ = ν, µ1 ̸= ν1, λ1 = ν1, κ1 = µ1, 1 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ2 −−→µ; ν1 ψ3 −−→µ; µ1 ψ4 −−→µ o = 1 m4 m2 v vs−1 −1 vs −1 + 2m2(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 +m2(m −1)2 v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (102) from the value of 2-2, case i-b) 1-1-1-1, i-e) (µ = ν, µ1 ̸= ν1, λ1 = ν1, κ1 ̸= (µ1, ν1), v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ2 −−→µ; ν1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 m3 v vs−1 −1 vs −1 vs−1 −2 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (103) from the value of 2-1-1, case i-c). 32 1-1-1-1, i-f) (µ = ν, µ1 ̸= ν1, λ1 ̸= (µ1, ν1), κ1 = µ1, v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→µ; µ1 ψ4 −−→µ o = 1 m4 m3 v vs−1 −1 vs −1 vs−1 −2 vs −2 + m3(m −1) v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 (104) from the value of 2-1-1, case i-c). 1-1-1-1, i-g) (µ = ν, µ1 ̸= ν1, λ1 ̸= (µ1, ν1), κ1 = (µ1, ν1, λ1), (v−2)(v−3) of the (v−1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→µ; κ1 ψ4 −−→µ o = 1 m4 m4 v vs−1 −1 vs −1 vs−1 −2 vs −2 vs−1 −3 vs −3 . (105) 1-1-1-1, ii-a) (µ ̸= ν, µ1 = ν1, λ1 = κ1, (v −1) of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; µ1 ψ3 −−→ν; λ1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2,ψ3̸=ψ1,ψ4̸=ψ2 P n µ1 ψ1 −−→µ; µ1 ψ3 −−→ν; λ1 ψ2 −−→µ; λ1 ψ4 −−→ν o = 1 m4 m2(m −1)2 v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (106) 1-1-1-1, ii-b) (µ ̸= ν, µ1 = ν1, λ1 ̸= κ1, (v −1)(v −2) of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; µ1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 X ψ1,ψ2,ψ3̸=ψ1,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; µ1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 m3(m −1) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (107) 1-1-1-1, ii-c) (µ ̸= ν, µ1 ̸= ν1, λ1 = κ1, v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→ν; λ1 ψ4 −−→ν o = 1 m4 m3(m −1) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 , (108) from the value of 1-1-1-1, case ii-b). 1-1-1-1, ii-d) (µ ̸= ν, µ1 ̸= ν1, λ1 = ν1, κ1 = µ1, 1 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ2 −−→µ; ν1 ψ3 −−→ν; µ1 ψ4 −−→ν o = 1 m4 m2(m −1)2 v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 , (109) from the value of 1-1-1-1, case ii-a). 33 1-1-1-1, ii-e) (µ ̸= ν, µ1 ̸= ν1, λ1 = ν1, κ1 ̸= (µ1, ν1), v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; ν1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 m3(m −1) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 , (110) from the value of 1-1-1-1, case ii-b). 1-1-1-1, ii-f) (µ ̸= ν, µ1 ̸= ν1, λ1 ̸= (µ1, ν1), κ1 = µ1, v −2 of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→ν; µ1 ψ4 −−→ν o = 1 m4 m3(m −1) v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 , (111) from the value of 1-1-1-1, case ii-b). 1-1-1-1, ii-g) (µ ̸= ν, µ1 ̸= ν1, λ1 ̸= (µ1, ν1), κ1 = (µ1, ν1, λ1), (v −2)(v −3) of the (v −1)2 choices of λ1, κ1) ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩= 1 m4 X ψ1,ψ2,ψ3,ψ4 P n µ1 ψ1 −−→µ; λ1 ψ2 −−→µ; ν1 ψ3 −−→ν; κ1 ψ4 −−→ν o = 1 m4 m4 v vs−1 −1 vs −1 vs−1 vs −2 vs−1 −1 vs −3 . (112) Total. Consider the factor multiplying (C(1))2 in the right-hand side of Eq. 84, F(µ, ν) := X µ1,ν1   p(µ\|µ1)2p(ν\|ν1)2 − 1 v −1 X κ1̸=ν1 p(µ\|µ1)2p(ν\|ν1)p(ν\|κ1) − 1 v −1 X λ1̸=µ1 p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)2 + 1 (v −1)2 X λ1̸=µ1,κ1̸=ν1 ⟨p(µ\|µ1)p(µ\|λ1)p(ν\|ν1)p(ν\|κ1)⟩  . (113) By organising the terms in the sum according to the classification of the previous paragraphs, F(µ, ν) = v F(2-2) a (µ, ν) + (v −1)F(2-2) b (µ, ν) −2v(v −1) v −1 F(2-1-1) a + F(2-1-1) b + (v −2)F(2-1-1) c + v(v −1) (v −1)2 h F(1-1-1-1) a + (v −2)F(1-1-1-1) b (114) +(v −2)F(1-1-1-1) c + F(1-1-1-1) d + (v −2)F(1-1-1-1) e + (v −2)F(1-1-1-1) f + (v −2)(v −3)F(1-1-1-1) g i . For ν = µ, by summing all the case i) terms, we get F(µ, µ) = vs mv3(v + 1) −v2+s(1 −v + mv + mv2) + (v + 1)v2s(6 −6v + mv + v2 + mv2) (vm)3(vs −1)(vs −2)(vs −3) v≫1 −−−→(m + 1) m3 m≫1 −−−→ 1 m2 . (115) 34 Summing, instead, all the case ii) terms, we get, for ν ̸= µ, F(µ, ν) = vs mv3(v + 1) + v2+s(v −1 −8m + 7mv −3mv2) (v −1)(vm)3(vs −1)(vs −2)(vs −3) = v3s(6 −10v + mv + 5v2 + 3mv2 −v3 −3mv3 + mv4) (v −1)(vm)3(vs −1)(vs −2)(vs −3) v≫1 −−−→ 1 m2 . (116) To sum up, as in the i1 ̸= j1 case (Eq. 82), for large vocabulary size v ≫1 and large m (e.g. m = fvs−1, with f ∈(0, 1]), C(2)(µ, ν) 2 v,m≫1 −−−−→ C(1)(µ1, ν1)2 m2 . (117) D.3 Level-l LCA By replacing C(1) with C(ℓ−1) and C(2) with C(ℓ), the recurrence relation Eq. 79 extends to the case where the LCA of the i-th and j-th tokens is a level-ℓsymbol. Asymptotically in m and v, and independently of µ and ν, C(1)(µ, ν) 2 =(1 −m/vs−1) v3m , C(ℓ)(µ, ν) 2 = DC(ℓ−1)(µ, ν) 2E m2 ⇒ C(ℓ)(µ, ν) 2 = (1 −m/vs−1) v3m2ℓ−1 . (118) E Sampling noise in the empirical correlation function In this appendix, we prove that the sampling noise on empirical correlation functions of RHM data has a characteristic size (v2P)−1/2. Let us denote, to ease notation, P {Xd−t = µ, Xd = ν} with p(µ, ν), P {Xd−t = µ} with p(µ) and P {Xd−t = µ} with p(ν). When measuring probabilities from the frequency of observations over P i.i.d. samples, ˆp(µ, ν) = 1 P P X k=1 δ(Xk,d−t = µ, Xk,d = ν), (119) whereˆ. denotes the empirical estimate and the indicator variable δ is 1 with probability p(µ, ν) and 0 otherwise. With δ having finite mean and variance, by the central limit theorem, ˆp(µ, ν) P →∞ −−−−→p(µ, ν) + r p(µ, ν)(1 −p(µ, ν)) P ξ, (120) where ξ is a Gaussian random variable with zero mean and unitary variance. Analogously, ˆp(µ) P →∞ −−−−→p(µ) + r p(µ)(1 −p(µ)) P ζ1, ˆp(ν) P →∞ −−−−→p(ν) + r p(ν)(1 −p(ν)) P ζ2, (121) where ζ1 and ζ2 are also Gaussian random variables with zero mean and unitary variance, correlated with each other and with ξ. As a result, the empirical estimation of Ct(µ, ν) reads ˆCt(µ, ν) P →∞ −−−−→p(µ, ν)−p(µ)p(ν) + r p(µ, ν)(1 −p(µ, ν)) P ξ −p(µ) r p(ν)(1 −p(ν)) P ζ2 −p(ν) r p(µ)(1 −p(µ)) P ζ1. (122) 35 In the limit of large v and m, where p(µ, ν) converges to 1/v2 plus vanishingly small fluctuations and p(µ), p(ν) converge to 1/v plus vanishingly small fluctuations, the dominant noise contribution is the one of ξ, with standard deviation r p(µ, ν)(1 −p(µ, ν)) P v,m≫1 −−−−→ r 1 v2P . (123) The correlation function ˜C(t) is the standard deviation of Ct(µ, ν) over vocabulary entries. Hence, the sampling noise on Ct(µ, ν) results in an additive factor of (v2P)−1/2. F Correlations between mask and tuples of observable tokens In this section, we generalise the results of App. D and App. E to the correlations between the last token and a s-tuple of observable tokens. Let us then replace µ and i with the s-tuple of input features µ and the multi-index i. This change only affects the level-1 rules probability p(1) in Eq. 17 and Eq. 38. Therefore, we can write the tuple-token correlation with LCA at level ℓas follows, C(ℓ)(µ, ν) = X µ1,ν1 p(1) i1 (µ\|µ1)p(1) j1 (ν\|ν1)C(ℓ−1)(µ1, ν1) = 1 m X ν1 p(1) j1 (ν\|ν1)C(ℓ−1)(µ1(µ), ν1), (124) where the second line is obtained by recalling that, for each available s-tuple of input features µ, there is a unique level-1 variable µ1(µ) that can generate it, with probability 1/m. The mean of C(ℓ)(µ, ν) vanishes together with that of C(ℓ−1)(µ1(µ), ν1). Te variance reads C(ℓ)(µ, ν) 2 = 1 m2 X ν1,κ1 D p(1) j1 (ν\|ν1)p(1) j1 (ν\|κ1) E D C(ℓ−1)(µ1(µ), ν1)C(ℓ−1)(µ1(µ), κ1) E = 1 m2 X ν1=κ1 1 v2 + σ2 p C(ℓ−1)(µ1, ν1) 2 + 1 m2 1 v2 −1 v2 v −1 vs −1 X ν1,κ1̸=ν1 D C(ℓ−1)(µ1(µ), ν1)C(ℓ−1)(µ1(µ), κ1) E = v m2 1 v2 + σ2 p − 1 v2 −1 v2 v −1 vs −1 C(ℓ−1)(µ1, ν1) 2 , (125) where we used Eq. 74. Using Eq. 29 and Eq. 34, C(ℓ)(µ, ν) 2 = v m2 v −1 v vs vs −1 1 vm C(ℓ−1)(µ1, ν1) 2 v≫1 −−−→ DC(ℓ−1)(µ1, ν1) 2E m3 , (126) which equals the token-token correlation divided by a factor of m. Correspondingly, ˜Cℓis reduced by a factor of √m. Crucially, since the average joint tuple-token probability p(µ, ν) is 1/(v2m), the sampling noise size, obtained via the calculations of App. E, is also reduced by a factor of √m, leaving the condition of Eq. 9 unaltered. G Experiments on deep CNNs and scaling of the loss steps In this section, we present empirical learning curves of Deep CNNs trained for last-token prediction (details in subsection A.1). In particular, we discuss discrepancies between these curves and those of Transformers (Fig. 2) in subsection G.1, verify the scaling with m of the first two steps of Eq. 12 in subsection G.2, then discuss the role of the context window size t in subsection G.3. 36 102 103 104 105 106 training set size P 100 test cross-entropy t=1 t=3 t=7 t=15 102 103 104 105 106 training set size P 100 test cross-entropy MLA CNN Figure 6: Left: Learning curves of deep CNNs trained on RHM data with L = 4, s = 2, v = 64 and m = 8 for different sizes t of the context window. The network’s depth is fixed to log sL/ log (t + 1) and the blacked dashed line represents predictions from Eq. 12 and Eq. 11. The finite context window causes saturation of the loss as predicted by our analysis. However, the third step occurs with less training data than P3. Right: This discrepancy is highlighted by the comparison of Transformer and deep CNN learning curves, here for L = 4, s = 2, v = 64 and m = 8. 101 102 103 104 105 106 training set size P 0 1 2 test cross-entropy L m=8 m=11 m=16 m=23 100 101 102 103 104 P/P1 0.00 0.25 0.50 0.75 1.00 L/ ¯L1 Figure 7: Learning curves of depth-3 CNNs trained on RHM data with L = 3, s = 3, v = 11 and m as in the key. Dashed curves highlight our prediction for the first step. In the right panel, the first step is made to collapse by rescaling P with P1 and L with L0 = log v. The collapse confirms our prediction on the behaviour of P1 with m. G.1 Differences between Transformers and deep CNNs The learning curves of deep CNNs are qualitatively similar to those of transformers, but also present apparent quantitative differences, as shown in Fig.6. Specifically, a noticeable difference is the sample complexity of the third step P3. This difference is possibly due to favourable implicit biases of CNNs, such as weight sharing. Indeed after learning the penultimate level-1 features in the second step, weight sharing would facilitate learning the other level-1 features along the entire data. As a result, the model can directly access the correlations between the last token and tuples of level-1 symbols. According to the discussion of App. F, these correlations are stronger than those with tuples of level-0 symbols by a factor of √m. Correspondingly, the sample complexity of the third step P3 is reduced by a factor m with respect to Eq. 12. In general, we can assume that, in the presence of weight sharing, after the ℓ-th step all level-(ℓ−1) features have been learnt, so that the (ℓ+ 1)-th step requires resolving correlations between the last token and tuples of level-(ℓ−1) features. The corresponding sample complexity scales like mℓ+1 instead of m2ℓ−1. However, the steps with ℓ≥3 occur for large values of the training set size, and we cannot investigate this issue systematically with our current numerical experiments. G.2 Scaling with the number of production rules m Fig. 7 and Fig. 8 show a scaling analysis of the behaviour of P1 and P2 from Eq. 12 in Deep CNNs. The collapse achieved when rescaling the number of data P by Pℓand the test loss by the value before the jump Lℓ−1 confirms this prediction. 37 103 105 107 training set size P 0 1 2 test cross-entropy L m=8 m=11 m=16 m=23 10−3 10−2 10−1 100 101 102 P/P2 0.00 0.25 0.50 0.75 1.00 L/ ¯L2 Figure 8: Same as Fig. 7 but focused on the second step, highlighted on the left by dashed curves. For the second step, collapse is achieved by rescaling P with P2 = vm3 and L with L1 from Eq. 11. 102 103 104 training set size P 100 test cross-entropy t=1 t=3 t=7 t=15 102 103 104 P/(t + 1) 2 × 100 3 × 100 4 × 100 Figure 9: Zoom of the learning curves in Fig. 6, left, on the first step. The zoom highlights the dependence of the sample complexity on the context size t. The collapse of the curves on the right panel, achieved after dividing P by (t + 1), reveals that P1 ∝(t + 1). This dependence is analogous to the sample complexity of regression of a target function depending on a low-dimensional linear projection of a large-dimensional input [55]. G.3 Scaling with the size of the context window t Similarly, Fig. 9 shows a scaling analysis (for CNNs) of the behaviour of P1 with the number of s-tuples in the input, proportional to (t + 1) with t the size of the context window. The figure highlights a linear behaviour P1 ∝(t + 1) that our analysis does not capture. Nevertheless, this behaviour is expected from the theory of regression with one-hidden-layer neural networks [55]: when the target function depends on a small number of variables among d, the sample complexity is generically proportional to d. Proving this result by considering a single or a few steps of gradient descent, as often done in this literature, is an interesting work for the future. 38 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The analytical study of correlations in our generative model is presented in section 3. In section 4 we build our prediction of the learning curve based on reconstructing the grammar’s hidden variables and compare it with the empirical learning curves of transformers trained on our model dataset. Our general conjecture is presented in section 5, together with experiments on real text data. 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4,435	ComBack: A Versatile Dataset for Enhancing Compiler Backend Development Efficiency Ming Zhong1,2, Fang Lyu1∗, Lulin Wang1, Hongna Geng1,2, Lei Qiu1,2 Huimin Cui1,2∗, Xiaobing Feng1,2 1 SKLP, Institute of Computing Technology, CAS, Beijing, China 2 UCAS, Beijing, China {zhongming21s, flv, wanglulin, genghongna, qiulei21b, cuihm, fxb}@ict.ac.cn Abstract Compiler backends are tasked with generating executable machine code for processors. With the proliferation of diverse processors, it is imperative for programmers to tailor specific compiler backends to accommodate each one. Meanwhile, compiler backend development is a laborious and time-consuming task, lacking effective automation methods. Although language models have demonstrated strong abilities in code related tasks, the lack of appropriate datasets for compiler backend development limits the application of language models in this field. In this paper, we introduce ComBack, the first public dataset designed for improving compiler backend development capabilities of language models. ComBack includes 178 backends for mainstream compilers and three tasks including statement-level completion, next-statement suggestion and code generation, representing common development scenarios. We conducted experiments by fine-tuning six pre-trained language models with ComBack, demonstrating its effectiveness in enhancing model accuracy across the three tasks. We further evaluated the top-performing model (CodeT5+) across the three tasks for new targets, comparing its accuracy with conventional methods (Fork-Flow), ChatGPT3.5-Turbo, and Code-LLaMA-34B-Instruct. Remarkably, fine-tuned CodeT5+ with only 220M parameters on ComBack outperformed Fork-Flow methods significantly and surpassed ChatGPT and Code-LLaMA, suggesting potential efficiency improvements in compiler development. ComBack is avaliable at https://huggingface.co/datasets/docz1105/ComBack. 1 Introduction A compiler is a fundamental computer software which translates source code from high-level programming language into low-level machine code, e.g., assembly code, for target machines (referred to as "target" for simplicity). As shown in Fig. 1, mainstream compilers like GCC [18] and LLVM [30] are divided into three parts: frontend, middle-end and backend. Specifically, the frontend related to programming languages, while the middle-end comprises of target-independent optimizations, and backend converts intermediate representation produced by the middle-end into machine code for various targets. The flourishing development of new processors nowadays demands continuous development for backends. Compiler backend development necessitates a profound understanding of the target characteristics and the compiler infrastructure [19]. Thus, it entails prolonged development cycles and substantial manual ∗Corresponding Author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. FrontEnd Middle-End (Gimple,LLVM IR ...) BackEnd Source Code (C/C++,Fortran ...) Input CPU Machine Code MicroProcessor GPU Output ... Code Optimization Instruction Selection Instruction Scheduling Code Emission Register Allocation Instruction Selection Instruction Scheduling Code Emission Register Allocation Code Generator Code Optimizer Target-Dependent Figure 1: Compiler backend structure. 0 50 100 150 200 250 AArch64 AMDGPU ARC ARM AVR BPF CSKY DirectX Hexagon Lanai LoongArch M68k Mips MSP430 NVPTX PowerPC RISCV X86 Sparc SPIRV SystemZ VE WebAssembly XCore Xtensa Avg 17.7% 82.3% 66.3% 33.7% BPF X86 57.6% 43.4% Average 25 Targets in LLVM 17.0 Number Y X 68.9 28.8 C/C++ Files Code(KLoc) Manual Effort Inherited Customized Proportion of Functions 219 118.5 124 110.3 Figure 2: Heavy manual efforts in backends development for 25 targets in LLVM 17.0.1. efforts. Data depicted in Fig. 2 underscores the magnitude of manual efforts and the distribution of functions across the development of compiler backends for 25 targets in LLVM 17.0.1 (latest released version). For instance, AMDGPU comprises 219 C++/C files, totaling 118.5 KLoC (Line of Code), while X86 comprises 124 files with 110.3 KLoC. On average, a LLVM backend in LLVM 17.0.1 consists of 68.9 files, encompassing 28.8 KLoC, indicating considerable manual efforts. The emergence of AI has spurred considerable interest in leveraging its techniques for code-related tasks, such as code completion and generation [34, 14, 22, 50, 49, 21, 23, 45, 56]. Models like Code-LLaMA [45] have shown promise in significantly reducing the burden on programmers by being pre-trained on extensive code datasets. However, their efficacy in tasks concerning compiler backends, as evidenced by experimental findings in Sec. 4.3, remains limited, indicating ample room for enhancement. Moreover, the compiler community currently lacks a publicly available large-scale backend dataset, which could enhance the efficiency of backend development across diverse targets. In this paper, we present ComBack, which is the first public dataset leading to a promising future for the application of language models for backend development. ComBack comprises 178 backends for mainstream compilers (77 from GCC and 101 from LLVM), sourced from open-source GitHub repositories. We also design three tasks to evaluate the performance of language models based on ComBack for three prevalent scenarios encountered in backend development, including 1) StatementLevel Completion, 2) Next-Statement Suggestion, 3) Code Generation. In the experiment, we selected 6 representative open-source language models [50, 23, 14, 49, 22, 7] and fine-tuned them with ComBack. The results indicate that ComBack effectively improves the accuracy of 6 language models across 3 tasks. Furthermore, we conducted research on executing three code tasks for three new targets within GCC and LLVM. Additionally, experimental findings show that fine-tuning a model with just 220M parameters based on ComBack significantly boosts programmers’ efficiency compared to Fork-Flow, ChatGPT and Code-LLaMA, demonstrating the value of ComBack in enhancing the language model’s performance with compiler backend development. 2 Background: Conventional Backend Development Process To develop a compiler backend for a new target, programmers are required to provide specific implementations for a series of compiler infrastructure provided function interfaces based on targetdependent information and characteristics, including instruction sets, registers, byte order, and similar attributes. Specifically, functions within a backend can be divided into two categories: Inherited Functions. This category includes compiler infrastructure function interfaces that carry out specific tasks in the backend process. Programmers must inherit these interfaces and provide implementations tailored to each target. For instance, the "getReloctype" function in LLVM maps relocation variants and immediate values in instruction sets. Differences in this function across targets mainly involve target-specific relocation variants and immediate values. It’s important to note that programmers need not to implement all provided interfaces but only a subset relevant to the target, resulting in variations in the implemented inherited functions across different targets. 2 ... ... Programmers A Similar Backend Language Models ComBack Existing Backends Info of New Target New Backend Fork Modify Fine-Tune Preprocess Debug Assisted with ComBack Conventional Backend Development Process (Fork-Flow) Assisted Assisted Assisted Figure 3: Conventional backend development process and assisted process with ComBack. Customized Functions. This category includes specialized functions designed specifically for certain targets. For example, the "isImm24bit" function in ARM target checks if the encoding length of an immediate value is 24 bits, unique to ARM and not found in other targets. Fig. 2 shows the proportion of two types of functions in LLVM 17.0.1. In BPF, 82.3% of functions, and in X86, 66.3%, are inherited from LLVM interfaces. On average across all 25 targets, inherited functions account for 57.6%. This prevalence highlights the significant presence of inherited functions across various targets, indicating a notable commonality among them. Fig. 3 depicts the conventional backend development process (Fork-Flow) [43, 32], where programmers must acquire knowledge of the unique characteristics of a new target, such as instruction formats and target-specific flags. They then fork an existing backend that shares similarities (e.g., both being CPU or GPU) and make modifications based on this knowledge to create a tailored backend for the new target. Despite its steep learning curve, similarities among backends of the same type result in redundant development efforts, causing inefficiencies in manual work. To mitigate this challenge, we propose ComBack, which can be utilized to fine-tune models and facilitate fine-tuned models to assist programmers with backend development, as shown in Fig. 3, thereby reducing redundant efforts and enhancing efficiency. 3 ComBack: A Dataset for Compiler Backend Development 3.1 Overview of ComBack To the best of our knowledge, ComBack is the first public dataset for compiler backend development. Notably, it comprises three features as outlined below: (1) Large-Scale. ComBack is sourced from 317 GitHub repositories and the official websites of GCC [20] and LLVM [33], covering versions 3.0 to 13.0 for GCC and 2.0.1 to 17.0.1 for LLVM. It includes 43,299 functions and 883.7 KLoC (Kilo lines of code) for GCC, and 138,940 functions and 4,847.5 KLoC for LLVM, shown in Table 1. Its large scale enhances model performance on common backends and facilitates generalization to less common ones. (2) Multi-Target. Mainstream compiler infrastructure now supports multiple backends for diverse targets, requiring ComBack to be inclusive of such diversity. As indicated in Table 1, there are a total of 77 targets for GCC backends and 101 for LLVM backends in ComBack. These targets cover various types including CPUs, MPUs (Micro-Processors), GPUs, etc. Among them, CPUs and MPUs are more abundant due to their wide applicability across various scenarios. In contrast, other types of processors such as GPUs and DSPs are fewer as they are usually designed for specific tasks, such as GPUs for deep learning workloads and parallel data computation. Leveraging commonalities among these targets, as discussed in Sec. 1, enables models to learn cross-target patterns, facilitating advanced research among various backends. For detailed target information, refer to Appendix A. 3 Table 1: Data statistics about targets and code in ComBack. (a) GCC Type Target Function KLoC CPU 30 35,147 647.2 MPU 33 6,010 183.9 GPU 2 457 11.2 VLIW 5 959 25.4 DSP 3 399 9.6 Virtual 4 327 6.5 Sum 77 43,299 883.7 (b) LLVM Type Target Function KLoC CPU 43 84,914 3,450.4 MPU 30 11,311 173.0 GPU 5 22,591 768.3 VLIW 4 2,048 24.3 DSP 7 9,646 263.2 Virtual 12 8,430 168.3 Sum 101 138,940 4,847.5 (3) Versatility. To tackle real-world challenges in compiler backend development, like code completion, ComBack focuses on enhancing model versatility. It covers three tasks: 1) Statement-Level Completion; 2) Next-Statement Suggestion; 3) Code Generation, aiding programmers in backend modification and customization. By analyzing diverse target backends, models can better assist with code completion and generation for both existing and new backends. This adaptable approach reduces programming workload, enabling ComBack to handle various scenarios. 3.2 Data Collection and Pre-processing The collection and pre-processing of data in ComBack adhere to the following steps: 1. Code Collection. We crawled GitHub using "GCC/LLVM+Backend" as keywords, filtering out incomplete repositories. This yielded 21 GCC repositories and 296 LLVM repositories. We also collected source code versions 3.0 to 13.0 from the official GCC website [20], and versions 2.0.1 to 17.0.1 from the official LLVM website [33]. The backend code from multiple repositories was aggregated and reorganized by targets to create the raw code data. 2. Function Description Collection. We collected function descriptions from two sources. Firstly, we extracted descriptions directly from comments within the source code associated with each function. Additionally, for LLVM, we obtained function descriptions from its official Doxygen website [31] using crawling techniques to analyze them further. 3. Code Extraction. We started by removing duplicate files and comments from the source code for each target to minimize their influence on fine-tuning. Then, we used the tree-sitter tool [47] to extract functions from the code after comment removal. Each line ending with ";", ":", "{", or "}" was partitioned into a single statement, allowing us to obtain all functions within the backend source code along with their internal statements. 4. Target-Specific Value Extraction. Backend code, unlike basic C/C++ programs, prominently includes target-specific values comprising information and characteristics of the instruction set architecture (ISA) of the corresponding target. Fig. 4(a)-(c) illustrates three typical target-specific values: instruction encodings (Fig. 4(c)), size (Fig. 4(c)), immediate values (Fig. 4(b)), and target-specific flags (Fig. 4(a)). Observations indicate that target-specific values can be categorized into 3 types: (1) numerical values (Fig. 4(b) and (c)); (2) strings in double quotation marks (Fig. 4(c)); (3) enumeration variable values with the target’s name prefix (Fig. 4(a)). However, some enumeration values may start with the target name abbreviation, like "PPC" for "PowerPC". We use a script to automatically filter out target-specific values based on these patterns, including enumeration values starting with abbreviations, like "PPC". Following the approach used in CodeXGlue [34], we replace target-specific values in the code with intermediate representations: "<ISA_LIT>" for enumeration variables, "<NUM_LIT>" for numerical values, and "<STR_LIT>" for strings. Moreover, we store each target-specific value corresponding to these intermediate representations. All target abbreviations are listed in Appendix B. 4 case RISCVII::MO_LO: Kind = RISCVMCExpr::VK_RISCV_LO; case CSKYII::MO_GOT32: Kind = CSKYMCExpr::VK_CSKY_GOT; (a) Target-Specific Flag and VariantKind ... return isImm(16, 31); ... return isImm(−8, 7); (b) Immediate Value ... OS.write("\0\0\x40\x03", 4); ... OS.write("\x20", 1); (c) Instruction Encoding and Size Figure 4: Examples of target-specific values in GCC and LLVM. Inputs: ... adjustReg(DL, SPReg, FPReg, −StackSize+RVFI−>getVarArgsSaveSize(), _______________ Ground Truth: MachineInstr::FrameDestroy); (a) Statement-Level Completion Inputs: ... maxCallFrameSize = (maxCallFrameSize + AlignMask) & ~AlignMask; Ground Truth: MFI −> setMaxCallFrameSize(maxCallFrameSize); (b) Next-Statement Suggestion Inputs: getPointerRegClass: Returns a TargetRegisterClass used for pointer values. Target−Specific Value: Sparc, SP::I64RegsRegClass, SP::IntRegsRegClass. Ground Truth: TargetRegisterClass *SparcRegisterInfo::getPointerRegClass(MachineFunction &MF ,unsigned Kind) { return Subtarget.is64Bit() ? &SP::I64RegsRegClass : &SP::IntRegsRegClass; } (c) Code Generation Figure 5: Examples of three tasks in ComBack. 3.3 Tasks in ComBack For two common scenarios in compiler backend development, we’ve outlined three tasks, depicted in Fig. 5. For on-the-fly programming, we’ve devised Statement-Level Completion (Fig. 5(a)) and Next-Statement Suggestion (Fig. 5(b)) [37], aiming to speed up the programming process. For situations where programmers provide function descriptions in natural language, we’ve introduced Code Generation (Fig. 5(c)), facilitating direct code generation for a given function. Data processing steps for each task are detailed in subsequent subsections. Language models fine-tuned with ComBack aid programmers in backend development by completing current statements (Statement-Level Completion), predicting next statements (Next-Statement Suggestion) based on the contextual information. Additionally, it can generate functions based on provided natural language descriptions and target-specific values (Code Generation), reducing repetitive tasks and enhancing efficiency. 3.3.1 Statement-Level Completion Following the data extraction method used in the code completion dataset of CodexGlue [34], we initially aimed to extract five consecutive statement sequences randomly from each function in every backend. We retained samples where the proportion of tokens in the sequence relative to the entire function exceeded 30%, aiming to capture more contextual semantics. Assuming each sample contains n statements, we used the first n −1 statements along with 50%-90% of tokens from the nth statement as input. The remaining 10%-50% of tokens from the nth statement served as ground truth, with this ratio chosen randomly. We treated tokens like ";", ":", "{", "}" in C/C++ as statement terminators, as described in Sec. 3.2. We maintained the intermediate representations from Sec. 3.2 in the task’s input and ground truth because target-specific values are sourced from ISA of the target, 5 making accurate prediction based solely on the code context challenging. Finally, we filtered out data with input lengths exceeding 512 tokens or output lengths exceeding 128 tokens, resulting in a total of 161,124 samples for Statement-Level Completion. 3.3.2 Next-Statement Suggestion Data processing for Next-Statement Suggestion mirrors that of Statement-Level Completion. We randomly extract five consecutive statement sequences from each function in every backend, retaining samples with over 30% of the function’s tokens. The main distinction is that, for a Next-Statement Suggestion sample with n statements, the preceding n −1 statements serve as input, while the nth statement serves as the ground truth, as shown in Fig. 5(b). We also retained the intermediate representation in code and filtered out samples with input lengths exceeding 512 tokens or ground truth lengths surpassing 128 tokens. Finally, we obtained the dataset comprising 216,315 samples for Next-Statement Suggestion. 3.3.3 Code Generation For Code Generation, we only kept functions with natural language descriptions (68.08% functions in LLVM and 48.12% functions in GCC), discarding those lacking such descriptions. Each function’s description, along with its internal target-specific values, was used as input (typically requiring extraction from ISA manuals), while the entire function (replacing each intermediate representation with corresponding target-specific value) served as the ground truth, as seen in Fig. 5(c). During filtering, samples with input exceeding 256 tokens or ground truth surpassing 512 tokens were removed, retaining 45,296 samples. 4 Experiment This section addresses the following research questions: • RQ.1: Can ComBack effectively enhance backend development capabilities of various language models? (Sec. 4.2) • RQ.2: Can ComBack facilitate fine-tuning a model to enhance backend development efficiency for new targets of existing types and new types? (Sec. 4.3) • RQ.3: Can ComBack support iterative expansion to improve backend development efficiency for customized targets? (Sec. 4.4) 4.1 Experimental Setup Fundamental Models. We selected six open-source language models pre-trained or fine-tuned on C or C++ language: 1) CodeBert (Fine-Tuned with C) [14, 16], 2) GraphCodeBert (Fine-Tuned with C) [23, 15], 3) UniXcoder-base-nine [22], 4) CodeT5-base [50], 5) CodeT5+-220M [49] and 6) NatGen [7]. We chose them for two reasons: 1) these models are representative open-source programming language models, suitable for various tasks in ComBack; 2) their relatively small model size helps reduce computational resources needed for training and deployment. All fine-tuned models and code are available at https://huggingface.co/docz1105/ComBack_Models. Baselines. For experiment in Sec. 4.3, we include Fork-Flow method as the baseline of conventional development efficiency. Additionally, we choose ChatGPT-3.5-Turbo and Code-LLaMA-34B-Instruct as baselines for mainstream large language models (LLMs). ChatGPT is the most widely used LLM globally, while Code-LLaMA, an open-source LLM designed specifically for code-related tasks, achieves state-of-the-art performance on many code related benchmarks. Evaluation Metrics. To evaluate the inference capability of models fine-tuned with ComBack, we use exact match accuracy (EM) and Levenshtein Edit Distance Similarity (ED) [22, 34] for StatementLevel Completion and Next-Statement Suggestion. For Code Generation, we use Levenshtein Edit Distance Similarity and BLEU-4 [38] as evaluation metrics.Exact Match was used for the two code completion tasks because it directly measures the correctness of the generated code, meeting developers’ needs in real-time programming. For Code Generation, we chose BLEU-4 to assess structural similarity between the generated code and the ground truth, the higher the BLEU-4 score, 6 Table 2: Comparison of accuracy across three tasks of six models fine-tuned by ComBack. Model Stmt. Comp. Next. Sugg. Code. Gen. Stmt. Comp. Next. Sugg. Code. Gen. EM (%) ED EM (%) ED BLEU-4 ED EM (%) ED EM (%) ED BLEU-4 ED Without Fine-Tuning Fine-Tuned CodeBert 0.00 0.97 0.00 1.31 0.00 0.44 53.84 77.44 52.67 70.82 23.54 54.63 GraphCodeBert 0.00 0.35 0.00 0.54 0.00 2.41 43.00 71.89 47.10 61.31 20.73 48.83 UniXcoder 0.07 27.56 15.93 29.11 0.00 31.81 67.84 85.06 58.51 75.31 56.24 73.45 CodeT5 0.65 21.45 7.23 23.50 0.00 13.57 66.47 84.34 58.52 76.03 70.87 80.45 NatGen 0.00 13.52 0.02 15.95 0.01 28.76 67.47 84.83 60.30 76.84 71.73 81.39 CodeT5+ 0.02 7.24 0.12 9.87 0.00 12.33 66.93 84.45 59.57 76.41 75.29 82.92 the greater the similarity. We also used edit distance for all tasks to measure the modifications needed to align the generated code with the ground truth, where a higher score indicates fewer required edits and closer alignment to the ground truth. Training Settings. All models are trained and evaluated on a server with a 64-core Intel Xeon Gold CPU and 8 NVIDIA Tesla V100 GPUs, each with 16GB of memory. We set the fine-tuning objective as: sequence-to-sequence prediction for three tasks. To ensure fairness, all hyperparameters are identical for the six models, detailed in Appendix C. 4.2 Accuracy Improvement across Various Models To evaluate accuracy improvement of different models across three tasks, we randomly split the backend data from all targets into train/validation/test sets in an 80%:10%:10% ratio, with details on the quantity of data and tokens in each set provided in Appendix D. Subsequently, we fine-tuned and tested six models with the dataset. Table 2 shows the accuracy improvement of six models across three tasks after fine-tuning with ComBack. The models exhibited improvements of 41.64 - 77.21 of ED across three tasks, 42.58%-67.77% in absolute terms of EM for Statement-Level Completion and Next-Statement Suggestion, and 20.73-75.29 of BLEU-4 for Code Generation. Answer to RQ.1: ComBack can effectively improve backend development capabilities of various language models. 4.3 Efficiency Enhancement for New Targets In Sec. 4.3.1 and Sec. 4.3.2, we simulate code completion and generation scenarios for new targets of existing types and new types. We select CodeT5+ for experiments in following sections, as it achieves the highest accuracy on average across three tasks (Sec. 4.2). 4.3.1 Targets of Existing Types We simulate code completion and generation scenarios for new targets of existing types. Therefore, we select RISC-V (CPU), ARC (MPU), and NVPTX (GPU) in GCC and LLVM as test sets. Other CPU, MPU, and GPU targets are split into train and validation sets at an 85%:15% ratio. RI5CY in LLVM is excluded since it’s a customized target based on RISC-V and shares most code with it. Further dataset details are provided in Appendix D. Next, we fine-tuned CodeT5+ with the dataset including CPU, MPU and GPU. We further compared the accuracy of fine-tuned CodeT5+, mainstream LLMs, and conventional backend development methods (Fork-Flow) for backend development of three targets. Mainstream LLMs. We evaluated the performance of ChatGPT-3.5-Turbo and Code-LLaMA-34BInstruct across three tasks for RISC-V, ARC, and NVPTX, as shown in Table 3. Inputs for both LLMs closely matched those in ComBack, with the addition of a unified prompt, detailed in Appendix F. CodeT5+ consistently outperforms two LLMs across three tasks. Specifically, in Statement-Level Completion, CodeT5+ surpasses 37.10%-40.82% for EM compared with ChatGPT and 49.72%54.57% compared with Code-LLaMA in absolute terms on three targets in GCC and LLVM. The significant improvement in accuracy indicates that fine-tuning small LLMs with ComBack exceeded large LLMs significantly. Therefore, ComBack holds significant importance in enhancing the performance of language models in backend development scenarios. 7 Table 3: Accuracy of code generated by ChatGPT, Code-LLaMA and CodeT5+ fine-tuned by ComBack for targets of existing types. Model Stmt. Comp. Next. Sugg. Code. Gen. RISC-V ARC NVPTX RISC-V ARC NVPTX RISC-V ARC NVPTX EM (%) ED EM (%) ED EM (%) ED EM (%) ED EM (%) ED EM (%) ED BLEU-4 ED BLEU-4 ED BLEU-4 ED GCC ChatGPT 10.34 38.41 15.35 42.94 12.01 41.47 6.44 12.90 9.75 20.79 7.97 17.79 1.37 24.12 1.67 28.26 1.57 26.97 Code-LLaMA 0.41 19.07 0.85 16.77 0.56 18.22 1.58 13.54 2.66 17.95 2.47 16.59 1.67 27.89 1.71 30.49 1.57 27.65 CodeT5+ 51.16 75.32 52.45 74.57 50.56 75.52 49.11 67.84 38.26 59.21 38.33 56.31 32.56 58.67 19.94 50.27 25.47 52.60 LLVM ChatGPT 12.08 41.39 16.77 42.02 14.73 43.72 9.80 21.86 10.81 20.66 11.39 22.82 1.23 25.12 1.30 27.19 1.43 25.45 Code-LLaMA 0.45 17.61 0.61 17.21 0.99 17.23 1.75 15.04 0.42 11.27 2.42 16.25 1.43 27.24 1.61 32.12 1.59 28.08 CodeT5+ 62.68 82.02 71.34 85.98 64.45 81.53 48.71 68.95 58.68 74.57 47.81 65.51 50.34 72.98 55.38 74.41 44.33 66.36 Targets in LLVM Edit. Dis. Y X 74.41 66.36 72.98 39.04 60.62 46.47 28.73 33.43 22.18 80 0 RISC-V ARC NVPTX Targets in LLVM BLEU-4 Y X 55.38 44.33 50.34 18.81 41.81 27.32 15.06 19.98 12.45 80 0 RISC-V ARC NVPTX Targets in GCC Edit. Dis. Y X 52.60 50.27 58.67 3.81 8.85 34.80 3.81 3.73 4.79 80 0 RISC-V ARC NVPTX FF-Avg FF-Max CodeT5+ Targets in GCC BLEU-4 Y X 25.47 19.94 32.56 4.70 4.94 28.77 4.70 1.77 3.48 80 0 RISC-V ARC NVPTX (a) GCC (b) LLVM FF-Max CodeT5+ FF-Avg FF-Max CodeT5+ FF-Max CodeT5+ FF-Avg FF-Max FF-Max CodeT5+ FF-Avg FF-Max FF-Max CodeT5+ Figure 6: Comparison of fine-tuned CodeT5+ and Fork-Flow for Code Generation, where "FF" is the abbreviation of Fork-Flow. Fork-Flow. Due to the similarity between the Fork-Flow process, which involves modifying complete functions, and scenarios in Code Generation where developers modify functions automatically generated by the model, we only compare Fork-Flow with fine-tuned CodeT5+ on Code Generation. To simulate the process of Fork-Flow, we used scripts to calculate the ED and BLEU-4 between functions with identical names on new targets (RISC-V, ARC, NVPTX) and their corresponding implementations on other targets. We aggregate their average and maximum values across these targets (excluding RISC-V, ARC, NVPTX) and compare them with values of functions generated by fine-tuned CodeT5+, as depicted in Fig. 6. It is evident that the accuracy of fine-tuned CodeT5+ exceeds both the average and maximum values of Fork-Flow, demonstrating that the CodeT5+ fine-tuned by ComBack can achieve higher efficiency compared to conventional development method. Details of Fork-Flow can be viewed in Appendix E. 4.3.2 Targets of New Types We further explore whether ComBack can facilitate code completion and generation for targets of new types. We fine-tune CodeT5+ with CPU data only, excluding MPU and GPU data from train and validation sets in Sec. 4.3.1. Next, we exclude CPU data and only retain MPU and GPU data in the test set, detailed in Appendix D. After fine-tuning CodeT5+ with the dataset only containing CPU, we explore whether it can generate functions for new types of targets (MPU and GPU) in the test dataset. Results in Table 4 indicate that CodeT5+ fine-tuned on existing types of targets (CPU) can indeed facilitate code completion and generation for new types of targets (MPU and GPU), as backends of different types of targets under the same compiler infrastructure (GCC or LLVM) adhering to unified programming standards (such as same function interfaces and classes). However, there tends to be a decrease in accuracy on most targets, as depicted in Table 4. Further analysis in Appendix H reveals that there are differences in functions required in the backend of different types of targets. Therefore, the fine-tuned model struggles to effectively complete and generate code corresponding to some functions for new types of targets. Answer to RQ.2: The model fine-tuned by ComBack can enhance backend development efficiency for new targets of both existing and new types. 4.4 Iterative Expansion Ability In this section, we explore ComBack’s iterative expansion ability. As application scenarios diversify, the field of processor design witnesses a proliferation of customized targets. These targets, often 8 Table 4: Accuracy across three tasks of targets of new types (MPU and GPU). Dataset Stmt. Comp. Next. Sugg. Code. Gen. ARC (MPU) NVPTX (GPU) ARC (MPU) NVPTX (GPU) ARC (MPU) NVPTX (GPU) EM (%) ED EM (%) ED EM (%) ED EM (%) ED BLEU-4 ED BLEU-4 ED GCC -w/o GPU and MPU 50.53 74.09 46.37 72.45 37.22 58.21 38.33 56.83 19.29 49.12 22.46 50.33 -w GPU and MPU 52.45 74.57 50.56 75.52 38.26 59.21 38.33 56.31 19.94 50.27 25.47 52.60 Diff -1.92 -0.48 -4.19 -3.07 -1.04 -1.00 0.00 +0.52 -0.65 -1.15 -3.01 -3.37 LLVM -w/o GPU and MPU 69.82 85.59 60.04 79.85 58.26 73.75 46.28 63.92 49.62 70.26 42.94 65.43 -w GPU and MPU 71.34 85.98 64.45 81.53 58.68 74.57 47.81 65.5 55.38 74.41 44.33 66.36 Diff -1.52 -0.39 -4.41 -1.68 -0.42 -0.82 -1.53 -1.58 -5.76 -4.15 -1.39 -0.93 Table 5: Improvement of accuracy across three tasks for RI5CY after iterative expansion. Dataset Stmt-Level. Comp. Next-Stmt. Sugg. Code. Gen. EM (%) ED EM (%) ED BLEU-4 ED -w RISC-V 74.06 87.91 67.25 81.28 79.46 89.92 -w/o RISC-V 66.16 83.79 57.29 74.73 54.41 75.41 Diff +7.90 +4.12 +9.96 +6.55 +25.05 +14.51 built upon existing targets, integrate customized instructions to swiftly cater to specific application scenarios. Consequently, their backends merely require extensions from the existing backend. We chose RI5CY in LLVM to test if ComBack can be iteratively expanded to improve backend development efficiency for customized targets. As a target based on RISC-V, RI5CY shares most backend code with RISC-V but includes customized instruction handling. Initially, we fine-tuned CodeT5+ with train and validation set in Sec. 4.3.2 (excluding RISC-V), then we add RISC-V into train and validation set and fine-tuned CodeT5+ with new data (detailed in Appendix D) and restart fine-tuning from scratch. Results in Table 5 show a notable accuracy improvement across three tasks after integrating RISC-V data, demonstrating ComBack’s iterative expansion ability. Answer to RQ.3: ComBack effectively enables backend development for customized targets by iterative data expansion. 5 Related Work Backend Development. Compiler backend development heavily relies on manual efforts. Some researchers have proposed Processor Design Languages (PDL) to describe ISA and hardware information for processors [40, 6, 13, 4, 12, 24, 35, 5]. While these methods mitigate manual efforts to some degree, programmers still need to invest significant effort in learning PDL rules and writing files. Dataset for Compiler. Datasets like CodeXGlue [34] and CodeSearchNet [25] have enhanced language models in programming. As AI extends into compilers, datasets like Compile [21], TenSet [56], and ANGHABENCH [11] focus on compiler optimization. However, there remains a dearth of datasets tailored for compiler backends within the community. ComBack is the first dataset designed to substantially augment the capabilities of language models in backend code generation. AI for Compilation. AI has driven the widespread adoption of machine-learning-based compilation techniques. These methods have found application in tasks such as developing cost and performance models [54, 44, 55, 42, 36], determining transformation order [48, 17, 39, 29, 8], and optimizing parallel programs [28, 27, 52, 51, 26]. Ongoing projects using transformer models for decompilation[1, 2, 46, 53] and code optimization [10] highlight the significant potential of AI for compilers. 9 6 Discussion Limitation. One limitation of ComBack is the absence of function descriptions for highly-customized functions in backends for specific targets. We plan to address this in future iterations of the dataset. Potential Societal Impact. ComBack does not contain any personally identifiable information or offensive content, thereby mitigating any potential negative societal impact. Conclusion. In this paper, we introduce ComBack, the first public dataset for compiler backend development. ComBack includes 178 backends for mainstream compilers and features three tasks, including statement-level completion, next-statement suggestion and code generation. It enables efficient backend code completion and generation after fine-tuning language models with ComBack. Our evaluation, conducted on six representative language models, shows that ComBack boosts language models’ performance across all three tasks. Notably, CodeT5+ with only 220M parameters significantly outperforms the efficiency of conventional backend development methods and even surpasses ChatGPT-3.5-Turbo and Code-LLaMA-34B-Instruct across three tasks, suggesting potential improvements in compiler development speed and efficiency. Acknowledgement We would like to thank all anonymous reviewers for their insightful feedback. This work was supported by National Key R&D Program of China, Grant No. 2023YFB3001502, the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.XDB0500102 and XDB0660102. It was also supported by the National Natural Science Foundation of China, Grant No.U23B2020, No. 62090024, No. 62302479 and the Innovation Funding of ICT, CAS under Grant No.E361010 and No.E261110. References [1] Jordi Armengol-Estapé and Michael F. P. O’Boyle. Learning c to x86 translation: An experiment in neural compilation, 2021. [2] Jordi Armengol-Estapé, Jackson Woodruff, Chris Cummins, and Michael F. P. O’Boyle. SLaDe: A Portable Small Language Model Decompiler for Optimized Assembler. https://arxiv. org/abs/2305.12520, 2023. [3] BigQuery. Github activity data. https://console.cloud.google.com/marketplace/ details/github/github-repos, 2024. 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Compiler ISA Target GCC CPU aarch64, arm, clipper, crx, csky, d30v, i370, i386, i860, i960, ia64, iq2000, loongarch, mep mips, mmix, moxie, mt, nds32, or1k, pa, powerpcspe, pru, riscv, rs6000, rx, sh, sparc stormy16, vax, bfin, c4x, fr30, gcn, nvptx MPU 1750a, a29k, alpha, arc, avr, cr16, cris, eco32, epiphany, ft32, h8300, lm32, m32c, m32r m68hc11, m68k, m88k, mcore, microblaze, mn10200, mn10300, msp430, nios2, ns32k pdp10, pdp11, rl78, romp, s390, spu, v850, xtensa, z8k Virtual bpf, mapip, visium, vms VLIW c6x, convex, frv, tilegx, tilepro LLVM CPU AArch64, ARM, ARM64, AZPR, CAHP, CJG, Comet2, Cpu0, CSKY, Dcpu16, Digital DLX, F2003f, FISC, FPGA, IA64, Kudeyar, Lanai, LC2200, LC3, LC3b, LEG LoongArch, Mandarin, MINA32, Mips, MMIX, OpenRISC, OR1K, PowerPC, RI5CY RISCV, SHUXI, SIC, Sparc, StackPU2, SystemZ, TeeRISC, TOY, UPT, VE, X86, XNCM DSP Blackfin, Hexagon, MDSP, SNES, Teak, Videocore, VideoCore4 GPU AMDGPU, NVPTX, Nyuzi, PTX, R600 MPU AAP, AGC, Alpha, ARC, ARCompact, AVR, CellSPU, ECLair, Epiphany, GBZ80, J2 LM32, M680x0, M68k, M88k, MBlaze, MCS51, MOS, MSP430, Nios2, P2, PIC16 TL45, TLCS900, TriCore, WDC65816, XCore, Xtensa, Z80, Z80old Virtual BPF, DirectX, HSAIL, JVM, mproc, NPEngine, RV16K, SPIRV, TGSI, TPC, TVM WebAssembly VLIW Patmos, rvex, Tile64, TMS320C64X B Appendix: Target abbreviation occurred during pre-processing. Table 7: Targets Abbreviation in ComBack. Target Abbreviation Target Abbreviation Target Abbreviation AMDGPU SI ARCompact ARC Mandarin MD Blackfin BF CellSPU SPU PowerPC PPC DirectX DXIL GBZ80 GB R600 SI RI5CY RISCV Sparc SP Tile64 T64 Videocore VC WDC65816 WDC In Table 7, we provide all abbreviations for targets in ComBack. Recording these abbreviations can assist us in accurately extracting target-specific values. C Appendix: Hyperparameters and Input/Output Sequence Length Settings. In Table 8, we provide all hyperparameter settings. For CodeBert and GraphCodeBert, the input sequence length is set to 384, with output lengths of 128 for Statement-Level Completion and NextStatement Suggestion, and 256 for both input and output for Code Generation, given the maximum token length of 512 for both models. For the other four models, the input sequence length is set to 512, with output lengths of 128 for Statement-Level Completion and Next-Statement Suggestion, and 256 for input and 512 for output for Code Generation. Table 8: Hyperparameter settings. Hyperparameter Value Hyperparameter Value Hyperparameter Value Training Batch Size 32 Beam Size 4 Learning Rate 5e-5 Evaluation Batch Size 16 Max Optimization Steps 3 14 D Appendix: Data Statistics about the Number and Token of Three Tasks. In Table 9, we provide all detailed data of train, validation and test set of experiments in Sec. 4.2 to Sec. 4.4. Table 9: Data statistics about the number and token of three tasks. (a) Data statistics about the number and token of three tasks for Sec. 4.2. Task Train validation Test Stmt. Comp. 128,899(11.36M Token) 16,112(1.43M Token) 16,113(1.43M Token) Next. Sugg. 173,052(15.69M Token) 21,631(1.99M Token) 21,632(1.98M Token) Code. Gen. 36,236(5.10M Token) 4,530(0.64M Token) 4,530(0.64M Token) (b) Data statistics about the number and token of three tasks for Sec. 4.3.1. Task Train validation Test Stmt. Comp. 114,016(10.20M Token) 20,121(1.81M Token) 6,645(0.58M Token) Next. Sugg. 152,114(14.10M Token) 26,844(2.49M Token) 9,313(0.83M Token) Code. Gen. 30,633(4.44M Token) 5,406(0.79M Token) 2,819(0.37M Token) (c) Data statistics about the number and token of three tasks for Sec. 4.3.2. Task Train validation Test Stmt. Comp. 87,018(7.78M Token) 15,357(1.37M Token) 2,764(0.26M Token) Next. Sugg. 113,684(10.65M Token) 20,063(1.87M Token) 4,029(0.38M Token) Code. Gen. 21,184(3.14M Token) 3,739(0.55M Token) 1,372(0.18M Token) (d) Data statistics about the number and token of three tasks for Sec. 4.4 (Excluding RISC-V in train and validation set). Task Train validation Test Stmt. Comp. 87,018(7.78M Token) 15,357(1.37M Token) 721(0.04M Token) Next. Sugg. 113,684(10.65M Token) 20,063(1.87M Token) 1,035(0.06M Token) Code. Gen. 21,184(3.14M Token) 3,739(0.55M Token) 219(0.02M Token) (e) Data statistics about the number and token of three tasks for Sec. 4.4 (Including RISC-V in train and validation set). Task Train validation Test Stmt. Comp. 90,316(8.06M Token) 15,940(1.42M Token) 721(0.04M Token) Next. Sugg. 118,175(11.04M Token) 20,856(1.94M Token) 1,035(0.06M Token) Code. Gen. 22,413(3.30M Token) 3,957(0.58M Token) 219(0.02M Token) 15 E Appendix : Fork-Flow Detailed Experimental Data. In Table 10, we provide all detailed data in Fork-Flow experiment. Table 10: Fork-Flow experimental data. Compiler Type Target BLEU4 ED EM Target BLEU4 ED EM GCC MPU z8k 0.32 1.33 0 m68k 1.27 2.84 0 GCC MPU a29k 0 0 0 m88k 0 0 0 GCC MPU avr 4.27 8.85 0.24 microblaze 1.39 3.53 0 GCC MPU lm32 1.89 3.68 0.24 mn10200 0 0 0 GCC MPU mcore 1.4 3.61 0 mn10300 2.73 5.47 0 GCC MPU msp430 0.94 1.89 0 nios2 3.35 7.07 0.48 GCC MPU v850 2.32 4.58 0 ns32k 0 0 0 GCC MPU xtensa 2.93 6.01 0.24 cris 2.43 6.27 0 GCC MPU cr16 1.49 3.86 0 pdp11 1.39 3.75 0 GCC MPU rl78 0.9 1.69 0 pdp10 0.02 0.25 0 GCC MPU m32c 1.35 4.07 0.24 1750a 0 0 0 GCC MPU ft32 2.23 4.14 0 s390 3.53 8.05 0 GCC MPU h8300 2.48 5.25 0 romp 0 0 0 GCC MPU alpha 3.69 7.5 0.24 spu 1.98 3.78 0 GCC MPU epiphany 4.94 7.84 0.24 eco32 1.36 2.74 0 GCC MPU m32r 4.31 7.85 0.95 GCC CPU aarch64 12.54 18.21 3.51 sparc 3.68 7.81 0.39 GCC CPU arm 4.28 7.97 0.39 mep 0.96 2.27 0.19 GCC CPU csky 3.77 7.76 0.19 vax 0.78 2.13 0 GCC CPU d30v 0.19 0.49 0 clipper 0 0 0 GCC CPU i370 0 0 0 iq2000 1.91 4.03 0.39 GCC CPU i386 0.26 0.68 0 crx 0.43 1.82 0 GCC CPU i860 0 0 0 moxie 1.05 2.77 0.19 GCC CPU i960 0 0 0 mt 1.01 2.81 0 GCC CPU ia64 2.16 5.71 0 nds32 1.88 4.24 0.19 GCC CPU loongarch 28.77 34.8 8.38 pru 2.15 5.28 0.19 GCC CPU mips 22.24 29.99 3.51 rs6000 3.41 7.25 0.19 GCC CPU mmix 1.75 4.27 0.19 rx 1.01 2.4 0 GCC CPU or1k 2.06 4.69 0.19 sh 2.49 5.71 0 GCC CPU pa 2.09 4.47 0 stormy16 0 0 0 GCC CPU powerpcspe 0.07 0.36 0 LLVM GPU AMDGPU 18.81 39.04 0.58 PTX 12.39 21.79 0.97 LLVM GPU Nyuzi 12.74 21.35 1.94 R600 16.31 32.72 0.39 LLVM MPU AVR 28.42 45.24 2.33 CellSPU 11.29 25.76 0 LLVM MPU LM32 12.55 18.37 3.1 ECLair 3.94 5.4 1.55 LLVM MPU MCS51 28.1 43.36 2.33 Epiphany 0.78 0.78 0.78 LLVM MPU MSP430 29.04 46.19 2.33 GBZ80 27.87 45.74 0.78 LLVM MPU P2 28.72 42.04 4.65 M680x0 24.2 39.33 4.65 LLVM MPU PIC16 12.21 26.22 0 M68k 25.49 42.28 5.43 LLVM MPU TriCore 18.83 25.93 6.2 M88k 23.26 41.2 5.43 LLVM MPU XCore 41.8 60.62 5.43 MBlaze 15.84 29.81 0 LLVM MPU Xtensa 22.1 41.71 6.98 Nios2 12.89 20.59 2.33 LLVM MPU AGC 13.11 22.84 3.88 Z80 24.64 43.71 2.33 LLVM MPU TL45 24.63 38.95 5.43 Z80old 21.75 38.27 3.1 LLVM MPU TLCS900 20.59 32.4 0 MOS 22.77 42.36 3.1 LLVM MPU J2 17.75 35.71 2.33 AAP 30.41 44.91 4.65 LLVM MPU Alpha 12.81 25.61 0 WDC65816 13.2 22.6 0 LLVM MPU ARCompact 10.4 21.48 0 LLVM CPU AArch64 27.32 46.47 1.5 OR1K 15.18 26.21 0.43 LLVM CPU ARM 23.93 42.38 2.14 PowerPC 21.42 39.99 0.75 LLVM CPU ARM64 15.33 27.04 0.75 SHUXI 11.21 19.73 1.71 LLVM CPU AZPR 2.92 5.72 0.21 Sparc 18.19 32.98 1.61 LLVM CPU CAHP 23.54 33.61 5.03 StackPU2 2.08 2.6 0.11 LLVM CPU CJG 11.08 19.17 1.61 SystemZ 21.85 38.97 1.39 LLVM CPU Cpu0 16.92 29.97 1.28 TOY 9.45 20.55 0.32 LLVM CPU CSKY 25.86 38.25 3.53 UPT 5.65 12.1 0.64 LLVM CPU DLX 12.13 24.55 1.39 X86 18.88 35.77 1.39 LLVM CPU IA64 4.16 9.52 0 XNCM 7.04 14.61 0.21 LLVM CPU Kudeyar 8.89 16.03 0.32 Comet2 3.87 7.21 0.96 LLVM CPU Lanai 16.7 30.37 1.28 Dcpu16 9.56 18.43 0 LLVM CPU LC2200 15.08 24.3 1.71 F2003f 9.24 16.72 0.54 LLVM CPU LC3 8.49 17.79 0.86 SIC 11.87 22.42 1.18 LLVM CPU LC3b 3.14 6.47 0.32 TeeRISC 8.39 15.64 0.32 LLVM CPU LoongArch 13.6 21.83 2.57 Digital 0.87 1.1 0.21 LLVM CPU Mandarin 9.24 16.75 0.54 FISC 12.9 25.27 0.96 LLVM CPU MINA32 9.96 19.04 1.07 FPGA 1.07 2.01 0.21 LLVM CPU Mips 22.96 40.35 2.25 LEG 8.94 18.63 0.86 LLVM CPU MMIX 16.42 25.04 3.75 VE 20.54 34.91 2.57 LLVM CPU OpenRISC 4.58 9.06 0.21 16 F Appendix: Prompt Example of Input for ChatGPT and Code-LLaMA. We provide prompt examples of Input for ChatGPT and Code-LLaMA in Fig. 7. //Prompt: Complete the last statement of this code snippet: ... adjustReg(MBB,LastFrameDestroy, DL, SPReg, FPReg, −StackSize+RVFI−>getVarArgsSaveSize() (a) Statement-Level Completion //Prompt: Predict the next statement of this code snippet: ... maxCallFrameSize = (maxCallFrameSize + AlignMask) & ~AlignMask; (b) Next-Statement Suggestion //Prompt: Create a function named "getPointerRegClass" for "Sparc" backend of LLVM Compiler. //The description of this function is "Returns a TargetRegisterClass used for pointer values". //It contains "Sparc", "SP::I64RegsRegClass", "SP::IntRegsRegClass" as target specific values. (c) Code Generation Figure 7: Prompt examples of tasks in ComBack. G Appendix : License of Assets. In Table 11, we provide all license of assets in experiment. Table 11: License of assets. Assets CodeBase License CodeBER [14] CodeSearchNet [25] MIT License GraphCodeBERT [23] CodeSearchNet [25] MIT License UnixCoder [22] CodeSearchNet [25], C4 [41] MIT License CodeT5 [50] CodeSearchNet [25], BigQuery1 [3] Apache-2.0 NatGen [7] CodeSearchNet [25], BigQuery1 [3] MIT License CodeT5+ [49] GitHub-Code Dataset [9] bsd-3-clause 17 H Appendix: Heatmap Analysis. 0 300 X Y Top 300 functions implemented by most targets ISA 100% 0% CPU GPU MPU Virtual DSP VLIW (a) GCC 0 300 X Y Top 300 functions implemented by most targets ISA CPU GPU MPU Virtual DSP VLIW 100% 0% (b) LLVM Figure 8: Heatmap analysis of top 300 functions implemented by most targets We analyzed the top 300 functions implemented by most targets in both GCC and LLVM backends, creating Fig. 8 based on target types. CPUs and MPUs showed high similarity, while CPUs and GPUs exhibited significant differences, making it challenging to generate accurate GPU code solely from CPU data. Additionally, VLIW and Virtual targets differed from mainstream CPUs due to variations in instruction sets, highlighting the need to use backend code from similar targets for training, as discussed in Sec. 4.3.1. 18 Checklist 1. For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] See Sec. 6 (c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 6 (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] 3. If you ran experiments (e.g. for benchmarks)... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?[Yes] See Supplemental Material and https://huggingface.co/datasets/docz1105/ComBack and https://huggingface.co/docz1105/ComBack_Models (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 4.1 (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Sec. 4.1 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes] See Sec. 4.1 (b) Did you mention the license of the assets? [Yes] See Appendix G (c) Did you include any new assets either in the supplemental material or as a URL?[Yes] See https://huggingface.co/datasets/docz1105/ComBack and https://huggingface.co/docz1105/ComBack_Models (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Sec. 6 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] 19	2024	3567
4,436	Visual Pinwheel Centers Act as Geometric Saliency Detectors Haixin Zhong1,2 hxzhong@fudan.edu.cn Mingyi Huang1,3 myhuang20@fudan.edu.cn Wei P. Dai1,5 weidai@fudan.edu.cn Haoyu Wang3 haoyuwang18@fudan.edu.cn Anna Wang Roe4 annawang@zju.edu.cn Yuguo Yu1,2,3,5,∗ yuyuguo@fudan.edu.cn 1. Research Institute of Intelligent Complex Systems, Fudan University. 2. State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science, Institutes of Brain Science, Fudan University. 3. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University. 4. MOE Frontier Science Center for Brain Science and Brain-machine Integration, School of Brain Science and Brain Medicine, Key Laboratory of Biomedical Engineering of Ministry of Education, College of Biomedical Engineering and Instrument Science, Zhejiang University. 5. Shanghai Artificial Intelligence Laboratory. ∗Corresponding author. Abstract During natural evolution, the primary visual cortex (V1) of lower mammals typically forms salt-and-pepper organizations, while higher mammals and primates develop pinwheel structures with distinct topological properties. Despite the general belief that V1 neurons primarily serve as edge detectors, the functional advantages of pinwheel structures over salt-and-peppers are not well recognized. To this end, we propose a two-dimensional self-evolving spiking neural network that integrates Hebbian-like plasticity and empirical morphological data. Through extensive exposure to image data, our network evolves from salt-and-peppers to pinwheel structures, with neurons becoming localized bandpass filters responsive to various orientations. This transformation is accompanied by an increase in visual field overlap. Our findings indicate that neurons in pinwheel centers (PCs) respond more effectively to complex spatial textures in natural images, exhibiting quicker responses than those in salt-and-pepper organizations. PCs act as first-order stage processors with heightened sensitivity and reduced latency to intricate contours, while adjacent iso-orientation domains serve as second-order stage processors that refine edge representations for clearer perception. This study presents the first theoretical evidence that pinwheel structures function as crucial detectors of spatial contour saliency in the visual cortex. 1 Introduction The seminal work of Hubel and Wiesel revealed orientation-selective columns in the visual cortex of higher mammals [1, 2]. In higher mammals’ primary visual cortex (V1), neurons cluster into "pinwheel" structures around singularities [3], unlike in some mammals like rodents, which display "salt-and-pepper" organizations [4] or mini-columns [5]. While there are established theories and experiments for studying the formation of topological organization maps in the visual cortex [6, 7, 8, 9, 10, 11], the functional significance of pinwheel-like columnar organization remains an unresolved question and is even debated [12, 13]. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Sophisticated visual analyses, such as image pattern extraction [14], pattern symmetry [15], material properties [16], and textures [17], are crucial for understanding complex visual inputs. Imaging and electrophysiological studies have shown that iso-orientation domains (IODs) undergo crossorientation suppression [18], reducing a neuron’s response to its preferred orientation when another orientation is also present in the stimulus [13, 19, 20]. This indicates IODs encoding the linear oriented stimuli, which is crucial for detecting edges and contours [21, 22]. Cross-orientation suppression is believed to facilitate the detection of local discontinuities, such as orientation discontinuities [23, 24, 25], leading to perceptual "pop-out" effects and the perception of illusory contours [24, 26, 27]. In contrast, neurons at pinwheel centers (PCs) exhibit greater selectivity for cross-orientation stimuli [12, 13]. This indicates that PCs respond more effectively to multi-orientation patterns, such as pattern symmetry than IODs [12]. This indicates PCs may contribute to encode more complex contour features. However, PCs are less selective but have longer response latency than IODs for stimulus orientation in the hierarchy process within OPMs when it comes to a single stimulus orientation [13, 19, 28]. Some studies indicate that colors [29], textures [30], darks and lights [31], luminance [32], and mirror symmetry [15] play a role in salient to visual processing. Despite these insights, the functional implications of how neurons within IODs and PCs of pinwheels process complex contour stimuli—potentially affecting stimulus salience for both IODs and PCs—from bottom-up visual inputs remain poorly understood, particularly from a temporal-spatial neural dynamics standpoint. In response to these challenges, our research contributes the following: • We propose a novel 2D self-evolving spiking neural network (SESNN) model that investigates the spiking mechanisms behind orientation preference maps (OPMs), spanning from salt-and-pepper organizations in mice to pinwheel structures in cats and macaques. The SESNN uniquely produces sparse codes through local synaptic plasticity during natural image learning, establishing a new benchmark for neural coding strategies. • PCs act as first-stage processors, detecting natural images and initiating spiking waves to neighboring IODs, which then process as second-stage neurons. This indicates that early processing involves complex contours, not just edge detection. • PCs react faster to a variety of orientation features than IODs, indicating their function in detecting complex orientations and serving as geometric saliency detectors. This suggests PCs have an evolutionary advantage due to self-organized pinwheel structures, which improves their ability to process complex contours. 2 Results 2.1 Visual overlap underlying pinwheels emergence Our SESNN model generates diverse OPMs, from salt-and-peppers to pinwheel structures, by adjusting the visual overlap metric ε. This metric, crucial for the variety of visual topologies across species, is shown in Fig. 1a to produce pinwheel structures at high overlap, akin to those in cats and macaques, while low overlap results in salt-and-pepper organizations, typical of mice or rats. High overlap also enables cortical neurons to sample natural scenes more frequently, aiding in generating high-resolution images during decoding [7, 33]. Fig. 1a shows how visual input overlap levels from 9 to 15 pixels affect V1 orientation selectivity maps in the model. The top panel illustrates a higher overlap (15 pixels), and the middle, a lower overlap (12 pixels). This comparison reveals the impact of stimulus overlap on pinwheel density and layout in the visual cortex. Below the threshold (10 pixels in our case), salt-and-pepper patterns form, as the bottom panel indicates. Thus, 9 pixels of overlap are excluded from pinwheel analysis, as shown in Fig. 1b-d. We quantitatively analyze the OPMs shown in Fig. 1a with several metrics [7, 34]: Pinwheel counts, defined as the number of PCs, can be measured by 2D fast Fourier transform [35], which are located at the intersection of the real and imaginary components that equal 0 [34]. It exhibits a decreasing trend as the visual input overlap increases (illustrated in Fig. 1b), suggesting that a greater overlap in the visual field may lead to a reduction in the number of discrete pinwheel structures. 2 a Δε < 0 Decrease overlap degree 1 mm e 140 160 180 200 Visual field (degree) 0 0.5 1 Anatomical IOD size vs. visual field IOD size (mm) R2 = 0.91 y = - 0.012x + 2.463 f ... ... ε1 = 15 pixels Overlap degree 15 pixels ... ... ... ... ε3 = 9 pixels Hypercolumn size (mm) d 0.4 0.6 0.8 1 1.2 10 11 12 13 14 15 16 Visual input overlap (ε) NNPD (mm) c 0.1 0.2 0.3 0.4 10 11 12 13 14 15 16 Visual input overlap (ε) Number of pinwheels 10 11 12 13 14 15 16 Visual input overlap (ε) 0 50 100 150 200 b ε2 = 12 pixels SESNN post-training SESNN post-training SESNN post-training Visual input to RFs 50 60 70 80 90 100 96 97 98 99 100 SESNN vs. anatomical data overlap y = 0.077x + 92.97 R2 = 0.97 Anatomical data overlap (%) SESNN model overlap (%) Figure 1: Receptive field (RF) visual overlaps underlying the emergence of OPM and the saltand-peppers are revealed via our SESNN model. a. Modifying the overlap parameter (ε) among neighboring neurons receiving (16 × 16 pixels) visual inputs from natural images influences the dimensions (e.g., b. Pinwheel counts, c. Nearest-neighbor pinwheel distance, d. Hypercolumn size) of pinwheel structures and salt-and-pepper organizations. (Lines: mean. Shaded area: SD.) e. Comparing the SESNN model overlap percentage ( ε sRF × 100%) with actual anatomical data overlap percentages (ε′ percentage) in various species (mice, cats, and macaques). f. Relationship between the IOD size and the extent of the visual field in anatomical data (mice, cats, and macaques). The nearest-neighbor pinwheel distance (NNPD) in millimeter (mm) unit is defined as the distance between the two nearest PCs. The increasing trend of visual input overlap expands the distance between neighboring pinwheels (Fig. 1c). The size of hypercolumns (mm) is defined with periodicity measured by 2D fast Fourier transform and also increases with the visual input overlap (shown in Fig. 1d). This paper does not account for left- and right-eye dominance columns, so the hypercolumn size is defined as the full 180° cycle of repeating column spacing (Λ) (mm). It’s noteworthy that pinwheel density is not included as a metric in our analysis. This omission is because the observed pinwheel density, irrespective of the hypercolumn size, approaches π pinwheels/Λ2, conforming to topological constraints [34, 35]. Our findings emphasize the importance of overlap degrees (Fig. 1a). Greater overlap (e.g., ε1 = 15 pixels) fosters stronger local clustering, leading to larger hypercolumn sizes, fewer pinwheels, and longer NNPDs, versus lower overlap (e.g., ε2 = 12 pixels). Minimal overlap (e.g., ε3 = 9 pixels), yields weak clustering, resembling salt-and-pepper organizations. This suggests that shared input among V1 neurons significantly influences OPM and salt-and-pepper formation. We obtain the anatomical data overlap using Eq. 3 and observe a strong positive correlation (R2 = 0.97) between the SESNN model and species’ visual RF overlaps (mouse, cat, macaque) (Fig. 1e). This relationship highlights the overlap index’s key role in spatial organization within orientation maps. The model’s predictions on IOD sizes and visual field extent (Fig. 1f) align with empirical data [7], confirming the SESNN model’s robustness in simulating neuroanatomical organization and the biological development of orientation maps. 2.2 Spatial-temporal distributed spiking waves propagate within pinwheels V1 neurons stimulated by natural images primarily fire within pinwheel structures, particularly within and around PCs (Fig. 2a-b). This pattern is especially pronounced in higher mammals with large IODs, such as macaques and cats. 3 a b c t0 ms t0+1 ms Neurons' positions in the box 20 40 60 80 100 120 140 160 0 180 Pinwheel centers Firing neuron Resting neuron Orientation map Spatial-temporal response 0 1 2 3 4 5 6 t0 t0+1 Distance in 2D grid Response time (ms) 0.2 0.6 0.8 1 Response Onset Latency (normalized) 0.5 1 1.5 2 2.5 3 3.5 4 R2 = 0.99 Distance from neuron to PC within a pinwheel 0.4 * Figure 2: Spatial-temporal response pattern within pinwheels. a. This figure displays the neuronal responses on OPM with a large IOD in a pinwheel structure. The neurons that fire at time t0 are shown as large black dots at PC, and they expand towards the periphery at time t0+1, also denoted as large black dots. The other small dots represent resting neurons. b. Distance between firing neurons and the PC at time t0 and t0+1. c. This panel shows the response onset latency of neurons and the mean distance (± SD) between these neurons within a pinwheel. The distance is measured as the Euclidean distance within a 2D grid, simulating the structure of a 2D V1 area. (Significance: p<0.001, Mann-Whitney U test.) We define the response onset latency as 1 ms for the initial discharge from pinwheel structures, with subsequent firings occurring at 2 ms, based on a 1 ms time unit. Stimulated by natural images, the discharges start at the PCs and exhibit pronounced diffusion within the IODs sequentially, depending on their distance from the center, as suggested in Fig. 2c. 2.3 Visual bottom-up saliency detection: functional role of pinwheel in geometric encoding In this section, we investigate whether pinwheel structures respond distinctly to salient features in input images. The ground truth boundary from the BSDS 500 dataset [36] used as binary input represents geometric complexity (edges and curves) (Fig. 3a). The complexity is measured by calculating the local pixel entropy using sliding windows, with a 15×15 pixel neighborhood to assess pixel value dispersion in the binary images. The computation adheres to the following equation: H(i, j) = − L−1 X k=0 p(mk) log2 p(mk), (1) where H(i, j) denotes the entropy at pixel position (i, j) in the entropy map, L the count of distinct gray levels within the local neighborhood around pixel (i, j), and mk the kth gray level within this specified neighborhood. A large entropy value reflects great unpredictability or complexity in the pixel values, signifying a highly variable pixel value distribution. Conversely, a low entropy value indicates a high degree of predictability, less variation, and reduced complexity in the contours of pixel values. In addition, the saliency map of images is generated based on the classical methodology [37]. Furthermore, we propose a bimodal ratio analysis to compute the orientation bimodal ratio (OBR) to indicate a neuron’s orientation tuning curve as either unimodal (single peak) or perfectly bimodal (two peaks of equal strength). This analysis focuses on identifying the peaks in the orientation tuning curve and quantifying their relative strengths. OBR = 2 · min(R1, R2) R1 + R2 , (2) 4 a b Saliency (normalized) 0.2 0.6 y = 1.39x - 0.55 R2 = 0.82 1 0.4 0.6 0.8 1 Complexity (normalized) BSDS500 saliency vs. complexity 0 0.5 1 BSDS500 image 0 1 BSDS500 binary input BSDS500 saliency map 0.4 1 BSDS500 entropy map 0 0.5 1 0.7 c 0 5 10 0.2 0.4 0.6 0.8 Pinwheels Salt-and-peppers ✱✱✱✱ Response to complexity of BSDS500 Response onset latency (ms) Complexity (normalized) Figure 3: Pinwheel structures in V1 exhibit geometric properties. a. A BSDS 500 grayscale image displays boundaries, saliency, and entropy maps. b. Natural images show a positive correlation between saliency and entropy. c. Neuronal response onset latency from pinwheels and salt-andpeppers relates to structural complexity, measured by local pixel entropy. (Data: mean ± SD, significance: **p<0.0001, Welch’s t-test.) where R1 and R2 represent the normalized firing rates corresponding to the strengths of the two most pronounced peaks in the orientation tuning curve. The OBR ranges from 0, denoting unimodality, to 1, indicating perfect bimodality in the neuron’s orientation tuning. A positive correlation is observed between the saliency map and the geometrical complexity of the BSDS500 dataset (Fig. 3b), demonstrating that higher geometrical complexity correlates with increased saliency. Significantly, in response to the stimulus shown in the BSDS500 image (Fig. 3a), pinwheel structures primarily activate in areas of high contour complexity (regions with the highest saliency in this binary image), which is a response pattern have not been observed in salt-and-peppers (Fig. 3c). To confirm the disparity in contour complexity responses between pinwheel and salt-and-pepper organizations, we design a star-like binary input (depicted in Fig. 4a), including four identical entities to negate the impact of neuronal positioning within the SESNN model. This approach reaffirms the saliency-complexity correlation (Fig. 4b) and the priority of pinwheel activation over salt-and-peppers in response to heightened complexity (Fig. 4c). Findings show that PCs exhibit enhanced saliency detection and significantly faster response times than IODs, indicating that PCs respond more quickly and sensitively to geometrically complex stimuli, while IODs are slower and react to simpler geometrical stimuli (see Fig. 4d). Both saliency and latency measurements are normalized to a 0-1 scale for comparison. The enhanced saliency detection of PCs is due to the complex orientation preference in RFs. As addressed in Fig. 4e, the ordinate represents the OBR, reflecting that neurons near PC generally exhibit bimodal orientation tuning curves with near-equal peak strengths while there is a primary peak and a secondary peak at a relatively far position from PC (x = 2). And the secondary peak is nearly absent at the IODs level (x = 3), leading to an OBR close to 0. Salt-and-peppers, however, show less variation, maintaining a consistent OBR. In conclusion, PCs demonstrate selectivity for more intricate orientations. This is experimentally supported by [13, 38], who suggest that PCs are particularly sensitive to specific geometric configurations, such as T junctions. Characterized by multiple orientations and an OBR nearing 1 (Fig. 4e), these neurons tend to initiate action potentials in response to complex orientations. Consequently, 5 c Star-like entropy map 0 0.5 1 Star-like binary input 0 1 a 1 2 3 0.5 Bimodal tuning within 2D cortical area 2D cortical distance (a.u.) Pinwheels Salt-and-peppers 0 1 e d 0 0.2 0.4 0.6 0.8 1 Latency (normalized) 0 0.2 0.6 1 Response to saliency of star-like shape Saliency (normalized) PCs IODs * b Response to complexity of star-like shape 0 2 4 6 8 0.5 Pinwheels Salt-and-peppers ✱✱✱✱ 1 0 Complexity (normalized) Response onset latency (ms) Star-like shape saliency vs. complexity y = 1.37x - 0.52 Saliency (normalized) 0.2 0.6 R2 = 0.80 1 0.4 0.6 0.8 1 Complexity (normalized) Figure 4: Geometric properties emergence in PCs of V1 on star-like patterns. a. We introduce artificial star-like patterns to assess neural response to complexity. b. Star-like images show a link between saliency and entropy. c. Neuron response times in PCs and salt-and-peppers reduce with lower entropy. d. The analysis compares PCs and IODs for saliency and response to star-like patterns; the inset details saliency and latency. e. OBR varies across cortical distance; the red line marks pinwheels, and the black line, salt-and-peppers (an arbitrary point for salt-and-peppers). (Data: mean ± SD, significance: p<0.01, p<0.001, *p<0.0001, Welch’s t-test.) this leads to pinwheels being the first to respond. In contrast, neurons in salt-and-peppers do not exhibit a similar responsiveness to complex orientations as observed in pinwheels. 3 Methods 3.1 The architecture of SESNN model Our SESNN model is a two-dimensional network of excitatory (E-) and inhibitory (I-) leaky integrateand-fire neurons (LIF) (5), stimulated by whitened natural images to mimic the LGN’s functions of contrast normalization and edge enhancement without complex modeling [43, 44, 45]. We use 160 whitened natural images as the training dataset, normalized to zero mean and uniform variance, derived from 20 base images (512×512 pixels) [45, 46]. To capture orientation details, each of the base images undergoes a 90-degree clockwise rotation and flip, creating 8 variations per original. The configuration features E- and I- neurons in recurrent networks with periodic boundary conditions (PBCs) (Fig. 5b), simulating a continuous 2D cortical surface. The neural connectivity at initialization is depicted in Fig. 5c (see Eq. 11). Moreover, the connection strengths in the well-trained model align closely with the experimental finding ([47]). Under natural image stimuli, the SESNN forms single neuron RFs and population-level pinwheel structures in the OPM (Fig. 5e-f). To validate the model, we compare its evolution from randomness to organized states against biological data from macaque pinwheel structures and a baseline model [42], using metrics such as pinwheel density (pinwheels/ Λ2), NNPD (mm), and hypercolumn size (mm) [7, 34, 35] (Fig. 5f and Table 1). Table 1: SESNN pinwheels (mean ± SD) vs. macaque pinwheels. Metric E-I baseline SESNN model Macaque Pinwheel density (pinwheels/Λ2) ∼2.941 3.175 ± 0.397 ∼3.327 NNPD (mm) N/A 0.277 ± 0.043 ∼0.242 Hypercolumn size (mm) N/A 0.839 ± 0.054 ∼0.760 6 Batch size: 100 Image set: 512×512×160 a E- and I- recurrent connections SESNN model b c d E-neuron I-neuron Excitatory connection Inhibitory connection HO rule: FF(im→E) and W(E→E) CM rule: W(E→I), W(I→E) , and W(I→I) 10-1 100 101 Connection strength f Populational spatial self-organization SESNN pinwheels Randomness training Macaque pinwheels Vs. 1 mm … … … … Feedforward connection, FF(im→E) Lateral connection, W(K→K) j k i RF visual overlap Select 100 patches HO1 … … … … I CM2 CM3 E HO2 CM1 Neuron groups Periodic boundary conditions Neural connectivity Excitatory connection Inhibitory connection 2D cortical Distance Connection strength Post-training connection strengths Single neuron RF emergence Randomness RF training -1 0 1 (100 ms) e SNN-based model (Baseline model) Figure 5: Architecture of proposed SESNN model. a. The SESNN model comprises 4,900 Eand 1,225 I- neurons [39, 40, 41]. It processes 160 natural images (100 patches each), presenting each 512×512 pixel patch to E-neurons for 100 ms with input overlap. The Feed-forward and E-E connections adhere to the Hebbian-Oja (HO) rule; others follow the Correlation Measuring (CM) rule. b. E- and I-neurons are spatially arranged with periodic boundaries, sharing coordinates with connected boundaries as per diagram arrows. Identical connections are marked by same-color arrows. c. Initial weights are Gaussian distributed. d. Post-training connection strengths are depicted, with medians in red. e. RF emerges after training. f. Post-training spatial organization is compared among the SESNN model’s OPM, macaque V1, and an SNN-based model [42], with color bars for orientation and a 1 mm scale bar on the cortical surface. 3.2 Experiment-data-justified overlapping visual fields among nearby neurons In each trial, E-neuron processes 100 different 16×16 patches for 100 milliseconds each, randomly selected from the training dataset to serve as RFs (see Fig. 5a middle and right panels). It is assumed that these visual inputs overlap on the retina (Fig. 5a, middle panel and its inset). To reflect biological conditions, we perform a statistical analysis based on data from cats, macaques, and mice (Table 2), calculating average overlaps of 99.90% for cats, 99.98% for macaques, and 97.23% for mice using (Eq. 3). These results closely align with our SESNN model’s configurations (refer to Fig. 1e). RF size in V1 is more related to resolution than orientation map formation. In macaque V1, RF size increases more than tenfold from fovea to periphery, while orientation map properties show little variation [48, 49]. Our study does not focus on RF size variations across the retina, as we expect minimal effects from these shifts across species, provided the overlap remains constant. Since the fovea is key for detailed visual information, we use V1 RFs in the area centralis to modeling. We propose the visual input overlap metric ε′ percentage, which is defined as follows: ε′ percentage = √ρV1Sunit −LunitM −1 s′ RF √ρV1Sunit −1 × 100% (3) where Sunit represents the unit cortical area mm2, s′ RF denotes the size of the RF in V1 with unit in degrees, ρV1 represents the density of neurons in V1, Lunit represents the unit of cortical spacing length in unit of mm, and M refers to the cortical magnification factor (CMF) (mm/degree). We consider only an effective cortical layer composed of output neurons. This is because the apparent overlap within a vertical cortical column primarily contributes to intermediary processing stages for the same input. Therefore, such overlaps should not be conflated with overlaps in the input space. 3.3 Neural dynamics E-neurons receive stimuli from natural images as well as noise N(0, 0.04) from other brain areas (noise term). I-neurons indirectly receive natural image stimuli by adjusting E-neurons. The neural spiking dynamics are modeled using biologically inspired LIF neurons, incorporating refractory 7 Table 2: Comparative anatomical data of the retina and V1 across three species. a. This table includes three diverse species, encompassing both primates (e.g., macaques) and non-primates (e.g., mice and cats). b. V1 neuron density (neurons/mm2) within 2D surface. c. Size of V1 RF in area centralis (deg). d. Peak CMF (mm/deg) of V1 in area centralis. (A)SPECIES (B)V1 NEURON DENSITY (C)V1 RF SIZE (D)PEAK CMF CAT ∼99, 200 [33] ∼1.0 [50] ∼1.90 [51] MACAQUE ∼243, 000 [33] ∼0.2 [52] ∼18.18 [52] MOUSE ∼86, 600 [33] ∼4.0 [53] ∼0.03 [54] periods and adaptive firing thresholds [55]. The neural dynamics are iteratively formulated as follows: u(K) i (t + 1) = u(K) i (t)e− η τ(K) + hK(i) X j FF(image→E) ij Xj (4) + X K∗ X j β(K∗→K) ij · W (K∗→K) ij · z(K∗) j (t) + noise, hK(i) = 1, if i is an E-neuron ID, 0, if i is an I-neuron ID, (5) ∆θ(K) i ∝pi(z(K∗) i = 1) −p(K) i , (6) where i = 1, 2, . . . , Nth (the neuron IDs of E-neurons and I-neurons). In neural dynamics equation, u(K) i (t) denotes the membrane potential of neuron i at time t, applicable to neurons of class K, which includes E- and I- neuron groups. The membrane time constant, symbolized by τ in the resistor-capacitor circuit, governs the decay rate of the membrane potential in individual neurons. Notably, inhibitory neurons are configured to fire more rapidly than excitatory neurons [44, 56]. This setup reduces reconstruction error and hastens system convergence, leading to a more efficient and accurate representation of input stimuli. See the section A.2 for detailed neural dynamics. 3.4 Hebbian Learning in SESNN The learning rules consist of the HO rule [43] for input weight adjustments and the CM rule [44, 45] for intra-network weight changes (Fig. 5a-c). These facilitate adaptive synaptic weight adjustment based on firing pattern correlations, emulating a key learning mechanism in biological neural networks. The formula for these adjustments is given by: HO : ∆W (K∗→K) ij ∝yixj −y2 i W (K∗→K) ij , (7) CM : ∆W (K∗→K) ij ∝yixj −⟨yi⟩⟨xj⟩ 1 + W (K∗→K) ij , (8) where x, y denote the spike rates of presynaptic and postsynaptic neurons, respectively, with ⟨·⟩ denotes the lifetime average. After each stimulus presentation of 100 ms, we calculate the network’s neuronal instantaneous spike rates using exponential moving averages (EMAs), which aggregate spikes over time to reflect recent activity (see section A.1). Lifetime averages, also computed as EMAs, are crucial for homeostatic stability, helping to modulate neuronal properties or synaptic strengths for consistent activity. See the section A.2 for the hyperparameters. Our SESNN model reflects experimental findings [57, 47] by representing V1 pyramidal neurons with weaker synaptic strengths, essential for preventing over-excitation and maintaining neural balance. We apply the HO rule [43] to E-E connections with a normalization factor to keep synaptic weights between 0 and 1, while stronger lateral E-I connections under the CM rule lack this normalization [45, 44]. Post-training synaptic strengths are depicted in Fig. 5d. Stabilizing neural network training requires careful learning rate adjustment. A slower rate for E-E connections compared to others is crucial to prevent E-neuron over-excitation, aligning with empirical data [57, 47, 58]. The HO and CM rules facilitate LTP and LTD mechanisms, common in rate learning rules that do not require precise spike timing. We selected these rules for their ease of tuning and ability to stabilize recurrent excitation. 8 4 Related works Functional roles of pinwheel structure can be revealed by SESNN model The classical selforganizing map model [8] and other computational approaches like on-off models [6, 7, 9, 59] and related ANNs [10, 60, 61] lack the dynamic and temporal fidelity needed to realistically simulate the emergence of pinwheel structures in the visual cortex. To address these shortcomings, we propose the novel SESNN model, integrating retinotopy data [33, 50, 52, 53], detailed morphological data [62, 63, 64], and CMF [51, 52, 54] to enhance biological fidelity. The SESNN model effectively simulates macaque cortical organization and pinwheel development within OPMs (Fig. 5f). Furthermore, our investigations reveal that the degree of overlap—reflecting similar feed-forward inputs from identical RGCs to neighboring neurons—positively correlates with the retino-cortical mapping ratio [6], aiding in distinguishing between different V1 organizational patterns. PCs and IODs in neural processing hierarchies Our findings show that PCs and IODs exhibit distinct neural activity waves, leading to varied responses to contour complexity from spatial-temporal dynamics: PCs react first to complex contours, having more multi-orientation selective neurons (Fig. 4e, x = 1) before activity spreads to IODs, which process simpler edges (Fig. 4d and e, x = 3). PCs display a stronger correlation with contour saliency, indicating a heightened role in processing visual stimuli over IODs. In rodents with salt-and-pepper organizations, contour saliency is less pronounced (Figs. 3c and 4c). While PCs are thought to indicate higher-order processing due to delayed response [13, 28], this is likely due to the nature of the stimuli. Studies reveal IODs show cross-orientation suppression under complex stimuli [12], unlike PCs with broader tuning. The SESNN model illustrates a preference for complex stimuli in PCs and simple stimuli in IODs, with activity propagating from PCs to IODs upon encountering complex contours (Fig. 4d and e). PCs as geometric saliency detectors The SESNN model reveals PCs have broader orientation tuning and less selectivity for complex contours, unlike IODs, which show sharper tuning and crossorientation suppression, preferring simpler edges (x = 3 in Fig. 4e) [12, 13, 19, 58, 65, 66, 67]. PCs’ excitation leads to reduced cross-orientation suppression. With binary input, PCs correlate more positively with contour complexity than IODs (Figs. 3b and 4b), making them more salient in processing visual stimuli. This differs from rodents with salt-and-pepper organizations that lack distinct contour complexity saliency (Figs. 3c and 4c). Prior studies [12, 13, 28] suggest PCs have delayed response latency, indicative of higher-order processing. This arises from using drifting grating stimuli that activate IODs more readily. Koch et al. [12] note that IODs show cross-orientation suppression under complex stimuli, narrowing their tuning, unlike PCs. However, these studies omit temporal neural data within pinwheel structures. The SESNN model supports physiological findings that IODs and PCs favor single and complex orientation stimuli, respectively. 5 Conclusion and limitations The advantages of pinwheel structures in visual representation and encoding are not fully understood. To address this, we develop a two-dimensional SESNN model that incorporates Hebbian-like plasticity and empirical morphological data. This model evolves to function as localized, bandpass filters, enhancing its responsiveness to a range of orientations and complex spatial textures in natural images. Our findings reveal that neurons within pinwheel structures respond more effectively to these textures, with stronger and quicker reactions than those in salt-and-pepper configurations. Specifically, PCs act as first-order stage processors with heightened sensitivity and reduced response latency to intricate contours, while IODs function as second-stage processors, refining edge representation for greater clarity. This advanced processing capability of pinwheel structures, particularly in detecting spatial contour saliency, not only deepens our understanding of visual processing in higher mammals but may also inform new strategies for visual saliency algorithms in computational models. Using sliding windows, local entropy assesses variation and complexity in spatial distributions by capturing local intensity changes, indirectly reflecting geometric complexity through edges, corners, and patterns. Since this method cannot directly measure geometric shapes, we verify the use of the Ramer-Douglas-Peucker algorithm to approximate and directly measure geometric structures (refer to section A.5) [68]. This algorithm simplifies shape contours by reducing vertices while preserving the overall form. The resulting polygon will allow us to calculate the distribution of edge lengths and angles, with geometric entropy defined as the sum of these entropy values. In future studies, we will utilize the Ramer-Douglas-Peucker algorithm to enhance our geometric analysis by identifying 9 and measuring the complexity of specific structural features, such as junctions, sharp corners, and textures, which are essential in complex visual scenes. 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Concurrently, lifetime average values are also calculated using EMAs, which are crucial for maintaining homeostatic stability. This method helps stabilize the neural network by adjusting neuronal properties or synaptic strengths to sustain consistent activity levels over time. xj(t) = (1 −ζ)xj(t −1) + ζ · zj(t), (9) where ζ = 1 −e−1 10 , indicating that the 10 ms is a temporal window of the moving average weighted with exponential decay. The initialization of xj is 0. The exponential moving average is calculated dynamically and updated along with synaptic weights. ⟨xj⟩:= (1 −ξ) · ⟨xj⟩+ ξ · xj, (10) where ξ = 1 −e−1. It is dynamically updated to ensure the sum of the weights remains constant over time. A.2 Detailed parameters and connectivity settings for the model Detailed neural dynamics: The feed-forward connection, labeled FF(image→E) ij , links pixel Xj of the whitened image patch to excitatory (E-) neuron i. W (K∗→K) ij signifies the synaptic weight from neuron j of neuron class K∗to neuron i of neuron class K, with its sign determined by the connection type, described as β(K∗→K) ij (the neuron receives E-connections, set as +1; conversely, the neuron receives inhibitory (I-) connections, the sign is set as -1). z(K∗) j (t) indicates the spike output of neuron j at time t . Upon reaching the spike threshold θ (initialized as 2), a spike is emitted, z(K∗) j (t) is set to 1, then the membrane potential is reset to 0 mV, remaining so until the refractory period (3 ms) concludes. Within primary visual cortex (V1), homeostatic plasticity [34, 55] ensures neural activity stability by dynamically adjusting the firing threshold θ. This adjustment is based on the deviation of the current firing rate pi (t) from the target rates p(K) i (p(E) = 2, p(I) = 4), as outlined in Eq. 6 [55]. We assign τ (E) = 10 ms for E-neurons and τ (I) = 5 ms for I-neurons. To enhance computational efficiency, we set the time step to 1 ms. Hyperparameters: For the synaptic plasticity, learning rates are ηFF = 0.2 (image to E-neurons), ηEE = 0.01 (E- to E-neurons), ηEI = 0.7 (I- to E-neurons), ηII = 1.5 (I- to I-neurons), and ηIE = 0.7 (E- to I-neurons), while the neural connectivity parameters are αmax,E = 1.0 (E- max weight), αmax,I = 0.5 (I- max weight), σEE = 3.5 (E-E coupling range), σEI = 2.9 (E-I coupling range), σIE = 2.6 (I-E coupling range), and σII = 2.1 (I-I coupling range). Neural connectivity within 2D cortical area: E- and I- neurons are arranged symmetrically on a two-dimensional lattice, as illustrated in Fig. 5b. Periodic boundary conditions are employed to mimic the large number of neurons in the actual V1 cortical surface. Specifically, neurons at the boundary are connected to neurons at corresponding symmetric positions on the opposite boundary. The initial connection weights between neurons are modeled by a Gaussian function of their distance (see Fig. 5c), which can be expressed as: W K∗→K 0 (i, j) = αK∗× exp −d (i, j)2 2σK∗2 ! . (11) In this equation, d(i, j) represents the Euclidean distance from neuron i to neuron j in a grid, α determines the maximum connection weight, which is set to αEE = 1, αEI = 1, αIE = 0.5, αII = 0.5, and σ governs the rate at which the weight decays with distance. The synaptic types predominantly determine the parameters for this connection weight distribution function. To accurately replicate the neuronal architecture of V1 in macaques. The connectivity radiuses, denoted by σ, are set to σEE = 3.5, σEI = 2.9, σIE = 2.6, σII = 2.1. These values are based on anatomical data indicating that the axon length scales of E- and I-neurons are approximately 200 µm and 100 µm, respectively, while the dendrite length scales are around 150 µm for E-neurons and 75 µm for I-neurons in the V1 [62, 63, 64]. We prune any connection strengths below a threshold of 0.01 to maintain computational efficiency and biological plausibility. 16 A.3 Anatomical data integration Neural connection data The experimental subjects include six adult cats with unknown genders, with data sourced from research by Armen Stepanyants et al.[63]; and eight macaques, aged 5-11 years, including six males and two females, with data sourced from research by Joseph Amatrudo et al.[64]. Neuronal synaptic plasticity The subjects are rats aged 14-16 days, with unknown gender and quantity, with data sourced from research by Holmgren et al.[57]; transgenic mice, with unknown quantity and gender, with data sourced from research by Hofer et al. [47]. Retinal-V1 topological projection data Receptive field (RF) data: V1 neuron counts for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels respectively come from Tehovnik et al. [69] (subjects: 3 macaques, unknown gender and age), Scholl et al. [50] (subjects: cats, unknown gender and age), Veit et al.[70] (subjects: 9 male and 7 female tree shrews, aged 3-8 years), Huberman et al.[71] (subjects: 8 ferrets, unknown gender and age), Niell et al.[53] (subjects: mice, aged 2–6 months, unknown gender), Foik et al.[72](subjects: 21 rats, unknown gender and age), and Hall et al.[73] (subjects: 17 gray squirrels, unknown gender and age). V1 neuron density: Neuron density data for macaques, cats, mice, rats, and gray squirrels come from Srinivasana et al.[52] (subjects: unknown gender and age); tree shrew, ferret, and gray squirrel density data respectively come from Weigand et al.[74]. Cortical magnification factor Cortical magnification factor (CMF) data for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels are sourced from Tehovnik et al.[69] (subjects: 3 macaques, unknown gender and age), Veit et al.[70](subjects: cats, unknown gender and age), Bosking et al.[48] (subjects: tree shrews, unknown gender and age), Rockland et al. [75] (subjects: 9 ferrets, female, unknown age), Beest et al.[54] (subjects: 28 mice, 11 males and 17 females, ages 2-14 months), Keller et al.[76] (subjects: male rats, age 3 months), and Hall et al.[73] (subjects: 17 gray squirrels, unknown gender and age). Additionally, the anatomical data concerning inter-ocular distances are obtained from Najafian et al. [7]. A.4 Unveiling species-specific factors distinguishing pinwheels and salt-and-peppers A.4.1 Anatomical data suggests RFs density underlying V1 organizations Table 3: Comparative anatomical data of the retina and V1 across species. a. Species (mean) b. Retina (mm2) c. V1 size (mm2) d. V1 neurons density (neurons/mm2) e. V1 RF size in area centralis (deg) f. RFD ((c) × (d)/(b)) (RFs/mm2) Macaque 636[6] 1,090[33] 243,000[33] 0.2[52] 416,462.26 Cat 510[6] 380[6, 33] 99,200[33] 1.0[50] 73,913.73 Tree shrew 122[6, 77] 73[6, 33] 192,800[74] 2.0[70] 115,363.93 Ferret 83[6, 75] 78[33] 95,813[74] 3.0[71] 90,041.13 Mouse 15[6] 2.5[33] 86,600[33] 4.0[53] 14,433.33 Rat 52[6, 78] 7.1[33] 90,800[33] 3.0[72] 12,397.69 Gray squirrel 205[6] 32[6] 84,213[74] 2.0[73] 13,145.44 We analyzed anatomical data from seven species, including primates (e.g., macaques) and nonprimates (e.g., mice, rats, cats, tree shrews, gray squirrels, and ferrets), as detailed in Table 3. We first find that V1 RFs density (RFD) (ρRF) acts as a linear classifier (y = 4.42 × 104x), effectively distinguishing species with pinwheel structures from those with salt-and-pepper organizations. In this classifier, species like macaques, cats, tree shrews, and ferrets, which have higher RFD, are 17 RF density ρRF (RFs/mm2) RFs density classifier 106 105 104 Species Mouse Gray squirrel Rat Cat Tree shrew Macaque Ferret Salt-and-pepper organizations Pinwheel structures a y = 4.42×104 Retina size (mm2) Species classified by V1 neurons-retina size ratio Number of V1 neuron 109 108 107 106 105 101 102 103 Salt-and-pepper organizations Pinwheel structures b y = 4.42×104x Figure 6: A linear classifier based on RFD (y = 4.42 × 104x) effectively differentiates species with salt-and-pepper organizations (rats, mice, gray squirrels) from those with pinwheel structures (macaques, ferrets, cats, tree shrews). a. This classifier reflects variations in V1 organizations across species. b. A plot categorizing species by the ratio of V1 neuron number to retina size acts as a divider, implying a critical ratio for the formation of pinwheel structures. associated with pinwheel structures (light red area in Fig. 6) and exceed the classification threshold. In contrast, species with lower RFD, such as mice, rats, and gray squirrels, are linked to salt-andpepper organizations (light blue area in Fig. 6). Thus, V1 RFD serves as a predictive metric for V1 organizational patterns across species. The ρRF is calculated as follows: ρRF = n′ s′r ∝ n [(sRF −ε) (√n −1) + sRF]2 , (12) where n′ denotes the total number of neurons in V1, s′ r indicates the retinal surface area. We have n′ = sV1 × ρV1, where sV1 corresponds to the V1 2D surface area, and ρV1 signifies the neuronal density within V1. The variable ε quantifies the degree of visual input overlap among adjacent neurons, n denotes the total number of neurons, and sRF represents the RF size in the self-evolving spiking neural network (SESNN). Referring to Eq. 12 and anatomical data (Table 3), the overlap ε positively correlates to V1 RFD ρRF and is a main factor influencing V1 RFD. We discuss the overlap as the variable of V1 organizations in the main text. A.4.2 SESNN reveals neuronal connection range influencing V1 clusters The anatomical data in Table 3d for seven species show variability in V1 neuronal density (ρV1), which influences inter-neuronal spacing and connection strength. We explore how V1 cortical orientation patterns form by adjusting the lateral connection range, impacting axon reach among E- and I-neurons, as depicted in Fig. 7. We modulate axonal arborization through parameter σ to adjust the connection range, allowing us to simulate neuronal connections in areas with varying densities. This setup enables the SESNN model to predict changes in cortical patterns (Fig. 7). Our observations indicate that increasing axon lengths, thereby extending the connection range, enlarges hypercolumn sizes within pinwheel structures (Fig. 7d), reduces the overall number of pinwheels (Fig. 7b), and increases the nearest-neighbor pinwheel distance (NNPD) (Fig. 7c). These findings underscore the critical role of neural synaptic connection range in organizing orientation maps. A.5 Relationship between maximum values of local pixel entropy and local geometrical entropy for various shapes To address the limitations of using local pixel entropy (LPE) with sliding windows alone to capture complex geometric properties, we conduct a new analysis comparing the maximum values of LPE with local geometrical entropy (LGE) across various shapes. These shapes include lines, angles, and junctions (L-, T-, X-junctions), as well as jagged edges. Both LPE and LGE values were normalized to the range [0,1] for consistency. Let P = {v1, v2, . . . , vn} be a polygon with vertices vi = (xi, yi), where i = 1, 2, . . . , n. The edges of the polygon are the line segments between consecutive vertices, denoted as ei = ∥vi+1 −vi∥, 18 a 1 mm σ1 1 mm σ2 σ2 = 2.9 σ1 = 1.7 b Number of pinwheels 1.7 2.0 2.3 2.6 2.9 3.2 3.5 0 50 100 150 200 Neuronal connection range (σ) c 1.7 2.0 2.3 2.6 2.9 3.2 3.5 0.1 0.15 0.2 0.25 0.3 0.35 NNPD (mm) Neuronal connection range (σ) d Neuronal connection range (σ) 1.7 2.0 2.3 2.6 2.9 3.2 3.5 0.4 0.6 0.8 1 1.2 Hypercolumn size (mm) Δσ = σ2 -σ1 Increase the connection range Figure 7: Neuronal connection range within V1 contributes to the formation of pinwheel structures. a. Modifying the synaptic connection range reshapes the dimensions of pinwheel structures. b-d. The relationship between the synaptic connection range (σ) and the number of pinwheels, NNPD (mm), and hypercolumn size (mm). The scale bar: 1 mm in V1 cortical surface. Color scheme: orientation preference. Lines: mean. Shaded area: SD. where ∥· ∥represents the Euclidean distance. The angle θi between two consecutive edges ei and ei+1 can be computed using the dot product: θi = cos−1 ei · ei+1 ∥ei∥∥ei+1∥ . (13) With the set of edge lengths {e1, e2, . . . , en} and angles {θ1, θ2, . . . , θn}, we calculate the entropy for both distributions. The entropy H of a discrete distribution X with probability mass function p(x) is given by: H(X) = − X x∈X p(x) log p(x). (14) For the edge lengths and angles, the probability mass function is estimated by normalizing the frequency of occurrence of each unique edge length and angle in the polygon: H(Lengths) = − n X i=1 p(ei) log p(ei), (15) H(Angles) = − n X i=1 p(θi) log p(θi). (16) To enhance the sensitivity of geometrical entropy to structural complexity, particularly in differentiating shapes that have similar edge lengths and angles but different structural arrangements, we introduce a scaling factor based on the logarithm of the number of vertices n. The defined geometrical entropy (GE) with the scaling factor is thus defined as: GE = (H(Lengths) + H(Angles)) × log(n). (17) 19 This modification allows GE to capture additional complexity arising from intersections and the global arrangement of vertices, providing a more comprehensive assessment of the shape’s structural intricacies. Our results, summarized in Table 4, show that while LPE can reflect the complexity of certain patterns, it does not fully capture the geometric variations seen in more intricate shapes. For instance, the LPE values for line structures remain relatively low compared to those for jagged edges, which have the highest LPE and LGE values due to their high structural complexity. This comparison highlights the added value of incorporating LGE to better characterize local geometric structures, providing a more nuanced measure of complexity that includes both intensity distribution and spatial organization. Table 4: Relationship between maximum values of LPE and LGE for various shapes. Both metrics are normalized to the range [0,1]. Various shapes Max local pixel entropy Max local geometrical entropy Line 1 0.56 0.43 Line 2 0.56 0.43 Angle 1 0.81 0.87 Angle 2 0.79 0.86 Angle 3 0.77 0.87 L-junction 0.78 0.74 T-junction 0.78 0.64 X-junction 0.78 0.84 Jagged edges 1.00 1.00 A.6 Pinwheel centers response to different orientation bandwidths Probability distribution of preferred adjusted acute angles in PCs 20 30 40 50 60 70 80 90 Adjusted acute angle (degrees) 0 0.1 0.2 0.3 0.4 0.5 Probability a b Figure 8: PCs in V1 prefer orientations and ablation study. a. Probability distribution of preferred acute angles in PCs. b. Ablation study on normalized complexity across response onset latencies. Data: mean ± SD. Understanding the tuning of pinwheel centers (PCs) in V1 to edges, corners, and junctions is essential. In Fig. 4e, we show that PCs exhibit broader orientation tuning curves than IODs when using star-like patterns as stimuli, potentially enabling the detection of T-junctions and corners, as demonstrated by Li et al. [13] and Koch et al. [12]. We further examine the distribution of PCs’ tuning curves using gratings as inputs, specifically analyzing acute angles formed by the primary and secondary peaks (Fig. 8a). This analysis reveals that PCs are more frequently associated with larger acute angles, closer to orthogonal (90°), suggesting a preference for orthogonal junctions. However, this result does not differentiate between L- and T- junctions based solely on angle. We propose that such high-order feature extraction be deferred to higher visual cortices, like V2 and V4, which are involved in texture detection, as noted by Wang et al. [79] and Roe et al. [80]. A.7 Ablation study We present a mechanism of multiple orientation tuning that is essential for processing complexity. Our analysis of PCs’ preferred acute angles (Fig. 8a) suggests that their broad tuning enables the detection 20 of complex junctions, such as T- and L-junctions, likely due to variations in local connectivity within and between iso-orientation domains. To test this, we conduct an ablation study by disrupting local connectivity and shuffling the spatial arrangement of orientation-tuned RFs in the pinwheel orientation map, while keeping other properties constant (Fig. 8b). The control group (red) maintains higher complexity over time, whereas shuffling connections—especially both feed-forward and lateral—resulted in a decline in complexity. This highlights the importance of structured connectivity in preserving complex neural responses in V1 and supports the conclusion that structured connectivity underlies enhanced saliency detection by pinwheels. A.8 Computing infrastructure Table 5: Computing infrastructure CPU Intel® Xeon® Gold 6348 CPU @ 2.60GHz GPU A100 Memory 512 GB Operating system Ubuntu 20.04.6 LTS Simulation platform MATLAB R2023a and Python 3.9 The simulations and analyses in this study are performed on a high-performance computing infrastructure to ensure efficient processing of large datasets and complex models. The system is powered by an Intel® Xeon® Gold 6348 CPU running at 2.60 GHz and an NVIDIA A100 GPU, providing robust computational power for intensive tasks. The system includes 512 GB of memory, which supports handling memory-intensive applications and large-scale simulations. The operating system used is Ubuntu 20.04.6 LTS, known for its stability and compatibility with scientific software. 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4,437	NeuralFuse: Learning to Recover the Accuracy of Access-Limited Neural Network Inference in Low-Voltage Regimes Hao-Lun Sun1, Lei Hsiung2, Nandhini Chandramoorthy3, Pin-Yu Chen3, Tsung-Yi Ho4 1 National Tsing Hua University 2 Dartmouth College 3 IBM Research 4 The Chinese University of Hong Kong s109062594@m109.nthu.edu.tw lei.hsiung.gr@dartmouth.edu {pin-yu.chen, nandhini.chandramoorthy}@ibm.com tyho@cse.cuhk.edu.hk Abstract Deep neural networks (DNNs) have become ubiquitous in machine learning, but their energy consumption remains problematically high. An effective strategy for reducing such consumption is supply-voltage reduction, but if done too aggressively, it can lead to accuracy degradation. This is due to random bit-flips in static random access memory (SRAM), where model parameters are stored. To address this challenge, we have developed NeuralFuse, a novel add-on module that handles the energy-accuracy tradeoff in low-voltage regimes by learning input transformations and using them to generate error-resistant data representations, thereby protecting DNN accuracy in both nominal and low-voltage scenarios. As well as being easy to implement, NeuralFuse can be readily applied to DNNs with limited access, such cloud-based APIs that are accessed remotely or non-configurable hardware. Our experimental results demonstrate that, at a 1% bit-error rate, NeuralFuse can reduce SRAM access energy by up to 24% while recovering accuracy by up to 57%. To the best of our knowledge, this is the first approach to addressing low-voltage-induced bit errors that requires no model retraining. 1 Introduction Energy-efficient computing is of primary importance to the effective deployment of deep neural networks (DNNs), particularly in edge devices and in on-chip AI systems. Increasing DNN computation’s energy efficiency and lowering its carbon footprint require iterative efforts from both chip designers and algorithm developers. Processors with specialized hardware accelerators for AI computing, capable of providing orders of magnitude better performance and energy efficiency for AI computation, are now ubiquitous. However, alongside reduced precision/quantization and architectural optimizations, endowing such systems with the capacity for low-voltage operation is a powerful lever for reducing their power consumption. The computer engineering literature contains ample evidence of the effects of undervolting and low-voltage operation on accelerator memories that store weights and activations during computation. Aggressive scaling-down of static random access memory’s (SRAM’s) supply voltage to below the rated value saves power, thanks to the quadratic dependence of dynamic power on voltage. Crucially, Project Page: https://trustsafeai-neuralfuse.static.hf.space Code: https://github.com/IBM/NeuralFuse 38th Conference on Neural Information Processing Systems (NeurIPS 2024). NeuralFuse NeuralFuse Disabled Nominal Volt 92.6% 89.8% 38.9% 87.8% Low Volt Bit Errors Input data 𝑿= {𝑥1,…,𝑥𝑛} On-chip Model Inference (a) The pipeline of the NeuralFuse framework at inference. 18 22 26 30 Energy Saving Percentage (%) 50 70 90 Perturbed Accuracy (%) Acc. 20.6% Sav. 29.1% Acc. 48.8% Sav. 19.0% Acc. 29.9% Sav. 27.5% Acc. 49.6% Sav. 21.2% Acc. 29.8% Sav. 28.9% Acc. 45.6% Sav. 24.0% ResNet18 (Base Model) 1% BER Energy/accuracy Tradeoff of CIFAR-10 Pre-trained ResNet18 NeuralFuse Architecture ConvS ConvL DeConvS DeConvL UNetS UNetL NeuralFuse Architecture ConvS ConvL DeConvS DeConvL UNetS UNetL (b) Energy/accuracy tradeoff example. Figure 1: (a) At inference, NeuralFuse transforms input samples x into robust data representations. The nominal voltage allows models to work as expected, whereas at low voltage, one would encounter bit errors (e.g., 1%) that cause incorrect inferences. The percentages reflect the accuracy of a CIFAR10 pre-trained ResNet18 with and without NeuralFuse in both those voltage cases. (b) On the same base model (ResNet18), we illustrate the energy/accuracy tradeoff of six NeuralFuse implementations. The x-axis represents the percentage reduction in dynamic-memory access energy at low-voltage settings (base model protected by NeuralFuse), as compared to the bit-error-free (nominal) voltage. The y-axis represents the perturbed accuracy (evaluated at low voltage) with a 1% bit-error rate. however, it also leads to an exponential increase in bit failures. Memory bit flips cause errors in the stored weight and activation values [Chandramoorthy et al., 2019, Ganapathy et al., 2017], leading to catastrophic accuracy loss. A recent wave of research has proposed numerous techniques for allowing low-voltage operation of DNN accelerators while preserving their accuracy. Most of these have been either hardware-based error-mitigation techniques or error-aware robust training of DNN models. On-chip error mitigation methods have significant performance and power overheads [Chandramoorthy et al., 2019, Reagen et al., 2016]. On the other hand, some have proposed to generate models that are robust to bit errors via a specific learning algorithm [Kim et al., 2018, Koppula et al., 2019, Stutz et al., 2021], thereby eliminating the need for on-chip error mitigation. However, error-aware robust training to find the optimal set of robust parameters for each model is time- and energy-intensive and may not be possible in all access-limited settings. In this paper, therefore, we propose a novel model-agnostic approach: NeuralFuse. This proof-ofconcept machine-learning module offers trainable input transformation parameterized by a relatively small DNN; and, by enhancing input’s robustness, it mitigates bit errors caused by very low-voltage operation, thus serving the wider goal of more accurate inferencing. The pipeline of NeuralFuse is illustrated in Figure 1. To protect the deployed models from making wrong predictions under low-power conditions, NeuralFuse accepts scenarios under access-limited neural networks (e.g., non-configurable hardware or cloud-based APIs). Specifically, we consider two access-limited scenarios that are common in the real world: 1) relaxed access, in which ‘black box’ model details are unknown, but backpropagation through those models is possible; and 2) restricted access, in which the model details are unknown and backpropagation is disallowed. To enable it to deal with relaxed access, we trained NeuralFuse via backpropagation, and for restricted-access cases, we trained it on a white-box surrogate model. To the best of our knowledge, this is the first study that leverages a learning-based method to address random bit errors as a means of recovering accuracy in low-voltage and access-limited settings. We summarize our main contributions as follows: • We propose NeuralFuse, a novel learning-based input-transformation module aimed at enhancing the accuracy of DNNs that are subject to random bit errors caused by very low voltage operation. NeuralFuse is model-agnostic, i.e., operates on a plug-and-play basis at the data-input stage and does not require any re-training of deployed DNN models. • We explore two practical limited-access scenarios for neural-network inference: relaxed access and restricted access. In the former setting, we use gradient-based methods for module training. In the latter one, we use a white-box surrogate model for training, which is highly transferable to other types of DNN architecture. • We report the results of an extensive program of experiments with various combinations of DNN models (ResNet18, ResNet50, VGG11, VGG16, and VGG19), datasets (CIFAR-10, CIFAR-100, 2 GTSRB, and ImageNet-10), and NeuralFuse implementations of different architectures and sizes. These show that NeuralFuse can consistently increase the perturbed accuracy (accuracy evaluated under random bit errors in weights) by up to 57%, while simultaneously saving up to 24% of the energy normally required for SRAM access, based on our realistic characterization of bit-cell failures for a given memory array in a low-voltage regime inducing a 0.5%/1% bit-error rate. • We demonstrate NeuralFuse’s transferability (i.e., adaptability to unseen base models), versatility (i.e., ability to recover low-precision quantization loss), and competitiveness (i.e., state-of-the-art performance) in various scenarios, establishing it as a promising proof-of-concept for energyefficient, resilient DNN inference. 2 Related Work and Background Software-based Energy-saving Strategies. Various recent studies have proposed software-based methods of reducing computing’s energy consumption. For instance, quantization techniques have been reported to reduce the precision required for storing model weights, and thus to decrease total memory storage [Gong et al., 2014, Rastegari et al., 2016, Wu et al., 2016]. On the other hand, Yang et al. [2017] - who proposed energy-aware pruning on each layer and fine-tuning of weights to maximize final accuracy - suggested several ways to reduce DNNs’ energy consumption. For example, they devised the ECC framework, which compresses DNN models to meet a given energy constraint [Yang et al., 2019a], and a method of compressing such models via joint pruning and quantization [Yang et al., 2020]. It is also feasible, during DNN training, to treat energy constraints as an optimization problem and thereby reduce energy consumption while maximizing training accuracy [Yang et al., 2019b]. However, unlike ours, all these methods imply changing either model architectures or model weights. Hardware-based Energy-saving Strategies. Prior studies have also explored ways of improving energy efficiency via specially designed hardware. Several of them have focused on the undervolting of DNN accelerators and proposed methods to maintain accuracy in the presence of bit errors. For instance, Reagen et al. [2016] proposed an SRAM fault-mitigation technique that rounds faulty weights to zero to avoid degradation of prediction accuracy. Srinivasan et al. [2016] recommended storing sensitive MSBs (most significant bits) in robust SRAM cells to preserve accuracy. Chandramoorthy et al. [2019] proposed dynamic supply-voltage boosting to improve the resilience of memory-access operations; and the learning-based approach proposed by Stutz et al. [2021] aims to find models that are robust to bit errors. The latter paper discusses several techniques for improving such robustness, notably quantization, weight-clipping, random bit-error training, and adversarial bit-error training. Its authors concluded from their experiments that a combination of quantization, weight-clipping, and adversarial bit-error training will yield excellent performance. However, they also admitted that the relevant training process was sensitive to hyperparameter settings, and hence, it might come with a challenging training procedure. However, we suggest that all the methods mentioned above are difficult to implement and/or unsuitable for use in real-world access-limited settings. For example, the weights of DNN models packed on embedded systems may not be configurable or updatable, making model retraining (e.g., Stutz et al. [2021]) non-viable in that scenario. Moreover, DNN training is already a tedious and time-consuming task, so adding error-aware training to it may further increase its complexity and, in particular, make hyperparameter searches more challenging. Özdenizci and Legenstein [2022] also reported that error-aware training was ineffective for large DNNs with millions of bits. NeuralFuse obviates the need for model retraining via an add-on trainable input-transformation function parameterized by a relatively small secondary DNN. SRAM Bit Errors in DNNs. Low voltage-induced memory bit-cell failures can cause bit-flips from 0 to 1 and vice versa. In practice, SRAM bit errors increase exponentially when the supply voltage is scaled below Vmin, i.e., the minimum voltage required to avoid them. This phenomenon has been studied extensively in the prior literature, including work by Chandramoorthy et al. [2019] and Ganapathy et al. [2017]. The specific increases in bit errors as voltage scales down, in the case of an SRAM array of 512 × 64 bits with a 14nm technology node, is illustrated in Figure 2. The corresponding dynamic energy per SRAM read access, measured at each voltage at a constant frequency, is shown on the right-hand side of the figure. In this example, accessing the SRAM at 3 Figure 2: The bit-error rates (left) and dynamic energy per memory access versus voltage for static random access memory arrays (right) as reported by Chandramoorthy et al. [2019]. The x-axis shows voltages normalized with respect to the minimum bit error-free voltage (Vmin). 0.83Vmin leads to a 1% bit-error rate, and at the same time, dynamic energy per access is reduced by approximately 30%. This can lead to DNNs making inaccurate inferences, particularly when bit-flips occur at the MSBs. However, improving robustness to bit errors can allow us to lower Vmin and exploit the resulting energy savings. It has been observed that bit-cell failures for a given memory array are randomly distributed and independent of each other. That is, the spatial distribution of bit-flips can be assumed to be random, as it generally differs from one array to another, within as well as between chips. Below, following Chandramoorthy et al. [2019], we model bit errors in a memory array of a given size by generating a random distribution of such errors with equal likelihood of 0-to-1 and 1-to-0 bit-flipping. More specifically, we assume that the model weights are quantized to 8-bit precision (i.e., from 32-bit floats to 8-bit integers), and generate perturbed models by injecting our randomly distributed bit errors into the two’s complement representation of weights. For more implementation details, please refer to Section 4.1. 3 NeuralFuse: Framework and Algorithms 3.1 Error-Resistant Input Transformation As illustrated in Figure 1, we propose a novel trainable input-transformation module, NeuralFuse, parametrized by a relatively small DNN, to mitigate the accuracy-energy tradeoff for model inference and thus overcome the drawback of performance degradation in low-voltage regimes. A specially designed loss function and training scheme are used to derive NeuralFuse and apply it to the input data such that the transformed inputs will become robust to low voltage-induced bit errors. Consider the input x sampled from the data distribution X and a set of models Mp with p% random bit errors on weights (i.e., perturbed models). When it is not manifesting any bit errors (i.e., at normal-voltage settings), the perturbed model operates as a nominal deterministic one, denoted by M0. NeuralFuse aims to ensure that a model Mp ∈Mp can make correct inferences on the transformed inputs while also delivering consistent results in its M0 state. To adapt to various data characteristics, NeuralFuse – designated as F in Eq. (1), below – is designed to be input-aware. This characteristic can be formally defined as F(x) = clip[−1,1] x + G(x) , (1) where G(x) is a “generator” (i.e., an input-transformation function) that can generate a perturbation based on input x. As transformed by NeuralFuse, i.e., as F(x), that input is passed to the deployed model (M0 or Mp) for final inference. Without loss of generality, we assume the transformed input lies within a scaled input range F(·) ∈[−1, 1]d, where d is the (flattened) dimension of x. 3.2 Training Objective and Optimizer To train our generator G(·), which ought to be able to ensure the correctness of both the perturbed model Mp and the clean model M0, we parameterized it with a neural network and apply our training objective function arg max WG log PM0(y\|F(x; WG)) + λ · EMp∼Mp[log PMp(y\|F(x; WG))], (2) 4 where WG is the set of trainable parameters for G; y is the ground-truth label of x; PM denotes the likelihood of y as computed by a model M being given a transformed input F(x; WG); Mp is the distribution of the perturbed models inherited from the clean model M0, under a p% random bit-error rate; and λ is a hyperparameter that balances the importance of the nominal and perturbed models. The training objective function can be readily converted to a loss function (L) that evaluates crossentropy between the ground-truth label y and the prediction PM(y\|F(x; WG). That is, the total loss function can be calculated as LTotal = LM0 + λ · LMp. (3) In particular, optimizing the loss function requires evaluation of the impact of the loss term LMp on randomly perturbed models. Our training process is inspired by expectation over transformation (EOT) attacks [Athalye et al., 2018], which aim to produce robust adversarial examples that are simultaneously adversarial over the entire transformation distribution. Based on that idea, we propose a new optimizer for solving Eq. (3), which we call expectation over perturbed models (EOPM). EOPM-trained generators can generate error-resistant input transformations and mitigate inherent bit errors. However, it would be computationally impossible to enumerate all possible perturbed models with random bit errors, and the number of realizations for perturbed models is constrained by the memory size of the GPUs used for training. In practice, therefore, we only use N perturbed models per iteration to calculate empirical average loss, i.e., LMp ≈ LMp1 + · · · + LMpN N , (4) where N is the number of perturbed models {Mp1, · · · , MpN } that are simulated to calculate the loss caused by random bit errors. Therefore, the gradient used to update the generator can be calculated as follows: ∂LTotal ∂WG = ∂LM0 ∂WG + λ N ∂LMp1 ∂WG + · · · + ∂LMpN ∂WG . (5) Through our implementation, we found that stable performance could be delivered when N = 10, and that there was little to be gained by using a larger value. The results of our ablation study for different values of N can be found in Appendix E. 3.3 Training Algorithm Algorithm 1 in Appendix A summarizes NeuralFuse’s training steps. Briefly, this involves splitting the training data X into B mini-batches for training the generator in each epoch. For each mini-batch, we first feed these data into F(·) to obtain the transformed inputs. Also, we simulate N perturbed models using a p% random bit-error rate, denoted by Mp1, · · · , MpN , from Mp. Then, the transformed inputs are fed into those N perturbed models as well as into the clean model M0, and their respective losses and gradients are calculated. Finally, NeuralFuse parameters WG are updated based on the gradient obtained by EOPM. 4 Experiments 4.1 Experiment Setups Datasets. We evaluate NeuralFuse on four different datasets: CIFAR-10 [Krizhevsky and Hinton, 2009], CIFAR-100 [Krizhevsky and Hinton, 2009], the German Traffic Sign Recognition Benchmark (GTSRB) [Stallkamp et al., 2012], and ImageNet-10 [Deng et al., 2009]. CIFAR-10 consists of 10 classes, with 50,000 training images and 10,000 testing images in total. Similarly, CIFAR-100 consists of 100 classes, with 500 training images and 100 testing images in each. The GTSRB contains 43 classes with a total of 39,209 training images and 12,630 testing images. Similar to CIFAR-10 and CIFAR-100, we resize GTSRB into 32×32×3 in our experiment. For ImageNet-10, we chose the same ten categories as Huang et al. [2022], in which there are 13,000 training images and 500 test images cropped into 224×224×3. Due to space limitations, our CIFAR-100 results are presented in Appendices F and G. Base Models. We selected several common architectures for our base models: ResNet18, ResNet50 [He et al., 2016], VGG11, VGG16, and VGG19 [Simonyan and Zisserman, 2015]. To replicate the deployment of models on chips, all our based models were given quantization-aware training that followed Stutz et al. [2021]. 5 NeuralFuse Generators. The architecture of the NeuralFuse generator (G) ) is based on an encoderdecoder structure. We designed and compared three types of generators, namely convolution-based, deconvolution-based, and UNet-based. We also considered large(L)/small(S) network sizes for each type. Further details can be found below and in Appendix B. • Convolution-based (Conv). Conv uses convolution with MaxPool layers for its encoder and convolution with UpSample layers for its decoder. This architecture has previously been shown to be efficient and effective at generating input-aware backdoor triggers [Nguyen and Tran, 2020]. • Deconvolution-based (DeConv). DeConv uses convolution with MaxPool layers for its encoder and deconvolution layers for its decoder. We expected this modification both to enhance its performance and to reduce its energy consumption. • UNet-based (UNet). UNet uses convolution with MaxPool layers for its encoder, and deconvolution layers for its decoder. UNet is known for its robust performance in image segmentation [Ronneberger et al., 2015]. Energy-consumption Calculation. The energy consumption reported in Figure 1 is based on the product of the total number of SRAM memory accesses in a systolic array-based convolution neural network (CNN) accelerator and the dynamic energy per read access at a given voltage. Research by Chen et al. [2016] previously showed that energy consumption by SRAM buffers and arrays accounts for a high proportion of total system energy consumption. We assume that there are no bit errors on NeuralFuse, given its status as an add-on data preprocessing module whose functions could also be performed by a general-purpose core. In this work we assume it is implemented on the accelerator equipped with dynamic voltage scaling and therefore NeuralFuse computation is performed at nominal error-free voltage. We report a reduction in overall weight-memory energy consumption (i.e., NeuralFuse + Base Model under a p% bit-error rate) with respect to the unprotected base model in the regular-voltage mode (i.e., 0% bit-error rate and without NeuralFuse). To quantify memory accesses, we used the SCALE-SIM simulator [Samajdar et al., 2020], and our chosen configuration simulated an output-stationary dataflow and a 32×32 systolic array with 256KB of weight memory. We collected data on the dynamic energy per read access of the SRAM both at Vmin and at the voltage corresponding to a 1% bit-error rate (Vber ≈0.83Vmin) from Cadence ADE Spectre simulations, both at the same clock frequency. Relaxed and Restricted Access Settings. In the first of our experiments’ two scenarios, relaxed access, the base-model information was not entirely transparent, but allowed us to obtain gradients from the black-box model through backpropagation. Therefore, this scenario allowed direct training of NeuralFuse with the base model using EOPM. In the restricted-access scenario, on the other hand, only the inference function was allowed for the base model, and we therefore trained NeuralFuse using a white-box surrogate base model and then transfering the generator to the access-restricted model. Computing Resources. Our experiments were performed using eight Nvidia Tesla V100 GPUs and implemented with PyTorch. NeuralFuse was found to generally take 150 epochs to converge, and its training time was similar to that of the base model it incorporated. On both the CIFAR-10 and CIFAR-100 datasets, average training times were 17 hours (ResNet18), 50 hours (ResNet50), 9 hours (VGG11), 13 hours (VGG16), and 15 hours (VGG19). For GTSRB, the average training times were 9 hours (ResNet18), 27 hours (ResNet50), 5 hours (VGG11), 7 hours (VGG16), and 8 hours (VGG19); and for ImageNet-10, the average training times were 32 hours (ResNet18), 54 hours (ResNet50), 50 hours (VGG11), 90 hours (VGG16), and 102 hours (VGG19). 4.2 Performance Evaluation, Relaxed-access Scenario Our experimental results pertaining to the relaxed-access scenario are shown in Figure 3. The biterror rate (BER) due to low voltage was 1% in the cases of CIFAR-10 and GTSRB, and 0.5% for ImageNet-10. The BER of ImageNet-10 was lower than that of the other two because, being pretrained, it has more parameters than either of them. For each experiment, we sampled and evaluated N = 10 perturbed models (independent from training), and below, we report the means and standard deviations of their respective accuracies. Below, clean accuracy (CA) refers to a model’s accuracy measured at nominal voltage, and perturbed accuracy (PA) to its accuracy measured at low voltage. In the cases of CIFAR-10 and the GTSRB, we observed that large generators like ConvL and UNetL recovered PA considerably, i.e., in the range of 41% to 63% on ResNet18, VGG11, VGG16, 6 Nominal (w/o NeuralFuse) Low voltage (w/o NeuralFuse) Nominal (w/ NeuralFuse) Low voltage (w/ NeuralFuse) w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 Accuracy (%) 38 .9 89 88 89 82 86 84 .8 .2 .6 .9 .6 .4 87 59 88 68 84 68 .8 .5 .5 .8 .6 .8 CIFAR-10 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 26 .1 85 85 87 82 86 77 .5 .2 .4 .4 .2 .3 53 34 63 42 75 56 .2 .6 .3 .2 .5 .2 CIFAR-10 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 88.4 42 .2 89 84 89 85 87 85 .6 .9 .3 .6 .1 .5 87 66 87 68 83 72 .2 .3 .2 .2 .6 .7 CIFAR-10 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.3 35 .7 90 87 89 86 87 87 .1 .4 .7 .8 .4 .4 86 59 85 66 83 71 .0 .6 .5 .5 .4 .2 CIFAR-10 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.5 36 .0 89 87 86 86 86 86 .8 .3 .3 .5 .3 .3 77 52 78 58 82 66 .7 .7 .4 .2 .1 .4 CIFAR-10 pre-trained VGG19 (a) CIFAR-10, 1% Bit-error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 Accuracy (%) 36 .9 95 94 95 95 96 95 .7 .4 .6 .7 .2 .9 91 68 91 78 93 85 .1 .6 .3 .1 .8 .1 GTSRB pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.0 29 .5 95 94 94 93 94 94 .6 .8 .9 .0 .5 .7 71 50 71 56 80 64 .6 .5 .6 .4 .6 .7 GTSRB pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 91.9 34 .9 94 91 95 92 92 94 .8 .1 .0 .4 .2 .7 85 62 84 67 83 73 .7 .2 .6 .5 .2 .4 GTSRB pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.2 15 .1 96 94 96 93 95 94 .3 .1 .4 .8 .8 .3 72 39 72 50 78 63 .4 .8 .0 .9 .6 .3 GTSRB pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 36 .6 96 93 95 94 95 94 .0 .8 .4 .5 .4 .6 88 69 87 73 88 80 .3 .0 .2 .1 .2 .6 GTSRB pre-trained VGG19 (b) GTSRB, 1% Bit-error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.2 Accuracy (%) 72 .3 94 91 94 92 94 93 .0 .8 .0 .8 .0 .2 88 83 89 87 88 86 .0 .6 .2 .5 .1 .4 ImageNet-10 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 89.8 39 .4 92 91 93 93 92 92 .2 .8 .0 .2 .2 .4 80 65 79 70 80 73 .0 .0 .4 .9 .5 .6 ImageNet-10 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 91.6 47 .8 92 89 91 89 92 86 .0 .4 .0 .0 .4 .2 86 66 86 72 83 73 .1 .4 .0 .5 .0 .5 ImageNet-10 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 94.6 38 .4 90 90 91 90 90 86 .8 .2 .2 .0 .6 .4 77 60 77 62 81 72 .1 .2 .2 .3 .1 .3 ImageNet-10 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.4 37 .2 91 88 91 88 89 87 .4 .8 .0 .8 .4 .6 75 56 75 64 77 65 .5 .5 .9 .0 .9 .9 ImageNet-10 pre-trained VGG19 (c) ImageNet-10, 0.5% Bit-error Rate Figure 3: Relaxed-access scenario test accuracies (%) of various pre-trained models with and without NeuralFuse, compared at nominal voltage (0% bit-error rate) or low voltage (with specified bit-error rates). The results demonstrate that NeuralFuse consistently recovered perturbation accuracy. and VGG19. ResNet50’s recovery percentage was slightly worse than those of the other base models, but it nevertheless attained up to 51% recovery on the GTSRB. On the other hand, the recovery percentages achieved when we used small generators like DeConvS were worse than those of their larger counterparts. This could be explained by larger-sized networks’ better ability to learn error-resistant generators (though perhaps at the cost of higher energy consumption). In the case of ImageNet-10, using larger generators also yielded better PA performance recovery, further demonstrating NeuralFuse’s ability to work well with large input sizes and varied datasets. 4.3 Performance Evaluation, Restricted-access Scenario (Transferability) The experimental results of our restricted-access scenario are shown in Table 1. We adopted ResNet18 and VGG19 as our white-box surrogate source models for training the generators under a 1.5% biterror rate. We chose ConvL and UNetL as our generators because they performed best out of the six we tested (see Figure 3). From Table 1, we can see that transferring from a larger BER such as 1.5% can endow a smaller one (e.g., 1%) with strong resilience. The table also makes it clear that using VGG19 as a surrogate model with UNet-based generators like UNetL can yield better recovery performance than other model/generator combinations. On the other hand, we observed in some cases that if we transferred between source and target models of the same type (but with different BERs for training and testing), performance results could exceed those we had obtained during the original relaxed-access scenario. For instance, when we transferred VGG19 with UNetL under a 1.5% BER to VGG19 or VGG11 under a 0.5% BER, the resulting accuracies were 85.86% (as against 84.99% for original VGG19) and 84.81% (as against 82.42% for original VGG11). We conjecture that generators trained on relatively large BERs can cover the error patterns of smaller BERs, and even help improve the latter’s generalization. These findings indicate the considerable promise of access-limited base models in low-voltage settings to recover accuracy. 4.4 Energy/Accuracy Tradeoff We report total dynamic energy consumption as the total number of SRAM-access events multiplied by the dynamic energy of a single such event. Specifically, we used SCALE-SIM to calculate total 7 Table 1: Restricted-access scenario: Transfer results on CIFAR-10 with 1.5% bit-error rate SM TM BER CA PA ConvL (1.5%) UNetL (1.5%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 1% 92.6 38.9 ± 12.4 89.8 89.0 ± 0.5 50.1 85.8 85.2 ± 0.5 46.3 0.5% 70.1 ± 11.6 89.6 ± 0.2 19.5 85.7 ± 0.2 15.6 ResNet50 1% 92.6 26.1 ± 9.4 89.2 36.1 ± 18 10.0 84.4 38.9 ± 16 12.8 0.5% 61.0 ± 10.3 74.1 ± 10 13.1 72.7 ± 4.6 11.7 VGG11 1% 88.4 42.2 ± 11.6 86.3 59.2 ± 10 17.0 82.3 69.8 ± 7.5 27.6 0.5% 63.6 ± 9.3 78.9 ± 4.9 15.3 77.0 ± 4.0 13.4 VGG16 1% 90.3 35.7 ± 7.9 89.4 62.2 ± 18 26.5 84.7 68.9 ± 14 33.2 0.5% 66.6 ± 8.1 83.4 ± 5.5 16.8 80.5 ± 5.9 13.9 VGG19 1% 90.5 36.0 ± 12.0 89.8 49.9 ± 23 13.9 85.0 55.1 ± 17 19.1 0.5% 64.2 ± 12.4 81.8 ± 8.5 17.6 78.5 ± 6.8 14.3 VGG19 ResNet18 1% 92.6 38.9 ± 12.4 88.9 62.6 ± 13 23.7 85.0 72.3 ± 11 33.4 0.5% 70.1 ± 11.6 84.2 ± 7.2 14.1 82.1 ± 2.2 12.0 ResNet50 1% 92.6 26.1 ± 9.4 88.8 37.9 ± 18 11.8 85.2 46.7 ± 17 20.6 0.5% 61.0 ± 10.3 76.6 ± 7.8 15.6 78.3 ± 3.7 17.3 VGG11 1% 88.4 42.2 ± 11.6 88.9 76.0 ± 6.1 33.8 85.5 81.9 ± 3.9 39.7 0.5% 63.6 ± 9.3 85.9 ± 2.6 22.3 84.8 ± 0.5 21.2 VGG16 1% 90.3 35.7 ± 7.9 89.0 76.5 ± 9.0 40.8 85.9 79.2 ± 7.5 43.5 0.5% 66.6 ± 8.1 87.7 ± 0.7 21.1 84.7 ± 0.9 18.1 VGG19 1% 90.5 36.0 ± 12.0 89.1 80.2 ± 12 44.2 86.3 84.3 ± 1.2 48.3 0.5% 64.2 ± 12.4 88.8 ± 0.4 24.6 85.9 ± 0.3 21.7 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Table 2: Energy saving (%) by NeuralFuse for 30 combinations of base models and generators. ConvL ConvS DeConvL DeConvS UNetL UNetS ResNet18 19.0 29.1 21.2 27.5 24.0 28.9 ResNet50 25.4 29.9 26.4 29.2 27.7 29.9 VGG11 6.6 27.5 11.2 24.1 17.1 27.2 VGG16 17.1 28.9 19.7 27.0 23.0 28.7 VGG19 20.3 29.7 22.3 27.8 24.8 29.1 weight-memory access (TWMA), specifics of which can be found in Appendix C’s Table 6. In Table 2, below, we report the percentages of energy saved (ES) at voltages that yield a 1% bit-error rate for various base-model and generator combinations. The formula for computing ES is ES = EnergyNV− EnergyLV+EnergyNeuralFuse at NV EnergyNV × 100%, (6) where NV denotes nominal voltage regime, and LV, a low-voltage one. Our results indicate that when ResNet18 is utilized as a base model, NeuralFuse can recover model accuracy by 20 −49% while reducing energy use by 19 −29%. In Appendix C, we provide more results on the tradeoff between energy and accuracy of different NeuralFuse and base-model combinations. Overall, it would appear that using NeuralFuse can effectively restore model accuracy when SRAM encounters low-voltage-induced random bit errors. Runtime and Latency. On the other hand, runtime and its corresponding energy consumption may also affect overall energy savings. For instance, previous research has shown that multiply-andaccumulate (MAC) operations account for more than 99% of all operations in state-of-the-art CNNs, dominating processing runtime and energy consumption alike Yang et al. [2017]. Therefore, we also report the results of our MAC-based energy-consumption estimation in Appendix C, and of our latency analysis in Appendix D. Here, it should also be noted that an additional latency overhead is an inevitable tradeoff for reducing energy consumption in our scenarios. Although neither runtime nor latency is a major focus of this paper, future researchers could usefully design a lighter-weight version of the NeuralFuse module, or apply model-compression techniques to it, to reduce these two factors. 4.5 Model Size and NeuralFuse Efficiency To arrive at a full performance characterization of NeuralFuse, we analyzed the relationship between the final recovery achieved by each base model in combination with generators of varying parameter counts. For this purpose, we defined efficiency ratio as the recovery percentage in PA divided by NeuralFuse’s parameter count. Table 3 compares the efficiency ratios of all NeuralFuse generators 8 Table 3: The efficiency ratio for all NeuralFuse generators. Base Model BER NeuralFuse ConvL ConvS DeConvL DeConvS UNetL UNetS ResNet18 1% 67.5 182 76.6 190.7 94.5 245.9 0.5% 24.7 73.3 30.7 62.5 33.6 88.3 ResNet50 1% 37.4 75.1 57.4 102.7 102.3 248.4 0.5% 35.2 108.7 40.4 92.5 47.4 124.6 VGG11 1% 62.3 212.9 69.5 165.8 92.0 251.7 0.5% 32.3 96.3 35.8 77.2 38.9 100.7 VGG16 1% 69.6 211.2 76.9 196.5 98.8 292.9 0.5% 30.3 98.1 33.0 75.3 40.6 113 VGG19 1% 57.6 147.5 65.5 141.6 95.4 250.8 0.5% 33.0 91.0 37.5 70.2 43.1 106.4 Original Accuracy w/o NeuralFuse NeuralFuse (ConvL) NeuralFuse (UNetL) 8 7 6 5 4 3 2 Bit Quantization Number 95.5 Accuracy (%) 95 95 95 95 92 67 .5 .5 .4 .4 .6 .7 93 93 93 93 93 92 .4 .3 .3 .3 .3 .0 96 96 96 96 96 95 .2 .2 .2 .2 .1 .7 3 .8 3 .8 3 .8 GTSRB pre-trained ResNet18 (Nominal voltage, No Bit Error) 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 75 69 67 48 24 5 .2 .5 .2 .6 .6 .3 89 89 88 79 52 9 .5 .9 .9 .3 .1 .7 93 94 93 87 62 12 .5 .5 .5 .5 .2 .6 3 .8 3 .8 3 .8 GTSRB pre-trained ResNet18 (Low voltage, 0.5% BER) Figure 4: Reduced-precision accuracy trained on CIFAR-10. Those results show that UNet-based generators had better efficiency per million parameters than either convolution-based or deconvolution-based ones. 4.6 NeuralFuse’s Robustness to Reduced-precision Quantization Lastly, we also explored NeuralFuse’s robustness to low-precision quantization on model weights. Uniform quantization is the usual method for quantizing model weights [Gholami et al., 2022]. However, it is possible for it to cause an accuracy drop due to lack of precision. Given our aim of offering protection from bit errors, we hoped to understand whether NeuralFuse could also recover model-accuracy drops caused by this phenomenon. We therefore uniformly quantized the model weights to a lower bit precision and measured the resulting accuracy. Specifically, we applied symmetric uniform quantization to our base models with various numbers of bits to induce precision loss, and defined the quantized weight Wq (integer) as Wq = ⌊W s ⌉, where W denotes the original model weight (full precision), s = max \|W\| 2b−1−1 is the quantization scale parameter, and b is the precision (number of bits) used to quantize the models. Bit errors induced by low voltage operation as previously described, are also applied to low precision weights. We used the GTSRB pre-trained ResNet18 as our example in an evaluation of two NeuralFuse generators, i.e., ConvL and UNetL trained with 0.5% BER, and varied precision b from 8 bits to 2 bits (integer). The results, shown in Figure 4, indicated that when b > 3 bits, NeuralFuse could effectively recover accuracy in both the low-voltage and low-precision scenarios. When b = 3, while NeuralFuse could still handle the bit-error-free model (Fig. 4 top), it exhibited a limited ability to recover the random bit-error case (Fig. 4 bottom). We find these results encouraging, insofar as NeuralFuse was only trained on random bit errors, yet demonstrated high accuracy in dealing with unseen bit-quantization errors. Further experimental results derived from other base models and datasets can be found in Appendix H. 4.7 Extended Analysis Here, we would like to highlight some key findings from the additional results in the Appendices. In Appendix E, we compare NeuralFuse against a simple baseline of learning a universal input perturbation. We found that such baseline performed much worse than NeuralFuse at that task, validating the necessity of adopting input-aware transformation if the goal is to learn error-resistant data representations in low-voltage scenarios. In Appendix G, we report that ensemble training of white-box surrogate base models could further improve the transferability of NeuralFuse in restrictedaccess scenarios. Appendices K and L present visualization results of NeuralFuse’s data embeddings and transformed inputs. In Appendix J, we show that NeuralFuse can further recover the accuracy of a base model trained with adversarial weight perturbation in a low-voltage setting. 5 Conclusion In this paper, we have proposed NeuralFuse, the first non-intrusive post hoc module that protects model inference against bit errors induced by low voltage. NeuralFuse is particularly well suited to 9 practical machine-deployment cases in which access to the base model is limited or relaxed. The design of NeuralFuse includes a novel loss function and a new optimizer, EOPM, that enable it to handle simulated randomness in perturbed models. Our comprehensive experimental results and analysis show that NeuralFuse can recover test accuracy by up to 57% while simultaneously enjoying an up to 24% reduction in memory-access energy requirements. Moreover, NeuralFuse demonstrates high transferability to access-constrained models and high versatility, e.g., robustness to low-precision quantization. In short, NeuralFuse is a notable advancement in mitigating neural-network inference’s energy/accuracy tradeoff in low-voltage regimes, and points the way to greener future AI technology. Our future work will include extending this study to other neural-network architectures and modalities, such as transformer-based language models. Limitations. We acknowledge the challenge of achieving significant power savings without accuracy loss and view NeuralFuse as a foundational, proof-of-concept step toward this goal. Future research could enhance this approach by optimizing the pre-processing module to adapt to specific error characteristics of low-voltage SRAM or by integrating lightweight hardware modifications to further improve the energy-accuracy trade-off. Broader Impacts. We see no ethical or immediate negative societal consequence of our work, and it holds the potential for positive social impacts, from environmental benefits to improved access to technology and enhanced safety in critical applications. Acknowledgments and Disclosure of Funding We thank the anonymous reviewers for their insightful comments and valuable suggestions. The research described in this paper was conducted in the JC STEM Lab of Intelligent Design Automation, which is funded by The Hong Kong Jockey Club Charities Trust in support of Tsung-Yi Ho. References Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok. Synthesizing robust adversarial examples. 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Theoretically principled trade-off between robustness and accuracy. In International Conference on Machine Learning (ICML), pages 7472–7482. PMLR, 2019. 12 Appendix The appendix provides more implementation details for our method, experimental results on more datasets and settings, ablation studies, and qualitative analysis. The appendices cover the following: • Implementation Details: NeuralFuse Training Algorithm (Sec. A), NeuralFuse Generator (Sec. B), Energy/Accuracy Tradeoff (Sec. C), Latency Reports (Sec. D) • Experimental Results: Ablation Studies (Sec. E), Relaxed Access (Sec. F), Restricted Access (Sec. G), Reduced-precision Quantization (Sec. H), Adversarial Training (Sec. I), Adversarial Weight Perturbation (Sec. J) • Qualitative Studies: Data Embeddings Visualization (Sec. K), Transformed Inputs Visualization (Sec. L) A Training Algorithm of NeuralFuse Algorithm 1 Training steps for NeuralFuse Input: Base model M0; Generator G; Training data samples X; Distribution of the perturbed models Mp; Number of perturbed models N; Total training iterations T Output: Optimized parameters WG for the Generator G 1: for t = 0, ..., T −1 do 2: for all mini-batches {x, y}B b=1 ∼X do 3: Create transformed inputs xt = F(x) = clip[−1,1] x + G(x) . 4: Sample N perturbed models {Mp1, ..., MpN } from Mp under p% random bit errors. 5: for all Mpi ∼{Mp1, ..., MpN } do 6: Calculate the loss Lpi based on the output of the perturbed model Mpi. Then calculate the gradients gpi for WG based on Lpi. 7: end for 8: Calculate the loss L0 based on the output of the clean model M0. Then calculate the gradients g0 for WG based on L0. 9: Calculate the final gradient gfinal using (5) based on g0 and gp1, ..., gpN . 10: Update WG using gfinal. 11: end for 12: end for B Implementation Details of NeuralFuse Generator We consider two main goals in designing the NeuralFuse Generator: 1) efficiency (so the overall energy overhead is decreased) and 2) robustness (so that it can generate robust patterns on the input image and overcome the random bit flipping in subsequent models). Accordingly, we choose to utilize an encode-decoder architecture in implementing the generator. The design of ConvL is inspired by Nguyen and Tran [2020], in which the authors utilize a similar architecture to design an inputaware trigger generator, and have demonstrated its efficiency and effectiveness. Furthermore, we attempted to enhance it by replacing the Upsampling layer with a Deconvolution layer, leading to the creation of DeConvL. The UNetL-based NeuralFuse draws inspiration from Ronneberger et al. [2015], known for its robust performance in image segmentation, and thus, we incorporated it as one of our architectures. Lastly, ConvS, DeConvS, and UNetS are scaled-down versions of the model designed to reduce computational costs and total parameters. The architectures of Convolutional-based and Deconvolutional-based are shown in Table 4, and the architecture of UNet-based generators is in Table 5. For the abbreviation used in the table, ConvBlock means the Convolution block, Conv means a single Convolution layer, DeConvBlock means the Deconvolution block, DeConv means a single Deconvolution layer, and BN means a Batch Normalization layer. We use learning rate = 0.001, λ = 5, and Adam optimizer. For CIFAR-10, GTSRB, and CIFAR-100, we set batch size b = 25 for each base model. For ImageNet-10, we set b = 64 for ResNet18, ResNet50 and VGG11, and b = 32 for both VGG16 and VGG19. 13 Table 4: Model architecture for both Convolution-based and Deconvolution-based generators. Each ConvBlock consists of a Convolution (kernel = 3 × 3, padding = 1, stride = 1), a Batch Normalization, and a ReLU layer. Each DeConvBlock consists of a Deconvolution (kernel = 4 × 4, padding = 1, stride = 2), a Batch Normalization, and a ReLU layer. ConvL ConvS DeConvL DeConvS Layers #CHs Layers #CHs Layers #CHs Layers #CHs (ConvBlock)×2, MaxPool 32 ConvBlock, Maxpool 32 (ConvBlock)×2, MaxPool 32 ConvBlock, Maxpool 32 (ConvBlock)×2, MaxPool 64 ConvBlock, Maxpool 64 (ConvBlock)×2, MaxPool 64 ConvBlock, Maxpool 64 (ConvBlock)×2, MaxPool 128 ConvBlock, Maxpool 64 (ConvBlock)×2, MaxPool, 128 ConvBlock, Maxpool 64 ConvBlock, UpSample, ConvBlock 128 ConvBlock, UpSample 64 ConvBlock 128 DeConvBlock 64 ConvBlock, UpSample, ConvBlock 64 ConvBlock, UpSample 32 DeConvBlock, ConvBlock 64 DeConvBlock 32 ConvBlock, UpSample, ConvBlock 32 ConvBlock, UpSample 3 DeConvBlock, ConvBlock 32 DeConv, BN, Tanh 3 Conv, BN, Tanh 32 Conv, BN, Tanh 3 Conv, BN, Tanh 3 [Note] #CHs: number of channels. Table 5: Model architecture for UNet-based generators. Each ConvBlock consists of a Convolution (kernel = 3 × 3, padding = 1, stride = 1), a Batch Normalization, and a ReLU layer. Other layers, such as the Deconvolutional layer (kernel = 2 × 2, padding = 1, stride = 2), are used in UNet-based models. For the final Convolution layer, the kernel size is set to 1. UNetL UNetS Layers #Channels Layers #Channels L1: (ConvBlock)×2 16 L1: (ConvBlock)×2 8 L2: Maxpool, (ConvBlock)×2 32 L2: Maxpool, (ConvBlock)×2 16 L3: Maxpool, (ConvBlock)×2 64 L3: Maxpool, (ConvBlock)×2 32 L4: Maxpool, (ConvBlock)×2 128 L4: Maxpool, (ConvBlock)×2 64 L5: DeConv 64 L5: DeConv 32 L6: Concat[L3, L5] 128 L6: Concat[L3, L5] 64 L7: (ConvBlock)×2 64 L7: (ConvBlock)×2 32 L8: DeConv 32 L8: DeConv 16 L9: Concat[L2, L8] 64 L9: Concat[L2, L8] 32 L10: (ConvBlock)×2 32 L10: (ConvBlock)×2 16 L11: DeConv 16 L11: DeConv 8 L12: Concat[L1, L11] 32 L12: Concat[L1, L11] 16 L13: (ConvBlock)×2 16 L13: (ConvBlock)×2 8 L14: Conv 3 L14: Conv 3 C NeuralFuse’s Energy/Accuracy Tradeoff SCALE-SIM. SCALE-SIM [Samajdar et al., 2020] is a systolic array-based CNN simulator that can calculate the number of memory accesses and the total time in execution cycles by giving the specific model architecture and accelerator architectural configuration as inputs. In this paper, we use SCALESIM to calculate the weights memory access of 5 based models (ResNet18, ResNet50, VGG11, VGG16, VGG19), and 6 generators (ConvL, ConvS, DeConvL, DeConvS, UNetL, UNetS). While SCALE-SIM supports both Convolutional and Linear layers, it does not yet support Deconvolution layers. Instead, we try to approximate the memory costs of Deconvolution layers by Convolution layers. We change the input and output from Deconvolution into the output and input of the Convolution layers. Besides, we also change the stride into 1 when we approximate it. We also add padding for the convolution layers while generating input files for SCALE-SIM. In this paper, we only consider the energy saving on weights accesses, so we only take the value “SRAM Filter Reads” from the output of SCALE-SIM as the total weights memory accesses (T.W.M.A.) for further energy calculation. In Table 6, we report the total weight memory access (T.W.M.A.) using SCALE-SIM. We then showed the energy/accuracy tradeoff between all of the combinations of NeuralFuse and base models under a 1% of bit error rate in Figure 5. Parameters and MACs Calculation. In addition to T.W.M.A., the number of parameters and MACs (multiply–accumulate operations) are also common metrics in measuring the energy consumption of machine learning models. Yang et al. [2017] have shown that the energy consumption of computation 14 Table 6: The total weights memory access calculated by SCALE-SIM. Base Model ResNet18 ResNet50 VGG11 VGG16 VGG19 T.W.M.A. 2,755,968 6,182,144 1,334,656 2,366,848 3,104,128 NeuralFuse ConvL ConvS DeConvL DeConvS UNetL UNetS T.W.M.A. 320,256 41,508 259,264 86,208 180,894 45,711 [Note] T.W.M.A.: total weight memory access. Figure 5: The energy/accuracy tradeoff of different NeuralFuse implementations with all CIFAR-10 pre-trained based models. The X-axis represents the percentage reduction in dynamic memory access energy at low-voltage settings (base model protected by NeuralFuse) compared to the bit-error-free (nominal) voltage; the Y-axis represents the perturbed accuracy (evaluated at low voltage) with a 1% bit error rate. and memory accesses are both proportional to MACs, allowing us to take computation energy consumption into account. Here, we use the open-source package ptflops [Sovrasov, 2018-2023] to calculate the parameters and MAC values of all the base models and the NeuralFuse generators, in the same units as Bejnordi et al. [2020] used. The results are shown in Table 7. Note that we modified the base model architectures for ImageNet-10, as it has larger input sizes. For example, we use a larger kernel size = 7 instead of 3 in the first Convolution layer in ResNet-based models to enhance the learning abilities. Therefore, the parameters of base models are different between different datasets. For NeuralFuse generators, we utilize the same architectures for implementation (including ImageNet-10). As a result, our proposed NeuralFuse generators are generally smaller than base models, either on total model parameters or MAC values. Table 7: Parameter counts and MACs for all base models and generators in this paper. Base Model ResNet18 ResNet50 VGG11 VGG16 VGG19 Parameter CIFAR-10 11,173,962 23,520,842 9,231,114 14,728,266 20,040,522 ImageNet-10 11,181,642 23,528,522 128,812,810 134,309,962 139,622,218 MACs CIFAR-10 557.14M 1.31G 153.5M 314.43M 399.47M ImageNet-10 1.82G 4.12G 7.64G 15.53G 19.69G NeuralFuse ConvL ConvS DeConvL DeConvS UNetL UNetS Parameter CIFAR-10 ImageNet-10 723,273 113,187 647,785 156,777 482,771 121,195 MACs CIFAR-10 80.5M 10.34M 64.69M 22.44M 41.41M 10.58M ImageNet-10 3.94G 506.78M 3.17G 1.1G 2.03G 518.47M 15 MACs-based Energy Consumption. We can then use the MAC values to further approximate the end-to-end energy consumption of the whole model. Assume that all values are stored on SRAM and that a MAC represents single memory access. The corresponding MACs-based energy saving percentage (MAC-ES, %) can be derived from Eq. 7 (c.f. Sec. 4.4), and results can be found in Table 8. We can observe that most combinations can save a large amount of energy, except that VGG11 with two larger NeuralFuse (ConvL and DeConvL) may increase the total energy. These results are consistent with the results reported in Table 2. In addition, we also showed the MACs-based energy/accuracy tradeoff between all of the combinations of NeuralFuse and base models under a 1% of bit error rate in Figure 6. MAC-ES = MACsbase model·Energynominal voltage− MACsbase model·Energylow-voltage-regime+MACsNeuralFuse·EnergyNeuralFuse at nominal voltage MACsbase model·Energynominal voltage × 100% (7) Table 8: The MACs-Based energy saving percentage (%) for different combinations of base models and NeuralFuse. ConvL ConvS DeConvL DeConvS UNetL UNetS ResNet18 16.2 28.7 19.0 26.6 23.2 28.7 ResNet50 24.5 29.8 25.7 28.9 27.4 29.8 VGG11 -21.8 23.9 -11.5 16.0 3.6 23.7 VGG16 5.0 27.3 10.0 23.5 17.4 27.2 VGG19 10.4 28.0 14.4 25.0 20.2 28.0 Figure 6: The Mac-based energy/accuracy tradeoff of different NeuralFuse implementations with all CIFAR-10 pre-trained based models. The X-axis represents the percentage reduction in dynamic memory access energy at low-voltage settings (base model protected by NeuralFuse), compared to the bit-error-free (nominal) voltage; the Y-axis represents the perturbed accuracy (evaluated at low voltage) with a 1% bit error rate. Although using ConvL or DeConvL along with base model VGG11 for CIFAR-10 implies an increase in energy consumption, using other smaller-scale generators, we can still save the overall energy and recover the base model’s accuracy. That said, developers can always choose smaller generators (with orders of magnitude fewer MAC operations than the original network) to restore model accuracy, further demonstrating the flexibility of choosing NeuralFuse generators of different sizes. D Inference Latency of NeuralFuse In Table 9, we report the latency (batch_size=1, CIFAR-10/ImageNet-10 testing dataset) of utilizing the different NeuralFuse generators with two different base models, ResNet18 and VGG19. While 16 NeuralFuse contributes some additional latency, we consider this an unavoidable tradeoff necessary to achieve reduced energy consumption within our framework. Although the primary focus of this paper is not on latency, we acknowledge its importance. Future research could explore designing a more lightweight version of the NeuralFuse module or applying model compression techniques to minimize latency. Additionally, we recognize that running NeuralFuse on a general-purpose CPU could lead to different latency and energy consumption figures due to various influencing factors like CPU architecture and manufacturing processes. Table 9: The Inference Latency of base model and base model with NeuralFuse. ResNet18 (CIFAR-10) VGG19 (CIFAR-10) ResNet18 (ImageNet-10) VGG19 (ImageNet-10) Base Model 5.84 ms 5.32 ms 6.21 ms 14.34 ms + ConvL 9.37 ms (+3.53) 8.96 ms (+3.64) 10.51 ms (+4.3) 17.66 ms (+3.32) + ConvS 7.86 ms (+2.02) 7.40 ms (+2.08) 8.28 ms (+2.07) 16.72 ms (+2.38) + DeConvL 9.18 ms (+3.34) 8.59 ms (+3.27) 10.07 ms (+3.86) 17.24 ms (+2.90) + DeConvS 7.49 ms (+1.65) 7.04 ms (+1.72) 7.79 ms (+1.58) 15.67 ms (+1.33) + UNetL 10.69 ms (+4.85) 10.06 ms (+4.74) 11.14 ms (+4.93) 18.54 ms (+4.20) + UNetS 10.63 ms (+4.79) 10.13 ms (+4.81) 11.36 ms (+5.15) 18.60 ms (+4.26) E Ablation Studies Study for N in EOPM. Here, we study the effect of N used in EOPM (Eq. 5). In Figure 7, we report the results for ConvL and ConvS on CIFAR-10 pre-trained ResNet18, under a 1% bit error rate (BER). The results demonstrate that if we apply larger N, the performance increases until convergence. Specifically, for ConvL (Figure 7a), larger N empirically has a smaller standard deviation; this means larger N gives better stability but at the cost of time-consuming training. In contrast, for the small generator ConvS (Figure 7b), we can find that the standard deviation is still large even trained by larger N; the reason might be that small generators are not as capable of learning representations as larger ones. Therefore, there exists a tradeoff between the stability of the generator performance and the total training time. In our implementation, choosing N = 5 or 10 is a good balance. (a) Using ConvL (b) Using ConvS Figure 7: The experimental results on using different sizes of N for EOPM. Tradeoff Between Clean Accuracy (CA) and Perturbed Accuracy (PA). We conducted an experiment to study the effect of different λ values, which balance the ratio of clean accuracy and perturbed accuracy. In Table 10, the experimental results showed that a smaller λ can preserve clean accuracy, but result in poor perturbed accuracy. On the contrary, larger λ can deliver higher perturbed accuracy, but with more clean accuracy drop. This phenomenon has also been observed in adversarial training [Zhang et al., 2019]. Comparison to Universal Input Perturbation (UIP). Moosavi-Dezfooli et al. [2017] has shown that there exists a universal adversarial perturbation to the input data such that the model will make wrong predictions on a majority of the perturbed images. In our NeuralFuse framework, the universal perturbation is a special case when we set G(x) = tanh (UIP) for any data sample x. The transformed data sample then becomes xt = clip[−1,1] x + tanh (UIP) , where xt ∈[−1, 1]d 17 Table 10: Experimental results based on λ value choosing. The results show that λ = 5 can balance the tradeoff between clean accuracy and perturbed accuracy. Base Model λ CA PA ConvL CA (NF) PA (NF) RP ResNet18 10 92.6 38.9 ± 12.4 90.1 88.0 ± 1.7 49.1 5 89.8 87.8 ± 1.7 48.8 1 90.0 84.2 ± 3.8 45.3 0.1 91.6 65.7 ± 9.3 26.8 0.01 92.2 43.6 ± 13 4.7 VGG19 10 90.5 36.0 ± 12.0 89.6 77.9 ± 19 41.9 5 89.8 77.7 ± 19 41.7 1 89.9 73.1 ± 19 37.1 0.1 89.1 51.2 ± 16 15.2 0.01 90.2 36.8 ± 12 0.8 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recover percentage of PA (NF) vs. PA and UIP is a trainable universal input perturbation that has the same size as the input data. The experimental results with the universal input perturbation are shown in Table 11. We observe that its performance is much worse than our proposed NeuralFuse. This result validates the necessity of adopting input-aware transformation for learning error-resistant data representations in low-voltage scenarios. Table 11: Performance of the universal input perturbation (UIP) trained by EOPM on CIFAR-10 pre-trained ResNet18. Base Model BER CA PA CA (UIP) PA (UIP) RP ResNet18 1% 92.6 38.9 ± 12.4 91.8 37.9 ± 11 -1.0 0.5% 70.1 ± 11.6 92.5 70.6 ± 11 0.5 ResNet50 1% 92.6 26.1 ± 9.4 80.7 21.0 ± 5.9 -5.1 0.5% 61.0 ± 10.3 91.9 62.4 ± 12 1.4 VGG11 1% 88.4 42.2 ± 11.6 86.9 43.0 ± 11 0.8 0.5% 63.6 ± 9.3 88.2 64.2 ± 8.8 0.6 VGG16 1% 90.3 35.7 ± 7.9 90.1 37.1 ± 8.5 1.4 0.5% 66.6 ± 8.1 90.4 67.3 ± 8.1 0.7 VGG19 1% 90.5 36.0 ± 12.0 89.9 35.3 ± 12 -0.7 0.5% 64.2 ± 12.4 90.1 64.4 ± 12 0.2 Note. BER: the bit-error rate of the base model; CA (%): clean accuracy; PA (%): perturbed accuracy; UIP: universal input transformation parameter; and RP: total recovery percentage of PA (UIP) vs. PA 18 F Additional Experimental Results on Relaxed Access Settings We conducted more experiments on Relaxed Access settings to show that our NeuralFuse can protect the models under different BER The results can be found in Sec. F.1 (CIFAR-10), Sec. F.2 (GTSRB), Sec. F.3 (ImageNet-10), and Sec. F.4 (CIFAR-100). For comparison, we also visualize the experimental results in the figures below each table. F.1 CIFAR-10 Table 12: Testing accuracy (%) under 1% and 0.5% of random bit error rate on CIFAR-10. Base Model NF CA 1% BER 0.5% BER PA CA (NF) PA (NF) RP PA CA (NF) PA (NF) RP ResNet18 ConvL 92.6 89.8 87.8 ± 1.7 48.8 90.4 87.9 ± 2.2 17.8 ConvS 88.2 59.5 ± 11 20.6 91.7 78.4 ± 8.3 8.3 DeConvL 38.9 89.6 88.5 ± 0.8 49.6 70.1 90.2 90.0 ± 0.2 19.9 DeConvS ± 12.4 82.9 68.8 ± 6.4 29.9 ± 11.6 84.1 79.9 ± 3.6 9.8 UNetL 86.6 84.6 ± 0.8 45.6 89.7 86.3 ± 2.4 16.2 UNetS 84.4 68.8 ± 6.0 29.8 90.9 80.7 ± 5.8 10.7 ResNet50 ConvL 92.6 85.5 53.2 ± 22 27.1 90.3 86.5 ± 3.2 25.5 ConvS 85.2 34.6 ± 14 8.5 90.8 73.3 ± 8.7 12.3 DeConvL 26.1 87.4 63.3 ± 21 37.2 61.0 89.5 87.2 ± 2.5 26.2 DeConvS ± 9.4 82.4 42.2 ± 17 16.1 ± 10.3 90.3 75.5 ± 8.1 14.5 UNetL 86.2 75.5 ± 12 49.4 89.9 83.9 ± 3.6 22.9 UNetS 77.3 56.2 ± 19 30.1 89.7 76.1 ± 7.2 15.1 VGG11 ConvL 88.4 89.6 87.2 ± 2.9 45.1 89.8 87.0 ± 1.3 23.3 ConvS 84.9 66.3 ± 7.5 24.1 88.2 74.5 ± 5.7 10.9 DeConvL 42.2 89.3 87.2 ± 2.6 45.0 63.6 89.6 86.9 ± 1.1 23.2 DeConvS ± 11.6 85.6 68.2 ± 7.1 26.0 ± 9.3 88.3 75.7 ± 4.6 12.1 UNetL 87.1 83.6 ± 1.3 41.4 88.0 82.4 ± 1.8 18.8 UNetS 85.5 72.7 ± 4.6 30.5 88.1 75.8 ± 4.3 12.2 VGG16 ConvL 90.3 90.1 86.0 ± 6.2 50.3 90.2 88.5 ± 0.9 21.9 ConvS 87.4 59.6 ± 12 23.9 89.9 77.8 ± 4.8 11.1 DeConvL 35.7 89.7 85.5 ± 6.8 49.8 66.6 89.7 88.2 ± 1.0 21.4 DeConvS ± 7.9 86.8 66.5 ± 11 30.8 ± 8.1 90.0 78.4 ± 4.7 11.8 UNetL 87.4 83.4 ± 4.4 47.7 89.0 86.2 ± 1.5 19.6 UNetS 87.4 71.2 ± 8.2 35.5 89.0 80.2 ± 3.5 13.7 VGG19 ConvL 90.5 89.8 77.7 ± 19 41.7 90.4 88.1 ± 1.8 23.9 ConvS 87.3 52.7 ± 17 16.7 89.6 74.5 ± 9.0 10.3 DeConvL 36.0 86.3 78.4 ± 18 42.4 64.2 90.4 88.5 ± 1.4 24.3 DeConvS ± 12.0 86.5 58.2 ± 18 22.2 ± 12.4 89.7 75.2 ± 8.6 11.0 UNetL 86.3 82.1 ± 4.8 46.0 89.1 85.0 ± 2.7 20.8 UNetS 86.3 66.4 ± 13 30.4 89.2 77.1 ± 7.3 12.9 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Nominal (w/o NeuralFuse) Low voltage (w/o NeuralFuse) Nominal (w/ NeuralFuse) Low voltage (w/ NeuralFuse) w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 Accuracy (%) 38 .9 89 88 89 82 86 84 .8 .2 .6 .9 .6 .4 87 59 88 68 84 68 .8 .5 .5 .8 .6 .8 CIFAR-10 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 26 .1 85 85 87 82 86 77 .5 .2 .4 .4 .2 .3 53 34 63 42 75 56 .2 .6 .3 .2 .5 .2 CIFAR-10 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 88.4 42 .2 89 84 89 85 87 85 .6 .9 .3 .6 .1 .5 87 66 87 68 83 72 .2 .3 .2 .2 .6 .7 CIFAR-10 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.3 35 .7 90 87 89 86 87 87 .1 .4 .7 .8 .4 .4 86 59 85 66 83 71 .0 .6 .5 .5 .4 .2 CIFAR-10 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.5 36 .0 89 87 86 86 86 86 .8 .3 .3 .5 .3 .3 77 52 78 58 82 66 .7 .7 .4 .2 .1 .4 CIFAR-10 pre-trained VGG19 (a) CIFAR-10, 1% Bit Error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 70 .1 90 91 90 84 89 90 .4 .7 .2 .1 .7 .9 87 78 90 79 86 80 .9 .4 .0 .9 .3 .7 CIFAR-10 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.6 61 .0 90 90 89 90 89 89 .3 .8 .5 .3 .9 .7 86 73 87 75 83 76 .5 .3 .2 .5 .9 .1 CIFAR-10 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 88.4 63 .6 89 88 89 88 88 88 .8 .2 .6 .3 .0 .1 87 74 86 75 82 75 .0 .5 .9 .7 .4 .8 CIFAR-10 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.3 66 .6 90 89 89 90 89 89 .2 .9 .7 .0 .0 .0 88 77 88 78 86 80 .5 .8 .2 .4 .2 .2 CIFAR-10 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 90.5 64 .2 90 89 90 89 89 89 .4 .6 .4 .7 .1 .2 88 74 88 75 85 77 .1 .5 .5 .2 .0 .1 CIFAR-10 pre-trained VGG19 (b) CIFAR-10, 0.5% Bit Error Rate Figure 8: Experimental results on CIFAR-10 19 F.2 GTSRB Table 13: Testing accuracy (%) under 1% and 0.5% of random bit error rate on GTSRB. Base Model NF CA 1% BER 0.5% BER PA CA (NF) PA (NF) RP PA CA (NF) PA (NF) RP ResNet18 ConvL 95.5 95.7 91.1 ± 4.7 54.2 93.4 89.5 ± 1.9 14.3 ConvS 94.4 68.6 ± 12 31.7 94.8 87.7 ± 4.2 12.4 DeConvL 36.9 95.6 91.3 ± 4.3 54.4 75.2 95.4 93.4 ± 1.1 18.1 DeConvS ± 16.0 95.7 78.1 ± 9.1 41.2 ± 12.7 95.8 90.1 ± 3.3 14.9 UNetL 96.2 93.8 ± 1.0 56.9 96.2 93.5 ± 1.6 18.3 UNetS 95.9 85.1 ± 6.9 48.2 95.5 91.4 ± 2.8 16.2 ResNet50 ConvL 95.0 95.6 71.6 ± 20 42.1 94.6 90.6 ± 3.7 16.6 ConvS 94.8 50.5 ± 22 21.0 95.4 84.5 ± 8.5 10.5 DeConvL 29.5 94.9 71.6 ± 21 42.0 74.0 94.7 91.6 ± 2.9 17.6 DeConvS ± 16.9 93.0 56.4 ± 17 26.9 ± 13.0 94.6 87.4 ± 5.9 13.5 UNetL 94.5 80.6 ± 15 51.1 96.5 93.7 ± 2.3 19.7 UNetS 94.7 64.7 ± 22 35.2 95.9 90.6 ± 4.8 16.7 VGG11 ConvL 91.9 94.8 85.7 ± 7.2 50.9 93.9 92.6 ± 0.7 27.7 ConvS 91.1 62.2 ± 11 27.3 90.9 80.5 ± 3.5 15.7 DeConvL 34.9 95.0 84.6 ± 7.6 49.7 64.9 93.6 91.9 ± 0.6 27.1 DeConvS ± 12.4 92.4 67.5 ± 11 32.6 ± 10.8 92.3 83.1 ± 3.7 18.2 UNetL 92.2 83.2 ± 6.0 48.3 94.8 90.6 ± 1.7 25.7 UNetS 94.7 73.4 ± 10 38.5 94.6 88.9 ± 2.2 24.1 VGG16 ConvL 95.2 96.3 72.4 ± 12 57.3 95.6 93.2 ± 1.8 34.4 ConvS 94.1 39.8 ± 13 24.6 94.3 82.2 ± 6.2 23.4 DeConvL 15.1 96.4 72.0 ± 12 56.9 58.8 95.6 93.1 ± 2.0 34.3 DeConvS ± 6.8 93.8 50.9 ± 13 35.8 ± 8.9 95.1 84.0 ± 5.3 25.2 UNetL 95.8 78.6 ± 11 63.5 96.0 92.8 ± 2.0 34.0 UNetS 94.3 63.3 ± 14 48.1 95.4 87.8 ± 3.6 29.0 VGG19 ConvL 95.5 96.0 88.3 ± 7.2 51.7 95.6 93.4 ± 2.1 24.2 ConvS 93.8 69.0 ± 14 32.4 94.9 87.0 ± 4.4 17.8 DeConvL 36.6 95.4 87.2 ± 7.5 50.6 69.1 95.5 92.4 ± 2.2 23.3 DeConvS ± 6.8 94.5 73.1 ± 12 36.5 ± 11.1 95.5 88.8 ± 3.7 19.7 UNetL 95.4 88.2 ± 6.7 51.7 94.9 91.7 ± 2.5 22.6 UNetS 94.6 80.6 ± 9.0 44.1 96.5 90.8 ± 3.4 21.6 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Nominal (w/o NeuralFuse) Low voltage (w/o NeuralFuse) Nominal (w/ NeuralFuse) Low voltage (w/ NeuralFuse) w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 Accuracy (%) 36 .9 95 94 95 95 96 95 .7 .4 .6 .7 .2 .9 91 68 91 78 93 85 .1 .6 .3 .1 .8 .1 GTSRB pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.0 29 .5 95 94 94 93 94 94 .6 .8 .9 .0 .5 .7 71 50 71 56 80 64 .6 .5 .6 .4 .6 .7 GTSRB pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 91.9 34 .9 94 91 95 92 92 94 .8 .1 .0 .4 .2 .7 85 62 84 67 83 73 .7 .2 .6 .5 .2 .4 GTSRB pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.2 15 .1 96 94 96 93 95 94 .3 .1 .4 .8 .8 .3 72 39 72 50 78 63 .4 .8 .0 .9 .6 .3 GTSRB pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 36 .6 96 93 95 94 95 94 .0 .8 .4 .5 .4 .6 88 69 87 73 88 80 .3 .0 .2 .1 .2 .6 GTSRB pre-trained VGG19 (a) GTSRB, 1% Bit Error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 75 .2 93 94 95 95 96 95 .4 .8 .4 .8 .2 .5 89 87 93 90 93 91 .5 .7 .4 .1 .5 .4 GTSRB pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.0 74 .0 94 95 94 94 96 95 .6 .4 .7 .6 .5 .9 90 84 91 87 93 90 .6 .5 .6 .4 .7 .6 GTSRB pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 91.9 64 .9 93 90 93 92 94 94 .9 .9 .6 .3 .8 .6 92 80 91 83 90 88 .6 .5 .9 .1 .6 .9 GTSRB pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.2 58 .8 95 94 95 95 96 95 .6 .3 .6 .1 .0 .4 93 82 93 84 92 87 .2 .2 .1 .0 .8 .8 GTSRB pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 95.5 69 .1 95 94 95 95 94 96 .6 .9 .5 .5 .9 .5 93 87 92 88 91 90 .4 .0 .4 .8 .7 .8 GTSRB pre-trained VGG19 (b) GTSRB, 0.5% Bit Error Rate Figure 9: Experimental results on GTSRB. 20 F.3 ImageNet-10 Table 14: Testing accuracy under 0.5% of random bit error rate on ImageNet-10. Base Model NF CA 0.5% BER PA CA (NF) PA (NF) RP ResNet18 ConvL 92.2 72.3 ± 7.0 94.0 88.0 ± 2.0 15.7 ConvS 91.8 83.6 ± 4.1 11.3 DeConvL 94.0 89.2 ± 1.3 16.9 DeConvS 92.8 87.5 ± 2.3 15.2 UNetL 94.0 88.1 ± 1.4 15.8 UNetS 93.2 86.4 ± 2.2 14.1 ResNet50 ConvL 89.8 39.4 ± 11 92.2 80.0 ± 5.8 40.6 ConvS 91.8 65.0 ± 11 25.6 DeConvL 93.0 79.4 ± 5.9 40.0 DeConvS 93.2 70.9 ± 9.1 31.5 UNetL 92.2 80.5 ± 5.8 41.1 UNetS 92.4 73.6 ± 8.9 34.2 VGG11 ConvL 91.6 47.8 ± 13 92.0 86.1 ± 3.7 38.3 ConvS 89.4 66.4 ± 7.1 18.6 DeConvL 91.0 86.0 ± 3.0 38.2 DeConvS 89.0 72.5 ± 7.8 24.7 UNetL 92.4 83.0 ± 3.5 35.2 UNetS 86.2 73.5 ± 6.0 25.7 VGG16 ConvL 94.6 38.4 ± 17 90.8 77.1 ± 11 38.7 ConvS 90.2 60.2 ± 14 21.8 DeConvL 91.2 77.2 ± 11 38.8 DeConvS 90.0 62.3 ± 14 23.9 UNetL 90.6 81.1 ± 5.9 42.7 UNetS 86.4 72.3 ± 8.8 33.9 VGG19 ConvL 92.4 37.2 ± 11 91.4 75.5 ± 8.8 38.3 ConvS 88.8 56.5 ± 13 19.3 DeConvL 91.0 75.9 ± 8.9 38.7 DeConvS 88.8 64.0 ± 11 26.8 UNetL 89.4 77.9 ± 6.1 40.7 UNetS 87.6 65.9 ± 10 28.7 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Nominal (w/o NeuralFuse) Low voltage (w/o NeuralFuse) Nominal (w/ NeuralFuse) Low voltage (w/ NeuralFuse) w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.2 Accuracy (%) 72 .3 94 91 94 92 94 93 .0 .8 .0 .8 .0 .2 88 83 89 87 88 86 .0 .6 .2 .5 .1 .4 ImageNet-10 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 89.8 39 .4 92 91 93 93 92 92 .2 .8 .0 .2 .2 .4 80 65 79 70 80 73 .0 .0 .4 .9 .5 .6 ImageNet-10 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 91.6 47 .8 92 89 91 89 92 86 .0 .4 .0 .0 .4 .2 86 66 86 72 83 73 .1 .4 .0 .5 .0 .5 ImageNet-10 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 94.6 38 .4 90 90 91 90 90 86 .8 .2 .2 .0 .6 .4 77 60 77 62 81 72 .1 .2 .2 .3 .1 .3 ImageNet-10 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 92.4 37 .2 91 88 91 88 89 87 .4 .8 .0 .8 .4 .6 75 56 75 64 77 65 .5 .5 .9 .0 .9 .9 ImageNet-10 pre-trained VGG19 Figure 10: Experimental results on ImageNet-10, 0.5% Bit Error Rate. 21 F.4 CIFAR-100 As mentioned in the previous section, larger generators like ConvL, DeConvL, and UNetL have better performance than small generators. For CIFAR-100, we find that the gains of utilizing NeuralFuse are less compared to the other datasets. We believe this is because CIFAR-100 is a more challenging dataset (more classes) for the generators to learn to protect the base models. Nevertheless, NeuralFuse can still function to restore some degraded accuracy; these results also demonstrate that our NeuralFuse is applicable to different datasets. In addition, although the recover percentage is less obvious on CIFAR-100 (the more difficult dataset), we can still conclude that our NeuralFuse is applicable to different datasets. Table 15: Testing accuracy (%) under 1%, 0.5% and 0.35% of random bit error rate on CIFAR-100. Base Model NF C.A. 1% BER 0.5% BER 0.35% BER PA CA (NF) PA (NF) RP PA CA (NF) PA (NF) RP PA CA (NF) PA (NF) RP ResNet18 ConvL 73.7 54.8 11.0 ± 7.7 6.4 65.2 39.0 ± 7.1 18.1 69.4 42.9 ± 6.2 11.4 ConvS 49.7 4.2 ± 2.2 -0.4 70.0 24.5 ± 7.6 3.6 72.1 35.1 ± 7.3 3.7 DeConvL 4.6 55.2 11.9 ± 8.2 7.3 20.9 66.3 38.2 ± 6.9 17.3 31.4 69.2 42.9 ± 5.5 11.4 DeConvS ± 2.9 32.7 4.0 ± 2.2 -0.6 ± 7.4 68.2 25.9 ± 6.8 5 ± 7.6 71.6 35.8 ± 5.5 4.4 UNetL 50.6 14.5 ± 8.9 10.0 66.2 40.1 ± 6.4 19.2 70.3 46.3 ± 5.5 14.9 UNetS 26.8 4.6 ± 2.5 -0.0 67.1 28.8 ± 6.8 7.9 70.9 38.3 ± 6.4 6.9 ResNet50 ConvL 73.5 63.5 3.2 ± 1.7 0.1 68.4 28.8 ± 6.7 7.6 72.0 40.8 ± 7.5 5.1 ConvS 65.5 3.2 ± 1.6 0.1 71.9 23.1 ± 6.9 1.9 73.0 37.4 ± 8.0 1.7 DeConvL 3.0 59.6 3.2 ± 1.7 0.2 21.3 68.1 28.6 ± 7.0 7.4 35.7 71.7 41.7 ± 7.7 6.1 DeConvS ± 1.8 61.1 3.2 ± 1.7 0.1 ± 7.0 70.3 25.0 ± 6.7 3.7 ± 8.6 72.8 38.9 ± 7.9 3.3 UNetL 39.0 5.0 ± 1.7 1.9 66.6 36.5 ± 6.2 15.3 70.8 45.3 ± 6.7 9.6 UNetS 47.7 3.4 ± 1.8 0.3 69.1 26.1 ± 6.6 4.8 72.6 39.6 ± 7.8 3.9 VGG11 ConvL 64.8 58.3 19.7 ± 11 11.5 63.1 38.8 ± 9.3 15.0 63.9 42.4 ± 9.0 11.1 ConvS 56.6 10.4 ± 7.4 2.2 62.7 27.9 ± 10 4.0 63.9 41.8 ± 8.3 10.5 DeConvL 8.2 60.3 21.2 ± 11 13.0 23.9 63.9 40.0 ± 9.0 16.2 31.3 64.0 42.8 ± 9.1 11.5 DeConvS ± 5.7 58.3 11.8 ± 7.9 3.5 ± 9.4 61.9 29.8 ± 9.9 5.9 ± 10 63.5 36.1 ± 10 4.8 UNetL 51.1 22.1 ± 8.2 13.9 61.8 37.8 ± 9.0 13.9 63.5 40.9 ± 9.3 9.6 UNetS 51.9 13.1 ± 7.9 4.9 61.7 29.8 ± 9.7 6.0 63.8 35.7 ± 9.9 4.5 VGG16 ConvL 67.8 51.4 19.2 ± 6.0 12.6 61.8 41.1 ± 5.6 18.7 64.9 44.9 ± 5.3 13.8 ConvS 44.3 6.7 ± 2.3 0.1 63.8 27.5 ± 6.8 5.1 66.0 36.3 ± 6.1 5.1 DeConvL 7.0 53.1 20.8 ± 6.2 14.2 22.4 62.8 42.1 ± 5.5 19.8 31.1 65.0 46.6 ± 5.2 15.5 DeConvS ± 3.5 23.5 4.8 ± 1.7 -1.8 ± 7.0 62.1 29.9 ± 6.7 7.5 ± 7.2 64.9 38.1 ± 6.3 7.0 UNetL 50.2 25.3 ± 1.7 18.7 61.7 41.3 ± 5.0 18.9 64.8 46.8 ± 4.6 15.7 UNetS 27.7 9.9 ± 2.1 3.3 61.6 31.3 ± 6.3 8.9 65.0 39.8 ± 5.9 8.7 VGG19 ConvL 67.8 59.4 29.2 ± 8.1 18.6 65.6 46.5 ± 6.8 12.5 66.9 49.2 ± 7.4 7.0 ConvS 63.7 14.4 ± 5.1 3.8 66.6 38.3 ± 6.8 4.2 67.7 45.3 ± 8.5 3.2 DeConvL 10.6 60.1 29.6 ± 8.5 19.0 34.0 65.7 46.9 ± 7.1 12.9 42.1 67.3 49.8 ± 7.6 7.6 DeConvS ± 4.3 60.9 16.1 ± 6.0 5.6 ± 9.6 66.5 39.0 ± 3.7 5.0 ± 9.4 67.7 45.7 ± 8.4 3.6 UNetL 58.7 30.2 ± 8.2 19.6 65.5 46.9 ± 6.5 12.9 67.4 50.0 ± 7.5 7.9 UNetS 59.1 18.0 ± 6.2 7.4 66.3 40.1 ± 8.0 6.1 67.5 46.6 ± 8.4 4.5 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Nominal (w/o NeuralFuse) Low voltage (w/o NeuralFuse) Nominal (w/ NeuralFuse) Low voltage (w/ NeuralFuse) w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.7 Accuracy (%) 4 .6 54 49 55 32 50 26 .8 .7 .2 .7 .6 .8 11 4 11 4 14 4 .0 .2 .9 .0 .5 .6 CIFAR-100 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.5 3 .0 63 65 59 61 39 47 .5 .5 .6 .1 .0 .7 3 3 3 3 5 3 .2 .2 .2 .2 .0 .4 CIFAR-100 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 64.8 8 .2 58 56 60 58 51 51 .3 .6 .3 .3 .1 .9 19 10 21 11 22 13 .7 .4 .2 .8 .1 .1 CIFAR-100 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 7 .0 51 44 53 23 50 27 .4 .3 .1 .5 .2 .7 19 6 20 4 25 9 .2 .7 .8 .8 .3 .9 CIFAR-100 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 10 .6 59 63 60 60 58 59 .4 .7 .1 .9 .7 .1 29 14 29 16 30 18 .2 .4 .6 .1 .2 .0 CIFAR-100 pre-trained VGG19 (a) CIFAR-100, 1% Bit Error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.7 20 .9 65 70 66 68 66 67 .2 .0 .3 .2 .2 .1 39 24 38 25 40 28 .0 .5 .2 .9 .1 .8 CIFAR-100 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.5 21 .3 68 71 68 70 66 69 .4 .9 .1 .3 .6 .1 28 23 28 25 36 26 .8 .1 .6 .0 .5 .1 CIFAR-100 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 64.8 23 .9 63 62 63 61 61 61 .1 .7 .9 .9 .8 .7 38 27 40 29 37 29 .8 .9 .0 .8 .8 .8 CIFAR-100 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 22 .4 61 63 62 62 61 61 .8 .8 .8 .1 .7 .6 41 27 42 29 41 31 .1 .5 .1 .9 .3 .3 CIFAR-100 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 34 .0 65 66 65 66 65 66 .6 .6 .7 .5 .5 .3 46 38 46 39 46 40 .5 .3 .9 .0 .9 .1 CIFAR-100 pre-trained VGG19 (b) CIFAR-100, 0.5% Bit Error Rate w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.7 31 .4 69 72 69 71 70 70 .4 .1 .2 .6 .3 .9 42 35 42 35 46 38 .9 .1 .9 .8 .3 .3 CIFAR-100 pre-trained ResNet18 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 73.5 35 .7 72 73 71 72 70 72 .0 .0 .7 .8 .8 .6 40 37 41 38 45 39 .8 .4 .7 .9 .3 .6 CIFAR-100 pre-trained ResNet50 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 64.8 31 .3 63 63 64 63 63 63 .9 .9 .0 .5 .5 .8 42 41 42 36 40 35 .4 .8 .8 .1 .9 .7 CIFAR-100 pre-trained VGG11 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 31 .1 64 66 65 64 64 65 .9 .0 .0 .9 .8 .0 44 36 46 38 46 39 .9 .3 .6 .1 .8 .8 CIFAR-100 pre-trained VGG16 w/o NF ConvL ConvS DeConvL DeConvS UNetL UNetS 67.8 42 .1 66 67 67 67 67 67 .9 .7 .3 .7 .4 .5 49 45 49 45 50 46 .2 .3 .8 .7 .0 .6 CIFAR-100 pre-trained VGG19 (c) CIFAR-100, 0.35% Bit Error Rate Figure 11: Experimental results on CIFAR-100. 22 G Additional Experimental Results on Restricted Access Settings (Transferability) We conduct more experiments with Restricted Access settings to show that our NeuralFuse can be transferred to protect various black-box models. The experimental results are shown in Sec. G.1 (CIFAR-10), Sec. G.2 (GTSRB), and Sec. G.3 (CIFAR-100). We find that using VGG19 as a white-box surrogate model has better transferability than ResNet18 for all datasets. In addition, we can observe that some NeuralFuse generators have downward applicability if base models have a similar architecture. In other words, if we try to transfer a generator trained on a large BER (e.g., 1%) to a model with a small BER (e.g., 0.5%), the performance will be better than that of a generator trained with the original BER (e.g., 0.5%). For example, in Table 16, we could find that if we use VGG19 as a source model to train the generator ConvL (1%), the generator could deliver better performance (in terms of PA (NF)) when applied to similar base models (e.g., VGG11, VGG16, or VGG19) under a 0.5% BER, compared to using itself as a source model (shown in Table 12). We conjecture that this is because the generators trained on a larger BER can also cover the error patterns of a smaller BER, and thus they have better generalizability across smaller B.E.Rs. To further improve the transferability to cross-architecture target models, we also conduct an experiment in Sec. G.4 to show that using ensemble-based training can help the generator to achieve this feature. G.1 CIFAR-10 The results of CIFAR-10 in which NeuralFuse is trained at 1% BER are shown in Table 16. Table 16: Transfer results on CIFAR-10: NeuralFuse trained on SM with 1% BER SM TM BER CA PA ConvL (1%) UNetL (1%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 0.5% 92.6 70.1 ± 11.6 89.8 89.5 ± 0.2 19.4 86.6 86.2 ± 0.3 16.1 ResNet50 1% 92.6 26.1 ± 9.4 89.5 36.0 ± 19 9.9 85.2 38.8 ± 19 12.7 0.5% 61.0 ± 10.3 75.1 ± 10 14.1 77.1 ± 5.0 16.1 VGG11 1% 88.4 42.2 ± 11.6 88.4 62.5 ± 8.4 20.3 76.8 61.1 ± 8.5 18.9 0.5% 63.6 ± 9.3 81.0 ± 4.6 17.4 73.7 ± 3.0 10.1 VGG16 1% 90.3 35.7 ± 7.9 89.6 63.3 ± 18 27.6 85.2 59.9 ± 16 24.2 0.5% 66.6 ± 8.1 85.0 ± 3.4 18.4 80.2 ± 4.5 13.6 VGG19 1% 90.5 36.0 ± 12.0 89.6 50.7 ± 22 14.7 85.3 51.1 ± 16 15.1 0.5% 64.2 ± 12.4 80.2 ± 8.7 16.0 76.5 ± 7.8 12.3 VGG19 ResNet18 1% 92.6 38.9 ± 12.4 89.8 61.0 ± 17 22.1 87.0 69.7 ± 11 30.8 0.5% 70.1 ± 11.6 86.1 ± 6.9 16.0 84.2 ± 3.0 14.1 ResNet50 1% 92.6 26.1 ± 9.4 89.9 34.0 ± 19 7.9 87.0 44.2 ± 17 18.1 0.5% 61.0 ± 10.3 76.5 ± 10 15.5 80.7 ± 4.2 19.7 VGG11 1% 88.4 42.2 ± 11.6 89.7 76.5 ± 7.0 34.3 87.1 79.9 ± 5.6 37.7 0.5% 63.6 ± 9.3 88.0 ± 2.1 24.4 85.4 ± 0.8 21.8 VGG16 1% 90.3 35.7 ± 7.9 89.6 75.5 ± 12 39.8 87.2 78.9 ± 7.8 43.2 0.5% 66.6 ± 8.1 88.9 ± 0.6 22.3 86.2 ± 0.3 19.6 VGG19 0.5% 90.5 64.2 ± 12.4 89.8 89.6 ± 8.7 25.4 87.4 86.8 ± 0.4 22.6 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA 23 G.2 GTSRB In Tables 17 and 18, we show the results on GTSRB in which NeuralFuse is trained at 1.5% and 1% BER, respectively. Table 17: Transfer results on GTSRB: NeuralFuse trained on SM with 1.5% BER SM TM BER CA PA ConvL (1.5%) UNetL (1.5%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 1% 95.5 36.9 ± 16.0 95.7 93.9 ± 1.9 57.0 94.9 94.4 ± 0.4 57.5 0.5% 75.2 ± 12.7 95.7 ± 0.2 20.5 94.8 ± 0.2 19.6 ResNet50 1% 95.0 29.5 ± 16.9 94.4 37.0 ± 22 7.5 94.4 47.1 ± 23 17.6 0.5% 74.0 ± 13.0 77.5 ± 13 3.5 84.8 ± 9.5 10.8 VGG11 1% 91.9 34.9 ± 12.4 92.8 45.2 ± 10 10.3 91.4 50.5 ± 13 15.6 0.5% 64.9 ± 10.8 79.4 ± 5.8 14.5 83.9 ± 4.2 19.0 VGG16 1% 95.2 15.1 ± 6.8 95.4 31.1 ± 13 15.9 94.6 36.8 ± 12 21.7 0.5% 58.8 ± 8.9 84.5 ± 8.3 25.8 86.0 ± 8.6 27.2 VGG19 1% 95.5 36.6 ± 6.8 95.0 56.4 ± 15 19.8 94.3 60.8 ± 15 24.2 0.5% 69.1 ± 11.1 86.9 ± 3.4 17.8 87.7 ± 3.8 18.6 VGG19 ResNet18 1% 95.5 36.9 ± 16.0 88.4 50.3 ± 12 13.4 92.8 63.7 ± 16 26.8 0.5% 75.2 ± 12.7 77.9 ± 7.4 2.7 87.5 ± 3.9 12.3 ResNet50 1% 95.0 29.5 ± 16.9 87.5 29.7 ± 17 0.2 92.5 40.4 ± 21 10.9 0.5% 74.0 ± 13.0 67.9 ± 17 -6.1 77.5 ± 15 3.5 VGG11 1% 91.9 34.9 ± 12.4 89.7 47.1 ± 11 12.2 93.5 60.0 ± 12 25.1 0.5% 64.9 ± 10.8 76.3 ± 5.1 11.4 86.0 ± 3.8 21.1 VGG16 1% 95.2 15.1 ± 6.8 93.0 29.2 ± 15 14.1 93.0 38.5 ± 16 23.4 0.5% 58.8 ± 8.9 75.7 ± 12 16.9 79.9 ± 8.3 21.1 VGG19 1% 95.5 36.6 ± 6.8 95.1 87.4 ± 6.0 50.8 94.6 88.7 ± 5.0 52.1 0.5% 69.1 ± 11.1 92.4 ± 2.4 23.3 92.4 ± 2.2 23.3 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Table 18: Transfer results on GTSRB: NeuralFuse trained on SM with 1% BER SM TM BER CA PA ConvL (1%) UNetL (1%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 0.5% 95.5 75.2 ± 12.7 95.7 95.3 ± 0.5 20.1 96.2 95.7 ± 0.3 20.5 ResNet50 1% 95.0 29.5 ± 16.9 94.5 35.6 ± 21 6.1 95.6 42.6 ± 23 13.1 0.5% 74.0 ± 13.0 78.8 ± 13 4.8 87.3 ± 9.0 13.3 VGG11 1% 91.9 34.9 ± 12.4 93.1 45.8 ± 11 10.9 94.0 47.1 ± 14 12.2 0.5% 64.9 ± 10.8 81.8 ± 5.0 16.9 84.2 ± 4.8 19.3 VGG16 1% 95.2 15.1 ± 6.8 95.5 26.5 ± 12 11.4 95.5 32.4 ± 11 17.3 0.5% 58.8 ± 8.9 82.2 ± 9.0 23.4 85.4 ± 6.7 26.6 VGG19 1% 95.5 36.6 ± 6.8 94.9 53.2 ± 14 16.6 95.6 60.9 ± 15 24.3 0.5% 69.1 ± 11.1 85.4 ± 4.5 16.3 87.5 ± 3.7 18.4 VGG19 ResNet18 1% 95.5 36.9 ± 16.0 93.7 53.1 ± 16 16.2 95.0 63.4 ± 18 26.5 0.5% 75.2 ± 12.7 83.9 ± 7.6 8.7 89.7 ± 4.8 14.5 ResNet50 1% 95.0 29.5 ± 16.9 92.8 30.6 ± 18 1.1 95.4 38.9 ± 22 9.4 0.5% 74.0 ± 13.0 74.7 ± 18 0.7 81.5 ± 16 7.5 VGG11 1% 91.9 34.9 ± 12.4 93.7 50.6 ± 11 15.7 95.1 58.9 ± 15 24.0 0.5% 64.9 ± 10.8 82.3 ± 5.1 17.4 87.5 ± 3.7 22.6 VGG16 1% 95.2 15.1 ± 6.8 95.2 27.8 ± 15 12.7 95.2 33.5 ± 14 18.4 0.5% 58.8 ± 8.9 79.0 ± 12 20.2 81.8 ± 7.8 23.0 VGG19 0.5% 95.5 69.1 ± 11.1 96.0 94.0 ± 2.2 24.9 95.4 93.9 ± 2.1 24.8 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA 24 G.3 CIFAR-100 In Tables 19 and 20, we show results on CIFAR-100 with NeuralFuse trained at 1% and 0.5% BER, respectively. Table 19: Transfer results on CIFAR-100: NeuralFuse trained on SM with 1% BER SM TM BER CA PA ConvL (1%) UNetL (1%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 0.5% 73.7 20.9 ± 7.4 54.8 35.8 ± 5.2 14.9 50.6 39.3 ± 2.8 18.4 0.35% 31.4 ± 7.6 41.7 ± 3.7 10.3 43.3 ± 1.4 11.9 ResNet50 1% 73.5 3.0 ± 1.8 44.9 2.2 ± 2.0 -0.8 41.5 2.4 ± 1.9 -0.6 0.5% 21.3 ± 7.0 15.9 ± 8.2 -5.4 17.1 ± 7.1 -4.2 0.35% 35.7 ± 8.6 23.7 ± 7.1 -12.0 26.2 ± 5.6 -9.5 VGG11 1% 64.8 8.2 ± 5.7 41.2 9.8 ± 5.6 1.6 37.5 10.2 ± 5.1 2.0 0.5% 23.9 ± 9.4 24.2 ± 5.9 0.3 24.5 ± 4.7 0.6 0.35% 31.3 ± 10.0 29.0 ± 5.4 -2.3 28.2 ± 4.5 -3.1 VGG16 1% 67.8 7.0 ± 3.5 44.0 7.9 ± 3.7 0.9 39.5 10.1 ± 4.5 3.1 0.5% 22.4 ± 7.0 22.4 ± 7.6 0.0 26.3 ± 5.3 3.9 0.35% 31.1 ± 7.2 28.1 ± 5.9 -3.0 30.6 ± 3.6 -0.5 VGG19 1% 67.8 10.6 ± 4.3 44.2 13.5 ± 6.1 2.9 40.7 15.6 ± 6.2 5.0 0.5% 34.0 ± 9.6 27.9 ± 4.8 -6.1 29.3 ± 4.6 -4.7 0.35% 42.1 ± 9.4 33.2 ± 48 -8.9 32.8 ± 3.9 -9.3 VGG19 ResNet18 1% 73.7 4.6 ± 2.9 55.5 5.8 ± 3.7 1.2 57.3 6.8 ± 4.4 2.2 0.5% 20.9 ± 7.4 24.6 ± 6.3 3.7 28.1 ± 5.9 7.2 0.35% 31.4 ± 7.6 31.1 ± 5.0 -0.3 36.4 ± 4.5 5.0 ResNet50 1% 73.5 3.0 ± 1.8 56.1 2.8 ± 2.1 -0.2 56.1 3.7 ± 2.4 0.7 0.5% 21.3 ± 7.0 18.9 ± 8.6 -2.4 22.8 ± 8.5 1.5 0.35% 35.7 ± 8.6 28.7 ± 8.2 -7.0 33.7 ± 7.0 -2.0 VGG11 1% 64.8 8.2 ± 5.7 52.8 12.3 ± 8.4 4.1 53.9 15.4 ± 9.4 7.2 0.5% 23.9 ± 9.4 30.0 ± 9.3 6.1 33.3 ± 7.2 9.4 0.35% 31.3 ± 10.0 36.5 ± 7.7 5.2 38.8 ± 6.5 7.5 VGG16 1% 67.8 7.0 ± 3.5 53.6 11.2 ± 4.4 4.2 55.2 13.6 ± 5.2 6.6 0.5% 22.4 ± 7.0 32.4 ± 7.3 10.0 35.9 ± 6.2 13.5 0.35% 31.1 ± 7.2 39.4 ± 6.3 8.3 42.4 ± 4.9 11.3 VGG19 0.5% 67.8 34.0 ± 9.6 59.4 50.2 ± 3.1 16.2 58.7 49.1 ± 3.5 15.1 0.35% 42.1 ± 9.4 53.1 ± 2.3 11.0 52.0 ± 3.1 9.9 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Table 20: Transfer results on CIFAR-100: NeuralFuse trained on SM with 0.5% BER SM TM BER CA PA ConvL (0.5%) UNetL (0.5%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 ResNet18 0.35% 73.7 31.4 ± 7.6 65.2 47.7 ± 4.9 16.3 66.2 49.2 ± 4.1 17.8 ResNet50 0.5% 73.5 21.3 ± 7.0 62.5 24.0 ± 9.9 2.8 63.5 26.4 ± 9.1 5.1 0.35% 35.7 ± 8.6 36.3 ± 8.9 0.6 39.4 ± 8.1 3.7 VGG11 0.5% 64.8 23.9 ± 9.4 59.2 33.0 ± 9.8 9.2 61.1 34.2 ± 9.8 10.3 0.35% 31.3 ± 10.0 40.4 ± 8.7 9.1 41.4 ± 9.0 10.1 VGG16 0.5% 67.8 22.4 ± 7.0 59.5 34.7 ± 8.0 12.3 61.4 37.5 ± 6.8 15.2 0.35% 31.1 ± 7.2 42.9 ± 6.0 11.8 45.3 ± 4.9 14.2 VGG19 0.5% 67.8 34.0 ± 9.6 61.6 43.7 ± 6.2 9.6 62.0 45.0 ± 6.3 11.0 0.35% 42.1 ± 9.4 49.0 ± 5.5 6.8 50.5 ± 5.3 8.3 VGG19 ResNet18 0.5% 73.7 20.9 ± 7.4 66.1 24.9 ± 6.7 4.0 67.8 27.7 ± 6.8 6.8 0.35% 31.4 ± 7.6 34.4 ± 5.4 3.0 38.1 ± 5.6 6.7 ResNet50 0.5% 73.5 21.3 ± 7.0 66.2 22.7 ± 7.8 1.4 66.7 25.4 ± 8.0 4.2 0.35% 35.7 ± 8.6 35.5 ± 7.7 -0.2 38.8 ± 7.5 3.2 VGG11 0.5% 64.8 23.9 ± 9.4 59.9 29.3 ± 10 5.4 61.0 31.2 ± 9.8 7.4 0.35% 31.3 ± 10.0 36.6 ± 9.5 5.3 38.1 ± 9.0 6.8 VGG16 0.5% 67.8 22.4 ± 7.0 62.5 30.8 ± 7.3 8.4 62.6 33.0 ± 7.3 10.7 0.35% 31.1 ± 7.2 40.0 ± 6.5 8.9 42.5 ± 5.9 11.3 VGG19 0.35% 67.8 42.1 ± 9.4 65.6 52.0 ± 6.2 9.8 65.5 52.6 ± 6.1 10.4 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA 25 G.4 Generator Ensembling To improve the transferability performance on cross-architecture cases (e.g., using ResNet-based models as surrogate models to train NeuralFuse and then transfer NeuralFuse to VGG-based target models), we try to adopt ensemble surrogate models to train our NeuralFuse. The experimental results are shown in Table 21. We use the same experimental settings mentioned in Table 1 but change one source model (e.g., ResNet18 or VGG19) into two (ResNet18 with VGG19) for training. The results show that the overall performance is better than the results shown in Table 1, which means ensemble-based training can easily solve the performance degradation on cross-architecture target models. Table 21: Transfer results on CIFAR-10: NeuralFuse trained on two SM with 1.5% BER SM TM BER CA PA ConvL (1.5%) UNetL (1.5%) CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 + VGG19 ResNet18 1% 92.6 38.9 ± 12.4 89.4 88.1 ± 1.0 49.2 86.3 85.4 ± 0.5 46.5 0.5% 70.1 ± 11.6 89.2 ± 0.2 19.1 86.1 ± 0.2 16.0 ResNet50 1% 92.6 26.1 ± 9.4 89.3 44.0 ± 22 17.9 86.1 50.9 ± 20 24.8 0.5% 61.0 ± 10.3 80.3 ± 6.7 19.3 78.6 ± 3.9 17.6 VGG11 1% 88.4 42.2 ± 11.6 89.1 77.0 ± 5.6 34.8 85.9 82.3 ± 4.1 40.1 0.5% 63.6 ± 9.3 87.5 ± 1.6 23.9 85.0 ± 0.6 21.4 VGG16 1% 90.3 35.7 ± 7.9 89.1 80.5 ± 8.6 44.8 85.7 81.4 ± 5.5 45.7 0.5% 66.6 ± 8.1 88.2 ± 0.7 21.6 85.0 ± 0.7 18.4 VGG19 1% 90.5 36.0 ± 12.0 89.2 75.1 ± 17 39.1 86.1 83.0 ± 3.4 47.0 0.5% 64.2 ± 12.4 89.0 ± 0.2 24.8 85.9 ± 0.4 21.7 Note. SM: source model, used for training generators; TM: target model, used for testing generators; BER: the bit-error rate of the target model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA H NeuralFuse on Reduced-precision Quantization and Random Bit Errors As mentioned in Sec. 4.6, we explore the robustness of NeuralFuse to low-precision quantization of model weights and consider the case of random bit errors. Here, we demonstrate that NeuralFuse can recover not only the accuracy drop due to reduced precision, but also the drop caused by low-voltageinduced bit errors (0.5% BER) under low precision. We selected two NeuralFuse generators (ConvL and UNetL) for our experiments, and these generators were trained with the corresponding base models (ResNet18 and VGG19) at 1% BER (CIFAR-10, GTSRB) and 0.5% BER (ImageNet-10). The experimental results are shown as follows: CIFAR-10 (Sec. H.1), GTSRB (Sec. H.2), and ImageNet-10 (Sec. H.3). Similarly, for ease of comparison, we visualize the experimental results in the figures below each table. Our results show that NeuralFuse can consistently perform well in low-precision regimes as well as recover the low-voltage-induced accuracy drop. H.1 CIFAR-10 Table 22: Reduced-precision Quantization and with 0.5% BER on CIFAR-10 pre-trained models. Base #Bits CA PA ConvL (1%) UNetL (1%) Model CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 8 92.6 70.1 ± 11.6 89.8 89.5 ± 0.2 19.4 86.6 86.2 ± 0.3 16.1 7 92.5 68.8 ± 10.4 89.8 89.5 ± 1.7 20.7 86.5 86.0 ± 0.5 17.2 6 92.6 68.4 ± 11.2 89.7 89.5 ± 0.2 21.1 86.6 85.9 ± 0.3 17.5 5 92.4 52.7 ± 14.1 89.7 90.0 ± 0.7 37.3 86.5 85.5 ± 0.8 32.8 4 91.8 26.3 ± 12.7 89.8 58.7 ± 24.5 32.4 86.6 64.9 ± 22.5 38.6 3 84.8 11.3 ± 1.8 89.8 12.8 ± 5.8 1.5 86.0 14.8 ± 10.0 3.5 2 10.0 10.0 ± 0.0 10.0 10.0 ± 0.0 0.0 10.0 10.0 ± 0.0 0.0 VGG19 8 90.5 64.2 ± 12.4 89.8 89.6 ± 8.7 25.4 87.4 86.8 ± 0.4 22.6 7 90.3 66.5 ± 8.5 89.8 89.6 ± 0.2 23.1 87.4 86.7 ± 0.3 20.2 6 90.1 59.8 ± 13.2 89.9 89.4 ± 3.8 29.6 87.4 86.4 ± 0.7 26.6 5 90.2 37.7 ± 14.1 89.8 78.0 ± 15.8 40.3 87.2 79.8 ± 0.8 42.1 4 87.5 14.7 ± 6.0 89.8 27.8 ± 18.9 13.1 87.2 34.4 ± 20.5 19.7 3 78.3 10.5 ± 1.5 89.7 10.9 ± 2.6 0.4 86.8 11.0 ± 2.9 0.5 2 10.0 10.0 ± 0.0 10.0 10.0 ± 0.0 0.0 10.0 10.0 ± 0.0 0.0 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA 26 Original Accuracy w/o NeuralFuse NeuralFuse (ConvL) NeuralFuse (UNetL) 8 7 6 5 4 3 2 Bit Quantization Number 92.6 Accuracy (%) 92 92 92 92 91 84 .6 .5 .6 .4 .8 .8 90 90 90 90 90 89 .4 .4 .4 .4 .0 .9 89 89 89 89 89 88 .7 .6 .7 .7 .7 .9 10 .0 10 .0 10 .0 CIFAR-10 pre-trained ResNet18 (Nominal voltage, No Bit Error) (a) Base Model: ResNet18, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 70 68 68 52 26 11 .1 .8 .4 .7 .3 .3 87 87 87 78 40 11 .9 .5 .0 .8 .0 .6 88 85 84 77 42 12 .1 .2 .9 .4 .4 .6 10 .0 10 .0 10 .0 CIFAR-10 pre-trained ResNet18 (Low voltage, 0.5% BER) (b) Base Model: ResNet18, 0.5% B.E.R. 8 7 6 5 4 3 2 Bit Quantization Number 90.5 Accuracy (%) 90 90 90 90 87 78 .5 .3 .1 .2 .5 .3 90 90 90 90 90 89 .4 .3 .4 .3 .1 .4 89 89 89 88 88 87 .1 .1 .2 .9 .6 .3 10 .0 10 .0 10 .0 CIFAR-10 pre-trained VGG19 (Nominal voltage, No Bit Error) (c) Base Model: VGG19, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 64 66 59 37 14 10 .2 .5 .8 .7 .7 .5 88 88 85 60 19 10 .1 .0 .5 .1 .5 .4 85 84 82 60 19 10 .0 .7 .0 .2 .8 .9 10 .0 10 .0 10 .0 CIFAR-10 pre-trained VGG19 (Low voltage, 0.5% BER) (d) Base Model: VGG19, 0.5% B.E.R. Figure 12: Results of Reduced-precision and bit errors (0.5%) on CIFAR-10 pre-trained base models. H.2 GTSRB Table 23: Reduced-precision Quantization and with 0.5% BER on GTSRB pre-trained models. Base #Bits CA PA ConvL (1%) UNetL (1%) Model CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 8 95.5 75.2 ± 12.7 95.7 95.3 ± 0.5 20.1 96.2 95.7 ± 0.3 20.5 7 95.5 69.5 ± 10.6 95.7 95.3 ± 0.3 25.8 96.2 95.9 ± 0.3 26.4 6 95.4 67.2 ± 14.4 95.7 95.2 ± 0.5 28.0 96.2 95.7 ± 0.5 28.5 5 95.4 48.6 ± 18.2 95.8 92.6 ± 5.1 44.0 96.2 94.8 ± 2.5 46.2 4 92.6 24.6 ± 9.8 95.9 75.6 ± 16.2 51.0 96.2 86.6 ± 9.5 62.0 3 67.7 5.3 ± 3.5 95.4 18.4 ± 15.3 13.1 96.2 25.3 ± 22.5 20.0 2 3.8 3.8 ± 0.0 4.1 3.8 ± 0.0 0.0 3.8 3.8 ± 0.0 0.0 VGG19 8 95.5 69.1 ± 11.1 96.0 94.0 ± 2.2 24.9 95.4 93.9 ± 2.1 24.8 7 95.6 66.1 ± 14.8 96.0 92.2 ± 5.7 26.1 95.4 92.6 ± 3.7 26.5 6 95.3 64.2 ± 8.4 96.0 92.2 ± 5.7 28.0 95.4 92.3 ± 2.3 28.1 5 95.2 48.2 ± 14.0 96.0 92.2 ± 5.7 44.0 95.4 86.2 ± 8.4 38.0 4 92.0 18.2 ± 14.3 93.0 92.2 ± 5.7 74.0 95.0 49.6 ± 22.8 31.4 3 60.0 2.0 ± 0.9 87.3 92.2 ± 5.7 90.2 87.2 1.7 ± 0.9 -0.3 2 5.9 3.8 ± 0.0 5.9 3.8 ± 0.0 0.0 5.9 3.8 ± 0.0 0.0 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Original Accuracy w/o NeuralFuse NeuralFuse (ConvL) NeuralFuse (UNetL) 8 7 6 5 4 3 2 Bit Quantization Number 95.5 Accuracy (%) 95 95 95 95 92 67 .5 .5 .4 .4 .6 .7 93 93 93 93 93 92 .4 .3 .3 .3 .3 .0 96 96 96 96 96 95 .2 .2 .2 .2 .1 .7 3 .8 3 .8 3 .8 GTSRB pre-trained ResNet18 (Nominal voltage, No Bit Error) (a) Base Model: ResNet18, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 75 69 67 48 24 5 .2 .5 .2 .6 .6 .3 89 89 88 79 52 9 .5 .9 .9 .3 .1 .7 93 94 93 87 62 12 .5 .5 .5 .5 .2 .6 3 .8 3 .8 3 .8 GTSRB pre-trained ResNet18 (Low voltage, 0.5% BER) (b) Base Model: ResNet18, 0.5% B.E.R. 8 7 6 5 4 3 2 Bit Quantization Number 95.5 Accuracy (%) 95 95 95 95 92 60 .5 .6 .3 .2 .0 .0 95 95 95 95 95 91 .6 .6 .6 .5 .1 .0 94 94 95 94 94 90 .9 .9 .0 .7 .3 .8 5 .9 5 .9 5 .9 GTSRB pre-trained VGG19 (Nominal voltage, No Bit Error) (c) Base Model: VGG19, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 69 66 64 48 18 2 .1 .1 .2 .2 .2 .0 93 90 91 80 37 1 .4 .4 .3 .2 .1 .8 91 90 89 77 39 1 .7 .0 .6 .6 .0 .8 3 .8 3 .8 3 .8 GTSRB pre-trained VGG19 (Low voltage, 0.5% BER) (d) Base Model: VGG19, 0.5% B.E.R. Figure 13: Results of Reduced-precision and bit errors (0.5%) on GTSRB pre-trained base models. 27 H.3 ImageNet-10 Table 24: Reduced-precision Quantization and with 0.5% BER on ImageNet-10 pre-trained models. Base #Bits CA PA ConvL (0.5%) UNetL (0.5%) Model CA (NF) PA (NF) RP CA (NF) PA (NF) RP ResNet18 8 92.2 72.3 ± 7.0 94.0 88.0 ± 2.0 15.7 94.0 88.1 ± 1.4 15.8 7 92.4 70.6 ± 13.0 94.2 86.7 ± 4.1 16.1 93.6 87.8 ± 3.5 17.2 6 92.4 68.9 ± 9.9 94.2 85.1 ± 4.8 16.2 93.6 86.4 ± 3.7 17.5 5 91.0 60.9 ± 13.0 94.2 82.5 ± 6.8 21.6 94.0 83.2 ± 5.9 22.3 4 91.4 47.4 ± 9.8 93.8 68.6 ± 9.8 21.2 92.6 68.7 ± 9.2 21.3 3 85.2 28.8 ± 11.8 89.2 44.1 ± 14.0 15.3 89.4 42.7 ± 14.2 13.9 2 10.0 10.0 ± 0.0 10.0 10.0 ± 0.0 0.0 10.0 10.0 ± 0.0 0.0 VGG19 8 92.4 37.2 ± 11.0 91.4 75.5 ± 8.8 38.3 89.4 77.9 ± 6.1 40.7 7 92.0 27.3 ± 6.6 91.2 59.3 ± 13.0 32.0 89.4 65.4 ± 10.0 38.1 6 92.4 27.9 ± 6.4 91.0 59.7 ± 11.8 31.8 89.4 64.9 ± 9.9 37.0 5 92.0 15.1 ± 4.4 91.6 23.1 ± 0.7 8.0 89.0 27.9 ± 8.8 12.8 4 89.4 12.2 ± 2.7 90.8 14.0 ± 4.3 1.8 89.6 14.6 ± 4.9 2.4 3 46.8 9.9 ± 0.5 83.2 10.4 ± 0.6 0.5 84.2 9.9 ± 0.7 0.0 2 10.0 10.0 ± 0.0 10.0 10.0 ± 0.0 0.0 10.0 10.0 ± 0.0 0.0 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA Original Accuracy w/o NeuralFuse NeuralFuse (ConvL) NeuralFuse (UNetL) 8 7 6 5 4 3 2 Bit Quantization Number 92.2 Accuracy (%) 92 92 92 91 91 85 .2 .4 .4 .0 .4 .2 94 94 94 94 93 89 .0 .2 .2 .2 .8 .2 94 93 93 94 92 89 .0 .6 .6 .0 .6 .4 10 .0 10 .0 10 .0 ImageNet-10 pre-trained ResNet18 (Nominal voltage, No Bit Error) (a) Base Model: ResNet18, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 72 70 68 60 47 28 .3 .6 .9 .9 .4 .8 88 86 85 82 68 44 .0 .7 .1 .5 .6 .1 88 87 86 83 68 42 .1 .8 .4 .2 .7 .7 10 .0 10 .0 10 .0 ImageNet-10 pre-trained ResNet18 (Low voltage, 0.5% BER) (b) Base Model: ResNet18, 0.5% B.E.R. 8 7 6 5 4 3 2 Bit Quantization Number 92.4 Accuracy (%) 92 92 92 92 89 46 .4 .0 .4 .0 .4 .8 91 91 91 91 90 83 .4 .2 .0 .6 .8 .2 89 89 89 89 89 84 .4 .4 .4 .0 .6 .2 10 .0 10 .0 10 .0 ImageNet-10 pre-trained VGG19 (Nominal voltage, No Bit Error) (c) Base Model: VGG19, no bit error. 8 7 6 5 4 3 2 Bit Quantization Number Accuracy (%) 37 27 27 15 12 9 .2 .3 .9 .1 .2 .9 75 59 59 23 14 10 .5 .3 .7 .1 .0 .4 77 65 64 27 14 9 .9 .4 .9 .9 .6 .9 10 .0 10 .0 10 .0 ImageNet-10 pre-trained VGG19 (Low voltage, 0.5% BER) (d) Base Model: VGG19, 0.5% B.E.R. Figure 14: Results of Reduced-precision and bit errors (0.5%) on ImageNet-10 pre-trained base models. I Additional Experiments on Adversarial Training Adversarial training is a common strategy to derive a robust neural network against certain perturbations. By training the generator using adversarial training proposed in Stutz et al. [2021], we report its performance against low voltage-induced bit errors. We use ConvL as the generator and ResNet18 as the base model, trained on CIFAR-10. Furthermore, we explore different K flip bits as the perturbation on weights of the base model during adversarial training, and then for evaluation, the trained-generator will be applied against 1% of bit errors rate on the base model. The results are shown in Table 25. After careful tuning of hyperparameters, we find that we are not able to obtain satisfactory recovery when adopting adversarial training. Empirically, we argue that adversarial training may not be suitable for training generator-based methods. J Additional Experiments on Robust Model Trained with Adversarial Weight Perturbation with NeuralFuse Previously, Wu et al. [2020] proposed that one could obtain a more robust model via adversarial weight perturbation. To seek whether such models could also be robust to random bit errors, we 28 Table 25: Performance of the generator trained by adversarial training under K flip bits on ResNet18 with CIFAR-10. The results show that the generator trained by adversarial training cannot achieve high accuracy against bit errors under a 1% bit error rate. K-bits CA PA CA (NF) PA (NF) RP 100 92.6 38.9 ± 12.4 92.4 38.3 ± 12.1 -0.6 500 92.1 38.7 ± 12.5 -0.2 5,000 92.6 38.9 ± 12.5 0 20,000 60.1 23.0 ± 8.1 -16 100,000 71.1 23.6 ± 6.6 -16 Note. CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA conducted an experiment on CIFAR-10 with the proposed adversarially trained PreAct ResNet18. The experimental results are shown in Table 26. We find that the average perturbed accuracy is 23% and 63.2% for PreAct ResNet18 under 1% and 0.5% BER, respectively. This result is lower than 38.9% and 70.1% from ResNet18 in Table 12, indicating their poor generalization ability against random bit errors. Nevertheless, when equipped NeuralFuse on the perturbed model, we could still witness a significant recover percentage under both 1% and 0.5% BER This result further demonstrates that NeuralFuse could be adapted to various models (i.e., trained in different learning algorithms). Table 26: Performance of NeuralFuse trained with rubust CIFAR-10 pre-trained PreAct ResNet18. The results show that NeuralFuse can be used together with a robust model and further improve perturbed accuracy under both 1% and 0.5% BER Base Model BER NF CA PA CA (NF) PA (NF) RP PreAct ResNet18 1% ConvL 89.7 23.0 ± 9.3 87.6 53.7 ± 26 30.7 ConvS 83.1 34.6 ± 15 11.6 DeConvL 87.7 55.4 ± 27 32.4 DeConvS 82.9 32.4 ± 14 9.4 UNetL 86.1 60.4 ± 28 37.4 UNetS 80.4 51.9 ± 24 28.9 0.5% ConvL 89.7 63.2 ± 8.7 89.2 87.8 ± 1.1 24.6 ConvS 89.2 74.0 ± 6.5 10.8 DeConvL 89.0 87.4 ± 1.1 24.2 DeConvS 89.9 74.4 ± 7.0 11.2 UNetL 87.5 85.9 ± 0.8 22.7 UNetS 88.2 80.4 ± 3.9 17.2 Note. BER: the bit-error rate of the base model; CA (%): clean accuracy; PA (%): perturbed accuracy; NF: NeuralFuse; and RP: total recovery percentage of PA (NF) vs. PA K Data Embeddings Visualization To further understand how our proposed NeuralFuse works, we visualize the output distribution from the final linear layer of the base models and project the results onto the 2D space using t-SNE [van der Maaten and Hinton, 2008]. Figure 15 shows the output distribution from ResNet18 (trained on CIFAR-10) under a 1% bit error rate. We chose two generators that have similar architecture: ConvL and ConvS, for this experiment. We can observe that: (a) The output distribution of the clean model without NeuralFuse can be grouped into 10 classes denoted by different colors. (b) The output distribution of the perturbed model under a 1% bit error rate without NeuralFuse shows mixed representations and therefore degraded accuracy. (c) The output distribution of the clean model with ConvL shows that applying NeuralFuse will not hurt the prediction of the clean model too much (i.e., it retains high accuracy in the regular voltage setting). (d) The output distribution of the perturbed model with ConvL shows high separability (and therefore high perturbed accuracy) as opposed to (b). (e)/(f) shows the output distribution of the clean/perturbed model with ConvS. For both (e) and (f), 29 we can see nosier clustering when compared to (c) and (d), which means the degraded performance of ConvS compared to ConvL. The visualization validates that NeuralFuse can help retain good data representations under random bit errors and that larger generators in NeuralFuse have better performance than smaller ones. (a) (b) (c) (d) (e) (f) Figure 15: t-SNE results for ResNet18 trained by CIFAR-10 under 1% of bit error rate. (a) Clean model. (b) Perturbed model. (c) Clean model with ConvL. (d) Perturbed model with ConvL. (e) Clean model with ConvS. (f) Perturbed model with ConvS. L Qualitative Analysis of Transformed Inputs In this section, we conduct a qualitative study to visualize the images transformed by NeuralFuse and present some properties and observations of these images. We utilize six different architectures of NeuralFuse generators trained with ResNet18 under a 1% bit error rate. Figure 16 (a) showcases several images from the truck class in CIFAR-10. Notably, images of the same class, when transformed by the same NeuralFuse, exhibit similar patterns, such as circles symbolizing the wheels of the trucks. In Figures 16 (b) and 16 (c), we observe analogous phenomena in the GTSRB and CIFAR-100 datasets. Transformed images of the same class using the same generator consistently display patterns. On GTSRB, NeuralFuse-generated patterns highlight the sign’s shape with a green background, even if the original images have a dark background and are under different lighting conditions. These results further underscore the efficacy and efficiency of NeuralFuse. Figure 17 presents more images from different classes in (a) CIFAR-10, (b) GTSRB, and (c) CIFAR100. The transformed images exhibit distinct patterns for each class, suggesting that NeuralFuse effectively transforms images into class-specific patterns, making associated features robust to random bit errors and easily recognizable by the base model in low-voltage settings. 30 Clean ConvL ConvS DeConvL DeConvS UNetL UNetS (a) Truck class in CIFAR-10. (b) No Passing sign in GTSRB. (c) Apple class in CIFAR-100. Figure 16: Visualization of transformed images from different NeuralFuse generators trained with ResNet18 at 1% bit error rate. Clean ConvL ConvS DeConvL DeConvS UNetL UNetS (a) 10 classes sampled from CIFAR-10 (b) 10 traffic signs sampled from GTSRB Clean ConvL ConvS DeConvL DeConvS UNetL UNetS (c) 20 classes sampled from CIFAR-100 Figure 17: Visualization of transformed images from different NeuralFuse generators trained by ResNet18 with 1% bit error rate. 31 NeurIPS Paper Checklist 1. 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4,438	MambaTree: Tree Topology is All You Need in State Space Model Yicheng Xiao1†∗, Lin Song2,3B∗, Shaoli Huang3, Jiangshan Wang1, Siyu Song4, Yixiao Ge2,3, Xiu Li1B, Ying Shan2,3 1Tsinghua Shenzhen International Graduate School, Tsinghua University 2ARC Lab, Tencent PCG 3Tencent AI Lab 4South China Normal University xiaoyc23@mails.tsinghua.edu.cn ronnysong@tencent.com Abstract The state space models, employing recursively propagated features, demonstrate strong representation capabilities comparable to Transformer models and superior efficiency. However, constrained by the inherent geometric constraints of sequences, it still falls short in modeling long-range dependencies. To address this issue, we propose the MambaTree network, which first dynamically generates a tree topology based on spatial relationships and input features. Then, feature propagation is performed based on this graph, thereby breaking the original sequence constraints to achieve stronger representation capabilities. Additionally, we introduce a linear complexity dynamic programming algorithm to enhance long-range interactions without increasing computational cost. MambaTree is a versatile multimodal framework that can be applied to both visual and textual tasks. Extensive experiments demonstrate that our method significantly outperforms existing structured state space models on image classification, object detection and segmentation. Besides, by fine-tuning large language models, our approach achieves consistent improvements in multiple textual tasks at minor training cost. Code is available at https://github.com/EasonXiao-888/GrootVL. 1 Introduction Mainstream fundamental models are primarily based on CNN [30, 62, 44, 32, 13] and Transformer architectures [15, 43, 42, 59, 14], which dominate in visual and language tasks. However, the small receptive field of CNNs and the high complexity of Transformers make it challenging to strike a good balance between effectiveness and efficiency. The state space models (SSMs) [22, 24, 52] attempt to disrupt this impasse, which model sequences in a recurrent form. Different from the previous recurrent neural networks [31, 7], these approaches draw inspiration from control systems, leveraging structural parameter initialization to attain stable optimization and superior computing performance. Nevertheless, it remains susceptible to the intrinsic flaw shared by recurrent neural networks, i.e., a deficiency in capturing long-range dependencies. Recently, an improved selection mechanism known as Mamba [19] is proposed to mitigate the challenges of SSMs. This approach introduces weight modulation during the propagation process, which substantially enlarges the effective receptive field and achieves impressive performance in NLP tasks. Besides, numerous studies aim to extend Mamba into computer vision, by employing various pre-defined strategies to map 2D image features into 1D sequences. ViM [77] and VMamba [41] utilize a multi-directional raster-scanning strategy, while LocalMamba [34] further confines its ∗Equal contribution. † Work done during an internship at Tencent. B Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). propagation range within a local window. They have successfully adapted Mamba to image inputs. Nevertheless, as shown in Fig. 1(a), both raster-scanning and local-scanning strategies introduce spatial discontinuities between adjacent pixels, and feature transformations in Mamba rely on the feature relationships, thereby impeding the effective information flow in a sequence. Additionally, PlainMamba [69] introduces a continuous scanning strategy, aiming to alleviate this issue by simply adjusting the propagation direction at discontinuous positions. However, all these methods rely on fixed propagation trajectories, which ignore the inherent spatial structure and cannot dynamically adjust the topology based on input. Therefore, this paper endeavors to explore a new perspective: introducing an input-aware topological network for feature propagation in state space models. To achieve it, we develop a tree state space model and propose a new framework, termed MambaTree, which adaptively generates a tree topology based on the input feature and then performs feature propagation on it. Specifically, two sub-networks, MambaTreeV and MambaTreeL, are designed for visual and language tasks respectively, which are illustrated in Fig. 1(b) and Fig. 1(d). For visual tasks, motivated by [71, 54], we first utilize the dissimilarity between adjacent features to construct a minimum spanning tree on a four-connected planner graph. This process can adaptively encode the spatial and semantic information into a tree graph [71, 54]. Then, we iteratively traverse each pixel, considering it as the root vertex, and aggregate the features of other pixels using the state transition function of Mamba. Intuitively, this operation requires two levels of traversal across the entire pixel set, resulting in an unacceptable quadratic complexity relative to the number of pixels. However, given that the tree graph is acyclic, we propose a dynamic programming algorithm to achieve linear complexity propagation. With such an input-aware tree topology, our approach enables more effective long-range interactions while maintaining consistent linear complexity with Mamba. Furthermore, our method can also be applied to language tasks by constructing a tree topology based on the dissimilarity between token features, which overcomes the geometrical constraints of the text sequence. Using a similar aggregation process as MambaTreeV, MambaTreeL can significantly enhance the language representation of a pre-trained Large Language Model [19]. We conduct extensive experiments to validate the effectiveness of MambaTreeV on multiple visual benchmarks, i.e. image classification on ImageNet [12], object detection and instance segmentation on MSCOCO [39] as well as semantic segmentation on ADE20K [75]. Results show that our method notably outperforms existing SSM-based methods for all benchmarks and achieves competitive performance with CNN and Transformer-based approaches. Moreover, with LoRA finetuning [33], MambaTreeL demonstrates consistent improvements for a pre-trained large language model at minor training cost. 2 Related Work 2.1 Conventional Vision Foundation Models The evolution of deep neural networks has been a significant catalyst in machine vision perception. CNN-based models [30, 51, 35, 25, 61, 72, 38, 55, 73] firstly emerge as pivotal landmarks, with ResNet [30] notably standing out for its inventive residual connection module, garnering widespread adoption across diverse domains of visual recognition. Furthermore, more efficient convolution operations are formulated, such as depth-wise convolutions introduced by MobileNet [32], paving the way for lightweight models. Additionally, deformable convolution [10] has been proposed to enhance the receptive field. Subsequently, ViT [15] has significantly improved the vision recognition paradigm. It reformulates the architecture design and training mechanism by combining transformer architecture in natural language processing, aiming to improve computational efficiency and broaden the scope of applications. After research discourse is centred on hierarchical ViTs [43, 42, 11, 63, 14, 56, 5] which design networks by decreasing feature resolution across the backbone gradually. Furthermore, recent research built on CNN serves to re-emphasize the capabilities of convolutional networks. For example, InternImage [62] presents a large model based on deformable CNN, while UniRepLKNet [13] exhibits significant performance through large kernel convolution. 2.2 Explorations about State Space Models State space models (SSMs) have emerged as a novel class of models within the deep learning paradigm, showing significant potential for sequence transforming [23, 22, 52]. These methods have attracted significant attention due to their linear scalability with sequence length. The early method, 2 Continuous Scan Local Scan Raster Scan Tree Topology Scan shortcrust …… ingredients do I need make a shortcrust for …… mini to What pies What ingredients do I need to make a for mini pies (a) Previous Visual SSMs (b) MambaTreeV (c) Previous Textual SSMs (d) MambaTreeL Figure 1: Comparison of different propagation strategies for multi-modal tasks. For visual tasks, the previous strategies (a) are based on fixed patterns, while our method can adaptively generate the propagation topology according to input features. For textual tasks, compared to previous methods (c), our approach (d) can break the inherent constraints of text sequences, facilitating the effective transmission of long-range information. LSSL [23], draws inspiration from continuous state space models in control systems and attempts to address the long-range dependency problem through a combination with HIPPO [20] initialization. S4 [22] proposes to normalize the parameters into a diagonal matrix, prompting a subsequent series of research on structured SSMs [24, 21, 26, 19]. Recently, the Selective State Space Model [19], known as Mamba, strikes a balance between effectiveness and efficiency through the design of an input-dependent parameter initialization strategy, which has emerged as a formidable competitor to both transformer and CNN structures. In addition to showcasing superior outcomes in sequence modeling, Mamba has been seamlessly incorporated into the visual domain [77, 41, 34, 69, 68]. These studies often rely on handcrafted fixed scanning mechanisms to mitigate the execution bias of the selective state space model on 2D non-causal images. However, such simplistic approaches cannot effectively capture spatial relationships in an input-dependent paradigm. To address this limitation, we propose an effective framework MambaTree in this work to enhance long-range modeling for both vision and language tasks by introducing an input-aware tree-based topological structure. 3 Method In this section, we first revisit the selective state space model [19] and then elaborate on our inputaware topology scanning algorithm for state space modeling. With this superior algorithm, we develop a tree SSM and propose a novel framework called MambaTree, which consists of two sub-networks: MambaTreeV for visual tasks and MambaTreeL for fine-tuning a pre-trained language model [19]. 3.1 Revisiting Selective State Space Model State Space Models (SSMs) are commonly regarded as continuous linear time-invariant systems [64] that map input stimulation x(t) ∈R1×D to output signal y(t) ∈R1×D through a state vector h(t) ∈R1×N, where t, D and N indicate the time step, channel number of the signal and state size, respectively. These models can be formulated as the following linear ordinary differential equations: h′(t) = Ah(t) + Bx(t), y(t) = Ch(t) + Dx(t), (1) where A ∈RN×N, B ∈RN×D, C ∈RN×D and feedthrough coefficient D ∈RD. Discretization. Although SSM serves as a powerful tool in systems and control engineering, its time-continuous nature poses challenges for integration into deep learning architectures. To alleviate this issue, most methods utilize the zero-order hold rule [19] to discretize the continuous system described by Eq. (1) and convert continuous variables (A, B, C, D) into corresponding discrete 3 TSA SiLU Building a 4-connected Graph ! !", !$, %, & Minimum Spanning Tree Linear Linear SiLU LN × Linear Pruning ( Parameters Generator !", !$, %, & Linear Embedding " SoftPlus !" = +', -" !$ = -$ $ % & Linear Linear Tree Scanning Algorithm Tree State Space Model DWConv Parameters Generator ' ' ( ' Figure 2: Illustration of Tree State Space Model. With an image feature map x, we perform Tree Scanning Algorithm (TSA) to construct a 4-connected graph with edge weights measured by dissimilarity between pixels. Then, we obtain an MST with vertices set Ωthrough a pruning algorithm and perform the state transition for each vertex in this topology (detailed in Sec. 3.2). Red arrows describe the propagation source of vertex i. parameters ( ¯A, ¯B, ¯C, ¯D) over the specified sampling time-scale ∆∈RD: ¯A = e∆A, ¯B = e∆A −I A−1B, ¯C = C, ¯D = D (2) In addition, many improved methods [41, 19] use an approximation of ¯B based on the first-order Taylor Series: ¯B = e∆A −I A−1B ≈(∆A)(∆A)−1∆B = ∆B (3) Selective Mechanism . Previous SSMs store information through finite states and inherent timeinvariance, which limits their effectiveness. Therefore, Mamba [19] introduces a dynamic mechanism to selectively filter out input into a sequential state. Specifically, it utilizes Linear Projection to calculate the parameters {Bi}L i=1, {Ci}L i=1 and {∆i}L i=1 from the input sequence {xi}L i=1 with xi ∈R1×D directly to improve the context-aware ability. Then the output sequence {yi}L i=1 can be computed with those input-adaptive discretized parameters as follows: hi = ¯Aihi−1 + ¯Bixi, yi = Cihi + Dxi (4) 3.2 Tree State Space Model Mamba [19] has showcased remarkable performance in modeling the dependencies of consecutive words in a sequence. However, its applicability in long-context tasks, especially visual modeling, still poses certain challenges. For visual tasks, many methods attempt to address this problem by employing fixed scanning strategies, such as multi-directional raster scan [41, 77], local scan [34], and continuous scan [69]. However, these handcrafted scanning methods fail to effectively preserve the 2D structural information of images. Following the design in Mamba [19], we construct a transform block as a tree state space model, which is presented in Fig. 2. The only difference between our block and Mamba lies in the replacement of the structured state space block with the proposed tree scanning algorithm. In the tree scanning algorithm, we generate a tree topology and then propagate the state of each vertex along the topological path to obtain strong feature representations. In addition, our algorithm can effectively enhance language representations by incorporating such a tree topology during text processing, which overcomes the geometrical constraints of text sequences. In the following, we elaborate on the proposed tree scanning algorithm and its applications for multi-modal tasks. Tree Scanning Algorithm. Given an input feature X = {xi}L i=1 where L is the sequence length (or the number of input pixels), we construct an undirected m-connected graph G = (V, E) for the 4 Stem !×#×3 STAGE 2 Basic Block ×"% STAGE 3 Basic Block ×"& STAGE 4 Basic Block ×"' Down Sampling Down Sampling Down Sampling STAGE 1 Basic Block ×"( ! 4 × # 4 ×* ! 8 × # 8 ×2* ! 16 × # 16 ×4* ! 32 × # 32 ×8* Head LN Tree State Space Model FFN LN + + Stem Conv: 3×3, S2, P1 LN + GELU Conv: 3×3, S2, P1 LN Basic Block Down Sampling Conv: 3×3, S2, P1 LN Head Cls. Det. Seg. Figure 3: Overview of MambaTreeV. LN means LayerNorm and FFN is a feed-forward network in the basic block. S2 and P1 denote stride of 2 and padding size of 1 in convolution, respectively. feature. m is a hyper-parameter that indicates the number of adjacent tokens. Following [71, 54], we set m = 4 for visual tasks, meaning each pixel is connected to its four neighboring pixels. For language tasks, we set m = 3 by default, meaning each token is connected to the previous three tokens. In addition, the vertices V represent the pixel (or token) embeddings, and the E indicates the edges of the graph. The edge weight is calculated by the feature dissimilarity between adjacent vertices. Besides, the metric of dissimilarity uses cosine distance by default, and the comparison with other metrics refers to Table 5. We use the Contractive Boruvka algorithm [2] to prune the edges with significant dissimilarity, which generates a minimum spanning tree (MST) GT whose sum of dissimilarity weights is minimum out of all spanning trees. In the propagation process, we iteratively traverse each vertex, treating it as the root, and aggregate the features of the remaining vertices. Intuitively, applying state propagation within such a geometric configuration makes its preferential interactions among vertices with small spatial and feature distances. Following the Mamba, we employ the data-dependent transition matrix for state propagation. For a vertex k, we denote the transition matrix with its parent as ¯Ak. Furthermore, following the Eq. (4), the state aggregation process for the i-th vertex can be formulated as: hi = X ∀j∈Ω S(Eij) ¯Bjxj, S(Eij) = Y k∈Nij ¯Ak, (5) where Ωdenotes the index set of all vertices in the tree. S(Eij) represents the path weight of hyperedge Eij traced from j-th vertex to i-th vertex in the tree GT , and Nij indicates the index set of all vertices on this hyperedge. For visual tasks, we iterate over each vertex, treating it as the root of the spanning tree GT , and aggregate the states from the other vertices, thereby obtaining the transformed states {hi}L i=1. For textual tasks, because of the causal prediction manner in large language models, we only take the last token as root and aggregate from other tokens. To achieve end-to-end training, we derive the derivative of the output hidden state hi to the input variables ¯Ak, ¯Bj and xj as follows: ∂hi ∂xj = S (Eij) ¯Bj, ∂hi ∂¯Bj = S (Eij) xj (6) ∂hi ∂¯Ak = X ∀j∈Ci k ¯BjxjS(Ekj)S(Ein), (7) where Ci k indicates the children of vertex k in tree GT whose root is the vertex i, and n denotes the parent of vertex k in Eq. (7). Finally, the output feature Y can be formulated as: Y = C ⊙Norm(H) + D ⊙X, (8) where Y , H and X indicate the stack of {yi}L i=1, {hi}L i=1 and {x}L i=1 respectively. ⊙denotes the element-wise multiplication. Efficient Implementation for Multi-Modality. For visual tasks, the tree scanning algorithm requires two levels of traversal across the entire pixel set, resulting in an unacceptable quadratic complexity relative to the number of pixels O(L2). To alleviate this issue, we utilize a dynamic 5 Algorithm 1 Vision Tree Scanning Input: Input feature {xi}L i=1; Input matrix { ¯Bi}L i=1; State matrix { ¯Ai}L i=1; Gradient of loss to hidden states { ∂Loss ∂hi }L i=1; Minimum Spanning Tree GT . Traverse Path: Root, . . . , Leaf ←BFS(GT ) ▷Breadth-first topological order of GT Forward: Initialization: {ξi}L i=1 ←{xi}L i=1 2: for i ←Leaf to Root do ξi = ¯Bixi + P ∀j∈{t\|Par(t)=i} ξj ¯Aj 4: end for for i ←Root to Leaf do 6: if i is Root then hi = ξi 8: else hi = ¯Ai(hPar(i) −¯Aiξi) + ξi = (1 −¯A2 i )ξi + ¯AihPar(i) 10: end if end for Backward: 12: Initialization: {ηi}L i=1 ←{ ∂Loss ∂hi }L i=1 for i ←Leaf to Root do 14: ηi = ¯Bi ∂Loss ∂hi + P ∀j∈{t\|Par(t)=i} ηj ¯Aj end for 16: for i ←Root to Leaf do if i is Root then 18: ∂Loss ∂xi = ηi ¯Bi , ∂Loss ∂¯Bi = ηixi , ∂Loss ∂¯Ai = 0 else 20: ∂Loss ∂xi = (1 −¯A2 i )ηi ¯Bi + ¯Ai ∂Loss ∂xPar(i) ¯Bi , ∂Loss ∂¯Bi = (1 −¯A2 i )ηixi + ¯Ai ∂Loss ∂¯BPar(i) xi ∂Loss ∂¯Ai = ηi ∗(hi −¯Aiξi) + ξi ∗( ∂Loss ∂xi −¯Aiηi) = ηihi + ξi ∂Loss ∂xi −2ηiξi ¯Ai 22: end if end for Output: Hidden states {hi}L i=1; Grad. of loss to input feature { ∂Loss ∂xi }L i=1; Grad. of loss to input matrix { ∂Loss ∂¯Bi }L i=1; Grad. of loss to state matrix { ∂Loss ∂¯Ai }L i=1. programming procedure to accelerate the inference and training processes as elaborated in Algorithm 1, which results in linear complexity O(L). For textual tasks, we perform a unidirectional aggregation approach (shown in Algorithm 2 of Appendix B) in adherence to the causal nature of language. Moreover, we provide the back-propagation process for both Vision Tree Scanning and Language Tree Scanning processes, whose detailed proofs refer to Appendix C. 3.3 Application for Vision and Language MambaTreeV Given an image with a shape of H × W × 3, our goal is to obtain high-quality visual features for downstream tasks. To this end, we propose an effective vision architecture MambaTreeV which consists of a stem module, several basic blocks and downsampling layers to generate hierarchical representations illustrated in Fig. 3. Overall, our MambaTreeV comprises four stages similar to previous general vision backbones [44, 43, 62, 41]. We integrate the stem module before the first stage to decrease the resolution of the input image signal by a factor of 4, resulting in a feature map with a shape of H 4 × W 4 ×C. It includes two convolutions, two Layer Normalization (LN) layers and one GELU activation function. The kernel size for both convolutions is 3 with a stride of 2 and padding of 1. Similarly, a downsampling layer consists of a 3 × 3 convolution with a stride of 2 and padding of 1 and an LN layer. Positioned between two stages, it serves to downsample the input feature map by a factor of 2. Motivated by [62, 41], we devise a residual block with skip connections to integrate our fundamental Tree State Space Model in Sec. 3.2. In detail, we first normalize the input features with LN layer. Spatial priors and long-range dependencies are then obtained through our tree scanning algorithm with residual connections established alongside the input features. Finally, a feedforward neural network is utilized to project the normalized features to output signals as shown 6 Method Type #Param. #FLOPs Top-1 Acc. Deit-S [59] T 22M 4.6G 79.9 Swin-T [43] T 28M 4.6G 81.3 CoAtNet-0 [11] T 25M 4.0G 81.6 SG-Former-S [50] T 23M 4.8G 83.2 ConvNeXt-T [44] C 29M 4.5G 82.1 SLaK-T [40] C 30M 5.0G 82.5 UniRepLKNet-T [13] C 31M 4.9G 83.2 InternImage-T [62] C 30M 5.0G 83.5 ViM-S [77] S 26M 5.1G 80.5 LocalViM-S [34] S 28M 4.8G 81.2 PlainMamba-L2 [69] S 25M 8.1G 81.6 Mamba-2D-S [37] S 24M 81.7 S4ND-ConvNeXt-T [48] S 30M 82.2 VMamba-T [41] S 31M 4.9G 82.5 LocalVMamba-T [34] S 26M 5.7G 82.7 MambaTreeV-T (Ours) S 30M 4.8G 83.4 Swin-S [43] T 50M 8.7G 83.0 CoAtNet-1 [11] T 42M 8.0G 83.3 Method Type #Param. #FLOPs Top-1 Acc. ConvNeXt-S [44] C 50M 8.7G 83.1 SLaK-S [40] C 55M 9.8G 83.8 UniRepLKNet-S [13] C 56M 9.1G 83.9 InternImage-S [62] C 50M 8.0G 84.2 HyenaViT-B [17] S 88M 78.5 S4ND-ViT-B [48] S 89M 80.4 PlainMamba-L3 [69] S 50M 14.4G 82.3 VMamba-S [41] S 50M 8.7G 83.6 LocalVMamba-S [34] S 50M 11.4G 83.7 MambaTreeV-S (Ours) S 51M 8.5G 84.2 Deit-B [59] T 86M 55.4G 83.1 Swin-B [43] T 88M 15.4G 83.5 CoAtNet-2 [11] T 75M 16.0G 84.1 ConvNeXt-B [44] C 89M 15.4G 83.8 SLaK-B [40] C 95M 17.0G 84.0 Mamba-2D-B [37] S 92M 83.0 VMamba-B [41] S 89M 15.4G 83.9 MambaTreeV-B (Ours) S 91M 15.1G 84.8 Table 1: Image classification performance on the ImageNet-1K validation set. T, C and S indicate the model type of Transformer, CNN and SSM, respectively. All models take a scale of 2242 as input. in Fig. 3. Based on the above origin components, we develop our MambaTreeV in three scales, i.e., MambaTreeV-Tiny, MambaTreeV-Small and MambaTreeV-Base. MambaTreeL Recurrent neural networks rely on fixed memory to preserve past information, which poses limitations when handling long contexts where relevant words are distant from the current moment. While Mamba [19] employs a selection mechanism to enhance context awareness, its fixed memory size cannot expand over time, resulting in restricted state space. Therefore, the ability to extrapolate decreases during scrolling as the prompt extends. To mitigate this issue, we propose an effective fine-tuning paradigm. Specifically, the tree-based topology branch is built upon one-way scrolling with a scaling factor, enabling state transitions within such a structure. This arrangement facilitates the preferential interaction of semantically related tokens. It is noteworthy that this paradigm does not introduce any additional training parameters. Instead, it utilizes pretrained state transformation parameters to conduct semantic aggregation by incorporating topological structures. Experimental results demonstrate the effectiveness of our approach. 4 Experiments We conduct extensive experiments to evaluate the effectiveness of MambaTreeV and compare it with advanced CNN-based, Transformer-based, and SSM-based models covering various downstream tasks, including image classification, object detection and semantic segmentation. Furthermore, we validate the capability of MambaTreeL in the field of natural language understanding. 4.1 Image Classification Settings. We assess the classification performance of MambaTreeV on the ImageNet-1k dataset [12]. Following previous practices [43, 44, 62, 41], all MambaTreeV models are trained for 300 epochs from scratch using AdamW optimizer with a warm-up strategy of 20 epochs. During training, we utilize a Cosine Scheduler with an initial learning rate of 1 × 10−3 and weight decay of 0.05. In addition, the exponential moving average (EMA) is also applied. Results. The comparison results summarized in Table 1 show MambaTreeV leading all SSM-based models and competitive with advanced CNNs and Transformers across tiny, small, and base scales. Specifically, MambaTreeV-T achieves 83.4% Top-1 Acc. boosting ViM-S by 2.9%, LocalVim-S by 2.2%, PlainMamba-L2 by 1.8% and VMamba-T by 0.9% with similar FLOPs. Additionally, it surpasses ConvNeXt-T by 1.3% and Swin-T by 2.2%, demonstrating the effectiveness of our method. 7 (a) Input Signal (b) TP Scan (c) Raster Scan Figure 4: Visualization of affinity maps in the specific position. The Location is marked by the red cross in each input (a). TP is our tree topology scanning algorithm (b), which captures more detailed structural information and has a larger receptive field compared to raster scanning (c). Method Type #FLOPs mIoU mIoU SS MS Swin-T [43] T 945G 44.5 45.8 ConvNeXt-T [44] C 939G 46.0 46.7 SLaK-T [40] C 936G 47.6 InternImage-T [62] C 944G 47.9 48.1 UniRepLKNet-T [13] C 946G 48.6 49.1 ViM-S [77] S 44.9 LocalViM-S [34] S 297G 46.4 47.5 PlainMamba-L2 [69] S 285G 46.8 VMamba-T [41] S 964G 47.3 48.3 LocalVMamba-T [41] S 970G 47.9 49.1 MambaTreeV-T (Ours) S 941G 48.5 49.4 Swin-S [43] T 1038G 47.6 49.5 ConvNeXt-S [44] C 1027G 48.7 49.6 SLaK-S [40] C 1028G 49.4 InternImage-S [62] C 1017G 50.1 50.9 UniRepLKNet-S [13] C 1036G 50.5 51.0 PlainMamba-L3 [69] S 419G 49.1 VMamba-S [41] S 1081G 49.5 50.5 LocalVMamba-S [34] S 1095G 50.0 51.0 MambaTreeV-S (Ours) S 1019G 50.7 51.7 Table 2: Semantic segmentation performance on ADE20K val set. The crop size is all set to 5122. SS and MS denote singlescale and multi-scale testing, respectively. 4.2 Object Detection Settings. We verify the detection performance of MambaTreeV on the MSCOCO 2017 dataset [39] with MMDetection library [3]. We follow previous works [41, 62, 43, 34, 53, 55, 74, 70, 6] to validate object detection and instance segmentation tasks with Mask-RCNN [29]. Specifically, We adopt the AdamW optimizer with a learning rate of 1 × 10−4 and batch size of 16 to optimize the model built upon our pre-trained classification backbones on ImageNet-1K. The training schedules include 1× (12 epochs) and 3× (36 epochs) with multi-scale data augmentation. Results. As depicted in Table 8 (in Appendix A.), our method outperforms existing methods on most evaluation metrics, especially for instance segmentation. Under 1× schedule, MambaTreeV-T achieves 47.0 in box mAP (APb), which is 1.1 points higher than ViM-S and 0.5 points higher than VMamba-T. It is worth noting that MambaTreeV-T outperforms ViM-S by 1.7 points with 1× schedule and LocalVMamba-T by 0.4 points with 3× schedule in mask mAP (APm). Moreover, the best APb 50.1 and APm 44.6 are obtained by MambaTreeV-S in 3× schedule with multi-scale training. 4.3 Semantic Segmentation Settings. To evaluate the semantic segmentation performance of our MambaTreeV series, we train our models with UperNet [65] initialized by pre-trained classification weights on ADE20K[75] for 160k iterations, following common practices without additional augmentations for fair comparison. Results. Our method performs exceptionally well on segmentation tasks shown in Table 2. MambaTreeV-T yields a clear improvement of +3.6 in single-scale mIoU compared to ViM-S and +1.9 in multi-scale mIoU compared to LocalViM-S. Furthermore, MambaTreeV-S boosts InternImage-S by 0.6 and 0.8 in single-scale and multi-scale respectively. We consider the preservation of intricate structural details through tree topology scanning to be particularly advantageous for segmentation tasks that require pixel-level perception. 8 Method PIQA ↑ Arc-E ↑ SST ↑ WG ↑ L-ppl ↓ Race ↑ BQA ↑ Average↑ Acc. Mamba [19] 64.5 48.0 65.6 51.8 16.1 27.4 16.8 45.7 + LoRA [33] 64.7 48.3 65.1 52.2 17.7 28.6 17.8 46.1 + MambaTreeL (Ours) 65.0 49.8 69.5 51.1 15.9 28.9 19.2 47.2 Table 3: Evaluation on language model benchmarks. Arc-E, WG, L-ppl and BQA indicate Arceasy [8], WinoGrande, LAMBADA [49] and Openbookqa [47] benchmark, respectively. Scanning Strategy Acc Raster Scan 82.6 Cross Scan 83.1 Tree Topology Scan 83.4 Table 4: Effectiveness of our algorithm. Distance Metric Acc. Manhattan 82.9 Euclidean 83.2 Cosine 83.4 Table 5: Impact of different distance Metrics. Root Setting Acc. First vertex 82.9 Last vertex 83.0 All vertices 83.4 Table 6: Superiority of traversing all vertices. 4.4 Language Understanding We regard Mamba [19] with 130M parameters as the base model. To verify the effectiveness of our MambaTreeL in nature language understanding, we first fine-tune pre-trained Mamba via LoRA [33] and MambaTreeL under the same setting with the Alpaca data [58], which contains 52000 instruction tuning data for supervised fine-tuning. Then we utilize popular language benchmarks provided in the open-sourced lm-evaluation-harness project [18] for evaluation, including PIQA [1], AI2-ARC [8], SST [60], WinoGrande, LAMBADA [49], Race [36] and Openbookqa [47]. The results in Table 3 demonstrate that our MambaTreeL provides a benefit of +1.1% in average Acc. compared to LoRA. Since the short prompt length of WinoGrande dataset, the performance degrades with a marginal gap. 4.5 Ablation Study & Qualitative Results In this section, we conduct analysis experiments on ImageNet-1K dataset and present some visual results to illustrate the effectiveness of our algorithm. Scanning Strategy. We conduct a head-to-head comparison of different scanning strategies, as shown in Table 4. The tree topology scanning outperforms previous strategies by 0.8% and 0.3%, highlighting the superiority of our algorithm in vision recognition. Distance Metric. Before generating a minimum spanning tree from a connected graph, it is important to measure the edge weights between vertices. Therefore, we validate several distance metrics as illustrated in Table 5. The results indicate that Cosine distance most effectively represents the relationship between vertices, performing 0.5% better than Manhattan and 0.2% better than Euclidean. Root Setting. We traverse all vertices, treating each as a root, and perform state transitions along the topological path from the other vertices toward the root. This traversal ensures that each vertex captures long-range dependencies. To verify the effectiveness of this operation, we consider only the first and last vertices as the root in Table 6. The results show reductions of 0.5% and 0.4%, respectively. Inference speed comparison. As shown in Table 7, we report the inference throughputs of our method on an Nvidia V100 GPU. MambaTreeV-T∗refers to each stage sharing the same tree topology structure, which enhances efficiency without compromising accuracy. To achieve better practical inference speed, we also introduce a cuda implementation optimized for GPUs. Compared with other counterparts, our approach exhibits superior effectiveness and faster inference speed. Qualitative Results. To better illustrate the superiority of our scanning strategy, we visualize the affinity maps of different positions marked by the red cross in each input image. For example, we 9 Method Throughput GPU Memory FLOPs #Params. Acc. (Top1.) (224x224) (img/s) PlainMamba-L2 [69] 363 4204M 8.1G 25M 81.6 VMamba-T [41] 374 8646M 4.9G 31M 82.5 LocalVMamba-T [34] 311 11298M 5.7G 26M 82.7 MambaTreeV-T(one root) 283 6012M 4.8G 30M 83.0 MambaTreeV-T 281 6471M 4.8G 30M 83.4 MambaTreeV-T∗ 392 4800M 4.8G 30M 83.4 Table 7: Runtime comparison on an Nvidia V100 GPU during inference. set the anchor point in the upper left corner of the sky as shown in the second row of in Fig. 4(a). Our method can easily identify white houses, flagpoles, and the sky, which raster scanning fails to achieve. This demonstrates the capability of our algorithm to preserve detailed structural information. More comparisons can be seen in Fig. 6 (in Appendix D.) 5 Conclusion & Limitations In this paper, we propose a tree state space model to perform feature propagation on an input-aware topology. Besides, we introduce a linear complexity dynamic programming algorithm to enhance long-range interactions without increasing computational cost. With the proposed techniques, we establish the general multi-modal networks to break the original sequence constraints and achieve stronger representation capabilities. Extensive experiments demonstrate the effectiveness of our method in both visual and language tasks. The limitation of our method is that the tree structure is not a common paradigm, and it needs to be specifically optimized according to the hardware device. Acknowledgments and Disclosure of Funding This work was supported by the STI 2030-Major Projects under Grant 2021ZD0201404. 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TNNLS (2023) 15 [77] Zhu, L., Liao, B., Zhang, Q., Wang, X., Liu, W., Wang, X.: Vision mamba: Efficient visual representation learning with bidirectional state space model. arXiv preprint arXiv:2401.09417 (2024) 1, 3, 4, 7, 8, 16 14 Appendix A Detailed Training Settings and Results A.1 Image Classification. We follow the previous works [62, 41, 43] to conduct the experiments. The models are trained with thirty-two 32GB V100 GPUs by default. We set betas and momentum of the AdamW [45, 76, 66] optimizer with (0.9, 0.999) and 0.9, respectively. During training, we utilize a Cosine Scheduler with an initial learning rate of 1×10−3 and weight decay of 0.05. We adopt the common training data augmentation strategies following [34, 62], including AutoAugment [9] with rand-m9-mstd0.5-inc1. A MixUp strategy with a ratio of 0.8 is also adopted in each batch. Horizontal flip and Random resized crop strategy are both used in the process of training. 6 8 10 12 14 FLOPs 81 82 83 84 85 Top-1 Accuracy MambaTreeV-T MambaTreeV-S MambaTreeV-B PlainMamba-L2 PlainMamba-L3 VMamba-T VMamba-S VMamba-B ViM-S LocalMamba-T LocalMamba-S MambaTreeV-T PlainMamba-L2 VMamba-T ViM-S LocalMamba-T Figure 5: Classification performance comparison among SSM-based vision foundation models. Performance Comparison. We compare various SSM-based visual foundation models as shown in Fig. 5, with different colors representing different models and different shapes indicating different model scales. The size of each shape indicates the number of model parameters. The horizontal axis denotes FLOPs and the vertical axis represents the Top-1 accuracy of the corresponding method on ImageNet-1K val dataset. Fig. 5 demonstrates that MambaTreeV is the best choice in terms of efficiency and effectiveness. A.2 Object Detection. For a fair comparison, we conduct the evaluation following common practice [62, 41, 43]. The models are trained with eight 32GB V100 GPUs by default. The input image is resized so that the shorter side is 800 pixels, while the longer side does not exceed 1333 pixels during the 1× schedule. The number of warmup steps is set to 500 in the 1× schedule. For 3× schedule, the shorter side is resized to 480-800 pixels and the longer side does not exceed 1333 pixels. The number of warmup steps is set to 1000 in 3× schedule. Results shown in Table 8 demonstrate the effectiveness of MambaTreeV in object detection and instance segmentation on COCO val2017. A.3 Semantic Segmentation. We optimize our MambaTreeV-T/S using AdamW optimizer with an initial learning rate of 6 × 10−5 which is decayed by a rate of 1.0 with the polynomial decay schedule following [62, 27, 57, 16, 28]. The number of warmup iters is set to 1600 with an initial learning rate of 1 × 10−6 [41, 34, 46, 67]. The default input resolution is 512 × 512 as well as FLOPs are calculated with an input size of 512 × 2048. The models are trained with eight 32GB V100 GPUs by default. 15 Method #FLOPs. Mask R-CNN 1× Schedule Mask R-CNN 3× MS Schedule APb APb 50 APb 75 APm APm 50 APm 75 APb APb 50 APb 75 APm APm 50 APm 75 Swin-T [43] 267G 42.7 65.2 46.8 39.3 62.2 42.2 46.0 68.1 50.3 41.6 65.1 44.9 ConvNeXt-T [44] 262G 44.2 66.6 48.3 40.1 63.3 42.8 46.2 67.9 50.8 41.7 65.0 44.9 CSWin-T [14] 279G 46.7 68.6 51.3 42.2 65.6 45.4 49.0 70.7 53.7 43.6 67.9 46.6 ViM-S [77] 218G 44.9 67.1 49.3 41.0 64.2 44.1 VMamba-T [41] 286G 46.5 68.5 50.7 42.1 65.5 45.3 48.5 69.9 52.9 43.2 66.8 46.3 L-Vmamba-T [34] 291G 46.7 68.7 50.8 42.2 65.7 45.5 48.7 70.1 53.0 43.4 67.0 46.4 MambaTreeV-T (Ours) 265G 47.0 69.4 51.5 42.7 66.4 46.0 49.0 70.8 54.0 43.8 67.6 47.1 Vit-Adapter-S [4] 403G 44.7 65.8 48.3 39.9 62.5 42.8 48.2 69.7 52.5 42.8 66.4 45.9 Swin-S [43] 354G 44.8 66.6 48.9 40.9 63.4 44.2 48.2 69.8 52.8 43.2 67.0 46.1 ConvNeXt-T [44] 348G 45.4 67.9 50.0 41.8 65.2 45.1 47.9 70.0 52.7 42.9 66.9 46.2 InternImage-S [62] 340G 47.8 69.8 52.8 43.3 67.1 46.7 49.7 71.1 54.5 44.5 68.5 47.8 VMamba-S [41] 400G 48.2 69.7 52.5 43.0 66.6 46.4 49.7 70.4 54.2 44.0 67.6 47.3 L-Vmamba-S [34] 414G 48.4 69.9 52.7 43.2 66.7 46.5 49.9 70.5 54.4 44.1 67.8 47.4 MambaTreeV-S (Ours) 341G 48.6 70.3 53.5 43.6 67.5 47.1 50.1 71.2 54.9 44.6 68.7 47.8 Table 8: Object detection and instance segmentation performance on COCO val2017. APb and APm indicate the mAP of detection and segmentation, respectively. MS indicates the multi-scale training strategy. B Language Tree Topology Scanning Operator Algorithm 2 Language Tree Scanning Input: Input feature {xi}L i=1; Input matrix { ¯Bi}L i=1; State matrix { ¯Ai}L i=1; Gradient of loss to hidden states { ∂Loss ∂hi }L i=1; Minimum Spanning Tree GT . Traverse Path: Root, . . . , Leaf ←BFS(GT ) ▷Breadth-first topological order of GT Forward: Initialization: {ξi}L i=1 ←{xi}L i=1 2: for i ←Leaf to Root do ξi = ¯Bixi + P ∀j∈{t\|Par(t)=i} ξj ¯Aj 4: end for Backward: for i ←Root to Leaf do 6: if i is Root then ∂Loss ∂xi = ηi ¯Bi , ∂Loss ∂¯Bi = ηixi, ∂Loss ∂¯Ai = 0 8: else ∂Loss ∂xi = ∂Loss ∂hi ¯Bi + ¯Ai ∂Loss ∂xPar(i) ¯Bi , ∂Loss ∂¯Bi = ∂Loss ∂hi xi + ¯Ai ∂Loss ∂¯BPar(i) xi 10: ∂Loss ∂¯Ai = ∂Loss ∂x′ P ar(i) hi end if 12: end for Output: Hidden states {hi}L i=1; Grad. of loss to input feature { ∂Loss ∂xi }L i=1; Grad. of loss to input matrix { ∂Loss ∂¯Bi }L i=1; Grad. of loss to state matrix { ∂Loss ∂¯Ai }L i=1. C Algorithm Proof In this section, we present detailed proofs for our tree scanning algorithm. The definitions of symbols are consistent with those in the main paper. 16 C.1 Proof for Algorithm 1. We randomly take a vertex in the MST GT as the root. According to the definition of the tree scanning algorithm introduced in Sec. 3.2, we can derive hroot as follows: hroot = X ∀j∈Croot S(Eroot,j) ¯Bjxj, S(Eroot,j) = Y k∈Nroot,j ¯Ak, (9) which shows a process of aggregation from all leaf vertices to the root. Therefore, each vertex is only related to its child in this period. Taking vertex m as an example, the Aggrm can be derived as: Aggrm(x) = ¯Bmxm + X ∀k∈{t\|Par(t)=i} Aggrk(x) ¯Ak. (10) We assume that one of the child of m is n and hn can be derived as following: hn = Aggrn(x) + ¯An] Aggrm(x), (11) where ] Aggrm(x) indicates the aggregation value from the vertices ∈Ω\Croot m to vertex m. Therefore, we can obtain the propagation relationship between the hidden state of parent m and child n: hn = Aggrn(x) + ¯An] Aggrm(x) = Aggrn(x) + ¯An(hm −¯AnAggrn(x)) = ¯Anhm + (1 −¯A2 n)Aggrn(x) (12) Through the above derivation, we can calculate {hi}L i=1 with only two traversals (i.e., the aggregation from leaf to root and the propagation from root to leaf) in the forward process as shown in Algorithm 1, thereby reducing the computational complexity from O(L2) to O(L). Next, we analyze the backpropagation process in Algorithm 1. According to the chain rule, we can easily calculate the derivative of loss with respect to xi: ∂loss ∂xi = X j∈Ω ∂loss ∂hj ∂hj ∂xi = ¯Bi X j∈Ω S (Eji) ∂loss ∂hj (13) Similarly, the derivative of loss with respect to ¯Bi is: ∂loss ∂¯Bi = X j∈Ω ∂loss ∂hj ∂hj ∂¯Bi = xi X j∈Ω S (Eji) ∂loss ∂hj (14) The above formulas are equivalent to replacing the input x with ∂loss ∂h during the forward process. 17 Subsequently, we assume that the vertex k is the child of vertex l and define Ck l indicates the children of vertex l with the root of vertex k. ∂loss ∂¯Ak is formulated as follows: ∂loss ∂¯Ak = X j∈Ω ∂loss ∂hj ∂hj ∂¯Ak = X j∈Ω ∂loss ∂hj X p∈Ω ∂S(Ejp) ¯Bpx′ p ∂¯Ak = X j∈Ck l ∂loss ∂hj X p∈Cl k S(Ekp)S(Ejl) ¯Bpx′ p + X j∈Cl k ∂loss ∂hj X p∈Ck l S(Ekj)S(Epl) ¯Bpx′ p = X j∈Ck l S(Ejl)∂loss ∂hj X p∈Cl k S(Ekp) ¯Bpx′ p + X j∈Cl k S(Ekj)∂loss ∂hj X p∈Ck l S(Epl) ¯Bpx′ p = (∂Loss ∂xk −¯AkAggrk(∂loss ∂h )) ∗Aggrk(x) + Aggrk(∂loss ∂h ) ∗(hk −¯AkAggrk(x)) = ∂Loss ∂xk Aggrk(x) + Aggrk(∂loss ∂h )hk −2 ¯AkAggrk(∂loss ∂h )Aggrk(x) ≜∂Loss ∂xk ξk + ηkhk −2 ¯Akηkξk (definition in Algorithm 1) (15) So far we have completed the proof of forward and back-propagation of Algorithm 1. C.2 Proof for Algorithm 2. We only take the last token as root and replace the transition source from Ωto Ci in sequence modeling tasks like nature language understanding to ensure causality. Therefore, only one traversal (from leaf to root) is required for the forward process, and another traversal (from root to leaf) is needed for the backpropagation process. The proof is similar to the Algorithm 1. (a) (b) (c) (d) Input TP Raster Cross TP Raster Cross Figure 6: Visualization of affinity maps in the specific position. The Location is marked by the red cross in each affinity map. TP represents our Tree Scanning Algorithm. 18 D More Qualitative Results Fig. 6 displays additional qualitative comparisons between our algorithm and previous scanning strategies (e.g., cross-scanning and raster-scanning), which shows our advanced capability to perceive detailed structural information and capture long-range dependencies. E Statistical Significance Method PIQA Arc-Easy SST WinoGrande LAM-ppl Race Openbookqa MambaTreeL (Ours) 0.011 0.010 0.016 0.014 0.553 0.014 0.018 Table 9: Standard error on language model benchmarks. LAM-ppl indicates LAMBADA [49]. We calculate the standard deviation of our MambaTreeL on language model benchmarks in the open-sourced lm-evaluation-harness project as shown in Table 9. 19 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The claims in the abstract and introduction accurately reflect our motivation and contribution. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,439	Achieving Near-Optimal Convergence for Distributed Minimax Optimization with Adaptive Stepsizes Yan Huang College of Control Science and Engineering Zhejiang University, China huangyan5616@zju.edu.cn Xiang Li Department of Computer Science ETH Zurich, Switzerland xiang.li@inf.ethz.ch Yipeng Shen College of Control Science and Engineering Zhejiang University, China 22332074@zju.edu.cn Niao He Department of Computer Science ETH Zurich, Switzerland niao.he@inf.ethz.ch Jinming Xu College of Control Science and Engineering Zhejiang University, China jimmyxu@zju.edu.cn Abstract In this paper, we show that applying adaptive methods directly to distributed minimax problems can result in non-convergence due to inconsistency in locally computed adaptive stepsizes. To address this challenge, we propose D-AdaST, a Distributed Adaptive minimax method with Stepsize Tracking. The key strategy is to employ an adaptive stepsize tracking protocol involving the transmission of two extra (scalar) variables. This protocol ensures the consistency among stepsizes of nodes, eliminating the steady-state error due to the lack of coordination of stepsizes among nodes that commonly exists in vanilla distributed adaptive methods, and thus guarantees exact convergence. For nonconvex-strongly-concave distributed minimax problems, we characterize the specific transient times that ensure timescale separation of stepsizes and quasi-independence of networks, leading to a near-optimal convergence rate of ˜O ϵ−(4+δ) for any small δ > 0, matching that of the centralized counterpart. To our best knowledge, D-AdaST is the first distributed adaptive method achieving near-optimal convergence without knowing any problem-dependent parameters for nonconvex minimax problems. Extensive experiments are conducted to validate our theoretical results. 1 Introduction Distributed optimization has seen significant research progress over the last decade, resulting in numerous algorithms (Nedic and Ozdaglar, 2009; Yuan et al., 2016; Lian et al., 2017; Pu and Nedi´c, 2021). However, the traditional focus of distributed optimization has primarily been on minimization tasks. With the rapid growth of machine learning research, various applications have emerged that go beyond simple minimization, such as Generative Adversarial Networks (GANs) (Goodfellow et al., 2014; Gulrajani et al., 2017), robust optimization (Mohri et al., 2019; Sinha et al., 2017), adversary training of neural networks (Wang et al., 2021), fair machine learning (Madras et al., 2018), and just 38th Conference on Neural Information Processing Systems (NeurIPS 2024). to name a few. These tasks typically involve a minimax structure as follows: min x∈X max y∈Y f (x, y) , where X ⊆Rp, Y ⊆Rd, and x, y are the primal and dual variables to be learned, respectively. One of the simplest yet effective methods for solving the above minimax problem is Gradient Descent Ascent (GDA) (Dem’yanov and Pevnyi, 1972; Nemirovski et al., 2009) which alternately performs stochastic gradient descent for the primal variable and stochastic gradient ascent for the dual variable. This approach has demonstrated its effectiveness in solving minimax problems, especially for convexconcave objectives (Hsieh et al., 2021; Daskalakis et al., 2021; Antonakopoulos et al., 2021), i.e., the function f(·, y) is convex for any y ∈Y, and f(x, ·) is concave for any x ∈X. Adaptive gradient methods, such as AdaGrad (Duchi et al., 2011), Adam (Kingma and Ba, 2014), and AMSGrad (Reddi et al., 2018), are often integrated with GDA to effectively solve minimax problems with theoretical guarantees in convex-concave settings (Diakonikolas, 2020; Antonakopoulos et al., 2021; Ene and Lê Nguyen, 2022). These adaptive methods are capable of adjusting stepsizes based on historical gradient information, making it robust to hyper-parameters tuning and can converge without requiring to know problem-dependent parameters (a characteristic often referred to as being “parameter-agnostic"). However, in the nonconvex regime, it has been shown by Lin et al. (2020); Yang et al. (2022b) that it is necessary to have a time-scale separation in stepsizes between the minimization and maximization processes to ensure the convergence of GDA and GDA-based adaptive algorithms. In particular, the stepsize ratio between primal and dual variables needs to be smaller than a threshold depending on the properties of the problem such as the smoothness and strong-concavity parameters (Li et al., 2022; Guo et al., 2021; Huang et al., 2021), which are often unknown or difficult to estimate in real-world tasks, such as training deep neural networks. Applying GDA-based adaptive methods into decentralized settings poses additional challenges due to the presence of inconsistency in locally computed adaptive stepsizes. In particular, it has been shown that the inconsistency of stepsizes can result in non-convergence in federated learning with heterogeneous computation speeds (Wang et al., 2020; Sharma et al., 2023). This is mainly due to the lack of a central node coordinating the stepsizes of nodes in distributed settings, making it difficult to converge, as observed in minimization problems (Liggett, 2022; Chen et al., 2023b). As a result, the following question arises naturally: “Can we design an adaptive minimax method that ensures the time-scale separation and consistency of stepsizes with provable convergence in fully distributed settings?" Contributions. In this paper, we aim to propose a distributed adaptive method for efficiently solving nonconvex-strongly-concave (NC-SC) minimax problems. The contributions are threefold: • We construct counterexamples showing that directly applying adaptive methods designed for centralized problems will lead to inconsistencies in locally computed adaptive stepsizes, resulting in non-convergence in distributed settings. To tackle this issue, we propose the first distributed adaptive minimax method, named D-AdaST, that incorporates an efficient stepsize tracking mechanism to maintain consistency across local stepsizes, which involves transmission of merely two extra (scalar) variables. The proposed algorithm exhibits timescale separation in stepsizes and parameter-agnostic capability in fully distributed settings. • Theoretically, we prove that D-AdaST is able to achieve a near-optimal convergence rate of ˜O ϵ−(4+δ) with arbitrarily small δ > 0 to find an ϵ-stationary point for distributed NC-SC minimax problems. In contrast, we also prove the existence of a constant steady-state error in both the lower and upper bounds for GDA-based distributed minimax algorithms when being directly integrated with the adaptive stepsize rule without the stepsize tracking mechanism. Moreover, we explicitly characterize the transient times that ensure time-scale separation and quasi-independence of network, respectively. • We conduct extensive experiments on real-world datasets to verify our theoretical findings and the effectiveness of D-AdaST on a variety of tasks, including robust training of neural networks and optimizing Wasserstein GANs. In all tasks, we demonstrate the superiority of D-AdaST over several vanilla distributed adaptive methods across various graphs, initial stepsizes and data distributions (see also additional experiments in Appendix A). 2 1.1 Related Works Distributed nonconvex minimax methods. In the realm of federated learning, Deng and Mahdavi (2021) introduce Local SGDA algorithm combining FedAvg/Local SGD with stochastic GDA and show an ˜O ϵ−6 sample complexity for NC-SC objective functions. Sharma et al. (2022) provide improved complexity result of ˜O ϵ−4 matching that of the lower bound of first-order algorithms for both NC-SC and nonconvex-Polyak-Lojasiewicz (NC-PL) settings (Li et al., 2021; Zhang et al., 2021a) . Yang et al. (2022a) integrate Local SGDA with stochastic gradient estimators to eliminate the data heterogeneity. More recently, Zhang et al. (2023) adopt compressed momentum methods with Local SGD to increase the communication efficiency of the algorithm. For decentralized nonconvex minimax problems, Liu et al. (2020) study the training of GANs using decentralized optimistic stochastic gradient and provide non-asymptotic convergence with fixed stepsizes. Tsaknakis et al. (2020) propose a double-loop decentralized SGDA algorithm with gradient tracking techniques (Pu and Nedi´c, 2021) and achieve ˜O ϵ−4 sample complexity. With a stronger assumption of average smoothness, some studies employ variance reduction techniques to accelerate convergence (Zhang et al., 2021b; Chen et al., 2022; Xian et al., 2021; Tarzanagh et al., 2022; Wu et al., 2023; Chen et al., 2024; Zhang et al., 2024), which require more memory and computational resources due to the need for larger batch-sizes or full gradient evaluations. However, all the above-mentioned methods use a fixed or uniformly decaying stepsize, requiring the prior knowledge of smoothness and concavity. (Distributed) adaptive minimax methods. For centralized nonconvex minimax problems, Yang et al. (2022b) show that, even in deterministic settings, GDA-based methods necessitate the timescale separation of the stepsizes for primal and dual updates. Many attempts have been made for ensuring the time-scale separation requirement (Lin et al., 2020; Yang et al., 2022c; Bo¸t and Böhm, 2023; Huang et al., 2023). However, these methods typically come with the prerequisite of having knowledge about problem-dependent parameters, which can be a significant drawback in practical scenarios. To this end, Yang et al. (2022b) introduce a nested adaptive algorithm named NeAda that achieves parameter-agnosticism by incorporating an inner loop to effectively maximize the dual variable, which can obtain an optimal sample complexity of ˜O ϵ−4 when the strong-concavity parameter is known. More recently, Li et al. (2023) introduce TiAda, a single-loop parameter-agnostic adaptive algorithm for nonconvex minimax optimization which employs separated exponential factors on the adaptive primal and dual stepsizes, improving upon NeAda on the noise-adaptivity. There has been few works dedicated to adaptive minimax optimization in federated learning settings. For instance, Huang et al. (2024) introduces a federated adaptive algorithm that integrates the stepsize rule of Adam with full-client participation, resembling the centralized counterpart. Ju et al. (2023) study a federated Adam algorithm for fair federated learning where the objective function is properly weighted to account for heterogeneous updates among nodes. To the best of our knowledge, it is still unknown how one can design an adaptive minimax method capable of fulfilling the time-scale separation requirement and being parameter-agnostic in fully distributed settings. Notations. Throughout this paper, we denote by E [·] the expectation of a random variable, ∥·∥the Frobenius norm, ⟨·, ·⟩the inner product of two vectors, ⊙the Hadamard product (entry wise), ⊗ the Kronecker product. We denote by 1 the all-ones vector, I the identity matrix and J = 11T /n the averaging matrix with n dimension. For a vector or matrix A and constant α, we denote Aα the entry-wise exponential operations. We denote Φ (x) := f (x, y∗(x)) as the primal function where y∗(x) = argmax y∈Y f (x, y), and PY (·) as the projection operation onto set Y. 2 Distributed Adaptive Minimax Methods We consider the distributed minimax problem collaboratively solved by a set of agents over a network. The overall objective of the agents is to solve the following finite-sum problem: min x∈Rp max y∈Y f (x, y) = 1 n n X i=1 Eξi∼Di [Fi (x, y; ξi)] \| {z } :=fi(x,y) , (1) where fi : Rp+d →R is the local private loss function accessible only by the associated node i ∈N = {1, 2, · · · , n}, Y ⊂Rd is closed and convex, and ξi ∼Di denotes the data sample locally stored at node i ∈N with distribution Di. We consider a graph G = (V, E), here, V = {1, 2, ..., n} represents the set of agents, and E ⊆V × V denotes the set of edges consisting of ordered pairs (i, j) 3 0.0 0.5 1.0 1.5 2.0 x 0 1 2 3 4 y stationary points D-SGDA DTiAda D-AdaST (a) trajectory 0 2000 4000 6000 8000 10000 Iterations 10 −5 10 −3 10 −1 10 1 10 3 \|\|∇ x f(x, y)\|\| 2 D-SGDA DTiAda D-AdaST (b) convergence of ∥∇xf (x, y)∥2 0 2000 4000 6000 8000 10000 Iterations 10 −8 10 −6 10 −4 10 −2 10 0 Inconsistency ζ 2 v D-SGDA DTiAda D-AdaST (c) convergence of ζ2 v Figure 1: Comparison among D-SGDA, D-TiAda and D-AdaST for NC-SC quadratic objective function (6) with n = 2 nodes and γx = γy. In (a), it shows the trajectories of primal and dual variables of the algorithms, the points on the black dash line are stationary points of f. In (b), it shows the convergence of ∥∇xf (xk, yk)∥2 over the iterations. In (c), it shows the convergence of the inconsistency of stepsizes, ζ2 v defined in (8), over the iterations. Notably, ζ2 v fails to converge for D-TiAda and ζ2 v = 0 for non-adaptive D-SGDA. representing the communication link from node j to node i. For node i, we define Ni = {j \| (i, j) ∈ E} as the set of its neighboring nodes. Before proceeding to the discussion of distributed algorithms, we first introduce the following notations for brevity: xk := [x1,k, x2,k, · · · , xn,k]T ∈Rn×p, yk := [y1,k, y2,k, · · · , yn,k]T ∈Rn×d, where xi,k ∈Rp, yi,k ∈Y denote the primal and dual variable of node i at each iteration k, and ∇xF (xk, yk; ξx k) := · · · , ∇xFi xi,k, yi,k; ξx i,k , · · · T , ∇yF (xk, yk; ξy k) := h · · · , ∇yFi xi,k, yi,k; ξy i,k , · · · iT are the corresponding partial stochastic gradients with i.i.d. samples ξx k, ξy k in a compact form. Next, we will first explain the pitfalls of directly applying centralized adaptive stepsize rules to decentralized settings, and then introduce our newly proposed solution to address the challenge. 2.1 Non-Convergence of Direct Extensions For the distributed minimax optimization problem as depicted in (1) involving NC-SC objective functions, we will show shortly that the Distributed Stochastic Gradient Descent Ascent (D-SGDA) method may not converge due to the inability of time-scale separation with constant stepsizes (c.f., Figure 1), which is also observed in centralized settings (Lin et al., 2020; Yang et al., 2022b). To address this issue, one can adopt the adaptive stepsize rule used in centralized TiAda (Li et al., 2023) for each individual node, which is renowned for its ability to adaptively fulfill the time-scale separation requirements. As a result, we arrive at the following Distributed TiAda (D-TiAda) algorithm. xk+1 = W xk −γxV −α k+1∇xF (xk, yk; ξx k) , (2a) yk+1 = PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) , (2b) where γx and γy are the stepsizes, W is a doubly-stochastic weight matrix induced by graph G (Xiao et al., 2006) (c.f., Assumption 4), and V −α k+1 = diag n v−α i,k+1 on i=1 , U −β k+1 = diag n u−β i,k+1 on i=1 , (3) with vi,k+1 = max n mx i,k+1, my i,k+1 o , ui,k+1 = my i,k+1, and mx i,k+1 = mx i,k + ∇xFi xi,k, yi,k; ξx i,k 2 , my i,k+1 = my i,k + ∇yFi xi,k, yi,k; ξy i,k 2 (4) are the local accumulated gradient norm. Note that we impose a maximum operator in the preconditioner vi,k, and employ different stepsize decaying rates, i.e., 0 < β < α < 1, for the primal and 4 dual variables, respectively. Such design allows to balance the updates of x and y, and achieves the desired time-scale separation without requiring any knowledge of parameters (Li et al., 2023). However, in the distributed setting, such direct extension may fail to converge to a stationary point because vi,k and ui,k can be inconsistent due to the difference of local objective functions fi, In particular, we can rewrite the above vanilla distributed optimization algorithm (2) in the sense of average system of primal variables as below, ¯xk+1 = ¯xk −γx¯v−α k 1T n ∇xF (xk, yk; ξx k) \| {z } adaptive descent −γx ˜v−α k+1 T n ∇xF (xk, yk; ξx k) \| {z } inconsistancy , (5) where ˜v−α k T := h · · · , v−α i,k −¯v−α k , · · · i , ¯xk := 1T xk/n and ¯vk := 1/n Pn i=1 vi,k. It is evident that, in comparison to centralized adaptive methods, an unexpected term (i.e., ˜vk) on the right-hand side (RHS) arises due to inconsistencies. This term introduces inaccuracies in the directions of gradient descent, degrading the optimization performance. The theorem presented below reveals a gap near the stationary points in a properly designed counterexample, indicating the non-convergence of D-TiAda. The proof is available in Appendix B.3. Theorem 1. There exists a distributed minimax problem in the form of Problem (1) and certain initialization such that after running D-TiAda with any 0 < β < 0.5 < α < 1 and γx, γy > 0, it holds that for any t = 0, 1, 2, . . . , we have, ∥∇xf(xt, yt) ∥=∥∇xf(x0, y0) ∥, ∥∇yf(xt, yt) ∥=∥∇yf(x0, y0) ∥, where ∥∇xf(x0, y0)∥and ∥∇yf(x0, y0)∥can be arbitrarily large depending on the initialization. Remark 1. The counterexample we constructed consists of three nodes, forming a complete graph. Without the stepsize tracking, D-TiAda will remain stationary, and the iterates will not progress if initiated along a specific line. In this counterexample, the only stationary point is at (0, 0), but initial points along the line (c.f., Eq. (72)) can be positioned arbitrarily far away from this stationary point, implying the non-convergence of D-TiAda with certain initialization. Apart from the counterexample discussed in Theorem 1, we also experimentally observe the divergence of of D-SGDA and D-TiAda even in a simple scenario involving only two connected agents. This phenomenon is illustrated in Figure 1 and the functions are depicted as follows: f1 (x, y) = −9 20y2 + 3 5y −x + xy −1 2x2, f2 (x, y) = −9 20y2 + 3 5y −x + 2xy −2x2. (6) It is not difficult to verify that the points on the line 3y = 5x + 2 are stationary points of f (x, y) = 1/2 (f1 (x, y) + f2 (x, y)). It follows from Figure 1(a) and 1(b) that D-SGDA does not converge to a stationary point because of the lack of time-scale separation, and D-TiAda also fails to converge due to stepsize inconsistency, as shown in Figure 1(c). In contrast, the utilization of the stepsize tracking protocol in D-AdaST ensures convergence to a stationary point, with the inconsistency in stepsizes gradually diminishing (c.f., Lemma 9). These two motivating examples effectively highlight the challenges associated with applying adaptive minimax algorithms to distributed settings. 2.2 The Proposed D-AdaST Algorithm To address the issue of stepsize inconsistency across different nodes, we propose the following Distributed Adaptive minimax optimization algorithm with Stepsize Tracking protocol, termed DAdaST, which allows us to asymptotically eliminate the stepsize inconsistency in a decentralized manner over networks. The pseudo-code for the algorithm is summarized in Algorithm 1, and can be rewritten in a compact form as follows: mx k+1 = W (mx k + hx k) , (7a) my k+1 = W (my k + hy k) , (7b) xk+1 = W xk −γxV −α k+1∇xF (xk, yk; ξx k) , (7c) yk+1 = PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) , (7d) 5 Algorithm 1 Distributed Adaptive Minimax Method with Stepsize Tracking (D-AdaST) Initialization: xi,0 ∈Rp, yi,0 ∈Y, buffers mx i,0 = my i,0 = c > 0, stepsizes γx, γy > 0, exponential factors 0 < β < α < 1 and weight matrix W. 1: for iteration k = 0, 1, · · · , each node i ∈[n], do 2: Sample i.i.d. gx i,k = ∇xFi xi,k, yi,k; ξx i,k and gy i,k = ∇yFi xi,k, yi,k; ξy i,k . 3: Accumulate the gradient norm: mx i,k+1 = mx i,k + ∥gx i,k∥2, my i,k+1 = my i,k + ∥gy i,k∥2. 4: Compute the ratio: ψi,k+1 = (mx i,k+1)α/ max n (mx i,k+1)α, (my i,k+1)αo ⩽1. 5: Update primal and dual variables locally: xi,k+1 = xi,k −γxψi,k+1 mx i,k+1 −α gx i,k, yi,k+1 = yi,k + γy(my i,k+1)−βgy i,k. 6: Communicate adaptive stepsizes and decision variables with neighbors: n mx i,k+1, my i,k+1, xi,k+1, yi,k+1 o ← X j∈Ni Wi,j n mx j,k+1, my j,k+1, xj,k+1, yj,k+1 o . 7: Projection of dual variable on the set Y: yi,k+1 ←PY (yi,k+1). 8: end for where mx k = [· · · , mx i,k, · · · ]T , my k = [· · · , my i,k, · · · ]T denote the tracking variables for the accumulated global gradient norm, i.e., for z ∈{x, y}, 1T n mz k+1 = 1 n Xn i=1 Xk t=0 gz i,t 2 + mz i,0 while hz k = [· · · , ∥gz i,k ∥2, · · · ]T , and Vk, Uk are diagonal matrices with vi,k = max n mx i,k, my i,k o and ui,k = mx i,k. Note that we also provide a variant of D-AdaST with coordinate-wise adaptive stepsizes in Algorithm 2, along with its convergence analysis in Appendix B.5. 3 Convergence Analysis In this section, we present the main convergence results for the proposed D-AdaST algorithm and compare it with D-TiAda to show the effectiveness of the proposed stepsize tracking protocol. To this end, letting ¯uk := 1/n Pn i=1 ui,k, we define the following metrics to evaluate the level of inconsistency of stepsizes among nodes, which are ensured to be bounded by Assumption 3. ζ2 v := sup i∈[n],k>0 v−α i,k −¯v−α k 2 / ¯v−α k 2 , ζ2 u := sup i∈[n],k>0 u−β i,k −¯u−β k 2 / ¯u−β k 2 . (8) 3.1 Assumptions We consider the NC-SC setting of Problem (1) with the following assumptions that are commonly used in the existing works (c.f., Remark 2 and Remark 3). Notably, for the function and algorithm class determined by the assumptions of this work, Li et al. (2021) derived a lower complexity bound of Ω ϵ−4 and proved that such a dependency on ϵ is optimal (c.f., Remark 2). Assumption 1 (µ-strong concavity in y). Each objective function fi (x, y) is µ-strongly concave in y, i.e., ∀x ∈Rp, ∀y, y′ ∈Y and µ > 0, fi (x, y) −fi (x, y′) ⩾⟨∇yfi (x, y) , y −y′⟩+ µ 2 ∥y −y′∥2 . (9) 6 Assumption 2 (Joint smoothness). Each objective function fi (x, y) is L-smooth in x and y, i.e., ∀x, x′ ∈Rp and ∀y, y′ ∈Y, there exists a constant L such that for z ∈{x, y}, ∥∇zfi (x, y) −∇zfi (x′, y′)∥2 ⩽L2 ∥x −x′∥2 + ∥y −y′∥2 . (10) Furthermore, fi is second-order Lipschitz continuous for y, i.e., for z ∈{x, y}, ∇2 zyfi (x, y) −∇2 zyfi (x′, y′) 2 ⩽L2 ∥x −x′∥2 + ∥y −y′∥2 . (11) Remark 2. Assumption 1 does not require the convexity in x and the objective function thus can be nonconvex. Assumption 1 and 2 ensure that y∗(·) is smooth (c.f., Lemma 2), which is essential for achieving (near) optimal convergence rate (Chen et al., 2021; Li et al., 2023). Besides, it can be verified that the constructed ‘hard’ examples for obtaining the lower complexity bound in Li et al. (2021) satisfy the above second-order Lipschitz continuity (11) on y, implying that the achievable optimal complexity for the function and algorithm class considered in this work is O ϵ−4 . Assumption 3 (Stochastic gradient). For i.i.d. sample ξi, the stochastic gradient of each i is unbiased, i.e., ∀x ∈Rp, y ∈Y, Eξi [∇zFi (x, y; ξi)] = ∇zfi (x, y), for z ∈{x, y}, and there is a constant C > 0 such that ∥∇zFi (x, y; ξi)∥⩽C. Remark 3. Assumption 3 on unbiased stochastic gradient is widely used for establishing convergence rates of both minimization and minimax optimization methods with AdaGrad (Kavis et al., 2022; Li et al., 2023) or Adam (Zou et al., 2019; Chen et al., 2023a; Huang et al., 2024) adaptive stepsize. We note that under Assumption 2, this assumption can be easily satisfied in many real-world tasks by imposing constraints on the compact domain of f, e.g., neural networks with rectified activation (Dinh et al., 2017) and GANs with projections on the critic (Gulrajani et al., 2017). Next, we make the following assumption on the underlying graph to ensure its connectivity. Assumption 4 (Graph connectivity). The weight matrix W induced by graph G is doubly stochastic, i.e., W1 = 1, 1T W = 1T and ρW := ∥W −J∥2 2 < 1. Note that one can always find a proper weight matrix W compliant to the graph that satisfies Assumption 4 once the underlying graph is undirected and connected. For instance, the weight matrix can be easily determined based on the Metropolis-Hastings protocol (Xiao et al., 2006). Moreover, this assumption is more general than that in Lian et al. (2017); Borodich et al. (2021) in the sense that W is not required to be symmetric, implying that certain directed graphs can be included in this assumption, e.g., directed ring and exponential graphs (Ying et al., 2021). 3.2 Main Results We are now ready to present the key convergence results in terms of the primal function Φ (x) := f (x, y∗(x)) with y∗(x) = argmax y∈Y f (x, y), whose proofs can be found in Appendix B.4. Theorem 2. Suppose Assumption 1-4 hold. Let 0 < β < α < 1 and the total iteration K satisfy Ω  max    γ2 xκ4 γ2y 1 α−β , 1 (1 −ρW )2 !max{ 1 α , 1 β}     (12) with κ := L/µ to ensure time-scale separation and quasi-independence of the network. For D-AdaST, we have 1 1 K K−1 X k=0 E h ∥∇Φ (¯xk)∥2i = ˜O 1 K1−α + 1 (1 −ρW )α Kα + ˜O 1 K1−β + 1 (1 −ρW ) Kβ . (13) Remark 4 (Near-optimal convergence). Theorem 2 implies that if the total number of iterations satisfies the conditions (12), the proposed D-AdaST algorithm converges to a stationary point exactly for Problem (1) with an ˜O ϵ−(4+δ) sample complexity for arbitrarily small δ > 0, e.g., letting 1The complete convergence result can be found in (75) in Appendix. 7 α = 0.5 + δ/ (8 + 2δ) and β = 0.5 −δ/ (8 + 2δ). It is worth noting that this rate is near-optimal compared to the existing lower bound of Ω ϵ−4 (Li et al., 2021) for a class of smooth NC-SC functions. Moreover, this result recovers the centralized TiAda algorithm (Li et al., 2023) as a special case, i.e., setting ρW = 0, without assuming the existence of interior optimal point (c.f., Assumption 3.3 Li et al. (2023)). To the best of our knowledge, there is no existing fully parameter-agnostic method that achieves a convergence rate of ˜O ϵ−4 , even in a centralized setting. Remark 5 (Parameter-agnostic property and transient times). The above results show that D-AdaST converges without requiring to know any problem-dependent parameters, i.e., L, µ and ρW , or tuning the initial stepsize γx and γy, and is thus parameter-agnostic. Moreover, we explicitly characterize the transient times (c.f., Eq. (12)) that ensure time-scale separation and quasi-independence of the network, respectively. Indeed, we can see that if α and β are close to each other, the time required for time-scale separation to occur increases significantly, which has been observed in (Li et al., 2023). On the other hand, if α and β are relatively large, then ˜O 1/K1−α + 1/K1−β dominates the other terms, indicating independence on the network. These observations highlight the trade-offs between the convergence rate and the required duration of the transition phase. For proper comparison, we also derive an upper bound for D-TiAda as follows. Together with the lower bound in Theorem 1, we demonstrate that without the stepsize tracking mechanism, the inconsistency among local stepsizes prevents D-TiAda from converging in the distributed setting. Corollary 1. Under the same conditions of Theorem 2. For the proposed D-TiAda, we have 1 K K−1 X k=0 E h ∥∇Φ (¯xk)∥2i = ˜O 1 K1−α + 1 (1 −ρW )α Kα + ˜O 1 K1−β + 1 (1 −ρW ) Kβ + ˜O ζ2 v + κ2ζ2 u C2 . (14) 4 Experiments In this section, we conduct experiments to validate the theoretical findings and demonstrate the effectiveness of the proposed algorithm on real-world machine learning tasks. We compare the proposed D-AdaST with the distributed variants of AdaGrad (Duchi et al., 2011), TiAda (Li et al., 2023) and NeAda (Yang et al., 2022b), namely D-AdaGrad, D-TiAda and D-NeAda, respectively. These experiments run across multiple nodes with different networks, and we consider heterogeneous distributions of local objective functions/datasets. For example, each node can only access samples with a subset of labels on MNIST and CIFAR-10 datasets, which is a common scenario in decentralized and federated learning tasks (Sharma et al., 2023; Huang et al., 2022). The experiments cover three main tasks: synthetic function, robust training of the neural network, and training of Wasserstein GANs (Heusel et al., 2017). For the exponential factors of stepsize, we set α = 0.6 and β = 0.4 for both D-TiAda and D-AdaST. More detailed settings and additional experiments with different initial stepsizes, data distributions and choices of α and β can be found in Appendix A. Synthetic example. We consider a distributed minimax problem with the following NC-SC local objective functions over exponential networks with n = 50 (ρW = 0.71) and n = 100 (ρW = 0.75). fi (x, y) = −1 2y2 + Lixy −L2 i 2 x2 −2Lix + Liy, (15) where Li ∼U (1.5, 2.5). The local gradient of each node is computed with an additive N (0, 0.1) Gaussian noise. It follows from Figure 2 (a) and 2 (b) that the proposed D-AdaST algorithm outperforms other distributed adaptive methods for both initial stepsize settings, especially in cases with a favorable initial stepsize ratio, as illustrated in plots (b) and (d) where γx/γy = 0.2. Similar observation can be found in Figure 2 (c) and 2 (d), demonstrating the effectiveness of D-AdaST. Robust training of neural networks. Next, we consider the task of robust training of neural networks, in the presence of adversarial perturbations on data samples (Sharma et al., 2022; Deng and Mahdavi, 2021). The problem can be formulated as min x max y 1/n Pn i=1 fi (x; ξi + y) −η ∥y∥2, where x denotes the parameters of the model, y denotes the perturbation and ξi denotes the data sample of node i. Note that if η is large enough, the problem is NC-SC. We conduct experiments on 8 0 1000 2000 3000 4000 # Gradient calls 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (a) γx = 0.1, n = 50 0 1000 2000 3000 4000 # Gradient calls 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (b) γx = 0.02, n = 50 0 1000 2000 3000 4000 # Gradient calls 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (c) γx = 0.1, n = 100 0 1000 2000 3000 4000 # Gradient calls 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (d) γx = 0.02, n = 100 Figure 2: Performance comparison of algorithms on quadratic functions over exponential graphs with node counts n = {50, 100} and different initial stepsizes (γy = 0.1). 0 1000 2000 3000 4000 # Gradient calls 10 −1 10 0 10 1 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (a) ring, ρW = 0.97 0 1000 2000 3000 4000 # Gradient calls 10 −1 10 0 10 1 \|\|∇ x f(x, y)\|\| 2 D-AdaGrad D-NeAda DTiAda D-AdaST (b) exp., ρW = 0.67 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (c) dense, ρW = 0.55 0 4000 8000 12000 16000 # Gradient calls 10 −1 10 0 10 1 \|\|∇ x f(x, y)\|\| 2 D-AdaST, n = 2 D-AdaST, n = 8 D-AdaST, n = 32 (d) scalability 0 1000 2000 3000 4000 5000 6000 # Gradient calls 10 −2 10 −1 10 0 10 1 ∇ x f(x, y) 2 D-Adam D-NeAda-Adam DTiAda-Adam D-AdaST -Adam (e) ring, ρW = 0.97 0 1000 2000 3000 4000 5000 6000 # Gradient calls 10 −2 10 −1 10 0 10 1 ∇ x f(x, y) 2 D-Adam D-NeAda-Adam DTiAda-Adam D-AdaST -Adam (f) exp., ρW = 0.67 0 1000 2000 3000 4000 5000 6000 # Gradient calls 10 −2 10 −1 10 0 10 1 ∇ x f(x, y) 2 D-Adam D-NeAda-Adam DTiAda-Adam D-AdaST -Adam (g) dense, ρW = 0.55 0 2000 4000 6000 8000 # Gradient calls 10 −1 10 0 10 1 \|\|∇ x f(x, y)\|\| 2 D-AdaST-Adam, n = 4 D-AdaST-Adam, n = 16 D-AdaST-Adam, n = 64 (h) scalability Figure 3: Comparison of the algorithms on training robust CNN on MNIST dataset. The first row shows the results of AdaGrad-like stepsize, and the second row is for Adam-like stepsize. For the first three columns, we compare the algorithms on different graphs with n = 20. For the last column, we show the scalability of D-AdaST in terms of number of nodes. Initial stepsizes are set as γx = 0.01, γy = 0.1 for AdaGrad-like stepsize, and γx = 0.1, γy = 0.1 for Adam-like stepsize. MNIST dataset over different networks, e.g., ring graph, exponential (exp.) graph (Ying et al., 2021) and dense graph with n/2 edges for each node. We consider a heterogeneous scenario in which each node possesses only two distinct classes of labeled samples, resulting in heterogeneity among the local datasets across nodes, while the data is i.i.d within each node. In Figure 3, we compare D-AdaST with D-AdaGrad, D-TiAda and D-NeAda, using adaptive stepsizes in AdaGrad (first row) and Adam (second row, name suffixed with Adam) respectively, it can be observed from the first three columns that the proposed D-AdaST outperforms the others on three different graphs and it is not very sensitive to the graph connectivity (i.e., ρW ), demonstrating the quasi-independence of network as indicated in Theorem 2. It should be noted that Adam-like algorithms exhibit more fluctuations in the later stages of optimization as the gradient norm vanishes, leading to an inevitable increase in the Adam stepsize as the optimization process converges (Kingma and Ba, 2014). In plots (d) and (h), we further demonstrate that D-AdaST can scale efficiently with respect to the number of nodes, while keeping a constant batch-size of 64 for each node. This showcases the algorithm’s ability to handle large-scale distributed scenarios effectively. Generative Adversarial Networks. We further illustrate the effectiveness of D-AdaST on another popular task of training GANs, which has a generator and a discriminator used to generate and distinguish samples respectively (Goodfellow et al., 2014). In this experiment, we train Wasserstein GANs (Gulrajani et al., 2017) on CIFAR-10 dataset in a decentralized setting where each discriminator is 1-Lipschitz and has access to only two classes of samples. We compare the inception score of D-AdaST with D-Adam and D-TiAda adopting Adam-like stepsizes in Figure 4. It can be observed from the figure that D-AdaST achieves higher inception scores in three cases with different initial stepsizes, and has a small score loss as the initial step size changes. We believe that this example shows the great potential of D-AdaST in solving real-world problems. 9 0 20000 40000 60000 80000 # Gradient calls 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Inception score D-Adam D-TiAda-Adam D-AdaST-Adam (a) γx = γy = 0.001 0 20000 40000 60000 80000 # Gradient calls 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Inception score D-Adam D-TiAda-Adam D-AdaST-Adam (b) γx = γy = 0.01 0 20000 40000 60000 80000 # Gradient calls 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Inception score D-Adam D-TiAda-Adam D-AdaST-Adam (c) γx = γy = 0.05 Figure 4: Training GANs on CIFAR-10 dataset over exponential graphs with n = 10 nodes. 5 Conclusion We introduced a new distributed adaptive minimax method, D-AdaST, designed to tackle the issue of non-convergence in nonconvex-strongly-concave minimax problems caused by the inconsistencies among locally computed adaptive stepsizes. Vanilla distributed adaptive methods could suffer from such inconsistencies, as highlighted by the carefully designed counterexamples for demonstrating their potential non-convergence. In contrast, our proposed method employs an efficient adaptive stepsize tracking protocol that not only ensures the time-scale separation, but also guarantees stepsize consistency among nodes and thus effectively eliminates steady-state errors. Theoretically, we showed that D-AdaST can achieve a near-optimal convergence rate of ˜O ϵ−(4+δ) with any arbitrarily small δ > 0. Extensive experiments on both real-world and synthetic datasets have been conducted to validate our theoretical findings across various scenarios. Acknowledgments The work of Huang, Shen and Xu has been supported by the National Key R&D Program of China under Grant No. 2022YFB3102100, and in parts by National Natural Science Foundation of China under Grants 62373323, 62088101. 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All experiments are deployed in a server with Intel Xeon E5-2680 v4 CPU @ 2.40GHz and 8 Nvidia RTX 3090 GPUs, and implemented using distributed communication package torch.distributed in PyTorch 2.0, where each process serves as a node, and we use inter-process communication to mimic communication between nodes. For the AdaGrad-like algorithms considered in the experiments of training neural networks, similar to the Adam-like stepsize, we adopt a coordinate-wise adaptive stepsize rule as commonly used in existing centralized adaptive methods (Yang et al., 2022b; Li et al., 2023). Moreover, since we attempt to develop a parameter-agnostic algorithm that does not need much effort in tuning hyper-parameters, we set α = 0.6 and β = 0.4 for all tasks in the main text, and evaluate the effect of the choices of α and β on the performance of D-AdaST individually in an additional experiment on the synthetic objective function as shown in Appendix A.4. A.1 Experimental details Communication topology. For the experiments in the main text, we utilize three commonly used communication topologies: indirect ring, exponential graph and dense graph. An indirect ring is a sparse graph in which each node is sequentially connected to form a ring, with only two neighbors per node. Exponential graph (Ying et al., 2021) is a directed graph where each node is connected to nodes at distances of 20, 21..., 2log n. Exponential graphs achieve a good balance between the degree and connectivity of the graph. A dense graph is an indirect graph where each node is connected to nodes at distances of 1, 2, 4, ..., n. We also consider directed ring and fully connected graphs, which are more sparsely and densely connected, respectively, in the additional experiments. Robust training of neural network. In this task, we train CNNs with three convolutional layers and one fully connected layer on MNIST dataset containing images of 10 classes. Each layer adopts batch normalization and ELU activation. The total batch-size is 1280, and the batch-size of each node during training is 1280/n. For Adam-like algorithms, we set the first and second moment parameters as β1 = 0.9, β2 = 0.999 respectively. Since NeAda is a double-loop algorithm, for fair comparison, we imply D-AdaGrad and D-Adam using 15 iterations of inner loop in this task. Generative Adversarial Networks. In this task, we train Wasserstein GANs on CIFAR-10 dataset, where the model used for discriminator is a four layer CNNs, and for generator is a four layer CNNs with transpose convolution layers. The total batch-size is 1280, and the batch-size of each node during training is 128 with 10 nodes. For Adam-like algorithms, we use β1 = 0.5, β2 = 0.9. To obtain the inception score, we use 8000 artificially generated samples to feed the previously trained inception network. A.2 Additional experiments on robust training of neural network. In this part, we conduct additional experiments on robust training of CNNs on MNIST dataset considering a variety of settings. We compare the convergence performance of D-AdaST with D-AdaGrad, D-TiAda and D-NeAda using adaptive stepsizes of AdaGrad and Adam. Unless otherwise specified, the total batch-size is set to 1280; the initial stepsizes for x and y are assigned as γx = 0.01, γy = 0.1 for AdaGrad-like algorithms, and γx = γy = 0.1 for Adam-like algorithms. Specifically, we consider two extra graphs that are more sparse and more dense, respectively in Figure 5, e.g., directed ring and fully-connected (fc) graphs. We consider more initial stepsizes settings for x and y respectively in Figure 6. Further, we also consider different data distributions where each node has samples from 4 of the 10 classes in Figure 7. Finally, we perform a comparison experiment with 40 nodes in Figure 8. Under all settings, the proposed D-AdaST outperforms the others, demonstrating the superiority of D-AdaST. A.3 Additional experiments on training GANs We provide additional experiments of training GANs on a more complicated dataset CIFAR-100 to further illustrate the effectiveness of the proposed D-AdaST, as shown in Figure 9. We use the entire training set of CIFAR-100 with coarse labels (20 classes) to train GANs over networks, where each node is assigned with four distinct classes of labeled samples. Under the same settings as in 14 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (a) directed-ring 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (b) fc 0 1000 2000 3000 4000 5000 6000 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (c) directed-ring 0 1000 2000 3000 4000 5000 6000 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (d) fc Figure 5: Performance comparison of training CNN on MNIST with n = 20 nodes over directed ring and fully connected graphs. 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (a) 0.01, 0.01 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (b) 0.001, 0.01 0 1000 2000 3000 4000 5000 6000 10−3 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (c) 0.01, 0.01 0 1000 2000 3000 4000 5000 6000 10−3 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (d) 0.01, 0.1 Figure 6: Performance comparison of training CNN on MNIST with n = 20 nodes with different initial stepsizes γx and γy. 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (a) exp. 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (b) dense 0 1000 2000 3000 4000 5000 6000 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (c) exp. 0 1000 2000 3000 4000 5000 6000 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (d) dense Figure 7: Performance comparison of training CNN on MNIST with n = 20 nodes over exponential and dense graphs where each node has 4 sample classes. 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (a) exp. 0 1000 2000 3000 4000 # Gradient calls 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-AdaGrad D-NeAda D-TiAda D-AdaST (b) dense 0 1000 2000 3000 4000 5000 6000 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (c) exp. 0 1000 2000 3000 4000 5000 6000 10−2 10−1 100 101 \|\|∇xf(x, y)\|\|2 D-Adam D-NeAda-Adam D-TiAda-Adam D-AdaST-Adam (d) dense Figure 8: Performance comparison of training CNN on MNIST with n = 40 nodes over exponential and dense graphs. Figure 4 (a), it can be observed that D-AdaST outperforms the others in terms of the inception score. Together with other experimental results in the main text, we believe that we have demonstrated the effectiveness of the proposed D-AdaST method and its potential for further real-world applications. 15 0 7500 15000 22500 30000 # Gradient calls 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Inception score D-Adam DTiAda-Adam D-AdaST -Adam Figure 9: Performance comparison of D-AdaST with D-Adam and D-TiAda adopting Adam-like stepsizes for training GANs on CIFAR-100 with coarse labels over the exponential graph consisting of n = 10 nodes under initial stepsizes γx = γy = 0.001. 0 2000 4000 6000 8000 # Gradient calls 0 20 40 60 80 100 ∇ x f(x, y) 2 D-AdaST: α = 0.6, β = 0.4 D-AdaST: α = 0.62, β = 0.38 D-AdaST: α = 0.64, β = 0.36 D-AdaST: α = 0.66, β = 0.34 Figure 10: Performance comparison of D-AdaST on quadratic functions over an exponential graph of n = 50 nodes with different choices of α and β. A.4 Additional experiments with different choices of α and β In this part, we evaluate the effect of the choices of α and β on the performance of D-AdaST. In particular, we provide an additional experimental result on the synthetic quadratic objective functions (15) with a larger ratio of initial stepsizes, i.e., γx/γy = 20 (indicating faster minimization and slower maximization processes at the beginning). As shown in Figure 10, it can be observed that the transient time (iteration before the inflection point) becomes longer as α −β decreases, while the convergence rate is relatively faster, which is consistent with Theorem 2 and the result in the centralized TiAda algorithm (c.f., Figure 5, Li et al., 2023). B Proof of the main results We recall here some definitions used in the main text. The averaged variables and the inconsistency are defined as follows: ¯xk := 1T n xk, ¯vk := 1 n n X i=1 vi,k, ˜v−α k T := h · · · , v−α i,k −¯v−α k , · · · i , ¯yk := 1T n yk, ¯uk := 1 n n X i=1 ui,k, ˜u−β k T := h · · · , u−β i,k −¯u−β k , · · · i . 16 The inconsistency of stepsizes of the primal and dual variables is defined as follows: ζ2 v := sup i∈[n],k>0 v−α i,k −¯v−α k 2 / ¯v−α k 2 , ζ2 u := sup i∈[n],k>0 u−β i,k −¯u−β k 2 / ¯u−β k 2 . Proof Sketch. The convergence analysis of the main results in Theorem 2 is mainly based on carefully analyzing the average system as shown in (5), and the difference between the distributed system and the averaged system. In general, under Assumption 1-4, we first give a telescoped descent lemma from 0 to K −1 iterations in Lemma 3, which is upper bounded by the following key error terms: • S1 := 1 nK PK−1 k=0 E h ¯v−α k+1 ∥∇xF (xk, yk; ξx k)∥2i : The asymptotically decaying terms by adopting adaptive stepsize; • S2 := 1 nK PK−1 k=0 E h ∥xk −1¯xk∥2 + ∥yk −1¯yk∥2i : The consensus error of x and y between the distributed system and the average system; • S3 := 1 K PK−1 k=0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)]: The optimality gap in dual variable y; • S4 := 1 K PK−1 k=0 E " (˜v−α k+1) T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2# : The inconsistency of stepsize of x. Next, we prove the contraction properties of these terms in Lemma 4-8 and Lemma 9 respectively. Finally, these results are integrated into the descent lemma to complete the proof. We note that the proof is not trivial in the sense that these terms are coupled and therefore are needed to be carefully analyzed. This proof can also be adapted to analyze the coordinate-wise adaptive stepsize variant of D-AdaST as explained in Appendix B.5, which is of independent interest. B.1 Supporting lemmas In this part, we provide several supporting lemmas that have been shown in the existing literature, which are essential to the subsequent convergence analysis. Lemma 1 (Lemma A.2 in Yang et al. (2022b)). Let {xt}T −1 t=0 be a sequence of non-negative real numbers, x0 > 0 and α ∈(0, 1). Then we have, T −1 X t=0 xt !1−α ⩽ T −1 X t=0 xt Pt k=0 xk α ⩽ 1 1 −α T −1 X t=0 xt !1−α . (16) When α = 0, we have T −1 X t=0 xt Pt k=0 xk α ⩽1 + log PT −1 t=0 xt x0 ! . (17) Lemma 2. Suppose Assumption 1 and 2 hold. Define Φ (x) := f (x, y∗(x)) as the envelope function and y∗(x) = argmax y∈Y f (x, y). Then, we have, • Φ (·) is LΦ-smooth with LΦ = L (1 + κ), and ∇Φ (x) = ∇xf (x, y∗(x)) (c.f., Lemma 4.3 in Lin et al. (2020)); • y∗(·) is κ-Lipschitz and ˆL-smooth with ˆL = κ (1 + κ)2(c.f., Lemma 2 in Chen et al. (2021)). B.2 Key Lemmas In this subsection, we give the key lemmas to help the analysis of the main results. For simplicity, we define ∆k := ∥xk −1¯xk∥2 + ∥yk −1¯yk∥2 as the consensus error for primal and dual variables. Then, we have the following lemmas. 17 Lemma 3 (Descent lemma). Suppose Assumption 1-4 hold. Then, we have 1 K K−1 X k=0 E h ∥∇Φ (¯xk)∥2i ⩽8C2α (Φmax −Φ∗) γxK1−α −4 K K−1 X k=0 E h ∥∇xf (¯xk, ¯yk)∥2i + 8γxLΦ 1 + ζ2 v 1 nK K−1 X k=0 E h ¯v−α k+1 ∥∇xF (xk, yk; ξx k)∥2i \| {z } S1 + 8L2 1 nK K−1 X k=0 E [∆k] \| {z } S2 + 8κL 1 K K−1 X k=0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)] \| {z } S3 + 16 1 K K−1 X k=0 E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2  \| {z } S4 , (18) where κ := L/µ is the condition number of the function in y, Φmax = max x Φ (x) , Φ∗= min x Φ (x). Proof. By the smoothness of Φ given in Lemma 2, i.e., Φ (¯xk+1) −Φ (¯xk) ⩽⟨∇Φ (¯xk) , ¯xk+1 −¯xk⟩+ LΦ 2 ∥¯xk+1 −¯xk∥2 , and noticing that the scalar ¯vk, ¯uk are random variables, we have E " Φ (¯xk+1) −Φ (¯xk) γx¯v−α k+1 # ⩽−E ∇Φ (¯xk) , 1T n ∇xF (xk, yk; ξk) −E "* ∇Φ (¯xk) , ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) +# + γxLΦ 2 E   1 ¯v−α k+1 ¯v−α k+11T n + ˜v−α k+1 T n ! ∇xF (xk, yk; ξx k) 2 , (19) where we have used the definition of ¯xk+1 as presented in (5). Then, we bound the inner-product terms on the RHS. Firstly, −E ∇Φ (¯xk) , 1T n ∇xF (xk, yk; ξx k) = −E ∇Φ (¯xk) , 1T n ∇xF (xk, yk) −1T n ∇xF (1¯xk, 1¯yk) + 1T n ∇xF (1¯xk, 1¯yk) ⩽1 4E h ∥∇Φ (¯xk)∥2i + E " 1T n ∇xF (xk, yk) −1T n ∇xF (1¯xk, 1¯yk) 2# + 1 2 E h ∥∇Φ (¯xk) −∇xf (¯xk, ¯yk)∥2i −E h ∥∇Φ (¯xk)∥2i −E h ∥∇xf (¯xk, ¯yk)∥2i ⩽−1 4E h ∥∇Φ (¯xk)∥2i + L2 n E [∆k] + L2 2 E h ∥¯yk −y∗(¯xk)∥2i −1 2E h ∥∇xf (¯xk, ¯yk)∥2i . (20) wherein the last inequality we have used the smoothness of the objective functions. Then, for the second inner-product in (19), using Young’s inequality we get −E "* ∇Φ (¯xk) , ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) +# ⩽1 8E h ∥∇Φ (¯xk)∥2i + 2E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2 . (21) 18 Then, for the last term on the RHS of (18), recalling the definition of stepsize inconsistency in (8), we have γxLΦ 2 E   1 ¯v−α k+1 ¯v−α k+11T n + ˜v−α k+1 T n ! ∇xF (xk, yk; ξx k) 2  ⩽γxLΦ 1 + ζ2 v n E h ¯v−α k+1 ∥∇xF (xk, yk; ξx k)∥2i . (22) Plugging the obtained inequalities into (18) and telescoping the terms, we get K−1 X k=0 E h ∥∇Φ (¯xk)∥2i ⩽8 K−1 X k=0 E Φ (¯xk) −Φ (¯xk+1) γx¯v−α k −4 K−1 X k=0 E h ∥∇xf (¯xk, ¯yk)∥2i + 4L2 K−1 X k=0 E h ∥¯yk −¯y∗∥2i + 8L2 n K−1 X k=0 E [∆k] + 8γxLΦ 1 + ζ2 v n K−1 X k=0 E h ¯v−α k ∥∇xF (xk, yk; ξx k)∥2i + 16 K−1 X k=0 E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2 . (23) Now it remains to bound the first term on the RHS of the above inequality. With the help of Assumption 3, we have K−1 X k=0 E " Φ (¯xk) −Φ (¯xk+1) γx¯v−α k+1 # = K−1 X k=0 E " Φ (¯xk) γx¯v−α k −Φ (¯xk+1) γx¯v−α k+1 + Φ (¯xk) 1 γx¯v−α k+1 − 1 γx¯v−α k !# ⩽E Φmax γx¯v−α 0 − Φ∗ γx¯v−α K + K−1 X k=0 E " Φmax 1 γx¯v−α k+1 − 1 γx¯v−α k !# ⩽(Φmax −Φ∗) γx E [¯vα K] ⩽(Φmax −Φ∗) KC2α γx . (24) Noticing that E h ∥¯yk −y∗(¯xk)∥2i ⩽2 µE [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)], we thus complete the proof. Next, we need to bound the last four terms S1-S4 in (18) respectively. For S1, we have the asymptotic convergence for both primal and dual variables in the following lemma. Lemma 4. Suppose Assumption 1-4 hold. Then, we have 1 nK K−1 X k=0 E h ¯v−α k+1 ∥∇xF (xk, yk; ξx k)∥2i ⩽ C2−2α (1 −α) Kα , (25) and 1 nK K−1 X k=0 E h ¯u−β k+1 ∥∇yF (xk, yk; ξy k)∥2i ⩽ C2−2β (1 −β) Kβ . (26) 19 Proof. With the help of Lemma 1 and Assumption 3, taking the primal variable x as an example, and noticing that vi,0 > 0, i ∈[n], we have 1 K K−1 X k=0 E h ¯v−α k+1 ∥∇xF (xk, yk; ξx k)∥2i = 1 K K−1 X k=0 1 n n X i=1 ∇xFi xi,k, yi,k; ξx i,k 2 ¯vα k+1 ⩽1 K K−1 X k=0 1 n n X i=1 ∇xFi xi,k, yi,k; ξx i,k 2 Pk t=0 1 n Pn j=1 ∇xFj xj,t, yj,t; ξx j,t 2α ⩽ 1 1 −α 1 K K−1 X k=0 1 n n X i=1 ∇xFi xi,k, yi,k; ξx i,k 2 !1−α ⩽ C2−2α (1 −α) Kα . The similar result can be obtained for dual variable y and we thus complete the proof. Next, we bound the the consensus error term S2 in the following lemma. Lemma 5. Suppose Assumption 1-4 hold. Then, we have 1 K K X k=0 E [∆k] ⩽ 2E [∆0] (1 −ρW ) K + 8nρW γ2 x 1 + ζ2 v (1 −ρW )2 C2−4α (1 −2α) K2α Iα<1/2 + 1 + log vK −log v1 K¯v2α−1 1 Iα⩾1/2 + 8nρW γ2 y 1 + ζ2 u (1 −ρW )2 C2−4β (1 −2β) K2β Iβ<1/2 + 1 + log uK −log u1 K¯u2β−1 1 Iβ⩾1/2 ! , (27) where I[·] ∈{0, 1} is the indicator for specific condition, and the initial consensus error ∆0 can be set to 0 with proper initialization. Proof. By the updating rule of the primal variable, we have E h ∥xk+1 −1¯xk+1∥2i = E h W xk −γxV −α k+1∇xF (xk, yk; ξx k) −J xk −γxV −α k+1∇xF (xk, yk; ξx k) 2i ⩽1 + ρW 2 E h ∥xk −1¯xk∥2i + 2γ2 x (1 + ρW ) ρW 1 −ρW E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i + 2γ2 x (1 + ρW ) ρW 1 −ρW E h V −α k+1 −¯v−α k+1I ∇xF (xk, yk; ξx k) 2i , (28) where we have used Young’s inequality. Then, by the definition of ζv in (8), we have E h V −α k+1 −¯v−α k+1I ∇xF (xk, yk; ξx k) 2i ⩽ζ2 vE h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i , (29) and thus K−1 X k=0 E h ∥xk+1 −1¯xk+1∥2i ⩽ 2 1 −ρW E h ∥xk −1¯xk∥2i + 8γ2 xρW 1 + ζ2 v (1 −ρW )2 K−1 X k=0 E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i . (30) 20 Then, we bound the last term on the RHS of the above inequality by Lemma 4. For the case α < 1/2, by Assumption 3 we have K−1 X k=0 E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i = K−1 X k=0 n X i=1 E   ∇xFi xi,k, yi,k; ξx i,k 2 ¯v2α k+1  ⩽n KC21−2α (1 −2α) . (31) For the case α ⩾1/2, with the help of Lemma 1, we have K−1 X k=0 E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i = K−1 X k=0 n X i=1 E   ∇xFi xi,k, yi,k; ξx i,k 2 ¯vk+1 · ¯v2α−1 k+1  ⩽n (1 + log vT −log v1) ¯v2α−1 1 . (32) For the dual variable, we have yk+1 = PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) = Wyk + γy∇y ˆG where ∇y ˆG = 1 γy PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) −Wyk . Then, using Young’s inequality with parameter λ, we have E h ∥yk+1 −1¯yk+1∥2i = E Wyk + γy∇y ˆG −J Wyk + γy∇y ˆG 2 ⩽(1 + λ) ρW E h ∥yk −Jyk∥2i + 1 + 1 λ E PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) −Wyk 2 ⩽1 + ρW 2 E h ∥yk −Jyk∥2i + 1 + ρW 1 −ρW E PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) −Wyk 2 . Noticing that Wyk = PY (Wyk) holds for convex set Y, we get E h ∥yk+1 −1¯yk+1∥2i ⩽1 + ρW 2 E h ∥yk −Jyk∥2i + 1 + ρW 1 −ρW E PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) −PY (Wyk) 2 ⩽1 + ρW 2 E h ∥yk −Jyk∥2i + 1 + ρW 1 −ρW E γyU −β k+1∇yF (xk, yk; ξy k) 2 ⩽1 + ρW 2 E h ∥yk −Jyk∥2i + 4γ2 y 1 + ζ2 u (1 −ρW ) E h ¯u−2β k+1 ∥∇yF (xk, yk; ξy k)∥2i , 21 where we have used the non-expansiveness of projection operator. Then, we have K−1 X k=0 E h ∥yk −1¯yk∥2i ⩽ 2 1 −ρW E h ∥y0 −Jy0∥2i + 8γ2 y 1 + ζ2 u (1 −ρW )2 K−1 X k=0 E h ¯u−2β k+1 ∥∇yF (xk, yk; ξy k)∥2i . Similar to the primal variable, we can bound the last term above, which completes the proof. Next, we need to bound the term S3 i.e., the optimality gap in dual variable. The intuition of the proof relies on the adaptive two time-scale protocol, that is, for given α and β, we try to find the threshold of the iterations k0, after which the inner sub-problem can be well solved (faster) to ensure that the computation of outer sub-problem can be solved accurately (slower). In specific, we suppose that there is a constant G such that ¯uk ⩽G hold for k = 0, 1, · · · , k0 −1, then the analysis is divided into two phases. Lemma 6 (First phase). Suppose Assumption 1-4 hold. If ¯uk ⩽G, k = 0, 1, · · · , k0 −1, then we have k0−1 X k=0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)] ⩽ k0−1 X k=0 E [E1,k] + γ2 xκ2 1 + ζ2 v G2β nµγ2y k0−1 X k=0 E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i + γy 1 + ζ2 u n k0−1 X k=0 E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + 4κL n k0−1 X k=0 E h ∥xk −1¯xk∥2i + 4 µ k0−1 X k=0 E   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + C k0−1 X k=0 E "r 1 n ∥yk −1¯yk∥2 # , (33) where E1,k := 1 −3µγy¯u−β k+1/4 2γy¯u−β k+1n ∥yk −1y∗(¯xk)∥2 − ∥yk+1 −1y∗(¯xk+1)∥2 2 + µγy¯u−β k+1 γy¯u−β k+1n . (34) Proof. Using Young’s inequality with parameter λk, we get 1 n ∥yk+1 −1¯y∗(¯xk+1)∥2 ⩽(1 + λk) n ∥yk+1 −1y∗(¯xk)∥2 + 1 + 1 λk ∥y∗(¯xk) −y∗(¯xk+1)∥2 . (35) Recalling that yk+1 = PY W yk + γyU −β k+1∇yF (xk, yk; ξy k) , we further define ˆyk+1 = W yk + γyU −β k+1∇yF (xk, yk; ξy k) . Then, for the first term on the RHS of (35), by the non-expansiveness property of projection operator PY(·) (c.f., Lemma 1 in (Nedic et al., 2010)), we have 1 n ∥yk+1 −1y∗(¯xk)∥2 ⩽1 n ∥ˆyk+1 −1y∗(¯xk)∥2 −1 n ∥yk+1 −ˆyk+1∥2 ⩽1 n ∥yk −1y∗(¯xk)∥2 + γ2 y n U −β k+1∇yF (xk, yk; ξy k) 2 −1 n n X i=1 2 D γy¯u−β k+1gy i,k, yi,k −y∗(¯xk) E −1 n n X i=1 2 D γy u−β i,k+1 −¯u−β k+1 gy i,k, yi,k −y∗(¯xk) E , (36) 22 wherein the last inequality we have used the fact ∥W∥2 2 ⩽1. Then, multiplying by 1/ γy¯u−β k+1 on both sides of (35) we get 1 nγy¯u−β k+1 ∥yk+1 −1y∗(¯xk)∥2 ⩽ 1 + λk λkγy¯u−β k+1 ∥¯y∗(¯xk) −¯y∗(¯xk+1)∥2 + (1 + λk) 1 nγy¯u−β k+1 ∥yk −1y∗(¯xk)∥2 + γy n¯u−β k+1 U −β k+1∇yF (xk, yk; ξy k) 2 ! −(1 + λk) 1 n n X i=1 2 D gy i,k, yi,k −y∗(¯xk) E −1 n n X i=1 2 * u−β i,k+1 −¯u−β k+1 ¯u−β k+1 ! gy i,k, yi,k −y∗(¯xk) +! . (37) For the inner-product terms on the RHS, taking expectation on both sides, we have 1 n n X i=1 E h −2 D gy i,k, yi,k −y∗(¯xk) Ei = 1 n n X i=1 E [−2 ⟨∇yfi (¯xk, yi,k) , yi,k −y∗(¯xk)⟩] + 1 n n X i=1 E [−2 ⟨∇yfi (xi,k, yi,k) −∇yfi (¯xk, yi,k) , yi,k −y∗(¯xk)⟩] ⩽1 n n X i=1 E h −2 (fi (¯xk, y∗(¯xk)) −fi (¯xk, yi,k)) −µ ∥yi,k −y∗(¯xk)∥2i + 1 n n X i=1 E 8 µ ∥∇yfi (xi,k, yi,k) −∇yfi (¯xk, yi,k)∥2 + µ 8 ∥yi,k −¯y∗(¯xk)∥2 ⩽E [−2 (f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk))] + 1 n n X i=1 E [−2 (fi (¯xk, ¯yk) −fi (¯xk, yi,k))] + 8κL n n X i=1 E h ∥xi,k −¯xk∥2i −7µ 8n n X i=1 E h ∥yi,k −y∗(¯xk)∥2i , (38) where we have used Young’s inequality and strong-concavity of fi, and 1 n n X i=1 E " −2 * u−β i,k+1 −¯u−β k+1 ¯u−β k+1 ! gy i,k, yi,k −y∗(¯xk) +# ⩽1 n n X i=1 E  8 µ u−β i,k+1 −¯u−β k+1 ¯u−β k+1 ! gy i,k 2 + µ 8 ∥yi,k −y∗(¯xk)∥2  . (39) For the consensus error of dual variable on the objective function, using strong-concavity of fi and Jensen’s inequality, we have 1 n n X i=1 −2 (fi (¯xk, ¯yk) −fi (¯xk, yi,k)) ⩽1 n n X i=1 2 ⟨∇yfi (¯xk, ¯yk) , yi,k −¯yk⟩−µ n ∥yk −1¯yk∥2 ⩽2C 1 n n X i=1 ∥yi,k −¯yk∥⩽2C r 1 n ∥yk −1¯yk∥2. (40) 23 Letting λk = µγy¯u−β k+1/2, we get E [f (¯xk, ¯y∗(¯xk)) −f (¯xk, ¯yk)] ⩽E  1 −3µγy¯u−β k+1/4 2γy¯u−β k+1n ∥yk −1y∗(¯xk)∥2 − ∥yk+1 −1y∗(¯xk+1)∥2 2 + µγy¯u−β k+1 γy¯u−β k+1n   + γ2 xκ2 1 + ζ2 v G2β nµγ2y E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i + γy 1 + ζ2 u n n X i=1 E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + 4κL n E h ∥xk −1¯yk∥2i + 4 µE   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + CE "r 1 n ∥yk −1¯yk∥2 # . (41) By the κ-smoothness of y∗, we have ∥y∗(¯xk+1) −y∗(¯xk)∥2 ⩽κ2 ∥¯xk+1 −¯xk∥2 = κ2 γx¯v−α k+1 1T n ∇xF (xk, yk; ξk) −γx ˜v−α k+1 T n ∇xF (xk, yk; ξx k) 2 ⩽2γ2 xκ2 1 + ζ2 v ¯v−2α k+1 n ∥∇xF (xk, yk; ξx k)∥2 . (42) Telescoping the obtained terms from 0 to k0 −1 and noticing that ¯uk ⩽G for k ⩽k0 −1 we complete the proof. For the second phase, i.e., k ⩾k0, we have the following lemma. Lemma 7 (Second phase). Suppose Assumption 1-4 hold. If ¯uk ⩽G, k = 0, 1, · · · , k0 −1, then we have K−1 X k=k0 E [f (¯xk, ¯y∗(¯xk)) −f (¯xk, ¯yk)] ⩽ K−1 X k=k0 E [E1,k] + 8γ2 xκ2 1 + ζ2 v µγ2yG2α−2β K−1 X k=k0 ∥∇xf (¯xk, ¯yk)∥2 + 8γ2 xκ2L2 1 + ζ2 v nµγ2yG2α−2β + 4κL n ! K−1 X k=k0 E [∆k] + γy 1 + ζ2 u n E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + C K−1 X k=k0 E "r 1 n ∥yk −1¯yk∥2 # + γ2 x 1 + ζ2 v γy¯vα−β 1 κ2 + 2γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 1 ! K−1 X k=k0 E " ¯v−α k+1 n ∥∇xF (xk, yk; ξx k)∥2 # + 4γxκ (1 + ζv) C2 µγy¯vα 1 E h ¯uβ K i + 4 µ K−1 X k=k0 E   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 . (43) 24 Proof. Firstly, by the non-expansiveness of projection operator, we have ∥yi,k+1 −y∗(¯xk+1)∥2 ⩽∥ˆyi,k+1 −y∗(¯xk+1)∥2 −∥yi,k+1 −ˆyi,k+1∥2 = ∥ˆyi,k+1 −y∗(¯xk)∥2 + ∥y∗(¯xk+1) −y∗(¯xk)∥2 −2 ⟨ˆyi,k+1 −y∗(¯xk) , y∗(¯xk+1) −y∗(¯xk)⟩ = ∥ˆyi,k+1 −y∗(¯xk)∥2 + ∥y∗(¯xk+1) −y∗(¯xk)∥2 −2 (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) (¯xk+1 −¯xk)T −2 (ˆyi,k+1 −y∗(¯xk))T y∗(¯xk+1) −y∗(¯xk) −∇y∗(¯xk) (¯xk+1 −¯xk)T . (44) Then, for the first inner-product term on the RHS, letting ∇x ˜Fk = ∇xF (xk, yk; ξk)−∇xF (xk, yk), we get −2 (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) (¯xk+1 −¯xk)T = 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) (∇xF (xk, yk))T 1¯v−α k+1 n + ˜v−α k+1 n ! + 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜Fk T 1¯v−α k+1 n + ˜v−α k+1 n ! ⩽2γxκ ∥ˆyi,k+1 −y∗(¯xk)∥ (∇xF (xk, yk))T 1¯v−α k+1 n + ˜v−α k+1 n ! + 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜Fk T 1¯v−α k+1 n + ˜v−α k+1 n ! . (45) wherein the last inequality we have used the fact that y∗is κ-Lipschitz. Then, using Young’s inequality with parameter λk, we get −2 (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) (¯xk+1 −¯xk)T ⩽λk ∥ˆyi,k+1 −y∗(¯xk)∥2 + 2γ2 x¯v−2α k+1 κ2 λk   1T n ∇xF (xk, yk) 2 + ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk) 2  + 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜Fk T 1¯v−α k+1 n + ˜v−α k+1 n ! . (46) For the second inner-product term on the RHS, noticing that y∗is ˆL = κ (1 + κ)2 smooth given in Lemma 2, we have 2 (ˆyi,k+1 −y∗(¯xk))T y∗(¯xk) −y∗(¯xk+1) + ∇y∗(¯xk) (¯xk+1 −¯xk)T ⩽2 ∥ˆyi,k+1 −y∗(¯xk)∥∥y∗(¯xk) −y∗(¯xk+1) + ∇y∗(¯xk) (¯xk+1 −¯xk)∥2 ⩽2 ∥ˆyi,k+1 −y∗(¯xk)∥ ˆL 2 ∥¯xk+1 −¯xk∥2 ⩽γ2 x ˆL ∥ˆyi,k+1 −y∗(¯xk)∥ ¯v−α k+11T n + ˜v−α k+1 T n ! ∇xF (xk, yk; ξx k) 2 ⩽γ2 x ˆL ∥ˆyi,k+1 −y∗(¯xk)∥2¯v−2α k+1 1 + ζ2 v C n ∥∇xF (xk, yk; ξx k)∥ ⩽τγ2 x¯v−2α k+1 1 + ζ2 v C2 ˆL ∥ˆyi,k+1 −y∗(¯xk)∥2 + γ2 x¯v−2α k+1 1 + ζ2 v ˆL τn ∥∇xF (xk, yk; ξx k)∥2 , (47) 25 wherein the last inequality we have used Young’s inequality with parameter τ. Plugging the obtained inequalities into (44), we get ∥yi,k+1 −y∗(¯xk+1)∥2 ⩽ 1 + λk + τγ2 x¯v−2α k+1 1 + ζ2 v C2 ˆL ∥ˆyi,k+1 −y∗(¯xk)∥2 + γ2 x¯v−2α k+1 1 + ζ2 v n 2κ2 + ˆL τ ! ∥∇xF (xk, yk; ξk)∥2 + 2γ2 x¯v−2α k+1 κ2 λk   1T n ∇xF (xk, yk) 2 + ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk) 2  + 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜F T 1¯v−α k+1 n + ˜v−α k+1 n ! . (48) Setting the parameters for Young’s inequalities we used as follows, λk = µγy¯u−β k+1 4 , τ = µγy¯v2α−β 0 4γ2x (1 + ζ2v) C2 ˆL , (49) then we get ∥yi,k+1 −y∗(¯xk+1)∥2 ⩽ 1 + µγy¯u−β k+1 2 ! ∥ˆyi,k+1 −y∗(¯xk)∥2 + γ2 x 1 + ζ2 v n 2κ2 + 4γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 0 ! ¯v−2α k+1 ∥∇xF (xk, yk; ξk)∥2 + 8γ2 x¯v−2α k+1 κ2 µγy¯u−β k+1   1T n ∇xF (xk, yk) 2 + ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk) 2  + 2γx (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜Fk T 1¯v−α k+1 n + ˜v−α k+1 n ! . (50) Recalling that 1 n n X i=1 E " 1 γy¯u−β k+1 ∥ˆyi,k+1 −¯y∗(¯xk)∥2 # ⩽1 n n X i=1 E " 1 −3µγy¯u−β k+1/4 γy¯u−β k+1 ∥yi,k −¯y∗(¯xk)∥2 # + 8κL n E h ∥xk −1¯yk∥2i + 2γy 1 + ζ2 u n E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i −E [2 (f (¯xk, ¯y∗(¯xk)) −f (¯xk, ¯yk))] + 8 µE   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + 2CE "r 1 n ∥yk −1¯yk∥2 # , 26 and multiplying by 2 (2+µγy ¯u−β k+1)γy ¯u−β k+1 on both sides of (50), we obtain that E [f (¯xk, ¯y∗(¯xk)) −f (¯xk, ¯yk)] ⩽E [E1,k] + γy 1 + ζ2 u n E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + 4κL n E h ∥xk −1¯yk∥2i + 4 µE   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + CE "r 1 n ∥yk −1¯yk∥2 # + E  4γ2 x¯v−2α k+1 κ2 µγ2y ¯u−2β k+1   1T n ∇xF (xk, yk) 2 + ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk) 2    \| {z } E[E2,k] + γ2 x 1 + ζ2 v n κ2 + 2γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 1 ! E " ¯v−2α k+1 γy¯u−β k+1 ∥∇xF (xk, yk; ξk)∥2 # \| {z } E[E3,k] + 1 n n X i=1 E " γx γy¯u−β k+1 (ˆyi,k+1 −y∗(¯xk))T ∇y∗(¯xk) ∇x ˜Fk T 1¯v−α k+1 n + ˜v−α k+1 n !# \| {z } E[E4,k] . (51) Telescoping the terms from t0 to K −1, we get K−1 X k=k0 E [f (¯xk, ¯y∗(¯xk)) −f (¯xk, ¯yk)] ⩽ K−1 X k=k0 E [E1,k] + K−1 X k=k0 E [E2,k] + K−1 X k=k0 E [E3,k] + K−1 X k=k0 E [E4,k] + γy 1 + ζ2 u n E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + 4κL n K−1 X k=k0 E h ∥xk −1¯yk∥2i + 4 µE   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + C K−1 X k=k0 E "r 1 n ∥yk −1¯yk∥2 # . (52) Next we need to further bound the running sums of E [E2,k], E [E3,k] and E [E4,k] respectively. For E [E2,k], with the help of Assumption 2 and noticing that ¯uk ⩽G, k = 0, 1, · · · , k0 −1, we get K−1 X k=k0 E [E2,k] ⩽ K−1 X k=k0 E  4γ2 x¯v−2α k+1 κ2 µγ2y ¯u−2β k+1   1T n ∇xF (xk, yk) 2 + ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk) 2    ⩽8γ2 xκ2 1 + ζ2 v µγ2yG2α−2β K−1 X k=k0 E ∥∇xf (¯xk, ¯yk)∥2 + L2 n ∆k . (53) 27 Then, for the term E [E3,k], noticing that ¯uk+1 ⩽¯vk+1 and ¯vk+1 ⩾¯v1, we have K−1 X k=k0 E [E3,k] ⩽ K−1 X k=k0 E " γ2 x 1 + ζ2 v nγy κ2 + 2γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 1 ! ¯v−2α k+1 ¯u−β k+1 ∥∇xF (xk, yk; ξx k)∥2 # ⩽γ2 x 1 + ζ2 v γy¯vα−β 1 κ2 + 2γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 1 ! K−1 X k=k0 E " ¯v−α k+1 n ∥∇xF (xk, yk; ξx k)∥2 # . (54) For the term E4,k, we denote ek := γx γy¯u−β k+1 1 n n X i=1 (ˆyi,k+1 −y∗(¯xk))T ! ∇y∗(¯xk) ∇x ˜Fk T 1 n + ˜v−α k+1 n¯v−α k+1 ! , then we have \|ek\| ⩽ γxκ γy¯u−β k+1 1 n n X i=1 ∥ˆyi,k+1 −y∗(¯xk)∥ ∇x ˜Fk T 1 n + ˜v−α k+1 n¯v−α k+1 ! ⩽γxκ (1 + ζv) γy √n¯u−β k+1 1 n n X i=1 1 µ ∥∇yf (¯xk, ˆyi,k+1) −∇yf (¯xk, y∗)∥ ! ∇x ˜F ⩽2γxκ (1 + ζv) C2¯uβ K µγy \| {z } M , (55) where we have used the Lipschitz continuity of y∗given in Lemma 2 and Assumption 3. Then, noticing that E h ∇x ˜Fk i = 0, we obtain K−1 X k=k0 E [E4,k] = K−1 X k=k0 E ek¯v−α k+1 = E ek0¯v−α k0+1 + K−1 X k=k0+1 E ek¯v−α k \| {z } 0 + K−1 X k=k0+1 E  −ek ¯v−α k −¯v−α k+1 \| {z } >0   ⩽E M¯v−α k0+1 + K−1 X k=k0+1 E M ¯v−α k −¯v−α k+1 ⩽2E M¯v−α k0+1 ⩽4γxκ (1 + ζv) C2 µγy¯vα 1 E h ¯uβ K i . (56) Therefore, combining the obtained inequalities, we complete the proof. Now, it remains to bound the term E1,k. Lemma 8. Suppose Assumption 1-4 hold. Then, we have K−1 X k=0 E [E1,k] ⩽ 1 2γy¯u−β 1 n ∥y0 −1y∗(¯x0)∥2 + 2 4βC22+ 1 1−β µ3+ 1 1−β γ 2+ 1 1−β y ¯u2−2β 1 . (57) 28 Proof. Recalling the definition of E1,k as given in (34), we have K−1 X k=0 E  1 −3µγy¯u−β k+1/4 2γy¯u−β k+1n ∥yk −1y∗(¯xk)∥2 − ∥yk+1 −1y∗(¯xk+1)∥2 2 + µγy¯u−β k+1 γy¯u−β k+1n   ⩽1 −3µγy¯u−β 1 /4 2γy¯u−β 1 n ∥y0 −1y∗(¯x0)∥2 + K−1 X k=1 E    1 −3µγy¯u−β k+1/4 2γy¯u−β k+1n − 1 2nγy¯u−β k 2 + µγy¯u−β k  ∥yk −1y∗(¯xk)∥2   ⩽1 −3µγy¯u−β 1 /4 2γy¯u−β 1 n ∥y0 −1y∗(¯x0)∥2 + K−1 X k=1 E          1 2γy¯u−β k+1 − 1 4γy¯u−β k −µ 8 + µ 2 2 + µγy¯u−β k −µ 2 \| {z } <0        1 n ∥yk −1y∗(¯xk)∥2   . (58) Next, we show that the term 1 2γy ¯u−β k+1 − 1 2γy ¯u−β k −µ 8 is positive for only a constant number of iterations. If the term is positive at iteration k, then we have 0 < ¯uβ k+1 2γy −¯uβ k 2γy −µ 8 ⩽¯uβ k 1 + ∥∇yF (xk, yk; ξy k)∥2 /n¯uβ k β 2γy −¯uβ k 2γy −µ 8 ⩽¯uβ k 1 + β ∥∇yF (xk, yk; ξy k)∥2 /n¯uk 2γy −¯uβ k 2γy −µ 8 = β ∥∇yF (xk, yk; ξy k)∥2 2γyn¯u1−β k −µ 8 , (59) wherein the last inequality we used Bernoulli’s inequality. Then we have the following two conditions,    1 n ∥∇yF (xk, yk; ξk)∥2 ⩾ γy ¯u1−β k+1 4β ⩾γy ¯u1−β 1 4β , 4βG2 µγy ⩾ 4β∥∇yF(xk,yk;ξy k)∥ 2 µγyn ⩾¯u1−β k+1, (60) which implies that we have at most 4βC2 µγy 1 1−β 4β µγy¯u1−β 1 (61) constant number of iterations when the term is positive. Furthermore, when the term is positive, by the inequality (59), we have 1 2γy¯u−β k+1 − 1 2γy¯u−β k −µ 8 ! 1 n ∥yk −1y∗(¯xk)∥2 ⩽β ∥∇yF (xk, yk; ξy k)∥2 2γyn¯u1−β 1 1 n ∥yk −1y∗(¯xk)∥2 ⩽ βC2 2µ2γy¯u1−β 1 1 n n X i=1 ∥∇yfi (¯xk, yi,k) −∇yfi (¯xk, y∗)∥2 ⩽ 2βC4 µ2γy¯u1−β 1 , (62) 29 where we have used the concavity of fi in y and Assumption 3. Then, we have K−1 X k=1 E " 1 2γy¯u−β k+1 − 1 2γy¯u−β k −µ 8 ! 1 n ∥yk −1y∗(¯xk)∥2 # ⩽ 2βC4 µ2γy¯u1−β 1 4βC2 µγy 1 1−β 4β µγy¯u1−β 1 ⩽ 2 4βC22+ 1 1−β µ3+ 1 1−β γ 2+ 1 1−β y ¯u2−2β 1 , (63) which completes the proof. Next, we show in the following lemma that the inconsistency terms, as described in (5), exhibit asymptotic convergence for the proposed D-AdaST algorithm. Lemma 9 (Convergence of inconsistency terms). Suppose Assumption 1-4 hold. For the proposed D-AdaST in Algorithm 1, we have 1 K K−1 X k=0 E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2 ⩽ v u u t 1 n1−α 4ρW (1 −ρW )2 !α (1 + ζv) ζvC2−α (1 −α) Kα , (64) and 1 K K−1 X k=0 E   ˜u−β k+1 T n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 ⩽ v u u t 1 n1−β 4ρW (1 −ρW )2 !β (1 + ζu) ζuC2−β (1 −β) Kβ . (65) Proof. By the definition of vi,k in (3), we have E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2  ⩽E  1 n2 n X i=1 ¯vα k+1 −vα i,k+1 2 gx i,k 2 v2α i,k+1   ⩽E  1 n2 n X i=1 ¯vα k+1 −vα i,k+1 2 ¯vα k+1 v2α i,k+1 gx i,k 2 ¯vα k+1  . (66) 30 Noticing that \|¯vα k+1−vα i,k+1\| vα i,k+1 ⩽ζv, we have E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2  ⩽E  1 n2 n X i=1 ¯vα k+1 −vα i,k+1 2 ¯vα k+1 −vα i,k+1 v2α i,k+1 + 1 vα i,k+1 ! gx i,k 2 ¯vα k+1   ⩽E  1 n2 n X i=1 ¯vα k+1 −vα i,k+1 2 v2α i,k+1 ¯vα k+1 −vα i,k+1 gx i,k 2 ¯vα k+1   + E  1 n2 n X i=1 ¯vα k+1 −vα i,k+1 vα i,k+1 ¯vα k+1 −vα i,k+1 gx i,k 2 ¯vα k+1   ⩽(1 + ζv) ζvE  1 n n X i=1 ¯vα k+1 −vα i,k+1 1 n n X i=1 gx i,k 2 ¯vα k+1  . (67) By Lemma 4, we get 1 K K−1 X k=0 E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2  ⩽(1 + ζv) ζvE  1 n n X i=1 ¯vα k+1 −vα i,k+1 1 K K−1 X k=0 1 n Pn i=1 gx i,k 2 ¯vα k+1   ⩽(1 + ζv) ζvE " 1 n n X i=1 ¯vα k+1 −vα i,k+1 # C2−2α (1 −α) Kα ⩽(1 + ζv) ζv r 1 nE h ∥vk+1 −1¯vk+1∥2αi C2−2α (1 −α) Kα . (68) Next, for the term of inconsistency of the stepsize ∥vk −1¯vk∥2, we consider two cases due to the max operator we used. At iteration k, for the case mx k ⩾my k with ∥mx 0 −1 ¯mx 0∥2 = 0, we have E h ∥vk+1 −1¯vk+1∥2i = E h mx k+1 −1 ¯mx k+1 2i = E h ∥(W −J) (mx k −1 ¯mx k) + ηk (W −J) hx k∥2i ⩽1 + ρW 2 E h ∥mx k −1 ¯mx k∥2i + (1 + ρW ) ρW 1 −ρW E h ∥hx k∥2i ⩽ 1 + ρW 2 k E h ∥mx 0 −1 ¯mx 0∥2i + nC2 (1 + ρW ) ρW 1 −ρW k X t=0 1 + ρW 2 k−t ⩽2nC2 (1 + ρW ) ρW (1 −ρW )2 . (69) For the case mx k < my k, with ∥my 0 −1 ¯my 0∥2 = 0, E h ∥vk+1 −1¯vk+1∥2i = E h my k+1 −1 ¯my k+1 2i ⩽2nC2 (1 + ρW ) ρW (1 −ρW )2 , (70) 31 Combining these two cases, and using Lemma 4 and the fact ∥vα k −1¯vα k ∥2 ⩽∥vk −1¯vk∥2α for α ∈(0, 1), we obtain the result for primal decision variable. Following the same proof, we can also derive the result for dual decision variable. We thus complete the proof. We further give the following lemma to show that the inconsistency of stepsize remains uniformly bounded for the vanilla D-TiAda algorithm as given in (2). Lemma 10 (Inconsistency for D-TiAda). Suppose Assumption 1-4 hold. Then, for D-TiAda, we have 1 K K−1 X k=0 E   ˜v−α k+1 T n¯v−α k+1 ∇xF (xk, yk; ξx k) 2 ⩽ζ2 vC2, 1 K K−1 X k=0 E   ˜u−β k+1 T n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 ⩽ζ2 uC2. (71) Proof. By the definition of inconsistency of stepsizes in (8) and Assumption 3 on bounded gradient, we immediately get the result. B.3 Proof of Theorem 1 Proof of Theorem 1. Consider a complete graph with 3 nodes where the functions corresponding to the nodes are as follows: f1(x, y) = −1 2y2 + xy −1 2x2, f2(x, y) = f3(x, y) = −1 2y2 −(1 + 1 a + 1 b )xy −1 2x2, where a = 2 −1 2α−1 and b = 2 −1 2β−1 . Notice that the only stationary point of f(x, y) = (f1(x, y) + f2(x, y) + f3(x, y))/3 is (0, 0). We denote gx i,k = ∇xfi(xk, yk) and gy i,k = ∇yfi(xk, yk). Now we consider points initialized in line y = −1 + a a + a b x, (72) where we have gx 1,0 = y0 −x0 = −2ab + a + b ab + a x0 gx 2,0 = gx 3,0 = − 1 + 1 b + 1 a y0 −x0 = 2ab + a + b a2(b + 1) x0 gy 1,0 = x0 −y0 = 2ab + a + b ab + a x0 gy 2,0 = gy 2,0 = −2ab + a + b ab(b + 1) x0. Note that by our assumptions of the range of α and β, we have a < b. Thus, we have \|gx 1,0\| = \|gy 1,0\| and \|gx 2,0\| > \|gy 2,0\|, 32 which means gx 2,0 would be chosen in the maximum operator in the denominator of TiAda stepsize for x. Therefore, after one step, we have x1 = x0 −ηx gx 1,0 \|gx 1,0\|2α + gx 2,0 \|gx 2,0\|2α + gx 3,0 \|gx 3,0\|2α ! \| {z } =0 y1 = y0 −ηy gy 1,0 \|gy 1,0\|2β + gy 2,0 \|gy 2,0\|2β + gy 3,0 \|gy 3,0\|2β ! \| {z } =0 . Next, we will use induction to show that x and y will stay in x0 and y0 for any iteration. Assuming for all iterations k in 1, . . . , t, xk = x0 and yk = y0, then we have in next step xt+1 = xt −ηx gx 1,0 t · \|gx 1,0\|2α + gx 2,0 t · \|gx 2,0\|2α + gx 3,0 t · \|gx 3,0\|2α ! . Note that gx 1,0 = −a · gx 2,0. Then, we get xt+1 = xt −ηx −p · gx 2,0 tα · a2α · \|gx 2,0\|2α + 2gx 2,0 tα · \|gx 2,0\|2α ! = xt − gx 2,0 tα · \|gx 2,0\|2α 2 −a1−2α \| {z } =0 (by definition of a) = xt. Similarly, we can show that yt+1 = yt. Therefore all iterates will stay at (x0, y0) if initialized at line y = −ab+b ab+ax, which implies that the initial gradient norm can be arbitrarily large by picking x0 to be large. B.4 Proof of Theorem 2 and Corollary 1 Proof of Theorem 2. Combining the results obtained in Lemma 6, 7 and 8, we get K−1 X k=0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)] = k0−1 X k=0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)] + K−1 X k=k0 E [f (¯xk, y∗(¯xk)) −f (¯xk, ¯yk)] ⩽ 1 2γy¯u−β 1 n E h ∥y0 −1y∗(¯x0)∥2i + 2 4βC22+ 1 1−β µ3+ 1 1−β γ 2+ 1 1−β y ¯u2−2β 1 + 2γ2 xκ2 1 + ζ2 v G2β nµγ2y k0−1 X k=0 E h ¯v−2α k+1 ∥∇xF (xk, yk; ξx k)∥2i + γy 1 + ζ2 u n K−1 X k=0 E h ¯u−β k+1 ∥∇yF (xk, yk; ξk)∥2i + C K−1 X k=0 E "r 1 n ∥yk −1¯yk∥2 # + 4 µ K−1 X k=0 E   ˜u−β k+1 n¯u−β k+1 ∇yF (xk, yk; ξy k) 2 + 8γ2 xκ2 1 + ζ2 v µγ2yG2α−2β K−1 X k=k0 ∥∇xf (¯xk, ¯yk)∥2 + 8γ2 xκ2L2 1 + ζ2 v nµγ2yG2α−2β + 4κL n ! K−1 X k=0 E [∆k] + 4γxκ (1 + ζv) C2 µγy¯vα 1 E h ¯uβ K i + γ2 x 1 + ζ2 v γy¯vα−β 1 κ2 + 2γ2 x 1 + ζ2 v C2 ˆL2 µγy¯v2α−β 1 ! K−1 X k=k0 E " ¯v−α k+1 n ∥∇xF (xk, yk; ξx k)∥2 # . (73) 33 Letting the separation point between the two phases discussed in Lemma 6 and 7 satisfy G = 16 1 + ζ2 v γ2 xκ4 γ2y ! 1 2α−2β , (74) then, plugging above inequality into (18), with the help of Lemma 4-8 and Lemma 9, we get 1 K K−1 X k=0 E h ∥∇Φ (¯xk)∥2i ⩽E0 + EG + EW + 8C2α (Φmax −Φ∗) γxK1−α + 32γxκ3 (1 + ζv) C2+2β γy¯vα 1 K1−β + 8κLγy 1 + ζ2 u C2−2β (1 −β) Kβ + γxLΦ + κ3Lγ2 x γy¯vα−β 1 + 2γ4 xκ2 1 + ζ2 v C2 ˆL2 γ2y¯v3α−2β 1 ! 8 1 + ζ2 v C2−2α (1 −α) Kα + v u u t 1 n1−α 4ρW (1 −ρW )2 !α 16 (1 + ζv) ζvC2−α (1 −α) Kα + v u u t 1 n1−β 4ρW (1 −ρW )2 !β 32κ2 (1 + ζu) ζuC2−β (1 −β) Kβ + 8κLC v u u t8ρW γ2y (1 + ζ2u) (1 −ρW )2 C2−4β (1 −2β) K2β Iβ<1/2 + 1 + log uK −log v1 K¯u2β−1 1 Iβ⩾1/2 ! , (75) where ˆL = κ (1 + κ)2 , LΦ = L (1 + κ), and E0 := 4κL Kγy¯u−β 1 n E h ∥y0 −1y∗(¯x0)∥2i + 16κ2 4βC22+ 1 1−β Kµ2+ 1 1−β γ 2+ 1 1−β y ¯u2−2β 1 , EG := 16γ2 xκ4 1 + ζ2 v G2β γ2y C2−4α (1 −2α) K2α Iα<1/2 + 1 + log vK −log v1 K¯v2α−1 1 Iα⩾1/2 , EW := 32 8κL + 3L2 ρW γ2 x 1 + ζ2 v (1 −ρW )2 C2−4α (1 −2α) K2α Iα<1/2 + 1 + log vK −log v1 K¯v2α−1 1 Iα⩾1/2 + 32 8κL + 3L2 ρW γ2 y 1 + ζ2 u (1 −ρW )2 C2−4β (1 −2β) K2β Iβ<1/2 + 1 + log uK −log v1 K¯u2β−1 1 Iβ⩾1/2 ! . Letting the total iteration K satisfy the conditions given in (12) such that the terms E0, EG and EW are dominated by the others, we thus complete the proof. Proof of Corollary 1. With the help of Lemma 10, we can directly adapt the proof of Theorem 2 to get the result in (14). B.5 Extend the proof to coordinate-wise stepsize In this subsection, we show how to extend our convergence analysis of D-AdaST to the coordinatewise adaptive stepsize (Zhou et al., 2018) variant. We first present this variant in Algorithm 2, which can be rewritten in a compact form with the Hadamard product denoted by ⊙. 34 Algorithm 2 D-AdaST with coordinate-wise adaptive stepsize Initialization: xi,0 ∈Rp, yi,0 ∈Y, buffers mx i,0, my i,0 > 0, stepsizes γx, γy > 0 and 0 < β < α < 1. 1: for iteration k = 0, 1, · · · , each node i ∈[n], do 2: Sample i.i.d ξx i,k and ξy i,k, compute: gx i,k = ∇xfi xi,k, yi,k; ξx i,k , gy i,k = ∇yfi xi,k, yi,k; ξy i,k . 3: Accumulate the gradient with Hadamard product: mx i,k+1 = mx i,k + gx i,k ⊙gx i,k, my i,k+1 = my i,k + gy i,k ⊙gy i,k 4: Compute the ratio: ψi,k+1 = mx i,k+1 2α max mx i,k+1 2α , my i,k+1 2α ⩽1. 5: Update primal and dual variables locally: xi,k+1 = xi,k −γxψi,k+1 mx i,k+1 −α ⊙gx i,k, yi,k+1 = yi,k + γy my i,k+1 −β ⊙gy i,k. 6: Communicate parameters with neighbors: n mx i,k+1, my i,k+1, xi,k+1, yi,k+1 o ← X j∈Ni Wi,j n mx j,k+1, my j,k+1, xj,k+1, yj,k+1 o . 7: Projection of dual variable on to set Y: yi,k+1 ←PY (yi,k+1). 8: end for mx k+1 = W (mx k + hx k) , (76a) my k+1 = W (my k + hy k) , (76b) xk+1 = W xk −γxV −α k+1 ⊙∇xF (xk, yk; ξx k) , (76c) yk+1 = PY W yk + γyU −β k+1 ⊙∇yF (xk, yk; ξy k) , (76d) where hx k = · · · , gx i,k ⊙gx i,k, · · · T ∈Rn×p, hy k = h · · · , gy i,k ⊙gy i,k, · · · iT ∈Rn×d, and the matrices U α k and V β k are redefined as follows: V −α k = h · · · , v−α i,k , · · · iT , [vi,k]j = max mx i,k j , h my i,k i j , j ∈[p] , U −β k = h · · · , u−β i,k , · · · iT , [ui,k]j = h my i,k i j , j ∈[d] , (77) where [·]j denotes the j-th element of a vector. Recalling the definitions of inconsistency of stepsize in (8), we give the following notations: ˜Vk = Vk −¯vk11T p , ¯vk = 1 np n X i=1 p X j Vij, ¯vi,k = 1 p p X j Vij, ¯vj,k = 1 n n X i=1 Vij, ˜Uk = Uk −¯uk11T p , ¯uk = 1 nd n X i=1 d X j Uij, ¯ui,k = 1 d d X j Uij, ¯uj,k = 1 n n X i=1 Uij, (78) 35 and ζ2 V = sup k⩾0 ( V −α k −¯v−α k 11T p 2 np ¯v−α k 2 ) , ˆζ2 v = sup k⩾0      V −α k −(VkJp)−α 2 np ¯v−α k 2      , ζ2 U = sup k⩾0      U −β k −¯u−β k 11T d 2 nd ¯u−β k 2      , ˆζ2 u = sup k⩾0      U −β k −(UkJd)−β 2 nd ¯u−β k 2      . Building upon the established definitions of coordinate-wise stepsize inconsistency, the subsequent lemma is presented to show the non-convergence of the inconsistency term compared to Lemma 9. Lemma 11 (Inconsistency, coordinate-wise). Suppose Assumption 1-4 hold. For the proposed D-AdaST algorithm, we have 1 K K−1 X k=0 E   1T n¯v−α k+1 ˜V −α k+1 ⊙∇xF (xk, yk; ξx k) 2  ⩽2 (1 + ζv) ζv v u u t 1 n1−α 4C2ρW (1 −ρW )2 !α C2−2α (1 −α) Kα + 2npˆζ2 vC2 (79) and 1 K K−1 X k=0 E   1T n¯u−β k+1 ˜U −β k+1 ⊙∇yF (xk, yk; ξy k) 2  ⩽2 (1 + ζu) ζu v u u t 1 n1−β 4C2ρW (1 −ρW )2 !β C2−2β (1 −β) Kβ + 2ndˆζ2 uC2. (80) In contrast, for D-TiAda, we have 1 K K−1 X k=0 E   1T n¯v−α k+1 ˜V −α k+1 ⊙∇xF (xk, yk; ξx k) 2 ⩽pζ2 V C2, 1 K K−1 X k=0 E   1T n¯u−β k+1 ˜U −β k+1 ⊙∇yF (xk, yk; ξy k) 2 ⩽dζ2 UC2. (81) Proof. For the coordinate-wise adaptive stepsize, with the definitions of Frobenius norm and Hadamard product, we have E   1T n¯v−α k+1 ˜V −α k+1 ⊙∇xF (xk, yk; ξx k) 2  = E   1T n¯v−α k+1 V −α k+1 −(Vk+1J)−α + (Vk+1J)−α −¯v−α k+111T p ⊙∇xF (xk, yk; ξx k) 2  ⩽2E   1T n¯v−α k+1 (Vk+1J)−α −¯v−α k+111T p ⊙∇xF (xk, yk; ξx k) 2  + 2E   1T n¯v−α k+1 V −α k+1 −(Vk+1J)−α ⊙∇xF (xk, yk; ξx k) 2 . (82) 36 For the first term on the RHS, according to the definitions given in (78), we have E   1T n¯v−α k+1 (Vk+1J)−α −¯v−α k+111T p ⊙∇xF (xk, yk; ξx k) 2  ⩽E " 1 n2¯v−2α k+1 n X i=1 ¯vα i,k+1 −¯vα k+1 2 ∇xfi xi,k, yi,k; ξx i,k 2 # . (83) Then, for the second part, we have E   1T n¯v−α k+1 V −α k+1 −(Vk+1J)−α ⊙∇xF (xk, yk; ξx k) 2  ⩽1 nE   V −α k+1 −(Vk+1J)−α ¯v−α k+1 2 ∥∇xF (xk, yk; ξx k)∥2   ⩽pˆζ2 vE h ∥∇xF (xk, yk; ξx k)∥2i . (84) where the term ˆζ2 v is not guaranteed to be convergent because the stepsizes between the different dimensions of each node are not consistent. Then, similar to the proof of Lemma 9, we can obtain the result presented in (79). Next, noticing that for D-TiAda, E   1T n¯v−α k+1 ˜V −α k+1 ⊙∇xF (xk, yk; ξx k) 2 ⩽1 nE   ˜V −α k+1 ¯v−α k+1 2 ∥∇xF (xk, yk; ξx k)∥2  ⩽pζ2 V C2, (85) and using Lemma 9, we complete the proof. Theorem 3. Suppose Assumption 1-4 hold. Let 0 < β < α < 1 and the total iteration satisfy K = Ω  max    γ2 xκ4 γ2y 1 α−β , 1 (1 −ρW )2 !max{ 1 α , 1 β}    . to ensure time-scale separation and quasi-independence of network. For D-AdaST with coordinatewise adaptive stepsize, we have 1 K K−1 X k=0 E h ∥∇Φ (¯xk)∥2i = ˜O 1 K1−α + 1 (1 −ρW )α Kα + 1 K1−β + 1 (1 −ρW ) Kβ + O n pˆζ2 v + κ2dˆζ2 u C2 . (86) Proof. With the help of Lemma 11 and the obtained result (75) in the proof of Theorem 2, we can derive the convergence results for D-AdaST with coordinate-wise adaptive stepsize. Remark 6. In Theorem 3, we show that the coordinate-wise variant of D-AdaST exhibits a steadystate error in its upper bound. This error depends on the number of nodes and the dimension of the problem, which stems from the stepsize inconsistency in each dimension of the local decision variables for each node (c.f., Line 3 of Algorithm 2). 37 NeurIPS Paper Checklist 1. 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4,440	DISP-LLM: Dimension-Independent Structural Pruning for Large Language Models Shangqian Gao ˚ Florida State University Chi-Heng Lin Samsung Research America Ting Hua Samsung Research America Tang Zheng Samsung Research America Yilin Shen Samsung Research America Hongxia Jin Samsung Research America Yen-Chang Hsu Samsung Research America Abstract Large Language Models (LLMs) have achieved remarkable success in various natural language processing tasks, including language modeling, understanding, and generation. However, the increased memory and computational costs associated with these models pose significant challenges for deployment on resource-limited devices. Structural pruning has emerged as a promising solution to reduce the costs of LLMs without requiring post-processing steps. Prior structural pruning methods either follow the dependence of structures at the cost of limiting flexibility, or introduce non-trivial additional parameters by incorporating different projection matrices. In this work, we propose a novel approach that relaxes the constraint imposed by regular structural pruning methods and eliminates the structural dependence along the embedding dimension. Our dimension-independent structural pruning method offers several benefits. Firstly, our method enables different blocks to utilize different subsets of the feature maps. Secondly, by removing structural dependence, we facilitate each block to possess varying widths along its input and output dimensions, thereby significantly enhancing the flexibility of structural pruning. We evaluate our method on various LLMs, including OPT, LLaMA, LLaMA-2, Phi-1.5, and Phi-2. Experimental results demonstrate that our approach outperforms other state-of-the-art methods, showing for the first time that structural pruning can achieve an accuracy similar to semi-structural pruning. 1 Introduction Large Language Models (LLMs) have revolutionized the field of natural language processing by leveraging deep learning techniques to process and generate human-like text. Compared to smaller models, LLMs exhibit unique characteristics and demonstrate remarkable abilities in tackling a wide range of complex tasks [40]. Despite their impressive capabilities, the vast number of parameters in LLMs often hinders their deployment on resource-constrained devices, such as mobile phones. Consequently, there is significant interest in reducing the computational and memory requirements of LLMs. Existing compression techniques for large language models (LLMs) include weight sparsification [9], structural pruning [30], and quantization [10]. In this work, we focus on structural pruning and ˚Part of this project was completed at Samsung Research America. Correspondence to sgao@cs.fsu.edu 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Activation 𝐖𝐢𝐧 𝐖𝐨𝐮𝐭 𝐗𝐨𝐮𝐭 𝐗𝐢𝐧 Structural Dependency from Residual Connections Regular Structural Pruning Methods with Structural Dependency Activation 𝐖𝐢𝐧 𝐖𝐨𝐮𝐭 𝐗𝐨𝐮𝐭 𝐗𝐢𝐧 Dimension-Independent Structural Pruning (Ours) Index Operations Index Operations No Dependency Figure 1: We use an MLP layer as an example. Left: Regular pruning methods have to follow structural dependence thus their flexibility is limited. Right: Our dimension-independent structural pruning method breaks the structural dependence via index operations and thus largely improves the flexibility for pruning. address the limitations of previous methods in this category. Structural pruning [30] is a generalpurpose compression solution that maintains LLM performance across various tasks, facilitates deployment on devices, and is computationally efficient. However, existing methods may restrict pruning flexibility or add significant overhead to the compressed model. For instance, LLM-Pruner [30] follows structural dependence during pruning, requiring different layers to use the same subset of feature maps, which limits pruning flexibility. SliceGPT [2] alleviates this issue by applying orthogonal projections for each layer but introduces a non-trivial number of additional parameters (e.g., 5% to 13% of the parameters of the original model for LLaMA-2 7B). Our approach aims to overcome these drawbacks and offer a better performance-cost trade-off for structural pruning. We aim to increase the flexibility of current structural pruning methods and consequently improve performance. Our method provides different sub-spaces or subsets of features to different layers, but unlike SliceGPT, it doesn’t introduce additional parameters. To achieve this, we break the structural dependence of regular structural pruning methods, allowing different layers to have different subsets of features along the embedding dimension and an example is given in Fig. 1. After pruning, we employ index selection and index addition operations to sample subsets of features from the residual connection and add them back after the computation of each layer. Furthermore, our method introduces an additional level of flexibility by learning different widths for each layer. Our approach significantly improves the flexibility of structural pruning without adding additional parameters. Extensive experimental results show that our method can outperform state-of-the-art structural pruning methods for LLMs while still maintaining low computational costs. Our method does not require recovery fine-tuning to obtain such performance. In addition, our method does not update the remained model weights during pruning which is a distinct departure from several other methods, such as SparseGPT [9] and LLM Surgeon [37]. Our contributions are as follows: • We break the structural dependence of regular structural pruning methods, significantly increasing the flexibility of structural pruning. This allows different layers to select their own subset of features from the embedding dimension. Importantly, our method achieves this without introducing additional parameters, unlike SliceGPT. • We propose to learn the widths of each layer using gradient-based optimization methods. A hypernetwork generates the column or row selection matrices, while the width of each layer is controlled globally. This approach allows for fine-grained control over the pruning process and enhances the adaptability of our method to various models and tasks. • Our method demonstrates superior performance compared to state-of-the-art structural pruning techniques for LLMs across a range of models, including OPT, LLaMA, LLaMA-2, Phi-1.5, and Phi-2. Notably, the resulting model from our method is a sub-network that exists within the original model, indicating the effectiveness of our method in discovering strong sub-networks. 2 Related Works Magnitude-based pruning is the most straightforward approach to reduce model size, where weights with the smallest magnitude are pruned. Han et al. [14] employ this strategy for pruning with L1 or L2 norm of weights. Filter pruning [24] extends this setting to structures of the model instead of performing weight-level sparsification. Although magnitude-based pruning methods are very efficient, they result in significant performance drops for LLMs, even for weight pruning [9]. Another 2 line of research, Optimal Brain Damage [23] and Optimal Brain Surgeon [15], utilize second-order information to remove connections. These methods require calculating the inverse of the Hessian matrix, which is computationally intensive for modern neural network architectures like Convolutional Neural Networks (CNNs) [22, 16], Transformers [38], or Large Language Models (LLMs) [35]. To reduce the cost of computing the Hessian inverse matrix, Optimal Brain Surgeon can be applied in a layer-wise fashion [7, 8], making the computation tractable. However, further scaling up these methods for LLMs remains challenging. Inputs 𝐒𝟏 𝐓𝐖𝐤 𝐒𝟏 𝐓𝐖𝐪 𝐒𝟏 𝐓𝐖𝐯 Multi-head Attention 𝐖𝐨𝐒𝟐 Activation Outputs 𝐒𝟑 𝐓𝐖𝟏𝐒𝟒 𝐒𝟒 𝐓𝐖𝟐𝐒𝟓 Layer Norm DISP-LLM (Ours) Layer Norm Index Add Index Select & Layer Norm When Pruning Layer Norm 𝐒𝟏, 𝐒𝟐, 𝐒𝟑, 𝐒𝟒, 𝐒𝟓 ෠𝐒𝟏, ෠𝐒𝟐, ෠𝐒𝟑, ෠𝐒𝟒, ෠𝐒𝟓 Add Index Selection with Layer Norm Replace addition with Index Add Prune weights with the actual selection matrices When Training the Hypernetwork Figure 2: Our method, DISP-LLM, applies different selection matrices to the input and output dimension of the Attention layer and MLP layer (S1{S2: Attention in/out; S3{S4{S5: MLP in/middle/out). When pruning the model, we add “Index Selection” before Layer Norm and we replace addition with “Index Add.” ˆS1, ¨ ¨ ¨ , ˆS5 are applied for pruning weight matrices. Recent methods like SparseGPT [9] or GPTQ [10] aim to minimize the squared error before and after pruning or quantization of a given layer. In this setting, the Hessian inverse matrix becomes easy to compute, as it is simply the multiplication between the feature map and its transpose for a given layer. GPTQ and SparseGPT then quantize or sparsify model weights in a column-by-column manner, and the unpruned or unquantized weights are updated to compensate for the error of pruning and quantization. Wanda [34] further avoids computing the inverse of the Hessian matrix by only considering the diagonal of the Hessian matrix. While SparseGPT and Wanda achieve good results, unstructured sparsity is known to be harder to achieve actual speedup. They also applied their methods on semi-structured settings [31], but the performance becomes much worse. Several researches [28, 19, 44, 13, 42, 12] apply learnable parameters for specific structures when pruning vision or language models. However, many of these methods cannot be scaled up to LLMs since they need to learn weights and structures together. In contrast, our method explores sub-networks within the original model without updating model weights. Additionally, our method mainly explores the regime of pruning without recovery fine-tuning, which is rarely presented in previous methods with learnable parameters on structures. Our method is also related to the unconstrained channel pruning for CNNs [39]. However, our method explores this idea from the perspective of breaking structural dependence and scales it to much larger models than [39]. Moreover, our method thoroughly explores the global allocation of parameters, where [39] fails to do. Recently, several works have been proposed to reduce the size of LLMs. LLM-Pruner [30] aims to remove connected structures using importance calculated from Taylor expansions. SliceGPT [2] offers more flexibility than regular pruning by projecting the feature maps to different spaces but introduces extra parameters in the residuals. LLM Surgeon [37] periodically updates model weights and structures, resulting in a higher cost than LLM-Pruner and SliceGPT. Our proposed DISP-LLM breaks the structural dependence relied on by LLM-Pruner, without additional transformation matrices in the residual connections like SliceGPT. Furthermore, in contrast to LLM Surgeon, which requires extensive computational resources, our method is significantly more efficient. 3 Preliminary 3.1 Notations To better understand our paper, we first define some notations. We use d to denote the model dimension or embedding dimension of LLMs. X P ℜbˆnˆd is used to represent feature maps, and b is the mini-batch size, n is the number of tokens. W P ℜd1ˆd2 is the model weights of size d1 ˆ d2. Let S denote a pseudo-index selection matrix of size d ˆ d, which is a diagonal matrix filled with 0 or 1 and the positions of the ones indicate the selected index. We further use ˆS of size d ˆ dsmall to 3 XQin XQout QinWin T WoutQout Unified selection matrices S: LLMPruner or Regular Structural Pruning Dense orthogonal matrices Qin or Qout XS S Win T WoutS XS XSin Sin Win T WoutSout XSout Selection matrices Sin and Sout: DISP-LLM (Ours) SliceGPT Figure 3: Comparison of the projection matrices for structural pruning. We use Win and Wout in Fig. 1 as an example. Left: SliceGPT employs orthogonal projection matrices, and it has to insert the projection matrices into the residual connections. Middle: Regular structural pruning methods remove structures based on their dependence, requiring to use the unified selection matrix S for all blocks, which limits flexibility. Right: Our method breaks the structural dependence, allowing the use of different selection matrices Sin and Sout for the embedding dimension, significantly improving the flexibility of pruning. represent the actual selection matrix by removing d ´ dsmall columns with all zeros from S. For any matrix A, nnz(A) represents the number of nonzero entries of A. 3.2 Revisit SliceGPT The core idea of SliceGPT [2] is to achieve computational invariance within the transformer architecture. It demonstrates that orthogonal projections can be applied to the output of each block and subsequently undone in the next block. This transformation is computed using Principal Component Analysis (PCA), allowing the feature maps between blocks to be projected into their principal components. A significant advantage of this approach is that it projects the feature maps of different blocks into distinct spaces, thereby introducing an additional degree of freedom for compression. This flexibility is not captured by regular structural pruning methods like LLM-Pruner [30], which rely on structural dependence. After slicing (pruning), the feature map and weight matrix of lth layer of SliceGPT become: ˜Xl “ XlQlˆS, ˜ Wl “ ˆSJQJ l Wl. (1) where ˆS is a dˆdsmall selection matrix, Xl is the output of the lth block, and Ql contains eigenvectors of Cl: Cl “ ÿ i XJ l,iXl,i and Xl,i is the i-th column of Xl (corresponding to the ith sequence in the calibration dataset). From Eq. 1, we can see that SliceGPT uses the same selection matrix ˆS for all layers, but the feature map Xl is firstly projected by Ql, and the pruning for different layers is along with different directions. One crucial drawback of SliceGPT also comes from the projection matrix Ql, since the residual connection must be multiplied by the linear transformation QJ l Ql`1 (shown in Fig. 7 left in the Appendix). These additional operations bring a non-trivial amount of additional parameters. For a model that has L blocks, with the model dimension d and the remaining percentage of parameters p P r0, 1s, it brings approximately Ld2p2 additional parameters to the model (more than 10% of model parameters in some cases, and more details are given in Fig 10 in the Appendix). 3.3 Residual Connections Limit the Flexibility of Structural Pruning SliceGPT offers significant flexibility, but achieving similar flexibility with regular structural pruning methods without adding extra parameters is challenging. This section explains the reasons behind this difficulty. To simplify our reasoning, we replace ˆS with its pseudo selection matrix S. Assume we follow the basic setting of dependence-based structural pruning but allow each layer the flexibility to have its own selection matrix, Sl, along the embedding dimension. Under this assumption, due to structural dependence, all layers will share the same width of nnzpS0S1 ¨ ¨ ¨ SLq. In order to prune different positions for different layers, we need to add a transformation matrix to align the width of layers l and l ` 1. Intuitively, if we have Sl and Sl`1, we can then insert SJ l Sl`1 in the residual connection to align consecutive layers. 4 Algorithm 1: Block inference after pruning. Input: Feature map of the previous block Xin. Preserved indices sets Ind1, Ind2, Ind3, Ind5. 1. ˆXin “ LayerNormpXinr:, Ind1sq. Ź Index Selection for Attention 2. Xatt “ MultiHeadpˆXinˆSJ 1 Wq, ˆXinˆSJ 1 Wk, ˆXinˆSJ 1 WvqWoˆS2. 3. Xres “ Index_AddpXin, Xattn, Ind2q. Ź Index Addition with the input 4. ˆXres “ LayerNormpXresr:, Ind3sq. Ź Index selection for MLP 5. Xmlp “ pσpˆXresˆSJ 3 W1ˆS4q d pˆXresˆSJ 3 W2ˆS4qqˆS4JW3ˆS5. 6. Xout “ Index_AddpXres, Xmlp, Ind5q Ź Index Addition with the residual Return Xout for the next block. With this setup, we can use XlSl to select subsets of features for different layers, mimicking QlS for SliceGPT. Although it seems promising, this formulation has issues with layer widths, as detailed in Proposition 1. Proposition 1 (Decreasing feature dimensions for deeper layers). Let the pseudo-selection matrices in layers l and l ` 1 be Sl and Sl`1, respectively. The number of nonzero entries in the residual adapter satisfies nnzpSJ l Sl`1q ď mintnnzpSlq, nnzpSl`1qu. For compression strategies that remove dependent structures for layer l ` 1 following SJ l Sl`1, this implies that the dimension in layer l ` 1 is less than or equal to that in layer l, with equality holding when the feature indices selected in layer l ` 1 are contained within those in layer l or vice versa. Remark. The proof of Proposition. 1 is straightforward and it is given in the Appendix A.1. From Proposition 1, we observe that if we naively apply Sl for different layers, the model width will progressively decrease as we go deeper into the network. It also fails to provide different sets of features for different layers; instead, it merely passes a subset of features from the previous layer to the next. To avoid this restriction, all blocks must share the same width and the same pruned columns or rows. And we then fall back to the regime of previous structural pruning methods such as LLM-Pruner [30], Shared LLaMA [43], etc. Proposition 1 highlights two significant obstacles. First, dependence-based structural pruning methods result in a uniform width along the embedding dimension. Second, inserting selection matrices in the residual connections causes the embedding dimension to decrease with depth. These challenges are unavoidable due to the residual connections linking structures across layers. To enhance flexibility along the embedding dimension, bypassing the residual connections is crucial. 4 Dimension-Independent Large Language Model 4.1 Break the Structural dependence Section 3.3 demonstrates that the residual connection is the primary barrier preventing pruning methods from achieving better flexibility. To avoid modifying the residual connection, we relocate the selection matrices inside the residual connection. This approach allows us to successfully create different subsets from the feature maps for different layers. Based on this idea, we propose a solution that involves pruning different positions in consecutive blocks and selecting or merging feature maps from or back to the residual connection. This approach breaks the structural dependence inherent in previous pruning methods. Formally, given a transformer block, we apply the following operations: AttentionpXq “ MultiHeadpXSJ 1 Wq, XSJ 1 Wk, XSJ 1 WvqWoS2, (2) MLPpXq “ pσpXSJ 3 W1S4q d pXSJ 3 W2S4qqS4 JW3S5, (3) where S1, . . . , S5 are pseudo selection matrices of size d ˆ d, and d denotes element-wise multiplication. Eq. 3 gives an example operation for gated MLP modules used in LLaMA [35]. For Phi models [1] or OPT [46], the MLP operation is defined as MLPpXq “ σpXSJ 3 W1S4qS4JW3S5. Fig 2 illustrates how to insert these selection matrices into a transformer block. 5 Given the operations defined in Eq. 2 and Eq. 3, we successfully remove the constraint in Proposition 1. The input and output of both the Attention layer and the MLP layer can be selected differently from the original feature maps for different layers, mimicking the function of Ql in SliceGPT. Additionally, our method eliminates the need for extra parameters in QJ l Ql`1 as it does not alter the residual connection. We also enhance flexibility by pruning the middle dimension of the MLP layer. Additionally, this flexibility can be further improved by allowing the query, key, and value weight matrices to use different selection matrices. Our current form is kept for two reasons: (1) SliceGPT uses one Ql per layer, and we followed this design for a fair comparison, and (2) adding separate selection matrices would increase indexing operations, potentially slowing down the inference. Fig 3 further compares the projection matrices for SliceGPT, regular structural pruning, and the proposed method. Once we have the final selection matrices S1, . . . , S5, the pruned model will use a combination of index selection and index addition for inference as shown in Algorithm 1, where Indi is a set containing all indices equal to one in the diagonal of Si: Indi “ tj \| if sirjs “ 1u, si “ diagpSiq. The same color is used to mark the index set Indi and its corresponding selection matrix ˆSi. Index_AddpA, B, Indq adds matrices A and B along the last dimension on selected positions from Ind, then returns A after index addition. With index selection and addition, the block dimension can be freely changed. Index selection and addition introduce some overhead, but as demonstrated in the experiment section, we still observe improvements in throughput. 4.2 Learning the Width for Dimension-Independent LLMs Building on the dimension-independent setting introduced in Section 4.1, our approach offers much greater flexibility in selecting sub-networks from the original dense model compared to the constrained settings in LLM-Pruner [30]. The next challenge is determining the width of each layer. Given the large search space of our dimension-independent structural pruning and the computationally intensive nature of LLMs, it is impractical to use reinforcement learning [17] or evolutionary search-based algorithms [27]. Therefore, we adopt gradient-based methods to address this challenge. Given the diagonal vector si P t0, 1ud from Si, the Straight-Through (ST) gradient estimator [3] is used to estimate the gradients with respect to learnable continuous latent parameters. More specifically, we use the recently proposed gradient estimator ReinMax [26] to estimate the gradients through the binary operation. A detailed explanation of ReinMax for the binary case is provided in Appendix A.2. Given the large search space of our method, we find that only using element-wise learnable parameters is insufficient. To address this issue, a hypernetwork is introduced to generate latent parameters for ReinMax, as detailed below: s “ ReinMaxpHyperNetworkpΘqq, (4) where Θ represents the parameters of the hypernetwork and s contains si from all blocks. The hypernetwork is composed of GRU [5] and fully connected layers, where the GRU captures blockwise relationships and the fully connected layers capture relationships between different dimensions. With the hypernetwork and ReinMax, we can effectively learn the width of each block. The details of the hypernetwork are provided in Appendix A.3. 4.3 Dimension-Independent Structural Pruning as an Optimization Problem With the methods described above, we can formulate dimension-independent structural pruning as an optimization problem, with regularization to control the number of remaining parameters. We insert s back into S as defined in section 4.1 for forward and backward calculations. The overall objective function is listed below: min Θ LpX; W, sq ` λRpTpsq, pTtotalq, (5) RpTpsq, pTtotalq “ logpmaxpTpsq, pTtotalq{ minpTpsq, pTtotalqq, (6) where L is the language modeling loss function of next word prediction, X represents the input tokens, W is the collection of model weights, s is defined in Eq. 4, and R is a parameter regularization loss function defined in Eq. 6. Here, Tpsq denotes the number of parameters controlled by the current structure s, Ttotal is the total number of parameters of the model, and p P p0, 1s is a user-defined 6 Table 1: Perplexities of different structural pruning methods on WikiText-2. Our method is the only one that does not update model weights. SliceGPT does not directly update model weights, however, it applies orthogonal transformation matrices to the weights. Method Pruning Ratio W Update? Test Performance (PPL) OPT 125M OPT 1.3B OPT 2.7B OPT 6.7B LLaMA-2 7B LLaMA-2 13B Dense 0% 27.65 14.62 12.47 10.86 5.12 4.57 SliceGPT [2] 10% ✓✗ 29.34 15.10 12.75 10.92 5.89 5.21 20% ✓✗ 34.26 16.43 13.73 11.48 6.64 5.81 25% ✓✗ 37.74 17.46 14.56 11.90 7.24 6.30 30% ✓✗ 43.98 19.09 15.83 12.51 8.12 6.99 K-OBD [34] 20% ✓ 29.89 15.63 12.47 11.28 9.14 6.29 30% ✓ 36.54 18.29 14.53 13.03 15.43 10.08 40% ✓ 47.54 24.65 18.09 16.21 28.03 13.06 50% ✓ 75.95 37.68 26.68 25.54 46.64 16.06 LLM Surgeon [34] 20% ✓ 28.73 15.12 12.27 11.02 6.18 5.29 30% ✓ 31.82 16.24 12.92 11.64 7.83 6.21 40% ✓ 38.47 18.45 14.23 12.58 10.39 7.25 50% ✓ 49.78 22.95 17.15 14.90 15.38 9.43 20% ✗ 25.21 13.12 11.72 9.89 6.10 5.21 30% ✗ 28.16 14.79 12.16 10.90 6.85 5.77 40% ✗ 34.31 17.77 14.11 12.18 8.11 6.59 DISP-LLM (Ours) 50% ✗ 39.87 21.70 17.07 14.06 9.84 7.11 Table 2: Comparison of our method against semi-structure pruning methods on WikiText-2. Method Pruning Ratio W Update? Structure? Test Performance (PPL) LLaMA 7B LLaMA 13B LLaMA-2 7B LLaMA-2 13B Dense 0% 5.68 5.09 5.12 4.57 Magnitude 2:4 ✗ ✗ 42.13 18.37 54.59 8.33 SparseGPT [9] 2:4 ✓ ✗ 11.00 9.11 10.17 8.32 Wanda [34] 2:4 ✗ ✗ 11.53 9.58 11.02 8.27 DISP-LLM (ours) 50% ✗ ✓ 11.47 8.15 9.84 7.11 parameter to control how many parameters should be preserved within the model. With the objective function in Eq. 5, the structures for dimension-independent pruning can be efficiently optimized. Moreover, the overhead of our method is minimal and comparable to parameter-efficient fine-tuning methods like LoRA [18], as it does not require storing gradients or the first and second-order momentum of model weights for the Adam optimizer [21]. 5 Experiments 5.1 Settings Models. We evaluate our DISP-LLM method using several LLMs with transformer blocks. Specifically, we choose the following models: OPT [46]: OPT-125M, OPT-1.3B, OPT-2.7B, OPT-6.7B; Phi-1.5 [25] and Phi-2 [20]; LLaMA 7B [35]; LLaMA-2 [36]: LLaMA-2 7B and LLaMA-2 13B. Implementations. We implemented our method using Pytorch [32] and Hugging Face transformer library [41]. We freeze the model weights W when training the hypernetwork. We use the AdamW [29] optimizer to optimize the hypernetwork. The hypernetwork is trained for 10,000 iterations for all models. For all experiments, we set λ in Obj. 5 to 6. Depending on the size of the base model, we use 1 to 4 NVIDIA A100 GPUs to train the hypernetwork. More implementation details can be found in the Appendix A.4. Datasets. Following previous papers [2, 30], we use WikiText-2 and Alpaca datasets to train the hypernetwork. Following SliceGPT [2], we evaluate our method and other methods on five wellknown zero-shot tasks: PIQA [4]; WinoGrande [33]; HellaSwag [45]; ARC-e and ARC-c [6]. We use llm-eval-harness [11] to evaluate the compressed models. Baselines. We compare our DISP-LLM across baselines from structural pruning like LLMPruner [30], SliceGPT [2] and LLMSurgeon [37]. We also include semi-structure pruning baselines like SparseGPT [9] and Wanda [34]. 5.2 Language Modeling In Table 1, we report the perplexity of pruned OPT and LLaMA-2 models. Our DISP-LLM, which does not update weights, consistently outperforms more complex pruning methods such as K-OBD and LLM Surgeon, which involve weight updates, across all pruning ratios and models. The 7 Table 3: Zero-shot performance of the compressed LLaMA 7B, LLaMA-2 7B and Phi models. The structure of DISP-LLM is based on the WikiText dataset, and the structure of DISP-LLM Alpaca is based on the Alpaca dataset. Pruning Ratio Method W Update? WinoGrande HellaSwag ARC-e ARC-c PIQA Avg acc acc-norm acc-norm acc-norm acc-norm 0% LLaMA 7B 69.85 76.21 72.81 44.71 79.16 68.55 LLM-Pruner [30] ✗ 61.33 65.34 59.18 37.12 75.57 59.71 +finetuning ✓ 65.11 68.11 63.43 37.88 76.44 62.19 DISP-LLM (Ours) ✗ 66.54 68.75 59.60 35.24 74.97 61.02 20% DISP-LLM Alpaca (Ours) ✗ 64.72 68.39 64.81 37.12 76.66 62.34 LLM-Pruner [30] ✗ 53.20 35.64 33.50 27.22 59.63 41.84 +finetuning ✓ 55.09 47.56 46.46 28.24 68.82 49.23 DISP-LLM (Ours) ✗ 58.41 47.71 44.40 28.50 64.09 48.62 50% DISP-LLM Alpaca (Ours) ✗ 56.91 48.76 48.91 31.57 67.46 50.72 0% LLaMA-2 7B 69.14 75.99 74.58 46.15 79.11 68.99 SliceGPT [2] ✓✗ 61.33 49.62 51.77 31.23 63.55 51.50 K-OBD [34] ✓ 56.83 53.07 51.05 33.11 71.82 53.18 LLM Surgeon [34] ✓ 61.09 60.72 63.09 36.69 73.56 59.03 DISP-LLM (Ours) ✗ 62.27 63.43 59.81 33.19 71.82 58.10 30% DISP-LLM Alpaca (Ours) ✗ 63.93 62.87 60.10 37.03 73.72 59.53 K-OBD [34] ✓ 53.04 36.84 36.11 26.71 60.66 42.67 LLM Surgeon [34] ✓ 52.57 40.29 44.91 26.28 64.36 45.68 DISP-LLM (Ours) ✗ 54.54 46.33 43.06 25.85 63.93 46.72 50% DISP-LLM Alpaca (Ours) ✗ 56.20 49.35 51.14 30.20 68.34 51.05 0% Phi-1.5 72.77 62.58 73.11 48.04 75.63 66.43 SliceGPT [2] ✓✗ 64.96 42.54 53.66 31.91 65.45 51.52 30% DISP-LLM (Ours) ✗ 61.48 47.97 57.66 33.01 71.08 54.24 0% Phi-2 75.61 73.86 78.24 54.01 79.11 72.17 SliceGPT [2] ✓✗ 63.14 47.56 53.03 30.29 65.94 51.99 30% DISP-LLM (Ours) ✗ 65.19 54.43 63.59 38.48 73.34 59.00 Model Dimension Depth Preserved Pruned Pruning Decisions along the Model Dimension and Depth Pruning Rate along the Model Dimension and the Overall Pruning Rate Figure 4: The pruned model architecture along the embedding dimension (model dimension) for the LLaMA-2 7B model when the pruning ratio equals 50%. performance gap is even larger when compared to SliceGPT. The advantage is particularly clear in better-trained models like LLaMA-2 7B and 13B. For instance, our method surpasses LLM Surgeon by margins of 5.54 and 2.22 when pruning 50% of parameters of LLaMA-2 7B and 13B, respectively. Against K-OBD, our performance advantage extends to 36.80 and 9.49 under the same setting. For consistency, we let the pruning ratio of SliceGPT equal the slicing ratio. However, the actual pruning ratio for SliceGPT is much lower than the slicing ratio. More details are given in Appendix A.5. In Table 2, we report the perplexity of pruned LLaMA and LLaMA-2 models and we compare our method with semi-structure pruning methods. From the table, we can see that our method outperforms both SparseGPT and Wanda on LLaMA 13B and LLaMA-2 7B/13B models. Our method performs on par with SparseGPT and Wanda with the LLaMA 7B model, and our DISP-LLM is a little bit worse than SparseGPT and is similar to Wanda. We are the first to show that structural pruning methods can have a better or similar performance than semi-structural pruning methods. 5.3 Zero-shot Performance In Tab. 3, we present the zero-shot performance of the pruned model. For the LLaMA 7B model, we compare our method against LLM-Pruner with and without recovery fine-tuning. Our method consistently outperforms LLM-Pruner without fine-tuning, and the gap ranges from 2.63 to 8.88 across different pruning rates for average task performance. After fine-tuning, the performance of LLM-Pruner is largely boosted, however, our method is still able to outperform it demonstrating the existence of strong sub-networks within the original model. For the LLaMA-2 7B model, we compare our method against SliceGPT, K-OBD, and LLM Surgeon. With weight updates, LLM 8 (a) Ablation Loss L (b) Ablation Loss R (c) Ablation p (d) Ablation Iterations (e) Given p Loss L (f) Given p Loss R Dense 𝟏. 𝟎𝟖× 𝟏. 𝟐𝟎× 𝟏. 𝟑𝟔× 𝟏. 𝟓𝟎× (g) Acceleration 𝟏𝟒. 𝟕𝟔× 𝟐𝟕. 𝟑𝟗× (h) Costs Figure 5: The training dynamics when learning the hypernetwork are shown in Figs. 5a, 5b, 5e, 5f. The results of different settings are in Figs. 5c, 5d, throughput is in Fig. 5g, and cost is in Fig. 5h. 𝐒𝟏 𝐒𝟐 𝐒𝟑 𝐒𝟒 𝐒𝟓 Figure 6: Model width after pruning for the LLaMA-2 7B model when the pruning ratio equals 50%. Surgeon performs well with a lower pruning ratio like 30%. At this pruning ratio, our method performs similarly to LLM Surgeon, and our method outperforms other comparison baselines. When increasing the pruning ratio to 50%, the advantage of our method becomes obvious, and the gap between our method and LLM Surgeon is 5.37 for average task performance. We further compare our method with SliceGPT on Phi-1.5 and Phi-2, and our method consistently achieves better performance. 5.4 Analysis Ablation Study. We visualize the results of ablation studies in Figs. 5a, 5b, 5c, 5d with Phi-1.5 model. The setting “DISP-LLM w/o HN” refers to using element-wise gates for learning subnetworks. The setting “Constrained LLM w HN” refers to pruning the embedding dimension following the structural dependence like LLM-Pruner. From Figs. 5a, 5b, we can see that using the hypernetwork largely accelerates the learning process for DISP-LLM, which is also verified in Figs. 5c, 5d. From Figs. 5c, 5d, we also see that our DISP-LLM always outperforms constrained structural pruning, demonstrating the value of added flexibility by breaking the dependence. To further study the impact of the HyperNetwork architecture, we provide more results in Tab. 4. “w/o HN” is equivalent to “DISP-LLM w/o HN”. The setting “w/o Bi-GRU” simply removes GRU and adds a fixed input (initialized in the same way as see Appendix A.3 for more details) for each linear layer. These results indicate that both GRU and linear layers within the HyperNetwork affect the final performance. One explanation is that linear layers connect different dimensions of the model, accelerating learning, while GRU layers capture inter-layer relationships, further reducing the difficulty of learning sub-network structures. Therefore, both GRU and linear layers are essential to the HyperNetwork. Different Pruning Ratios, Costs and Throughput. In Figs. 5e, 5f, we show the language modeling loss L and regularization loss R in Obj 5 given different pruning ratios p with Phi-1.5 model. We can see that our method consistently minimizes the language modeling loss across different p. In addition, 9 Table 4: The impact of the Hypernetwork architecture on the Phi-1.5 model. Performance is measured by PPL (perplexity). Compression Rate Settings 0% 10% 20% 30% 40% 50% w/o HN 21.82 20.37 22.30 28.66 34.33 47.29 w/o Bi-GRU 19.90 21.65 26.11 30.88 37.43 Full HyperNetwork 18.75 20.23 22.81 25.49 32.89 our method quickly pushes the regularization loss R to near 0 values within 200 iterations. In Fig. 5g, the pruned model from LLaMA-13B improves the throughput of the dense model by 1.08ˆ to 1.50ˆ. In Fig. 5h, we compare the costs of our method against LLM Surgeon. Our method is 27.39ˆ and 14.76ˆ cheaper compared to LLM Surgeon with LLaMA-2 7B and LLaMA-2 13B models. Every Embedding Dimension is Important. In Fig. 4, we visualize the pruning decisions along the embedding dimension and depth for the LLaMA-2 7B model, we can see that all embedding dimensions have been used across different layers. This becomes more obvious in the second right figure of Fig. 4, where we sum all pruning decisions along the depth dimension, and we can see that every embedding dimension is kept around 80% given all layers. We further visualize the model width after pruning for the LLaMA-2 7B model in Fig. 6, where we can see that several layers are more severely pruned than other layers. 6 Conclusion In this paper, we proposed dimension-independent structural pruning for Large Language Models. By breaking the structural dependence imposed by previous compression methods, our method creates sub-networks with a lot more flexibility than regular structural pruning methods and also avoids the overhead brought by SliceGPT. The flexibility of our method is reflected in two perspectives. Firstly, our method provides different subsets of the feature maps for different layers. Secondly, our method freely selects the width for each layer without considering architecture dependence. With dramatically increased flexibility, our method outperforms other structural pruning and semi-structural pruning methods given similar pruning ratios. 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Opt: Open pre-trained transformer language models. arXiv preprint arXiv:2205.01068, 2022. 13 A Appendix Inputs RMS Norm 𝐐𝟏 𝐓𝐖𝐤′ 𝐐𝟏 𝐓𝐖𝐪′ 𝐐𝟏 𝐓𝐖𝐯′ Multi-head Attention 𝐖𝐨′𝐐𝟐 𝐐𝟏 𝐓𝐐𝟐 Activation 𝐐𝟐 𝐓𝐐𝟑 Outputs RMS Norm 𝐐𝟐 𝐓𝐖𝟏 ′ 𝐖𝟐 ′𝐐𝟑 SliceGPT Inputs 𝐒𝟏 𝐓𝐖𝐤 𝐒𝟏 𝐓𝐖𝐪 𝐒𝟏 𝐓𝐖𝐯 Multi-head Attention 𝐖𝐨𝐒𝟐 Activation Outputs 𝐒𝟑 𝐓𝐖𝟏𝐒𝟒 𝐒𝟒 𝐓𝐖𝟐𝐒𝟓 Layer Norm Ours Layer Norm Index Add Index Select & Layer Norm When Pruning Layer Norm 𝐒𝟏, 𝐒𝟐, 𝐒𝟑, 𝐒𝟒, 𝐒𝟓 ෠𝐒𝟏, ෠𝐒𝟐, ෠𝐒𝟑, ෠𝐒𝟒, ෠𝐒𝟓 Figure 7: Left: SliceGPT inserts QJ l Ql`1 to the residual connection and brings additional parameters. It also modifies the weights and Layer Norms within the original model. The selection matrix S is omitted for consistency. Right: Our method, DISP-LLM, applies different selection matrices to the input and output dimension of the Attention layer and MLP layer (S1{S2: Attention in/out; S3{S4{S5: MLP in/middle/out). Model Dimension Depth Preserved Pruned Pruning Decisions along the Model Dimension and Depth Figure 8: The pruned model architecture along the embedding dimension (model dimension) for the LLaMA-2 13B model when the pruning ratio equals 50%. Figure 9: Model width after pruning for the LLaMA-2 13B model when the pruning ratio equals 50%. A.1 Proof of Proposition 1 Proposition 1 (Decreasing feature dimensions for deeper layers). Let the pseudo-selection matrices in layers l and l ` 1 be Sl and Sl`1, respectively. The number of nonzero entries in the residual 14 Algorithm 2: Binary ReinMax Input: x: sigmoid input; τ: temperature; c: constant bias. Output: x: binary vector. 1. π0 “ sigmoidpx ` cq, 2. B “ sample_binarypπ0q, 3. π1 “ B`sigmoidppx`cq{τq 2 , 4. π1 “ sigmoidpstop_gradientplnpπ1q ´ px ` cqq ` px ` cqq, 5. π2 “ 2π1 ´ 1 2π0, 6. x “ π2 ´ stop_gradientpπ2q ` B Return x. adapter satisfies nnzpSJ l Sl`1q ď mintnnzpSlq, nnzpSl`1qu. For compression strategies that remove dependent structures for layer l ` 1 following SJ l Sl`1, this implies that the dimension in layer l ` 1 is less than or equal to that in layer l, with equality holding when the feature indices selected in layer l ` 1 are contained within those in layer l or vice versa. Proof. Consider the pseudo-selection matrices Sl and Sl`1, both of size d ˆ d, and both diagonal matrices. The number of nonzero entries in Sl and Sl`1 are given by nnzpSlq “ kl and nnzpSl`1q “ kl`1, respectively. The product SJ l Sl`1 is also a diagonal matrix of size d ˆ d. Each diagonal entry pi, iq in SJ l Sl`1 is the product of the i-th diagonal entry of Sl and the i-th diagonal entry of Sl`1. For an entry pi, iq to be nonzero, both Slpi, iq and Sl`1pi, iq must be nonzero. Thus, the number of nonzero entries in SJ l Sl`1, nnzpSJ l Sl`1q, is the number of indices i where both Slpi, iq and Sl`1pi, iq are nonzero. This count cannot exceed the smaller of the total number of nonzero entries in Sl and Sl`1. Hence, nnzpSJ l Sl`1q ď mintnnzpSlq, nnzpSl`1qu. This implies that the effective feature dimension will be smaller or equal to the previous layer. Equality holds if and only if the set of indices corresponding to nonzero entries in Sl`1 is a subset of those in Sl, or vice versa. This concludes the proof. Figure 10: Expected compression rate vs. actual compression rate of our method and Slice-GPT on the LLaMA-7B model. In Proposition 1, “remove dependent structures for layer l ` 1 following SJ l Sl`1” means that the actual selection matrix for layer l ` 1 becomes S1 l`1 “ SJ l Sl`1, and the structure dependence is cut off by the next residual connection. The pruning for layer l ` 1 will based on S1 l`1 instead of Sl`1. Although this setting partially breaks the structural dependence, it has the limitation that the embedding dimensions will be reduced when going deeper. A.2 Binary ReinMax In this section, we provide details for ReinMax when handling binary variables. The ReinMax in our work can be written as shown in Algorithm. 2. We add a constant bias c to x so that we can control binary vectors to have all one value at the beginning when learning the sub-network architecture for DISP-LLMs. Through all experiments, we set c to 3.0 and τ to 1.0. A.3 Details of the Hypernetwork 15 Table 5: The architecture of hypernetwork. Input z Bi-GRU(32,64)Ñ LayerNormÑ GeLU Linearl(128, Nl)ÑOutputs sl, l “ 1, ¨ ¨ ¨ , L Table 6: Time costs of our method. Model Time / GPUs LLaMA/LLaMA-2 7B 2.41 Hours / 2 NVIDIA A100 80G LLaMA/LLaMA-2 13B 8.83 Hours / 4 NVIDIA A100 80G (a) WikiText Loss L (b) WikiText Loss R (c) Alpaca Loss L (d) Alpaca Loss R Figure 11: The training dynamics when learning the hypernetwork for LLaMA-2 7B model with WikiText and Alpaca datasets. As we discussed in the paper, the Hypernetwork is composed of linear layers and Bi-GRUs, and now we present the architecture of the HN in Tab. 5. z is initially sampled from a normal distribution, and it is then fixed during training. Outputs sl are continuous values and it is then fed to ReinMax to produce the binary vector: s “ ReinMaxpsq, where s is the collection of sl from all layers. Nl is the original model width, and it equals the embedding dimension for S1, S2, S3 and S5. A.4 More Implementation Details Figure 12: Preserved rates of the LLaMA-2 13B model across different dimensions. The result is accumulated across all the layers. Additional Training Details. During training the hypernetwork, we use AdamW optimizer to optimize it with a constant learning rate 10´3 and weight decay 0.05. We train the hypernetwork for different models, we always set the mini-batchsize to 1 on each GPU. For OPT 6.7B, LLaMA 7B, and LLaMA-2 7B models, we use 2 NVIDIA A100 GPUs, and for LLaMA 13B and LLaMA-2 13B models, we use 4 NVIDIA A100 GPUs. For all the rest models, we use 1 NVIDIA A100 GPU. We set p “ t0.5, 0.4, 0.3, 0.2, 0.1u when the pruning ratios equals to t50%, 40%, 30%, 20%, 10%u. For the Alpaca dataset 2, we use the ‘text’ column within the dataset which combines the columns of ‘instruction’ and ‘output’. When training the hypernetwork, we again minimize the language modeling loss on the Alpaca dataset instead of applying the training process of instruction fine-tuning. Details of Eq. 6. The parameter regularization loss function in Eq. 6 is defined as follows: Rpx, yq “ log ˆmaxpx, yq minpx, yq ˙ “ $ ’ & ’ % log ´ x y ¯ if x ą y, 0 if x “ y, log ` y x ˘ if x ă y . Since y is fixed, when x ą y, Eq. 6 will decrease x, making it closer to y. Conversely, when x ă y, Eq. 6 will increase x, also making it closer to y. Thus, the parameter regularization loss always tries to push the current sub-network to achieve the pre-defined parameter budget. 2https://huggingface.co/datasets/tatsu-lab/alpaca 16 Table 7: Zero-shot performance of the compressed LLaMA-2 13B model. Pruning Ratio Method W Update? WinoGrande HellaSwag ARC-e ARC-c PIQA Avg acc acc-norm acc-norm acc-norm acc-norm 0% LLaMA-2 13B 72.22 79.39 77.48 49.23 80.47 71.76 SliceGPT [2] ✓✗ 65.11 52.69 51.77 31.23 66.10 55.16 K-OBD [34] ✓ 64.96 64.18 56.23 36.01 74.43 59.16 LLM Surgeon [34] ✓ 68.67 71.52 69.74 40.27 76.50 65.34 DISP-LLM (Ours) ✗ 66.85 70.86 63.80 39.42 74.43 63.07 30% DISP-LLM Alpaca (Ours) ✗ 67.32 70.04 68.98 44.28 77.31 65.59 K-OBD [34] ✓ 60.46 55.52 49.62 32.68 70.24 53.70 LLM Surgeon [34] ✓ 65.75 65.04 63.80 37.12 73.01 60.94 DISP-LLM (Ours) ✗ 62.67 65.86 60.31 37.63 73.39 59.97 40% DISP-LLM Alpaca (Ours) ✗ 64.25 67.52 66.79 42.75 75.30 63.32 K-OBD [34] ✓ 57.46 48.39 46.59 30.72 66.54 49.94 LLM Surgeon [34] ✓ 63.22 56.19 56.19 31.83 68.77 55.24 DISP-LLM (Ours) ✗ 59.27 58.63 52.57 33.28 68.77 54.50 50% DISP-LLM Alpaca (Ours) ✗ 59.59 62.39 55.72 37.54 72.20 57.49 Table 8: Zero-shot performance of the compressed Phi-2 given more pruning rates and settings. Pruning Ratio Method W Update? WinoGrande HellaSwag ARC-e ARC-c PIQA Avg acc acc-norm acc-norm acc-norm acc-norm 0% Phi-2 75.61 73.86 78.24 54.01 79.11 72.17 SliceGPT [2] ✓✗ 67.80 57.76 58.00 35.32 71.87 58.15 +fine-tuning ✓ 67.17 54.86 56.61 38.91 71.27 57.76 20% DISP-LLM (Ours) ✗ 67.09 62.93 68.18 44.11 74.86 63.43 SliceGPT [2] ✓✗ 65.35 52.40 53.70 31.66 69.21 54.46 +fine-tuning ✓ 65.19 52.48 52.78 35.49 69.91 55.17 25% DISP-LLM (Ours) ✗ 65.11 59.95 65.93 43.34 74.27 61.72 SliceGPT [2] ✓✗ 63.14 47.56 53.03 30.29 65.94 51.99 +fine-tuning ✓ 63.54 49.72 46.38 32.68 66.16 51.70 30% DISP-LLM (Ours) ✗ 65.19 54.43 63.59 38.48 73.34 59.00 Ablation Study Settings. In the ablation study 5.4, we removed the hyperntwork, and we revise Eq. 4: s “ ReinMaxpΘq, where Θ now has the same size of s, and the parametrization space becomes much smaller. For the constrained setting used in the ablation study 5.4, we simply let S1 “ S2 “ S3 “ S5. In section 5.4, we calculate the costs of our method and LLM-Surgeon, and the price comes from the official website of Lambda Cloud3. We also list the detailed time costs of our method in Tab. 6. In Fig. 5b, 5f and also in Fig. 11, we normalized the parameter regularization loss R with its maximum value to make plots more consistent. Licenses. The licenses for various models and datasets are as follows: LLaMA and LLaMA 2: Licensed under the LLAMA 2 Community License. Phi 1.5 and Phi 2: Licensed under the MIT License. WikiText dataset: Licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA 4.0). Alpaca dataset: Licensed under the Creative Commons AttributionNonCommercial License (CC BY-NC 4.0). A.5 Additional Results SliceGPT compression rates: In Fig. 10, we show the expected compression rate and the actual compression rate for our method and SliceGPT given the LLaMA-2 7B model. It can be seen that SliceGPT adds 5% to 13% parameters of the original model across different pruning rates, which is non-trivial for most LLMs. Notably, SliceGPT with 10% slicing actually adds 3% more parameters to the original model. LLaMA-2 13B Results. In Tab. 7, we show the results of the LLaMA-2 13B model given different pruning rates. From the table, we can see that our method consistently outperforms LLM Surgeon under different pruning rates. The advantage of our method becomes more obvious compared to other methods like K-OBD and SliceGPT. Phi-2 Results. In Tab. 8, we present a more comprehensive comparison of our method compared to SliceGPT. Our method outperforms SliceGPT by 5.28 to 7.01 giving SliceGPT with or without 3https://lambdalabs.com/service/gpu-cloud#pricing 17 Table 9: Zero-shot performance of the compressed LLaMA 13B model. Pruning Ratio Method W Update? WinoGrande HellaSwag ARC-e ARC-c PIQA Avg acc acc-norm acc-norm acc-norm acc-norm 0% LLaMA 13B 72.53 79.06 74.62 47.78 80.41 70.88 Magnitude ✗ 57.54 52.90 50.13 31.14 67.57 51.86 +fine-tuning ✓ 64.64 71.86 67.59 39.93 76.77 64.15 LLM Pruner [30] Channel ✗ 58.96 49.17 49.62 31.83 66.87 51.29 +fine-tuning ✓ 66.38 68.89 62.08 38.99 76.55 62.58 LLM Pruner [30] Block ✗ 65.11 73.41 68.35 38.40 77.15 64.48 +fine-tuning ✓ 67.88 75.16 71.09 42.41 77.91 66.89 DISP-LLM (Ours) ✗ 68.75 75.28 70.16 44.80 76.61 67.12 20% DISP-LLM Alpaca (Ours) ✗ 66.54 74.80 69.73 44.71 78.07 66.77 DISP-LLM (Ours) ✗ 60.85 57.81 52.51 32.51 68.44 54.42 50% DISP-LLM Alpaca (Ours) ✗ 59.80 58.63 56.44 34.85 71.27 56.20 Table 10: Average results with 5 different runs. The result is evaluated on WikiText-2. Test Performance (PPL), Phi-1.5 Dense: 21.82 10% 20% 30% 40% 50% 18.72 ˘ 0.07 20.48 ˘ 0.24 22.62 ˘ 0.31 25.44 ˘ 0.39 32.72 ˘ 0.37 Table 11: Throughput of the pruned model. Model Pruning Ratio Tokens/seconds LLaMA-2 13B 0% 227.99 20% 245.65 30% 273.60 40% 310.07 50% 342.09 fine-tuning on the WikiText dataset. These observations again demonstrate the effectiveness of our method, and our method outperforms methods with recovery fine-tuning in several settings. LLaMA-2 3B Results. In Table 9, we present a comparison of our method against LLM Pruner and magnitude pruning. At a pruning ratio of 20%, our method surpasses the performance of LLM Pruner both with and without fine-tuning. Remarkably, even when the pruning ratio is increased to 50%, our method continues to outperform the LLM Pruner Channel and the Magnitude pruning baselines at a 20% pruning rate. These results further illustrate our method’s ability to identify strong sub-networks within the original dense model. Architecture of the pruned LLaMA-2 13B. In Fig. 8 and Fig. 9, we visualize the pruned architecture of the LLaMA-2 13B model pruned with the WikiText dataset. We have similar observations as in section. 5.4. In Fig. 9, we can see that middle to late layers have large pruning rates, especially for the attention layer. In Fig. 8 and Fig. 12, we can also see that the preserved rates for different dimensions are similar, and all embedding dimensions are effectively utilized. More interestingly, similar pruning rates across different dimensions are achieved without adding any regularization or constraints. Training loss for LLaMA-2 7B. In Fig. 11, we further visualize the training dynamics of the LLaMA-2 7B model on WikiText and Alpaca datasets, respectively. The regularization loss with the Alpaca dataset decreases a little bit faster than the WikiText dataset, probably because the loss value on the Alpaca dataset is smaller. From Fig. 11c, we can also see that the language modeling loss continues to decrease when training longer, especially when the pruning ratio is higher. Lastly, we evaluate our method on the Phi-1.5 model with 5 runs, and we report our result with mean and standard deviation in Tab. 10. We also measure the throughput of our method on the LLaMA-2 13B model, and the result is shown in Tab. 11 A.6 Additional Analysis Table 12: Impact of λ on Phi-1.5 Test Performance (PPL), Phi-1.5 Dense: 21.82 λ=1.0 λ=2.0 λ=4.0 λ=6.0 λ=8.0 λ=10.0 NC 36.39 33.71 32.89 33.31 33.20 Impact of λ on Phi-1.5 when pruning 50% of parameters. We show the result in Tab. 12. ‘NC’ means the loss R does not converge, and it is much larger than zero, thus it can not prune the model to the target budget. From the table, we can see that the PPL of the model becomes quite stable if it is larger or equal to 6. If λ is not large enough, it takes longer to push the loss R to reach near 0 values and thus leaves less time for the model to explore different configurations of subnetworks. On the other hand, if λ is large enough, the loss R will reach zero in several hundred iterations and leave enough time to find the desirable subnetwork. Due to this reason, our method is quite stable across larger values of λ as shown in the table. 18 Table 13: PPL vs. pruning ratio trade-off for the Phi-2 model. Test Performance (PPL), Phi-2 Dense: 10.98 10% 20% 30% 40% 50% 10.22 10.94 14.46 16.02 20.05 Table 14: Zero-shot task performance vs pruning ratio trade-off for the LLaMA-2 7B model with the WikiText dataset Avg task acc, LLaMA-2 7B 0% 20% 30% 40% 50% 68.99 62.54 58.10 52.63 46.72 Table 15: Zero-shot performance of the compressed LLaMA-2 7B model with LoRA fine-tuning. Pruning Ratio Method W Update? WinoGrande HellaSwag ARC-e ARC-c PIQA Avg acc acc-norm acc-norm acc-norm acc-norm 0% LLaMA 7B 69.85 76.21 72.81 44.71 79.16 68.55 50% DISP-LLM Alpaca ✗ 56.20 49.35 51.14 30.20 68.34 51.05 +LoRA ft ✓ 56.83 53.74 57.49 32.42 70.78 54.25 More results on pruning ratio vs. performance trade-offs. We provide the trade-off between the pruning ratio and performance for the Phi-2 and LLaMA-2 7B model below in Tab. 13 and Tab. 14. LoRA fine-tuning [18] of the compressed LLaMA-2 7B model. We follow similar settings of the SliceGPT and the model is fine-tuned on Alpaca. The result is shown in Table. 15. We can see that our method can be further boosted by using parameter-efficient fine-tuning techniques. Since the performance without fine-tuning is already good enough, we prefer not to involve this additional process in our method to save time and computational costs. A.7 Generation Samples We show the generated text given DISP-LLM and SliceGPT in Tab. 16. The examples are obtained based on removing 20% of the model weights with DISP-LLM and 20% slicing of SliceGPT ( 10% compression rate). Both models are compressed from LLaMA-2 7B and they are not finetuned. From these two examples, we can see that SliceGPT only generates a small part of meaningful content and then starts repeating itself. On the other hand, our method tends to generate more relevant content, and the self-repeating behavior is much less obvious. These observations comply with the performance gap between DISP-LLM and SliceGPT. A.8 Limitations Our method explores how to break the structural dependency of pruning for LLMs. Although our method achieves competitive performance, there are still some limitations that are not solved in our current version. The throughput improvements of our method are not consistent across different models, which is probably because the current implementation of index add or index select operations with PyTorch is not efficient enough. An alternative implementation of our method is to select weight matrices instead of feature maps. In this approach, we perform matrix multiplication with the full feature map and fill the weight matrices with zeros. If we have a custom matrix multiplication implementation that ignores zero rows or columns, it may achieve further speed-up. However, this approach primarily focuses on the engineering perspective and is beyond the scope of this work. We leave the exploration of this alternative implementation to future research. A.9 Border Impact Our dimension-independent structural pruning method significantly reduces the computational and memory requirements of LLMs, enabling their deployment on resource-limited devices and lowering energy consumption, thus contributing to environmental sustainability. By facilitating the use of efficient LLMs on affordable hardware, our approach democratizes access to advanced AI technologies. However, it is crucial to address potential ethical concerns when reducing the size of LLMs, ensuring privacy and fairness in AI applications with compressed LLMs. Overall, our method fosters a more accessible and sustainable use of AI. 19 Table 16: Generation samples of LLaMA-2 7B with 20% compression rate SliceGPT AI is a very important part of the game. The game is played in a 3D environment, the player controls a character that can move freely around the environment. The player can interact with the environment by picking up items, using them to solve puzzles. The player can also interact with other characters in the game. The player can also interact with the environment by picking up items and using them to solve puzzles. The player can also interact with other characters in the game. The player can also interact with the environment by picking up items and using them to solve puzzles. The player can also interact with other characters in the game. The player can also interact with the environment by picking up items and using them to solve puzzles. The player can also interact with other characters in the game. The player can also interact with the environment by picking up items and using them to solve puzzles. The player can also interact with The room-temperature superconductor is a material that exhibits superconductivity at room temperature.(100 °C). The first room temperature superconductor was discovered in 1970 by Peter C. Edwards and his colleagues at the University of Michigan. The material was a mixture of bismuth and tin, and was named " bismuth- tin." The discovery of bismuth tin was followed by the discovery of other room temperature superconductors, including bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin, bismuth tin DISP-LLM AI is a branch of computer science that seeks to create intelligent machines that can think and learn like humans. The field was developed in the 1950s and 1960s by researchers who focused on making machines that could learn and solve problems like people do. The field was commercially successful in the 1980s and 1990s with the development of the first generation of intelligent agents. The term " artificial intelligence " was coined by John McCarthy in 1956. He was inspired by the idea of creating a machine that could learn and solve problems like humans do. The room-temperature superconductor is a class of superconductors that exhibit zero resistance at room temperature. ## History The room-temperature superconductor was discovered in 1986 by the Japanese scientist K. Masamichi Aoki and his colleagues at the University of Tokyo. The discovery was made possible by the use of a new technique called " zero temperature transport measurement " ( ZTM ), which allowed them to measure the resistance of the superconductor at temperatures as low as 0.05 K. The discovery was made possible by the use of a new technique called " zero temperature transport measurement " ( ZTM ), which allowed them to measure the resistance of the superconductor at temperatures as low as 0.05 K. ## Discovery The discovery of the room-temperature superconductor was made possible by the use of a new technique called " zero temperature transport measurement " ( ZTM ), NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and precede the (optional) supplemental material. The checklist does NOT count towards the page limit. Please read the checklist guidelines carefully for information on how to answer these questions. 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4,441	Federated Graph Learning for Cross-Domain Recommendation Ziqi Yang1,2, Zhaopeng Peng1,2, Zihui Wang1,2, Jianzhong Qi3, Chaochao Chen4, Weike Pan5, Chenglu Wen1,2, Cheng Wang1,2, Xiaoliang Fan1,2∗ 1Fujian Key Laboratory of Sensing and Computing for Smart Cities, Xiamen University, China 2Key Laboratory of Multimedia Trusted Perception and Efficient Computing, Ministry of Education of China, Xiamen University, China 3School of Computing and Information Systems, The University of Melbourne, Australia 4College of Computer Science and Technology, Zhejiang University Hangzhou, China 5College of Computer Science and Software Engineering, Shenzhen University Shenzhen, China {yangziqi,pengzhaopeng,wangziwei}@stu.xmu.edu.cn {clwen,cwang,fanxiaoliang}@xmu.edu.cn jianzhong.qi@unimelb.edu.au, zjuccc@zju.edu.cn, panweike@szu.edu.cn Abstract Cross-domain recommendation (CDR) offers a promising solution to the data sparsity problem by enabling knowledge transfer between source and target domains. However, many recent CDR models overlook crucial issues such as privacy as well as the risk of negative transfer (which negatively impact model performance), especially in multi-domain settings. To address these challenges, we propose FedGCDR, a novel federated graph learning framework that securely and effectively leverages positive knowledge from multiple source domains. First, we design a positive knowledge transfer module that ensures privacy during inter-domain knowledge transmission. This module employs differential privacy-based knowledge extraction combined with a feature mapping mechanism, transforming source domain embeddings from federated graph attention networks into reliable domain knowledge. Second, we design a knowledge activation module to filter out potential harmful or conflicting knowledge from source domains, addressing the issues of negative transfer. This module enhances target domain training by expanding the graph of the target domain to generate reliable domain attentions and fine-tunes the target model for improved negative knowledge filtering and more accurate predictions. We conduct extensive experiments on 16 popular domains of the Amazon dataset, demonstrating that FedGCDR significantly outperforms state-of-the-art methods. We open source the code at https://github.com/LafinHana/FedGCDR. 1 Introduction Cross-domain recommendation (CDR) has emerged as an effective solution for mitigating data sparsity in recommender systems [1; 2; 3; 4; 5]. CDR operates by integrating auxiliary information from source domains, thereby enhancing recommendation relevance in the target domain. Recently, to address data privacy constraints, many privacy-preserving CDR frameworks have been proposed [6; 7; 8; 9], which achieve strong performance under the assumptions of data sparsity and a dual-domain model (i.e., typically involving a single source domain and a single target domain). In this paper, we focus on a more generic scenario of Broader-Source Cross-Domain Recommendation (BS-CDR), which integrates knowledge from more than two source domains while ∗The corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). (a) The BS-CDR scenario. (b) The performance affected by the number of domains Figure 1: (a) In order to obtain accurate recommendations in the Books domain, we aim to exploit user preferences (i.e., knowledge of external domains should be fully utilized, e.g. Movie, Toys, and Games domains). However, with the influence of lossy privacy-preserving techniques, the results of the transfer could be negative (e.g., the Music domain with low-quality data). (b) There is a diminishing marginal effect on the growth rate of the model performance with pure positive knowledge, while NT accumulates with an increasing number of source domains. Consequently, the performance of existing methods declines and is worse than that of a single domain model. preserving privacy. Given the diverse nature of user preferences, it is essential to gain a more holistic understanding of user interests by incorporating user behaviors from diversified domains [10; 11]. For example, in Figure 1a, a user who enjoys certain types of books might also enjoy movies, toys, and games in similar genres. However, incorporating more domains while preserving privacy poses challenges to counteract negative transfer (NT), which is a phenomenon of transferring knowledge from a source domain that negatively impacts the performance of the recommender model in the target domain [12]. Suppose that the Books domain in Figure 1a is the target domain. The Clothing domain causes NT because of the domain discrepancy. While the Music domain is supposed to transfer positive knowledge, it might also lead to NT because of lossy privacy-preserving techniques applied to broader source domains. As a result, the influx of negative knowledge accumulated from the source domains will poison the model performance of the target domain in BS-CDR scenarios. To mitigate the NT issue, attention mechanisms have been widely leveraged, either in an explicit (e.g., determine domain attentions by predefined domain features [13; 14]) or implicit manner (e.g., employ hyper-parameters [7; 15]). Several other studies [16; 17; 18], ensure positive transfer by passing only domain-shared features. However, existing methods cannot be directly applied to BS-CDR due to two major challenges. First, inadequate privacy preservation (CH1). Both intra-domain and inter-domain privacy must be carefully considered in BS-CDR. As depicted in Figure 1a, BS-CDR relies on extensive knowledge transfer, risking simultaneous privacy leakages across broader source domains (inter-domain privacy) [9; 19; 20; 21]. Additionally, concerns over centralized data storage may prevent users from sharing sensitive rating data (intra-domain privacy). Second, accumulative negative transfer (CH2). Adjusting attention-related hyper-parameters for a large number of source domains in BS-CDR scenarios is extremely difficult, as well as predefined or domain-shared features cannot accommodate complex domain diversities. In addition, the use of various lossy privacypreserving techniques can further degrade the quality of the transferred knowledge, complicating the achievement of positive transfer. Consequently, the impact of NT can inevitably intensify with an increasing number of source domains [2] and the performance of CDR models can decline to levels lower than those of single-domain models, as shown in Figure 1b. To address the challenges of privacy (CH1) and NT (CH2) in BS-CDR, we propose Federated Graph learning for Cross-Domain Recommendation (FedGCDR). It follows a horizontal-verticalhorizontal pipeline [6] and consists of two key modules. First, the positive knowledge transfer module aims to safeguard inter-domain privacy and mitigate potential NT before transfer. This module adopts differential privacy (DP) [22] with a theoretical guarantee and aligns the feature spaces to facilitate positive knowledge transfer. Second, the positive knowledge activation module is engaged to further alleviate NT. Specifically, it expands the local graph of the target domain by incorporating virtual social links, enabling the generation of domain attentions. Additionally, it 2 performs target model fine-tuning to optimize the broader-source CDR. Extensive experiments on 16 popular domains of the Amazon benchmarks demonstrate that FedGCDR outperforms all baseline methods in terms of recommendation accuracy. Our contributions are summarized as follows: • We introduce FedGCDR, a novel federated graph learning framework for CDR that provides high-quality BS-CDR recommendations while safeguarding both user privacy and domain confidentiality; • We propose two key model, i.e., the positive knowledge transfer module and the positive knowledge activation module. The first transfer module ensure privacy and positive knowledge flows via privacy-preserving knowledge extraction and feature mapping. The second activation module filter harmful information via graph expansion, target domain training and target model fine-tuning; • We conduct extensive experiments on 16 domains of the Amazon datasets that confirm the effectiveness of FedGCDR in terms of recommendation accuracy. 2 Related work 2.1 Cross-domain recommendation CDR utilizes auxiliary information from external domains to alleviate the data sparsity problem and effectively improve recommendation quality. Li et al. [23] enrich domain knowledge by transferring user-item rating patterns from source domains to target domains. Man et al. [15] and Elkahky et al. [24] augment entities’ embeddings in the target domain by employing a linear or multilayer perceptron (MLP)-based nonlinear mapping function across domains. Liu et al. [25] address the review-based non-overlapped recommendation problem by attribution alignment. Zhao et al. [18] improve the recommendation quality of multi-sparse-domains by mining domain-invariant preferences. Liu et al. [26] achieve knowledge transfer without overlapping users by mining joint preferences. Chen et al. [19] and Liu et al. [27] avoid intermediate result privacy leakage during cross-domain knowledge transfer by employing DP. In these works, the NT problem is often ignored because most of them assume a carefully selected dual-domain scenarios or limited multi-domain scenarios where NT is not evident. We aim to solve the NT problem in complex BS-CDR scenarios. 2.2 Federated recommendation Recently, federated learning (FL) [28; 29; 30; 31; 32] has been widely adopted to tackle the privacy issue in recommender system. Chai et al. [33] adopt FL to classic matrix factorization algorithm and utilize homomorphic encryption to avoid the potential threat of privacy disclosure. Later, Wu et al. [34] explore the application of federated graph neural networks (GNN) models to improve the recommendation quality and ensure user privacy. To utilize sensitive social information, Liu et al. [8] adopt local differential privacy (LDP) and negative sampling. More recent studies use vertical federated learning (VFL) to protect company’s privacy in recommender system. Mai et al. [35] utilize random projection and ternary quantization to ensure privacy preservation in VFL. In CDR, Chen et al. [9] design a dual-target VFL CDR model with orthogonal mapping matrix and LDP for organizations’ privacy preservation. Liu et al. [36] design a graph convolutional networks (GCN)-based federated framework to learn user preference distributions for more accurate recommendations. To ensure user privacy in CDR, Liu et al. [6] utilize a VAE-based federated model to mine user preference with data stored locally. Wu et al. [7] design a personal module and a transfer module to provide personalized recommendation while preserving user privacy. These existing works, especially federated CDR frameworks, consider only one type of privacy (intra- or inter-domain). We aim to provide both intra-domain and inter-domain privacy. 3 Figure 2: An overview of FedGCDR. It consists of two key modules and follows a HVH pipeline: (1) Source Domain Training (Horizontal FL): 1⃝Each source domain maintains its graph attention network (GAT)-based federated model. (2) Positive Knowledge Transfer Module (Vertical FL): 2⃝ Source domain embeddings are extracted from GAT layers and perturbed with Gaussian noise. 3⃝ The multilayer perceptron aligns the feature space of source domain embeddings and target domain embeddings. (3) Positive Knowledge Activation Module (Horizontal FL): 4⃝Local graph is expanded with source domain embeddings. 5⃝Enhanced federated training of the target domain is achieved through the expanded graph. 6⃝The target domain maintains its GAT-based federated model. 7⃝The target domain freezes the GAT layer and fine tunes the model. 3 Methodology 3.1 Problem definition We consider M (M>3) domains participating in the CDR process. The domains are divided into M-1 source domains DS1, DS2, ..., DSM−1 and one target domain DT . Each domain is assigned a domain server to conduct intra-domain model training. U is the user set across all the domains, U = U1 S U2 S ... S UM, where Ui denotes the user set of domain i. We assume that users partially overlap between domains. Each user is treated as an individual client. User space refers to the virtual space in the user’s device containing domain models distributed from each domain server. Meanwhile, Vi is the item set of domain i. Let Ri ∈R\|Ui\|×\|Vi\| be the observed rating matrix of the i-th domain. We consider top-K recommendation, i.e., we learn a function to estimate the scores of unobserved entries in the rating matrix, which are later used for item ranking and recommendations. Our goal is to achieve highly accurate recommendations in the target domain. 3.2 Framework of FedGCDR 3.2.1 Overview The overall framework of FedGCDR is shown in Figure 2. FedGCDR follows a Horizontal-VerticalHorizontal (HVH) pipeline, and its two horizontal FL stages ensure the intra-domain privacy. Our two key modules focus on the vertical stage and the second horizontal stage: (1) The positive knowledge transfer module preserves the inter-domain privacy by DP and alleviates NT by feature mapping. (2) The positive knowledge activation module filters out potential harmful or conflicting knowledge from the source domains. Specifically, we expand the local graph of the target domain by virtual social links, such that the target domain graph attention network (GAT) model could generate reliable domain attention based on the expanded graph. After target domain GAT model training, we further mitigate NT by adopting a fine-tuning stage. Horizontal-Vertical-Horizontal pipeline The HVH pipeline contains three stages with switching federated settings. The first horizontal stage refers to the source domain training in which source domain servers individually interact with its domain users (clients). The private rating information is stored within each client, while the clients exchange model and gradients to train a domain-specific global model. The next two stages correspond to our two key modules (vertical positive knowledge transfer module and horizontal positive knowledge activation module), which we will cover in detail in the following subsections. It’s important to note that the vertical positive knowledge transfer 4 module is completely computed in each client’s user space (their personal devices), thus reducing communication overheads. This is because the needed source domain knowledge can be extracted from local source models on each client which are distributed during the first horizontal source domain training stage. Following the HVH pipeline, we achieve: (1) Privacy enhancement. The two horizontal stages can provide intra-domain privacy preservation, while we further ensure inter-domain privacy by applying DP to the vertical stage. In the mean time, servers are not involved in the knowledge transfer process (i.e., the positive knowledge transfer module), making them unaware of user interactions in other source domains. (2) Communication efficiency. Cross-domain knowledge transfer does not require additional communication overhead. Intra-domain GAT-based federated model We adopt a GAT-based [37; 38; 39] federated framework as the underlying model for our intra-domain recommender system. The horizontal paradigm avoids centralized storage of user ratings to ensure intra-domain privacy (CH1). In the initial step, each user and item is offered an ID embedding of size d, denoted by e0 u, e0 v ∈Rd respectively. The embedding is passed through L message propagation layers [40; 41; 42]. For the l-th layer: el+1 u = σ(Wl(al uuel u + X v∈Nu al uvel v)), (1) where Nu is the neighbor set of u, Wl is a learnable weight matrix, al uu and al uv are the importance coefficients computed by the attention mechanism: al uv = exp(LeakyReLU(α(Wel u\|\|Wel v)) P v′∈Nu S u exp(LeakyReLU(α[Welu\|\|Wel v′]), (2) where α is the weight vector. Inspired by LightGCN [43], we discard feature transformation and nonlinear activation for better model efficiency and learning effectiveness: el+1 u = al uuel u + X v∈Nu al uvel v, (3) auv = exp(α(el u\|\|el v)) P v′∈Nu S u exp(α(elu\|\|el v′). (4) In each source domain, the domain server and corresponding users collaboratively train a GAT-based federated model. The training process follows the horizontal federated learning (HFL) paradigm in which only the model and gradients are exchanged considering intra-domain privacy. We will not detail the horizontal federation model (e.g., further privacy guarantee and more high-order information) as it is a well established FL model and not our novel contribution. This model can be replaced by other GAT-based FL models [34; 44] as well. 3.2.2 Positive knowledge transfer module After the source domain training, we obtain a series of source models in individual client’s user space. Our positive knowledge transfer module then prepares positive knowledge to be transferred from each source domains DS to the target domain DT , while protecting inter-domain privacy (CH1). Specifically, suppose an individual user (client) u and a source domain DSi, we transfer the user u’s embedding matrix XSi ∈RL×d . Take the row l of the matrix (i.e., xl Si) as an example, it is the user u’s embedding output by the l-th message propagation layer (el u). In an ideal scenario (i.e., we transfer totally positive knowledge without taking inter-domain privacy into account) [6], embedding matrices from different source domains can be directly used to enhance target domain local training in client u. By utilizing the source domain embeddings, u’s final target domain embedding el T of layer l is: el T = fT (xl T , xl S1, ..., xl SM−1), l ∈[1, L] (5) where fT (·) is the function that the target domain aggregates the knowledge of the source domains and we will give its final expression in Subsection 3.2.3. In this process, the transfer of knowledge between domains takes place entirely in the user u’s local space. Such a fully localized mode of knowledge transfer avoids additional communication overhead and potential privacy issues [6]. However, this direct embeddings transfer does not meet the privacy and NT constraints in BS-CDR scenarios. 5 Privacy-preserving knowledge extraction In existing CDR frameworks, the user or item embedding was shared as knowledge [9; 15; 6], which neglects inter-domain privacy. In a GNN-based approach, such direct transfers are subject to privacy attacks. Each message propagation layer can be viewed as a function with user and item embeddings as input. An attacker can easily obtain the user’s private rating matrix based on these embeddings. We apply DP to the source domain embeddings xSi [22; 45] to safeguard inter-domain privacy. THEOREM 1. By introducing Gaussian noise into the source domain embeddings, the reconstructed data from the ideal attack deviates from the actual data, therefore preventing a perfect reconstruction. In FedGCDR, we incorporate the Gaussian mechanism with the source domain embeddings xSi to obtain ˆxSi for knowledge transfer. Detailed privacy analysis is included in Appendix A. Feature mapping User features could represent personal preferences and are influenced by domain features. The discrepancy of domains leads to the heterogeneity of feature space between domains which means that source domain embeddings cannot be utilized directly by the target domain. Man et al. [15] show that there exists an underlying mapping relationship between the latent user matrix of different domains, which can be captured by a mapping function. In order to alleviate NT, we adopt a series of MLP to explore mapping functions for each source domain. Adding Gaussian noise and feature mapping, Equation (5) becomes: el T = fT (xl T , MLP1(ˆxl S1), ..., MLPM−1(ˆxl SM−1)). (6) To learn more effective mapping function, we adopt a mapping loss term: lm = M−1 X i=1 L X l=1 \|\|xl T −MLPi(ˆxl Si)\|\|2, (7) 3.2.3 Positive knowledge activation module After the aforementioned operations, the target domain obtains a list of source domain matrices ˆXS1, ˆXS2, ..., ˆXSM−1. The rows of the matrices represent MLPi(ˆxl Si). It is worth noting that for source domains where a user has no rating, ˆXSi is a Gaussian noise matrix and our motivation is: (1) no rating may also suggest a preference; (2) this is beneficial for enhancing the model’s capability to filter noise and identify NT. With knowledge from the source domains, the purpose of the positive knowledge activation module is to alleviate NT following the knowledge transfer. (CH2). Although we have aligned the feature space in the previous module, the Gaussian noise that has been fed to the target domain with source domain embedding matrices leads to potential NT. How to utilize the transferred knowledge remains a great challenge. Figure 3: Illustration of target domain graph expansion. The virtual users are constructed with the source domain embeddings from the Movie domain and the Music domain. The attentions generated by social links to the virtual user can be regarded as the domain attentions. Graph expansion and target domain training To alleviate NT, common approaches are to generate domain attention by predefined domain features [13; 46; 47] or to control the transfer ratio of source domains by Writefull [7]. These methods are only applicable to a limited number of domains and have excessive human intervention. In FedGCDR, we take an attention-based approach. First, we expand u’s (Mary’s) local graph of the target domain as shown in Figure 3. 6 For the source domain embedding matrices ˆXS1, ˆXS2, ..., ˆXSM−1, we represent them as M −1 virtual users. Since the virtual users constructed from source domain embeddings represent the same individual u, they share correlated preferences, with their features (i.e., embeddings) characterizing u’s preferences. Inspired by social recommendation [48; 49; 50], we consider that there is an implicit social relationship between virtual users and the actual user u, because of the correlation in their preferences. Then, we build virtual social links between them to expand the original target domain graph. Second, by incorporating this expanded graph into target domain training, the GAT model generates corresponding attention coefficients for the virtual users, which can be interpreted as domain-specific attentions. Leveraging the domain attention coefficients, the target domain can focus on domains that transfer positive knowledge and we can finally give fT (·): fT (xl T , MLP1(ˆxl S1), ..., MLPM−1(ˆxl SM−1)) = al uuxl T + X v∈Nu al uvel v + M−1 X i=1 al iMLPi(ˆxl Si), (8) where al i is the domain attention of source domain i generated by the l-th layer. Beside, we introduce a social regularization term to strengthen the virtual social links: ls = L X l=1 ∥xl T − PM−1 i=1 Sim(xl T , ˆxl Si) × ˆxl Si PM−1 i=1 Sim(xl T , ˆxl Si) ∥2, (9) the function Sim(·) calculates the cosine similarity [48]. Through the graph expansion, we achieve: (1) dynamic domain attentions that focus on positive source domain knowledge to alleviate NT; (2) attention generation by GAT, eliminating the need for additional interventions such as hyper-parameter tuning or feature engineering. For top-k recommendation, we adopt a widely-used inner product model to estimate the value of target domain rating RT uv , which is the interaction probability between a pair of user u and item v: ˆRT uv = Sigmoid(eu T · ev T ), (10) where eu T and ev T are the final user and item embeddings output by GAT. Our objective function consists of three terms as follows: LGAT = BCELoss( ˆRT uv, RT uv) + α 2 lm + β 2 ls, (11) where α and β are Writefull, and BCELoss(·) is the binary cross-entropy loss [51]. The target domain federated GAT training with the expanded graph following the HFL paradigm. Target model fine-tuning After target domain training with the expanded graph, the target domain GAT model assimilates knowledge from the source domains. However, NT may still be unavoidable, potentially leading to the accumulation of negative knowledge in the target domain. An example of this is the Gaussian noise matrices transferred from source domains where the user has no interactions. On the basis of this consideration, we adopt an additional fine-tuning stage: First, we freeze the message propagation layers of GAT to isolate the influence of source domains preventing negative information from permeating through the transfer process. Second, we directly train the well-informed embeddings generated by the target domain GAT. Adapting the learned external knowledge through these steps enables more accurate prediction of ratings in the target domain. In this process, we use the loss of prediction in Equation (11) as the object function: Lft = BCELoss( ˆRT uv, RT uv). (12) We provide a computational analysis and a communication analysis of FedGCDR in Appendix B. 4 Experiments 4.1 Experimental setup Datasets We study the effectiveness of FedGCDR with 16 popular domains of a real-world dataset Amazon [52]. To study the impact of the number of domains on model performance, we divide these 7 Table 1: Statistics on the Amazon Dataset. (min-median-max) values are provided for \|Ud\|, \|Id\| and \|Rd\|. Dataset \|U\| \|Ud\| \|Id\| \|Rd\| avg (sparsity) Amazon-4 55,518 6,632 - 12,626 - 27,402 53,082 - 134,438 - 501,153 623,420 - 646,266 - 5,481,801 0.0802% Amazon-8 99,506 6,632 - 13,978 - 27,402 53,082 - 106,985 - 501,153 186,016 - 618,539 - 5,481,801 0.0399% Amazon-16 117,672 1,036 - 9,038 - 27,402 17,209 - 64,624 - 501,153 41,427 - 379,657 - 5,481,801 0.0928% Amazon-Dual 2,500 2,500 - 2,500 - 2,500 17,889 - 28,649 - 39,510 106,741 - 128,601 - 150,461 0.1955% domains into three subsets containing 4, 8, and 16 domains respectively and denote them as Amazon4, Amazon-8, and Amazon-16 respectively. We randomly selected 2500 overlapping users in the Books domain and CDs domain to construct the dataset Amazon-Dual to validate the performance of our FedGCDR in the conventional dual-domain scenarios where users full-overlap.The statistics of sub-datasets are shown in Table 1. We filter the original data in different ways, and more details are given in Appendix C.1. In our experiments, Books and CDs are selected as the target domains. For the ratings in each domain, we first convert them to implicit data, where entries corresponding to existing user-item interactions are marked as 1 and others are marked as 0. Baselines We compare FedGCDR with the following state-of-the-art models: (1) FedGNN [34] is an attempt to adopt FL graph learning to recommender systems. Its recommendation performance could represent the data quality of the target domain and reflect negative transfer. In Tables 2 and 3, in order to distinguish FedGNN from the CDR baselines, we denote it by Single Domain. (2) EMCDR [15] is a conventional embedding-mapping CDR framework. We adjust it to the HFL framework following [6]. (3) PriCDR [19] is a privacy-preserving CDR framework, which adopted DP on the rating matrices to ensure privacy. (4) FedCT [6] is a VAE-based federated framework that is the first attempt to protect intra-domain privacy in cross-domain recommendations. (5) FedCDR [7] is a dual-target federated CDR framework, where the user embeddings are transferred as knowledge to enhance the other domain’s model training. To adapt to the BS-CDR scenarios, we modify FedCDR by applying embedding averaging when receiving source domain embeddings. We provide implement details in Appendix C.2. 4.2 Recommendation performance We report the model performance results in Tables 2 and 3. Single domain shows that the Book domain has better single-domain recommendation accuracy than the Music domain, which represents higher data quality and quantity. Under BS-CDR settings, FedGCDR outperforms all CDR baselines on all three sub-datasets, which confirms the effectiveness of the proposed model on real-world data. To further study our model capacity in alleviating negative transfer, we first define two types of negative transfer: (1) Soft Negative Transfer (SNT), where recommender models’ performance under the multi-domain setting is worse than that under the single-domain setting. This means that the knowledge from source domains poisoning the target domain’s model training. (2) Hard Negative Transfer (HNT), where recommended performance of a large number of source domains is lower than that of a small number of source domains. This means that the newly added domains are not conducive to the training of the target domain or conflict with the already added source domain. Taking the Books domain as the target domain, EMCDR, PriCDR, FedCT and FedCDR both have serious negative transfer problems and lower performance on the three data subsets. From the SNT perspective, their performances is much worse than that of Single Domain as shown in Figure 4. From the HNT perspective, their performances under 16-domain settings is worse than that under the 8-domain and 4-domain settings, which suggests it is not appropriate to recklessly transfer knowledge to a well-informed domain. Our FedGCDR model successfully alleviates NT with consistently best and stable performance results. For the CDs domain, the performance of the CDR models greatly improves with less NT in Figure 4. From the SNT perspective, information-poor domains have a lower probability of negative transfer, as they are inherently less well-trained. From the HNT perspective, on the Amazon-8 dataset, the performance of all models declines, which we attribute to the strong negative knowledge introduced by the four additional domains compared to the Amazon-4 dataset. On the Amazon-16 dataset, all methods achieve best performance which indicates 8 Table 2: The recommendation performance on Amazon@Books. Single Domain represents FedGNN and its performance is exactly the same on three sub-datasets. FedGCDR-DP is a complete implementation of our method while FedGCDR does not incorporate Gaussian noise. (The best result for the same setting is marked in bold and the second best is underlined. These notes are the same to others.) Model Amazon-4@Books Amazon-8@Books Amazon-16@Books HR@5 NDCG@5 HR@10 NDCG@10 HR@5 NDCG@5 HR@10 NDCG@10 HR@5 NDCG@5 HR@10 NDCG@10 Single Domain 0.4693 0.3188 0.6067 0.3634 0.4693 0.3188 0.6067 0.3634 0.4693 0.3188 0.6067 0.3634 EMCDR 0.4633 0.3075 0.6179 0.3191 0.4678 0.3268 0.5990 0.3518 0.3140 0.2184 0.4207 0.2348 PriCDR 0.4061 0.3159 0.5275 0.3550 0.4409 0.3196 0.5913 0.3681 0.3699 0.2650 0.4914 0.3042 FedCT 0.2911 0.2044 0.4276 0.2482 0.4665 0.3516 0.6002 0.3939 0.2779 0.2335 0.3580 0.2593 FedCDR 0.4115 0.3153 0.5415 0.3570 0.4791 0.3538 0.6182 0.3967 0.3926 0.2907 0.5626 0.3403 FedGCDR-DP 0.4903 0.3417 0.6717 0.3733 0.5224 0.3608 0.6727 0.3973 0.4928 0.3509 0.6510 0.3742 FedGCDR 0.4941 0.3592 0.6732 0.3920 0.5300 0.3686 0.6752 0.3985 0.5016 0.3600 0.6516 0.3854 Table 3: The recommendation performance on Amazon@CDs. Model Amazon-4@CDs Amazon-8@CDs Amazon-16@CDs HR@5 NDCG@5 HR@10 NDCG@10 HR@5 NDCG@5 HR@10 NDCG@10 HR@5 NDCG@5 HR@10 NDCG@10 Single Domain 0.4119 0.2751 0.5031 0.3040 0.4119 0.2751 0.5031 0.3040 0.4119 0.2751 0.5031 0.3040 EMCDR 0.4074 0.2651 0.5591 0.2972 0.2882 0.1828 0.4361 0.2199 0.4704 0.3683 0.5740 0.3937 PriCDR 0.3987 0.2838 0.5114 0.3202 0.2946 0.1988 0.4229 0.2400 0.4405 0.3689 0.5399 0.4011 FedCT 0.2681 0.1603 0.3774 0.1956 0.1801 0.1282 0.3001 0.1681 0.3522 0.2963 0.4326 0.3219 FedCDR 0.4299 0.2949 0.5636 0.3381 0.3088 0.2109 0.4620 0.2600 0.4823 0.3983 0.5808 0.4297 FedGCDR-DP 0.4359 0.2960 0.5779 0.3520 0.4122 0.2983 0.5064 0.3106 0.4963 0.4061 0.6135 0.4453 FedGCDR 0.4588 0.3282 0.5819 0.3679 0.4276 0.3142 0.5270 0.3464 0.5267 0.4382 0.6208 0.4684 Figure 4: Illustrations of negative transfer on HR@5 and NDCG@5. Metric values lower than single-domain (dotted line and red area) mean severe negative soft negative transfer. The figure on HR@10 and NDCG@10 is shown in Appendix D.1. Figure 5: Ablation study on Amazon-16@CDs and Amazon-16@Books. more knowledge from the source domains can improve the model performance in the CDs domain. Overall, the capability of EMCDR, PriCDR and FedCDR to alleviate negative transfer is much higher than that of FedCT. This is because the proportion of target domain features in the final feature is guaranteed by tuning Writefull to control the transfer ratio of the source domain. Meanwhile, FedGCDR avoids this kind of human involvement and maintains performance optimality on three sub-datasets. In conclusion, our experiments show the superiority of FedCDR in recommendation performance and the effectiveness of alleviating NT. 4.3 Ablation study To study the contribution of each module of FedGCDR, we implement two model variants, FedGCDR-M and FedGCDR-T. FedGCDR-T transfers the source domain embeddings with9 Table 4: Dual-domain CDR performance. Model Books →CDs Books ←CDs HR@10 NDCG@10 HR@10 NDCG@10 Single Domain 0.2713 0.1429 0.2594 0.1524 EMCDR 0.2816 0.1409 0.2596 0.1540 PriCDR 0.2903 0.1446 0.2662 0.1583 FedCT 0.2384 0.1239 0.2570 0.1551 FedCDR 0.2566 0.1376 0.2657 0.1554 FedGCDR-DP 0.3076 0.1552 0.2749 0.1602 FedGCDR 0.3323 0.1838 0.2958 0.1797 out mapping. FedGCDR-M replaces the attention graph expansion with the average sum of source domain embeddings and omits the fine-tuning stage. We experiment with Books and CDs as target domains on the Amazon-16 dataset. The experimental results are shown in Figure 5. We make the following observations: (1) The two variants perform differently on different target domains. On the Books domain, FedGCDR-T performs better than FedGCDR-M, which indicates that for domains with higher data quality, preventing the transfer of negative knowledge from other domains is more important than mapping this knowledge better (in other words, the quality of external information holds greater significance than its quantity), and the Positive Knowledge Activation module meets the requirements of such domains. On the CDs domain, FedGCDR-M performs better than FedGCDRT, which indicates that for domains that are deficient in information, mapping knowledge correctly is more important than preventing inter-domain negative knowledge (in other words, the quantity of external information holds greater significance than its quality), and the Positive Knowledge Transfer module meets these requirements. (2) Compared to FedGCDR, the absence of either module can cause a significant drop in performance. This indicates that in cross-domain recommendation, we should not only focus on transferring positive knowledge, but also control the spread of negative knowledge to the target domain, especially when a large number of domains. 4.4 Dual-domain scenario According to the experimental results shown in Table 4, our FedGCDR achieved the best experimental metrics in both knowledge transfer directions. This shows that our approach is also suitable for dual-domain scenarios where users full-overlap and have only a single source domain and a single target domain. We provide experimental results on privacy budget in Appendix D.2. 5 Limitations Our experiments were conducted on 16 domains of the Amazon dataset. While this extensive dataset covers broader source domains, relying on a single dataset may limit the generalizability of our model to data from other sources. Our approach uses overlapping users as a cross-domain bridge. Indeed, there are no widely-recognized cross-domain recommendation datasets with more than three domains, aside from the Amazon dataset. Despite this limitation, we believe that the improvements in privacy preservation and model performance demonstrated by FedGCDR underscore its superiority. 6 Conclusion We proposed FedGCDR, a federated graph learning framework designed for BS-CDR. FedGCDR addresses the critical challenge of privacy preservation and negative transfer by employing a positive knowledge transfer module and a positive knowledge activation module. Our method achieves best recommendation quality results on 16 domains of the Amazon dataset. In the future, we aim to extend FedGCDR to improve the recommendation performance of both the target and the source domains. 7 Acknowledgment The research was supported by Natural Science Foundation of China (62272403). 10 References [1] Weiming Liu, Jiajie Su, Chaochao Chen, and Xiaolin Zheng. 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Feature inference attack on model predictions in vertical federated learning. In 2021 IEEE 37th International Conference on Data Engineering, pages 181–192. IEEE, 2021. [55] Kalervo Järvelin and Jaana Kekäläinen. Cumulated gain-based evaluation of ir techniques. ACM Transactions on Information Systems, 20(4):422–446, 2002. 14 A Privacy analysis Due to the algorithmic nature of GNN, the source domain embeddings we pass are a function result on the user embeddings and item embeddings. This means that in the event of a successful inference attack, our user item interaction matrix is exposed to the threat of privacy disclosure. We apply differential privacy (DP) to further safeguard embeddings, following the approach of Cai et al. [53]. Threat model In this paper, we assume the threat model to be semi-honest (honest-but-curious). Under this threat model, the participants adhere strictly to the FL protocol for collaborative model training. However, they are interested in the sensitive rating data and may attempt to extract as much information as possible from the transferred embeddings. Specifically, these semi-honest parties, i.e. the target domain, may employ inference attacks [54] on the embeddings to reconstruct or infer sensitive user-item interaction matrix of other domains. DEFINITION 1 (THE GAUSSIAN MECHANISM). Given a function f : D →Rd over a dataset D, the Gaussian mechanism is defined as: FG(x, f(·), ϵ) = f(x) + (r1, ...rk), (13) where ri is the random noise drawn from N ∼(0, σ2∆2f 2) and σ = √ 2ln(1.25/δ) ϵ . In FedGCDR, the intra-domain GAT-based federated model is considered as the function f(·). THEOREM 2. The Gaussian mechanism defined in Definition 1 preserves (ϵ, δ)-DP for each publication step [22]. First, we give the definition of the inverse function: DEFINITION 2 (INVERSE FUNCTION). Given a function f : D →Rd over a dataset D, the inverse function f −1 is defined as: f −1 = argming X i∈u S v ∥ei −g(f(eu, ev)) ∥2, (14) where eu, ev = Embedding(x), x ∈D. (15) For the target domain, the embeddings received form source domains can be regarded as the functional result of their models. Let the function be f(·, ·) and the input eu, ev is the user embedding and item embedding respectively. The embeddings is the output f(eu, ev). The target domain attempts to find a inference attack function I(·) which is as close to the inverse function as possible. DEFINITION 3 (PRIVACY LEAKAGE). Given a function f : E →Rd over a Embedding set E and an inference function I, the privacy leakage Λ is defined as: Λ = 1 1 + 1 \|U\| P Leaku + 1 \|V \| P Leakv . (16) where Leaku =∥ei −Iu(f(eu, ev)) ∥2, i ∈U, (17) Leakv =∥ej −Iv(f(eu, ev)) ∥2, j ∈V . (18) ∥e −I(f(eu, ev)) ∥2 reflects the closeness of the reconstructed input to the true input. Therefore, privacy leakage (PLeak) Λ is able to reflect privacy leakage of FedGCDR with the inference function I(·). PLeak equal to 1 means a perfect reconstruction, and being close to zero means a bad reconstruction. DP on the embeddings further ensures that attackers cannot perfectly reconstruct the raw data. THEOREM 2. If PLeak equals to 1 with the inference function I(·), the function f(·, ·) is bijection. Proof. If the function f(·, ·) is not bijection, there are i, j ∈D and i ̸= j, but f(ei u, ei v) = f(ej u, ej v) and I(f(ei u, ei v)) and I(f(ej u, ej v)). This is a contradiction as the perfect reconstruction requires both ei u, ei v = I(f(ei u, ei v)) and ej u, ej v = I(f(ej u, ej v)) to achieve Λ = 1. Therefore, the function f(·, ·) must be bijection. 15 THEOREM 3. Given the lipschiz constant L of the function f at x ∈D with the noise generated by Gaussian mechanism N on embeddings. If f(eu, ev) + N ∈f , the distance between x to the reconstructed data of the attack I(·) which achieves Λ = 1 is bounded by \|N \| L . Proof. By Theorem 2, we have for x ∈D and v ∈f, I(f(eu, ev)) = (eu, ev) and f(I(v)) = v, From the Lipschitz continuous, \|e −I(f(eu, ev) + N)\| ≥\|f(eu, ev) −(f(eu, ev) + N)\| L = \|N\| L . (19) Therefore, by perturbing the source domain embedding with Gaussian mechanism, the reconstructed data of the ideal attack deviates from the real data and prevents a perfect reconstruction (i.e., Λ = 1). B Cost analysis Due to the complexity of the FedGCDR pipeline, we perform a theoretical analysis of the computational and communication cost of FedGCDR accordance with the HVH pipeline including horizontal source domain training, vertical positive knowledge transfer module and horizontal positive knowledge activation module. B.1 Computational cost Given a GAT model, let V be the number of nodes, E be the number of edges, and F the embedding size. The computational cost of one propagation layer of classic GAT framework is O(V FF ′ + EF ′) [38]. In the horizontal source domain training, our model is a simplified GAT variants which discard feature transformation and non-linear activation. For a N K layer model, the simplified computational cost is O(N KEF). In the vertical positive knowledge transfer module, space mapping is carried by a N m layers’ MLP with computational cost O(N mF 2). In the horizontal positive knowledge activation module, the first part is the simplified GAT model and the second part is the fine-tuning model with computational cost O(F 2). In conclusion, for the FedCDR framework with N D domains, T G GAT-based federated model training epochs, and T F fine-tuning epochs, the total computational cost is O(T G(N DN KEF + N mF 2) + T F F 2)). Cause N mF 2 ≪N DN KEF, we get the final computational cost O(T GN DN KEF + T F F 2)). B.2 Communication cost In FedGCDR, the global model and item embeddings are held by the domain server. Let I be the number of items and F be the embedding size. The space complexity of global model and item embeddings are O(F) and O(IF) respectively. In the horizontal source domain training, the domain server distributes the global model and item embeddings and get the gradient with the same size. The communication cost is O(F + IF). In the vertical positive knowledge transfer module, the N m layers’ MLP and its gradients are transmitted with the communication cost O(N mF 2). In the horizontal positive knowledge activation module, the target domain additionally perform a fine-tuning stage with communication cost O(IF). In conclusion, for the FedCDR framework with N u users, T G federated model training epochs, and T F fine-tuning epochs, the total communication cost is O(T GN u(N mF 2+F +IF)+T F N uIF). Cause N mF 2+F ≪IF, we get O(N uIF(T G+T F )). According to the expression, the communication cost of our FedGCDR is basically equivalent to the cost of two HFL progress. The cost is reduced because knowledge transfer totally takes place in user space, thus avoiding large-scale information exchange. C Experimental details C.1 Dataset details The Amazon dataset we used is the 2018 version and can be easily accessed in https://cseweb. ucsd.edu/~jmcauley/datasets/amazon_v2/. The basis of our domain selection strategy is the amount of data before performing data filtering. Thus, we sorted the domains contained in the Amazon dataset based on the amount of data in descending order and selected the top 16 domains. Similarly, the Amazon-4 and Amazon-8 datasets were selected accordingly. The only exception is 16 that we prioritized the Movie domain, which has a relatively small amount of source data, based on popularity. In addition to multi-domain experiments, we randomly selected 2500 overlapping users in the Books domain and CDs domain to construct the dataset Amazon-Dual, so as to validate the performance of our FedGCDR in the conventional dual-domain scenarios where users full-overlap. The processing details are shown in Table 5. The bottleneck time of FedGCDR is the federated-GAT training time in each domain, and we also show it in Table 5. Table 5: Processing details on Amazon Dataset. Domain \|U\| \|I\| user core item core time per epoch (mm:ss) Clothing, Shoes and Jewelry 11,558 197,677 24 10 03:31 Amazon-4 Books 27,402 501,153 96 10 11:02 CDs and Vinyl 13,694 71,199 24 10 04:02 Movies and TV 6,632 53,082 48 10 01:55 Home and Kitchen 15,772 135,182 48 10 04:36 Amazon-8 Electronics 16,836 120,876 32 10 04:40 Sports and Outdoors 14,262 93,095 32 10 03:41 Cell Phones and Accessories 9,312 55,312 24 10 02:40 Tools and Home Improvement 9,899 65,378 16 10 02:47 Amazon-16 Toys and Games 5,267 63,870 32 10 01:10 Automotive 6,135 62,188 32 10 01:32 Pet Supplies 4,280 31,853 32 10 00:55 Kindle Store 8,756 82,874 48 10 02:06 Office Products 1,266 17,209 32 10 00:16 Patio, Lawn and Garden 1,036 17,605 32 10 00:16 Grocery and Gourmet Food 3,415 36,292 32 10 00:50 C.2 Implement details We provide the implemented details of our proposed model and baselines. We set batch size = 256 and latent dim = 8 for all domains. The number of propagation layer of GAT-base federated model is set to 2. The MLP has two hidden layers with size={16, 4}.Considering the trade-off between recommendation performance and privacy preservation, we set ϵ to 8 and σ to 10−5. We set α=0.01 and β=0.01 which are the two Writefull of the objective function LGAT (·). When training our models, we choose Adam as the optimizer, and set the learning rate to 0.01 both in GAT-based federated model training and the fine-tuning stage. To evaluate the recommendation performance, we use the leave-one-out method which is widely used in recommender systems [51]. Specifically, we held out the latest interaction as the test set and utilized the remaining data for training. Then, we follow the common strategy which randomly samples 99 negative items that are not interacted with by the user for the rank list generation of the test set. We consider the top-k recommendation task as the main experiment so we choose metrics including Hit Ratio (HR)@K score and the Normalized Discounted Cumulative Gain (NDCG)@K [55] of the top-K ranked items with K=5, 10. We conduct the experiments on three groups of random seeds and report the average results. We conduct all the experiments on NVIDIA 3090 GPUs. D Additional experimental results D.1 Neagtive transfer on HR@10 and NDCG@10. Figure 6: Illustrations of negative transfer on HR@10 and NDCG@10 For HR@10 and NDCG@10 in Figure 6, our method and baselines show similar trends to the previous HR@5 and NDCG@5. Compared to Figure 6, the slight difference is that FedCT’s HR@10 17 performance is better on Amazon-8@CDs than on Amazon-4@CDs. We believe that the reason is the poor performance of FedCT on Amzon-4@CDs lowers the threshold for negative transfer of the newly added source domain. D.2 Privacy budget Figure 7: The effect of ϵ in DP on model performance. To study the effects of privacy budget ϵ on the model performance, we vary the privacy budget ϵ = {4, 8, 16, 32, 64} to affect the σ. We experimented on the Amazon-16 with CDs as the target domain and fix δ = 10−5. We report the results Figure 7. From that we can observe that the model’s performance decreases as ϵ decreases. The degradation in model performance suggests that our approach struggles to counteract the effects of high-intensity noise in a large number of domains, but the model performance is not completely destroyed by Gaussian noise. Thus, there is a trade-off between accuracy and privacy, where a smaller ϵ value adds more noise to embeddings for stronger privacy preservation but leads to more prediction error. Therefore, to balance the data privacy preservation capacity and the model performance, we set it as ϵ = 8. E Broader impacts Our proposed FedGCDR is tailored for BS-CDR, focusing on the privacy and negative transfer problems. CDR is widely uesd, while BS-CDR is generic and close to the reality. Our approach can better mine user preferences and effectively protect privacy. On the one hand, users will benefit from more accurate recommendations and thus have a better experience in shopping, watching movies, etc. On the other hand, various economic entities can gain more profits. 18 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We have claimed the contributions and scope in lines 5-7 and 69-75. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,442	On the Saturation Effects of Spectral Algorithms in Large Dimensions Weihao Lu Department of Statistics and Data Science Tsinghua University Beijing, China 100084 luwh19@mails.tsinghua.edu.cn Haobo Zhang Department of Statistics and Data Science Tsinghua University Beijing, China 100084 zhang-hb21@mails.tsinghua.edu.cn Yicheng Li Department of Statistics and Data Science Tsinghua University Beijing, China 100084 liyc22@mails.tsinghua.edu.cn Qian Lin∗ Department of Statistics and Data Science Tsinghua University Beijing, China 100084 qianlin@tsinghua.edu.cn Abstract The saturation effects, which originally refer to the fact that kernel ridge regression (KRR) fails to achieve the information-theoretical lower bound when the regression function is over-smooth, have been observed for almost 20 years and were rigorously proved recently for kernel ridge regression and some other spectral algorithms over a fixed dimensional domain. The main focus of this paper is to explore the saturation effects for a large class of spectral algorithms (including the KRR, gradient descent, etc.) in large dimensional settings where n ≍dγ. More precisely, we first propose an improved minimax lower bound for the kernel regression problem in large dimensional settings and show that the gradient flow with early stopping strategy will result in an estimator achieving this lower bound (up to a logarithmic factor). Similar to the results in KRR, we can further determine the exact convergence rates (both upper and lower bounds) of a large class of (optimal tuned) spectral algorithms with different qualification τ’s. In particular, we find that these exact rate curves (varying along γ) exhibit the periodic plateau behavior and the polynomial approximation barrier. Consequently, we can fully depict the saturation effects of the spectral algorithms and reveal a new phenomenon in large dimensional settings (i.e., the saturation effect occurs in large dimensional setting as long as the source condition s > τ while it occurs in fixed dimensional setting as long as s > 2τ). 1 Introduction Let’s assume we have n i.i.d. samples (xi, yi) from a joint distribution supported on Rd × R. The regression problem, one of the most fundamental problems in statistics, aims to find a function ˆf based on these samples such that the excess risk, ∥ˆf −f⋆∥2 L2 = Ex[(f⋆(x) −ˆf(x))2], is small, where f⋆(x) = E[Y \|x] is the regression function. Many non-parametric regression methods are proposed to solve the regression problem by assuming that f⋆falls into certain function classes, including polynomial splines Stone (1994), local polynomials Cleveland (1979); Stone (1977), the spectral algorithms Caponnetto (2006); Caponnetto and De Vito (2007); Caponnetto and Yao (2010), etc. ∗Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Spectral algorithms, as a classical topic, have been studied since the 1990s. Early works treated certain types of spectral algorithms in their theoretical analysis (Caponnetto (2006); Caponnetto and De Vito (2007); Raskutti et al. (2014); Lin et al. (2020)). These works often consider d as a fixed constant and impose the polynomial eigenvalue decay assumption under a kernel (i.e., there exist constants 0 < c ≤C < ∞, such that the eigenvalues of the kernel satisfy cj−β ≤λj ≤Cj−β, j ≥1 for certain β > 1 depending on the fixed d). They further assume that f⋆belongs to the reproducing kernel Hilbert space (RKHS) H associated with the kernel. Under the above assumptions, they then showed that the minimax rate of the excess risk of regression over the corresponding RKHS is lower bounded by n−β/(β+1) and that some (regularized) spectral algorithms, e.g., the kernel ridge regression (KRR) and the kernel gradient flow, can produce estimators achieving this minimax optimal rate. However, subsequent studies have revealed that when higher regularity (or smoothness) of f⋆is assumed, KRR fails to achieve the information-theoretical lower bound on the excess risk, while kernel gradient flow can do so. Specifically, let’s assume that f⋆belongs to the interpolation space [H]s of the RKHS H with s > 0 (see, e.g., Steinwart et al. (2009); Dieuleveut et al. (2017); Dicker et al. (2017); Pillaud-Vivien et al. (2018); Lin et al. (2020); Fischer and Steinwart (2020); Celisse and Wahl (2021)). It is then shown that the information-theoretical lower bound on the excess risk is n−sβ/(sβ+1). When 0 < s ≤2, Caponnetto and De Vito (2007); Yao et al. (2007); Lin et al. (2020); Zhang et al. (2023) have already shown that the upper bound of the excess risks of both KRR and the kernel gradient flow is n−sβ/(sβ+1), and hence they are minimax optimal. On the contrary, when s > 2, Yao et al. (2007); Lin et al. (2020) showed that the upper bound of the excess risks of kernel gradient flow is n−sβ/(sβ+1) while the best upper bound of the excess risks of KRR is n−2β/(2β+1) (Caponnetto and De Vito (2007)). Bauer et al. (2007); Gerfo et al. (2008); Dicker et al. (2017) conjectured that the convergence rate of KRR is bounded below by n−2β/(2β+1) and Li et al. (2022) rigorously proved it. The above phenomenon is often referred to as the saturation effect of KRR: KRR is inferior to certain spectral algorithms, such as kernel gradient flow, when s > 2. In recent years, neural network methods have gained tremendous success in many large-dimensional problems, such as computer vision He et al. (2016); Krizhevsky et al. (2017) and natural language processing Devlin (2018). Several groups of researchers tried to explain the superior performance of neural networks on large-dimensional data from the aspects of "lazy regime" (Arora et al. (2019); Du et al. (2019, 2018); Li and Liang (2018)). They noticed that, when the width of a neural network is sufficiently large, its parameters/weights stay in a small neighborhood of their initial position during the training process. Later, Jacot et al. (2018); Arora et al. (2019); Hu et al. (2021); Suh et al. (2021); Lai et al. (2023); Li et al. (2024) proved that the time-varying neural network kernel (NNK) converges (uniformly) to a time-invariant neural tangent kernel (NTK) as the width of the neural network goes to infinity, and thus the excess risk of kernel gradient flow with NTK converges (uniformly) to the excess risk of neural networks in the ‘lazy regime’. Inspired by the concepts of the "lazy regime" and the uniform convergence of excess risk, the machine learning community has experienced a renewed surge of interest in large-dimensional spectral algorithms. The earliest works focused on the consistency of two specific types of spectral algorithms: KRR and kernel interpolation (Liang and Rakhlin (2020); Liang et al. (2020); Ghorbani et al. (2020, 2021); Mei et al. (2021, 2022); Misiakiewicz and Mei (2022); Aerni et al. (2023); Barzilai and Shamir (2023)). In comparison, results on large-dimensional kernel gradient flow were somewhat scarce, and these results largely mirrored those associated with KRR (e.g., Ghosh et al. (2021)). Recently, Lu et al. (2023) proved that large-dimensional kernel gradient flow is minimax optimal when s = 1. Then, Zhang et al. (2024) provided upper and lower bounds on the convergence rate on the excess risk of KRR for any s > 0. Surprisingly, they discovered that for s > 1, the convergence rate of KRR did not match the lower bound on the minimax rate. Unfortunately, they didn’t prove that certain spectral algorithms can reach the lower bound on the minimax rate they provided, and hence they didn’t rigorously prove that the saturation effect of KRR occurs in large dimensions. Instead, Zhang et al. (2024) only conjectured that certain spectral algorithms (e.g., kernel gradient flow) can provide minimax optimal estimators after their main results. If Zhang et al. (2024)’s conjecture is true, then we can safely conclude that: when the regression function f⋆is smooth enough, KRR is inferior to kernel gradient flow in large dimensions as well. Consequently, previous results on large-dimensional KRR may not be directly extendable to large2 dimensional neural networks, even if the neural networks are in the ‘lazy regime’. The main focus of this paper is to prove this conjecture by showing that kernel gradient flow is minimax optimal in large dimensions. 1.1 Related work Saturation effects of fixed-dimensional spectral algorithms. When the dimension d of the data is fixed, the saturation effect of KRR has been conjectured for decades and is rigorously proved in the recent work Li et al. (2022). Suppose f⋆∈[H]s with s > 2. It is shown that: (i) the minimax optimal rate is n−sβ/(sβ+1) (Rastogi and Sampath (2017); Yao et al. (2007); Lin et al. (2020)); and (ii) the convergence rate on the excess risk of KRR is n−2β/(2β+1) (Li et al. (2022)). More recently, Li et al. (2024) determined the exact generalization error curves of a class of analytic spectral algorithms, which allowed them to further show the saturation effect of spectral algorithms with finite qualification τ (see, e.g., Appendix C): suppose f⋆∈[H]s with s > 2τ, then the convergence rate on the excess risk of the above spectral algorithms is n−2τβ/(2τβ+1). New phenomena in large-dimensional spectral algorithms. In the large-dimensional setting where n ≍dγ with γ > 0, new phenomena exhibited in spectral algorithms are popular topics in recent machine-learning research. A line of work focused on the polynomial approximation barrier phenomenon (e.g., Ghorbani et al. (2021); Donhauser et al. (2021); Mei et al. (2022); Xiao et al. (2023); Misiakiewicz (2022); Hu and Lu (2022)). They found that, for the square-integrable regression function, KRR and kernel gradient flow are consistent if and only if the regression function is a polynomial with a low degree. Another line of work considered the benign overfitting of kernel interpolation (i.e., kernel interpolation can generalize) (e.g., Liang and Rakhlin (2020); Liang et al. (2020); Aerni et al. (2023); Barzilai and Shamir (2023); Zhang et al. (2024)). Moreover, two recent work (Lu et al. (2023); Zhang et al. (2024)) discussed two new phenomena exhibited in large-dimensional KRR and kernel gradient flow: the multiple descent behavior and the periodic plateau behavior. The multiple descent behavior refers to the phenomenon that the curve of the convergence rate ( with respect to n ) of the optimal excess risk is non-monotone and has several isolated peaks and valleys; while the periodic plateau behavior refers to the phenomenon that the curve of the convergence rate ( with respect to d ) of the optimal excess risk has constant values when γ is within certain intervals. Finally, Zhang et al. (2024) conjectured that the saturation effect of KRR occurs in large dimensions. The above works imply that these phenomena occur in many spectral algorithms in large dimensions, hence encouraging us to provide a unified explanation of these new phenomena. 1.2 Our contributions In this paper, we focus on the large-dimensional spectral algorithms with inner product kernels, and we assume that the regression function falls into an interpolation space [H]s with s > 0. We state our main results as follows: Theorem 1.1 (Restate Theorem 4.1 and 4.2, non-rigorous). Let s > 0, τ ≥1, and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Then under certain conditions, the excess risk of large-dimensional spectral algorithm with qualification τ satisfies E ˆfλ⋆−f⋆ 2 L2 X = ( ΘP d−min{γ−p,s(p+1)} · poly (ln(d)) , s ≤τ ΘP d−min{γ−p, τ(γ−p+1)+p˜s τ+1 ,˜s(p+1)} · poly (ln(d)) , s > τ, where ˜s = min{s, 2τ}. More specifically, we list the main contributions of this paper as follows: (1) In Theorem 3.1, we show that the convergence rate on the excess risk of (optimally-tuned) kernel gradient flow in large dimensions is ΘP(d−min{γ−p,s(p+1)}) · poly(ln(d)), which matches the lower bound on the minimax rate given in Theorem 3.3 (up to a logarithmic factor). We find that kernel gradient flow is minimax optimal for any s > 0 and any γ > 0, and KRR is not minimax optimal for s > 1 and for certain ranges of γ (We provide a visual illustration in Figure 2). Consequently, we rigorously prove that the saturation effect of KRR occurs in large dimensions. 3 (2) In Theorem 3.3, we enhanced the previous minimax lower bound results given in Lu et al. (2023) and Zhang et al. (2024). Specifically, we show that the minimax lower bound is Ω(d−min{γ−p,s(p+1)})/poly(ln(d)). In comparison, the previous minimax lower bound is Ω(d−min{γ−p,s(p+1)})/dε for any ε > 0, and the additional term dε changes the desired convergence rate. (3) In Section 4, we determine the convergence rate on the excess risk of large-dimensional spectral algorithms. From our results, we find several new phenomena exhibited in spectral algorithms in large-dimensional settings. We provide a visual illustration of the above phenomena in Figure 1: i) The first phenomenon is the polynomial approximation barrier, and as shown in Figure 1(a), when s is close to zero, the curve of the convergence rate of spectral algorithm drops when γ ≈p for any integer p and will stay invariant for most of the other γ; ii) The second one is the periodic plateau behavior, and as shown in Figure 1(b) and Figure 1(c), when 0 < s < 2τ and γ ∈[p(s + 1) + s + (max{s, τ} −τ)/τ, (p + 1)(s + 1)) for an integer p ≥0, the convergence rate does not change when γ varies; iii) The final one is the saturation effect, and as shown in Figure 1(c) and Figure 1(d), when s > τ, the convergence rate of spectral algorithm can not achieve the minimax lower bound for certain ranges of γ. A detailed discussion about the above three phenomena can be found in Section 4. (a) (b) (c) (d) Figure 1: Convergence rates of spectral algorithm with qualification τ = 2 in Theorem 4.1, Theorem 4.2, and corresponding minimax lower rates in Theorem 3.3 with respect to dimension d. We present four graphs corresponding to four kinds of source conditions: s = 0.01, 1, 3, 5. The x-axis represents asymptotic scaling, γ : n ≍dγ; the y-axis represents the convergence rate of excess risk, r : Excess risk ≍dr. 2 Preliminaries Suppose that we have observed n i.i.d. samples (xi, yi), i ∈[n] from the model: y = f⋆(x) + ϵ, (1) where xi’s are sampled from ρX , ρX is the marginal distribution on X ⊂Rd+1, y ∈Y ⊂R, f⋆is some function defined on a compact set X, and E(x,y)∼ρ h ϵ2 x i ≤σ2, ρX -a.e. x ∈X, for some fixed constant σ > 0, where ρ is the joint distribution of (x, y) on X × Y. Denote the n × 1 data vector of yi’s and the n × d data matrix of xi’s by Y and X respectively. 2.1 Kernel ridge regression and kernel gradient flow In this subsection, we introduce two specific spectral algorithms, kernel ridge regression and kernel gradient flow, which produce estimators of the regression function f⋆. A further discussion on general spectral algorithms will be provided in Section 4. Throughout the paper, we denote H as a separable RKHS on X with respect to a continuous and positive definite kernel function K(·, ·) : X × X →R and there exists a constant κ satisfying max x∈X K(x, x) ≤κ2. 4 Kernel ridge regression Kernel ridge regression (KRR) constructs an estimator ˆf KRR λ by solving the penalized least square problem ˆf KRR λ = arg min f∈H 1 n n X i=1 (yi −f (xi))2 + λ∥f∥2 H ! , where λ > 0 is referred to as the regularization parameter. The representer theorem (see, e.g., Steinwart and Christmann (2008)) gives an explicit expression of the KRR estimator, i.e., ˆf KRR λ (x) = K(x, X)(K(X, X) + nλI)−1Y. (2) Kernel gradient flow The gradient flow of the loss function L = 1 2n P i(yi −f(xi))2 induced a gradient flow in H which is given by d dt ˆf GF t (x) = −1 nK(x, X)( ˆf GF t (X) −Y ). (3) If we further assume that ˆf GF 0 (x) = 0, then we can also give an explicit expression of the kernel gradient flow estimator ˆf GF t (x) = K(x, X)K(X, X)−1(I −e−1 n K(X,X)t)Y. (4) 2.2 The interpolation space Define the integral operator TK as TK(f)(x) = R K(x, x′)f(x′) dρX (x′). It is well known that TK is a positive, self-adjoint, trace-class, and hence a compact operator (Steinwart and Scovel (2012)). The celebrated Mercer’s theorem further assures that K(x, x′) = X j λjϕj(x)ϕj(x′), (5) where the eigenvalues {λj, j = 1, 2, ...} is a non-increasing sequence, and the corresponding eigenfunctions {ϕj(·), j = 1, 2, ...} are orthonormal in L2(X, ρX ) function space. The interpolation space [H]s with source condition s is defined as [H]s := n X j ajλs/2 j ϕj : (aj)j ∈ℓ2 o ⊆L2(X, ρX ), (6) with the inner product deduced from ∞ X j=1 ajλs/2 j ϕj [H]s = ∞ X j=1 a2 j 1/2 . (7) It is easy to show that [H]s is also a separable Hilbert space with orthonormal basis {λs/2 j ϕj}j. Generally speaking, functions in [H]s become smoother as s increases (see, e.g., the example of Sobolev RKHS in Edmunds and Triebel (1996); Zhang et al. (2023). 2.3 Assumptions In this subsection, we list the assumptions that we need for our main results. To avoid potential confusion, we specify the following large-dimensional scenario for kernel regression where we perform our analysis: suppose that there exist three positive constants c1, c2 and γ, such that c1dγ ≤n ≤c2dγ, (8) and we often assume that d is sufficiently large. In this paper, we only consider the inner product kernels defined on the sphere. An inner product kernel is a kernel function K defined on Sd such that there exists a function Φ : [−1, 1] →R independent of d satisfying that for any x, x′ ∈Sd, we have K(x, x′) = Φ(⟨x, x′⟩). If we further 5 assume that the marginal distribution ρX is the uniform distribution on X = Sd, then the Mercer’s decomposition for K can be rewritten as K(x, x′) = ∞ X k=0 µk N(d,k) X j=1 Yk,j(x)Yk,j (x′) , (9) where Yk,j for j = 1, · · · , N(d, k) are spherical harmonic polynomials of degree k and µk’s are the eigenvalues of K with multiplicity N(d, 0) = 1; N(d, k) = 2k+d−1 k · (k+d−2)! (d−1)!(k−1)!, k = 1, 2, · · · . For more details of the inner product kernels, readers can refer to Gallier (2009). Remark 2.1. We consider the inner product kernels on the sphere mainly because the harmonic analysis is clear on the sphere ( e.g., properties of spherical harmonic polynomials are more concise than the orthogonal series on general domains). This makes Mercer’s decomposition of the inner product more explicit rather than several abstract assumptions ( e.g., Mei and Montanari (2022)). We also notice that very few results are available for Mercer’s decomposition of a kernel defined on the general domain, especially when the dimension of the domain is taking into consideration. e.g., even the eigen-decay rate of the neural tangent kernels is only determined for the spheres. Restricted by this technical reason, most works analyzing the spectral algorithm in large-dimensional settings focus on the inner product kernels on spheres (Liang et al., 2020; Ghorbani et al., 2021; Misiakiewicz, 2022; Xiao et al., 2023; Lu et al., 2023, etc.). Though there might be several works that tried to relax the spherical assumption (e.g., Liang et al. (2020); Aerni et al. (2023); Barzilai and Shamir (2023), we can find that most of them (i) adopted a near-spherical assumption; (ii) adopted strong assumptions on the regression function, e.g., f⋆(x) = x[1]x[2] · · · x[L] for an integer L > 0, where x[i] denotes the i-th component of x; or (iii) can not determine the convergence rate on the excess risk of the spectral algorithm. To avoid unnecessary notation, let us make the following assumption on the inner product kernel K. Assumption 1. Φ(t) ∈C∞([−1, 1]) is a fixed function independent of d and there exists a nonnegative sequence of absolute constants {aj ≥0}j≥0, such that we have Φ(t) = X∞ j=0 ajtj, where aj > 0 for any j ≤⌊γ⌋+ 3. The purpose of Assumption 1 is to keep the main results and proofs clean. Notice that, by Theorem 1.b in Gneiting (2013), the inner product kernel K on the sphere is semi-positive definite for all dimensions if and only if all coefficients {aj, j = 0, 1, 2, ...} are non-negative. One can easily extend our results in this paper when certain coefficients ak’s are zero (e.g., one can consider the two-layer NTK defined as in Section 5 of Lu et al. (2023), with ai = 0 for any i = 3, 5, 7, · · · ). In the next assumption, we formally introduce the source condition, which characterizes the relative smoothness of f⋆with respect to H. Assumption 2 (Source condition). Suppose that f⋆(x) = P∞ i=1 fiϕi(x). (a) f⋆∈[H]s for some s > 0, and there exists a constant Rγ only depending on γ, such that ∥f⋆∥[H]s ≤Rγ. (10) (b) Denote q as the smallest integer such that q > γ and µq ̸= 0. Define Id,k as the index set satisfying λi ≡µk, i ∈Id,k. Further suppose that there exists an absolute constant c0 > 0 such that for any d and k ∈{0, 1, · · · , q} with µk ̸= 0, we have X i∈Id,k µ−s k f 2 i ≥c0. (11) Assumption 2 is a common assumption when one is interested in the tight bounds on the excess risk of spectral algorithms (e.g., Caponnetto and De Vito (2007); Fischer and Steinwart (2020), Eq.(8) in Cui et al. (2021), Assumption 3 in Li et al. (2024), and Assumption 5 in Zhang et al. (2024)). Assumption 2 implies that the regression function exactly falls into the interpolation space [H]s, that is, f⋆∈[H]s and f⋆/∈[H]t for any t > s. For example, from the proof part I of Lemma D.14, one can check that f⋆with P i∈Id,p µ−s p f 2 i = P i∈Id,p+1 µ−s p+1f 2 i = 0 can have a faster convergence rate on the excess risk. 6 Notations. Let’s denote the norm in L2(X, ρX ) as ∥· ∥L2. For a vector x, we use x[i] to denote its i-th component. We use asymptotic notations O(·), o(·), Ω(·) and Θ(·). For instance, we say two (deterministic) quantities U(d), V (d) satisfy U(d) = o(V (d)) if and only if for any ε > 0, there exists a constant Dε that only depends on ε and the absolute positive constants σ, κ, s, γ, c0, c1, c2, C1, · · · , C8 > 0, such that for any d > Dε, we have U(d) < εV (d). We also write an = poly(bn) if there exist a constant θ ≥0, such that an = Θ(bθ n). We use the probability versions of the asymptotic notations such as OP(·), oP(·), ΩP(·), ΘP(·). For instance, we say the random variables Xn, Yn satisfying Xn = OP(Yn) if and only if for any ε > 0, there exist constants Cε and Nε such that P (\|Xn\| ≥Cε\|Yn\|) ≤ε, ∀n > Nε. 2.4 Review of the previous results The following two results are restatements of Theorem 2 and Theorem 5 in Zhang et al. (2024). Proposition 2.2. Let s ≥1 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose that Assumption 1 and Assumption 2 hold for s and γ. Let ˆf KRR λ be the function defined in (2). Define ˜s = min{s, 2}, then there exists λ⋆> 0, such that we have E ˆf KRR λ⋆−f⋆ 2 L2 X = ΘP d−min{γ−p, γ−p+p˜s+1 2 ,˜s(p+1)} · poly (ln(d)) , where ΘP only involves constants depending on s, σ, γ, c0, κ, c1 and c2. In addition, the convergence rates of the generalization error can not be faster than above for any choice of regularization parameter λ = λ(d, n) →0. Proposition 2.3 (Lower bound on the minimax rate). Let s > 0 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s+1), (p+1)(s+1)). Let P consist of all the distributions ρ on X × Y such that Assumption 1 and Assumption 2 hold for s and γ. Then for any ε > 0, we have: min ˆ f max ρ∈P E(X,Y )∼ρ⊗n ˆf −f⋆ 2 L2 = Ω d−min{γ−p,s(p+1)} · d−ε , where Ωonly involves constants depending on s, σ, γ, c0, κ, c1, c2 and ε. From the above two propositions, we can find that when s > 1, the convergence rate on the excess risk of KRR does not always match the lower bound on the minimax optimal rate. Zhang et al. (2024) further conjectured that the lower bound on the minimax optimal rate provided in Proposition 2.3 is tight (ignoring the additional term d−ε). Hence, they believed that the saturation effect exists for large-dimensional KRR. 3 Main results In this section, we determine the convergence rate on the excess risk of kernel gradient flow as d−min{γ−p,s(p+1)}poly (ln(d)), which differs from the lower bound on the minimax rate provided in Proposition 2.3 by dε for any ε > 0. We then tighten the lower bound on the minimax rate to d−min{γ−p,s(p+1)}/poly (ln(d)). Based on the above results, we find that KRR is not minimax optimal for s > 1 and for certain ranges of γ. Therefore, we show that the saturation effect of KRR occurs in large dimensions. 3.1 Exact convergence rate on the excess risk of kernel gradient flow We first state our main results in this paper. Theorem 3.1 (Kernel gradient flow). Let s > 0 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose that Assumption 1 and Assumption 2 hold for s and γ. Let ˆf GF t be the function defined in (4). Then there exists t⋆> 0, such that we have E ˆf GF t⋆−f⋆ 2 L2 X = ΘP d−min{γ−p,s(p+1)} · poly (ln(d)) , (12) where ΘP only involves constants depending on s, σ, γ, c0, κ, c1 and c2. 7 Theorem 3.1 is a direct corollary of Theorem 4.1 and Example 2. Combining with the previous results in Proposition 2.3, or our modified minimax rate given in Theorem 3.3, we can conclude that large-dimensional kernel gradient flow is minimax optimal for any s > 0 and any γ > 0. More importantly, the convergence rate of kernel gradient flow is faster than that of KRR given in Proposition 2.2 when (i) 1 < s ≤2 and γ ∈(p(s + 1) + 1, p(s + 1) + 2s −1) for some p ∈N, or (ii) s > 2 and γ ∈(p(s + 1) + 1, (p + 1)(s + 1)) for some p ∈N. Therefore, we have proved the saturation effect of KRR in large dimensions. Remark 3.2. When p ≥1, the logarithm term poly(ln(d)) in (12) can be removed. When p = 0, we have poly(ln(d)) = (ln(d))2 in (12). See Appendix D.4 for details. 3.2 Improved minimax lower bound Recall that Proposition 2.3 gave a lower bound on the minimax rate as d−min{γ−p,s(p+1)} · d−ε. The following theorem replaces the additional term d−ε (which has changed the convergence rate) into a logarithm term poly−1 (ln(d)) (which does not change the desired convergence rate). Theorem 3.3 (Improved minimax lower bound). Let s > 0 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Let P consist of all the distributions ρ on X × Y such that Assumption 1 and Assumption 2 hold for s and γ. Then we have: min ˆ f max ρ∈P E(X,Y )∼ρ⊗n ˆf −f⋆ 2 L2 = Ω d−min{γ−p,s(p+1)}. poly (ln(d)) , (13) where Ωonly involves constants depending on s, σ, γ, c0, κ, c1, and c2. 4 Exact convergence rate on the excess risk of spectral algorithms In this section, we will give tight bounds on the excess risks of certain types of spectral algorithms, such as kernel ridge regression, iterated ridge regression, kernel gradient flow, and kernel gradient descent. Given an analytic filter function φλ(·) with qualification τ ≥1 (refer to Appendix C for the definitions of analytic filter function and its qualification), we can define a spectral algorithm in the following way (see, e.g., Bauer et al. (2007)). For any y ∈R, let Kx : R →H be given by Kx(y) = y ·K(x, ·), whose adjoint K∗ x : H →R is given by K∗ x(f) = ⟨K(x, ·), f⟩H = f(x). Moreover, we denote by Tx = KxK∗ x and TX = 1 n Pn i=1 Txi. We also define the sample basis function ˆgZ = 1 n Xn i=1 Kxi(yi) = 1 n Xn i=1 yi · K(xi, ·). (14) Now, the estimator of the spectral algorithm is defined by ˆfλ = φλ(TX)ˆgZ. (15) Many commonly used spectral algorithms can be constructed by certain analytic filter functions. We provide two examples (kernel ridge regression and kernel gradient flow) as follows, and put two more examples (iterated ridge regression and kernel gradient descent) in Appendix C. We provide rigorous proof for these examples in Lemma C.3. Example 1 (Kernel ridge regression). The filter function of kernel ridge regression (KRR) is wellknown to be φKRR λ (z) = 1 z + λ, ψKRR λ (z) = λ z + λ, τ = 1. (16) Example 2 (Kernel gradient flow). The filter function is φGF λ (z) = 1 −e−tz z , ψGF λ (z) = e−tz, t = λ−1, τ = ∞. (17) For any analytic filter function φλ with qualification τ ≥1 and the corresponding estimator of the spectral algorithm defined in (15), the following two theorems provide exact convergence rates on the excess risk when (i) the regression function is less-smooth, i.e., we have s ≤τ, and (ii) s > τ, where s is the source condition coefficient of the regression function given in Assumption 2. 8 Theorem 4.1. Let 0 < s ≤τ and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose that Assumption 1 and Assumption 2 hold for s and γ. Let φλ(z) be an analytic filter function and ˆfλ be the function defined in (15). Suppose one of the following conditions holds: (i) τ = ∞, (ii) s > 1/(2τ), (iii) γ > ((2τ + 1)s)/(2τ(1 + s)); then there exists λ⋆> 0, such that we have E ˆfλ⋆−f⋆ 2 L2 X = ΘP d−min{γ−p,s(p+1)} · poly (ln(d)) , where ΘP only involves constants depending on s, σ, γ, c0, κ, c1 and c2. Theorem 4.2. Let s > τ and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose that Assumption 1 and Assumption 2 hold for s and γ. Let φλ(z) be an analytic filter function and ˆfλ be the function defined in (15). Define ˜s = min{s, 2τ}, then there exists λ⋆> 0, such that we have E ˆfλ⋆−f⋆ 2 L2 X = ΘP d−min{γ−p, τ(γ−p+1)+p˜s τ+1 ,˜s(p+1)} · poly (ln(d)) , where ΘP only involves constants depending on s, σ, γ, c0, κ, c1 and c2. In addition, the convergence rates of the generalization error can not be faster than above for any choice of regularization parameter λ = λ(d, n) →0. Remark 4.3. These theorems substantially generalize the results on exact generalization error bounds of analytic spectral algorithms under the fixed-dimensional setting given in Li et al. (2024). Although the “analytic functional argument” introduced in their proof is still vital for us to deal with the general spectral algorithms, their proof has to rely on the polynomial eigendecay assumption that λj ≍j−β (Assumption 1), which does not hold in large dimensions since the hidden constant factors in the assumption vary with d11 (Lu et al. (2023)). Hence, their proof is not easy to generalize to large-dimensional spectral algorithms. We provide some graphical illustrations of Theorem 4.1 and Theorem 4.2 in Figure 1 (with τ = 2) and in Appendix A (with τ = 1, τ = 2, τ = 4, and τ = ∞, corresponding to KRR, iterated ridge regression in Example 3 and kernel gradient flow). As a direct consequence of Theorem 3.3, Theorem 4.1, and Theorem 4.2, we find that for the spectral algorithm with estimator defined in (15), it is minimax optimal if s ≤τ and the conditions in Theorem 4.1 hold. Moreover, these results show several phenomena for large-dimensional spectral algorithms. Saturation effect of large-dimensional spectral algorithms with finite qualification. In the large-dimensional setting and for the inner product kernel on the sphere, our results show that the saturation effect of spectral algorithms occurs when s > τ. As shown in Figure 1(c) and Figure 1(d), when s > τ, no matter how carefully one tunes the regularization parameter λ, the convergence rate can not be faster than d−min{γ−p, τ(γ−p+1)+p˜s τ+1 ,˜s(p+1)}, thus can not achieve the minimax lower bound d−min{γ−p,s(p+1)}. Periodic plateau behavior of spectral algorithms when s ≤2τ. When 0 < s ≤2τ and γ ∈[p(s + 1) + s + max{s, τ}/τ −1, (p + 1)(s + 1)) for an integer p ≥0, from Theorem 4.1 and Theorem 4.2, the convergence rate on the excess risk of spectral algorithm d−s(p+1). The above rate does not change when γ varies, which can also be found in Figure 1(b) and Figure 1(c). BIn other words, if we fix a large dimension d and increase γ (or equivalently, increase the sample size n), the optimal rate of excess risk of a spectral algorithm stays invariant in certain ranges. Therefore, in order to improve the rate of excess risk, one has to increase the sample size above a certain threshold. Polynomial approximation barrier of spectral algorithms when s →0. From Theorem 4.1, when s is close to zero, the convergence rate d−min{γ−p,s(p+1)} is unchanged in the range γ ∈ [p(s + 1) + s, (p + 1)(s + 1)), and increases in the short range γ ∈[p(s + 1), p(s + 1) + s). In other words, the excess risk of spectral algorithms will drop when γ exceeds p(s + 1) ≈p for any integer p and will stay invariant for most of the other γ. We term the above phenomenon as the polynomial approximation barrier of spectral algorithms (borrowed from Ghorbani et al. (2021)), and it can be illustrated by Figure 1(a) with s = 0.01. 9 Remark 4.4. Ghorbani et al. (2021) discovered the polynomial approximation barrier of KRR. As shown by Figure 5 and Theorem 4 in Ghorbani et al. (2021), if s = 0 and the true function falls into L2 = [H]0, then with high probability we have E ˆf KRR λ⋆−f⋆ 2 L2 − P>pf⋆ 2 L2 ≤ε f⋆ 2 L2 + σ2 , (18) where p is the integer satisfying γ ∈[p, p + 1), λ⋆is defined as in Theorem 4 in Ghorbani et al. (2021), P>ℓmeans the projection onto polynomials with degree > ℓ, and ε is any positive real number. Notice that (18) implies that the excess risk of KRR will drop when γ exceeds any integer and will stay invariant for other γ, and is consistent with our results for spectral algorithms. 5 Conclusion In this paper, we rigorously prove the saturation effect of KRR in large dimensions. Let s > 0 and γ > 0 be fixed real numbers, denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Given that the kernel is an inner product kernel defined on the sphere and that f⋆falls into the interpolation space [H]s, we first show that the convergence rate on the excess risk of large-dimensional kernel gradient flow is ΘP d−min{γ−p,s(p+1)} · poly (ln(d)) (Theorem 3.1), which is faster than that of KRR given in Zhang et al. (2024). We then determine the improved minimax lower bound as Ω d−min{γ−p,s(p+1)} /poly (ln(d)) (Theorem 3.3). Combining these results, we know that kernel gradient flow is minimax optimal in large dimensions, and KRR is inferior to kernel gradient flow in large dimensions. Our results suggest that previous results on large-dimensional KRR may not be directly extendable to large-dimensional neural networks if the regression function is over-smooth. In Section 4, we generalize our results to certain spectral algorithms. We determine the convergence rate on the excess risk of large-dimensional spectral algorithms (Theorem 4.1 and Theorem 4.2). From these results, we find several new phenomena exhibited in large-dimensional spectral algorithms, including the saturation effect, the periodic plateau behavior, and the polynomial approximation barrier. In this paper, we only consider the convergence rate on the excess risk of optimal-tuned largedimensional spectral algorithms with uniform input distribution on a hypersphere. We believe that several results in fixed-dimensional settings with input distribution on more general domains (e.g., Haas et al. (2024); Li et al. (2024)) can indeed be extended to large-dimensional settings, although we must carefully consider the constants that depend on d. Furthermore, we believe that by considering the learning curve of large-dimensional spectral algorithms (i.e., the convergence rate on the excess risk of spectral algorithms with any regularization parameter λ > 0) or the convergence rate on the excess risk of large-dimensional kernel interpolation (i.e., KRR with λ = 0), further research can find a wealth of new phenomena compared with the fixed-dimensional setting. Acknowledgments and Disclosure of Funding Lin’s research was supported in part by the National Natural Science Foundation of China (Grant 92370122, Grant 11971257). The authors are grateful to the reviewers for their constructive comments that greatly improved the quality and presentation of this paper. References Aerni, M., M. Milanta, K. Donhauser, and F. Yang (2023). Strong inductive biases provably prevent harmless interpolation. arXiv preprint arXiv:2301.07605. Arora, S., S. Du, W. Hu, Z. Li, and R. Wang (2019). 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Lin (2023). On the optimality of misspecified kernel ridge regression. In International Conference on Machine Learning, pp. 41331–41353. PMLR. Zhang, H., Y. Li, W. Lu, and Q. Lin (2024). Optimal rates of kernel ridge regression under source condition in large dimensions. arXiv preprint arXiv:2401.01270. Zhang, H., W. Lu, and Q. Lin (2024). The phase diagram of kernel interpolation in large dimensions. arXiv preprint arXiv:2404.12597. 13 A Graphical illustration and numerical experiments of main results A.1 Graphical illustration of Theorem 3.1, Theorem 4.1, and Theorem 4.2 Recall that Theorem 3.1, Theorem 4.1, and Theorem 4.2 determined the convergence rate on the excess risk of: (i) large-dimensional kernel gradient flow with s > 0; (ii) large-dimensional spectral algorithm with τ ≥1 and s ≤τ; and (iii) large-dimensional spectral algorithm with τ ≥1 and s > τ. In Figure 1, we have provided a visual illustration of Theorem 4.1 and Theorem 4.2 when τ = 2. Now, in Figure 2, we provide more visual illustrations of the results of spectral algorithms with τ = 1, τ = 2, τ = 4, and τ = ∞, which correspond to kernel ridge regression (KRR), iterated ridge regression in Example 3, and kernel gradient flow. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Figure 2: Convergence rates of spectral algorithms with qualification τ = 1 (KRR), τ = 2 (iterated ridge regression), τ = 4 (iterated ridge regression), and τ = ∞(kernel gradient flow) in Theorem 4.1, Theorem 4.2, and corresponding minimax lower rates in Theorem 3.3 with respect to dimension d. We present four graphs corresponding to four kinds of source conditions: s = 0.01, 1, 3, 5. The x-axis represents asymptotic scaling, γ : n ≍dγ; the y-axis represents the convergence rate of excess risk, r : Excess risk ≍dr. 14 A.2 Numerical experiments We conducted two experiments using two specific kernels: the RBF kernel and the NTK kernel. Experiment 1 was designed to confirm the optimal rate of kernel gradient flow and KRR when s = 1. Experiment 2 was designed to illustrate the saturation effect of KRR when s > 1. Experiment 1: We consider the following two inner product kernels: (i) RBF kernel with a fixed bandwidth: Krbf(x, x′) = exp −∥x −x′∥2 2 2 , x, x′ ∈Sd. (ii) Neural Tangent Kernel (NTK) of a two-layer ReLU neural network: Kntk(x, x′) := Φ(⟨x, x′⟩), x, x′ ∈Sd, where Φ(t) = [sin (arccos t) + 2(π −arccos t)t] /(2π). The RBF kernel satisfies Assumption 1. For the NTK, the coefficients of Φ(·), {aj}∞ j=0, satisfy aj > 0, j ∈{0, 1} ∪{2, 4, 6, . . .} and aj = 0, j ∈{3, 5, 7, . . .} (see, e.g., Lu et al. (2023)). As noted after Assumption 1, our results can be extended to inner product kernels with certain zero coefficients aj. Specifically, for any γ > 0, as long as aj > 0 for j = ⌊γ⌋, ⌊γ⌋+ 1, the proof and convergence rate remain the same. Therefore, for γ < 2 in our experiments, the convergence rates for NTK will be the same as for the RBF kernel. We used the following data generation procedure: yi = f∗(xi) + ϵi, i = 1, . . . , n, where each xi is i.i.d. sampled from the uniform distribution on Sd, and ϵi i.i.d. ∼N(0, 1). We selected the training sample sizes n with corresponding dimensions d such that n = dγ, γ = 0.5, 1.0, 1.5, 1.8. For each kernel and dimension d, we consider the following regression function f∗: f∗(x) = K(u1, x) + K(u2, x) + K(u3, x), for some u1, u2, u3 ∈Sd. (19) This function is in the RKHS H, and it is easy to prove that, for any u0 ∈Sd, Assumption 2 (b) holds for K(u0, ·) with s = 1. Therefore, Assumption 2 holds for s = 1. We used logarithmic least squares to fit the excess risk with respect to the sample size, resulting in the convergence rate r. As shown in Figure 3 and Figure 4, the experimental results align well with our theoretical findings. Experiment 2: We use most of the settings from Experiment 1, except that the regression function is changed to f∗(x) = p µs 2N(d, 2)P2(< ξ, x >) with s = 1.9, P2(t) := (dt2 −1)/(d −1) the Gegenbauer polynomial, and ξ ∈Sd. Notice that the addition formula P2(< ξ, x >) = 1 N(d,2) PN(d,2) j=1 Y2,j(ξ)Y2,j(x) implies that ∥f∗∥2 [H]s = 1 N(d, 2) N(d,2) X j=1 Y 2 2,j(ξ) = P2(1) = 1, hence f∗∈[H]s and satisfies Assumption 2. Our experiment settings are similar to those on page 30 of Li et al. (2022). We choose the regularization parameter for KRR and kernel gradient flow as λ = 0.05 · d−θ. For KRR, since Corollary D.16 suggests that the optimal regularization parameter is λ ≍d−0.7, we set θ = 0.7. Similarly, based on Corollary D.16, we set θ = 0.5 for kernel gradient flow. Additionally, we set γ = 1.8. The results indicate that the best convergence rate of KRR is slower than that of kernel gradient flow, implying that KRR is inferior to kernel gradient flow when the regression function is sufficiently smooth. B Proof of Theorem 3.3 We first restate Theorem 3.3. 15 2.00 2.05 2.10 2.15 2.20 2.25 2.30 log10 n 2.10 2.05 2.00 1.95 1.90 1.85 1.80 1.75 1.70 log10 Err NTK, = 0.5, theretical rate=-1 KRR, parameters chosen by CV log10Err 1.03log10n + 0.37 Kernel regression log10Err 1.09log10n + 0.42 (a) 2.70 2.75 2.80 2.85 2.90 2.95 3.00 log10 n 2.1 2.0 1.9 1.8 1.7 log10 Err NTK, = 0.8, theretical rate=-1 KRR, parameters chosen by CV log10Err 0.84log10n + 0.38 Kernel regression log10Err 0.84log10n + 0.59 (b) 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 log10 n 2.2 2.1 2.0 1.9 1.8 1.7 log10 Err NTK, = 1.5, theretical rate=-2/3 KRR, parameters chosen by CV log10Err 0.63log10n + 0.23 Kernel regression log10Err 0.66log10n + 0.21 (c) 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 log10 n 2.0 1.9 1.8 1.7 1.6 log10 Err NTK, = 1.8, theretical rate=-5/9 KRR, parameters chosen by CV log10Err 0.54log10n + 0.03 Kernel regression log10Err 0.56log10n + 0.04 (d) Figure 3: Results of Experiment 1. We repeated each experiment 50 times and reported the average excess risk for (a) kernel gradient flow (labeled as "kernel regression" in our reports) and (b) kernel ridge regression (KRR) on 1000 test samples. We randomly selected u1, u2, u3 and kept them fixed for each repeat. We choose the stopping time t in kernel gradient flow as C1n0.5, where C1 ∈{0.001, 0.01, 0.1, 1, 10, 100, 1000}. We use 5-fold cross-validation to select the regularization parameter λ in kernel ridge regression. The alternative values of λ in cross-validation are C2n−C3, where C2 ∈{0.001, 0.005, 0.01, 0.1, 0.5, 1, 2, 5, 10, 40, 100, 300, 1000}, C3 ∈{0.1, 0.2, . . . , 1.5}. Theorem B.1 (Restate Theorem 3.3). Let s > 0 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Let P consist of all the distributions ρ on X × Y such that Assumption 1 and Assumption 2 hold for s and γ. Then for any d ≥C, a sufficiently large constant only depending on s, γ, c1, and c2, we have the following claims: (i) When γ ∈(p(s + 1), p + ps + s], we have min ˆ f max ρ∈P E(X,Y )∼ρ⊗n ˆf −f⋆ 2 L2 ≥ ln ln(d) 50(γ −p(s + 1))(ln(d))2 dp−γ. (ii) When γ ∈(p + ps + s, (p + 1)(s + 1)], we have min ˆ f max ρ∈P E(X,Y )∼ρ⊗n ˆf −f⋆ 2 L2 = Ω d−s(p+1) , where Ωonly involves constants depending on s, σ, γ, c0, κ, c1, and c2. Proof of Theorem B.1. The item (ii) is a direct corollary of Theorem 5 in Zhang et al. (2024). Now we begin to proof the item (i). We need the following lemma. 16 2.00 2.05 2.10 2.15 2.20 2.25 2.30 log10 n 2.3 2.2 2.1 2.0 1.9 1.8 log10 Err RBF, = 0.5, theretical rate=-1 KRR, parameters chosen by CV log10Err 1.23log10n + 0.65 Kernel regression log10Err 1.31log10n + 0.73 (a) 2.70 2.75 2.80 2.85 2.90 2.95 3.00 log10 n 2.40 2.35 2.30 2.25 2.20 2.15 2.10 2.05 log10 Err RBF, = 0.8, theretical rate=-1 KRR, parameters chosen by CV log10Err 1.06log10n + 0.76 Kernel regression log10Err 0.90log10n + 0.32 (b) 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 log10 n 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 log10 Err RBF, = 1.5, theretical rate=-2/3 KRR, parameters chosen by CV log10Err 0.65log10n + -0.09 Kernel regression log10Err 0.72log10n + -0.09 (c) 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 log10 n 2.5 2.4 2.3 2.2 2.1 2.0 1.9 log10 Err RBF, = 1.8, theretical rate=-5/9 KRR, parameters chosen by CV log10Err 0.59log10n + -0.13 Kernel regression log10Err 0.61log10n + -0.22 (d) Figure 4: A similar plot as Figure 3, but with the RBF kernel. Lemma B.2 (Restate Lemma 4.1 in Lu et al. (2023)). For any δ ∈(0, 1) and any 0 < ˜ε1, ˜ε2 < ∞ only depending on n, d, {λj}, c1, c2, and γ and satisfying VK(˜ε2, D) + n˜ε2 2 + ln(2) V2(˜ε1, B) ≤δ, (20) we have min ˆ f max ρ∈P E(X,Y )∼ρ⊗n ˆf −f⋆ 2 L2 ≥1 −δ 4 ˜ε2 1, (21) where ρf⋆is the joint-p.d.f. of x, y given by (1) with f = f⋆, B := f ∈H, ∥f∥[H]s ≤Rγ D := ρf joint distribution of (y, x) where x ∼ρX , y = f(x) + ϵ, ϵ ∼N(0, σ2), f ∈B , and V2, VK are the ε-covering entropies ( as defined in Yang and Barron (1999); Lu et al. (2023)) of (B, d2 = ∥· ∥2 L2) and (D, d2 = KL divergence ). Suppose γ ∈(p(s + 1), p + ps + s]. Let C(p) = C12/10 be a constant only depending on γ, where C12 are given in Lemma D.13. Then we introduce ˜ε2 1 ≜dp−γ/ ln(d) and ˜ε2 2 ≜C(p)dp n ln ln(d). (22) 17 3.0 3.1 3.2 3.3 3.4 log10 n 4.5 4.0 3.5 3.0 2.5 2.0 1.5 log10 Err NTK, = 1.8 Kernel gradient flow, = 0.5 log10Err 0.88log10n + -1.37 KRR, = 0.7 log10Err 0.52log10n + 0.09 Figure 5: Results of Experiment 2. It can be seen that the best rate of excess risk for KRR is slower than that of kernel gradient flow. Let us further assume that d ≥C, where C is a sufficiently large constant only depending on γ, s, and c1. By Lemma D.11 and Lemma D.13 we have ˜ε2 1 = dp−γ/ ln(d) < C9 dps ≤µs p µs p+1 < ˜ε2 2 = C(p)dp n ln ln(d) ≤C(p) c1 dp−γ ln ln(d) < µs p n˜ε2 2 Definition of C12 ≤ 1 10N(d, p) ln ln(d). (23) Therefore, for any d ≥C, where C is a sufficiently large constant only depending on s, γ, and c1, we have V2(˜ε1, B) Lemma A.5 in Lu et al. (2023) ≥ K (˜ε1) ≥1 2N(d, p) ln µs p ˜ε2 1 Definition of ˜ε2 1 ≥ 1 2N(d, p) ln C9dγ−p(s+1) ln(d) ≥1 2N(d, p) (γ −p(s + 1)) ln(d) + 1 2 ln ln(d) . (24) On the other hand, from Lemma D.11, Lemma D.13, and Lemma D.12, one can check the following claim: Claim 1. Suppose γ ∈(p(s + 1), p + ps + s]. For any d ≥C, where C is a sufficiently large constant only depending on s, γ, c1, and c2, we have K √ 2σ˜ε2/6 ≤1 2N(d, p) ln 18µs p σ2˜ε2 2 ln ln(d) . 18 Therefore, for any d ≥C, where C is a sufficiently large constant only depending on s, γ, c1, and c2, we have VK(˜ε2, D) = V2( √ 2σ˜ε2, B) Lemma A.5 in Lu et al. (2023) ≤ K √ 2σ˜ε2/6 Claim 1 ≤ 1 2N(d, p) ln 18µs p σ2˜ε2 2 ln ln(d) Definition of ˜ε2 2 ≤ 1 2N(d, p) ln 18C10σ−2[C(p)]−1c2dγ−p(s+1) ≤1 2N(d, p) (γ −p(s + 1)) ln(d) + 1 5 ln ln(d) . (25) Combining (23), (24), and (25), we finally have: VK(˜ε2, D) + n˜ε2 2 + ln(2) V2(˜ε1, B) ≤[10(γ −p(s + 1)) ln(d) + 4 ln ln(d)] [10(γ −p(s + 1)) ln(d) + 5 ln ln(d)] < 1, and from Lemma B.2, we get min ˆ f max f⋆∈B E(X,y)∼ρ⊗n f⋆ ˆf −f⋆ 2 L2 ≥ ln ln(d) 4 ln(d) [10(γ −p(s + 1)) ln(d) + 5 ln ln(d)]dp−γ ≥ ln ln(d) 50(γ −p(s + 1))(ln(d))2 dp−γ, finishing the proof. ■ C Definition of analytic filter functions We first introduce the following definition of analytic filter functions (Bauer et al. (2007); Li et al. (2024)). Definition C.1 (Analytic filter functions). Let φλ : [0, κ2] →R≥0 \| λ ∈(0, 1) be a family of functions indexed with regularization parameter λ and define the remainder function ψλ(z) := 1 −zφλ(z). (26) We say that {φλ \| λ ∈(0, 1)} (or simply φλ(z)) is an analytic filter function if: (1) zφλ(z) ∈[0, 1] is non-decreasing with respect to z and non-increasing with respect to λ. (2) The qualification of this filter function is τ ∈[1, ∞] such that ∀0 ≤τ ′ ≤τ (and also τ ′ < ∞), there exist positive constants Ci only depending on τ ′, i = 1, 2, 3, 4, 5, such that we have φλ(z) ≥C1z−1, ψλ(z) ≤C2(z/λ)−τ ′, ∀λ ∈(0, 1), z > λ (27) C3 ≤λφλ(z) ≤C4, ψλ(z) ≥C5, ∀λ ∈(0, 1), z ≤λ. (28) (3) If τ < ∞, then there exists a positive constant C6 only depending on τ and λ1, such that we have ψλ(λ1) ≥C6λτ, (29) where λ1 is the largest eigenvalue of K defined in (5); and there exist positive constants C7 and C8 only depending on τ, such that we have (z/λ)2τψ2 λ(z) ≥C7, ∀λ ∈(0, 1), z > λ (30) (z/λ)2τψ2 λ(z) ≤C8zφλ(z), ∀λ ∈(0, 1), z ≤λ. (31) (4) Let Dλ = z ∈C : Re z ∈[−λ/2, κ2], \|Im z\| ≤Re z + λ/2 ∪ z ∈C : z −κ2 ≤κ2 + λ/2, Re z ≥κ2 ; Then φλ(z) can be extended to be an analytic function on some domain containing Dλ and the following conditions holds for all λ ∈(0, 1): 19 (C1) \|(z + λ)φλ(z)\| ≤˜E for all z ∈Dλ; (C2) \|(z + λ)ψλ(z)\| ≤˜Fλ for all z ∈Dλ; where ˜E, ˜F are positive constants. Remark C.2. We remark that some of the above properties are not essential for the definition of filter functions in the literature (Bauer et al., 2007; Gerfo et al., 2008), but we introduce them to avoid some unnecessary technicalities in the proof. The requirements of analytic filter functions are first considered in Li et al. (2024) and used for their “analytic functional argument”, which will also be vital in our proof. The following examples show many commonly used analytic filter functions and their proofs can be found in Lemma C.3, see also Li et al. (2024). Example 3 (Iterated ridge regression). Let q ≥1 be fixed. We define φIT,q λ (z) = 1 z 1 − λq (z + λ)q , ψIT,q λ (z) = λq (z + λ)q , τ = q. (32) Example 4 (Kernel gradient descent). The gradient descent method is the discrete version of gradient flow. Let η > 0 be a fixed step size. Then, iterating gradient descent with respect to the empirical loss t steps yields the filter function φGD λ (z) = η t−1 X k=0 (1 −ηz)k = 1 −(1 −ηz)t z , λ = (ηt)−1, (33) ψGD λ (z) = (1 −ηz)t, τ = ∞. (34) Moreover, when η is small enough, say η < 1/(2κ2), we have Re(1−ηz) > 0 for z ∈Dλ, so we can take the single-valued branch of (1 −ηz)t even when t is not an integer. Therefore, we can extend the definition of the filter function so that λ can be arbitrary and t = (ηλ)−1. Lemma C.3. φKRR λ , φIT,q λ , φGF λ , and φGD λ are analytic filter functions. Proof. Notice that (i) z ≤z + λ ≤2z when z > λ; and that (ii) λ ≤z + λ ≤2λ when z ≤λ. Hence, the constants C1, C2, C3, C4, and C6 are given in Li et al. (2024). For C5, when z ≤λ, we can take C5 = min{1/2, 2−q, e−1, e−1} > 0. For C7, when z > λ, we have (z/λ)2τ(ψKRR λ (z))2 = z z + λ 2 ≥1/4 (z/λ)2τ(ψIT,q λ (z))2 = z z + λ 2q ≥2−2q. For C8, when z ≤λ, we have z2τ−1(ψKRR λ (z))2 λ2τφKRR λ (z) = z z + λ ≤1 2 z2τ−1(ψIT,q λ (z))2 λ2τφIT,q λ (z) = z2q (z + λ)2q −[λ(z + λ)]q ≤ 1 22q −2q . ■ D Proof of Theorem 4.1 and Theorem 4.2 D.1 Bias-variance decomposition We first apply a standard bias-variance decomposition on the excess risk of spectral algorithms, and readers can also refer to Zhang et al. (2023, 2024) for more details. 20 Recall the definition of ˆgZ and ˆfλ in (14) and (15). Let’s define their conditional expectations as ˜gZ := E (ˆgZ\|X) = 1 n n X i=1 Kxif⋆(xi) ∈H; (35) and ˜fλ := E ˆfλ\|X = φλ (TX) ˜gZ ∈H. (36) Let’s also define their expectations as g = EˆgZ = Z X K(x, ·)f⋆(x) dρX (x) ∈H, (37) and fλ = φλ (T) g. (38) Then we have the decomposition ˆfλ −f⋆= 1 nφλ (TX) n X i=1 Kxiyi −f⋆ = 1 nφλ (TX) n X i=1 Kxi(f ∗ ρ (xi) + ϵi) −f⋆ = φλ (TX) ˜gZ + 1 n n X i=1 φλ (TX) Kxiϵi −f⋆ = ˜fλ −f⋆ + 1 n n X i=1 φλ (TX) Kxiϵi. (39) Taking expectation over the noise ϵ conditioned on X and noticing that ϵ\|X are independent noise with mean 0 and variance σ2, we obtain the bias-variance decomposition: E ˆfλ −f⋆ 2 L2 X = Bias2(λ) + Var(λ), (40) where Bias2(λ) := ˜fλ −f⋆ 2 L2 , Var(λ) := σ2 n2 n X i=1 ∥φλ (TX) K(xi, ·)∥2 L2 . (41) Given the decomposition (40), we next derive the upper and lower bounds of Bias2(λ) and Var(λ) in the following two subsections. Before we close this subsection, let’s introduce some quantities and an assumption that will be used frequently in our proof later. Denote the true function as f⋆= ∞ P i=1 fiϕi(x), let’s define the following quantities: N1,φ(λ) = ∞ X j=1 [λjφλ(λj)] ; N2,φ(λ) = ∞ X j=1 [λjφλ(λj)]2 ; M1,φ(λ) = ess sup x∈X ∞ X j=1 (ψλ(λj)fjϕj(x)) ; M2,φ(λ) = ∞ X j=1 (ψλ(λj)fj)2 ; (42) moreover, when φλ = φKRR λ , we denote Nk(λ) = Nk,φKRR(λ) and Mk(λ) = Mk,φKRR(λ) for simplicity, where k = 1, 2. Assumption 3. Suppose that ess sup x∈X ∞ X j=1 [λjφλ(λj)]2 ϕ2 j(x) ≤N2,φ(λ); (43) 21 and ess sup x∈X ∞ X j=1 [λjφλ(λj)] ϕ2 j(x) ≤N1,φ(λ); (44) and ess sup x∈X ∞ X j=1 λjφKRR λ (λj) ϕ2 j(x) ≤N1(λ). (45) For simplicity of notations, we denote hx(·) = K(x, ·), x ∈X in the rest of the proof. Moreover, we denote Tλ := (T + λ)−1 and TXλ := (TX + λ)−1. D.2 Variance term The following proposition rewrites the variance term using the empirical semi-norm. Proposition D.1 (Restate Lemma 9 in Zhang et al. (2024)). The variance term in (41) satisfies that Var(λ) = σ2 n Z X ∥φλ (TX) hx(·)∥2 L2,n dρX (x). (46) The operator form (46) allows us to apply concentration inequalities and establish the following two-step approximation. Z X ∥φλ (TX) hx∥2 L2,n dρX (x) A≈ Z X ∥φλ (T) hx∥2 L2,n dρX (x) B≈ Z X ∥φλ (T) hx∥2 L2 dρX (x). (47) Approximation B The following lemma characterizes the magnitude of Approximation B in high probability. Recall the definitions of N1,φ(λ) and N2,φ(λ) in (42). Lemma D.2 (Approximation B). Suppose that (43) in Assumption 3 holds. Then, for any fixed δ ∈(0, 1), with probability at least 1 −δ, we have 1 2 Z X ∥φλ (T) hx∥2 L2 dρX (x) −R2 (48) ≤ Z X ∥φλ (T) hx∥2 L2,n dρX (x) (49) ≤3 2 Z X ∥φλ (T) hx∥2 L2 dρX (x) + R2, (50) where R2 = 5N2,φ(λ) 3n ln 2 δ . (51) Proof. Define a function f(z) = Z X (φλ (T) hx(z))2 dρX (x) = Z X ∞ X j=1 (λjφλ(λj))2 ϕ2 j(x)ϕ2 j(z)dρX (x) = ∞ X j=1 (λjφλ(λj))2 ϕ2 j(z). (52) Since (43) in Assumption 3 holds, we have ∥f∥L∞≤N2,φ(λ); ∥f∥L1 = N2,φ(λ). 22 Applying Proposition 34 in Zhang et al. (2024) for √f and noticing that ∥√f∥L∞= p ∥f∥L∞= N2,φ(λ) 1 2 , we have 1 2 p f 2 L2 −5N2,φ(λ) 3n ln 2 δ ≤ p f 2 L2,n ≤3 2 p f 2 L2 + 5N2,φ(λ) 3n ln 2 δ , (53) with probability at least 1 −δ. On the one hand, we have p f 2 L2,n = Z X f(z)dPn(z) = Z X Z X (φλ (T) hx(z))2 dρX (x) dPn(z) = Z X Z X (φλ (T) hx(z))2 dPn(z) dρX (x) = Z X ∥φλ (T) hx∥2 L2,n dρX (x). On the other hand, we have p f 2 L2 = Z X f(z)dρX (z) = Z X Z X (φλ (T) hx(z))2 dρX (x) dρX (z) = Z X ∥φλ (T) hx∥2 L2 dρX (x). Therefore, (53) implies the desired results. ■ Approximation A Lemma D.3. Suppose that (43) and (45) in Assumption 3 hold. Suppose that there exists a constant ϵ only depending on s and γ, such that λ = λ(n, d) satisfies nϵ−1N1(λ) →0. Then there exists an absolute constant C1, such that for any fixed δ ∈(0, 1), when n is sufficiently large, with probability at least 1 −δ, we have Z X ∥φλ (TX) hx∥2 L2,n dρX (x) − Z X ∥φλ (T) hx∥2 L2,n dρX (x) (54) ≤C1 q N2,φ(λ) + C1 p vN1(λ) ln λ−1 · p vN1(λ) ln λ−1, (55) where v = N1(λ) n ln n. Remark D.4. The proof of Lemma D.3 is mainly based on Lemma 4.18 in Li et al. (2024). Notice that we replace the Assumption 2 in Li et al. (2024) by (45) in Assumption 3 (borrowed from Zhang et al. (2024)), since both of them can deduce same results given by Lemma 4.2 in Li et al. (2024) or Lemma 37 in Zhang et al. (2024). Proof. We start with D = \|∥φλ(TX)hx∥L2 −∥φλ(T)hx∥L2\| ≤ T 1 2 [φλ(T) −φλ(TX)] hx H. Using operator calculus, we get T 1 2 [φλ(T) −φλ(TX)] hx = T 1 2 1 2πi I Γλ RTX(z)(T −TX)RT (z)φλ(z)dz hx = 1 2πi I Γλ T 1 2 (TX −z)−1(T −TX)(T −z)−1hxφλ(z)dz = 1 2πi I Γλ T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ hxφλ(z)dz. 23 Therefore, taking the norms yields D ≤1 2π T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ hx H I Γλ \|φλ(z)dz\| = 1 2π · I · II · III · IV · V · I Γλ \|φλ(z)dz\| ≤1 2π · 1 · √ 6C · r N1(λ) n ln n · C · p N1(λ) I Γλ \|φλ(z)dz\|, where in the second estimation, we use (I) operator calculus, (II and IV) Proposition E.8, (III) Lemma E.7, and (V) Lemma 37 in Zhang et al. (2024) for each term respectively. Finally, from (63) in Li et al. (2024), we get I Γλ \|φλ(z)dz\| ≤C ln λ−1, (56) and thus there exists an absolute constant C1, such that we have D = \|∥φλ(TX)hx∥L2 −∥φλ(T)hx∥L2\| ≤C1 p vN1(λ) ln λ−1. On the other hand, combining (52) and (43) in Assumption 3, we have ∥φλ(T)hx∥2 L2 ≤N2,φ(λ), and hence ∥φλ(TX)hx∥L2 + ∥φλ(T)hx∥L2 ≤2∥φλ(T)hx∥L2 + D ≤ q N2,φ(λ) + C1 p vN1(λ) ln λ−1. Finally, ∥φλ(TX)hx∥2 L2 −∥φλ(T)hx∥2 L2 = \|∥φλ(TX)hx∥L2 −∥φλ(T)hx∥L2\| (∥φλ(TX)hx∥L2 + ∥φλ(T)hx∥L2) ≤C1 q N2,φ(λ) + C1 p vN1(λ) ln λ−1 · p vN1(λ) ln λ−1, and hence Z X ∥φλ (TX) hx∥2 L2,n dρX (x) − Z X ∥φλ (T) hx∥2 L2,n dρX (x) ≤1 n n X i=1 ∥φλ(TX)hxi∥2 L2 −∥φλ(T)hxi∥2 L2 ≤sup x∈X ∥φλ(TX)hx∥2 L2 −∥φλ(T)hx∥2 L2 ≤C1 q N2,φ(λ) + C1 p vN1(λ) ln λ−1 · p vN1(λ) ln λ−1, ■ Final proof of the variance term Now we are ready to state the theorem about the variance term. Theorem D.5. Suppose that (43) and (45) in Assumption 3 hold. Suppose there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ) · nϵ−1 →0, (57) N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 →0; (58) then we have Var(λ) = [1 + oP(1)] σ2 n N2,φ(λ). (59) 24 Proof. Recall that Var(λ) = σ2 n R X ∥φλ (TX) hx∥2 L2,n dρX (x). Hence, when n is large enough, with probability at least 1 −δ we have Z X ∥φλ (TX) hx∥2 L2,n dρX (x) − Z ∥φλ(T)hx∥2 L2dρX (x) ≤ Z X ∥φλ (TX) hx∥2 L2,n dρX (x) − Z X ∥φλ (T) hx∥2 L2,n dρX (x) + Z X ∥φλ (T) hx∥2 L2,n dρX (x) − Z X ∥φλ(T)hx∥2 L2dρX (x) Lemma D.2 ≤ Z X ∥φλ (TX) hx∥2 L2,n dρX (x) − Z X ∥φλ (T) hx∥2 L2,n dρX (x) + 5N2,φ(λ) 3n ln 2 δ Lemma D.3 ≤ q N2,φ(λ) · C1 p vN1(λ) ln λ−1 + C2 1vN1(λ)(ln λ−1)2 + 5N2,φ(λ) 3n ln 2 δ Definition of v = r N2,φ(λ) n N1(λ) · C1 p ln(n) ln λ−1 + N 2 1 (λ) n · C2 1 ln(n)(ln λ−1)2 + N2,φ(λ) n · 5 3 ln 2 δ = I · C1 p ln(n) ln λ−1 + II · C2 1 ln(n)(ln λ−1)2 + III · 5 3 ln 2 δ . When n ≥C, a sufficiently large constant only depending on γ and C1, we have I · C1 p ln(n) ln λ−1 ≤1 6N2,φ(λ). Furthermore, when N 2 1 (λ) nN2,φ(λ) · nϵ →0, we have I · C1 p ln(n) ln λ−1/N2,φ(λ) →0 and II · C2 1 ln(n)(ln λ−1)2/N2,φ(λ) →0. Finally, from (52) we have ∥φλ(T)hx∥2 L2 = ∞ X i=1 (λjφλ(λj))2 ϕ2 i (z), and thus the deterministic term writes Z X ∥φλ(T)hx∥2 L2dρX (x) = N2,φ(λ). ■ D.3 Bias term In this subsection, our goal is to determine the upper and lower bounds of bias under some approximation conditions. The triangle inequality implies that Bias(λ) = ˜fλ −f⋆ L2 ≥∥fλ −f⋆∥L2 − ˜fλ −fλ L2 Bias(λ) ≤∥fλ −f⋆∥L2 + ˜fλ −fλ L2 . (60) The following lemma characterizes the dominant term of Bias(λ). Lemma D.6. For any λ > 0, we have ∥fλ −f⋆∥L2 = M2,φ(λ) 1 2 . (61) 25 Proof. We have ∥fλ −f⋆∥2 L2 = ∞ X i=1 λiφλ(λi)fiϕi(x) − ∞ X i=1 fiϕi(x) 2 L2 = ∞ X i=1 ψλ(λi)fiϕi(x) 2 L2 = ∞ X i=1 (ψλ(λi)fi)2 = M2,φ(λ). ■ The following lemma bounds the remainder term of Bias(λ) when s ≥1. Lemma D.7. Suppose that (45) in Assumption 3 holds. Suppose that there exist constants ϵ and C only depending on s and γ, such that λ = λ(n, d) satisfies nϵ−1N1(λ) →0, (62) N1(λ)M2 1,φ(λ) n2 = o M2,φ(λ) + σ2 n N2,φ(λ) , (63) N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i = o M2,φ(λ) + σ2 n N2,φ(λ) ; (64) then we have ˜fλ −fλ 2 L2 = oP M2,φ(λ) + σ2 n N2,φ(λ) . (65) Proof. Do the decomposition, ˜fλ −fλ = φλ(TX)˜gX −(ψλ(TX) + φλ(TX)TX)fλ = φλ(TX)(˜gX −TXfλ) −ψλ(TX)Tφλ(T)f⋆ = φλ(TX)(˜gX −TXfλ) −φλ(TX)ψλ(T)g + φλ(TX)ψλ(T)g −ψλ(TX)Tφλ(T)f⋆ = φλ(TX) [˜gX −TXfλ −ψλ(T)g] + [φλ(TX)ψλ(T)Tf⋆−ψλ(TX)Tφλ(T)f⋆] = φλ(TX)(˜gX −TXfλ −g + Tfλ) + (φλ(TX)Tψλ(T) −ψλ(TX)Tφλ(T))f⋆ = I + II. (66) Bound on I: For the first term in (66), we have ∥I∥L2 = ∥φλ(TX)(˜gX −TXfλ −g + Tfλ)∥L2 = T 1 2 φλ(TX)(˜gX −TXfλ −g + Tfλ) H ≤ T 1 2 T −1 2 λ · T 1 2 λ φλ(TX)T 1 2 λ · T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] H (72) in Zhang et al. (2024) ≤ T 1 2 λ φλ(TX)T 1 2 λ · T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] H Proposition E.1 ≤ 4 T 1 2 λ T −1 XλT 1 2 λ · T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] H (62) and (73) in Zhang et al. (2024) ≤ 12 T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] H, 26 Denote ξi = ξ(xi) = T −1 2 λ (Kxif⋆(xi) −Txifλ). To use Bernstein inequality, we need to bound the m-th moment of ξ(x): E ∥ξ(x)∥m H = E T −1 2 λ Kx(f⋆−fλ(x)) m H ≤E T −1 2 λ K(x, ·) m H E \|(f⋆−fλ(x))\|m x . (67) Note that Lemma 37 in Zhang et al. (2024) shows that T −1 2 λ K(x, ·) H ≤N1(λ) 1 2 , µ-a.e. x ∈X; By definition of M1,φ(λ), we also have ∥fλ −f⋆∥L∞= ∞ X i=1 ψλ(λi)fiϕi(x) L∞ = M1,φ(λ). (68) In addition, we have proved in Lemma D.6 that E\|(fλ(x) −f⋆(x))\|2 = M2,φ(λ). So we get the upper bound of (67), i.e., (67) ≤N1(λ) m 2 · ∥fλ −f⋆∥m−2 L∞· E\|(fλ(x) −f⋆(x))\|2 = N1(λ) m 2 M1,φ(λ)m−2M2,φ(λ) = N1(λ) 1 2 M1,φ(λ) m−2 N1(λ) 1 2 M2,φ(λ) 1 2 2 . Using Lemma 36 in Zhang et al. (2024) with therein notations: L = N1(λ) 1 2 M1,φ(λ) and σ = N1(λ) 1 2 M2,φ(λ) 1 2 , for any fixed δ ∈(0, 1), with probability at least 1 −δ, we have ∥I∥L2 ≤12 · 4 √ 2 log 2 δ N1(λ) 1 2 M1,φ(λ) n + N1(λ) 1 2 M2,φ(λ) 1 2 √n ! . (69) Bound on II: For the second term in (66), we have ∥II∥L2 = ∥(φλ(TX)Tψλ(T) −ψλ(TX)Tφλ(T))f⋆∥L2 ≤ T 1 2 (φλ(TX)Tψλ(T) −ψλ(T)Tφλ(T))f⋆ H + T 1 2 (ψλ(TX)Tφλ(T) −ψλ(T)Tφλ(T))f⋆ H. (70) For the first term in (70), we still employ the analytic functional argument: T 1 2 (φλ(TX)Tψλ(T) −ψλ(T)Tφλ(T))f⋆ = T 1 2 (φλ(TX) −φλ(T))Tψλ(T)f⋆ = 1 2πi I Γλ T 1 2 (TX −z)−1(TX −T)(T −z)−1φλ(z)Tψλ(T)f⋆dz = 1 2πi I Γλ T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ T 1 2 · T 1 2 ψλ(T)f⋆φλ(z)dz. 27 Therefore, 2π∥T 1 2 (φλ(TX)Tψλ(T) −ψλ(T)Tφλ(T))f⋆∥H ≤ I Γλ T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ T 1 2 · T 1 2 ψλ(T)f⋆ H\|φλ(z)dz\| (72) in Zhang et al. (2024) ≤ I Γλ T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T 1 2 ψλ(T)f⋆ H\|φλ(z)dz\| (45) and Proposition E.8 ≤ √ 6C2 I Γλ T −1 2 λ (T −TX)T −1 2 λ · T 1 2 ψλ(T)f⋆ H\|φλ(z)dz\| Lemma E.7 ≤ √ 6C2√v I Γλ T 1 2 ψλ(T)f⋆ H\|φλ(z)dz\| Definition of M2,φ(λ) = √ 6C2√vM1/2 2,φ(λ) I Γλ \|φλ(z)dz\| (56) ≤ √ 6C3√vM1/2 2,φ(λ) ln λ−1, (71) where v = N1(λ) n ln n. For the second term in (70), we have T 1 2 (ψλ(TX)Tφλ(T) −ψλ(T)Tφλ(T))f⋆ = T 1 2 1 2πi I Γλ RTX(z)(T −TX)RT (z)ψλ(z)dz Tφλ(T)f⋆ = 1 2πi I Γλ T 1 2 (TX −z)−1(T −TX)(T −z)−1ψλ(z)Tφλ(T)f⋆dz = 1 2πi Z Γλ T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ Tφλ(T)f⋆ψλ(z)dz. Hence, similar to (71), we have 2π T 1 2 (ψλ(TX)Tφλ(T) −ψλ(T)Tφλ(T))f⋆ H ≤ Z Γλ T 1 2 T −1 2 λ · T 1 2 λ (TX −z)−1T 1 2 λ · T −1 2 λ (T −TX)T −1 2 λ · T 1 2 λ (T −z)−1T 1 2 λ · T −1 2 λ Tφλ(T)f⋆ H\|ψλ(z)dz\| ≤ √ 6C2√v T −1 2 λ Tφλ(T)f⋆ H Z Γλ \|ψλ(z)dz\| Definition of analytic filter functions ≤ √ 6C2√v T −1 2 λ Tφλ(T)f⋆ HC ˜Fλ ln λ−1. (72) 28 Combining (66), (69), (70), (71), and (72), there exists a constant C1 only depending on δ and ˜F, such that we have ˜fλ −fλ L2 ≤C1 N1(λ) 1 2 M1,φ(λ) n + N1(λ) 1 2 M2,φ(λ) 1 2 √n ! + C1 √vM1/2 2,φ(λ) ln λ−1 + C1 √v T −1 2 λ Tφλ(T)f⋆ Hλ ln λ−1 (62) ≤ n−1N1(λ) 1/2 · C1C1/2 · (M2,φ(λ))1/2 + n−1N1(λ) 1/2 · C1 · (M2,φ(λ))1/2 + nϵ−1N1(λ) 1/2 · C1 · (M2,φ(λ))1/2 + o M2,φ(λ) + σ2 n N2,φ(λ) 1/2 . (73) ■ When s < 1, we can use the following lemma to bound the remainder term of Bias(λ). This lemma is a modification of Lemma D.7, and its proof is partly based on Lemma 26 in Zhang. Lemma D.8. Suppose that (45) in Assumption 3 holds. Suppose that there exist constants ϵ and C only depending on s and γ, such that λ = λ(n, d) satisfies nϵ−1N1(λ) →0, (74) N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) ; (75) n−1N1(λ) 1 2 ∥fλ∥L∞+ n 1−s 2 +ϵ = o M2,φ(λ) + σ2 n N2,φ(λ) 1/2 ; (76) then we have ˜fλ −fλ 2 L2 = oP M2,φ(λ) + σ2 n N2,φ(λ) . (77) Proof. Similar to the proof in Lemma D.7, we have the decomposition ˜fλ −fλ = I + II, with ∥I∥2 L2 ≤122 T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] 2 H, ∥II∥2 L2 = o M2,φ(λ) + σ2 n N2,φ(λ) . Denote ξi = ξ(xi) = T −1 2 λ (Kxif⋆(xi) −Txifλ). Further consider the subset Ω1 = {x ∈X : \|f⋆(x)\| ≤t} and Ω2 = X\Ω1, where t will be chosen appropriately later. Decompose ξi as ξiIxi∈Ω1 + ξiIxi∈Ω2 and we have the following decomposition: T −1 2 λ [(˜gX −TXfλ) −(g −Tfλ)] H = 1 n n X i=1 ξi −Eξx H (78) ≤ 1 n n X i=1 ξiIxi∈Ω1 −EξxIx∈Ω1 H + ∥1 n n X i=1 ξiIxi∈Ω2∥H + ∥EξxIx∈Ω2∥H := I + II + III. (79) Next we choose t = n 1−s 2 +ϵt, q = 2 1−s −ϵq such that ϵt < ϵ; and 1 −s 2 + ϵt > 1/ 2 1 −s −ϵq . (80) 29 Then we can bound the three terms in (78) as follows: (i) For the first term in (78), denoted as I, notice that ∥(fλ −f⋆) Ixi∈Ω1∥L∞≤∥fλ∥L∞+ n 1−s 2 +ϵt. (81) Imitating (67) in the proof of Lemma D.7, we have I = oP M2,φ(λ) + σ2 n N2,φ(λ) 1/2 . (82) (ii) For the second term in (78), denoted as II. Since q = 2 1−s −ϵq < 2 1−s, Lemma 42 in Zhang et al. (2024) shows that, [H]s ,→Lq(X, µ), (83) with embedding norm less than a constant Cs,κ. Then Assumption 2 (a) implies that there exists 0 < Cq < ∞only depending on γ, s and κ such that ∥f⋆∥Lq(X,µ) ≤Cq. Using the Markov inequality, we have P(x ∈Ω2) = P \|f⋆(x)\| > t ≤E\|f⋆(x)\|q tq ≤(Cq)q tq . Further, since (80) guarantees tq ≫n, we have τn := P (II > 0) (84) ≤P ∃xi s.t. xi ∈Ω2, = 1 −P xi /∈Ω2, ∀xi, i = 1, 2, · · · , n = 1 −P x /∈Ω2 n = 1 −P \|f⋆(x)\| ≤t n ≤1 − 1 −(Cq)q tq n →0. (85) (iii) For the third term in (78), denoted as III. Since Lemma 37 in Zhang et al. (2024) implies that ∥T −1 2 λ k(x, ·)∥H ≤N1(λ) 1 2 , µ-a.e. x ∈X, so III ≤E∥ξxIx∈Ω2∥H ≤E h ∥T −1 2 λ k(x, ·)∥H · f⋆−fλ(x) Ix∈Ω2 i ≤N1(λ) 1 2 E f⋆−fλ(x) Ix∈Ω2 ≤N1(λ) 1 2 ∥f⋆−fλ∥ 1 2 L2 · P (x ∈Ω2) 1 2 ≤N1(λ) 1 2 M2,φ(λ) 1 2 t−q 2 , (86) where we use Cauchy-Schwarz inequality for the third inequality and Lemma D.6 for the fourth inequality. Recalling that the choices of t, q satisfy t−q ≪n−1 and we have assumed nϵ−1N1(λ) → 0, we have III = o M2,φ(λ) 1 2 . (87) Plugging (82), (84) and (87) into (78), we finish the proof. ■ Final proof of the bias term Now we are ready to state the theorem about the bias term. Theorem D.9 (s ≥1). Suppose that (45) in Assumption 3 holds. Suppose that there exist constants ϵ and C only depending on s and γ, such that λ = λ(n, d) satisfies nϵ−1N1(λ) →0, N1(λ)M2 1,φ(λ) n2 ≪ M2,φ(λ) + σ2 n N2,φ(λ) , N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) ; then we have Bias2(λ) −M2,φ(λ) = oP M2,φ(λ) + σ2 n N2,φ(λ) . (88) 30 Theorem D.10 (s < 1). Suppose that (45) in Assumption 3 holds. Suppose that there exist constants ϵ and C only depending on s and γ, such that λ = λ(n, d) satisfies nϵ−1N1(λ) →0, N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) ; n−1N1(λ) 1 2 ∥fλ∥L∞+ n 1−s 2 +ϵ = o M2,φ(λ) + σ2 n N2,φ(λ) 1/2 ; then we have Bias2(λ) −M2,φ(λ) = oP M2,φ(λ) + σ2 n N2,φ(λ) . (89) D.4 Quantity calculations and conditions verification for the inner product kernels In the previous two sections, we have successfully bounded the bias and the variance terms by the quantities M2,φ(λ) and N2,φ(λ). In this subsection, we will focus on the inner product kernels on the sphere. We will (i) determine the rates for the above quantities, and (ii) verify all the conditions in Theorem D.5, Theorem D.9 and Theorem D.10. Recall that µk and N(d, k), defined in (9), are the eigenvalues of the inner product kernel K defined on the sphere and the corresponding multiplicity. The following three lemmas (mainly cited from Lu et al. (2023)) give concise characterizations of µk and N(d, k), which is sufficient for the analysis in this paper. Lemma D.11. For any fixed integer p ≥0, there exist constants C, C9 and C10 only depending on p and {aj}j≤p+1, such that for any d ≥C, we have C9d−k ≤µk ≤C10d−k, k = 0, 1, · · · , p + 1. (90) Lemma D.12. For any fixed integer p ≥0, there exist constants C only depending on p and {aj}j≤p+1, such that for any d ≥C, we have µk ≤C10 C9 d−1µp, k = p + 1, p + 2, · · · where C9 and C10 are constants given in Lemma D.11. Lemma D.13. For any fixed integer p ≥0, there exist constants C11, C12 and C only depending on p, such that for any d ≥C, we have C11dk ≤N(d, k) ≤C12dk, k = 0, 1, · · · , p + 1. (91) With these lemmas, we can begin to bound the quantities M2,φ(λ) and N2,φ(λ). Lemma D.14. Suppose that Assumption 1 and Assumption 2 hold for s and an integer p. Suppose ℓ≤p, t = λ−1 ∈(dℓ, dℓ+1]. Then we have the following bound. M2,φ(λ) =    Θ d−s(ℓ+1) τ = ∞ Θ t−2τdℓ(2τ−s) + d−s(ℓ+1) s ≤2τ < ∞ Θ λ2τ s > 2τ N2,φ(λ) n = Θ dℓ n + t2 ndℓ+1 ∞ X k=0 λ2µkφ2 λ(µk) λ + µk N(d,k) X j=1 f 2 k,j = O λ2dmax{p(2−s),0} + d−s(ℓ+1) ; (92) and thus Assumption 3 holds. Moreover, when s ≥1, We have M2 1,φ(λ) =    O d−(ℓ+1)(s−1) τ = ∞ O λ2τ−1dℓ(2τ−s) + d−(ℓ+1)(s−1) s ≤2τ < ∞ O λ2τ−1 s > 2τ (93) 31 Proof. I. We begin with M2,φ(λ). If s ≤2τ and τ < ∞, then we have M2,φ(λ) = ∞ X k=0 ψ2 λ(µk) N(d,k) X j=1 f 2 k,j ≤ ℓ X k=0 C2 2(tµk)−2τ(µk)s N(d,k) X j=1 (µk)−sf 2 k,j + ∞ X k=ℓ+1 ψ2 λ(µk) N(d,k) X j=1 f 2 k,j ≤ ℓ X k=0 C2 2(tµk)−2τ(µk)s N(d,k) X j=1 (µk)−sf 2 k,j + ∞ X k=ℓ+1 (µk)s N(d,k) X j=1 (µk)−sf 2 k,j ≤C2 2t−2τ(C9d−ℓ)s−2τ ℓ X k=0 N(d,k) X j=1 (µk)−sf 2 k,j + (C10d−ℓ−1)s ∞ X k=ℓ+1 N(d,k) X j=1 (µk)−sf 2 k,j = O t−2τdℓ(2τ−s) + d−s(ℓ+1) ; and when τ = ∞, a similar argument ( taking τ ′ < τ and let τ ′ →∞, then we have (td−ℓ)−2τ ′ →0) shows that M2,φ(λ) = O(d−s(ℓ+1)). Similarly, if s ≤2τ, then we have M2,φ(λ) ≥1 {τ < ∞} ℓ X k=0 C2 7(tµk)−2τ(µk)s N(d,k) X j=1 (µk)−sf 2 k,j + ∞ X k=ℓ+1 ψ2 λ(µk) N(d,k) X j=1 f 2 k,j ≥1 {τ < ∞} Ω t−2τdℓ(2τ−s) + ∞ X k=ℓ+1 C2 5(µk)s N(d,k) X j=1 (µk)−sf 2 k,j ≥1 {τ < ∞} Ω t−2τdℓ(2τ−s) + C2 5(C10d−ℓ−1)s N(d,ℓ) X j=1 (µℓ)−sf 2 ℓ,j = 1 {τ < ∞} Ω t−2τdℓ(2τ−s) + Ω d−s(ℓ+1) . If 2τ < s, then M2,φ(λ) = ∞ X k=0 ψ2 λ(µk) N(d,k) X j=1 f 2 k,j Lemma E.3 ≤ κ2(s−2τ)λ2τ ∞ X k=0 N(d,k) X j=1 µ−s k f 2 k,j = O λ2τ . Similarly, if 2τ < s, then we have M2,φ(λ) ≥ψ2 λ(µ0)f 2 0,1 ≥C2 6f 2 0,1 · λ2τ = Ω λ2τ . 32 II. Now let’s bound the second term N2,φ(λ)/n. We have N2,φ(λ) n = 1 n ∞ X k=0 N(d, k) [µkφλ(µk)]2 ≤1 n ℓ X k=0 N(d, k) + 1 n ∞ X k=ℓ+1 N(d, k) [µkφλ(µk)]2 ≤1 n ℓ X k=0 N(d, k) + C2 4t2 n ∞ X k=ℓ+1 N(d, k)(µk)2 ≤ℓN(d, ℓ) n + C2 4t2 n µℓ+1 = O dℓ n + t2 ndℓ+1 . (94) Similarly, we have N2,φ(λ) n ≥C2 1 n ℓ X k=0 N(d, k) + C2 3t2 n ∞ X k=ℓ+1 N(d, k)(µk)2 ≥C2 1 N(d, ℓ) n + C2 3t2 n µℓ+1 = Ω dℓ n + t2 ndℓ+1 . (95) III. For the third term, we have ∞ X k=0 λ2µkφ2 λ(µk) λ + µk N(d,k) X j=1 f 2 k,j ≤λ2R2 γ   p X k=0 µs kφ2 λ(µk) + λ−1 ∞ X k=p+1 µs+1 k C2 4λ−2   = O λ2dmax{p(2−s),0} + λ−1d−(s+1)(ℓ+1) = O λ2dmax{p(2−s),0} + d−s(ℓ+1) IV. Now we show that Assumption 3 holds. Notice that (45) has been verified in Lemma 20 of Zhang et al. (2024). Similarly, one can prove (43) and (44) hold using a similar proof as that for Lemma 20 of Zhang et al. (2024). V. For the final term, when s ≥1, we have M2 1,φ(λ) = ess sup x∈X ∞ X i=1 (ψλ(λi)fiei(x)) 2 ≤ ∞ X i=1 ψλ(λi) λiφλ(λi)f 2 i ! · ess sup x∈X ∞ X i=1 λiφλ(λi)ei(x)2 Assumption 3 ≤ ∞ X i=1 ψλ(λi) λiφλ(λi)f 2 i ! · ∞ X i=1 λiφλ(λi) := Q1,φ(λ) · N1,φ(λ). (96) 33 For Q1,φ(λ), when τ ≥s/2 and τ < ∞, we have Q1,φ(λ) = ∞ X k=0 ψ2 λ(µk)µs−1 k φλ(µk) N(d,k) X j=1 µ−s k f 2 k,j ≤C2 2 C1 ℓ X k=0 λ2τµ−2τ+s k N(d,k) X j=1 µ−s k f 2 k,j + (C3)−1λ ∞ X k=ℓ+1 µs−1 k N(d,k) X j=1 µ−s k f 2 k,j = O λ2τdℓ(2τ−s) + λd−(ℓ+1)(s−1) . (97) Similarly, when τ = ∞, we can show that Q1,φ(λ) = O(λd−(ℓ+1)(s−1)). And when τ < s/2, we have Q1,φ(λ) = ∞ X k=0 ψ2 λ(µk)µs−1 k φλ(µk) N(d,k) X j=1 µ−s k f 2 k,j Lemma E.3 ≤ C2 2κ2(s−2τ) C1 λ2τ p X k=0 N(d,k) X j=1 µ−s k f 2 k,j + ∞ X k=p+1 ψ2 λ(µk)µs−1 k φλ(µk) N(d,k) X j=1 µ−s k f 2 k,j (30) ≤C2 2κ2(s−2τ) C1 λ2τ p X k=0 N(d,k) X j=1 µ−s k f 2 k,j + ∞ X k=p+1 C8λ2τ N(d,k) X j=1 µ−s k f 2 k,j = O λ2τ . For N1,φ(λ), we have N1,φ(λ) = ∞ X k=0 N(d, k) [µkφλ(µk)] ≤ ℓ X k=0 N(d, k) + ∞ X k=ℓ+1 N(d, k) [µkφλ(µk)] ≤ ℓ X k=0 N(d, k) + C4t ∞ X k=ℓ+1 N(d, k)µk ≤ℓN(d, ℓ) + C4t = O dℓ+ λ−1 = O λ−1 . (98) Therefore, when s ≥1, we have M2 1,φ(λ) =    O d−(ℓ+1)(s−1) τ = ∞ O λ2τ−1dℓ(2τ−s) + d−(ℓ+1)(s−1) s ≤2τ < ∞ O λ2τ−1 s > 2τ (99) ■ 34 From Lemma D.14, we have the following three corollaries. Corollary D.15. Let 1 ≤s ≤τ and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose one of the following cases holds for λ⋆= d−ℓor λ⋆= d−ℓ· poly (ln(d)): (1) p ≥1, p(s + 1) ≤γ < ps + p + s, ℓ= p + 1/2 (2) p ≥1, ps + p + s ≤γ < ps + p + s + 1, ℓ= (γ −(p + 1)(s −1))/2 (3) γ < s, ℓ= min{γ, 1}/2 (4) s ≤γ < s + 1, ℓ= (γ −(s −1))/2 Then we have M2,φ(λ⋆) ≲N2,φ(λ⋆) n = Θ d−s(p+1) + dp n , (100) or M2,φ(λ⋆) ≲N2,φ(λ⋆) n = Θ d−s(p+1) + dp n · poly (ln(d)) . (101) Corollary D.16. Let τ < s ≤2τ and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Denote ∆= γ −p(s + 1). Suppose one of the following cases holds for λ⋆= d−ℓor λ⋆= d−ℓ· poly (ln(d)): (1) γ ≥1, 0 ≤∆≤τ, ℓ= ℓ1 := p + ∆/(2τ) (2) γ ≥1, τ ≤∆≤s + s/τ −1, ℓ= ℓ2 := p + (∆+ 1)/(2τ + 2) (3) γ ≥1, ∆≥s + s/τ −1, ℓ= ℓ3 := p + (∆+ 1 −s)/2 (4) γ < 1, ℓ= γ/2 Then we have M2,φ(λ⋆) ≍N2,φ(λ⋆) n = Θ d−min{γ−p, τ(γ−p+1)+ps τ+1 ,s(p+1)} , (102) or M2,φ(λ⋆) ≍N2,φ(λ⋆) n = Θ d−min{γ−p, τ(γ−p+1)+ps τ+1 ,s(p+1)} · poly (ln(d)) . (103) Proof. Denote I = −2ℓτ + 2pτ −ps, II = −sp −s, III = p −γ, and IV = 2ℓ−γ −p −1. From Lemma D.14 we have M2,φ(λ⋆) ≍dI + dII, N2,φ(λ⋆) n ≍dIII + dIV. We can verify that: (1) When 0 ≤∆≤τ and ℓ= p + ∆/(2τ), we have II ≤I = III ≥IV and min γ −p, τ(γ −p + 1) + ps τ + 1 , s(p + 1) = γ −p; (2) When τ ≤∆≤s + s/τ −1 and ℓ= p + (∆+ 1)/(2τ + 2), we have II ≤I = IV ≥III and min γ −p, τ(γ −p + 1) + ps τ + 1 , s(p + 1) = τ(γ −p + 1) + ps τ + 1 ; (3) When ∆≥s + s/τ −1 and ℓ= p + (∆+ 1 −s)/2, we have I ≤II = IV ≥III and min γ −p, τ(γ −p + 1) + ps τ + 1 , s(p + 1) = s(p + 1); 35 (4) When γ < 1 and ℓ= γ/2, we have III ≥max{I, II, IV}. ■ Corollary D.17. Let s < 1 and γ > 0 be fixed real numbers. Denote p as the integer satisfying γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose one of the following cases holds for λ⋆= d−ℓor λ⋆= d−ℓ· poly (ln(d)): (1) τ = ∞, p ≥1, p(s + 1) ≤γ < ps + p + s, ℓ= p + s/2 (2) τ = ∞, p ≥1, ps + p + s ≤γ < ps + p + s + 1, ℓ= (γ + p(1 −s))/2 (3) τ = ∞, γ < s, ℓ= min{γ, 1, 2γs}/2 (4) τ = ∞, s ≤γ < s + 1, ℓ= min{(γ + (1 −s))/2, γ(1 + s) −s, γ/2} (5) τ < ∞, p(s + 1) ≤γ < ps + p + s, ℓ= (γ + 2τp −sp −p)/(2τ) (6) τ < ∞, ps + p + s ≤γ < ps + p + s + 1, ℓ= p + s/(2τ) Then we have M2,φ(λ⋆) + N2,φ(λ⋆) n = Θ d−s(p+1) + dp n , (104) or M2,φ(λ⋆) + N2,φ(λ⋆) n = Θ d−s(p+1) + dp n · poly (ln(d)) . (105) D.4.1 Verification of variance conditions Lemma D.18 (Verification of variance conditions for inner-product kernels). Suppose n ≍dγ and s ≥1, for γ ∈[p(s + 1), (p + 1)(s + 1)). For any given ℓ≥0, if λ ≥    d−ℓ1 + ln2(d)1{γ = 2, s = 1} p ≥1, 2ℓ≤max{2p + 1, γ −(p + 1)(s −1)} d−ℓln2(d) p = 0, γ ≥1, 2ℓ≤max{1, γ −(s −1)} d−ℓ p = 0, γ < 1, 2ℓ≤γ; then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ) · nϵ−1 →0, N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 →0. Proof. From Lemma 21 in Zhang et al. (2024), we have N1(λ) ≍λ−1. When p = 0, we have γ −ℓ> 0. When p ≥1, we have γ −p −1/2 ≥ps −1/2 > 0. Therefore, there exists a constant ϵ > 0 only depending on s and γ, such that we have N1(λ) · nϵ−1 →0. Denote q := ⌊ℓ⌋. From Lemma D.14, we further have N2,φ(λ) = Ω dq + λ−2d−q−1 . Hence, we have N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 = O (ln(d))3 n(λ2dq + d−q−1) . Denote ∆:= (ln(d))3 nλ2dq , ∆′ := (ln(d))3 dγ−q−1 , then when ∆= o(1) or ∆′ = o(1), we have: N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 →0. Now we show that ∆= o(1): 36 • When p ≥3 and p = 2, s > 1, since γ −2ℓ+ q ≥(γ −ℓ−1) + (q + 1 −ℓ) > 0, we have ∆= o(1). • When p = 2, s = 1, since 2ℓ−q < ℓ+ 1 < 4 ≤γ, we have ∆= o(1). • When p = 2, s = 1, since 2ℓ−q < ℓ+ 1 < 4 ≤γ, we have ∆= o(1). • When p = 1, γ > 2s + 1, since ℓ< 2 and hence 2ℓ−q < 3 ≤γ, we have ∆= o(1). • When p = 1, s > 1, γ ≤2s + 1, or p = 1, s = 1, γ > 2, since 2ℓ−q ≤2 < γ, we have ∆= o(1). • When p = 1, s = 1, γ = 2, since 2ℓ−q ≤2 ≤γ, we have ∆= O((ln(d))−1). • When p = 0, since γ −2ℓ≥0, we have ∆= O((ln(d))−1). ■ Lemma D.19 (Verification of variance conditions for inner-product kernels: saturation case). Suppose τ < s ≤2τ. Suppose n ≍dγ, for γ ∈[p(s+1)+τ, p(s+1)+s+s/τ −1]. For any given ℓ≥0, if λ ≥d−ℓ, ℓ≤p + (γ −p(s + 1) + 1)/(2τ + 2); then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ) · nϵ−1 →0, N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 →0. Proof. From Lemma 21 in Zhang et al. (2024), we have N1(λ) ≍λ−1. Notice that we have 2(τ + 1)(γ −p) ≥ ps −1 p ≥1 2τ 2 + (τ −1) p = 0 > 0; Therefore, there exists a constant ϵ > 0 only depending on τ, s, and γ, such that we have N1(λ) · nϵ−1 →0. Denote q := ⌊ℓ⌋. From Lemma D.14, we further have N2,φ(λ) = Ω dq + λ−2d−q−1 . Hence, we have N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 = O (ln(d))3 n(λ2dq + d−q−1) = O (ln(d))3 nλ2dq + O (ln(d))3 dγ−q−1 . Denote ∆:= (ln(d))3 nλ2dq , ∆′ := (ln(d))3 dγ−q−1 . We have: • When p ≥1, since 2(τ + 1)[γ −2ℓ+ q] ≥2(τ + 1)[(γ −ℓ−1) + (q + 1 −ℓ)] ≥ ps −2 p ≥2 2(τ + 1)(τ −1) + 2[τs + s −1] p = 1 > 0, we have ∆= o(1). • When p = 0, since γ > 1, we have ∆′ = o(1). ■ 37 Lemma D.20 (Verification of variance conditions for inner-product kernels: misspecified case). Suppose n ≍dγ and 0 < s < 1, for γ ∈[p(s + 1), (p + 1)(s + 1)). For any given ℓ≥0, if λ ≥    d−ℓ p ≥1, 2ℓ≤max{2p + s, γ + p(1 −s)} d−ℓ p = 0, γ > s, 2ℓ≤γ d−ℓln(d) p = 0, γ ≤s, 2ℓ≤γ; then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ) · nϵ−1 →0, N 2 1 (λ) nN2,φ(λ) · ln(n)(ln λ−1)2 →0. Proof. When p ≥1, it is a direct result of step 2 (the verification of the second condition in (146) of Zhang et al. (2024)) in the proof of Theorem 3 in Zhang et al. (2024) and the fact that N2,φ(λ) ≍N2(λ). When p = 0, a similar argument as the proof for Lemma D.18 give the desired results. ■ D.4.2 Verification of bias conditions Lemma D.21 (Verification of bias conditions). Suppose 1 ≤s ≤τ. Suppose n ≍dγ, for γ ∈ [p(s + 1), (p + 1)(s + 1)). For any given ℓ≥0, if λ ≥    d−ℓ1 + ln2(d)1{γ = 2, s = 1} p ≥1, 2ℓ≤max{2p + 1, γ −(p + 1)(s −1)} d−ℓln2(d) γ ∈[1, s + 1), 2ℓ≤max{1, γ −(s −1)} d−ℓ γ ∈(0, 1), 2ℓ≤γ; then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ)M2 1,φ(λ) n2 ≪ M2,φ(λ) + σ2 n N2,φ(λ) , N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) . (106) Proof. When 1 ≤s ≤τ, from Lemma D.14, we have n M2,φ(λ) + σ2 n N2,φ(λ) = Ω dγ−s(q+1) + dq N1(λ)M2 1,φ(λ) n = O λ2(s−1)d−γ+qs + λ−1d−γ−(q+1)(s−1) N1(λ) ln(n)(ln λ−1)2 · ∞ X j=1 (λ)2λiφ2 λ(λi) λ + λi f 2 i = O (ln(d))3 · O λdmax{q(2−s),0} + λ−1d−s(q+1) , Denote I = λ2(s−1)d−γ+qs, II = λ−1d−γ−(q+1)(s−1), III = λdmax{q(2−s),0}(ln(d))3, and IV = λ−1d−s(q+1)(ln(d))3. For any p ≥0 and any s ≥1: • From Lemma D.18, we have IV ≪dγ−s(q+1). • When γ ≥1, we have γ ≥p + 1, and hence II ≪IV ≪dγ−s(q+1); when γ < 1, we have II ≪dq with q = 0. • When p ≥1 or γ ∈(s, s + 1), since −ℓs + qs ≤0, we have I/dγ−s(q+1) = O(d−2(γ−ℓ−s/2)) ≪1; when γ ∈(0, s], we have I = O(d−2sℓ+2ℓ−γ) = O(d−2sℓ) ≪dq with q = 0. 38 • When s ≥2, we have III ≪dq; when s < 2 and p = 0, we have III ≪dq; when s < 2 and p ≥1 and q ≥1, since γ −ℓ−s > min{(s + 1)q −ℓ, ps −1/2} > 0, we have III/dγ−s(q+1) = d−(γ−ℓ−s)−2(ℓ−q) ≪1 or III/dq ≪1; when s < 2 and p ≥1 and q = 0, we have III ≪dq. Combining all these, we get the desired results. ■ Lemma D.22. [Verification of bias conditions: saturation case] Suppose τ < s ≤2τ. Suppose n ≍dγ, for γ ∈[p(s + 1), (p + 1)(s + 1)). For any given ℓ≥0, if λ ≥    d−ℓ p ≥1, ℓ≤max{ℓ1, ℓ2, ℓ3} d−ℓln2(d) γ ∈[1, s + 1), ℓ≤max{ℓ1, ℓ2, ℓ3} d−ℓ γ ∈(0, 1), 2ℓ≤γ, where τ, ∆, ℓ1, ℓ2, and ℓ3 are given in Lemma D.16; then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ)M2 1,φ(λ) n2 ≪ M2,φ(λ) + σ2 n N2,φ(λ) , N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) . (107) Proof. When τ < s ≤2τ, from Lemma D.14, we have n M2,φ(λ) + σ2 n N2,φ(λ) = Ω λ2τdq(2τ−s) + dγ−s(q+1) + dq N1(λ)M2 1,φ(λ) n = O λ2(τ−1)d−γ+q(2τ−s) + λ−1d−γ−(q+1)(s−1) N1(λ) ln(n)(ln λ−1)2 · ∞ X j=1 (λ)2λiφ2 λ(λi) λ + λi f 2 i = O (ln(d))3 · O λdmax{q(2−s),0} + λ−1d−s(q+1) . Denote I′ = λ2(τ−1)d−γ+q(2τ−s), II = λ−1d−γ−(q+1)(s−1), III = λdmax{q(2−s),0}(ln(d))3, and IV = λ−1d−s(q+1)(ln(d))3. For any p ≥0 and any 1 ≤τ < s ≤2τ: • From Lemma D.18 and Lemma D.19, since N1(λ) · nϵ−1 →0, we have IV ≪dγ−s(q+1). • When γ ≥1, we have γ ≥p + 1, and hence II ≪IV ≪dγ−s(q+1); when γ < 1, we have II ≪dq with q = 0. • When p ≥1, since −ℓτ + qτ ≤0 and γ −ℓ−s/2 ≥max s(2p −1) 2 , (2τ + 1)(τ + ps) −(τ + 1)s + ps −1 2(τ + 1 , ps + s(τ + 1) 2τ −1 > 0, we have I′/dγ−s(q+1) ≪1; when p = 0, we have I′ = O(d−2τℓ+2ℓ−γ) ≪dq with q = 0. • When γ −p −ps ∈[0, τ] ∪[s + s/τ −1, s + 1], we have ℓ≤max{ℓ1, ℓ3}. Similar to the proof in Lemma D.21, we can show that III ≪dγ−s(q+1) + dq. • Finally, consider the case γ −p −ps ∈[τ, s + s/τ −1]. When s ≥2, we have III ≪dq; when s < 2, since s > 1, we have III/dq = λd−q(s−1) ≪0. Combining all these, we get the desired results. ■ 39 Lemma D.23 (Verification of bias conditions: misspecified case). Suppose 0 < s < 1. Suppose n ≍dγ, for γ ∈[p(s + 1), (p + 1)(s + 1)). Suppose one of the following holds: (1) τ = ∞. (2) s > 1/(2τ), (3) γ > ((2τ + 1)s)/(2τ(1 + s)). Suppose one of the following cases holds for λ = d−ℓor λ = d−ℓ(ln(d))2: (1) τ = ∞, p(s + 1) ≤γ ≤ps + p + s, ℓ∈[p, p + min{1/2, γs}] (2) τ = ∞, ps + p + s < γ < ps + p + s + 1, ℓ∈[p, min{(γ −(p + 1)(s −1))/2, γ(1 + s) −s(p + 1)}] (3) τ < ∞, p(s + 1) ≤γ ≤ps + p + s, ℓ= (γ + 2τp −sp −p)/(2τ) (4) τ < ∞, ps + p + s < γ < ps + p + s + 1, ℓ= p + s/(2τ). then there exists a constant ϵ > 0 only depending on s and γ, such that λ = λ(n, d) satisfies N1(λ) n ln(n)(ln λ−1)2 · ∞ X j=1 λ2λiφ2 λ(λi) λ + λi f 2 i ≪ M2,φ(λ) + σ2 n N2,φ(λ) ; n−2N1(λ) ∥fλ∥L∞+ n 1−s 2 +ϵ2 = o M2,φ(λ) + σ2 n N2,φ(λ) . Proof. When 0 < s < 1, from Lemma D.14, we have n M2,φ(λ) + σ2 n N2,φ(λ) = Ω dγ−s(p+1) + dp n−1N1(λ)n1−s = O λ−1d−γs N1(λ) ln(n)(ln λ−1)2 · ∞ X j=1 (λ)2λiφ2 λ(λi) λ + λi f 2 i = O (ln(d))3 · O λdmax{p(2−s),0} + λ−1d−s(p+1) , and the convergence rate of ∥fλ∥L∞can be attained similar to Lemma 25 in Zhang et al. (2024). Since τ ≥1, similar to the proof of Theorem 3 of Zhang et al. (2024), when 1/2 < s < 1, we have n−2N1(λ) ∥fλ∥L∞+ n 1−s 2 +ϵ2 = o M2,φ(λ) + σ2 n N2,φ(λ) , and when s ≤1/2, we have n−2N1(λ) ∥fλ∥2 L∞= o M2,φ(λ) + σ2 n N2,φ(λ) . Denote I = λ−1d−γs, II = λdp(2−s)(ln(d))3, and III = λ−1d−s(p+1)(ln(d))3. For any p ≥0 and any 0 < s < 1: • From Lemma D.20, we have III ≪dγ−s(p+1), • When γ ≤ps + p + s, we can show I ≪dp when: (1) p ≥1, or (2) p = 0 and s > 1/(2τ) > 0, or (3) τ = ∞, • When γ > ps + p + s, we can show I ≪dγ−s(p+1) holds if and only if τ = ∞or γ > (2τ + 1)s + 2τ(1 + s)p 2τ(1 + s) , τ = τ < ∞; 40 and the above inequality holds when (1) p > 0 or (2) p = 0, s > 1/(2τ) > 0, or (3) p = 0, γ > ((2τ + 1)s)/(2τ(1 + s)); • When γ ≤ps + p + s, since ℓ≥p > p −ps, we have II ≪dp; • When γ > ps + p + s, since ℓ≥p > p −ps, we have II ≪dγ−s(p+1). Combining all these, we get the desired results. ■ D.5 Final proof of Theorem 4.1 and Theorem 4.2 For each case, the proof can be done in the following steps: (i) When λ ≥λ⋆and s ≤2τ, where the definition of the balanced parameter λ⋆can be found in Corollary D.15 and Corollary D.16, we have M2,φ(λ⋆) + σ2 n N2,φ(λ⋆) = ΘP d−β⋆ · poly (ln(d)) M2,φ(λ) + σ2 n N2,φ(λ) = ΘP d−β · poly (ln(d)) , where d−β⋆is the desired convergence rate given in Theorem 4.1 or Theorem 4.2 and β ≤β∗. Similarly, when s > 2τ, by taking s = 2τ in Corollary D.16, we also have M2,φ(λ⋆) + σ2 n N2,φ(λ⋆) = ΘP d−β⋆ · poly (ln(d)) M2,φ(λ) + σ2 n N2,φ(λ) = ΘP d−β · poly (ln(d)) . (ii) When λ ≥λ⋆, from Lemma D.14, Lemma D.18, Lemma D.19, Lemma D.20, Lemma D.21, Lemma D.22, and Lemma D.23, we know that conditions in Theorem D.5, Theorem D.9, and Theorem D.10 are satisfied. Therefore, we have E ˆfλ⋆−f⋆ 2 L2 X = ΘP d−β⋆ · poly (ln(d)) E ˆfλ −f⋆ 2 L2 X = ΘP d−β · poly (ln(d)) . (iii) Finally, when s > τ, we can further show that: the convergence rates of the generalization error can not be faster than above for any choice of regularization parameter λ = λ(d, n) → 0. Notice that, when s ≥1, for any λ < λ⋆, from the monotonicity of Var(λ) (see, e.g., Li et al. (2024); Zhang et al. (2024)), we have E ˆfλ −f⋆ 2 L2 X ≥Var(λ) ≥Var(λ⋆) ≍E ˆfλ⋆−f⋆ 2 L2 X , and hence E ˆfλ −f⋆ 2 L2 X = ΩP d−β⋆ · poly (ln(d)) . E Auxiliary lemmas Proposition E.1. For any analytic filter function φλ, we have (z +λ)φλ(z) ≤4 and (z +λ)ψλ(z) ≤ 4λ. Proof. From (28), we have (z + λ)φλ(z) ≤2 max{z, λ}φλ(z) ≤2 max{1, C4} ≤4. From (27), we have (z + λ)ψλ(z) ≤2 max{z, λ}ψλ(z) ≤2 max{C2, 1}λ ≤4λ. ■ Lemma E.2. Let φλ be an analytic filter function defined in Definition C.1. Then, for any s ∈[0, 1], we have sup z∈[0,κ2] φλ(z)zs ≤4λs−1. 41 Proof. For any z ∈[0, κ2], from Proposition E.1, we have (z + λ)φλ(z) ≤4. Therefore, from Proposition B.3 in Li et al. (2024), we have φλ(z)zs ≤ 4zs z + λ ≤4λs−1. ■ Lemma E.3. Let ψλ be defined in Definition C.1. Then, for any s > 2τ, we have sup z∈[0,κ2] zsψ2 λ(z) ≤C2 2κ2(s−2τ)λ2τ. Proof. For any z, we have ψλ(z) ≤C2(z/λ)−τ1{z > λ} + 1{z ≤λ} ≤C2(z/λ)−τ, hence zsψ2 λ(z) ≤C2 2zsz−2τλ2τ ≤C2 2κ2(s−2τ)λ2τ. ■ E.1 Analytic functional calculus The “analytic functional argument” introduced in Li et al. (2024) is vital in our proof for Theorem 4.1. For readers’ convenience, we collect some of the main ingredients here, see Li et al. (2024) for details. Definition E.4. Let A be a linear operator on a Banach space X. The resolvent set ρ(A) is given by ρ(A) := {λ ∈C \| A −λ is invertible} , and we denote RA(λ) := (A −λ)−1. The spectrum of A is defined by σ(A) := C\ρ(A). A simple but key ingredient in the analytic functional calculus is the following resolvent identity: RA(λ) −RB(λ) = RA(λ)(B −A)RB(λ) = RB(λ)(B −A)RA(λ). (108) The resolvent allows us to define the value of f(A) in analog to the form of Cauchy integral formula, where A is an operator and f is an analytic function. The following two propositions are well-known results on operator calculus. Proposition E.5 (analytic functional calculus). Let A be an operator on a Hilbert space H and f be an analytic function defined on Df ⊂C. Let Γ be a contour contained in Df surrounding σ(A). Then, f(A) = 1 2πi I Γ f(z)(z −A)−1dz = −1 2πi I Γ f(z)RA(z)dz, (109) and it is independent of the choice of Γ. Now, let Γ be a contour contained in Df surrounding both σ(A) and σ(B). Using (108), we get f(A) −f(B) = −1 2πi I Γ f(z) [RA(z) −RB(z)] dz = 1 2πi I Γ RB(z)(A −B)RA(z)f(z)dz. (110) Proposition E.6 (Spectral mapping theorem). Let A be a bounded self-adjoint operator and f be a continuous function on σ(A). Then σ(f(A)) = {f(λ) \| λ ∈σ(A)} . (111) Consequently, ∥f(A)∥= supλ∈σ(A) \|f(λ)\| ≤∥f∥∞. 42 Let us define the contour Γλ considered in Li et al. (2024) by Γλ = Γλ,1 ∪Γλ,2 ∪Γλ,3 Γλ,1 = {x ± (x + η)i ∈C \| x ∈[−η, 0]} Γλ,2 = x ± (x + η)i ∈C \| x ∈(0, κ2) Γλ,3 = z ∈C \| z −κ2 = κ2 + η, Re(z) ≥κ2 , (112) where η = λ/2. Then, since T and TX are positive self-adjoint operators with ∥T∥, ∥TX∥≤κ2, we have σ(T), σ(TX) ⊂[0, κ2]. Therefore, Γλ is indeed a contour satisfying the requirement in Proposition E.5. Proposition E.7. Suppose that (45) in Assumption 3 holds. Suppose that λ = λ(n, d) satisfies v := N1(λ) n ln n = o(1). Then for any fixed δ ∈(0, 1), when n is sufficiently large, with probability at least 1 −δ, we have ∥T −1 2 λ (T −TX)T −1 2 λ ∥≤√v. T −1 2 λ T 1 2 Xλ 2 ≤2 (113) T 1 2 λ T −1 2 Xλ 2 ≤3. (114) Proof. These inequalities are direct results of (56), (58), and (59) in Zhang et al. (2024). ■ Proposition E.8 (Restate Proposition 4.13 in Li et al. (2024) with only the constant modified). When (113) holds, there is an absolute constant that for any z ∈Γλ, ∥T 1 2 λ (T −z)−1T 1 2 λ ∥≤C ∥T 1 2 λ (TX −z)−1T 1 2 λ ∥≤ √ 6C. 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4,443	Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow Chen-Hao Chao∗1,2 Chien Feng∗1 Wei-Fang Sun2 Cheng-Kuang Lee2 Simon See2 Chun-Yi Lee†1 1 Elsa Lab, National Tsing Hua University, Hsinchu City, Taiwan 2 NVIDIA AI Technology Center, NVIDIA Corporation, Santa Clara, CA, USA Abstract Existing Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) methods for continuous action spaces are typically formulated based on actor-critic frameworks and optimized through alternating steps of policy evaluation and policy improvement. In the policy evaluation steps, the critic is updated to capture the soft Q-function. In the policy improvement steps, the actor is adjusted in accordance with the updated soft Q-function. In this paper, we introduce a new MaxEnt RL framework modeled using Energy-Based Normalizing Flows (EBFlow). This framework integrates the policy evaluation steps and the policy improvement steps, resulting in a single objective training process. Our method enables the calculation of the soft value function used in the policy evaluation target without Monte Carlo approximation. Moreover, this design supports the modeling of multi-modal action distributions while facilitating efficient action sampling. To evaluate the performance of our method, we conducted experiments on the MuJoCo benchmark suite and a number of high-dimensional robotic tasks simulated by Omniverse Isaac Gym. The evaluation results demonstrate that our method achieves superior performance compared to widely-adopted representative baselines. 1 Introduction Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) [1–17] has emerged as a prominent method for modeling stochastic policies. Different from standard RL, MaxEnt RL integrates the entropy of policies as rewards, which leads to a balanced exploration-exploitation trade-off during training. This approach has demonstrated improved robustness both theoretically and empirically [17–19]. Building on this foundation, many studies leveraging MaxEnt RL have shown superior performance on continuous-control benchmark environments [8, 9] and real-world applications [20–22]. An active research domain in MaxEnt RL concentrates on the learning of the soft Q-function [8–15]. These methods follow the paradigm introduced in soft Q-learning (SQL) [8]. They parameterize the soft Q-function as an energy-based model [23] and optimize it based on the soft Bellman error [8] calculated from rewards and the soft value function. However, this approach presents two challenges. First, sampling from an energy-based model requires a costly Monte Carlo Markov Chain (MCMC) [24, 25] or variational inference [26] process, which can result in inefficient interactions with environments. Second, the calculation of the soft value function can involve computationally ∗Equal contribution. †Corresponding author. Email: cylee@cs.nthu.edu.tw 38th Conference on Neural Information Processing Systems (NeurIPS 2024). infeasible integration, which requires an effective approximation method. To tackle these issues, various methods [8–15] were proposed, all grounded in a common design philosophy. To address the first challenge, these methods suggest operating on an actor-critic framework and optimizing it through alternating steps of policy evaluation and policy improvement. For the second challenge, they resort to Monte Carlo methods to approximate the soft value function using sets of random samples. Although these two issues can be circumvented, these methods still have their drawbacks. The actor-critic design introduces an additional optimization process for training the actor, which may lead to optimization errors in practice [27]. Moreover, the results of Monte Carlo approximation may be susceptible to estimation errors and variances when there are an insufficient number of samples [28–30]. Instead of using energy-based models to represent MaxEnt RL frameworks, this paper investigates an alternative method employing normalizing flows (i.e., flow-based models), which offer solutions to the aforementioned challenges. Our framework is inspired by the recently introduced Energy-Based Normalizing Flows (EBFlow) [31]. This design facilitates the derivation of an energy function from a flow-based model while supporting efficient sampling, which enables a unified representation of both the soft Q-function and its corresponding action sampling process. This feature allows the integration of the policy evaluation and policy improvement steps into a single objective training process. In addition, the probability density functions (pdf) of flow-based models can be calculated efficiently without approximation. This characteristic permits the derivation of an exact representation for the soft value function. Our experimental results demonstrate that the proposed framework exhibits superior performance on the commonly adopted MuJoCo benchmark [32, 33]. Furthermore, the evaluation results on the Omniverse Isaac Gym environments [34] indicate that our framework excels in performing challenging robotic tasks that simulate real-world scenarios. 2 Background and Related Works In this section, we walk through the background material and the related works. We introduce the objective of MaxEnt RL in Section 2.1, describe existing actor-critic frameworks and soft value estimation methods in Section 2.2, and elaborate on the formulation of Energy-Based Normalizing Flow (EBFlow) in Section 2.3. 2.1 Maximum Entropy Reinforcement Learning In this paper, we consider a Markov Decision Process (MDP) defined as a tuple (S, A, pT , R, γ, p0), where S is a continuous state space, A is a continuous action space, pT : S × S × A →R≥0 is the pdf of a next state st+1 given a current state st and a current action at at timestep t, R : S × A →R is the reward function, 0 < γ < 1 is the discount factor, and p0 is the pdf of the initial state s0. We adopt rt to denote R(st, at), and use ρπ(st, at) to represent the state-action marginals of the trajectory distribution induced by a policy π(at\|st) [8]. Standard RL defines the objective as π∗= argmaxπ P t E(st,at)∼ρπ[rt] and has at least one deterministic optimal policy [35, 36]. In contrast, MaxEnt RL [4] augments the standard RL objective with the entropy of a policy at each visited state st. The objective of MaxEnt RL is written as follows: π∗ MaxEnt = argmax π X t E(st,at)∼ρπ [rt + αH(π( · \|st))] , (1) where H(π( · \|st)) ≜Ea∼π(·\|st)[−log π(a\|st)] and α ∈R>0 is a temperature parameter for determining the relative importance of the entropy term against the reward. An extension of Eq. (1) defined with γ is discussed in [8]. To obtain π∗ MaxEnt described in Eq. (1), the authors in [8] proposed to minimize the soft Bellman error for all states and actions. The solution can be expressed using the optimal soft Q-function Q∗ soft : S × A →R and soft value function V ∗ soft : S →R as follows: π∗ MaxEnt(at\|st) = exp 1 α(Q∗ soft(st, at) −V ∗ soft(st)) , where (2) Q∗ soft(st, at) = rt+γEst+1∼pT [V ∗ soft(st+1)] , V ∗ soft(st) = α log Z exp 1 αQ∗ soft(st, a) da. (3) In practice, a policy can be modeled as πθ(at\|st) = exp( 1 α(Qθ(st, at) −Vθ(st)) with parameter θ, where the soft Q-function and the soft value function are expressed as Qθ(st, at) and Vθ(st) = 2 α log R exp 1 αQθ(st, at) dat, respectively. Given an experience reply buffer D that stores transition tuples (st, at, rt, st+1), the training objective of Qθ (which can then be used to derive Vθ and πθ) can be written as the following equation according to the soft Bellman errors: L(θ) = E(st,at,rt,st+1)∼D 1 2 (Qθ(st, at) −(rt + γVθ(st+1)))2 . (4) Nonetheless, directly using the objective in Eq. (4) presents challenges for two reasons. First, drawing samples from an energy-based model (i.e., πθ(at\|st) ∝exp(Qθ(st, at)/α)) requires a costly MCMC or variational inference process [26, 37], which makes the interaction with the environment inefficient. Second, the calculation of the soft value function involves integration, which require stochastic approximation methods [28–30] to accomplish. To address these issues, the previous MaxEnt RL methods [8–14] adopted actor-critic frameworks and introduced a number of techniques to estimate the soft value function. These methods are discussed in the next subsection. 2.2 Actor-Critic Frameworks and Soft Value Estimation in MaxEnt RL Previous MaxEnt RL methods [8–15] employed actor-critic frameworks, in which the critic aims to capture the soft Q-function, while the actor learns to sample actions based on this soft Q-function. Available choices for modeling the actor include Gaussian models [9], Gaussian mixture models [38], variational autoencoders (VAE) [15, 13, 39], normalizing flows [10, 11], and amortized SVGD (A-SVGD) [8, 40], all of which support efficient sampling. The separation of the actor and the critic prevents the need for costly MCMC processes during sampling. However, this design induces additional training steps aimed to minimize the discrepancy between them. Let πθ(at\|st) ∝exp( 1 αQθ(st, at)) and πϕ(at\|st) denote the pdfs defined through the critic and the actor, respectively. The objective of this additional training process is formulated according to the reverse KL divergence DKL[πϕ( · \|st)\|\|πθ( · \|st)] between πϕ and πθ, and is typically reduced as follows [9]: L(ϕ) = Est∼D −Eat∼πϕ[Qθ(st, at) −α log πϕ(at\|st)] . (5) The optimization processes defined by the objective functions L(θ) and L(ϕ) in Eqs. (4) and (5) are known as the policy evaluation steps and policy improvement steps [9], respectively. Through alternating updates according to ∇θL(θ) and ∇ϕL(ϕ), the critic learns directly from the reward signals to estimate the soft Q-function, while the actor learns to draw samples based on the distribution defined by the critic. Although the introduction of the actor enhances sampling efficiency, calculating the soft value function in Eq. (3) still requires Monte Carlo approximations for the computationally infeasible integration operation. Prior soft value estimation methods can be categorized into two groups: soft value estimation in Soft Q-Learning (SQL) and that in Soft Actor-Critic (SAC), with the former yielding a larger estimate than the latter, derived from Jensen’s inequality (i.e., Proposition A.1 in Appendix A.1). These two soft value estimation methods are discussed in the following paragraphs. Soft Value Estimation in SQL. Soft Q-Learning [8] leverages importance sampling to convert the integration in Eq. (3) into an expectation, which can be estimated using a set of independent and identically distributed (i.i.d.) samples. To ensure the estimation variance is small, the authors in [8] proposed to utilize samples drawn from πϕ. Let {a(i)}M i=1 be a set of M samples drawn from πϕ. The soft value function is approximated based on the following formula: Vθ(st) = α log Z exp (Qθ(st, a)/α) da = α log Z πϕ(a\|st)exp (Qθ(st, a)/α) πϕ(a\|st) da = α log Ea∼πϕ exp (Qθ(st, a)/α) πϕ(a\|st) ≈α log 1 M M X i=1 exp Qθ(st, a(i))/α πϕ(a(i)\|st) ! . (6) Eq. (6) has the least variance when πϕ(· \|st) ∝exp(Qθ(st, ·)/α) [29]. In addition, as M →∞, the law of large numbers ensures that this estimation converge to Vθ(st) [41]. Soft Value Estimation in SAC. Soft Actor-Critic [9] and its variants [10, 11, 13, 14, 12] reformulated the soft value function Vθ(st) = α log R exp (Qθ(st, a)/α) da as its equivalent form Ea∼πθ[Qθ(st, a) −α log πθ(a\|st)] based on the relationship that πθ(a\|st) = exp( 1 α(Qθ(st, a)) − 3 Vθ(st)). By assuming that the policy improvement loss L(ϕ) is small (i.e., πθ ≈πϕ), the soft value function Vθ can be estimated as follows: Vθ(st) = Ea∼πθ[Qθ(st, a) −α log πθ(a\|st)] ≈Ea∼πϕ[Qθ(st, a) −α log πϕ(a\|st)] ≈1 M M X i=1 Qθ(st, a(i)) −α log πϕ(a(i)\|st) . (7) An inherent drawback of the estimation in Eq. (7) is its reliance on the assumption πϕ ≈πθ. In addition, the second approximation involves Monte Carlo estimation with M samples {a(i)}M i=1, where the computational cost increases with the number of samples M. 2.3 Energy-Based Normalizing Flows Normalizing flows (i.e., flow-based models) are universal representations for pdf [42]. Given input data x ∈RD, a latent variable z ∈RD with prior pdf pz, and an invertible function gθ = gL θ ◦· · ·◦g1 θ modeled as a neural network with L layers, where gi θ : RD →RD, ∀i ∈{1, · · · , L}. According to the change of variable theorem and the distributive property of the determinant operation, a parameterized pdf pθ can be described as follows: pθ(x) = pz (gθ(x)) L Y i=1 det Jgi θ(xi−1) , (8) where x0 ≜x is the input, xi = gi θ ◦· · · ◦g1 θ(x) is the output of the i-th layer, and Jgi θ(xi−1) ≜ ∂ ∂xi−1 gi θ(xi−1) represents the Jacobian of the i-th layer of gθ with respect to xi−1. To draw samples from pθ, one can first sample z from pz and then derive g−1 θ (z). To facilitate efficient computation of the pdf and the inverse of gθ, one can adopt existing architectural designs [43–48] for gθ. Popular examples involve autoregressive layers [43–45] and coupling layers [46–48], which utilizes specially designed architectures to speed up the calculation. Energy-Based Normalizing Flows (EBFlow) [31] were recently introduced to reinterpret flow-based models as energy-based models. In contrast to traditional normalizing flow research [46, 47, 49, 50] that focuses on the use of effective non-linearities, EBFlow emphasizes the use of both linear and non-linear transformations in the invertible transformation gθ. Such a concept was inspired by the development of normalizing flows with convolution layers [48, 51–54] or fully-connected layers [55, 56], linear independent component analysis (ICA) models [57, 58], as well as energy-based training techniques [58–60]. Let Sl = {i \| gi θ is linear} and Sn = {i \| gi θ is non-linear} represent the sets of indices of the linear and non-linear transformations in gθ, respectively. As shown in [31], the Jacobian determinant product in Eq. (8) can be decomposed according to Sn and Sl. This decomposition allows a flow-based model to be reinterpreted as an energy-based model, as illustrated in the following equation: pθ(x) = pz (gθ(x)) Y i∈Sn det Jgi θ(xi−1) \| {z } Unnormalized Density Y i∈Sl det Jgi θ \| {z } Const. ≜exp (−Eθ(x)) \| {z } Unnormalized Density Z−1 θ . \| {z } Const. (9) In EBFlow, the energy function Eθ(x) is defined as −log(pz (gθ(x)) Q i∈Sn \| det(Jgi θ(xi−1))\|) and the normalizing constant Zθ = R exp(−Eθ(x))dx = Q i∈Sl \| det Jgi θ\|−1 is independent of x. The input-independence of Zθ holds since gi θ is either a first-degree or zero-degree polynomial for any i ∈Sl, and thus its Jacobian is a constant to xi−1. This technique was originally proposed to reduce the training cost of maximum likelihood estimation for normalizing flows. However, we discovered that EBFlow is ideal for MaxEnt RL. Its unique capability to represent a parametric energy function with an associated sampler g−1 θ , and to calculate a normalizing constant Zθ without integration are able to address the challenges mentioned in Section 2.2. We discuss our insights in the next section. 3 Methodology In this section, we introduce our proposed MaxEnt RL framework modeled using EBFlow. In Section 3.1, we describe the formulation and discuss its training and inference processes. In Section 3.2, 4 we present two techniques for improving the training of our framework. Ultimately, in Section 3.3, we offer an algorithm summary. 3.1 MaxEnt RL via EBFlow We propose a new framework for modeling MaxEnt RL using EBFlow, which we call MEow. This framework possesses several unique features. First, as EBFlow enables simultaneous modeling of an unnormalized density and its sampler, MEow can unify the actor and the critic previously separated in MaxEnt RL frameworks. This feature facilitates the integration of policy improexvement steps with policy evaluation steps, and results in a single objective training process. Second, the normalizing constant of EBFlow is expressed in closed form, which enables the calculation of the soft value function without resorting to the approximation methods mentioned in Eqs. (6) and (7). Third, given that normalizing flow is a universal approximator for probability density functions, our policy’s expressiveness is not constrained, and can model multi-modal action distributions. In MEow, the policy is described as a state-conditioned EBFlow, with its pdf presented as follows: πθ(at\|st) = pz (gθ(at\|st)) Y i∈Sn det Jgi θ(ai−1 t \|st) \| {z } Unnormalized Density Y i∈Sl det(Jgi θ(st)) \| {z } Norm. Const. ≜exp 1 αQθ(st, at) \| {z } Unnormalized Density exp −1 αVθ(st) \| {z } Norm. Const. , (10) where the soft Q-function and the soft value function are selected as follows: Qθ(st, at) ≜α log pz (gθ(at\|st)) Y i∈Sn det Jgi θ(ai−1 t \|st) , Vθ(st) ≜−α log Y i∈Sl det(Jgi θ(st)) . (11) Such a selection satisfies Vθ(st) = α log R exp(Qθ(st, a)/α)da based on the property of EBFlow. In addition, both Qθ and Vθ have a common output codomain R, which enables them to learn to output arbitrary real values. These properties are validated in Proposition 3.1, with the proof provided in Appendix A.2. The training and inference processes of MEow are summarized as follows. Proposition 3.1. Eq. (11) satisfies the following statements: (1) Given that the Jacobian of gθ is non-singular, Vθ(st) ∈R and Qθ(st, at) ∈R, ∀at ∈A, ∀st ∈S. (2) Vθ(st) = α log R exp (Qθ(st, a)/α) da. Training. With Qθ and Vθ defined in Eq. (11), the loss L(θ) in Eq. (4) can be calculated without using Monte Carlo approximation of the soft value function target. Compared to the previous MaxEnt RL frameworks that rely on Monte Carlo estimation (i.e., Eqs. (6) and (7)), our framework offers the advantage of avoiding the errors induced by the approximation. In addition, MEow employs a unified policy rather than two separate roles (i.e., the actor and the critic), which eliminates the need for minimizing an additional policy improvement loss L(ϕ) to bridge the gap between πθ and πϕ. This simplifies the training process of MaxEnt RL, and obviates the requirement of balancing between the two optimization loops. Inference. The sampling process of πθ can be efficiently performed by deriving the inverse of gθ, as supported by several normalizing flow architectures [43–48]. In addition, unlike previous actor-critic frameworks susceptible to discrepancies between πθ and πϕ, the distribution established via g−1 θ (z\|st), where z ∼pz, is consistently aligned with the pdf defined by Qθ. As a result, the actions taken by MEow can precisely reflect the learned soft Q-function. 3.2 Techniques for Improving the Training and Inference Processes of MEow In this subsection, we introduce a number of training and inference techniques aimed at improving MEow while preserving its key features discussed in the previous subsection. For clarity, we refer to the MEow framework introduced in the last section as ‘MEow (Vanilla)’. 5 (b) (a) 5.0e13 4.0e13 3.0e13 2.0e13 1.0e13 0.0 0.0 0.2 1.0 0.4 0.6 0.8 Steps (1e5) 0.0 0.2 1.0 0.4 0.6 0.8 1.0e-1 1.0e0 1.0e-2 1.0e-3 1.0e-4 1.0e-5 Steps (1e5) Jacobian determinant product of the non-linear transformations Jacobian determinant product of the linear transformations +LRS Vanilla Figure 1: The Jacobian determinant products for (a) the non-linear and (b) the linear transformations, evaluated during training in the Hopper-v4 environment. Subfigure (b) is presented on a log scale for better visualization. This experiment adopt the affine coupling layers [47] as the nonlinear transformations. Learnable Reward Shifting (LRS). Reward shifting [61–65] is a technique for shaping the reward function. This technique enhances the learning process by incorporating a shifting term in the reward function, which leads to a shifted optimal soft Q-function in MaxEnt RL. Inspired by this, this work proposes modeling a reward shifting function bθ : S →R with a neural network to enable the automatic learning of a reward shifting term. For notational simplicity, the parameters are denoted using θ, and the details of the architecture are presented in Appendix A.5.1. The soft Q-function Qb θ augmented by bθ is defined as follows: Qb θ(st, at) = Qθ(st, at) + bθ(st). (12) The introduction of Qb θ results in a corresponding shifted soft value function V b θ (st) ≜ α log R exp(Qb θ(st, a)/α)da = Vθ(st) + bθ(st) (i.e., Proposition A.3 in Appendix A.2), which can be calculated without Monte Carlo estimation. Moreover, with the incorporation of bθ, the policy πθ remains invariant since exp( 1 α(Qb θ(st, at) −V b θ (st))) = exp( 1 α((Qθ(st, at) + bθ(st)) −(Vθ(st) + bθ(st)))) = exp( 1 α(Qθ(st, at) −Vθ(st))), which allows the use of g−1 θ for efficiently sampling actions. As evidenced in Fig. 1, this method effectively addresses the issues of the significant growth and decay of Jacobian determinants of gθ (discussed in Appendix A.3). In Section 4.4, we further demonstrate that the performance of MEow can be significantly improved through this technique. Shifting-Based Clipped Double Q-Learning (SCDQ). As observed in [66], the overestimation of value functions often occurs in training. To address this issue, the authors in [66] propose clipped double Q-learning, which employs two separate Q-functions and uses the one with the smaller output to estimate the value function during training. This technique is also used in MaxEnt RL frameworks [9–13]. Inspired by this and our proposed learnable reward shifting, we further propose a shifting-based method that adopts two learnable reward shifting functions, b(1) θ and b(2) θ , without duplicating the soft Q-function Qθ and soft value function Vθ defined by gθ. The soft Q-functions Q(1) θ and Q(2) θ with corresponding learnable reward shifting functions b(1) θ and b(2) θ can be obtained using Eq. (12), while the soft value function V clip θ is written as the following formula: V clip θ (st) = min Vθ(st) + b(1) θ (st), Vθ(st) + b(2) θ (st) = Vθ(st)+min b(1) θ (st), b(2) θ (st) . (13) This design also prevents the production of two policies in MEow, as having two policies can complicate the inference procedure. In our ablation analysis presented in Section 4.4, we demonstrate that this technique can effectively improve the training process of MEow. Deterministic Policy for Inference. As observed in [8], deterministic actors typically performed better as compared to its stochastic variant during the inference time. Such a problem can be formalized as finding an action a that maximizes Q(st, a) for a given st. Since A is a continuous space, finding such a value would require extensive calculation. In the MEow framework, this value can be derived by making assumptions on the model architecture construction. Our key observation is that if the Jacobian determinants of the non-linearities (i.e., gi θ ∈Sn) are constants with respect to its inputs, and that argmaxz pz(z) can be directly obtained, then the action a that maximizes Q(st, a) can be efficiently derived according to the following proposition. Proposition 3.2. Given that \| det(Jgi θ(ai−1\|st))\| is a constant with respect to ai−1, then g−1 θ (argmaxz pz(z)\|st) = argmaxa Qθ(st, a). The proof is provided in Appendix A.2. It is important to note that gi θ can still be a non-linear transformation, given that \| det(Jgi θ(ai−1\|st))\| is a constant. To construct such a model, a Gaussian prior with the additive coupling transformations [46] can be used as non-linearities. Under such a design, an action can be derived by calculating g−1 θ (µ\|st), where µ represents the mean of the 6 Algorithm 1 Pseudo Code of the Training Process of MEow Input: Learnable parameters θ and shadow parameters θ′. Target smoothing factor τ. Learning rate β. Neural networks gθ(· \| ·), b(1) θ (·), and b(2) θ (·). Temperature parameter α. Discount factor γ. 1: for each training step do 2: ▷Extend the Replay Buffer. 3: at = g−1 θ (z\|st), z ∼pz(·). 4: st+1 ∼pT (·\|st, at). 5: D ←D ∪{(st, at, rt, st+1)}. 6: ▷Update Policy. 7: (st, at, rt, st+1) ∼D. 8: Qθ(st, at) = α log(pz (gθ(at\|st)) Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\|). ▷Eq. (11) 9: Vθ′(st+1) = −α log Q i∈Sl \| det(Jgi θ′ (st+1))\|. ▷Eq. (11) 10: Q(1) θ (st, at) = Qθ(st, at) + b(1) θ (st) and Q(2) θ (st, at) = Qθ(st, at) + b(2) θ (st). ▷Eq. (12) 11: V clip θ′ (st+1) = Vθ′(st+1) + min b(1) θ′ (st+1), b(2) θ′ (st+1) . ▷Eq. (13) 12: L(θ) = 1 2(Q(1) θ (st, at) −(rt + γV clip θ′ (st+1)))2 + 1 2(Q(2) θ (st, at) −(rt + γV clip θ′ (st+1)))2.▷Eq. (4) 13: θ ←θ + β∇θL(θ). 14: θ′ ←(1 −τ)θ′ + τθ. 15: end for Gaussian distribution. We elaborate on our model architecture design in Appendix A.5.1, and provide a performance comparison between MEow evaluated using a stochastic policy (i.e., at ∼πθ(· \|st)) and a deterministic policy (i.e., at = argmaxa Qθ(st, a)) in Section 4.4. 3.3 Algorithm Summary We summarize the training process of MEow in Algorithm 1. The algorithm integrates the policy evaluation steps with the policy improvement steps, resulting in a single loss training process. This design differs from previous actor-critic frameworks, which typically perform two consecutive updates in each training step. In Algorithm 1, the learning rate is denoted as β. A set of shadow parameters θ′ is maintained for calculating the delayed target values [67], and is updated according to the Polyak averaging [68] of θ, i.e., θ′ ←(1 −τ)θ′ + τθ, where τ is the target smoothing factor. 4 Experiments In the following sections, we first present an intuitive example of MEow trained in a two-dimensional multi-goal environment [8] in Section 4.1. We then compare MEow’s performance against several continuous-action RL baselines in five MuJoCo environments [32, 33] in Section 4.2. Next, in Section 4.3, we evaluate MEow’s performance on a number of Omniverse Isaac Gym environments [34] simulated based on real-world robotic application scenarios. Lastly, in Section 4.4, we provide an ablation analysis to inspect the effectiveness of each proposed technique. Among all experiments, we maintain the same model architecture, while adjusting inputs and outputs according to the state space and action space for each environment. We construct gθ using the additive coupling layers [46] with element-wise linear transformations, utilize a unit Gaussian as pz, and model the learnable adaptive reward shifting functions bθ as multi-layer perceptrons (MLPs). For detailed descriptions of the experimental setups, please refer to Appendix A.5. 4.1 Evaluation on a Multi-Goal Environment In this subsection, we present an example of MEow trained in a two-dimensional multi-goal environment [8]. The environment involves four goals, indicated by the red dots in Fig. 2 (a). The reward function is defined by the negative Euclidean distance from each state to the nearest goal, and the corresponding reward landscape is depicted using contours in Fig. 2 (a). The gradient map in Fig. 2 (a) represents the soft value function predicted by our model. The blue lines extending from the center represent the trajectories produced using our policy. 7 7500 6000 4500 3000 1500 0 -1500 0.0 1.0 2.0 3.0 4.0 Ant-v4 HalfCheetah-v4 Hopper-v4 6000 5000 4000 3000 2000 1000 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 Walker2d-v4 Humanoid-v4 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 MEow SAC SQL PPO TD3 DDPG Figure 3: The results in terms of total returns versus the number of training steps evaluated on five MuJoCo environments. Each curve represents the mean performance, with shaded areas indicating the 95% confidence intervals, derived from five independent runs with different seeds. Estimation Error M (b) 0.0 2.0 4.0 6.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 -6.0 -4.0 -2.0 (a) -6.0 -4.0 -8.0 -2.0 8,000 10,000 2,000 4,000 6,000 19.900 20.700 20.300 \|(I) - (II)\| 21.645 21.630 21.615 \|(I) - (III)\| 8,000 10,000 2,000 4,000 6,000 Figure 2: (a) The soft value function and the trajectories generated using our method on the multi-goal environment. (b) The estimation error evaluated at the initial state under different choices of M. As illustrated in Fig. 2 (a), our model’s soft value function predicts higher values around the goals, suggesting successful learning of the goal positions through rewards. In addition, the trajectories demonstrate our agent’s correct transitions towards the goals, which validates the effectiveness of our learned policy. To illustrate the potential impact of approximation errors that might emerge when employing previous soft value estimation methods, we compare three calculation methods for the soft value function: (I) Our approach (i.e., Eq. (11)): Vθ(st), (II) SQL-like (i.e., Eq. (6)): α log( 1 M PM i=1 exp(Qθ(st,a(i))/α) πϕ(a(i)\|st) ), and (III) SAC-like (i.e., Eq. (7)): 1 M PM i=1(Qθ(st, a(i)) −α log πϕ(a(i)\|st)), where {a(i)}M i=1 is sampled from πϕ. The approximation errors of the soft value functions at the initial state are calculated using the Euclidean distances between (I) and (II), and between (I) and (III), for various values of M. As depicted in Fig. 2 (b), the blue line and the orange line decreases slowly with respect to M. These results suggest that Monte Carlo estimation converges slowly, making approximation methods such as Eqs. (6) and (7) challenging to achieve accurate predictions. 4.2 Performance Comparison on the MuJoCo Environments In this experiment, we compare MEow with several commonly-used continuous control algorithms on five MuJoCo environments [32] from Gymnasium [33]. The baseline algorithms include SQL [8], SAC [9], deep deterministic policy gradient (DDPG) [69], twin delayed deep deterministic policy gradient (TD3) [66], and proximal policy optimization (PPO) [70]. The results for SAC, DDPG, TD3, and PPO were reproduced using Stable Baseline 3 (SB3) [71], utilizing SB3’s refined hyperparameters. The results for SQL were reproduced using our own implementation, as SB3 does not support SQL and the official code is not reproducible. Our implementation adheres to SQL’s original paper. Each method is trained independently under five different random seeds, and the evaluation curves for each environment are presented in the form of the means and the corresponding confidence intervals. As depicted in Fig. 3, MEow performs comparably to SAC and outperforms the other baseline algorithms in most of the environments. Furthermore, in environments with larger action and state dimensionalities, such as ‘Ant-v4’ and ‘Humanoid-v4’, MEow offers performance improvements over SAC and exhibits fewer spikes in the evaluation curves. These results suggest that MEow is capable of performing high-dimensional tasks with stability. To further investigate the performance difference between MEow and SAC, we provide a thorough comparison between MEow, SAC [9], Flow-SAC [10, 11], and their variants in Appendix A.4.2. The results indicate that the training process involving policy evaluation and policy improvement steps may be inferior to our proposed training process with a single objective. In the next subsection, we provide a performance examination using the simulation environments from the Omniverse Isaac Gym [34]. 8 FrankaCabinet Ant (Isaac) MEow SAC Humanoid (Isaac) Ingenuity 5000 1000 -1000 0.0 2.5 7.5 10.0 5.0 3000 9000 7000 4000 1000 -500 0.0 2.5 7.5 10.0 5.0 2500 5500 4500 1500 0 0.0 1.25 3.75 5.0 2.5 3000 6000 3000 1000 0.0 2.5 7.5 10.0 5.0 2000 4000 0 Steps (1e5) Steps (1e5) AllegroHand 350 -50 0.0 2.5 7.5 10.0 5.0 150 750 550 -250 Steps (1e5) ANYmal 45 15 0.0 2.5 7.5 10.0 5.0 30 60 0 Steps (1e5) Steps (1e5) Return Return Steps (1e5) (N=128) (N=128) (N=128) (N=128) (N=512) (N=512) Figure 4: A comparison on six Isaac Gym environments. Each curve represents the mean performance of five runs, with shaded areas indicating the 95% confidence intervals. ‘Steps’ in the x-axis represents the number of training steps, each of which consists of N parallelizable interactions with the environments. AllegroHand FrankaCabinet Ant (Isaac) ANYmal Humanoid (Isaac) Ingenuity state dim. = 48 action dim. = 12 state dim. = 72 action dim. = 16 state dim. = 23 action dim. = 9 state dim. = 60 action dim. = 8 state dim. = 108 action dim. = 21 state dim. = 13 action dim. = 6 Figure 5: A demonstration of the six Isaac Gym environments introduced in Section 4.3. The dimensionalities of the state and action for each environment are denoted below each subfigure. 4.3 Performance Comparison on the Omniverse Issac Gym Environments In this subsection, we examine the performance of MEow on a variety of robotic tasks simulated by Omniverse Isaac Gym [34], a GPU-based physics simulation platform. In addition to ‘Ant’ and ‘Humanoid’, we employ four additional tasks: ‘Ingenuity’, ‘ANYmal’, ‘AllegroHand’, and ‘FrankaCabinet’. All of them are designed based on real-world robotic application scenarios. ‘Ingenuity’ and ‘ANYmal’ are locomotion environments inspired by NASA’s Ingenuity helicopter and ANYbotics’ industrial maintenance robots, respectively. On the other hand, ‘AllegroHand’ and ‘FrankaCabinet’ focus on executing specialized manipulative tasks with robotic hands and arms, respectively. A demonstration of these tasks is illustrated in Fig. 5. In this experimental comparison, we adopt SAC as a baseline due to its excellent performance in the MuJoCo environments. The evaluation results are presented in Fig. 4. The results demonstrate that MEow exhibits superior performance on ‘Ant (Isaac)’ and ‘Humanoid (Isaac)’. In addition, MEow consistently outperforms SAC across the four robotic environments (i.e., ‘Ingenuity’, ‘ANYmal’, ‘AllegroHand’, and ‘FrankaCabinet’), indicating that our algorithm possesses the ability to perform challenging robotic tasks simulated based on real-world application scenarios. 4.4 Ablation Analysis In this subsection, we provide an ablation analysis to examine the effectiveness of each technique introduced in Section 3.2. Training Techniques. Fig. 6 compares the performance of three variants of MEow: ‘MEow (Vanilla)’, ‘MEow (+LRS)’, and ‘MEow (+LRS & SCDQ)’, across five MuJoCo environments. The results show that ‘MEow (Vanilla)’ consistently underperforms, with its total returns demonstrating negligible or no growth throughout the training period. In contrast, the variants incorporating translation functions demonstrate significant performance enhancements. This observation highlights the importance of including bθ in the model design. In addition, the comparison between ‘MEow (+LRS)’ and ‘MEow (+LRS & SCDQ)’ suggests that our reformulated approach to clipped double Q-learning [66] improves the final performance by a noticeable margin. Inference Technique. Fig. 7 compares the performance of two variants of MEow: ‘MEow (Stochastic)’ and ‘MEow (Deterministic)’. The former samples action based on at ∼πθ(· \|st) while the latter derive action according to at = argmaxa Qθ(st, a) = g−1 θ (µ\|st). As shown in the figure, MEow with a deterministic policy outperforms its stochastic variant, suggesting that a deterministic policy may be more effective for MEow’s inference. 9 Ant-v4 HalfCheetah-v4 Hopper-v4 Walker2d-v4 Humanoid-v4 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return MEow (+LRS & SCDQ) MEow (+LRS) MEow (Vanilla) 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 7500 6000 4500 3000 1500 0 0.0 1.0 2.0 3.0 4.0 Figure 6: The performance comparison of MEow’s variants (i.e., ‘MEow (Vanilla)’, ‘MEow (+LRS)’, and ‘MEow (+LRS & SCDQ)’) on five MuJoCo environments. Each curve represents the mean performance of five runs, with shaded areas indicating the 95% confidence intervals. Ant-v4 HalfCheetah-v4 Hopper-v4 Walker2d-v4 Humanoid-v4 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return MEow (Deterministic) MEow (Stochastic) 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 7500 6000 4500 3000 1500 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 Figure 7: Performance comparison between MEow with a deterministic policy and MEow with a stochastic policy on five MuJoCo environments. Each curve represents the mean performance of five runs, with shaded areas indicating the 95% confidence intervals. 5 Conclusion In this paper, we introduce MEow, a unified MaxEnt RL framework that facilitates exact soft value calculations without the need for Monte Carlo estimation. We demonstrate that MEow can be optimized using a single objective function, which streamlines the training process. To further enhance MEow’s performance, we incorporate two techniques, learnable reward shifting and shiftingbased clipped double Q-learning, into the design. We examine the effectiveness of MEow via experiments conducted in five MoJoCo environments and six robotic tasks simulated by Omniverse Isaac Gym. The results validate the superior performance of MEow compared to existing approaches. Limitations and Discussions As discussed in Section 3.2, deterministic policies typically offer better performance compared to their stochastic counterparts. Although our implementation of MEow supports deterministic inference, this capability is based on the assumptions that the Jacobian determinants of the non-linear transformations are constants with respect to their inputs, and that argmaxz pz(z) can be efficiently derived. These assumptions may not hold for certain types of flow-based models. Therefore, exploring effective architectural choices for MEow represents a promising direction for further investigation. On the other hand, the training speed of MEow is around 2.3× slower than that of SAC, even though updates according to L(ϕ) are bypassed in MEow. According to our experimental observations, the computational bottleneck of MEow may lie in the inference speed of the flow-based model during interactions with environments. While this speed is significantly faster than many iterative methods, such as MCMC or variational inference, it is still slower compared to the inference speed of Gaussian models. As a result, enhancing the inference speed of flow-based models represents a potential avenue for further improving the training efficiency of MEow. Finally, our hyperparameter sensitivity analysis, as presented in A.4.5, indicates that our current approach requires different values of τ to achieve optimal performance. Since hyperparameter tuning often demands significant computational resources, establishing a more generalized parameter setting or developing an automatic tuning mechanism for τ presents an important direction for future exploration. 10 Acknowledgement The authors gratefully acknowledge the support from the National Science and Technology Council (NSTC) in Taiwan under grant numbers MOST 111-2223-E-002-011-MY3, NSTC 113-2221-E-002212-MY3, and NSTC 113-2640-E-002-003. The authors would also like to express their appreciation for the computational resources from NVIDIA Corporation and NVIDIA AI Technology Center (NVAITC) used in this work. Furthermore, the authors extend their gratitude to the National Center for High-Performance Computing (NCHC) for providing the necessary computational and storage resources. References [1] H J Kappen. Path Integrals and Symmetry Breaking for Optimal Control Theory. Journal of Statistical Mechanics: Theory and Experiment, 2005. 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Next, we discuss the issue of numerical instability in Section A.3. Then, we present additional experimental results in Section A.4, and summarize the experimental setups in Section A.5. Finally, we elaborate on the potential impacts of this work in Section A.6. A.1 The Soft Value Estimation Methods in SAC and SQL In this section, we elaborate on the soft value estimation methods mentioned in Section 2.2. We first show that Vθ(st) approximated using Eq. (6) is greater than that approximated using Eq. (7) for any given state st in Proposition A.1 and Remark A.2. Then, we discuss their practical implementation. Proposition A.1. For any st ∈S and α ∈R>0, the following inequality holds: α log Ea∼πϕ exp (Qθ(st, a)/α) πϕ(a\|st) ≥Ea∼πϕ[Qθ(st, a) −α log πϕ(a\|st)]. (A1) Proof. α log Ea∼πϕ exp (Qθ(st, a)/α) πϕ(a\|st) (i) ≥αEa∼πϕ log exp (Qθ(st, a)/α) πϕ(a\|st) = αEa∼πϕ [Qθ(st, a)/α −log πϕ(a\|st)] = Ea∼πϕ [Qθ(st, a) −α log πϕ(a\|st)] , where (i) is due to Jensen’s inequality. Remark A.2. The inequality in Eq. (A1) preserves after applying the Monte Carlo estimation. Namely, α log 1 M M X i=1 exp Qθ(st, a(i))/α πϕ(a(i)\|st) ! ≥1 M M X i=1 Qθ(st, a(i)) −α log πϕ(a(i)\|st) , (A2) where {a(i)}M i=1 represents a set of samples drawn from πϕ. Unlike the estimation in Eq. (7), the estimation in Eq. (6) is guaranteed to converge to Vθ(st) as M →∞. However, empirically, the estimation method in Eq. (7) is preferred and widely used in the contemporary MaxEnt framework. One potential reason could be the required number of samples needed for effective approximation. According to [8], M = 32 is an effective choice for Eq. (6), whereas M = 1 works well for Eq. (7), as adopted by many previous works. [9–12, 14]. A.2 Theoretical Properties of MEow In this section, we examine a number of key properties of the MEow framework. We begin by presenting a proposition to verify MEow’s capability in modeling the soft Q-function and the soft value function. Then, we present a proposition to derive a deterministic policy in MEow. Finally, we offer a discussion of the impact of incorporating learnable reward shifting functions. Proposition 3.1 Eq. (11) satisfies the following statements: (1) Given that the Jacobian of gθ is non-singular, Vθ(st) ∈R and Qθ(st, at) ∈R, ∀at ∈A, ∀st ∈S. (2) Vθ(st) = α log R exp (Qθ(st, a)/α) da. Proof. (1) Given that the Jacobian of gθ is non-singular, Vθ(st) ∈R and Qθ(st, at) ∈R, ∀at ∈ A, ∀st ∈S. If the Jacobian of gθ is non-singular, then both Q i∈Sl \| det(Jgi θ(st))\| ∈ R>0 and Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\| ∈R>0. This suggests that α log Q i∈Sl \| det(Jgi θ(st))\| ∈R and α log pz (gθ(at\|st)) Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\| ∈R. As a result, Vθ(st) ∈R and Qθ(st, at) ∈R. 16 (2) Vθ(st) = α log R exp (Qθ(st, a)/α) da. 1 = Z π(a\|st)da = Z pz (gθ(a\|st)) Y i∈Sn det Jgi θ(ai−1\|st) Y i∈Sl det(Jgi θ(st)) da. ⇔ Y i∈Sl det(Jgi θ(st)) !−1 = Z pz (gθ(a\|st)) Y i∈Sn det Jgi θ(ai−1\|st) da. ⇔α log Y i∈Sl det(Jgi θ(st)) !−1 = α log Z pz (gθ(a\|st)) Y i∈Sn det Jgi θ(ai−1\|st) da. ⇔ −α log Y i∈Sl det(Jgi θ(st)) = α log Z pz (gθ(a\|st)) Y i∈Sn det Jgi θ(ai−1\|st) da. ⇔Vθ(st) = α log Z exp (Qθ(st, a)/α) da. In Section 3.2, we demonstrate that argmaxa Qθ(st, a) can be efficiently obtained through g−1 θ (argmaxz pz(z)\|st). To provide theoretical support for this result, we include a proof for Proposition 3.2. Proposition 3.2 Given that \| det(Jgi θ(ai−1\|st))\| is a constant with respect to ai−1, then g−1 θ (argmaxz pz(z)\|st) = argmaxa Qθ(st, a). Proof. Let c ≜Q i∈Sn \| det(Jgi θ(ai−1\|st))\|. argmax a Qθ(st, a) = argmax a α log pz (gθ(a\|st)) Y i∈Sn det Jgi θ(ai−1\|st) ! = argmax a α log (pz (gθ(a\|st)) c) (i) = argmax a pz (gθ(a\|st)) c = argmax a pz (gθ(a\|st)) (ii) = argmax a pz (z) (iii) = g−1 θ (argmax z pz(z)\|st), where (i) is because logarithm is strictly increasing, (ii) is due to z = gθ(a\|st), and (iii) is due to a = g−1 θ (z\|st). In Section 3.2, we incorporate the learnable reward shifting function bθ in Qθ and Vθ. This incorporation results in redefined soft Q-function Qb θ and soft value function V b θ . In Proposition A.3, we verify that V b θ (st) = α log R exp(Qb θ(st, a)/α)da = Vθ(st) + bθ(st). Proposition A.3. Given that Q and V satisfy V (st) ≜α log R exp (Q(st, a)/α) da. The augmented functions, Qb(st, at) = Q(st, at) + b(st) and V b(st) ≜α log R exp(Qb(st, a)/α)da, where b(st) is the the reward shifting function, satisfy V b(st) = V (st) + b(st). 17 Proof. V b(st) = α log Z exp Qb(st, a)/α da. = α log Z exp ((Q(st, a) + b(st)) /α) da = α log exp(b(st)/α) Z exp (Q(st, a)/α) da = α log Z exp (Q(st, a)/α) da + α log (exp(b(st)/α)) = α log Z exp (Q(st, a)/α) da + b(st) = V (st) + b(st). A.3 The Issue of Numerical Instability In this section, we provide the motivation for employing the learnable reward shifting function described in Section 3.2. We show that while Qθ and Vθ defined in Eq. (11) have the theoretical capability to learn arbitrary real values (i.e., Proposition 3.1), they may experience numerical instability in practice. This instability arises due to the exponential growth of Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\| and the exponential decay of Q i∈Sl \| det(Jgi θ(st))\|. We first examine the relationship between Vθ(st) and Q i∈Sl \| det(Jgi θ(st))\| according to the following equations: Vθ(st) = −log Y i∈Sl det Jgi θ(st) ⇔ exp(−Vθ(st)) = Y i∈Sl det Jgi θ(st) . The above equation suggests that the value of Q i∈Sl \| det(Jgi θ(st))\| decreases exponentially with respect to Vθ(st), which may lead to numerical instability during training. On the other hand, the relationship between Qθ(st, at) and Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\| can be expressed according to the following equations: Qθ(st, at) = log pz (gθ(at\|st)) + log Y i∈Sn det Jgi θ(ai−1 t \|st) . ⇔Qθ(st, at) −log pz (gθ(at\|st)) = log Y i∈Sn det Jgi θ(ai−1 t \|st) . ⇔ exp (Qθ(st, at) −log pz (gθ(at\|st))) = Y i∈Sn det Jgi θ(ai−1 t \|st) . The equations indicate that Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\| increases exponentially with respect to Qθ(st, at) −log pz (gθ(at\|st)). Therefore, increasing Qθ may also lead to an exponential growth of Q i∈Sn \| det(Jgi θ(ai−1 t \|st))\|. LRS makes our model less susceptible to numerical calculation errors since the learnable reward shifting function bθ, unlike Qθ and Vθ, is not represented in logarithmic scale. Consider a case where FP32 precision is in use, ‘MEow (Vanilla)’ could fail to learn a target Vθ∗(st) > 38, ∀st, since Q i∈Sl \| det(Jgi θ(st))\| = exp(−Vθ∗(st)) < 2−126 cannot be represented using FP32 precision. Therefore, without shifting the reward function, the loss sometimes becomes undefined values, and can lead to ineffective training (e.g., the green lines in Fig. 6). The reward shifting term can be designed as a (state-conditioned) function or a (non-state-conditioned) value. It can also be learnable or non-learnable. All of these designs (i.e., bθ(st), b(st), bθ, and b) can be directly applied to MEow since none of them influences the action distribution. Based on our preliminary experiments, we identified that a learnable state-conditioned reward shifting delivers the best performance. 18 Ant-v4 HalfCheetah-v4 Hopper-v4 Walker2d-v4 Humanoid-v4 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return MEow (Additive) MEow (Affine) 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 7500 6000 4500 3000 1500 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 Figure A1: Performance comparison between MEow with additive coupling transformations in gθ and MEow with affine coupling transformations in gθ on five MuJoCo environments. Each curve represents the mean performance, with shaded areas indicating the 95% confidence intervals, derived from five independent runs with different seeds. Ant-v4 HalfCheetah-v4 Hopper-v4 Walker2d-v4 Humanoid-v4 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return MEow Energy Critic+Gaussian Actor Flow Critic+Gaussian Actor Flow Critic+Flow Actor Energy Critic+Flow Actor 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 7500 6000 4500 3000 1500 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 8000 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 Figure A2: Performance comparison between ‘MEow’, ‘Energy Critic+Gaussian Actor’ (ECGA), ‘Energy Critic+Flow Actor’ (ECFA), ‘Flow Critic+Gaussian Actor’ (FCGA), and ‘Flow Critic+Flow Actor’ (FCFA) on five MuJoCo environments. Each curve represents the mean performance, with shaded areas indicating the 95% confidence intervals, derived from five independent runs with different seeds. A.4 Supplementary Experiments In this section, we provide additional experimental results. In Section A.4.1, we offer a comparison between MEow with gθ modeled using additive coupling layers and that using affine coupling layers. In Section A.4.2, we compare the performance of MEow with four distinct types of actor-critic frameworks formulated based on prior works [9–11]. In Section A.4.3, we provide an example illustrating the ability of flow-based models to represent multi-modal distributions as policies. In Section A.4.4, we present a performance comparison between SAC and its variant with LRS. Finally, in Section A.4.5, we provide a sensitivity examination for the target smoothing parameter. A.4.1 Comparison of Additive and Affine Transformations In this section, we evaluate the performance of MEow with two commonly-adopted non-linear transformations, additive [46] and affine [47] coupling layers, for constructing gθ. The results are presented in Fig. A1. The results show that MEow with additive coupling layers achieves better performance than that with affine coupling layers. Based on this observation, we adopt additive coupling layers for constructing gθ throughout the experiments in Section 4 of the main manuscript. A.4.2 Influences of Parameterization in MaxEnt RL Actor-Critic Frameworks In this section, we compare the performance of MEow against four different actor-critic frameworks formulated based on prior works [9–11]. The first framework is the same as SAC [9], with the critic modeled as an energy-based model and the actor as a Gaussian. The second framework follows the approaches of [10, 11], where the critic is also an energy-based model, but the actor is a flow-based model. The third and fourth frameworks both utilize a flow-based model for the critic, with the actor modeled as a Gaussian and a flow-based model, respectively. These frameworks are denoted as: ‘Energy Critic+Gaussian Actor’ (ECGA), ‘Energy Critic+Flow Actor’ (ECFA), ‘Flow Critic+Gaussian Actor’ (FCGA), and ‘Flow Critic+Flow Actor’ (FCFA), respectively. Regarding the 19 Ant-v4 HalfCheetah-v4 Hopper-v4 Walker2d-v4 Humanoid-v4 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Return SAC (+LRS) SAC 7500 6000 4500 3000 1500 0 0.0 1.0 2.0 3.0 4.0 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 Figure A4: Performance comparison between ‘SAC’ and ‘SAC (+LRS)’. Each curve represents the mean performance, with shaded areas indicating the 95% confidence intervals, derived from five independent runs with different seeds. soft value calculation during training, the first and second frameworks adopt the value estimation method in SAC (i.e., Eq. (7)). For the third and the fourth frameworks, their training adopts the exact value calculation (i.e., Eq. (11)), which is the same as MEow. The results are presented in Fig. A2. As depicted in Fig. A2, MEow exhibits superior performance and stability compared to other actor-critic frameworks in the ‘Hopper-v4’, ‘Ant-v4’, and ‘Walker2d-v4’ environments, and shows comparable performance with ECGA in most environments. In addition, the results that compare the frameworks with flow-based models as actors (i.e., ECFA and FCFA) to those with Gaussians as actors (i.e., ECGA and FCGA) suggest that Gaussians are more effective for modeling actors. This finding is similar to that in [10]. On the other hand, the comparisons between ECGA and FCGA, and between FCGA and FCFA, do not show a clear trend. These findings suggest that both flow-based and energy-based models can be suitable for modeling the soft Q-function. Furthermore, the comparison between FCFA and MEow reveals that the training process involving alternating policy evaluation and improvement steps may be inferior to our proposed training process with a single objective. A.4.3 Modeling Multi-Modal Distributions using Flow-based Models (b) (a) Action State Action State Figure A3: (a) The reward landscape of the one-step environment described in Section A.4.3. (b) The conditional pdf prediction using an NSF model. In this section, we use a one-dimensional example to demonstrate that flow-based models are capable of learning multi-modal action distributions. We employ a state-conditioned neural spline flow (NSF) [50] as the model, and train it in a single-step environment with one-dimensional state and action spaces. Fig. A3 (a) illustrates the reward landscape with the state and action denoted on the xaxis and y-axis, respectively. Fig. A3 (b) illustrates the probability density function (pdf) predicted by the model. The result demonstrates the capability of flow-based models to effectively learn multi-modal distributions. A.4.4 Applying Learnable Reward Shifting to SAC In this section, we examine the performance of SAC with the proposed LRS technique. Since the original implementation of SAC involves the clipped double Q-Learning technique, SAC with LRS is equivalent to SAC with the shifting-based clipped double Q-Learning (SCDQ) technique discussed in Section 3.2 of the main manuscript. The performance of ‘SAC’ and ‘SAC (+LRS)’ is presented in Fig. A4. The results indicate that applying LRS does not improve SAC’s performance. Therefore, for a fair evaluation, the original implementation of SAC is adopted in the comparison in Section 4 of the main manuscript. A.4.5 Sensitivity Examination for the Target Smoothing Parameter In this section, we provide a performance comparison of SAC and MEow trained with different target smoothing parameter values (i.e., τ = 0.005, 0.003, 0.0005, and 0.0001). The results shown in 20 7500 6000 4500 3000 1500 0 -1500 0.0 1.0 2.0 3.0 4.0 Ant-v4 Hopper-v4 6000 5000 4000 3000 2000 1000 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 HalfCheetah-v4 Walker-v4 Humanoid-v4 0.0005 0.005 0.003 0.0001 Return 7500 6000 4500 3000 1500 0 -1500 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 0 0.0 1.0 2.0 3.0 4.0 6000 5000 4000 3000 2000 1000 00.0 1.0 2.0 3.0 4.0 5.0 7000 13500 11000 8500 6000 3500 1000 -1500 0.0 0.5 1.0 1.5 Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) Steps (1e6) 3000 2500 2000 1500 1000 500 00.0 0.5 1.0 1.5 3500 Return SAC MEow Figure A5: A performance comparison between MEow and SAC under different trained τ on five MuJoCo environments. Each curve represents the mean performance, with shaded areas indicating the 95% confidence intervals, derived from five independent runs with different seeds. Fig. A5 indicate that SAC performs the best when τ = 0.005, while MEow requires different τ values to achieve good performance across different tasks. Although both algorithms exhibit significant performance variations with different τ values, SAC demonstrates a more consistent trend in terms of the total returns among the tested values of τ. A.5 Experimental Setups In this section, we elaborate on the experimental configurations and provide the detailed hyperparameter setups for the experiments presented in Section 4 of the main manuscript. The code is implemented using PyTorch [72] and is available in the following repository: https: //github.com/ChienFeng-hub/meow. A.5.1 Model Architecture Among all experiments presented in Section 4 of the main manuscript, we maintain the same model architecture, while adjusting inputs and outputs according to the state space and action space for each environment. An illustration of this architecture is presented in Fig. A6. The model architecture comprises three main components: (I) normalizing flow, (II) hypernetwork, and (III) reward shifting function. For the first component, the transformation gθ includes four additive coupling layers [46] followed by an element-wise linear layer. The prior distribution pz is modeled as a unit Gaussian. For the second component, the hypernetwork involves two types of multi-layer perceptrons (MLPs), labeled as (a) and (b) in Fig. A6, which produce weights for the non-linear and linear transformations, respectively. Both MLPs employ swish activation functions [73] and have a hidden layer size of 64. The MLPs labeled as (a) incorporate layer normalization [74] and a dropout layer [75] with a dropout rate of 0.1. For the third component, the reward shifting functions (i.e., b(1) θ and b(2) θ ) are implemented using MLPs with swish activation and a hidden layer size of 256. The parameters used in these components are collectively referred to as θ, and are optimized using the same objective function L(θ) defined in Eq. (4), with the soft Q-function and the soft value function replaced by Qb θ and V b θ , respectively. Please note that, for the sake of notational simplicity and conciseness, the parameters of each network are all represented using θ instead of distinct symbols (e.g., θ(II)-(a), θ(II)-(b), θ(III)-(1), and θ(III)-(2)). A.5.2 Experiments on the Multi-Goal Environment The experiments in Section 4.1 are performed on a two-dimensional multi-goal environment [8]. This environment consists of four goals positioned at [0, 5], [0, −5], [5, 0], and [−5, 0], denoted as g1, g2, g3, and g4, respectively. The reward is the sum of two components, r1(st) and r2(at), which are 21 Additive Additive Additive Additive Linear (I) Normalizing Flow (II) Hypernetwork (III) Reward Shifting Functions Prior Linear Linear Linear swish swish Linear Linear Linear swish swish ... (a) ... (a) ... (a) ... (a) ... (b) 256 256 256 256 (b) Linear Linear Linear swish swish 64 64 (a) Linear Linear swish L. Norm Dropout Linear swish L. Norm Dropout 64 64 + + min + Figure A6: The architecture adopted in MEow. This architecture consists of three primary components: (I) normalizing flow, (II) hypernetwork, and (III) reward shifting function. The hypernetwork includes two distinct types of networks, labeled as (a) and (b), which are responsible for generating weights for the non-linear and linear transformations within the normalizing flow, respectively. Layer normalization is denoted as ‘L. Norm’ in (a). Table A1: Shared hyperparameters of MEow. Parameter Value optimizer Adam [78] learning rate (β) 0.001 gradient clip value 30 discount (γ) 0.99 buffer size 106 Table A2: Shared hyperparameters of SAC. Parameter Value optimizer Adam [78] learning rate (β) 0.0003 gradient clip value discount (γ) 0.99 buffer size 106 formulated as follow: r1(st) = max i −∥st −gi∥and r2(at) = −30 × ∥at∥. (A3) According to Eq. (A3), r1(st) encourages policies to reach states near the goals. On the other hand, r2(at) encourages policies to produce actions with small magnitudes. In this experiment, we adopt a temperature parameter α = 2.5, a target smoothing factor τ = 0.0005, a learning rate β = 0.001, a discount factor γ = 0.9, and a total of 4, 000 training steps. The computation was carried out on NVIDIA TITAN V GPUs equipped with 12GB of memory. The training takes approximately four minutes. A.5.3 Experiments on the MuJoCo Environments Software and Hardware Setups. For the experiments on the MuJoCo environments, our implementation is built on CleanRL [76], with the normalizing flow component adapted from [77]. The computation was carried out on NVIDIA V100 GPUs equipped with 16GB of memory. The training takes approximately 13 hours per 1 million steps, with each GPU capable of executing four training sessions simultaneously. Hyperparameter Setups. The shared and the environment-specific hyperparameters of MEow are summarized in Tables A1 and A3, respectively. The hyperparamers for the baseline methods are directly borrowed from Stable Baseline 3 (SB3) [71]. 22 Table A3: A list of environment-specific hyperparameters used in MEow. Environment Target Smoothing Parameter (τ) Temperature Parameter (α) MuJoCo Hopper-v4 0.005 0.25 HalfCheetah-v4 0.003 0.25 Walker2d-v4 0.005 0.1 Ant-v4 0.0001 0.05 Humanoid-v4 0.0005 0.125 Omniverse Isaac Gym Ant 0.0005 0.075 Humanoid 0.00025 0.25 Ingenuity 0.0025 0.025 ANYmal 0.025 0.00075 AllegroHand 0.001 0.1 FrankaCabinet 0.075 0.1 Table A4: A list of environment-specific hyperparameters used in SAC. Environment Target Smoothing Parameter (τ) Temperature Parameter (α) Omniverse Isaac Gym Ant 0.0025 0.4 Humanoid 0.0025 0.025 Ingenuity 0.0025 0.1 ANYmal 0.0025 0.01 AllegroHand 0.0025 0.1 FrankaCabinet 0.025 0.1 A.5.4 Experiments on the Omniverse Isaac Gym Environments Software and Hardware Setups. For the experiments performed on Omniverse Isaac Gym, the implementation is built on SKRL [79] due to its compatibility with Omniverse Issac Gym [34]. The computation was carried out on NVIDIA L40 GPUs equipped with 48GB of memory. The training takes approximately 22 hours per 1 million training steps, with each GPU capable of executing three training sessions simultaneously. For ‘Ant’, ‘Humanoid’, ‘Ingenuity’, and ‘ANYmal’, each training step consists of 128 parallelizable interactions with the environments. For ‘AllegroHand’ and ‘FrankaCabinet’, each training step consists of 512 parallelizable interactions with the environments. Hyperparameter Setups. The shared and the environment-specific hyperparameters of MEow are summarized in Tables A1 and A3, respectively. Those of SAC are summarized in Tables A2 and A4, respectively. Both SAC and MEow were tuned using the same search space for τ and α to ensure a fair comparison. Specifically, a grid search was conducted with τ values ranging from 0.1 to 0.00025 and α values from 0.8 to 0.0005 for both algorithms. The setups with the highest average return were selected for each environment. A.6 Broader Impacts This work represents a new research direction for MaxEnt RL. It discusses a unified method that can be trained using a single objective function and can avoid Monte Carlo estimation in the calculation of the soft value function, which addresses two issues in the existing MaxEnt RL methods. From a practical perspective, our experiments demonstrate that MEow can achieve superior performance compared to widely adopted representative baselines. In addition, the experimental results conducted in the Omniverse Isaac environments show that our framework can perform robotic tasks simulated based on real-world application scenarios. These results indicate the potential for deploying MEow in real robotic tasks. Given MEow’s potential to be extended to perform challenging tasks, it is unlikely to have negative impacts on society. 23 NeurIPS Paper Checklist 1. 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4,444	Gene-Gene Relationship Modeling Based on Genetic Evidence for Single-Cell RNA-Seq Data Imputation Daeho Um∗ Samsung Advanced Institute of Technology (SAIT) daeho.um@samsung.com Ji Won Yoon Chung-Ang University jiwonyoon@cau.ac.kr Seong Jin Ahn Korea Advanced Institute of Science and Technology (KAIST) sja1015@kaist.ac.kr Yunha Yeo Korea University serinahyeo@korea.ac.kr Abstract Single-cell RNA sequencing (scRNA-seq) technologies enable the exploration of cellular heterogeneity and facilitate the construction of cell atlases. However, scRNA-seq data often contain a large portion of missing values (false zeros) or noisy values, hindering downstream analyses. To recover these false zeros, propagation-based imputation methods have been proposed using k-NN graphs. However they model only associating relationships among genes within a cell, while, according to well-known genetic evidence, there are both associating and dissociating relationships among genes. To apply this genetic evidence to gene-gene relationship modeling, this paper proposes a novel imputation method that newly employs dissociating relationships in addition to associating relationships. Our method constructs a k-NN graph to additionally model dissociating relationships via the negation of a given cell-gene matrix. Moreover, our method standardizes the value distribution (mean and variance) of each gene to have standard distributions regardless of the gene. Through extensive experiments, we demonstrate that the proposed method achieves exceptional performance gains over state-of-the-art methods in both cell clustering and gene expression recovery across six scRNA-seq datasets, validating the significance of using complete gene-gene relationships in accordance with genetic evidence. The source code is available at https: //github.com/daehoum1/scCR. 1 Introduction Figure 1: Within a cell, there are two types of relationships among genes. Single-cell RNA sequencing (scRNA-seq) has become one of the most widely used technologies in biomedical research due to its ability to measure genome-wide gene expression at the single-cell level [1–3]. ScRNA-seq enables us to discover novel cell types [4], analyze cellular trajectories [5], and improve understanding human disease [6, 7]. However, scRNA-seq analysis encounters significant challenges due to the high rate of zero values in scRNA-seq data represented by a cell-gene matrix. Specifically, owing to the low RNA capture rate, scRNA-seq data often contain zero values. These zero values represent unobserved gene expression resulting from both technical omissions (referred to as dropouts [8]) and true biological absence. Moreover, even non-zero values in scRNA-seq data suffer from various sources of noise, such as cell cycle effects and batch effects [9, 10]. ∗Corresponding author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). To deal with the missing or noisy gene expression in scRNA-seq data, diverse imputation methods have been proposed, which can be categorized into non-graph-based, graph neural network (GNN)based, and propagation-based methods. Among these methods, propagation-based methods [11, 12] have been favored due to their outstanding performance. The propagation-based methods construct a k-nearest neighbor (k-NN) graph on scRNA-seq data represented as a cell-gene matrix, and fill in missing values by propagating nonzero values on the k-NN graph. Despite their effectiveness, they overlook well-known genetic evidence [13, 14], which means that there are two types of relationships between genes: associating relationship and dissociating one. As shown in Figure 1, associating relationships represent genes that co-occur, whereas dissociating relationships represent genes that avoid co-occurrence. However, the existing methods cannot model the dissociating gene-gene relationship by constructing a simple k-NN graph to connect only the associating genes with similar occurrence patterns. Consequently, these methods fail to connect dissociating genes. Within a cell, when considering the value to be imputed for gene Q, the value for its associating gene can assist in inferring the value for gene Q. However, its dissociating gene can also provide crucial information: if its dissociating gene has a high value, the value for gene Q may be low, as they tend to avoid each other. Additionally, the value distribution of a gene often differs significantly from that of other genes [15]. Therefore, the sum of propagated values from other genes may lead to the mixing of values at various scales, which is not suitable for data recovery. To resolve the aforementioned problems, we propose a novel propagation-based imputation scheme called Single-Cell Complete Relationship (scCR) for scRNA-seq data, which models both associating and dissociating gene-gene relationships. scCR concatenates a given cell-gene matrix and its negation, then standardizes the value distribution (mean and variance) in each column (i.e., gene) of the concatenated matrix. Subsequently, we construct a k-NN graph on this concatenated and standardized matrix to connect both associating and dissociating genes within a cell. Through a propagation process on this k-NN graph, scCR effectively denoises scRNA-seq data by capturing complete genegene relationships. Extensive experimental results demonstrate that scCR significantly outperforms state-of-the-art methods in both gene expression recovery and cell clustering. Through experiment, we further confirm that scCR can model dissociating gene-gene relationships inherent in scRNA-seq data. The main contributions of our work are summarized as follows: (1) We newly propose an effective imputation method for scRNA-seq data, which is based on the genetic evidence. Our method can model complete gene-gene relationships, including both associating and dissociating relationships; (2) We employ a standardization step before propagation among genes for additional performance improvement in downstream tasks on scRNA-seq data; (3) By modeling dissociating gene-gene relationships and utilizing the standardization step, our scCR significantly improves performance in various downstream tasks, outperforming the state-of-the-art methods by a large margin. 2 Related Work Handling noise in scRNA-seq data. Approaches for handling noise in scRNA-seq data can be categorized into non-graph-based, GNN-based, and propagation-based methods. As pioneering efforts to impute zero values, non-graph-based methods predominantly employ either statistical techniques [16, 17] or autoencoder frameworks [17, 18]. Building on this foundation, graph-based approaches, including GNN-based and propagation-based methods, have received significant attention due to their ability to model relationships among cells and genes through graph structures. scGNN [19] leverages cell-cell relationships by constructing a cell-cell similarity matrix within a graph autoencoder framework. scGCL [20] is a graph autoencoder framework that exploits contrastive learning to capture cell-cell relationships. scTAG [21] is a clustering method that employs a graph autoencoder framework using a cell-cell k-NN graph, which jointly optimizes clustering loss and reconstruction loss. Propagation-based imputation in scRNA-seq data. Propagation-based imputation methods have shown their superiority in scRNA-seq data imputation. They promote greater similarity in gene expression among cells that are already similar through iterative propagation steps. While MAGIC [22] utilizes a diffusion mechanism to denoise scRNA-seq data, updating values through the diffusion of both zero values and observed nonzero values may be significantly affected by false zero values (i.e., dropouts). To address this issue, Feature Propagation (FP) [23] can be a good solution because FP preserves observed values during diffusion while updates unobserved values through 2 Figure 2: A brief overview of Single-Cell Complete Relationship (scCR). diffusion. Propagation-based imputation methods have shown their superiority in scRNA-seq data imputation. scFP [11] adopts FP developed for graph-structured data to resolve imputation for scRNA-seq data. scFP constructs a cell-cell k-NN graph and applies FP for the imputation of zero values. Very recently, [12] proposes scBFP to utilize gene-gene relationships as well as cell-cell relationships. scBFP consists of two stages, and in each stage, it applies FP using a gene-gene k-NN graph and a cell-cell k-NN graph, respectively. Although scBFP is designed to leverage gene-gene relationships, the simple addition of FP using a gene-gene k-NN graph cannot effectively exploit gene-gene relationships due to the following two reasons: (1) a gene-gene k-NN graph can connect only associating genes which have co-occurrence relationships while overlooking the presence of dissociating gene-gene relationships; (2) Since the distributions for each gene significantly varies, propagation without additional processing will degrade recovery performance. 3 Preliminaries Notation. A graph can be represented as G = (V, E), where V = {v1, . . . , vN} is the set of N nodes and E is the set of edges. The connectivity of G can be represented by the adjacency matrix A ∈{0, 1}N×N with Ai,j = 1 iff (vi, vj) ∈E and Ai,j = 0 otherwise. Given an arbitrary matrix B ∈Ra×b, we let kNN(·) : Ra×b →{R}a×a be a function that generates a normalized adjacency matrix A of the row-row k-NN graph based on cosine similarity. Here, the normalized adjacency matrix A is obtained by A = D−1/2AD−1/2 where A is the adjacency matrix of the k-NN graph and D is a degree matrix with diagonal entries Di,i = P j Ai,j. Consequently, while kNN(X) yields A cell ∈{0, 1}C×C of the cell-cell k-NN graph from X, kNN(X⊤) produces A gene ∈{0, 1}G×G of the gene-gene k-NN graph from X. We let Bi,: and B:,j denote the i-th row vector of B and the j-th column vector of B, respectively. Feature Propagation. FP-based algorithms [23, 24] are proposed to impute missing features in graph-structured data. The core idea of FP is to impute missing values by diffusing observed values while preserving these observed values. Assume that a given graph G = (V, E) has a feature matrix X ∈RN×F with missing values, where rows and columns correspond to nodes and F feature channels, respectively. We use A ∈RN×N to denote a normalized adjacency matrix of the graph. To preserve known features during the diffusion process, we mark the positions of the features to be 3 preserved with 1 in the mask matrix M ∈{0, 1}N×F ; here, values of 1 in M indicate the location of observed features. We express FP as a function by X = FP(X, A, M) where X ∈RN×F is an output matrix. A detailed explanation of FP is provided in Appendix A. In summary, FP fills in missing values in X through diffusion using A while preserving features corresponding to values of 1 in M. It is noteworthy that FP(X, A, M) performs propagation among the rows of X. In scRNA-seq data imputation, FP-based imputation methods treat zero values as missing values to be imputed via features diffused from non-zero values. 4 Proposed Method 4.1 Overview of scCR In this paper, we design a novel imputation framework for scRNA-seq data, namely scCR, which utilizes complete gene-gene relationships. Unlike existing work, scCR exploits both associating and dissociating relationships, which contain valuable biological information. Given highly noisy scRNA-seq data, especially having a high number of false zeros, the goal of scCR is to recover scRNA-seq data by imputing zero values. As shown in Figure 2, our proposed framework consists of three stages: pre-imputation, complete relation, and denoising. Throughout these three stages, we enhance a gene expression matrix by gradually integrating complete gene-gene and cell-cell relationships. 4.2 Pre-Imputation Stage We consider a cell-gene matrix X ∈RC×G, where C and G represent the number of cells and genes, respectively. We let Acell ∈{0, 1}C×C denote an adjacency matrix of a cell-cell graph. Similarly, we let Agene ∈{0, 1}G×G denote an adjacency matrix of a gene-gene graph. Building a k-NN graph directly on X can lead to performance degradation in downstream tasks due to the noisy nature of X. Therefore, scCR begins with the pre-imputation stage, which creates a pre-imputed matrix to be used for k-NN graph construction in the complete relationship stage. In this stage, scCR imputes zero values through intercellular (i.e., cell-cell) propagation. Cell-cell FP. We first construct a cell-cell k-NN graph by A cell(1) = kNN(X). scCR then employs FP to impute zero values by the diffusion of nonzero values among cells. We let MX ∈{0, 1}C×G be a mask matrix with MX i,j = 1 iff Xi,j ̸= 1 and MX i,j = 0 otherwise, which indicates the positions of the nonzero features in X to be preserved during the diffusion. Cell-cell FP using A cell(1) is performed as follows, X (1) = FP(X, A cell(1), MX) (1) where X (1) ∈RC×G is an output of the pre-imputation stage, which is utilized in the following complete relationship stage. 4.3 Complete Relationship Stage Figure 3: A subset of the gene expression matrix in the Baron Human dataset. In the complete relationship stage, we refine X (1) through gene-gene and cell-cell propagation. ScBFP [12] adopts gene-gene FP on the k-NN graph constructed based on cosine similarity. However, it overlooks two key issues: (1) The similarity-based gene-gene k-NN graph can connect only associating (or co-occurring) genes, excluding highly correlated dissociating (or avoiding) genes, which can offer important biological information for imputation. This occurs because associating genes may have high cosine similarity due to their co-occurrence. These genes with high cosine similarity may become connected in the cosinesimilarity-based k-NN graph . (2) As shown in Figure 3, since each gene has the distinct distribution within a gene expression matrix [15], the value distribution for a gene varies significantly among genes. Although imputation methods for scRNA-seq generally normalize a gene expression matrix in a cell-wise manner, varying scales across genes still remain after the cell-wise normalization. Therefore, propagation across genes may degrade accurate recovery by mixing values with different scales. 4 Figure 4: An illustration of the concatenation and standardization processes in the complete relationship stage. Std denotes standard deviation. Concatenation. To address these key issues, we propose a novel propagation scheme called gene-gene standardized FP. The gene-gene standardized FP first produces Xcom ∈RC×2G by concatenating X (1) and its negative matrix along columns, i.e., Xcom = [X (1), −X (1)]. Standardization. Subsequently, to enable every gene to have the same scale during propagation among genes, we standardize Xcom in a column-wise manner. For b ∈{1, . . . , 2G}, we standardize Xcom to eXcom ∈RC×2G as follows, eXcom a,b = (Xcom a,b −µb) σb where µb = C X a=1 Xcom a,b , σb = v u u t 1 C −1 C X a=1 (Xcom a,b −µb)2. (2) Here, µb and σb denote the mean and standard deviation of the b-th column (i.e., gene) of Xcom. Since all the columns in eXcom are standardized, SFP can effectively perform propagation-based imputation without the mixing of values at various scales, addressing the aforementioned issue (2). Furthermore, by concatenating X (1) with its negative matrix before the construction of eXcom, SFP can connect not only associating but also dissociating gene-gene relationships. As demonstrated in Figure 4, assume that the i-th gene and the j-th gene have strong dissociating relationships, where i, j ∈{1, . . . , G}. Within any a-th cell (a ∈{1, . . . , C}), X (1) a,i will be very high when X (1) a,j is very low and vice versa. After the standardization, the cosine similarity between the i-th gene and the j-th gene will have a large negative value, which cannot be connected in a k-NN graph. However, through the concatenation, eXcom :,(G+j) corresponds to −X (1) :,j . Thus, the cosine similarity between eXcom :,i and eXcom :,(G+j) has a large positive value, which will be connected in a k-NN graph. Therefore, through gene-gene propagation using this k-NN graph constructed by using eXcom, we can effectively exploit complete gene-gene relationships, including associating and dissociating relationships. Gene-gene FP. We build a gene-gene k-NN graph on ( eXcom)⊤by A gene = kNN(( eXcom)⊤) where A gene ∈R2G×2G. We then perform the gene-gene standardized FP using A gene as follows: X com = (FP(( eXcom)⊤, Agene, [MX, MX]⊤))⊤ (3) where X com ∈RC×2G. Since each gene in X com does not have its original scale due to the standardization, we return all the columns in X com to their original scale as follows: ˇXcom a,b = σbX com a,b + µb (4) where ˇXcom ∈RC×2G is the rescaled matrix. We then reduce ˇXcom by retaining the first G columns, and we denote this reduced matrix as X ∗∈RC×G. X ∗is a final output of the gene-gene standardized FP. Cell-cell FP. X ∗containing information from complete gene-gene relationships plays a crucial role in scCR by contributing the formation of all subsequent cell-gene matrices. To perform additional 5 Table 1: Performance on cell clustering, measured by ARI, NMI, and CA. Standard deviation errors are given. Figures highlighted in green indicate performance improvements over the most competitive baseline at each setting. Dataset Baron Mouse Pancreas Mouse Bladder Method ARI NMI CA ARI NMI CA ARI NMI CA scTAG 0.565±0.016 0.689±0.023 0.526±0.163 0.678±0.141 0.789±0.011 0.69±0.108 0.604±0.149 0.734±0.047 0.605±0.011 DCA 0.447±0.022 0.710±0.010 0.562±0.002 0.566±0.002 0.786±0.001 0.727±0.002 0.447±0.022 0.710±0.010 0.562±0.002 AutoClass 0.408±0.002 0.699±0.002 0.525±0.004 0.564±0.020 0.795±0.009 0.724±0.030 0.506±0.02 0.732±0.009 0.613±0.029 scGNN 2.0 0.441±0.021 0.734±0.029 0.575±0.019 0.562±0.054 0.793±0.049 0.728±0.061 0.488±0.041 0.717±0.015 0.595±0.033 scGCL 0.478±0.001 0.720±0.000 0.645±0.003 0.645±0.061 0.755±0.042 0.747±0.026 0.529±0.002 0.725±0.005 0.598±0.008 MAGIC 0.419±0.007 0.712±0.007 0.557±0.015 0.595±0.007 0.803±0.004 0.765±0.022 0.565±0.004 0.754±0.001 0.651±0.005 scFP 0.613±0.000 0.817±0.000 0.763±0.000 0.802±0.001 0.872±0.000 0.878±0.000 0.655±0.002 0.767±0.000 0.730±0.001 scBFP 0.660±0.000 0.813±0.001 0.763±0.001 0.864±0.000 0.900±0.001 0.918±0.006 0.694±0.000 0.761±0.002 0.779±0.001 scCR (Ours) 0.827±0.139 0.847±0.034 0.846±0.084 0.812±0.000 0.855±0.000 0.873±0.000 0.704±0.000 0.778±0.000 0.765±0.000 (+25.3%) (+3.7%) (+10.9%) (+1.7%) (+1.4%) Dataset Zeisel Worm Neuron Baron Human Method ARI NMI CA ARI NMI CA ARI NMI CA scTAG 0.723±0.010 0.716±0.013 0.712±0.029 0.532±0.134 0.641±0.007 0.439±0.003 0.612±0.029 0.718±0.028 0.610±0.158 DCA 0.693±0.005 0.739±0.005 0.764±0.004 0.502±0.017 0.690±0.016 0.700±0.031 0.545±0.001 0.763±0.004 0.558±0.001 AutoClass 0.673±0.006 0.714±0.009 0.746±0.008 0.488±0.002 0.668±0.001 0.699±0.001 0.523±0.02 0.743±0.005 0.553±0.023 scGNN 2.0 0.533±0.050 0.657±0.063 0.666±0.041 0.453±0.061 0.637±0.03 0.653±0.051 0.525±0.031 0.744±0.025 0.569±0.014 scGCL 0.663±0.003 0.715±0.116 0.717±0.001 0.601±0.014 0.676±0.005 0.754±0.012 0.593±0.027 0.744±0.056 0.671±0.077 MAGIC 0.696±0.003 0.747±0.002 0.765±0.002 0.512±0.016 0.719±0.009 0.770±0.013 0.562±0.012 0.788±0.007 0.596±0.012 scFP 0.848±0.000 0.812±0.000 0.886±0.000 0.524±0.330 0.731±0.014 0.766±0.031 0.676±0.000 0.826±0.000 0.732±0.000 scBFP 0.835±0.000 0.792±0.000 0.869±0.000 0.608±0.000 0.715±0.000 0.792±0.000 0.677±0.000 0.827±0.000 0.733±0.000 scCR (Ours) 0.902±0.000 0.863±0.000 0.952±0.000 0.520±0.014 0.711±0.006 0.746±0.012 0.823±0.000 0.858±0.000 0.827±0.000 (+6.4%) (+6.3%) (+7.5%) (+21.6%) (+3.8%) (+12.8%) cell-cell FP using complete gene-gene relationships inherent in X ∗, we construct cell-cell a k-NN graph by A cell(2) = kNN(X∗). We perform cell-cell FP using X and A cell(2) as follows, X (2) = FP(X, A cell(2), MX) (5) where X (2) is an output of this cell-cell FP. Weighted sum. X (3), which is a final output of complete relationship stage is produced by the weighted sum of X (2) and X (1) as follows: X (3) = αX (1) + (1 −α)X (2) (6) where 0 < α < 1 is a hyperparameter. X (3) can incorporate valuable biological information since complete gene-gene relationships are delivered by X (2). 4.4 Denoising Stage Cell-cell Soft FP. While the pre-imputation and complete relationship stages focus on imputing zero values, the denoising stage aims to remove noise in the overall values of X via propagation-based smoothing. To exploit complete gene-gene relationships, the denoising stage utilizes X (3) containing them. We first build a cell-cell k-NN graph by A cell(3) = kNN(X (3)). To denoise X, we adopt Soft FP [11] that does not maintain zero values during propagation. We apply Soft FP [11] to X as follows, X (4)(t) = βA cell(3)X (4)(t −1) + (1 −β)X, t = 1, · · · , K, (7) where K is the total number of propagation steps, X (4)(0) = X, X (4)(t) ∈RC×G is the updated cell-gene matrix after t propagation steps, and 0 < β < 1 is a hyperparameter. An output of the denoising stage, denoted by X (4)(K), is obtained after K steps. Weighted sum. The final output of our scCR, bX, is the weighted sum of X (3) and X (4)(K) as follows: bX = γX (3) + (1 −γ)X (4)(K). (8) where and 0 < γ ≤1 is a hyperparameter. In summary, unlike existing propagation-based methods, our scCR enables the use of complete gene-gene relationships in denoising scRNA-seq. 6 Figure 5: Performance on dropout recovery, measured by RMSE. Figures highlighted in green indicate reduction rates from the most competitive baseline at each setting. 5 Experiments 5.1 Experimental Setup We performed comparative evaluation of scCR on six widely used scRNA-seq datasets with goldstandard cell type information: Baron Mouse [25], Pancreas [26], Mouse Bladder [27], Zeisel [2], Worm Neuron [28], and Baron Human [25]. We compared our scCR with eight state-of-the-art methods handling noise in scRNA-seq data: (1) non-graph-based methods: DCA [17] and AutoClass [18]; (2) GNN-based methods: scTAG [21], scGNN 2.0 [19], and scGCL [20]; (3) propagation-based methods: MAGIC [22], scFP [11], and scBFP [12]. We evaluated scTAG only on cell clustering, since scTAG is a clustering method. To evaluate the cell clustering of scCR and baselines, we utilized three standard evaluation metrics: Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and Clustering Accuracy (CA). We used KMeans clustering to compare the clustering performance of various imputation methods, including our scCR. For dropout recovery, we employed two standard evaluation metrics: Root Mean Square Error (RMSE) and Median L1 Distance. Implementation. For fair comparisons, we set the hyperparameters of baselines according to specifications in the papers and official codes. We reported the average performance across three independent runs. Experimental details regarding datasets, baselines, evaluation metrics, and hyperparameter settings are provided in Appendix G. 5.2 Results scCR enables improved cell clustering. To validate the effectiveness of scCR in cell clustering, we evaluated its clustering performance. Table 1 presents the performance comparison. While FP-based baselines, including scFP and scBFP, outperformed other baselines, scCR delivered the best or competitive cell clustering performance across all datasets. Specifically, scCR improved ARI by 25.3%, 1.7%, 6.4%, and 21.6% compared to previous state-of-the-art results on Baron Mouse, Mouse Bladder, Zeisel, and Baron Human, respectively. scCR effectively recovers dropout values. Since dropouts can occur at various rates, we generated false zeros (i.e., dropouts) at non-zero values in datasets by applying varying dropout rates. As shown in Figure 5, scCR significantly improved dropout recovery performance in various dropout rates across the datasets. We confirmed that scCR effectively reduced RMSE between imputed values and their original values with large reduction rates, only except for the Pancreas dataset with 40% dropout. We provided a recovery performance comparison, measured by the median L1 distance in Appendix H.2, which also demonstrates the effectiveness of scCR. 7 Figure 6: UMAP visualization using the Baron Human dataset, comparing scCR with the three most competitive imputation methods. The first row shows the visualization of the raw data and their imputed results. The second row displays the visualization of data subjected to an 80% dropout rate and their imputed results. Figure 7: The first row indicates the percentages of associating and dissociating gene-gene relationships in datasets. The second and third rows represents the percentages of associating and dissociating relationships within a gene-gene k-NN graph in each method. scCR identifies rare cell types well. To verify the effectiveness of scCR in identifying rare cell types, we conducted in-depth analysis using the Baron Human dataset. We visualized the two-dimensional UMAP [29] repesentations of the raw data and the data with an 80% dropout rate applied. As shown in the first row in Figure 6, scCR effectively identified rare cell types with few cells, such as ‘gamma’ and ‘Macrophage’. It is noteworthy that even under severe dropouts, scCR separated cell types well (in the second row), whereas compared methods failed. Does scCR really model dissociating relationships? To show that scCR models both associating and dissociating gene-gene relationships, we first investigated the ratio of dissociating relationships to associating relationships. For this investigation, we define that associating or dissociating relationships exist when the absolute values of the correlation coefficients between genes exceed the top 25%. We determine that a positive sign in the correlation coefficients indicates associating relationships, while a negative sign indicates dissociating relationships. We then measured the percentages of 8 Figure 8: Running time comparison of scCR and baselines according to the number of cells. associating and dissociating relationships within gene-gene k-NN graphs in scBFP and our scCR. As shown in Figure 7, scBFP hardly models dissociating gene-gene relationships. In contrast, scCR effectively models dissociating relationships, enabling the use of complete gene-gene relationships in its imputation. scCR is even faster than existing imputation methods. To show the advantage of scCR in terms of imputation time, we measured the running time of scCR and imputation methods on datasets. As shown in Figure 8, scCR showed the lowest running time across all the datasets, regardless of the number of cells. An ablation study (See Appendix C), further experimental results (See Appendix H), and the proof of convergence of FP (See Appendix B) are provided in Appendix. 6 Conclusion In this paper, we proposed a novel imputation framework called Single-Cell Complete Relationship (scCR) for scRNA-seq data imputation. scCR utilized complete gene-gene relationships by concatenating a given cell-gene matrix with its negation and facilitated effective gene-gene propagation through the standardization of the cell-gene matrix in a gene-wise manner. These processes, grounded in genetic evidence, led to significant performance improvements over state-of-the-art methods in various downstream tasks on scRNA-seq data, with fast imputation times. Furthermore, our work is not limited to simply utilizing genetic evidence to design a framework. We validated this evidence within real scRNA-seq datasets, and confirmed that our scCR effectively leveraged this insight. Like other scRNA-seq imputation methods, scCR is specifically designed for scRNA-seq data. However, since scRNA-seq data are inherently matrix-formatted, scCR can be extended to general tabular data imputation. We expect that the concatenation process will be effective even for general tabular data, as there are often both positive and negative correlation coefficients between channels in such data. The extension of scCR to other domains is left for future work. Broader Impacts Our work provides an important insight that, when applying machine learning to the biomedical domain, it is crucial to approach with biological grounding rather than focusing solely on applying existing cutting-edge machine learning techniques. Since scRNA-seq has opened a new frontier for understanding biological systems [15, 30, 31], we believe that our work will contribute to the biomedical domain by enhancing the analysis of human diseases and the discovery of new genetic observations. 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Here, values of 1 in M indicate the location of observed features, where feature values will be preserved. To formally explain the diffusion process of FP in detail, we temporarily reorder nodes for notational convenience. Since observed values may differ across channels, we reorder the nodes for each channel. When we consider the a-th channel, based on the values of 1 in M:,a, we let Va k be the set of nodes whose a-th values are known (observed). Similarly, by examining zero values in M:,a, Va u denotes the set of nodes whose a-th values are unknown (missing). By reordering the nodes in the order of Va k and Va u, the entire feature values and the adjacency matrix for the a-th channel can be expressed as xa = xa k xa u , A (a) = " A (a) kk A (a) ku A (a) uk A (a) uu # , (9) where xa is a column vector representing features for the a-th channel in X in the order of Va k and Va u. Here, xa k ∈R\|Va k \| and xa u ∈R\|Va u\|. Similarly, A (a) consists of four sub-matrices related to Va k and Va u. It is noteworthy that although A (a) ∈RN×N and A ∈RN×N is different due to reordering, they represent the same graph structure. To preserve observed values during the diffusion process, we replace the first \|Va k\| rows in A (a) with one-hot vectors indicating Va k. Consequently, we obtain a transition matrix eA(a) ∈RN×N expressed by eA(a) = Ikk 0ku A (a) uk A (a) uu , (10) where Inn ∈R\|Va n\|×\|Va n\| is an identity matrix and 0nz ∈R\|Va n\|×\|Va z \| is a zero matrix. The diffusion process of FP is implemented by iterative propagation steps utilizing eA(a) as ¯xa(t) = eA(a)¯xa(t −1), t = 1, · · · , K; ¯xa(0) = xa k 0a u , (11) where ¯xa(t) is an imputed feature vector after t propagation steps and 0a u denotes a zero vector of the same length as \|Va k\|. As K →∞, this recursion converges and ¯xa(t) reaches a steady state (the proof can be found in Appendix B). We use ¯xa(K) with large enough K to approximate the steady state. After this diffusion process for the entire channels, we attain {¯xa(K)}F a=1. Since these vectors have different ordering due to channel-wise reordering, we rearrange {¯xa(K)}F a=1 in the original order and construct X ∈RN×F by stacking the originally ordered vectors in {¯xa(K)}F a=1 along the channels. In summary, FP fills in missing values in X through diffusion using A while preserving features corresponding to values of 1 in M. B Proof of Convergence of Feature Propagation Feature propagation (FP) [23] utilize symmetrically normalized transition matrix for the diffusion process implemented by iterative propagation steps. We prove the convergence of this diffusion process as follows. Proposition 1. The transition matrix for the a-th channel is defined by eA(a) = Ikk 0ku A (a) uk A (a) uu , 13 where eA(a) is symmetrically normalized. Using eA(a), the diffusion process in the a-th channel is defined by ¯xa(t) = eA(a)¯xa(t −1), t = 1, · · · , K; ¯xa(0) = xa k 0a u , Then, lim K→∞¯x(a)(K) converges. The proof of Proposition 1 refers to [23]. We begin with two lemmas. Lemma 1. A (a) is the symmetrically normalized matrix calculated by A (a) = (D(a))−1/2A(a)(D(a))−1/2 where D(a) is a diagonal matrix that has diagonal entities D(a) ii = P j A(a) i,j . A (a) uu is the \|¯x(a) u \| × \|¯x(a) u \| bottom-right submatrix of A (a) and let ρ(·) denote spectral radius. Then, ρ(A (a) uu ) < 1. Proof. Consider A (a) uu0 ∈RN×N, where the bottom right submatrix is equal to A (a) uu and all other elements are zero. That is, A (a) uu0 = 0kk 0ku 0uk A (a) uu where 0kk ∈{0}\|¯x(a) k \|×\|¯x(a) k \|, 0ku ∈{0}\|¯x(a) k \|×\|¯x(a) u \|, and 0uk ∈{0}\|¯x(a) u \|×\|¯x(a) k \|. Given that A (a) represents the weighted adjacency matrix of the connected graph G, A (a) uu0 ≤A (a) element-wise and A (a) uu0 ̸= A (a). Furthermore, considering that A (a) uu0 + A (a) constitutes the weighted adjacency matrix of a strongly connected graph, we can conclude that A (a) uu0 + A (a) is irreducible based on Theorem 2.2.7 in [32]. Consequently, applying Corollary 2.1.5 in [32], ρ(A (a) uu0) < ρ(A (a)). Furthermore, ρ(A (a)) ≤1 since we can write A (a) = I −(D(a))−1/2A(D(a))−1/2, where (D(a))−1/2A(D(a))−1/2 has eigenvalues in the range [0, 2] [33]. Note that since both A (a) uu0 and A (a) uu share the same non-zero eigenvalues, it follows that ρ(A (a) uu0) = ρ(A (a) uu ). Ultimately, combining these inequalities leads to the result ρ(A (a) uu ) = ρ(A (a) uu0) < ρ(A (a)) = 1. Lemma 2. Iuu −A (a) uu is invertible where Iuu is the \|¯x(a) u \| × \|¯x(a) u \| identity matrix. Proof. Since 1 is not an eigenvalue of A (a) uu by Lemma 1, 0 is not an eigenvlaue of Iuu −A (a) uu . Thus Iuu −A (a) uu is invertible. We now prove Propostion 1 as follows. Proof. The recursive relation can be written as ¯x(a)(t) = " ¯x(a) k (t) ¯x(a) u (t) # = Ikk 0ku A (a) uk A (a) uu " ¯x(a) k (t −1) ¯x(a) u (t −1) # = " ¯x(a) k (t −1) A (a) uk ¯x(a) k (t −1) + A (a) uu ¯x(a) u (t −1) # . Since ¯x(a) k (t) = ¯x(a) k (t −1) in the first \|¯x(a) k \| rows, it follows that ¯x(a) k (K) = . . . = ¯x(a) k . That is, ¯x(a) k (K) retains the values of x(a) k . Therefore, lim K→∞¯x(a) k (K) converges to x(a) k . 14 Table 2: Ablation study. Con and Sta denote the concatenation and standardization, respectively, which enable capturing dissociating gene-gene relationships. Con Sta Baron Mouse Zeisel Baron Human ARI NMI CA ARI NMI CA ARI NMI CA ✗ ✗ 0.625±0.001 0.777±0.001 0.719±0.003 0.789±0.000 0.738±0.000 0.851±0.000 0.805±0.000 0.837±0.000 0.813±0.000 ✗ ✓ 0.623±0.000 0.788±0.001 0.710±0.001 0.898±0.000 0.856±0.000 0.948±0.000 0.816±0.000 0.846±0.002 0.819±0.001 ✓ ✗ 0.629±0.001 0.791±0.000 0.731±0.002 0.821±0.000 0.794±0.000 0.839±0.000 0.808±0.002 0.841±0.001 0.818±0.002 ✓ ✓ 0.827±0.093 0.847±0.034 0.846±0.084 0.902±0.000 0.863±0.000 0.952±0.000 0.823±0.000 0.858±0.000 0.827±0.000 Table 3: Further ablation study of scCR on cell clustering measured by ARI. PRE, COM, and DEN denote the pre-imputation stage, the complete relation stage, and the denosing stage, respectively. PRE COM DEN Baron Mouse Zeisel Baron Human ✓ ✗ ✗ 0.437 ± 0.061 0.682 ± 0.024 0.580 ± 0.036 ✓ ✓ ✗ 0.409 ± 0.008 0.732 ± 0.001 0.571 ± 0.007 ✓ ✗ ✓ 0.584 ± 0.000 0.822 ± 0.000 0.681 ± 0.000 ✓ ✓ ✓ 0.827 ± 0.093 0.902 ± 0.000 0.823 ± 0.000 We now focus solely on the convergence of lim K→∞¯x(a) u (K). When we unroll the recursion for the last \|¯x(a) u \| rows, ¯x(a) u (K) = A (a) uk x(a) k + A (a) uu ¯x(a) u (K −1) = A (a) uk x(a) k + A (a) uu (A (a) uk x(a) k + A (a) uu ¯x(a) u (K −2)) = . . . = ( K−1 X t=0 (A (a) uu )t)A (a) uk x(a) k + (A (a) uu )K ¯x(a) u (0) By Lemma 1, lim K→∞(A (a) uu )K = 0. Therefore, lim K→∞(A (a) uu )K ¯x(a) u (0) = 0, regardless of the initial state for ¯x(a) u (0). (we replace ¯x(a) u (0) with a zero column vector for simplicity.) Hence, our focus shifts to lim K→∞(PK−1 t=0 (A (a) uu )t)A (a) uk x(a) k . Given that Lemma 1 establishes ρ(A (a) uu ) < 1, and Lemma 2 affirms the invertibility of (Iuu − A (a) uu )−1, the geometric series converges as follows lim K→∞¯x(a) u (K) = lim K→∞( K−1 X t=0 (A (a) uu )t)A (a) uk x(a) k = (Iuu −A (a) uu )−1A (a) uk x(a) k . In conclusion, the recursion in FP converges. C Ablation Study We conducted an ablation study to analyze the effectiveness of each component in scCR. We performed cell clustering on the Baron Mouse, Zeisel, and Baron Human datasets. As shown in Table 2, both concatenation and standardization contributed to performance improvement, and the combination of the two components led to significant performance improvement. We conducted an additional ablation study to assess the effectiveness of each stage of scCR. Table 3 presents the results of the ablation study in terms of cell clustering, measured by ARI. As shown in the table, adding the complete relation stage and the denoising stage significantly improved performance compared to using only the pre-imputation stage. These results confirm that the complete relation and denoising stages contributed substantially to the high performance of scCR, underscoring the well-founded design of our approach. 15 Table 4: Performance on dropout recovery under Missing Not At Random (MNAR) settings, measured by RMSE. Dataset scFP scBFP scCR (ours) Baron Mouse 0.517 ± 0.000 0.465 ± 0.000 0.304 ± 0.000 Pancreas 0.537 ± 0.001 0.506 ± 0.001 0.352 ± 0.000 Mouse Bladder 0.374 ± 0.000 0.374 ± 0.001 0.170 ± 0.000 Zeisel 0.580 ± 0.001 0.538 ± 0.000 0.489 ± 0.000 Worm Neuron 0.330 ± 0.000 0.190 ± 0.000 0.049 ± 0.000 Baron Human 0.493 ± 0.000 0.475 ± 0.000 0.328 ± 0.000 D Missing Not at Random Settings Existing studies [11, 12] simulate dropout by randomly sampling non-zero values in a cell-gene matrix from a uniform distribution and setting them to zero (i.e., missing completely at random (MCAR)). However, in real scRNA-seq data, dropouts occur more frequently in genes with low expression levels rather than those with high variance [34]. This is because the probability of capturing RNA transcripts of low-expression-level genes during sequencing is lower. Based on this dropout pattern, we selected the 1,000 genes with the lowest expression levels and simulated dropout only in these genes. We randomly sampled non-zero values of these genes from a uniform distribution and replaced the sampled values with zero (i.e., missing not at random (MNAR)). Table 4 presents the performance comparison under the aforementioned MNAR settings in terms of data recovery, measured by RMSE. We compared our scCR to the two most competitive baselines, scFP [11] and scBFP [12]. The dropout rate was set to 20% of the total number of values in the cell-gene matrix. As shown in the table, scCR outperformed the compared methods by significant margins in the realistic dropout settings, demonstrating the robustness of scCR in realistic scenarios. Considering realistic dropout simulation can help pre-assess the generalizability of techniques in practical scRNA-seq applications. E Memory Usage Analysis Table 5: Comparison of input and memory complexity. X ∈RG×C denote a cell-gene matrix, where C and G represent the number of cells and genes, respectively. Acell ∈RC×C and Agene ∈RG×G denote cell-cell and gene-gene adjacency matrices, respectively. θ denotes trainable parameters. B represents the batch size for batch-wise k-NN graph construction. Method Input Big-O scTAG X, Acell, θ O(GC) + O(Ecell) + O(θ) DCA X, θ O(GC) + O(θ) AutoClass X, θ O(GC) + O(θ) scGNN 2.0 X, θ O(GC) + O(Ecell) + O(θ) scGCL X, Acell, θ O(GC) + O(Ecell) + O(θ) MAGIC X, Acell O(GC) + O(Ecell) scFP X, Acell O(BC) + O(Ecell) scBFP X, Acell, Agene O(BC) + O(Egene) + O(BG) + O(Ecell) scCR (Ours) X, Acell, Agene O(BC) + O(Egene) + O(BG) + O(Ecell) We analyzed the memory complexity of all methods used in this paper and conducted additional experiments to examine the memory usage of our scCR. Table 5 compares the input and memory complexity of scCR with other state-of-the-art methods. To alleviate the high memory demands during k-NN graph construction, we adopted the batch-wise k-NN graph construction strategy from [12]. When constructing k-NN graphs among genes, we divided the genes into batches of size B and computed k-nearest neighbors for each batch. We applied the same batch-wise strategy when constructing k-NN graphs among cells. This approach reduces memory requirements by avoiding the need to store distances between all points in the entire dataset simultaneously. Specifically, in the memory complexity of scCR, batch-wise k-NN graph construction changes O(G2) and O(C2) to O(BG) and O(BC), respectively. Consequently, batch-wise k-NN graph construction enables 16 Table 6: Memory usage of scCR for different datasets, measured by gigabytes (GB). Baron Mouse Pancreas Mouse Bladder Zeisel Worm Neuron Baron Human Memory usage (GB) 1.811 1.837 1.957 2.037 1.927 3.861 processing of large datasets that would otherwise be infeasible due to memory constraints. Moreover, scCR does not require any trainable parameters, unlike other deep-learning-based methods. We further measured the memory usage of scCR across various datasets, as shown in Table 6. The results in the table indicate that the advantages of scCR extend beyond its superior performance and time efficiency, showcasing its scalability as well. F Hyperparameter Sensitivity Table 7: Performance of scCR on cell clustering measured by ARI for different values of α. α Baron Mouse Zeisel Baron Human 0.01 0.627 ± 0.000 0.903 ± 0.000 0.823 ± 0.000 0.05 (used) 0.827 ± 0.093 0.902 ± 0.000 0.823 ± 0.000 0.1 0.727 ± 0.141 0.904 ± 0.000 0.824 ± 0.000 0.5 0.701 ± 0.000 0.901 ± 0.000 0.681 ± 0.000 0.9 0.448 ± 0.001 0.825 ± 0.000 0.683 ± 0.001 Table 8: Performance of scCR on cell clustering measured by ARI for different values of β. β Baron Mouse Zeisel Baron Human 0.1 0.440 ± 0.046 0.659 ± 0.058 0.553 ± 0.040 0.5 0.476 ± 0.049 0.724 ± 0.000 0.560 ± 0.001 0.9 0.509 ± 0.004 0.740 ± 0.000 0.657 ± 0.000 0.95 0.498 ± 0.002 0.910 ± 0.000 0.666 ± 0.011 0.99 (used) 0.827 ± 0.093 0.902 ± 0.000 0.823 ± 0.000 0.999 0.925 ± 0.000 0.900 ± 0.000 0.819 ± 0.000 Table 9: Performance of scCR on cell clustering measured by ARI for different values of γ. γ Baron Mouse Zeisel Baron Human 0.001 0.927 ± 0.001 0.902 ± 0.000 0.824 ± 0.000 0.01 (used) 0.827 ± 0.093 0.902 ± 0.000 0.823 ± 0.000 0.05 0.635 ± 0.000 0.903 ± 0.000 0.822 ± 0.000 0.1 0.595 ± 0.000 0.903 ± 0.000 0.667 ± 0.001 0.5 0.469 ± 0.003 0.740 ± 0.000 0.627 ± 0.011 0.9 0.409 ± 0.009 0.749 ± 0.000 0.586 ± 0.006 Table 10: Performance of scCR on cell clustering measured by ARI for different values of k. k Baron Mouse Zeisel Baron Human 1 0.628 ± 0.000 0.905 ± 0.000 0.827 ± 0.000 2 (used) 0.827 ± 0.093 0.902 ± 0.000 0.823 ± 0.000 3 0.631 ± 0.000 0.903 ± 0.000 0.819 ± 0.000 5 0.625 ± 0.002 0.902 ± 0.001 0.818 ± 0.000 10 0.621 ± 0.000 0.903 ± 0.000 0.818 ± 0.000 15 0.630 ± 0.000 0.902 ± 0.000 0.817 ± 0.000 We conducted additional experiments to provide a comprehensive analysis of the impact of different hyperparameters, including α, β, γ, and k, on the performance of scCR. We report ARI in cell clustering on three datasets by varying α, β, γ, and k within the ranges of {0.01, 0.05, 0.1, 0.5, 17 0.9}, {0.1, 0.5, 0.9, 0.95, 0.99, 0.999}, {0.001, 0.01, 0.05, 0.1, 0.5, 0.9}, and {1, 2, 3, 5, 10, 15}, respectively. When varying a target parameter, other hyperparameters were fixed to their default settings. Table 7, Table 8, Table 9, and Table 10 illustrate how these hyperparameter choices impact scCR performance. As shown in the tables, the values of α, β, γ, and k used in this study generally resulted in strong performance. In terms of sensitivity, when the runner-up ARI scores are 0.660±0.00, 0.848±0.00, and 0.677±0.00 for Baron Mouse, Zeisel, Baron Human, respectively, scCR demonstrated robustness against hyperparameter variations. Specifically, α ∈{0.05, 0.1, 0.5}, β ∈{0.99, 0.999}, and γ ∈{0.001, 0.01} yielded state-of-the-art performance across the datasets. For k, scCR showed strong performance across all values, except in the case of Baron Human. Additionally, some parameter adjustments led to performance surpassing that of the default settings. However, given the unsupervised nature of single-cell analysis, we retain default hyperparameter settings that generally perform well. G Experimental Details G.1 Implementation Details We conducted all the experiments on a single NVIDIA GeForce RTX 2080 Ti GPU and an Intel Core I5-6600 CPU at 3.30 Hz. The number of neighbors (k) in cell-cell and gene-gene k-NN graphs were set to 15 and 2, respectively. The total number of propagation steps K was set to 40 for both cell-cell and gene-gene FP. We set α, β, and γ to 0.05, 0.99, and 0.01, respectively. We found that the choice between row-stochastic normalization and symmetric normalization applied to A cell(3) within Soft FP [11] affected performance, and we reported the best result. For dropout recovery, we excluded the denoising stage (i.e., γ = 1). G.2 Datasets For our experiments, we utilized six real scRNA-seq datasets, including Baron Mouse [25], Pancreas [26], Mouse Bladder [27], Zeisel [2], Worm Neuron [28], and Baron Human [25]. Table 11 summarizes the dataset statistics. We downloaded all the datasets from the GitHub repository2 for [11]. These publicly available datasets and the repository have no public declaration of license. We leveraged a commonly used pre-processing procedure for scRNA-seq data as described in recent studies [11, 12]. We performed minimal quality control (QC) using SCANPY [35], a toolkit for scRNA-seq analyses. Cells and genes exhibiting no gene expression (i.e., with all zero values) were removed from a given cell-gene matrix. We retained the 2,000 genes with the highest variance in each dataset. We then normalized each cell by total counts over all genes to ensure that every cell had an equal total count of 1.0. That is, every row vector was divided by its library size, which is the sum of its values. After scaling by the median library size, log(x + 1) transformation was applied to all values in the cell-gene matrix, resulting in a pre-processed cell-gene matrix. Table 11: Dataset statistics. Dataset Protocol #Cells #Genes #Cell Type Baron Mouse inDrop 1,886 14,861 13 Pancreas inDrop 1,937 15,575 14 Mouse Bladder Microwell-seq 2,746 19,771 16 Zeisel STRT-seq UMI 3,005 19,972 7 Worm Neuron sci-RNA-seq 4,186 13,488 10 Baron Human inDrop 8,569 17,499 14 2https://github.com/Junseok0207/scFP 18 G.3 Baselines For all the baselines, we used the code released by the author of the respective papers. Table 12 shows the URL links for the baselines. scTAG and scGNN 2.0 are under the MIT license. The licenses of DCA, AutoClass, and MAGIC are Apache-2.0, GPL-3.0, and GPL-2.0, respectively. The code for scGCL, scFP, and scBFP has no public declaration of license. For each baseline, we adhered the hyperpameter/parameter setting in the released code or its respective paper. Table 12: URL links for baselines. Baseline URL Link scTAG https://github.com/Philyzh8/scTAG DCA https://github.com/theislab/dca AutoClass https://github.com/datapplab/AutoClass scGNN 2.0 https://github.com/OSU-BMBL/scGNN2.0 scGCL https://github.com/zehaoxiong123/scGCL MAGIC https://github.com/KrishnaswamyLab/MAGIC scFP https://github.com/Junseok0207/scFP scBFP https://github.com/Junseok0207/scBFP G.4 Evaluation Metrics G.4.1 Clustering Higher ARI, NMI, and CA indicate better performance in cell clustering. ARI. The Adjusted Rand Index (ARI) is the corrected-for-chance version of the Rand Index (RI). RI is computed as follows: RI = TP + TN N 2 (12) where TP is the number of true positives and TN is the number of true negatives. True positive indicates the number of cell pairs correctly assigned to the same cluster, and TP indicates the number of cell pairs correctly assigned to different clusters. ARI can be calculated as follows: ARI = RI −E(RI) max(RI) −E[RI] (13) While RI produces a value between 0 and 1, ARI can produce negative values if the index is less than the expected index. NMI. Normalized Mutual Information (NMI) is a normalization of the Mutual Information (MI) score to scale the scores between 0 and 1. NMI is computed as follows: NMI = 2 × I(S; C) H(S) + H(C) (14) where S is ground-truth cell types, I(·, ·) is the mutual information between two input distributions, and H(·) is the entropy function. Here, all logs are base-2. Higher NMI indicates the distribution of predicted cluster distribution is more similar to ground-truth cell type distribution. CA. Clustering Accuracy is calculated as follows: CA = max m PN i=1 1si=m(ci) N (15) where si is the ground-truth cell type of the i-th cell, ci is predicted cluster assignment of the i-th cell, and m(·) is the matching function responsible for mapping predicted cluster assignments to ground-truth cell types. G.4.2 Recovery Lower Median L1 Distance and RMSE indicate better performance in data recovery. Consider two set X = {x1, . . . , xn} and Y = {y1, . . . , yn}, where X is the set of imputed values and Y is the set of their ground-truth values. 19 Median L1 Distance. Median L1 Distance is calculated as follows: Median L1 Distance = median(\|x1 −y1\|, . . . , \|xn −yn\|). (16) RMSE. Root Mean Square Error (RMSE) is computed as follows: RMSE(X, Y) = sPN i=1(xi −yi)2 N (17) H Additional Experimental Results H.1 Varying Scales across Genes Figure 9: An heatmap of the cell-gene matrix in the Baron Human dataset. We randomly selected 10 genes (columns). 20 H.2 Dropout Recovery Figure 10: Performance on dropout recovery, measured by L1 Median Distance. Figures highlighted in green indicate reduction rates from the most competitive baseline at each setting. 21 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We clarified the scope of our work in the abstract and Sec. 1. The contributions of our paper are summarized in Sec. 1. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,445	The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks Andrea Bonfanti BMW AG, Digital Campus Munich Basque Center for Applied Mathematics University of the Basque Country abonfanti001@ikasle.ehu.eus Giuseppe Bruno BMW AG, Digital Campus Munich Giuseppe.GB.Bruno@bmw.de Cristina Cipriani Technical University of Munich Munich Center for Machine Learning Munich Data Science Institute cristina.cipriani@ma.tum.de Abstract The Neural Tangent Kernel (NTK) viewpoint is widely employed to analyze the training dynamics of overparameterized Physics-Informed Neural Networks (PINNs). However, unlike the case of linear Partial Differential Equations (PDEs), we show how the NTK perspective falls short in the nonlinear scenario. Specifically, we establish that the NTK yields a random matrix at initialization that is not constant during training, contrary to conventional belief. Another significant difference from the linear regime is that, even in the idealistic infinite-width limit, the Hessian does not vanish and hence it cannot be disregarded during training. This motivates the adoption of second-order optimization methods. We explore the convergence guarantees of such methods in both linear and nonlinear cases, addressing challenges such as spectral bias and slow convergence. Every theoretical result is supported by numerical examples with both linear and nonlinear PDEs, and we highlight the benefits of second-order methods in benchmark test cases. 1 Introduction PINNs have became ubiquitous in the scientific research community as a meshless and practical alternative tool for solving PDEs. The first attempts to exploit machine learning models for PDE solutions can be traced back to two articles from the 90s [3, 20], while the model acquired its name and popularity through a later publication [31]. Due to the flexible structure of the architecture, PINNs can be used for forward and inverse problems [42] and efficiently exploited for more complex engineering practice such as constrained shape and topology optimization, and surrogate modeling [35, 16]. However, the usability of PINNs for such applications is often hindered by their slow training and occasional failure to converge to acceptable solutions. Due to the black-box nature of PINNs, it is challenging to analyze their training dynamics and convergence properties mathematically [19]. Nonetheless, rapid training and reliable convergence are crucial aspects of any PDE solver intended for engineering applications. Related works. In this context, the NTK [15] viewpoint has yielded intriguing insights, particularly in the realm of linear PDEs [40]. Although based on the assumption of overparameterized networks, this perspective has proven valuable in highlighting various intrinsic pathologies in PINN training, 38th Conference on Neural Information Processing Systems (NeurIPS 2024). such as spectral bias [39, 2, 29], the complexity of the loss landscape generated by the PDE residuals [19] and the nuanced interplay among components of the loss function [38]. The salient characteristics of the NTK in the infinite-width limit are the fact that is deterministic at initialization, constant during training, and it linearizes the training dynamics due to the sparsity of the Hessian of PDE residuals [22, 23]. Our contributions. In this paper, we delineate the profound theoretical distinctions between the application of PINNs to linear versus nonlinear PDEs, elucidating the differences in their NTK behavior. We show that, even under the idealistic assumption of the infinite-width limit, the NTK framework fails in the nonlinear domain. Our novel contribution lies in demonstrating that the NTK is stochastic at initialization, it is dynamic during training, and is accompanied by a non-vanishing Hessian. Given the evolution of the Hessian throughout training, we emphasize the need of employing second-order methods for nonlinear PDEs. Furthermore, we analyze their convergence guarantees, revealing that even in linear scenarios, the utilization of second-order methods proves advantageous in mitigating the issue of spectral bias. As a second-order method, we employ Levenberg-Marquardt algorithm, a stabilized version of the well-known Gauss-Newton algorithm, which approximates the Hessian to make it computationally feasible even for large networks. It is important to note that our goal is not to propose a novel training algorithm but to demonstrate the benefits of using any second-order method. The reason is twofold: in the nonlinear regime, we achieve faster and better convergence, while in the linear regime, where fast convergence can be achieved by first-order methods, the advantage of second-order methods lies in their ability to alleviate spectral bias. Our work is organized as follows: Section 2 introduces PINNs, and Section 3 covers the NTK theory, comparing its dynamics in linear and nonlinear PDEs. Section 4 examines the convergence guarantees of second-order optimization methods. Finally, Section 5 presents numerical experiments that validate our theoretical insights. 2 Physics-Informed Neural Networks We address the following PDE formulated on a bounded domain Ω⊂Rdin, Ru(x) = f(x), x ∈Ω, u(x) = g(x), x ∈∂Ω. (1) Here, the PDE is defined with respect to the differential operator R, while the boundary and initial conditions are collected in the function g. Notice that Ωcan be either a spatial or spatio-temporal domain, depending on whether the PDE is time-dependent or not. PINNs aim to approximate the PDE solution u : Ω→Rdout with a neural network uθ parametrized by θ, which is a vector containing all the parameters of the network. The “Physics-Informed” nature of the neural network uθ lies in the choice of the loss function employed for training L(θ) = 1 2 Z Ω \|Ruθ(x) −f(x)\|2dx + 1 2 Z ∂Ω \|uθ(x) −g(x)\|2dσ(x), where σ denotes a measure on the surface ∂Ω. In this work, we specifically focus on scenarios where the PDE involves a nonlinear differential operator. Moreover, without loss of generality we consider the case where f(x) = 0. Since the function f(x) does not depend on the parametrization, all of our results hold also for the case when it is nonzero. Moreover, we express (1) as R(Φ[u](x)) = 0, x ∈Ω, u(x) = g(x), x ∈∂Ω, (2) where Φ[u] : Rdin →Rk×dout, defined as Φ[u](x) = [u(x), ∂xu(x), ∂2 xu(x), . . . , ∂k xu(x)], (3) denotes a vector encompassing all (possibly mixed) derivatives of u until order k, while R : Rk×dout →R represents a differentiable function of the components of Φ[u]. Remark 2.1. The importance of the function R lies in its ability to completely encode the nonlinearity of the PDE, while the term Φ remains linear. Furthermore, for numerous well-known nonlinear PDEs (such as Burgers’ or Navier-Stokes equations), the function R exhibits a distinctive structure as it takes the form of a second-order polynomial. 2 To illustrate this, we consider the example of the inviscid Burgers’ equation, which for (τ, x) ∈Ωis expressed as ∂τu + u ∂xu = 0, where τ represents time and x the space variable. It follows that Φ[u](τ, x) = [u(τ, x), ∂τu(τ, x), ∂xu(τ, x)], R(z1, z2, z3) = z2 + z1z3. 3 Neural Tangent Kernel for PINNs We now introduce and develop the NTK for PINNs, inspired by the definition in [40]. We employ a fully-connected neural network featuring a single hidden layer, as follows uθ(x) := 1 √mW 1 · σ(W 0x + b0) + b1, (4) for any x ∈Rdin. Here, W 0 ∈Rm×din and b0 ∈Rm denote the weights matrix and bias vector of the hidden layer, while W 1 ∈Rdout×m and b1 ∈Rdout are the corresponding parameters of the outer layer. Additionally, σ : R →R is a smooth coordinate-wise activation function, such as the hyperbolic tangent, which is a common choice for PINNs. Furthermore, we adopt the NTK rescaling 1 √m to adhere to the methodology introduced in the original work [15]. This is crucial for achieving a consistent asymptotic behavior of neural networks as the width of the hidden layer approaches infinity. In the following, for brevity, we denote with θ the collection of all the trainable parameters of the network, i.e. W 1, W 0, b1, b0. Remark 3.1. For the sake of brevity, we focus on the case of neural networks with a single hidden layer. However, the outcomes derived in this scenario may be extended to deep networks. We leave this extension to future works and refer to [33, 34] for results on finite networks with multiple hidden layers. We consider the discrete loss on the collocation points xr i ∈Ωand the boundary points xb i ∈∂Ω, L(θ) = 1 2Nr Nr X i=1 \|rθ(xr i )\|2 + 1 2Nb Nb X i=1 \|uθ(xb i) −g(xb i)\|2, (5) where rθ(xr i ) = R(Φ[uθ](xr i )) indicates the residual term. Furthermore, Nr and Nb denote the batch size of, respectively, the collection of xr = {xr i }Nr i=1 and xb = {xb i}Nb i=1, which are the discrete data used for training. We now consider the minimization of (5) as the gradient flow ∂tθ(t) = −∇L(θ(t)). (6) Using the following notation uθ(xb) = uθ(t)(xb i) Nb i=1 , rθ(xr) = rθ(t)(xr i ) Nr i=1 , (7) we can characterize how these quantities evolve during the gradient flow, through the NTK perspective. Lemma 3.2. Given the data (7) and the gradient flow (6), then uθ and rθ satisfy the following ∂tuθ(t)(xb) ∂trθ(t)(xr) = −K(t) uθ(t)(xb) −g(xb) rθ(t)(xr) , (8) where K(t) = J(t)J(t)T and J(t) = ∂θuθ(t)(xb) ∂θrθ(t)(xr) . (9) Proof. The proof is presented in [40]. We provide more details about the construction of J(t) in Appendix A. The matrix K is also referred to as Gram matrix. The analysis of Gram matrices and their behavior in the infinite-width limit [4, 5] yields results akin to the NTK analysis. It is important to note that Lemma 3.2 is applicable to any type of sufficiently regular differential operator. 3 3.1 The difference between linear and nonlinear PDEs In the work [40], PINNs have been thoroughly investigated using the NTK, but only in the case of linear PDEs. Additionally, [22] extensively explores the similar case of standard neural networks with linear output. In particular, they show that in the infinite-width limit, the NTK is deterministic under proper random initialization and stays constant during training. Thereby, the dynamics in (8) is equivalent to kernel regression and has an analytical solution expressed in terms of the kernel. As noted in [22], the constancy of the NTK during training is equivalent to the linearity of the model. This characteristic is related to the vanishing of the (norm of the) Hessian of the network’s output in the infinite-width limit. These well-known results are reported in Appendix B. In [43], the same convergence results for Gram matrices hold for nonlinear PDEs when using networks as in (4) with a scaling of 1 ms , where s > 1 2. However, this scaling is inconsistent with the NTK model, so we focus on the unexplored case where s = 1 2. The novel contribution of our paper lies in demonstrating that in this regime this phenomenon does not hold true when dealing with nonlinear PDEs, which we prove in this section. The network architecture and its associated assumptions are relatively standard, so we refer to Assumption B.2 in Appendix B. However, it is essential to delineate the specific assumptions related to the nonlinear PDE. Assumption 3.3 (on R). The differential operator R is nonlinear, hence the function R is nonlinear. Moreover, the gradient ∇R is continuous. The first distinction with linear PDEs arises in the convergence as m →∞of the NTK at initialization. Theorem 3.4. Consider a fully-connected neural network given by (4) satisfying Assumption B.2. Moreover, the PDE satisfies Assumption 3.3. Then, under a Gaussian random initialization θ(0), it holds K(0) D →¯K as m →∞, where the limit is in distribution and ¯K is not deterministic, but its law can be explicitly characterized. Proof. A detailed proof is in Appendix C. However, the basic idea is to reformulate the kernel as K(0) = ΛR(0) KΦ(0) ΛR(0)T , where the matrix KΦ(0) enclose the linear components of R, hence the derivatives of the network’s output, while the matrix ΛR(0) depends on the gradient of R (so its contribution is relevant just in the nonlinear case). We can establish the convergence in probability of KΦ(0) to a deterministic matrix by taking advantage of the linearity of the operator Φ and commuting Φ and ∂θ (see Lemma C.2). The matrix ΛR(0) only converges in distribution, since it is a function of the network output and its derivatives, whose limits are Gaussian Processes at initialization by Proposition C.1. Next, we focus on the NTK behavior during training. Proposition 3.5. Under Assumption B.2 on the network, and Assumption 3.3 on the PDE, assume additionally that R is a real analytic function. Let u∗be a solution of the corresponding PDE and suppose that for every m ∈N there exists tm such that ∥uθ(tm) −u∗∥Ck ≤εm, with εm →0 as m →∞. (10) Finally, let θ(t) be obtained through gradient flow as defined in (6) and denote by K(t) the corresponding NTK. For θ(0) ∼N(0, Im), the following holds: lim m→∞sup t∈[0,T ] ∥K(t) −K(0)∥> 0 a.s. Proof. The proof can be found in Appendix D. Remark 3.6. It is worth noticing that our result holds under the assumption that a neural network with m →∞can adequately approximate the solution u⋆of the PDE (1), and that the training process is successful in achieving this approximation. The first assumption is justified by results such as the universal approximation theorem for neural networks [1]. Despite this optimistic training scenario, as demonstrated in Proposition 3.5, the constancy of the kernel is unattainable. 4 Linear PDEs Nonlinear PDEs NTK at initialization Deterministic Random (Theorem 3.4) NTK during training Constant Dynamic (Proposition 3.5) Hessian Hr Sparse Not sparse (Proposition 3.7) First-order convergence bound ∼λmin( ¯ K) ∼0 or λmin(K(t)) Second-order convergence bound ∼1 ∼0 or 1 (Theorem 4.2) Table 1: Comparison of the theoretical results for linear and nonlinear PDEs. In the context of nonlinear PDEs, converging to a linear regime is unattainable, even in the infinitewidth limit, and this inability stems from the spectral norm of Hr, which is the Hessian of the residuals rθ with respect to the parameters θ. Indeed, in the linear scenario, the convergence of ∥Hr∥ to 0 as m →∞is crucial for demonstrating convergence to the linear regime, as established in Proposition B.3. Similar conclusions have been drawn in [22] for various deep learning architectures. However, we now show that this property does not hold for nonlinear PDEs. Proposition 3.7. Under Assumptions B.2 and 3.3 on the network and on the PDE, let us further assume that R is a second-order polynomial. Then, the Hessian of the residuals Hr is not sparse and lim m→∞∥Hr∥≥˜c, where the constant ˜c does not depend on m. Proof. The proof can be found in Appendix E, together with an explicit formula for ˜c. Remark 3.8. For the latter result, we additionally require that R is a second-order polynomial, which includes many classic nonlinear PDEs like Burgers’ or Navier-Stokes equations. We summarize all our results and provide a comparison with the linear case in Table 1. Motivated by the fact that the Hessian is not negligible, we shift our attention to second-order optimization methods and explore their convergence capabilities. 4 Convergence results Before delving into second-order methods, let us revisit a convergence result for first-order ones. Traditional analyses of the gradient descent (6) often rely on the smoothness and convexity of the loss, assumptions that may not hold in the context of deep learning. As an alternative, numerous results concentrate on the infinite-width limit, particularly in connection with the NTK analysis. While we refrain from presenting a formal proof, we highlight the notable result below. Theorem 4.1. Under Assumption B.1 on the PDE and Assumption B.2 on the network defined by (4), consider the scenario where m is sufficiently large. With high probability on the random initialization, there exists a constant µ > 0, depending on the eigenvalues of K, such that gradient descent, employing a sufficiently small step size η, converges to a global minimizer of (5) with an exponential convergence rate, i.e. L(θ(t)) ≤(1 −ηµ)tL(θ(0)). Proof. See [8], [4], [22], and others. It is noteworthy that this result is presented at the level of gradient descent, i.e. the discretization of the gradient flow (6), which explains the constant η representing its step size. Theorem 4.1 has also been extended to various types of architectures in [5]. We emphasize that this convergence result is rooted in the applicability of the Polyak-Lojasiewicz condition which, in turn, is linked to the smallest eigenvalue of the tangent kernel (denoted with λmin). In the case of linear PDEs, the tangent kernel K(t) is positive definite [8] for any t ∈[0, T], leading to positive eigenvalues. The key finding in this context is that if m is sufficiently large, K(t) ≈¯K, where ¯K is a deterministic matrix, which only 5 depends on the training input and not on the network’s parameters θ. As a result, in the infinite-width regime, the dynamics (8) can be approximated by ∂tuθ(xb) ∂trθ(xr) ≈−¯K uθ(xb) −g(xb) rθ(xr) . (11) In the linear case, the key steps (i.e. the fact that the NTK is deterministic and constant) of the convergence proof of Theorem 4.1 cannot be adapted to nonlinear PDEs. Indeed, the stochasticity of the matrix and its dynamic behavior during training make the reasoning of [8] or [4] inapplicable, and it is challenging to show that the eigenvalues of K(t) in the nonlinear case are uniformly bounded away from zero over training time. Nevertheless, we believe this question warrants further investigation. Another issue linked to the NTK’s eigenvalues is the phenomenon recognized as spectral bias by [39, 2, 29]. This is related to the fast decay of the NTK’s eigenvalues, which characterize the rate at which the training error diminishes. The presence of small or unbalanced eigenvalues leads to slow convergence, particularly for high-frequency components of the PDE solution, or even to training failure. This occurs regardless of the linearity of the PDE differential operator R. In the next section, we show that under certain assumptions, second-order methods can help mitigate both problems. 4.1 Second-Order Optimization Methods Due to all the aforementioned reasons and Proposition 3.7, our focus turns to the investigation of second-order optimization methods. These are powerful algorithms that leverage both the gradient and the Hessian of the loss function. Within this category, Quasi-Newton methods stand out as the most natural and widely known, relying on the Newton update rule θ(t + 1) = θ(t) − ∇2L(θ(t)) −1 ∇L(θ(t)). (12) However, the application of this update step relies on second-order derivatives, which are prohibitively expensive to compute as the number of parameters in the model increases. Indeed, the core idea behind Quasi-Newton methods involves utilizing an approximation of the Hessian as follows ∇2L(θ) = JT (t)J(t) + Hrrθ(t) ≈JT (t)J(t) (13) in the formula (12). Here, J(t) ∈Rn×p represents the Jacobian of the loss at the training time t, and it aligns with the definition in (9). Since the Jacobian J(t) is part of the evaluation of the gradient, the approximation (13) does not necessitate the computation of higher-order derivatives. We now tackle the issues of spectral bias and slow convergence by presenting a result applicable to the Gauss-Newton method. In practice, when the number of parameters p is larger than the number of samples n, the matrix JT (t)J(t) is surely singular. In this case, we consider the generalized inverse (JT (t)J(t))†, instead of the inverse. Theorem 4.2. Consider the parameter θ(t) obtained by the Gauss-Newton flow below ∂tθ(t) = −(JT (t)J(t))†∇L(θ(t)). (14) Then, the following holds ∂tuθ(t)(xb) ∂trθ(t)(xr) = −U(t)D(t)U(t)T uθ(t)(xb) rθ(t)(xr) , (15) where U(t) ∈Rn×n is a unitary matrix and D ∈Rn×n is a diagonal matrix with entries 0 or 1. In particular, if J(t) is full-rank for any t ∈[0, T], then convergence to a global minimum is attained. Proof. The proof is presented in Appendix F. This result is significant as it indicates that when utilizing second-order methods via (14), convergence no longer depends on the eigenvalues of K(t) as in (11), but rather on the elements of the diagonal matrix D(t). Consequently, the training process becomes nearly spectrally unbiased, as the nonzero eigenvalues of the controlling matrix in (15) are all 1s. Let us now compare the cases of linear and nonlinear PDEs, in relation to the assumption of full-rankness of J(t) and, consequently, the NTK. 6 • Linear PDEs: recent research [8] has theoretically confirmed that the NTK has full-rank in this case. Hence, convergence of second-order methods is achieved with all eigenvalues equal to 1, offering a notable advantage over (11) since the training method is unaffected by the spectral bias. • Nonlinear PDEs: showing theoretically the full-rankness is a complicated task, particularly in light of Theorem 3.5, which highlights the stochastic and dynamic nature of the NTK. Similarly, verifying numerically the full-rankness of J(t) is impractical due to the matrix’s ill-conditioning, as mentioned in [40]. However, even if J(t) is not full-rank, it holds that, although some singular values are zero, fast convergence for the remaining ones is attained. Moreover, let us stress that the result in Theorem 4.2 applies to any network, including those with finite width. Thus, while the NTK model motivates the use of second-order methods, the key insights about spectral bias and convergence hold without assuming infinite width. Remark 4.3. In practice, the Gauss-Newton method becomes less computationally expensive when combined with inexact techniques such as Krylov subspace methods, conjugate gradient, BFGS, or LBFGS [27]. It has been shown that BFGS and LBFGS asymptotically approach the exact Hessian under certain conditions [21]. To extend our findings to more practical inexact methods, we can leverage these asymptotic convergence properties. However, while this approach is theoretically sound, the speed of convergence of quasi-Newton methods to the exact Newton method — specifically their matrix approximation accuracy — depends on the minimum eigenvalue of the Hessian [21][Theorem 6]. As discussed in our paper, the Hessian in PINNs is typically very poorly conditioned. As a result, quasi-Newton methods may require an impractically large number of training steps to converge to the true inverse Hessian and, thus, to begin training higher modes. 5 Numerical Experiments 102 103 104 105 m 1.25 1.50 1.75 2.00 2.25 2.50 2.75 K(0) 102 103 104 105 m 0 1 2 3 4 K(0) 0 5000 10000 15000 Iterations 0 1 2 3 4 K(t) 1e3 0 5000 10000 15000 Iterations 0.0 0.5 1.0 1.5 2.0 1e4 m= 100 m= 500 m= 2000 m= 5000 m= 10000 m= 100000 (a) (b) Figure 1: (a) Mean and standard deviation of the spectral norm of K(0) as a function of the number of neurons m for 10 independent experiments. Left: linear case. Right: nonlinear case. (b) Mean and standard deviation of ∆K(t) := ∥K(t)−K(0)∥ ∥K(0)∥ over the network’s width m, for 10 independent experiments. Left: linear case. Right: nonlinear case. 5.1 Empirical validation of our NTK results First of all, we aim at numerically validate the results presented above, by comparing the NTK in case of linear and nonlinear PDEs. Our experiments are conducted on the following linear equation: ∂2 xu(x) = 16 π2 sin( 4 πx). Meanwhile, as nonlinear PDE, we consider u(x)∂xu(x) = 16 π2 sin( 4 πx). Notably, these results exhibit consistency across various equations and experimental setups. The result in Theorem 3.4 is confirmed by the numerical experiments depicted in Figure 1, part (a): in the linear case the NTK at initialization converges to a deterministic matrix when m →∞, while this does not happen in the nonlinear case. The statement of Proposition 3.5 is confirmed in part (b) of Figure 1 by showing that the constancy of the NTK during training is not attainable in the nonlinear case. Moreover, the result in Proposition 3.7 is supported by part (a) of Figure 2, where we compare the sparsity of the Hessian at initialization Hr(0) in both the linear and nonlinear case. Moreover, we observe that in the linear scenario ∥Hr∥decays as m grows, contrarily to the nonlinear example. Similarly, we refer to Figure 2, part (b) for a comparison of the eigenvalues when training with first-order or second-order methods on Burgers’ equation. 7 Hr(0) linear Hr(0) nonlinear 500 1000 1500 2000 m 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Hr 500 1000 1500 2000 m 1.4 1.5 1.6 1.7 1.8 1.9 Hr 100 101 102 103 104 10 17 10 14 10 11 10 8 10 5 10 2 101 first-order second-order (a) (a) (b) Figure 2: (a) Left: in yellow the non-zero components of the Hessian matrix at initialization (up in the linear case, down the nonlinear one). Center: mean and standard deviation of the spectral norm of the Hr(0) over m in the linear case (for 10 independent experiments). Right: same as Center, but for a nonlinear PDE. (b) Eigenvalues of K(0) for a first-order optimizer and D(0) for a second-order method applied to Burgers’ equation. 5.2 Employment of second-order methods Among all second-order methods, in our numerical experiments we make use of an existing variant of the Levenberg-Marquardt (LM) algorithm, as it offers further stability through the update rule θ(t + 1) = θ(t) − JT (t)J(t) + λIdp −1 ∇L(θ(t)), where λ is a damping parameter adjusted by the algorithm. In practice, the iterative step of LM can be considered as an average, weighted by λ, between the Gradient Descent step and a Gauss-Newton method. This aspect of the LM algorithm represents its crucial advantage over other Quasi-Newton methods such as Gauss-Newton or BFGS. Indeed, Quasi-Newton methods show good performance when the initial guess of the solution uθ is close to the correct one. The update rule of LM avoids this issue by relying on simil-gradient descent steps at early iteration. Moreover, the parameter λ typically decreases during training, in order to converge to a Quasi-Newton method when close to the optimum. Our primary aim is to showcase the effectiveness of second-order methods for nonlinear PINNs, a point which has been supported by findings such as those in [26]: their approach also employs a second-order method, akin to a Gauss-Newton method in function spaces. For details on the modified LM algorithm, along with pseudocode, we refer to Appendix G. Details on the Networks The neural network architectures adopted in the experiments are standard Vanilla PINNs with hyperbolic tangent as activation function. All of the PINNs trained in our analysis are characterized by 5 hidden layers with 20 neurons each. Every training is performed for 10 independent neural networks initialized with Xavier normal distribution [10]. All models are implemented in PyTorch [28] and trained on a single NVIDIA A10 GPU. Test Cases We assess our theoretical findings on the following equations: • Wave/Poisson/Convection Equation: despite being linear PDEs, they represent a suitable scenario to showcase the detrimental effect of the spectral bias on the training of PINNs, due to the presence of high-frequency components in the solution. • Burgers’ Equation: this nonlinear PDE is commonly used to test PINNs, and usually they reach a valid solution even with a first-order optimizer, due to the PDE’s simplicity. • Navier-Stokes Equation: it poses challenges for both PINNs and classical methods, being a difficult nonlinear PDEs. We test the case of the fluid flow in the wake of a 2D cylinder [17]. For the sake of compactness, we refer to Appendix G for detailed descriptions of the mentioned PDEs, and to Appendix H for supplementary numerical experiments not included in the main text. We compare results obtained by the LM algorithm with those from commonly used optimizers for training PINNs, such as Adam [18] and L-BFGS [24]. Where not stated otherwise, Adam is trained for 105 iterations and LM for 103 iterations. Additionally, we provide a comparison with other methods that are ad-hoc enhancements of PINNs, such as loss balancing [40] (also known as NTK rescaling), Random Fourier Features (RFF) [39], and curriculum training (CT) [19]. Our performance metric is the relative L2 loss on the test set, detailed in Appendix H formula (28). 8 102 103 104 105 m 1.25 1.50 1.75 2.00 2.25 2.50 2.75 K(0) 102 103 104 105 m 0 1 2 3 4 K(0) (a) (b) 10 2 10 1 100 101 Convection Coefficient 10 6 10 5 10 4 10 3 10 2 10 1 100 101 Final L 2 Loss Adam Adam + CT LM LM + CT 0 20000 40000 60000 80000 100000 Iterations 10 8 10 6 10 4 10 2 100 102 L2 Loss during training Adam Adam + RFF 0 1000 2000 3000 4000 5000 Iterations 10 8 10 6 10 4 10 2 100 102 LM LM + RFF 0.0 0.2 0.4 0.6 0.8 1.0 x 0 1 2 3 4 5 6 0.88 0.66 0.44 0.22 0.00 0.22 0.44 0.66 0.88 10 2 10 1 100 101 Convection Coefficient 10 6 10 5 10 4 10 3 10 2 10 1 100 101 Final L 2 Loss Adam + CT LM LM + CT Figure 3: (a) Poisson equation: median and standard deviation of the relative L2 loss for different optimizers over training iterations (repetitions over 10 independent runs). (b) Convection equation: median and standard deviation of the L2 loss after 1000 iterations achieved over 5 independent runs with and without CT for different values of the convection coefficient β (left) and solution obtained with LM (and no other enhancement) after 5000 iterations with β = 100 (right). 0 100 200 300 400 500 GPU time (s) 10 5 10 4 10 3 10 2 10 1 100 relative L2 loss Adam LBFGS LM (a) 0 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 relative L 2 error on u 0 10 0.0 0.2 0.4 0.6 0.8 1.0 relative L 2 error on v 0 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 relative L 2 error on p Adam + Causality LM + Causality (b) Figure 4: (a) Burgers’ equation: mean and standard deviation of the relative L2 loss for various optimizers over wall time (repetitions over 10 independent runs). (b) Navier-Stokes equation: mean and standard deviation of the relative L2 loss over the PDE time τ for PINNs trained with Adam and LM (10 independent runs). Both optimization methods are enhanced with causality training. Linear PDEs affected by spectral bias In Figure 3, we demonstrate the effectiveness of secondorder methods in handling equations with high spectral bias. Part (a) of Figure 3 focuses on the Poisson equation with high-frequency components, for which is common to use RFF [39]. On the left, we show that Adam requires RFF to converge to a reasonable solution. On the right, we observe that LM not only significantly outperforms Adam combined with RFF, but also that incorporating RFF with LM leads to remarkable loss reduction from the very first iterations. In Part (b) of Figure 3, we investigate the effect of high convection coefficients β in the convection equation as discussed in [19], where it is shown that a PINN trained with Adam necessitates of curriculum training to achieve meaningful results on such a spectrally biased PDE. However, we show on the left Figure 3, part (b), that the LM optimizer can handle higher values of β, especially when curriculum training is introduced. Remarkably, on the right of Figure 3(b), we show that a PINN trained with LM, without any other enhancements, achieves high accuracy with β values up to 100. This level of accuracy is not feasible with Adam and curriculum training alone, which, as noted in [19], manages coefficients only up to 20. Nonlinear PDEs Firstly, we consider the case of Burgers’ equation, where convergence is achievable even with first-order methods. To address concerns about the additional computational time required by second-order methods, in Figure 4, part (a), we display the relative L2 loss over wall time when training on Burgers’ equation. All training methods can reach a reasonable solution, however, while the precision of PINNs trained with Adam and L-BFGS is approximately 10−3, PINNs trained with LM can consistently attain precision around 10−5 in few iterations and very short GPU time. Figure 4 also provides a qualitative estimate of the runtime of LM in comparison to Adam and L-BFGS. The intermediate performance of L-BFGS, falling between first- and second-order methods, is explained in Remark 4.3. Lastly, a similar outcome can be seen in part (b) of Figure 4, where we demonstrate that employing the LM optimizer makes it possible to obtain a reasonable solution even for Navier-Stokes equation in terms of relative L2 loss over PDE time. Notice that in this case, we employ causality training [41] for both Adam and LM. 9 5.3 Limitations and possible solutions The major limitation of our findings is related to scalability. Traditionally, second-order methods have been avoided for machine learning models due to their poor scaling with an increasing number of parameters. However, one can adopt classical PDE solution approaches, such as domain decomposition, to utilize a collection of smaller networks instead of a single large one. Similarly, one can embrace machine learning-based solutions such as ensemble models [12] or mixture of experts [11]. We advocate that existing models such as [14, 25, 37] could already be strongly enhanced with the usage of second-order methods for training. In the scenario where these approaches are impractical, one could also resort to techniques in the field of optimization to enable the scalability of the method. For medium to large-sized networks, the challenge of storing the matrix JT J in GPU memory becomes infeasible. This can be addressed through an inexact LM method, which involves solving the equivalent system ∥Jθ −rθ∥= 0 using a Krylov subspace iterative method (LSQR or LSMR) [27, 7]. These methods only require Jacobian-vector products, which can be efficiently computed through backpropagation. 6 Conclusion In this paper, we conduct an in-depth analysis of PINNs training utilizing the NTK framework. We elucidate the distinction between linear and nonlinear cases, and reveal that even in the optimistic infinite-width limit, favorable outcomes observed with NTK in linear cases do not extend to nonlinear PDEs. Motivated by the NTK anaylsis, we emphasize the significant advantage of employing second-order methods. These seem to mitigate the spectral bias issue and to improve convergence even for challenging nonlinear PDEs. Second-order methods, such as LM, consistently achieve a precision comparable or even better than the state-of-the-art presented in [13]. 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Physica D: Nonlinear Phenomena, 457:133987, 2024. 12 Supplemental Material This supplemental material is divided into the following eight appendices. • Appendix A: Details about the NTK Matrix • Appendix B: Standard NTK results for linear PDEs • Appendix C: Proof of Theorem 3.4 • Appendix D: Proof of Proposition 3.5 • Appendix E: Proof of Proposition 3.7 • Appendix F: Proof of Theorem 4.2 • Appendix G: Details about the Numerical Experiments • Appendix H: Further Numerical Experiments In the following we denote with ∥· ∥2 and ⟨·, ·⟩the Euclidean and scalar product on Rd, respectively. The Euclidean ball centered in x with radius R is indicated with B(x, R). We denote with ∥· ∥the spectral norm of a matrix and with In the identity matrix of dimension n × n. We abbreviate with i.i.d. independently and identically distributed random variables. E[X] denotes the mean of the random variable X ∈Rd, while Cov[X] is its covariance matrix. Convergence of Xn to X in distribution is indicated with Xn D →X, while convergence in probability with Xn P→X. GP denotes a Gaussian Process. The operator ∇denotes the gradient of a function on Rd, while ∂xf(x, y) the partial derivative of f with respect to the variable x. A Details about the NTK Matrix We define the following matrices ∂θuθ(x) = [∂θ1uθ(x) · · · ∂θmuθ(x)] , ∂θrθ(x) = [∂θ1rθ(x) · · · ∂θmrθ(x)] , ∂θΦ[uθ](x) =   ∂θ1Φ1[uθ](x) · · · ∂θmΦ1[uθ](x) ... · · · ... ∂θ1Φk[uθ](x) · · · ∂θmΦk[uθ](x)  . By ∂θuθ(xb), ∂θrθ(xr) and ∂θΦ[uθ](xr) we mean the same matrices as before, calculated in each xb i (xr i respectively) and stacked vertically, e.g.: ∂θΦ[uθ](xb) =   ∂θΦ[uθ](xb 1) ... ∂θΦ[uθ](xb Nb)  . The only exception is given by: ∇R(Φ[uθ](xr)) =   ∇R(Φ[uθ](xr 1)) 0 · · · · · · 0 0 ∇R(Φ[uθ](xr 2)) 0 · · · 0 ... · · · · · · · · · ... 0 · · · · · · 0 ∇R(Φ[uθ](xr Nr))  . (16) 13 While the Hessians have the following structure: Hu(x) :=   ∂2 θ1θ1uθ(x) · · · ∂2 θ1θmuθ(x) ... · · · ... ∂2 θmθ1uθ(x) · · · ∂2 θmθmuθ(x)  , Hr(x) :=   ∂2 θ1θ1rθ(x) · · · ∂2 θ1θmrθ(x) ... · · · ... ∂2 θmθ1rθ(x) · · · ∂2 θmθmrθ(x)  , HΦi(x) :=   ∂2 θ1θ1Φi[uθ](x) · · · ∂2 θ1θmΦi[uθ](x) ... · · · ... ∂2 θmθ1Φi[uθ](x) · · · ∂2 θmθmΦi[uθ](x)  . (17) B NTK for linear PDEs First of all, we list here all the assumptions needed on the differential operator R and the neural network in (4). Assumption B.1 (on R). The differential operator R is linear, which implies that R is linear. Assumption B.2 (on the network). Given the network (4), we assume the following properties: (i) there exists a constant C > 0 such that all parameters of the network are uniformly bounded for t ∈[0, T], sup t∈[0,T ] \|\|θ(t)\|\|∞≤C with C independent from m. (ii) there exists a constant C > 0 such that Z T 0 Nb X i=1 (uθ(τ)(xb i) −g(xb i)) dτ ≤C, Z T 0 Nr X i=1 (Φ[uθ(τ)](xr i )) dτ ≤C. (iii) the activation function σ and as well as its derivatives σ(i) up to power k + 1 are smooth and \|σ(i)\| ≤C for i = 1, . . . , k, where σ(i) denotes the i-th order derivative of σ. In order to present the results, we denote with Hr the Hessian of the residuals rθ with respect to the parameters θ. The Hessian plays an important role in Proposition B.3, which aims to list all the prior results that can be derived by combining Theorem 4.4 of [40], Theorem 3.2 of [22]. Proposition B.3. Consider a fully-connected neural network given by (4), under the Assumption B.2 on the network and Assumption B.1 on the PDE. For the minimization of the loss function (5) through gradient flow, starting from a Gaussian random initialization θ(0), it holds that for any T > 0, • the randomly initialized tangent kernel K(0) converges in probability to a deterministic kernel ˜K as m →∞; • the Hessian matrix Hr of the residuals is sparse and \|\|Hr\|\| = O 1 √m , hence the spectral norm converges to 0 as m →∞; • as a consequence, the NTK is nearly constant during training, i.e. lim m→∞sup t∈[0,T ] ∥K(t) −K(0)∥2 = 0; Proof. The proof can be found in the papers mentioned above or as a special (linear) case in Appendix C-E. 14 C Proof of Proposition 3.4. First of all, we derive a result about the behavior of the vector of partial derivatives Φ[u]. The Proposition C.1 below is a generalization of Theorem 4.1 in [40] for any derivative of order k. This means that there are no nonlinearities involved, since these are encoded in the function R. Moreover we study the full vector and not each component separately as it is done in [40]. This is needed in the following proofs. Proposition C.1. Consider a fully-connected neural network of one hidden layer as in (4), under Assumption B.2. Then, starting from θ(0) i.i.d. from N(0, Id), it holds that Φ[uθ(0)](x) D →GP(0, Σ(x, x′)) for any x, x′ ∈Ω, as m →∞, where D means convergence in distribution and Σ is explicitly calculated. Proof. To ease the notation, we omit the initial time 0 and denote uθ(0) with uθ. Similarly, all the weights matrices and biases W 1(0), W 0(0), b1(0), b0(0) are indicated with W 1, W 0, b1, b0. Now according to the definition of Φ and the fact that it is linear, we obtain that Φ[uθ](x) = 1 √mW 1 · Φ[σ(W 0x + b0)] = 1 √m m X j=1 W 1 j Φ[σ(W 0 j x + b0 j)]. According to our assumptions, W 1 j Φ[σ(W 0 j x + b0 j)] are i.i.d. random variables. We prove below that their moments are finite, hence by the multidimensional Central Limit theorem (CLT) we can conclude that, for every x ∈Ω, Φ[uθ](x) D →N(0, Γ(x)), with covariance matrix: Γ(x) = Covu,v∼N(0,1) [Φ[σ(ux + v)]] . Now we compute the covariance of the limit gaussian process. In order to do so, we first need to show that Φi[uθ](x) are uniformly integrable with respect to m for every i = 1, ..., k. It follows from: sup m E[\|Φi[uθ](x)\|2] = sup m E  1 m m X j,l=1 W 1 j W 1 l Φi[σ(W 0 j x + b0 j)]Φi[σ(W 0 l x′ + b0 l )]   = sup m E  1 m m X j=0 (W 1 j )2Φi[σ(W 0 k x + b0 k)]2  = E Φi[σ(W 0 j x + b0 j)]2 ≤C2τ 2, where C = max1≤i≤k \|\|σ(i)\|\|∞and σ(i) indicates the i-th order derivative of σ, while τ = max1≤i≤k Ey∼N(0,1)[\|y\|i] < ∞. Now, for any given point x, x′ ∈Ωwe have that Σ(x, x′)i,j = lim m→∞E [Φi[uθ](x)Φj[uθ](x′)] = = lim m→∞E  1 m m X l1,l2=1 W 1 l1W 1 l2Φi[σ(W 0 l1x + b0 l1)]Φj[σ(W 0 l2x′ + b0 l2)]  = = lim m→∞E " 1 m m X l=1 (W 1 l )2 Φi[σ(W 0 l x + b0 l )]Φj[σ(W 0 l x′ + b0 l )] # = = Eu,v∼N(0,1) [Φi[σ(ux + v)]Φj[σ(ux′ + v)]] . 15 Lemma C.2. Consider a fully-connected neural network of one hidden layer as in (4), under Assumption B.2. Let us define KΦ(0) = ∂θuθ(0)(xb) ∂θΦ[uθ(0)](xr) ∂θuθ(0)(xb)T ∂θΦ[uθ(0)](xr)T , where Φ is the collection of all the partial derivatives of u, as in (3), and θ(0) ∼N(0, Id) i.i.d.. It follows that KΦ(0) converges in probability to a deterministic limiting kernel as m →∞. Proof. The component ∂θuθ(0) is linear, hence it is standard as in [40], Lemma 3.1. While the rest of the matrix needs to be generalized to any derivative Φi for i = 1, . . . , k. For any i, j = 1, . . . , k and every x, x′ ∈Ωconsider each entry ∂θΦi[uθ(0)](x) ∂θΦj[uθ(0)](x′)T = 4m X l=1 ∂θlΦi[uθ(0)](x) ∂θlΦj[uθ(0)](x′) = 4m X l=1 Φi ∂θluθ(0) (x) Φj ∂θluθ(0) (x′) where the second equality follows from Schwarz theorem (because of the smoothness of the derivatives of u), and the linearity of the operator Φ. This sum has to be split in 4 parts, one for each possible type of θl (in W 1, W 0, b0 or b1). Here we present the case when θl = W 1 l , while the other cases are analogous: m X l=1 Φi h ∂W 1 l uθ(0) i (x) Φj h ∂W 1 l uθ(0) i (x′) = = 1 m m X l=1 Φi σ(W 0 l (0)x + b0 l (0)) Φj σ(W 0 l (0)x′ + b0 l (0)) P→Eu,v∼N(0,1) [Φi[σ(ux + v)] Φj[σ(ux′ + v)]] , and the limit in probability in the last line comes from the law of Large Numbers. Lemma C.3. Suppose that there exist R > 0 and ϵ > 0 such that ∀θ ∈B(θ(0), R) it holds ∥Hu(xb)∥< ϵ, ∥HΦj(xb)∥< ϵ ∀j = 1, ..., k. Then maxθ∈B(θ0,R) ∥KΦ(t) −KΦ(0)∥= O(ϵR). Proof. Using the properties of the spectral norm, we just need to bound each block of J(t) as follows ∥J(t) −J(0)∥≤ Nr X i=0 ∥∂θΦ[uθ(t)](xr i ) −∂θΦ[uθ(0)](xr i )∥+ Nb X i=0 ∥∂θuθ(t)(xb i) −∂θuθ(0)(xb i)∥ ≤kNr max i,j ∥∂θΦj[uθ(t)](xr i ) −∂θΦj[uθ(0)](xr i )∥ + Nb max i ∥∂θuθ(t)(xb i) −∂θuθ(0)(xb i)∥ ≤kNr max i,j max θ∈B(θ(0),R) ∥HΦj(xr i )∥ ∥θ −θ0∥ + Nb max i max θ∈B(θ(0),R) ∥Hu(xb i)∥ ∥θ −θ0∥ ≤max(kNr, Nb)ϵR Hence: ∥KΦ(t) −KΦ(0)∥= ∥J(t)J(t)T −J(0)J(0)T ∥≤∥J(t) −J(0)∥· (∥J(t)∥+ ∥J(0)∥) ≤max(kNr, Nb)ϵR(∥J(t)∥+ ∥J(0)∥) and the last norm is bounded on B(θ(0), R) by smoothness of the model. 16 Lemma C.4. Under Assumption B.1 on the PDE and Assumption B.2 on the network, then KΦ is nearly constant during training, i.e. lim m→∞sup t∈[0,T ] ∥KΦ(t) −KΦ(0)∥= 0. Proof. The statement follows by combining Lemma C.3 and Lemma E.1. Now we are in position to prove Theorem 3.4: Proof. (of Theorem 3.4) By using the chain rule on the residual term, we can explicitly compute: K(0) = ∂θuθ(xb) ∂θrθ(xr) ∂θuθ(xb)T ∂θrθ(xr)T = = ∂θuθ(xb) ∇R(Φ[uθ](xr))∂θΦ[uθ](xr) ∂θuθ(xb)T ∇R(Φ[uθ](xr))∂θΦ[uθ](xr)T = = Id 0 0 ∇R(Φ[uθ](xr)) \| {z } ΛR(0) ∂θuθ(xb) ∂θΦ[uθ](xr) ∂θuθ(xb)T ∂θΦ[uθ](xr)T \| {z } KΦ(0) Id 0 0 ∇R(Φ[uθ](xr))T \| {z } ΛR(0)T , where we have denoted θ(0) with θ and omitted the initial time step and ∇R(Φ[uθ](xr)) is defined in (16). Let us first observe that the linear part, i.e. KΦ(0), converges in probability to a deterministic limit by Lemma C.2. Moreover, Φ[uθ(0)] converges in distribution to a gaussian process by Proposition C.1. Regarding the nonlinear part denoted with ΛR(0), we know by assumption that ∇R is a continuous function, hence we can apply the Continuous Mapping Theorem and conclude that ∇R(Φ[uθ](x)) D →∇R (GP(0, Σ(x, x′))) for x, x′ ∈Ω. From this, the convergence of K(0) follows by Slutsky’s theorem. D Proof of Proposition 3.5 Proof. Recall that we denote with K(t) the NTK obtained with θ(t), evolving according to the gradient flow (6). Similarly, K(0) is the NTK at initialization, i.e. with θ(0) ∼N(0, Idm). We can rewrite the kernels in terms of their linear and nonlinear part as we did for the proof of Theorem 3.4, and obtain lim m→∞sup t∈[0,T ] ∥K(t) −K(0)∥≥lim m→∞∥K(tm) −K(0)∥ = lim m→∞∥ΛR(tm)KΦ(tm)ΛR(tm)T −ΛR(0)KΦ(0)ΛR(0)T ∥ ≥lim m→∞ ∥ΛR(tm)KΦ(0)ΛR(tm)T −ΛR(0)KΦ(0)ΛR(0)T ∥ −∥ΛR(tm)[KΦ(t) −KΦ(0)]ΛR(tm)T ∥ , where the last is obtained by applying the inverse triangular inequality, after summing and subtracting the needed terms. Moreover, by considering that supt∈[0,T ] ∥KΦ(t) −KΦ(0)∥→0 as m →∞by Lemma C.4, we obtain that lim m→∞sup t∈[0,T ] ∥K(t) −K(0)∥≥lim m→∞∥ΛR(tm)KΦ(0)ΛR(tm)T −ΛR(0)KΦ(0)ΛR(0)T ∥= = lim m→∞ Id 0 0 ∇R(Φ[u(tm)]) KΦ(0) Id 0 0 ∇R(Φ[u(tm)]) T − Id 0 0 ∇R(Φ[uθ(0)]) KΦ(0) Id 0 0 ∇R(Φ[uθ(0)]) T . 17 Observe that (10) implies that Φ[u(tm)] →Φ[u⋆], hence ∇R(Φ[uθ(t)]) →∇R(Φ[u⋆]) as m →∞ by continuity of ∇R. Combining this and Lemma C.2, we find lim m→∞ Id 0 0 ∇R(Φ[u(tm)]) KΦ(0) Id 0 0 ∇R(Φ[u(tm)]) T − Id 0 0 ∇R(Φ[uθ(0)]) KΦ(0) Id 0 0 ∇R(Φ[uθ(0)]) T = Id 0 0 ∇R(Φ[u⋆]) KΦ(0) Id 0 0 ∇R(Φ[u⋆]) T − Id 0 0 ∇R(Φ[uθ(0)]) KΦ(0) Id 0 0 ∇R(Φ[uθ(0)]) T . Finally, to prove our statement, we just need to show that the matrix above is not 0 almost surely, or at least one of its components. Let us fix a collocation point x ∈Ωand let us define the function f : Rk →R: f(w) := ∇R(Φ[u⋆](x))KΦ(0)(x,x)∇R(Φ[u⋆](x))T −∇R(w)KΦ(0)(x,x)∇R(w)T , (18) where KΦ(0)(x,x) denotes the kernel evaluation at a fixed collocation point. The first term on the right hand side of (18) is a deterministic vector, so f is a well defined deterministic analytic function. Moreover, if R is nonlinear, f is not identically zero. By the properties of analytic functions we can conclude that Leb({w ∈Rk\|f(w) = 0}) = 0, where Leb denotes the Lebesgue measure. Notice that Φ[uθ(0)](x) ∼N(0, Σ(x)) in the infinite-width limit as proven in Proposition C.1 and a consequence of that proof is that Σ(x) is not singular. This implies that P(f(Φ[uθ(0)](x)) = 0) = 0. E Proof of Proposition 3.7 We present here some preparatory results. Lemma E.1. For any i = 1...k and any x ∈Ω, the Hessian HΦi(x) as defined in (17) is such that ∥HΦi(x)∥= O( 1 √m). Proof. Recall that (HΦi(x))jl = ∂2 θjθlΦi[uθ](x), where l, j = 1, . . . , m. By the linearity of the operator Φi and the smoothness of the activation function as in Assumption B.2, it holds that ∂2 θjθlΦi[uθ] = Φi h ∂2 θjθluθ i . For a specific choice, e.g. first parameter is θj = W 1 j and the second is θl = W 0 l , it holds that ∂2 W 1 j W 0 l Φi[uθ](x) = Φi h ∂2 W 1 j W 0 l uθ i = 1 √mΦi[σ′(W 0 l x + b0 l )x] 1l=j ≤C 1 √m, (19) where the last inequality follow from Assumption B.1, Assumption B.2 and the boundedness of the domain Ω. Since the calculations of (19) are similar for every combination of parameters W 1, W 0, b0, we do not report them here. Furthermore, we notice that the derivatives involving b1 are zeros and hence we obtain that HΦi is composed by 9 blocks (3 × 3 combinations of parameters). Each block is a diagonal matrix, whose elements are bounded by C 1 √m. By considering that the spectral norm of a diagonal matrix is equal to the maximum of its components, we can bound the spectral norm of each 18 block by C 1 √m. Moreover the spectral norm of a matrix can be bounded by the sum of the spectral norm of its blocks, hence: ∥HΦi(x)∥≤9C 1 √m = O( 1 √m). We can now prove Proposition 3.7. Proof. In the nonlinear case the Hessian of the residuals is (Hr(x))j,l =∂2 θlθjrθ(x) = ∂θl(∇R(Φ[uθ](x)])∂θjuθ(x) = = ⟨∂θlΦ[uθ](x), ∇2R(Φ[uθ](x))∂θjΦ[uθ](x)⟩ \| {z } Aij + ∇R(Φ[uθ](x))HΦ(x) \| {z } Bij . for every collocation point x ∈Ω. The matrix HΦ is defined in (17). Moreover, Lemma E.1 provides that the spectral norm of B goes to 0 in the infinite-width limit. Moreover, by making use of the inverse triangular inequality, we obtain that for any x ∈Ω, it holds lim m→∞∥Hr(x)∥≥lim m→∞\|∥A∥−∥B∥\| = lim m→∞∥A∥. According to the definition of spectral norm, we have that lim m→∞∥A∥= lim m→∞max ∥z∥2≤1 ∥Az∥2 ≥lim m→∞∥A¯z∥2, where ¯z := h 1 √m 1 √m . . . 1 √m i . Let us now focus on the term ∥A¯z∥2. By using some standard inequalities and taking advantage of the fact that each entry of ¯z is 1 √m, we obtain that \|\|Az\|\|2 ≥ 1 √m\|\|Az\|\|1 ≥ 1 √m m X i=1 (Az)i = 1 m m X i,j=1 Aij = 1 m m X i,j=1 ⟨∂θiΦ[uθ](x), ∇2R(Φ[uθ](x))∂θjΦ[uθ](x)⟩= = * 1 √m m X i=1 ∂θiΦ[uθ](x), ∇2R(Φ[uθ](x)) 1 √m m X j=1 ∂θjΦ[uθ](x) + Without loss of generality, we can restrict our focus to θi = W 1 i and θj = W 1 j , since the spectral norm of a matrix is greater or equal then the norm of its submatrix, and study the term lim m→∞ 1 √m m X i=1 ∂θiΦ[uθ](x) = lim m→∞ 1 m m X i=1 Φ[σ(W 0 i · +b0 i )](x) = = Eu,v∼N(0,1) [Φ[σ(u · +v)](x)] =: w (20) by the law of large numbers. In particular, w is deterministic. Notice that here we have considered a generic θ since, according to Lemma C.4, ∂θiΦ is constant. By combining this result with the previous one, we obtain that lim m→∞∥Hr(x)∥≥wT ∇2R(Φ[uθ](x))w ≥˜c where ˜c is a deterministic constant that does not depend on m, but only on the value of ∇2R (which is constant because R is a second-order polynomial) and on the vector w defined in (20). F Proof of Theorem 4.2 Proof. The gradient flow equation in case of Gauss-Newton methods has been defined in (14) for J(t) ∈Rn×p where t ∈[0, T]. It follows that ∂tuθ(t) ∂trθ(t) = ∂θuθ(t) ∂θrθ(t) ∂tθ(t) = J(t)∂tθ(t) = −J(t)(JT (t)J(t))†JT uθ(t) rθ(t) , 19 where the last equality comes from plugging in (14) into the equation. Now, let us consider the case when p >> n, then the singular value decomposition of J(t) is as follows J(t) = U ˜Σn 0p−n \| {z } Σ V T , where U ∈Rn×n, Σ ∈Rn×p, V ∈Rp×p and ˜Σn ∈Rn×n is a diagonal matrix with elements given by the square roots of the eigenvalues of the NTK. We drop the dependence on time t of U, Σ and V to ease the notation. Let us now study the term J(t)(JT (t)J(t))†JT (t) = UΣT V T (V ΣT U T UΣV T )†V ΣT U T = UΣV T V (ΣT Σ)†V T V ΣT U T = UΣ(ΣT Σ)†ΣT U T = U ˜Σn 0p−n ˜Σn 0p−n ˜Σn 0p−n † ˜Σn 0p−n U T = U ˜Σn 0p−n ˜Σ2 n 0p−n 0p−n 0n † ˜Σn 0p−n U T = U ˜Σn(˜Σ2 n)† ˜ΣnU T = UDU T where D is obtained from ˜Σn by replacing the non-zero components with 1. In particular we can rewrite the Gauss-Newton flow as: ∂tuθ(t) ∂trθ(t) = −UDU T uθ(t) rθ(t) . Notice that it has the same form of the gradient flow in Lemma 3.2 but the Neural Tangent Kernel is replace by a matrix with non-zeroes eigenvalues 1. This can be translated as: second-order optimizers are almost spectrally unbiased. Moreover if J(t) stays full rank during the training, we can obtain the result of convergence regardless of the singular values of J(t), i.e.: ∂tuθ(t) ∂trθ(t) = − uθ(t) rθ(t) . . G Details about the Numerical Experiments G.1 The LM Algorithm In the following, we provide a more detailed description of the version of the Levenberg-Marquardt algorithm along with its pseudocode and the details of the experiments whom results are shown in Section 5. The main difference between the Levenberg-Marquardt algorithm and other Quasi-Newton method is that general Quasi-Newton methods are line-search approaches, while LM is a trust region approach. In practice, line search approaches determine a descent direction of the loss function and thereinafter determine a suitable step size in such direction. On the other hand, a trust region method determines an area where the solution lies and computes the optimal step. If this step does not provide enough improvement in the objective function, the search area is reduced and the search is performed once more. We refer to [27] for a thorough description of trust region and line search methods. In the following part, we drop the dependence on training time as a continuous function and identify f(tk) = fk for some discrete time tk. As already mentioned in Section 5, the update step vk of the LM algorithm is computed follows: vk = − JT k Jk + λDk −1 ∇L(θk), (21) where Dk is a diagonal matrix of size n × n. In the classical LM algorithm, this matrix Dk is given by the identity matrix. Another viable alternative recommended in [6] is to use the diagonal of JT k Jk. 20 For our model, we choose Dk to be simply the identity matrix, which appears to be more stable when JT k Jk is singular. Another typical modification to the Levenberg-Marquardt algorithm is the introduction of the geodesic acceleration [36]. ak = − JT k Jk + λDk −1 vkHrvk. (22) The goal of the geodesic acceleration is to introduce a component which does consider all the components of the Hessian of the loss when the residuals are not small and when the Hessian of the residuals is not negligible. Moreover, at every iteration, one has to specify a criterion Ck whose objective is to evaluate the relative improvement of the model parameterized by θk with respect to the update step vk. The criterion depends on the modification of the LM algorithm chosen. For our algorithm we use the same condition as [9] i.e. Ck < toll where Ck is defined as Ck = L(θk)2 −L(θk + vk)2 ⟨vk, λkDkvk + ∇L(θk)⟩. (23) We provide in Algorithm 1 the pseudocode of the modified LM algorithm that we chose for our numerical experiments, inspired by the implementation of [9] and modifying it by adding the component of the geodesic acceleration. Algorithm 1 Modified Levenberg-Marquardt Algorithm Input: Maximum region area Λ > 0, Region Radius 0 < λ0 < Λ, Tollerance tol ∈[0, 1 4), α ∈[0, 1) for k = 0, 1, 2, . . . do Compute vk as in Equation (21) Compute criterion Ck as in Equation (23) while Ck < tol do λ = min(2λ, Λ) Compute vk with the new value of λ Compute criterion Ck as in Equation (23) end while θk+1 = θk + vk λk+1 = max( 1 3λ, Λ−1) Compute ak as in Equation (22) if 2\|\|ak\|\| ≤α\|\|vk\|\| then θk+1 = θk+1 + 1 2ak end if end for The main focus of the Levenberg-Marquardt method is to decide the size of the trust region. In practice, at every iteration, one wants to find a better solution and afterwards reduce the size of the trust region. When this does not happen, the solution is to enlarge the trust region in order to look for a better solution. In our method we choose to include the region search as part of the inner loop, as for line search approaches. This means that the iteration itself can be slower, but more accurate, which is why we include in the numerical evaluation also the computational time. G.2 Poisson Equation The Poisson equation that we choose for our study is a monodimensional instance of the PDE defined in [39] for x ∈Ω= [0, 1] and we try to find the solution u : Ω→R. In particular, we want to solve the following equation: ∂2 xu = f(x), x ∈Ω, u(0) = u(1) = 0. (24) As in [39], the function f is constructed in such a way that the exact solution of Equation (24) is given by: u(x) = sin(2πx) + 1 10 sin(50πx). 21 This approach is done to evaluate the behavior of PINNs when the target solution presents a high frequency and a low frequency component. We then train the PINN model by sampling Nr = 103 points in Ωwith latin hypercube sampling. G.3 Wave Equation We opt to solve the wave equation below for each (x, τ) ∈Ω= [0, 1]2 and aim to find the solution u : Ω→R . In particular, we aim to solve the following equation: ∂2 τu = −C2∂2 xu, (x, τ) ∈Ω, u(x, 0) = sin(πx) + 1 2 sin(4πx), x ∈[0, 1], ∂τu(x, 0) = 0 x ∈[0, 1], u(0, τ) = u(1, τ) = 0, τ ∈[0, 1]. (25) With C being equal to 2 for our case. It is straightforward to obtain the correct solution of this equation through Fourier transform. In particular, the exact solution of Equation (25) is given by: u(x, τ) = sin(πx) cos(2πτ) + 1 2 sin(4πx) cos(8πτ). We then train a PINN by sampling Nr = 104 training points in Ωfor the PDE residuals with latin hypercube sampling, and Nb = 3 · 103 points for training the model against the correct solution at ∂Ω. G.4 Burgers’ Equation Burgers’ equation is a 1D version of Navier-Stokes equations. Its solution at high times present a discontinuity, which makes it challenging for spectrally biased architectures. The specific instance chosen in our numerics for Burgers’ equation is the same as in [30]. In particular, we refer to the exact same data provided by the authors. In particular, given (x, τ) ∈Ω= [−1, 1] × [0, 1], we solve for u : Ω→R the following equation: ∂τu + u∂xu −ν∂2 xu = 0, (x, τ) ∈Ω, u(x, 0) = −sin(πx), x ∈[−1, 1], u(−1, τ) = u(1, τ) = 0, τ ∈[0, 1], (26) with the diffusivity ν being equal to 0.01 π for this specific instance. The correct solution is provided publicly by the authors of [30]. Training is performed with Nr = 104 collocation points for training the PDE residuals, sampled with latin hypercube sampling, and Nb = 3 · 103 points for training the boundary and initial condition in ∂Ω. G.5 Navier-Stokes Equation The most interesting scenario taken in consideration for our experiments is that of Navier-Stokes equations. In particular, we aim to solve the fluid flow in the wake of a cylinder in 2D tackled in [17]<. In particular, we have (x, y, t) ∈Ω= [2.5, 7.5] × [−2.5, 2.5] × [0, 16] and we wish to find ⃗u : Ω→R3 which is defined as ⃗u(x, y, t) = [u(x, y, t), v(x, y, t), p(x, y, t)]T . In particular u and v are respectively the horizontal and vertical components of the fluid velocity and p is the pressure at a point. Navier-Stokes equations are then expressed in vectorizer form as follows: ∂τu + u∂xu + v∂yu −1 Re ∂2 xu + ∂2 yu + ∂xp = 0, (x, y, τ) ∈Ω, ∂τv + u∂xv + v∂yv −1 Re ∂2 xv + ∂2 yv + ∂yp = 0, (x, y, τ) ∈Ω, ∂xu + ∂yv = 0, (x, y, τ) ∈Ω, u(x, y, 0) = gu0(x, y), (x, y) ∈[2.5, 7.5] × [−2.5, 2.5], v(x, y, 0) = gv0(x, y), (x, y) ∈[2.5, 7.5] × [−2.5, 2.5], u(2.5, y, τ) = 1, (y, τ) ∈[−2.5, 2.5] × [0, 16], v(2.5, y, τ) = 0, (y, τ) ∈[−2.5, 2.5] × [0, 16], (27) 22 0 2500 5000 7500 10000 12500 15000 17500 20000 Iterations 10 1 100 relative L2 loss Adam LBFGS LM 0 500 1000 1500 2000 10 1 100 Figure 5: Mean and standard deviation of the relative L2 loss on the test set on the Wave equation for Adam, L-BFGS and LM optimizer over iterations (repetition over 10 independent runs). where Re represents the Reynolds’ number, which is an adimensional quantity defined by the problem and is set to 100 for our case. The initial conditions (gu0, gv0) can be found in the repository published by the authors of [32], as well as the correct solution. The conditions at x = −2.5 represents the fluid velocity imposed at the inlet, and further conditions are given by the presence of a cylinder centered in (x, y) = (0, 0) with radius 0.25. Furthermore, an additional condition appears at the borders, namely where y = ±2.5, where the no-slip condition can be chosen (u = v = 0) or the correct solution can be given as boundary condition. Since the simulation provided in [32] refers to a free-flow stream, we use the correct solution at the boundaries. To train our PINNs, we use Nr = 5 · 105 collocation points for training the PDE residuals, sampled with latin hypercube sampling, and Nb = 2 · 104 points for training the boundary and initial condition in ∂Ω. Morever, at every iteration, we minimize the loss on random batches of the training data, respectively 104 points for the residuals and 5 · 103 for boundary and initial condition. H Further Numerical Experiments In this Appendix we present some additional numerical experiments. Notice that as a performance measure we utilize the L2 relative loss, defined as follows N X i=1 \|u(xi) −ˆu(xi)\| \|u(xi)\| , (28) where u is the exact solution and ˆu the approximated one. In Figure 5, we showcase the relative L2 loss obtained on the test set during training on the Wave equation with the aforementioned optimizers. While Adam and L-BFGS get stuck relatively fast in a local minima, the LM algorithm is able to decrease the loss consistently, despite the complexity of the problem. The poor performance of L-BFGS can be motivated by two factors. On one hand, the Hessian computed during BFGS iterations is merely an approximation of the true Hessian; on the other hand, convergence to the true solution is heavily hindered since the initial guess is typically not close to the correct one. In Figure 6 and Figure 7, it is possible to notice the effect of the spectral bias: the PINN trained with Adam can capture only the lower frequency components of the true solution, while the model trained with LM performs better as the spectral bias is alleviated in accordance with Theorem 4.2. It is worth noticing that the same holds even when introducing the loss balancing suggested in [40]: its performance is showed in Figure 7. Finally, in Figure 8, we show that by employing the LM optimizer, it is possible to obtain a reasonable solution even for a PDE as complex as Navier-Stokes with relatively small architectures. Notice that the scale in the two plots are different. 23 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.6 1.2 0.8 0.4 0.0 0.4 0.8 1.2 1.6 Figure 6: Experiments on the Wave equation. Left: Prediction of the parametrized solution of a PINN trained with Adam (Left) and LM (Center) alongside with the true solution (Right). 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.0 0.5 1.0 Exact LM Adam+LB Figure 7: Experiments on the prediction of the solution of Poisson equation with LM and Adam (with loss balancing), both compared with the exact solution. 0 500 1000 Iterations 10 4 10 3 10 2 10 1 100 Training Loss LBFGS LM 0 10000 20000 30000 40000 Iterations 10 4 10 3 10 2 10 1 100 Adam Figure 8: Mean and standard deviation of the training loss over the iterations for Adam, LBFGS and LM on Navier-Stokes equation (for 10 independent runs). NeurIPS Paper Checklist 1. 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4,446	FouRA: Fourier Low Rank Adaptation Shubhankar Borse∗§ Shreya Kadambi∗§ Nilesh Prasad Pandey† Kartikeya Bhardwaj Viswanath Ganapathy† Sweta Priyadarshi† Risheek Garrepalli Rafael Esteves Munawar Hayat§ Fatih Porikli§ Qualcomm AI Research‡ §{sborse, skadambi, mhayat, fporikli}@qti.qualcomm.com Abstract While Low-Rank Adaptation (LoRA) has proven beneficial for efficiently finetuning large models, LoRA fine-tuned text-to-image diffusion models lack diversity in the generated images, as the model tends to copy data from the observed training samples. This effect becomes more pronounced at higher values of adapter strength and for adapters with higher ranks which are fine-tuned on smaller datasets. To address these challenges, we present FouRA, a novel low-rank method that learns projections in the Fourier domain along with learning a flexible input-dependent adapter rank selection strategy. Through extensive experiments and analysis, we show that FouRA successfully solves the problems related to data copying and distribution collapse while significantly improving the generated image quality. We demonstrate that FouRA enhances the generalization of fine-tuned models thanks to its adaptive rank selection. We further show that the learned projections in the frequency domain are decorrelated and prove effective when merging multiple adapters. While FouRA is motivated for vision tasks, we also demonstrate its merits for language tasks on commonsense reasoning and GLUE benchmarks. 1 Introduction Figure 1: Distribution collapse with LoRA. Visual results generated by the Realistic Vision 3.0 model trained with LoRA and FouRA, for "Blue Fire" and "Origami" style adapters across four seeds. While LoRA images suffer from distribution collapse and lack diversity, we observe diverse images generated by FouRA. Parameter-Efficient FineTuning (PEFT) [27] methods such as Low-Rank Adaptation [17] provide a promising solution to quickly adapt large foundation models, including large vision models (LVMs) and large language models (LLMs) to new tasks [26, 22, 3]. The LoRA module has an elegant design, allowing quick adaptation to new styles or concepts without changing the underlying base model, thus effectively retaining previous knowledge and preventing catastrophic forgetting. ∗These authors contributed equally to this work. †Work done while employed at Qualcomm AI Research. ‡Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). While LoRAs are highly effective in quickly adapt to new styles, they exhibit multiple challenges, with the rank of LoRA modules being a highly sensitive parameter. As LoRA is built for adapting to new tasks using a small training set, it tends to overfit to the distribution of small training set when the rank is high. Recent works [39, 40] observed that when diffusion models overfit to a small training set, they demonstrate a tendency to repeatedly "copy" few samples from the training set. LoRAs trained on smaller data therefore tend to generate data copying artifacts, also known as distribution collapse. The generated images lack diversity, and the phenomenon is very similar to mode collapse observed in GANs. We illustrate this tendency in Fig. 1, specially at high values of adapter strength α across different seeds. Additionally, as the rank reduces, the strength of the adapter reduces, and LoRA has a reduced ability to generate diverse images due to underfitting. Hence, the rank is a very sensitive parameter. Gating mechanisms have been proposed [3] to produce a dynamic rank at every layer, to provide flexibility to the adapter in LLM tasks. However, we argue that dynamic rank reduction is still not flexible for vision tasks as the rank is computed during training and does not vary at inference. We observe that text-to-image diffusion models greatly benefit from a rank adaptation mechanism which can also vary during inference, along the diffusion time steps. Furthermore, while all the previous works apply low-rank adaptation in the feature space, we argue that there is a transform domain over which fine-tuning low-rank adaptation modules generates much richer representations. We provide theoretical and analytical evidence to show that low-rank adaptation in the frequency domain produces a highly compact representation, effectively reducing the generalization error. Hence, this can potentially push the adaptive rank selection mechanism to generalize better, not only reducing the risk of underfitting when rank reduces, but also overfitting at higher ranks. Additionally, there have been attempts to merge multiple LoRA concepts and/or styles as a linear weighted combination of multiple LoRAs [34]. Recent works [45, 12, 23] empirically show that this approach is prone to noisy and inaccurate outputs, and propose joint finetuning the adapters with learnable gates in the low rank subspace. However, we argue that jointly training multiple LoRA modules is highly restrictive and equally tedious for practical use-cases requiring flexibility in combining multiple different LoRAs. Our developed approach of gating in frequency domain enables flexible mixing of multiple adapters. In this paper, we propose FouRA (Fourier Low Rank Adaptation), a PEFT technique to address the aforementioned challenges of LoRA. We transform the input features to the frequency domain, and apply both the down-projection (to a lower rank) and the up-projection (back to the higher rank) in this frequency domain. During inference, we fold the adapter strength α into the low rank subspace. FouRA learns an adaptive mask inside the low-rank subspace to dynamically drop certain frequency transformed basis, effectively varying the rank for each layer. The adaptive mask selection is input dependant, and varies during the diffusion process. Through rigorous analysis, we show that FouRA provides clear benefits over LoRA (and other adaptive gating methods), and generates high quality diverse images.We show for lower ranks increasing the effect of adapter weights in FouRA does not deteriorate the representation power of original model. Additionally, we show that FouRA provides a rich disentangled orthogonal basis to Low Rank Adapters in the frequency domain, making it beneficial for merging multiple styles. Our contributions are summarized as: • We introduce FouRA, the first low-rank adapter module that performs the low rank transforms in the frequency domain along pixel or channel dimensions of the feature space. • We propose an adaptive learnable masking strategy in the frequency domain that flexibly varies the effective rank for every FouRA layer in the network, thus enabling the model to generalize well, even when the size of training set is very small. • We demonstrate that FouRA successfully provides a decorrelated orthonormal basis to Low Rank Adapters in the frequency domain, making it highly beneficial for merging two styles or concepts, without the need for joint training. • Through extensive experiments and theoretical analysis, we demonstrate how FouRA consistently produces a diverse set of aesthetically improved images compared to LoRA, and is equally effective for LLM tasks. 2 Related Work Text-to-Image Diffusion Models: Multiple diffusion based image generative models have been proposed recently [33, 31, 6], [32, 29, 36, 30]. These models exhibit excellent text-to-image generation ability and can be adapted to new styles using LoRA [17]. 2 Fourier Transforms in Generative Literature: Recent work [21] shows that the latents of the denoising models trained on sufficient data lie on adaptive basis with oscillating patterns. Other works have shown that we can use fourier operators for non parametric regression tasks and cast self attention as a kernel regression problem. [28] shows that it offers smoother representations over the input and better captures the correlations between query and keys. [24] has shown that Fourier spectral filters operate in the continuous domain and work well in representing images as continuous functions. Further convolutions in spatial domain can be represented as multiplications in the Fourier space thus spectral filters can act as global convolution operator. A concurrent work on language models [10] has proposed parameter-efficient fine-tuning in the Fourier Domain. Many works have analysed the eigen spread of signal transformed to harmonic basis. [1], analysed the effect of applying these transforms on a signal sampled from a Markovian process and show that Fourier transforms decorrelates such as signal in least mean square setting. Low Rank Adaptation: LoRAs [17] suffer from a tradeoff between fidelity and diversity of generated images. [3] tried to alleviate this problem by sparse regularization. SVDiff [14] explicitly only updates the singular values while retaining the subspaces. In a high rank setting this method is acceptable. However, in FouRA we are learning in a low rank subspace. Other works like AdaLORA [48], [46] applied to language models, further parameterized the weight matrices using SVD and jointly optimized for eigen vectors and the singular values through importance scoring metric. O-lora [42] computes orthogonal gradient spaces between different tasks letting the model sequentially adapt to new tasks without catastrophic forgetting. [3] applies proximal gradient gating in the loss function to learn important subspaces and mask out the remaining ones. While all these papers directly operate by constraining the subspace of the weight matrices, we show in our paper that the Fourier domain implicitly enforces these properties without any constraints in the optimization. We show that applying gating in the frequency domain provides a more compact representation with stable generalization error bounds. In addition results in lower effective rank for each layer. We also show that the learnt spaces across different adapters also have decorrelated basis. MoLE [45], ZipLoRA[37] and Mix of Show [12, 50] explore various strategies to merge LoRAs. This is done using either supervised or self-supervised objectives for jointly training weights corresponding to both adapters. As the number of adapters grow, we argue that the two-stage method to merge adapters is not flexible and quite tedious. FouRA on the other hand does not require any fine-tuning, and is truly a training-free approach to merge multiple adapters. Disentangled spaces for editing [43] [13] have explored diffusion models for disentangled interpretable latent representation. While LoRAs have been proposed for personalization, [9] proposed a way to do fine-grained editing of images while still preserving the features of the original image. They identify semantic directions and traverse on the latent space on these directions. Concept sliders have been applied to real applications such as fixing distortions in diffusion generated images. We show in our work that our method identifies more compact disentangled representations over LoRA, thus providing more performance improvements over fine-grained edits. 3 Proposed Approach 3.1 Formulation of Low Rank Adaptation We illustrate the base LoRA module in Fig. 2. Consider the original set of pre-trained weights W0 ∈Rk1×k2 where k1 and k2 represent the input and output embedding dimensions respectively. LoRA modules consist of the down layer A ∈Rk1×r and the up layer B ∈Rr×k2, projecting the input features to and from the low-rank subspace of rank r. Consider an input feature zin ∈Rd×k1, where d is the number of input tokens, the output after the low-rank adaptation zout ∈Rd×k2 is given as zout = zog + αzlora = W0zin + αBAzin. Here, zog and zlora are the outputs from the original and low-rank branches respectively, and α is a scalar to blend the two branches. We denote the learned adapter matrices as ∆Wlora = BA as in [17]. 3.2 Low Rank Adaptation in the Frequency Domain The projection to and from a low-rank subspace is prone to information loss. To mitigate this, we propose to transform the inputs to a domain which contains an inherently compact representation, i.e. the frequency domain. We are motivated by the fact that transforming to the frequency domain 3 Figure 2: LoRA v/s FouRA. For FouRA, we transform feature maps to frequency domain, where we learn up and down adapter projections along-with our proposed adaptive rank gating module. preserves valuable information, due to its inherent de-correlation capabilities [11, 16]. We validate this further by analyzing the effects of the frequency transform on the model weights in Sec. 4.1. Given the pre-trained weight matrix W0, we apply the low rank transforms B and A in the frequency domain. Inspired by [38], we fold the blending parameter α inside the low-rank subspace, effectively acting as a scaling factor in the frequency domain. We apply the frequency transforms as follows. zout = zog + zfoura = W0zin + F−1(BαAF(zin)) (1) Here, F(·) and F−1(·) are the normalized forward and inverse frequency transforms respectively. 3.3 Frequency Transforms We investigate the properties of Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT) in the low rank space. We apply 1D DFT to the embedding dimension k1 ∈(0, K) before the subspace decomposition. Given input zin ∈Rd×k1 to the adapter branch , we expand F in Eq. (5) as, Zk1(f) = F(zin)d×k1 = 1 √k1 k1−1 X k=0 e−j 2πfrk k1 zin(k), fr : ∀r ∈(0, 1...k1 −1). (2) Where fr is the frequency of the basis represented by DFT. As we do not apply any padding, the dimension of the transform preserves the dimension of zin. In our experiments, we apply the 1-D transform on the embedding dimension k1 for each token on both self and cross attention layers. To motivate the idea of generalizing FouRA across tasks such as targeted editing [9], where disentangled latent space is required to gain control over generated images, we further explored Discrete Cosine Transform (DCT) with compact subspaces (eigen spread), which leads to less overfitting. We later show in App. B.1 and Fig. 4 that the subspaces of FouRA are more uncorrelated from each other. We observe that for certain tasks, DCT provides a smoother representation as the implicit window is twice that of DFT signals. For a given a finite length signal zin ∈Rd×k1, we compute DCT as follows. We first construct a double length even signal by ˜ zin(d, k1) = zin(d, k1), 0 ≤k1 ≤K zin(d, 2K −k1 −1), K ≤k1 ≤2K −1, (3) The DCT is then computed as the DFT of ˜ zin. 3.4 Adaptive Rank Gating Method LoRA methods pre-define the rank for all layers. Recent method [3] has an adaptive rank during training, which is however fixed at inference time, thus lacking flexibility. In our approach, we propose a learned adaptive gating mechanism, which can vary each layers rank during training and inference, dependent upon the inputs. We introduce our learnable gating mechanism G(·) inside the low-rank subspace within the frequency domain. Consider the low-rank representation denoted as zlr ←AF(zin) ∈Rd×r, our gating operation is defined as, G(zlr) = 1, if S(H(Gzlr)) == 1 0, otherwise (4) 4 Figure 3: Operational diagram of FouRA. Illustrating the components of Eq. 5. Here, H(·) and S(·) represent entropy and sigmoid functions respectively, G represents the weights of a learnable multi-layer perceptron (MLP), G is a function to learn a weighting for every singular value in the low-rank subspace. The FouRA output, illustrated in Fig. 3, is then given by, zout = zog + zfoura = W0zin + F−1(BαG(zlr) · AF(zin)) (5) The learned FouRA adapter weights are ∆Wfoura = F−1(BG(zlr)F(A)), as per notation in Sec. 3.1. We conduct further analysis of our proposed gating function in Sec. 4.2, analyzing its behaviour across diffusion time-steps and various resolutions. Further, we demonstrate its efficacy over both fixed LoRA and recent Adaptive Rank selection methods which are fixed at inference (SoRA [3]). 3.5 Combining multiple adapters Merging of LoRA adapters has multiple practical use-cases [34]. The method we use to merge two adapters varies according to the task. Text-to-Image Style Transfer: Following the standard method, we merge two FouRA style based adapters using a linear combination of the outputs of adapter ∆W1.zin and ∆W2.zin during inference. Image editing using Concept Sliders: Similar to [9], we perform concept slider evaluations for text based editing using FouRA in Sec. 5.3. Given n concept sliders, we define cn,j concept for nth slider (e.g "very old") and ˜cn,i as the negative concept (e.g " very young"). We composite the adapters in the epsilon ϵ space, with composed score function ˆϵ, and sample from the factorized distribution p(x/(˜ci, cj)) ˆϵ(x) = ϵθ(x) + X n wn(ϵθ(x, cn,j) −ϵθ(x, cn,i)) (6) For merging of two styles, as well as composition of two concept adapters across different strengths α, we notice that the feature spaces of FouRA adapters are less entangled as compared to LoRA. Further analysis is present in Appendix B.4 and B.2. 4 Theoretical Analysis 4.1 Frequency Domain Fine Tuning Figure 4: Singular value spread for FouRA v/s LoRA. Frequency domain transforms decorrelate input representations, minimize spectral redundancy [47], and are effective in compression since they concentrate most of the energy in a few coefficients [16]. Learning in the spectral domain is shown to enable faster convergence and sparser weight matrices [11]. Motivated by these advantages, we propose to fine-tune adapters in the frequency domain. Singular Value Distribution Analysis: Consider a weight matrix W. The singular value decomposition of this matrix is represented as UDVT, where U ∈Rk1×k1, V ∈Rk2×k2 are orthonormal matrices and D ∈Rk1×k2 is a matrix, containing the singular values of W, σi∀i ∈{Nmin(k1,k2)}. Considering an r rank approximation of W, we denote the singular values as {σ1, σ2...σr}, arranged in descending order, and the corresponding diagonal matrix as Dr. The r-rank approximation of W is hence computed as LRr(W) = UDrVT. 5 Figure 5: Average Effective Rank of FouRA. Figure a. and b. shows plots for the average effective rank for various layers of the FouRA U-Net (Darker lines correspond to higher resolutions) and Figure c. compares the average effective rank of FouRA to SoRA. FouRA’s effective rank reduces with the feature resolution, and it also reduces as the diffusion process proceeds, owing to lesser changes required towards the end. Lemma 4.1. Considering two adapters ∆W1 and ∆W2 and their corresponding sets of singular values {σ1,i} and {σ2,i}. The adapter ∆W1, will admit r rank approximation with lower error than ∆W2 if σ1,i < σ2,i for all i ≥r. We provide a proof for the above lemma in Appendix B.1. We empirically analyze the distribution of singular values for r rank approximations of ∆Wlora and ∆Wfoura (without adaptive masking) for the last layer of our trained UNet model in Fig. 4. FouRA has a more compact spread of singular values as compared to LoRA. Hence, using Lemma 4.1, we can say that the accumulated error for a LoRA adapter with a low-rank approximation will be greater than the a FouRA adapter with the same rank. 4.2 Gated Frequency Domain Fine Tuning Motivated by observations in [3, 25], our proposed rank gating mechanism intends to vary the effective rank of each low-rank adapter in the network. We describe effective rank per layer as the number of singular values which are not masked out by the learned gating function. Using observations from [7, 25], we propose the following Lemma: Lemma 4.2. Consider an adapter ∆W with a rank higher than the required rank to fit a training data distribution. The upper-bound of generalization error R for fine-tuning this adapter reduces as the effective rank of the adapter reduces. After reducing to a certain value of effective rank, the upper-bound of generalization error will increase as rank reduces further. Corollary 4.2.1. Additionally, the generalization bound is more stable when the singular value distribution of adapter weights ∆W is more compact. We provide a proof in Appendix B.2. The effectiveness of variable rank selection can be justified using Lemma 4.2. As LoRA rank reduces, the model tends to underfit. However, increasing the rank above the required rank to fit a training distribution leads to overfitting, which reduces the models performance. Dynamically determining the effective rank in every layer produces promising results, as it provides a learnable trade-off between generalization and overfitting. In Fig. 5, we plot FouRA average effective ranks for a denoising UNet over 20 iterations of the reverse diffusion process. Our analysis reveals that the effective rank learnt for high-resolution layers is higher than low-resolution layers. Furthermore, the effective rank reduces as the denoising process continues. This essentially means that noisy inputs require more singular values to update. We further observe in Fig. 9 that our proposed adaptive masking (which varies in inference time) significantly outperforms methods such as SoRA (which freezes its masks after training). Furthermore, from Corollary 4.2.1 and a consequence of the property observed in Fig. 4, as FouRA obtains compact spread of singular values, we can determine that the generalization bound is more stable in the frequency domain for lower effective ranks, as compared to the feature space. We verify this in Fig. 9 as FouRA outperforms SoRA and LoRA with our proposed adaptive masking. The data copying artifacts observed for LoRA model in Fig. 1 are a consequence of overfitting. This was observed by recent works targeting Digital Forgery [39, 40]. As FouRA significantly reduces the generalization error, it can generate a diverse set of images. Additionally, we also observe in App. E.1.1 that FouRA is able to generalize better on unseen concepts, as compared to LoRA. 6 Figure 6: FouRA v/s LoRA: The prompt on the left is "a football in a field" and on the right is "man in a mythical forest". While staying more faithful to the adapter style, FouRA outputs look aesthetically better than LoRA, which have obvious artifacts at high values of α. Additional results are in Appendix E. 4.3 Subspace Learning In App. B, we provide a subspace perspective to verify empirically and theoretically that FouRA learns subspaces which are more decorrelated from the base model weights, as compared to LoRA. A higher emphasis on the set of learnt subsapces enables FouRA to learn new tasks without catastrophic forgetting. Additionally, we attribute the strong merging capabilities of different FouRA adapters to their disentangled and decorrelated subspaces learned by respecitve FouRAs. 5 Experiments 5.1 Experimental setup Datasets: For style transfer, we evaluate FouRA on four datasets collected from public domains, including Bluefire , Paintings, 3D and Origami styles, see Appendix C.1.3 for details. Our results are averaged over 30 random seeds, and a total of 1530 images. For evaluations on composite sliders, similar to [9], we train 3 sliders "Age", "Hair" "Surprised’ and composite experiments combining both "Age" and "Hair" . While our approach is motivated for vision tasks, we also evaluate FouRA on language tasks and assess the performance of our adapter on MNLI, CoLA, SST2, STSB, MRPC and QNLI tasks from the GLUE benchmarks. We also evaluate on Commonsense Reasoning benchmarks BoolQ, PIQA, SIQA, HellaSwag, WinoGrande, ARC and OBQA. See App. C.1 for details. Models: For text-to-image generation experiments, we employ Stable Diffusion-v1.5 [33], using both the base model weights and RealisticVision-v3.0 checkpoints for style transfer tasks. For concept editing, we train on Stable Diffusion-v1.5 [33] base weights. We use DeBERTAv3-Base [15] for General Language Understanging tasks and Llama3-8B [4] for Commonsense Reasoning tasks. See App. C for additional implementation details. Metrics: For quantifying the quality of images generated by FouRA and LoRA finetuned diffusion models, we report HPSv2.1 [44] and LPIPS diversity [49] scores. The HPSv2 metric evaluates the measure of the image quality, and alignment with the prompt/style. LPIPS diversity score captures the diversity within all possible pairs of generated images across seeds. We provide an in-depth analysis of these metrics in Appendix D. For the image editing task, we compare edited images using LPIPS similarity (compared to the base image). For language models, we report on the General Language Understanding Evaluation (GLUE) benchmarks [41], see details in App. C.2. On commonsense reasoning tasks, we report Accuracy. 5.2 Text-to-Image Stylized Generation In Fig. 6, we show visual results of LoRA and FouRA on the Paintings and Bluefire style tasks. FouRA is able to generate high quality images as compared to LoRA over a range of adapter strengths α. We observe that LoRA suffers from artifacts at high values of α in case of the Paintings adapter. Tab. 2 compares LPIPS Diversity and HPSv2 scores for all models, showing that FouRA significantly outperforms LoRA on both the metrics. Our analysis in App. D shows that this gap in LPIPS-diversity and HPS scores is quite significant, specially for higher α values, FouRA shows significant gains compared to LoRA. This is likely because at lower α values, the adapter effect would be reduced and 7 Dataset Base Model Adapter LPIPS Diversity(↑) HPSv2 score(↑) α = 1 α = 0.8 α = 0.6 α = 1 α = 0.8 α = 0.6 Paintings (630 Images) Stable Diffusion-v1.5 LoRA 38.3 ± 3.6 43.0 ± 3.2 43.6 ± 3.6 22.3 ± 1.7 25.3 ± 1.9 27.2 ± 2.9 FouRA 43.9 ± 3.7 44.1 ± 3.8 45.7 ± 3.8 25.2 ± 1.6 27.1 ± 1.8 28.0 ± 2.4 Realistic Vision-v3.0 LoRA 38.3 ± 3.5 37.8 ± 3.6 39.2 ± 3.7 24.6 ± 1.8 27.7 ± 1.8 30.3 ± 1.7 FouRA 44.2 ± 3.7 44.5 ± 4.0 44.6 ± 3.9 28.4 ± 1.8 30.6 ± 1.5 32.0 ± 1.4 Blue-Fire (900 Images) Stable Diffusion-v1.5 LoRA 47.8 ± 3.7 48.4 ± 3.9 49.5 ± 4.2 28.6 ± 2.1 30.4 ± 2.0 30.6 ± 2.2 FouRA 50.3 ± 3.0 50.8 ± 3.2 51.5 ± 3.6 29.7 ± 1.9 30.9 ± 1.9 30.9 ± 2.2 Realistic Vision-v3.0 LoRA 46.8 ± 4.0 48.5 ± 4.0 49.8 ± 4.2 32.7 ± 1.6 33.8 ± 1.4 34.0 ± 1.5 FouRA 50.4 ± 3.0 51.6 ± 3.3 52.2 ± 3.5 33.6 ± 1.5 34.1 ± 1.2 34.0 ± 1.4 Table 2: Evaluation of LoRAs on Text-to-Image tasks. Adapters are rank 64. Results are averaged over 30 seeds. Figure 7: Multi-Adapter Fusion in LoRA v/s FouRA. Sample images for style transfer on various prompts (e.g., bird, car, fox) for Paintings, Bluefire, 3D and Merged adapters. Observe the highlighted merged images. FouRA does a much better job in preserving both styles, compared to LoRA. thus both images look more realistic. These results demonstrate that FouRA images are both diverse (even at high adapter strengths) as well as aesthetically coherent. See App. E for more experiments. Adapter αb αp HPSv2 score LoRA 0.4 0.4 33.4 FouRA 0.4 0.4 33.5 LoRA 0.6 0.6 32.7 FouRA 0.6 0.6 33.5 LoRA 0.8 0.8 31.2 FouRA 0.8 0.8 33.6 LoRA 1.0 1.0 30.3 FouRA 1.0 1.0 33.1 Table 1: Merging two adapters for Blue Fire and Paintings with strengths αb and αp. Multi-Adapter: Fig. 7 shows images for style transfer merging for various prompts (e.g., bird, car, fox) for three styles: Paintings, Bluefire and 3D. We also provide the outputs of the linear combination of LoRA and FouRA for both these tasks. We see that merged LoRA images sometimes lose one of the concepts (e.g., the blue fire is lost for Panda and Dog) or have severe artifacts (e.g., the fox with multiple tails and the bird without a head). In comparison, FouRA images for merged adapters preserve the concepts and do not display any distortions. This property of FouRA is a direct consequence of our analysis in App. B.3 and is also evident from the HPSv2 reported in Tab. 1, where for higher adapter strengths, FouRA shows gains upto 3% over LoRA. 5.3 Text-to-Image Concept Editing We establish the performance of our approach on nuanced editing tasks for specific target images by training FouRA using the disentangled objective proposed in concept sliders [9]. We train LoRA and FouRA modules using pairs of prompts describing the editing concepts. Fig. 8 shows results of editing the Age and Hair concepts. As observed, although the Age adapters are trained using a disentangled objective, LoRA changes the gender of the subject, and produces artifacts at high scales. FouRA is elegantly able to age them while retaining their original features. Similarly, the Hair FouRA produces a smoother representation. We provide quantitative evaluations in App. 5.3, and observe that at higher strengths, FouRA consistently outperforms LoRA in terms of the LPIPS score. Composite Sliders: We qualitatively evaluate the composite ’hair’ and ’age’ adapter between LoRA and FouRA in Appendix 5.3. We show the results on two target prompts "A female Indian person" and 8 Figure 8: LoRA v/s FouRA . Age (Left) and Hair (right) concept slider examples where as the scale increases the effect of disentanglement in FouRA is more prominent. For larger scales the gender of the person changes in Age LoRA, and the structure of the face changes in Hair LoRA. "A male white person" respectively. Overall, we observe that FouRA does a better job at compositing both sliders, as it produces a smooth transition between the concepts. In comparison, LoRA distorts the subjects faces at high adapter scales, and interferes with other facial features. We also show that the LPIPS diversity is much lower for generated images between different strength for FouRA F.4 at higher scales of the adapter. 5.4 Commonsense Reasoning Tasks While our design choices for FouRA are primarily motivated for vision tasks, we evaluate its efficacy on eight commonsense reasoning tasks using the split from [18] in Tab. 3. We trained LoRA and FouRA adapters over a LLaMA3-8B [4] model. Our analysis shows that employing FouRA at rank 16 and 32 both outperform LoRA at the rank 32 setting. Adapter Rank Trainable Params BoolQ PIQA SIQA HellaSwag WinoGrande ARC-e ARC-c OBQA Average LoRA 32 56.60 M 71.3 87.1 79.9 92.7 84.5 87.9 77.2 82.4 82.9 FouRA 16 28.31 M 74.4 89.1 79.8 94.9 86.7 90.2 80.1 85.2 85.1 FouRA 32 56.63 M 74.8 89.0 79.9 95.3 85.9 90.9 80.8 85.6 85.3 Table 3: Performance on Commonsense Reasoning benchmarks: Evaluation on eight Commonsense Reasoning benchmarks with the Llama-3(8B) model. 5.5 Computational Analysis Table 4 provides the computational analysis for FouRA, as compared to LoRA. We provide the #parameters during inference along with the training time for FouRA. Along with this, we show the HPS-v2.1 scores on the Blue Fire validation set. Additionally, we provide the results for a FouRA variant with a fixed gating strategy during inference. FouRA layers with inference-adaptive masking produce an overhead of 0.02% more than LoRA, as compared to base model weights. However, FouRA with frozen masking can essentially reduce the computational overhead by a factor of 2, and still retain a higher performance than the base LoRA. Adapter Training Time Epoch Time GPU Memory Inference Time HPS (Paintings) (↑) LoRA 1.87 sec/iter 22.0 sec 53.69 GB 14.9 step/sec 27.7 FouRA (Inference-Adaptive Mask) 2.09 sec/iter 24.5 sec 53.89 GB 11.1 step/sec 30.6 FouRA (Frozen Mask) 2.07 sec/iter 24.3 sec 53.81 GB 14.9 step/sec 30.3 Table 4: Computational and Runtime Complexity. The training measurements are performed on Tesla A-100 GPU with a batch-size of 8. The adapters are all rank=64, and HPS-v2 is computed at α = 0.8. 5.6 Ablation Studies Individual gain of every component We show individual contributions from FouRA modules in Table 5. We fix rank=64 and α=0.8, and provide results on the paintings validation set. As evident from LPIPS-Diversity and HPS scores, the adaptive mask selection strategy performs better than the dynamic fixed mask selection strategy. For the case without frequency transform, Inference-Adaptive masking improves the HPS score from 28.2 to 28.7. When accompanied with Frequency transform, the HPS increases from 30.3 for frozen dynamic masking to 30.6 for inference-adaptive masking. 9 Adapter Fourier Frozen Dynamic Mask Inf-Adaptive Mask HPS (↑) LPIPS-Diversity (↑) LoRA 27.7 37.8 Frozen Mask ✓ 28.2 38.9 Inference-Adaptive Mask ✓ 28.7 39.7 FouRA (No Mask) ✓ 30.0 43.2 FouRA (Frozen Mask) ✓ ✓ 30.3 44.0 FouRA (Inference-Adaptive Mask) ✓ ✓ 30.6 44.5 Table 5: Individual gain with FouRA components. Gains from each individual component of FouRA. All results are with rank 64 and α = 0.8 on the paintings adapter. Varying the Adaptive Rank Selection Strategy in Text-to-Image Stylized Generation Figure 9: Comparison of different rank selection methods. Fig. 9 shows the HPS-v2.1 curves obtained for evaluating LoRA, SoRA [3] and FouRA on the Paintings validation set for different adapter strength α. Additionally, we also show the performance of our inference-adaptive rank selection method directly on LoRA. All the numbers are for base rank=64 adapters. As observed, SoRA outperforms LoRA at higher ranks. However, our inference-adaptive rank selection strategy improves performance over SoRA, indicating that in vision models, varying the effective-rank across time steps of the diffusion process is ideal. FouRA outperforms all methods, indicating the benefits of training our proposed rank selection strategy in the frequency domain. Varying the Rank in Text-to-Image Stylized Generation In Fig. 10, we investigate the impact of FouRA over varying values of input rank, and compare with LoRA. We observe that rank is a highly sensitive parameter for LoRA. However, the HPS scores across ranks for FouRA are higher than the highest HPS score acheived at any rank by LoRA, highlighting the effect of gating in frequency domain. This helps FouRA to avoid underfitting as the rank reduces and overfitting as it increases. Furthermore, FouRA generates a diverse set of images across all ranks. Figure 10: HPS-v2.1 scores for each adapter across ranks. FouRA continues to outperform LoRA as the rank increases for both Paintings and Blue Fire datasets. 6 Conclusion In this paper, we proposed FouRA, a parameter efficient fine-tuning method within the frequency domain. Through extensive experiments and rigorous analysis, we showed that FouRA successfully solves the problems related to data copying and distribution collapse while significantly improving the generated image quality over LoRA. We also present an intensive study on the impact of compact representation of Low rank subspaces in transformed domain. Further, we showed that FouRA can leverage our proposed adaptive mask ranking approach and further push the generalization capabilities of PEFT models without under-fitting. 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Multi-lora composition for image generation. arXiv preprint arXiv:2402.16843, 2024. 13 Appendices A Contents As part of the supplementary materials for this paper, we share our Implementation details, show extended qualitative and quantitative results and provide additional theoretical analysis for our proposed approach. The supplementary materials contain: • Extended Theoretical Analysis – Proof of Singular Value Decomposition Analysis Lemma 4.1 – Proof of Sparsity Lemma 4.2 – Subspace Analysis – Merging of Adapters – Learning disentangled representations • Implementation details and hyperparameters for all experiments – Datasets – Hyperparameters • Interpretations for learnt metrics (HPS-v2.1 and LPIPS diversity) • Additional experiments for text-to-image stylization. – Performance on Unseen Concepts for Text-to-Image Stylization – Effect of varying the frequency transform – Comparisons: 2D FFT on the tokens vs 1D FFT on token embeddings – Plots for quantiative metrics in Text-to-Image Stylization – Effect on data-copying artifacts after early stopping LoRA training – Additional Computational Analysis – Additional Visual Results on Text-to-Image Stylization • Additional Experiments for Text-to-Image Editing using Concept Sliders • Societal Impacts B Theoretical Analysis B.1 Proof for Lemma 4.1 In this section, we provide the proof for Lemma 4.1 of the main text. Lemma 4.1. Considering two adapters ∆W1 and ∆W2 and their corresponding sets of singular values {σ1,i} and {σ2,i}. The adapter ∆W1, will admit r rank approximation with lower error than ∆W2 if σ1,i < σ2,i for all i ≥r. Proof. Let D1,r and D2,r be diagonal matrices corresponding a rank r approximation of ∆W1 and ∆W2 respectively. The reconstruction errors E1,r and E2,r for these approximations are computes as follows: 14 E1,r = ∆W1 −LRr(∆W1) = U1D1VT 1 −U1D1,rVT 1 (7) E2,r = ∆W2 −LRr(∆W2) = U2D2VT 2 −U2D2,rVT 2 (8) A matrix ∆W can be written as the sum of it’s right and left 1-D singular vectors u and v as follows: ∆W = UDVT = min(k1,k2) X i=1 σiuvT (9) Hence, we rewrite the reconstruction errors E1,r and E2,r as a sum of the product of their 1-D singular vectors. E1 = min(k1,k2) X i=1 σ1,iu1vT 1 − r X i=1 σ1,iu1vT 1 = min(k1,k2) X i=r+1 σ1,iu1vT 1 (10) ∴E2 = min(k1,k2) X i=r+1 σ2,iu2vT 2 (11) Following the Eckart-Young theorem [5] and theorem 4.95 in Mathematics for Machine Learning [2], the value of the norm of reconstruction error is given as: ∥E1∥= min(k1,k2) X i=r+1 σ1,iu1vT 1 = σ1,r+1 (12) Hence the difference of reconstruction errors is computed as follows: ∥E2,r∥−∥E1,r∥= σ2,r+1 −σ1,r+1 (13) We know σ2,r+1 > σ1,r+1. Hence, we prove that ∥E2,r∥> ∥E1,r∥. Here it is important to note an adapter with lesser eigenvalue spread there will exist an r rank approximation such it has a lower approximation error than adapter with wider eigenvalue spread. However, the rank r should follow in lemma above. Further, it is important note the low rank adapter with a lower approximation error would estimate the noise closer to optimal estimate and will converge to de-noised image with improved perception scores. B.2 Proof for Lemma 4.2 In this section, we provide a proof for Lemma 4.2 and Corollary 4.2.1 of the main text. Lemma 4.2. Consider an adapter ∆W with a rank higher than the required rank to fit a training data distribution. The upper-bound of generalization error R for fine-tuning this adapter reduces as the effective rank of the adapter reduces. After reducing to a certain value of effective rank, the upper-bound of generalization error will increase as rank reduces further. Corollary 4.2.1. Additionally, the generalization bound is more stable when the singular value distribution of adapter weights ∆W is more compact. Proof. Consider A as a learning algorithm for finetuning our adaptation weights ∆W, and S is our training set of length n. Additionally, consider the ratio of effective rank to original rank as p (where 1 −p is a sparsity parameter). The LoRA Generalization error upper-bound for A can be computed from Pointwise Hypothesis Stability equations (Theorem 2 of [7]). We have for a constant C with a probability 1 −δ, 15 R(A, S) < ˆR(A) + s C2 + 24Cρ2 λmin+2(1−p) 2nδ (14) Here, ˆR(A, S) represents the emperical error, and λmin represents the minimum eign-value of the loss Hermitian matrix. For finetuning tasks, λmin ≈0 for a loss Hermitian matrix which is well behaved as the model has been trained, as observed by [35]. Based on the observations of [25, 7], and the above equation, we can observe that the generalization error reduces as the sparsity increases when the effective rank ratio p is low, and sparsity (1 −p) is relatively high. As effective rank increases and sparsity(1 −p) reduces, if the length of data distribution is small, there is a high risk of overfitting. However, as effective rank reduces and sparsity increases, there will come a point when the number of trainable parameters are much lower than what is required for representing the training data distribution, leading to underfitting. Hence, there exists an optimal effective rank, proving Lemma 4.2. The optimal effective rank is driven by the generalization error. For highly sparse representations, the empirical error ˆR(A, S) dominates over the second term, as it increases significantly. From Lemma 4.1, we know that if the singular value spread of LRr(∆W) contains a more compact representation, the reconstruction error from the r-rank subspace is reduced. Hence, the training objective ˆR(A, S) reduces. A consequence of this reduction in error signifies that the weights can potentially achieve higher generalization capability by even further sparsification, before ˆR(A, S) starts dominating the generalization error bound. Hence, model weights which can be represented in compact singular value representations can achieve a lower generalization error by further increasing sparsity, proving Corollary 4.2.1. B.3 Subspace analysis In Section 5, we demonstrate that the fine tuned FouRA adapter performs significantly better than LoRA. In this Section, we attempt to analyze the performance of adapters in terms of the correlation of the subspaces of the base model and that of the adapter. The analysis follows the approach discussed in [17]. We project the base model weights W0 onto the r-dimensional subspace of our finetuned adapters ∆W. The projection of base matrix W0 on to the subspace of the adapter is UT W0VT , where U/V are the left and right top-r singular vectors of ∆W. As defined in [17], ∥∆W∥F ∥UT W0VT ∥F is the amplification factor, a measure of the subspaces emphasised in the adapter ∆W when compared with base weights W0. Between two adapters of the same rank, a higher amplification factor effectively corresponds to the amount of information learned by the adapter, which is orthogonal to the model weights. In table B.1, we analyze the amplification factors of FouRA and LoRA at rank=32. This is an average over all the adaptors of finetuned UNet model. Observe that FouRA Amplifies the learnt subspaces by factor >2x as compared to LoRA. Hence, FouRA weights are more de-correlated from the pretrained base model weights. Additionally, higher emphasis on the set of learnt subsapces enables the learning of new tasks without catastrophic forgetting. Figure B.1 shows further analysis of learnt subspaces over multiple ranks. \|\| ∆w \|\|F \|\| U T W V T \|\|F (↓) \|\|∆w\|\|F \|\|UT W V T \|\|F (↑) LoRA 1.07 0.95 1.2 FouRA 0.32 0.81 2.8 Table B.1: Amplification Factor Analysis. Average amplification factor components over all layers of the diffusion UNet with Rank=32 LoRA and FouRA. 16 Figure B.1: Amplification Factor of FouRA v/s LoRA: As the computed Amplification Factor referred to in B.3 is higher in case of FouRA, we justify the learnt representations are more de-correlated from the base weights. B.3.1 Merging adapters Recent works [37] demonstrate joint-adapter training for effectively merging multiple low-rank adapters. In Section 5, we demonstrate the ability of the FouRA module to merge multiple adaptors in a way which retains both their capabilities with high fidelity. Proposition 1. Considering two adapters ∆W1 and ∆W2. The linear combination of both these adaptors tends to generate results which retain the capabilities of both the adapters, if the norm of the projection of ∆W1 on the subspace of ∆W2, computed as ∥U2 T ∆W1V2 T ∥is lower. Here, U2/V2 are the singular vectors of ∆W2. We provide analysis in Table B.2 complementing Proposition 1, and demonstrating how FouRA has a greater tendency to disentangle two adapters, making it highly effective for multi-adaptor fusion without joint training. We computed the Norm of the projections FouRA adapter weights trained on one subtask, onto the weights trained on another subtask, and compared it to LoRA projection norms. We analyzed the correlation between weights of three tasks: BlueFire, Paintings and 3D. As observed from the numbers, FouRA projection norms are much lower, suggesting a higher number of orthogonal subspaces for FouRA projections. This aligns with Table 1 and Figure 7 of the main text, where we observe that FouRA is successfully able to retain the capabilities of both adapters after the merge. Dataset 1 Dataset 2 LoRA Projection Norm(↓) FouRA Projection Norm (↓) BlueFire Paintings 0.40 0.25 BlueFire 3D 0.39 0.27 3D Paintings 0.47 0.32 Table B.2: Norm of projection of adapter weights trained on task 1, over adapter weights trained on task 2, calculated as ∥U2 T ∆W1V2 T ∥. Observe that FouRA has a lower Projection Norm, B.4 Learning disentangled representations Given zin, zout ∈Rd×k1 from (5), and let the input have three attributes that can be represented as zin = [zrace, zage, zgender], the autocorrelation matrix at the output of FouRA layer can be written as Rd×d = zoutzT out = zin(W0 + ∆W)(W0 + ∆W)T zT in = zinW0W0 T zT in + zin∆W∆W T zT in + F(W0∆W T , zin) (15) From B.1, we established that the overlap of subspaces between low rank in transform domain ∆W and base matrix W is smaller at lower rank. In addition, in frequency domain, the term in the middle (in blue) computes the autocorrelation between the subspaces. From [1], this term is almost diagonal making the dot product < zrace out , zgender out >≈0 or < zrace out , zage out >≈0. Thus the weights for each attribute is poised to be learned independently. To verify this, In the experiments section, we motivate the idea of using foura to edit concepts while preserving the attributes of an image using concept sliders [9] 17 C Implementation Details C.1 Datasets C.1.1 Commonsense Reasoning We use the commonsense reasoning datasets which comprise of 8 sub-tasks, each with a predefined training and testing set as shown in table C.1. We follow the setting of [19] for training. The common sense reasoning training dataset is a combination of the training datasets provided by [20], while we evaluate each evaluation dataset separately. Dataset #Train #Val Test PiQA 16K 2K 3K BoolQ 9.4K 2.4K 2.4K SIQA 33.4K 1.9K 1.9K OBQA 4.9K 0.5K 0.5K Winogrande 9.2K 1.3K 1.8K HellaSwag 39.9K 10K 10K Arc_easy 2.25K 570 2.36K Arc_challenge 1.12K 299 1.12K Table C.1: Commonsense Benchmark C.1.2 GLEU We have performed the LLM study on six of the GLUE benchmarks - CoLA, SST-2, MRPC, STS-B, MNLI, and QNLI. GLEU benchamrk has been widely used for natural language understanding. All the dataset and task described in the Table C.2 is being utilized from Huggingface Datasets and each task has its own respective evaluation metric. We have described the train and test split of each of the task along with the respective evaluation metric in Table C.2. Dataset #Train #Val Metric CoLA 8.5K 1043 Mcc SST-2 67K 872 Acc MRPC 3.7K 408 Acc STS-B 5.7K 1.5K Corr MNLI 393K 9.8K Acc(m/mm) QNLI 105K 5.5K Acc Table C.2: GLUE Benchmark C.1.3 Style Transfer Datasets In this section, we provide more details on the four style transfer datasets we use for vision adaptation experiments. We followed the licensing terms for every dataset which was curated. BlueFire (Training): The BlueFire dataset is created by collecting images from open public domain and consist of 6 concepts - car, dragon, bird, fox, man and castle. The dataset has a total of 54 images covering all the concepts. BlueFire (Validation): The Bluefire validation set consists of 30 curated text prompts, of which 9 prompts contain one of 6 categories on which the model was trained, and the remaining 21 prompts correspond to categories which the low-rank adapter has not been fine-tuned on. These contain categories such as: (football, monster, sword, chess rook, lion, tiger, dog, cat, koala, panda). For all training experiments validating on this dataset, we produce 30 images per prompt, varying the input seed. Hence, the HPS analysis is over 900 image and LPIPS-diversity analysis is over 14500 image pairs. Paintings: On similar lines, the Paintings dataset is also a collection of images from public domain (CC0 license). The dataset has a total of 90 images cover 9 concepts - fire, bird, elephants, ship, horse, flower, woman, man and tiger. Paintings (Validation): The Paintings validation set consists of 21 curated text prompts, of which 9 prompts contain one of 9 categories on which the model was trained, and the remaining 12 prompts 18 correspond to categories which the low-rank adapter has not been fine-tuned on. These contain categories such as: (lion, tiger, dog, cat, koala, panda, and other landscapes) Paintings merged with BlueFire (Validation): The evauation set for merging Paintings and Bluefire consists of 18 curated text prompts. These contain categories such as: (fox, bird, lion, tiger, dog, cat, koala, panda, and other landscapes) For all training experiments validating on this dataset, we produce 30 images per prompt, varying the input seed. Hence, the HPS analysis is over 440 image and LPIPS-diversity analysis is over 8750 image pairs. Origami: The Origami dataset is also a collection of origami images from public domains. The dataset has a total of 52 images covering 7 concepts - bird, boat, flower, cat, dog, fox and house. 3D: The 3D dataset is also a collection of images from public domains. These images are animated images showing 3D concepts. The dataset has a total of 30 images covering 6 concepts - boy, girl, astronaut, cat, dog, elephant, dog and building. Concept Sliders: For concept sliders, we train and evaluate on three different concepts as shown in Table C.3. The evaluation set for each concept consists of 400 examples, over 10 seeds, essentially validating over 4000 images per concept. We follow the method in [8] Concept Positive prompt Negative prompt # Training Attributes # Val. Attributes Age very old, wrinkly, gray hair, aged skin very young, smooth skin, youthful 20 400 Surprise looking surprised, wide eyes, open mouth looking calm, neutral expression 20 400 Hair curly hair, wavy hair straight hair 20 400 Table C.3: Dataset statistics for Concept Slider Experiments C.2 Hyper-parameters and Implementation details for all experiments Text-to-image style transfer We used the kohya-ss4 repository for finetuning models for the text-to-image stylization task. For the masking we follow the approach for soft gating in 5. For each task, we trained both LoRA and FouRA adapters with the same set of hyperparameters. We trained using 4 NVIDIA A100 GPUs, for 100 epochs at at batch size of 8. Our initial learning rate was set to 1e−4 for UNet and 5e−5 for the text encoder. LoRA and FouRA modules are applied in the default places for stable-diffusion-v1.5 backbone, same as in HuggingFace Diffusers. We trained using two sets of weights, the base sd-1.56 from runwayML, and RealisticVision3.07. For some ablation studies, we varied the rank between 16, 32, 48, 64. In all the remaining experiments, we set the rank at 64 unless stated otherwise. Additionally, we set the Realistic Vision weights as our default for all experiments. For quantitative evaluation, we observed the HPS-v2.1 and LPIPS-Diversity metrics at a range of values between [0, 1] for adapter strength α. In all quantitative evaluations, we averaged over the same set of 30 seeds {0, 1, 2, ....29}. Image editing using Concept Sliders Single slider: The training data used in these experiments were curated from [9] . We used the repository 8 for finetuning the adapters. We train across 20 different attributes spanning different genders and races and other person attributes for each concept. The learning rate and other hyperparameters are re-used from the repository. For all the experiments we fix a rank of 8 and with 50 denoising steps. For evaluations, we tested across 400 different examples for 10 seeds on each prompt including unseen categories such as ’doctor’ , ’barista’, ’cowboy’ . For qualitative analysis, we compare across strengths ∈[−6, 6]). We also evaluated the inference across different 3 different edit times [750, 800, 850]. 4https://github.com/kohya-ss/sd-scripts 5https://github.com/prachigarg23/Memorisation-and-Generalisation-in-Deep-CNNs-Using-Soft-GatingMechanisms 6https://huggingface.co/runwayml/stable-diffusion-v1-5 7https://huggingface.co/spaces/Thafx/sdrv30 8https://github.com/rohitgandikota/sliders 19 Composite slider: For compositing we use similar setup as in the single slider. We compose the score functions using additive guidance. Specifically we weight each score function based on the relative strengths of the adapter during inference. GLUE benchmark experiments We trained the LoRA and SoRA [3] baselines on the GLUE benchmark using the code and default set of hyper-parameters provided by the authors9. For training FouRA, we used the same set of hyper-parameters as the LoRA baseline. These are provided in this issue in their repository. For all the experiments, we trained using 1 NVIDIA A100 GPU. For each task, and each baseline, we evaluated on all the samples of the validation set, the size of which is mentioned in Appendix C.2. This is slightly different from the evaluation in [3], as the authors originally ran inference only on a subset of the validation set, indicated here. Additionally, we used the set of three seeds {100, 81, 20}, chosen at random, to run all experiments. D Interpretations for Metrics In the main text, we used two metrics to validate style transfer on text-to-image diffusion models. Both are learnt metrics, i.e. HPS-v2.1 [44] and LPIPS-Diversity [49]. In this section, we provide reference ranges for both metrics, and how they can be interpreted. D.1 LPIPS Diversity We compute the LPIPS diversity δlpips of a dataset of n images as the average of the LPIPS pairwise distance between nC2 image pairs. In Figure D.1, we provide reference ranges for LPIPS distance between pairs of images. Notice the images in D.1a. are very similer. Hence, they generate a low LPIPS score (0.35). Hence in Table 2, we observe for high values of α, as the average LPIPS scores reflect that LoRA produces close to identical images in many case, but FouRA successfully gets rid of this data copying problem. Figures D.1b. and c. are lesser correlated from each other and hence produce a higher distance. Figures D.1d.-f. and g.-i. similarly vary from one another with in ascending order of LPIPS diversity scores, which is reflected in the image (The pose of the fox and variations in the fire in car images). The scores in Table 2 reflect a gain of 2-6 points in LPIPS diversity between LoRA and FouRA. These are significant improvements in the diversity of generated samples as observed from Figure D.1. Figure D.1: Interpretation of the LPIPS Diversity metric. This figure illustrates the interpretation of LPIPS Diversity, which we used to detect mode collapse. Images which look similar (i.e. sharing the same pose or similar characteristics) tend to generate a lower LPIPS distance. 9https://github.com/TsinghuaC3I/SoRA 20 D.2 Human Preference Scores For computing Human Preference Score, we utilized to the v2.1 HPS model provided by the authors [44]. Please refer to Figure D.2 for reference HPS-v2.1 values. Please note that in the Figure D.2 the "prompt" corresponds to the input prompt to HPS model, and may or may not be the prompt used to generate the image. Figure D.2: Interpretation of the HPS-v2.1 metric. This figure illustrates the interpretation of HPS scores, which we used to track three key aspects of generated images: 1.Alignment with the prompt, 2.Alignment with the adapter style and 3.Aesthetic quality. Observe that the HPS-v2.1 metric is able to effectively quantify these key aspects of generated images. The "Prompt" in this figure corresponds to the input prompt to HPS model for text and image alignment, and may or may not be the prompt used to generate the image We used HPS as a metric to track a combination of three key aspects of generated images. Alignment with the Prompt: Observe the first row in Figure D.2. For the wrong prompt (e.g. "Origami" for a cat image), the model produces a low HPS score (21.6). However, this score increases as the prompt and image alignment improves. 21 Strength of the adapter: Observe the second row in Figure D.2. The prompt we fed into HPS is the name of the adapter(blue fire). Notice how the HPS values increase for increase in the adapter strength. Image Quality: Observe the third row in Figure D.2. HPS scores can successfully differentiate between images with high and low aesthetic quality. Thus the, HPS provides us with a quantifiable metric for all the three aspects over we wish to evaluate our finetuned adapters. Moreover, the fourth row in Figure D.2 shows how the HPS can effectively track all these three aspects at once. Hence, the prompt we feed to the HPS model to evaluate an image is a combination of the name of the adapter and the prompt used for generating the image. E.g. the prompt used to evaluate image generated by "dog in space" with the adapter BlueFire, is "blue fire dog in space." This method also works well for evaluating the merging of two adapters. We simply add both the adapter names in the prompts while evaluating their HPS scores. E Additional Experiments on Text-to-Image stylization E.1 Additional Ablation Studies E.1.1 Performance on Unseen Concepts for Text-to-Image Stylization Section C.1.3 details the distribution of both our validation sets, Bluefire and Paintings. We split the validation set in seen and unseen concepts during training of the adapter. Bluefire contains 21 unseen categories (630 generated images), and Paintings contains 12 unseen categories (360 generated images). From Table E.1, we can observe that FouRA has a better generalization capability on unseen classes, as compared to LoRA. This result supplements our Proof for Corollary 4.2.1, essentially confirming that FouRA is able to reduce the upper bound of generalization error. Adapter Dataset HPSv2 score(↑) α = 1.0 α = 0.8 α = 0.6 LoRA Paintings (Unseen) 24.1 27.0 29.7 FouRA Paintings (Unseen) 28.5 30.4 31.7 LoRA Bluefire (Unseen) 32.5 33.6 33.8 FouRA Bluefire (Unseen) 33.2 34.4 34.4 Table E.1: Performance on unseen classes. Shows that on unseen classes FouRA generalizes better on unseen categories. E.1.2 Effect of varying the frequency transform Finally, we evaluate the effect of changing the frequency transform between DFT and DCT for our proposed FouRA (see Table E.2). First, we observe that both DFT- and DCT-based FouRA models significantly outperform LoRA. Also, both DFT and DCT achieve comparable scores in terms of HPSv2 which means our approach is robust to the type of frequency transforms being used. Transform LPIPS Diversity(↑) HPSv2 score(↑) α = 1.0 α = 0.8 α = 0.6 α = 1.0 α = 0.8 α = 0.6 LoRA 38.3 37.8 39.1 24.6 27.7 30.3 FouRA DFT 44.2 44.7 44.8 29.1 30.9 32.2 FouRA DCT 46.7 45.5 45.0 28.9 30.6 31.9 Table E.2: Effect of varying the frequency transform in FouRA E.1.3 Comparisons: 2D FFT on the tokens vs 1D FFT on token embeddings As illustrated in Fig. E.1, we proposed two variants of our approach: (1) FouRAemb that computes the frequency transform across the embedding dimension, and (2) FouRAtoken that computes the frequency transform along the token dimension. Table E.3, we compare FFT applied on token embeddings with LoRA. We hypothesize that transform done this way might capture variations in local patches of the image. Further as LoRA on vision adaptors generally apply rank reduction in the embedding dimension, applying the same in fourier dimension translates to spectral filtering in the embedding space. For the sake of completeness, we also run experiments to apply transform in the 2D token space, we call this FouRAtoken. In 22 Figure E.1: Two directions of the proposed Frequency Transform. FouRAemb computes the frequency transform along the embedding dimension (top), whereas FouRAtoken computes the frequency transform across all the tokens (bottom). Table E.3, we empirically observe that FouRAemb performs better than FouRAtoken. Hence, unless stated otherwise, we set FouRAemb as the default variant of FouRA for our experiments. Style Base Model Adapter LPIPS Diversity(↑) HPSv2 score(↑) α = 1 α = 0.8 α = 0.6 α = 1 α = 0.8 α = 0.6 Painting RealisticVision LoRA 38.3 ± 3.5 37.8 ± 3.6 39.2 ± 3.7 24.6 ± 1.8 27.7 ± 1.8 30.3 ± 1.7 FouRAtoken 44.2 ± 3.7 44.5 ± 4.0 44.6 ± 3.9 28.4 ± 1.8 30.6 ± 1.5 32.0 ± 1.4 FouRAemb 44.2 ± 3.8 44.7 ± 3.9 44.8 ± 3.9 29.1 ± 1.9 30.9 ± 1.6 32.2 ± 1.5 Blue Fire RealisticVision LoRA 46.8 ± 4.0 48.5 ± 4.0 49.8 ± 4.2 32.7 ± 1.6 33.8 ± 1.4 34.0 ± 1.5 FouRAtoken 50.4 ± 3.0 51.6 ± 3.3 52.2 ± 3.5 33.6 ± 1.5 34.1 ± 1.2 34.0 ± 1.4 FouRAemb 50.9 ± 3.1 52.3 ± 3.2 53.3 ± 3.8 33.4 ± 1.7 34.6 ± 1.3 34.5 ± 1.2 Table E.3: FouRAemb vs FouRAtoken vs LoRA E.2 Plots for quantiative metrics in Text-to-Image Stylization In Fig. E.2, we provide HPS and LPIPS-diversity scores at ranks {16, 32, 48, 64} and adapter strengths α = {0.2, 0.4, 0.6, 0.8, 1.0} for LoRA and FouRA. These plots are using the base weights of Realistic Vision-3.0. These scores are an extenstion to Table 2 of the main text. Observe FouRA outperforms LoRA on both metrics, at all ranks. Figure E.2: Quantitative Evaluations for LoRA v/s FouRA on text-to-image stylization. We provide plots at ranks {16, 32, 48, 64} and adapter strengths α = {0.2, 0.4, 0.6, 0.8, 1.0} E.3 Effect on data-copying artifacts after early stopping LoRA training We study the data-copying(distribution collapse) phenomenon in more detail in Figure E.3. We tracked the LPIPS-diversity as a measure of data-copying and HPS-v2 scores as a measure of adapter quality. We do notice lesser data copying artifacts in the initial phase of training. However, the adapter quality and strength are sub-par due to inadequate training (i.e. the style is not visible in the image). This is visible in HPS-v2 alignment scores. The images produced are similar to those from the base model, and hence lesser artifacts exist. As the training epochs increase, images start to represent the adapter style (represented by HPS scores). Once we reach this point, the number of data-copying artifacts increase significantly in LoRA, as tracked by the LPIPS-diversity. FouRA can achieve the adapter style while being able to produce a diverse range of images, as seen in Fig. 1. 23 Figure E.3: Studying the training curves for signs of data-copying artifacts: We analyzed the effect of early stopping of training by measuring the performance. All results are with rank 64 and α = 0.8 on the paintings adapter. E.4 Additional Computational Analysis In Section 5.5, we compared LoRA v/s FouRA in terms of training memory and inference time. In this Section, we provide additional computational analysis of our approach. As shown in Figure E.4, we analyzed performance of FouRA v/s LoRA with varying training complexity (training time, memory usage). To vary time, we report HPS scores of FouRA v/s LoRA at intermediate epochs. To vary the memory, we use rank. We observe that FouRA consistently achieves better performance v/s compute operating points compared to LoRA. Figure E.4: Training complexity v/s performance: We perform an analysis of training complexity v/s performance. This follows two settings: Varying the training epoch (left) to measure training time and Varying the rank (right) to measure peak training GPU memory. We measure HPS as the performance metric. All results are with α = 0.8 on the paintings validation set. Additionally, we showed how the training memory overhead scales with batch-size in Table E.4. We observe that the FouRA memory overhead during training time is negligible and only 0.3-0.4% over LoRA. Batch Size 8 6 4 2 LoRA 53687 MB 40872 MB 28151 MB 15499 MB FouRA 53894 MB 41020 MB 28255 MB 15448 MB Table E.4: Memory Overhead/Scaling with batch size: We report the scaling of training memory based on batch size. E.5 Additional Visual Results on Text-to-Image Stylization In Figure E.5, we provide additional visual results for FouRA and LoRA finetuning on the Bluefire dataset at varying adapter strengths. Within the generated images, the concepts ’Football’ and ’Dog’ are unseen. As observed, FouRA produces aesthetically appealing images as compared to LoRA in all cases. This is more evident in the ’Football’ example. As observed, FouRA can generalize better to new concepts, as compared to LoRA. In Figure E.6, we show additional results obtained by finetuning the Realistic Vision Model with FouRA adapters on our curated style datasets, 3d, Origami and Paintings. As observed, FouRA is capable of generating a diverse set of aesthetically appealing images. 24 Figure E.5: Visual Results using BlueFire adapters comparing LoRA and FouRA at varying values of α. Figure E.6: Images generated by FouRA trained on 3D, Paintings and Origami datasets. 25 F Additional Experiments for Text-to-Image Editing using Concept Sliders Concept sliders provide a framework to train LoRA adapters on single (image, prompt) pair (for example: "very old, wrinkly, gray hair, aged skin") in conjunction with multiple attributes (for example: Male person, very old etc). The disentanglement objective operates on the semantic space of diffusion models constraining the edit to occur only along the direction of the concept without changing the attributes. From 4 we learnt that ∆W has a small eigen spread leading to more compact representation. Our method favous lower effective rank and the trained model naturally converges to decorrelated subspaces from the base model weights B.3 . In addition in an informal proof B.4 we show that one can leverage the properties of FouRA to learn composition of concepts with less interference with the subspace of other concepts. We compare the performance of FouRA with LoRA when trained on explicit pairs of prompts across 20 different attributes acting as guidance. We train 3 sliders "curly hair", "surprise face" and "Age slider" on both the baseline LoRA and our adapter for upto 1000 steps. We trained the model on rank = 8. We show that despite explicit training on pairs, low rank adapter space is still prone to changes in gender and race for strong adapter scales especially strength ≥4. Below we show results on Single Adapter and Composite adapter. Single Concept We follow the SDEdit style inference where the adapter kicks in after T ∈ (750, 800, 850) timesteps. We notice that the effect of adapter in FouRA-DCT is far less below 800. Refer to figures below for more examples. For our results we fixed the T = 800. We evaluate our results on LPIPS F.4. While our adapter is far more stable compared to LoRA adapter between the strengths [−6, 6]. We also note that FouRA on DCT slightly better performance over FFT and for brevity we only show results on DCT. We note that FouRa maintains the balance between prompt and style fidelity and the quality of generated images. Below are some of the examples of Age, Figure F.1: Age Slider, LoRA (top) vs FouRA (bottom). We find that as the strength increases there are more prominent skin tone variations in LoRA. Figure F.2: Age FouRA Slider, "Portrait of a doctor" (top) and "Photo of an Hispanic man" (bottom). 26 In general Age sliders shows a good improvement on LPIPS score for strength above 3 as shown in figure F.4. We notice that as the strength increases FouRA disentangles from other attributes better. We also train an adapter to change the strength of curls in hair. Below we show more examples for curly hair. We notice that the both LoRA and FouRA adapters are sensitive to increasing strength. As can be observed LPIPS score are higher for Hair than for Age. As the strength increases the LoRA adapter tend move in the direction of increased prompt fidelity and removing the face of the person or crunching the face to add more details of hair in LoRA. We show the quanitative results for the same using LPIPS. We observe that across strengths 1 ≤5 the FouRA has much smaller LPIPS score. Please refer to the right figure in 8. Below we share more examples of FouRA on other prompts. Figure F.3: Hair Slider: We find that as the strength of the adapter increases the curls increase. In the top image we also see minor variations in the facial details of the person. Figure F.4: Perceptual metric drops for LoRA compared to FouRA for the sliders on "age" and "hair". These were tested across 10 scales from (-5, 5). Similarity score was computed across 1000 images and 500 prompts of 10 seeds each. Composite LoRA : Below we show the results for combining adapters. To combine adapters, we varied the strengths of Adapter 1 between strengths ∈(−8, 8) and Adapter 2 between strengths ∈ (−8, 8). We show some examples of only FouRA F.5 for combined hair and Age adapter. We show the images for when the adapter strengths are equal i.e increase from (−6, 6) to (6, 6). Below we show comparison between LoRA and FouRA across different adapter strengths. We emphasize the effect when one slider for e.g "Age" has a very high adapter strength on the second slider when the strength is low (bottom left image). We observe that for LoRA the facial distortions when both adapter strengths are high (bottom right) are very evident. The Age adapter in general seems to interfere more with the Hair for higher strengths. 27 Figure F.5: Composite FouRA . Composite surprised, age slider. Here we show the combined adapter as the strengths of each adapter are jointly incremented in each step in the image. The adapter strengths are (-6 6) for left most image and (6,6) for the right most image. The positive prompt for surprised face prompt: "looking surprised, wide eyes, open mouth" Figure F.6: Composite LoRA . Composite hair, age slider. We find that for higher strength of Age adapter as we increase the strength of Hair, adapter seems to interfere with the facial features and almost distort the face. However for lower values of Hair adapter. Here we show scales between -6 to 8 G FouRA on General Language Understanding Tasks While our design choices for FouRA are primarily motivated for vision tasks, we evaluate its efficacy on langauge tasks in Tab. G.1, and compare FouRA against another adaptive rank selection approach, SoRA, designed specifically for language tasks [3]. Results show that FouRA’s rank selection in frequency domain outperforms SoRA on four out of the six GLUE benchmarks we evaluated on, demonstrating that the feature disentanglement induced by FouRA can be used beyond vision tasks. 28 Figure F.7: Composite FouRA . Composite hair, age slider. We note that the adapter is stable for many prompts and seeds upto scale of 8. There are artifacts at large scales strength upto scale=8 of positive slider, however we find that artifacts are fewer and don’t distort the facial features. Adapter MNLI CoLA SST2 STSB MRPC QNLI LoRA 90.2 ± 0.2 67.3 ± 0.8 94.9 ± 0.3 89.9 ± 0.3 90.3 ± 0.6 93.6 ± 0.6 SoRA 90.5 ± 0.1 69.9 ± 0.8 95.2 ± 0.4 91.4 ± 0.1 90.6 ± 0.8 93.9 ± 0.3 FouRA 90.5 ± 0.1 70.6 ± 0.7 95.5 ± 0.4 91.6 ± 0.1 90.4 ± 0.5 94.2 ± 0.5 Table G.1: Evaluation of DeBERTa-V3 on the GLUE benchmarks, averaged over 3 seeds. H Societal Impacts In this section, we discuss the societal impacts of our work. While there are benefits of training FouRA modules as highlighted in the main text, we consider that it can potentially have larger societal impacts. One of the major challenges of text-to-image models is digital forgery, highlighted in previous works [39, 40]. We observed that finetuning low-rank adapters on various tasks in image generation can lead to replication of the input image. This is due to the overfitting of LoRA on a small training set. However, we demonstrate in the paper how FouRA can push the generalization error bound further, hence resolving the data forgery problem to a great extent. Hence, we propose to utilize FouRA in applications where it is imperative to hide the training set, such that it can’t be replicated. I Limitations FouRA, as demonstrated in the main text, is a highly effective parameter efficient fine-tuning method. However, as it makes use of frequency transforms (dft, dct), one potential limitation is that current Deep Learning hardware systems are not as optimal for frequency transform operations, as they are for matrix multiplies and convolutions. However, with astute recent works such as [38, 24, 28], their 29 popularity has increased in the field of Deep Learning. Hence, we foresee that it is only a matter of time before DL hardware systems get heavily optimized for frequency transforms. J Future Work We have demonstrated that FouRA achieves great performance on tasks such as image generation, Image concept and style editing on Vision tasks in diffusion framework. A good extension of FouRA would be to explore the generalization capabilities to reuse the learnt basis on other adapters trained on different datasets. Additionally, for the FouRA module we would like to explore direct token masking in the frequency domain, as we observed some initial indicators, effectively correlating bands of frequencies and various characteristics of generated images. Seeing the performance of FouRA, we feel encouraged to think that frequency domain fine-tuning of adapters will potentially be a popular research direction in the coming years. 30 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The paper provides detailed experimentation results and related theory which accuracy reflects the paper’s contributions. 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4,447	ContextGS: Compact 3D Gaussian Splatting with Anchor Level Context Model Yufei Wang1 Zhihao Li1 Lanqing Guo1 Wenhan Yang2 Alex C. Kot1 Bihan Wen1 1 Nanyang Technological University, Singapore 2 PengCheng Laboratory, China {yufei001, zhihao.li, lanqing.guo, eackot, bihan.wen}@ntu.edu.sg yangwh@pcl.ac.cn Homepage: https://github.com/wyf0912/ContextGS (c) Cosine similarity (d) Bit saving (a) Rendered image (b) Anchor division Level 0 Level 1 Level 2 Scaffold-GS (PSNR 24.50dB / Size: 248MB) ContextGS (Ours) (PSNR 25.08dB / Size: 21.81MB) 10× Compression! Figure 1: An illustration of the necessity of using autoregressive model in the anchor level. While Scaffold-GS [20] greatly reduces the spatial redundancy among adjacent 3D Gaussians by grouping them and introducing a new data structure anchor to capture their common features, spatial redundancy still exists among anchors. Our method, ContextGS, first proposes to reduce the spatial redundancy among anchors using an autoregressive model. We divide anchors into levels as shown in Fig. (b) and the anchors from coarser levels are used to predict anchors in finer levels, i.e., • predicts • then •• together predict •. Fig. (c) verifies the spatial redundancy by calculating the cosine similarity between anchors in level 0 and their context anchors in levels 1 and 2. Fig. (d) displays the bit savings using the proposed anchor-level context model evaluated on our entropy coding based strong baseline built on Scaffold-GS [20]. Compared with Scaffold-GS, we achieve better rendering qualities, faster rendering speed, and great size reduction of up to 15 times averaged over all datasets we used. Abstract Recently, 3D Gaussian Splatting (3DGS) has become a promising framework for novel view synthesis, offering fast rendering speeds and high fidelity. However, the large number of Gaussians and their associated attributes require effective compression techniques. Existing methods primarily compress 3D Gaussians individually and independently, i.e., coding all the 3D Gaussians at the same time, with little design for their interactions and spatial dependence. Inspired by the effectiveness of the context model in image compression, we propose the first autoregressive model at the anchor level for 3DGS compression in this work. We divide anchors into different levels and the anchors that are not coded yet can be predicted based on the already coded ones in all the coarser levels, leading to more 38th Conference on Neural Information Processing Systems (NeurIPS 2024). accurate modeling and higher coding efficiency. To further improve the efficiency of entropy coding, e.g., to code the coarsest level with no already coded anchors, we propose to introduce a low-dimensional quantized feature as the hyperprior for each anchor, which can be effectively compressed. Our work pioneers the context model in the anchor level for 3DGS representation, yielding an impressive size reduction of over 100 times compared to vanilla 3DGS and 15 times compared to the most recent state-of-the-art work Scaffold-GS, while achieving comparable or even higher rendering quality. 1 Introduction Over the past few years, novel view synthetic has rapidly progressed. As a representative work, Neural Radiance Field (NeRF) [22] uses a Multilayer Perceptron (MLP) to predict the attributes of quired points in the 3D scene. While good rendering qualities are achieved, the dense querying process results in slow rendering, which greatly hinders their applications in practical scenarios. Significant efforts have been made to enhance training and rendering speeds, achieving notable progress through various techniques, such as factorization [4, 8, 10, 13] and hash grids [24, 12]. However, they still face challenges in the real-time rendering of large-scale scenes due to the intrinsic limitations of volumetric sampling. Recently, 3D Gaussian Splatting (3DGS) [15] has achieved state-of-the-art (SOTA) rendering quality and speed. As an emerging alternative strategy for representing 3D scenes, 3DGS represents a 3D scene using a set of 3D Gaussians initiated from Structure-from-Motion (SfM) with learnable attributes such as color, shape, and opacity. The 2D images can be effectively rendered using differentiable rasterization and end-to-end training is enabled. Meanwhile, benefiting from efficient CUDA implementation, real-time rendering is achieved. Despite its success, 3DGS still encounters limitations in storage efficiency. Representing large scenes requires millions of 3D Gaussian points, which demand several GBs of storage, e.g., an average of 1.6 GB for each scene in the BungeeNerf [32] dataset. The huge storage overhead greatly hinders the applications of 3DGS [15], thus an efficient compression technique is required. However, the unorganized and sparse nature of these 3D Gaussians makes it highly challenging to effectively reduce data redundancy. To address this issue, various techniques have been proposed to enhance the storage efficiency of 3D Gaussian models. For example, [7, 18, 25, 26] proposed to discretize continuous attributes of 3D Gaussians to a cluster of attributes stored in the codebooks; [7, 18] proposed to prune neural Gaussians with little effect. Entropy coding is also used to reduce the storage overhead by further encoding neural Gaussian features into bitstream [11, 5, 18, 23]. Although space utilization has greatly improved, they focus more on individually compressing each Gaussian point and do not well explore the relationship and reduce the spatial redundancy among neural Gaussians. To further reduce the spatial redundancy, most recently, [20] proposed to divide anchors into voxels and introduced an anchor feature for each voxel to grasp the common attributes of neural Gaussians in the voxel, i.e., the neural Gaussians are predicted by the anchor features. While the spatial dependency has been significantly reduced, as shown in Fig. 1 (c), the similarity among anchors remains high in certain areas, indicating that spatial redundancy still exists. To further enhance the coding efficiency of 3D scenes, we propose a novel framework named ContextGS for 3DGS compression. Inspired by the effectiveness of context models [31] in image compression [21], we introduce an autoregressive model at the anchor level into 3DGS. Specifically, building on Scaffold-GS [20], we divide anchors into hierarchical levels and encode them progressively. Coarser level anchors are encoded first, and their decoded values are used to predict the distribution of nearby anchors at finer levels. This approach leverages spatial dependencies among adjacent anchors, allowing already decoded anchors to better predict the distribution of subsequent anchors, leading to significant improvements in coding efficiency. Additionally, anchors decoded at coarser levels can be directly used in the final fine-grained level, reducing storage overhead. To further enhance coding efficiency, especially for encoding the coarsest level anchors without already decoded ones, we employ a quantized hyperprior feature as an additional prior for each anchor. It is worth noting that the proposed method can also support vanilla 3DGS renderers by simply removing the view-dependent features from the anchor feature. Our contributions can be summarized as follows: • We propose the first context model for 3DGS at the anchor level. By predicting the properties of anchors that are not coded yet given already coded ones, we greatly eliminate the spatial redundancy among anchors. 2 • We propose a unified compressing framework with the factorized prior, enabling end-to-end entropy coding of anchor features. Besides, a strategy for anchor layering is proposed, which allows already decoded anchors to quickly locate adjacent anchors that are to be decoded. Meanwhile, the proposed method avoids redundant coding of anchors by the proposed anchor forward. • The experimental results on real-world datasets demonstrate the effectiveness of the proposed method compared with SOTA and concurrent works. On average across all datasets, our model achieves a compression ratio of 15× compared to the Scaffold-GS model we used as the backbone and 100× compared to the standard 3DGS model, while maintaining comparable or even enhanced fidelity. 2 Related works 2.1 Neural radiance field and 3D Gaussian splatting Early 3D scene modeling often employs the Neural Radiance Field (NeRF) [22] as a global approximator for 3D scene appearance and geometry. These approaches [2, 29, 30] use a multi-layer perceptron (MLP) to implicitly represent the 3D scene by predicting attributes of queried points. However, the dense querying process results in extremely slow rendering. Various methods have been developed to speed up the rendering process significantly [6, 9, 27], such as plane factorization-based techniques like K-Planes [8] and TensoRF [4], and the use of hash grid features in InstantNGP [24]. While these methods enable high-quality rendering with a much smaller MLP compared to the vanilla NeRF, rendering a single pixel still requires numerous queries. This can lead to increased storage requirements for the grid-based features and difficulties in efficiently rendering empty space or large-scale scenes. To achieve real-time and efficient rendering while maintaining high fidelity, 3DGS [15] introduces an innovative approach by representing the scene explicitly with numerous learnable 3D Gaussians. By employing differentiable splatting and tile-based rasterization [17], 3DGS [15] optimizes these Gaussians during training in an end-to-end manner. 2.2 Deep compression Despite the effectiveness of 3DGS [15] in rendering speed and fidelity, the large number of Gaussians and their associated attributes result in significant storage overhead. Many techniques are proposed to reduce the storage requirements of 3DGS. For example, [7, 18] proposes to prune neural Guassians (aka 3D Gaussians) with minimal impact. [7, 18, 25, 26] propose to utilize codebooks to cluster Gaussian parameters. Entropy coding is also used in [11, 5, 18, 23] to encode neural Gaussians into bit streams by modeling their distributions. While remarkable performances are achieved, they mainly focus on improving the efficiency of a single neural Gaussian and neglect the spatial redundancy among neighbor neural Gaussians. Most recently, Scaffold-GS [20] proposes to introduce an anchor level to capture common features of nearby neural Gaussians in the same voxel, and successive work [5] demonstrates its effectiveness by further introducing hash-feature as a prior for entropy coding. However, [5] codes all the anchors at the same time, and its spatial redundancy can be further reduced. In the image compression task, an important category of methods to improve the coding efficiency is the context model [21, 31], which greatly reduces the spatial redundancy by predicting the distribution of latent pixels based on an already coded one. Inspired by the context models used in compression, we propose to encode the anchor features in an autoregressive way, i.e., predict the anchor points from already coded ones at coarser levels. As far as we know, we are the first to reduce the storage redundancy of 3DGS using a context model at the anchor level. 3 Preliminary 3DGS [15] utilizes a collection of anisotropic 3D neural Gaussians to depict the scene so that the scene can be efficiently rendered using a tile-based rasterization technique. Beginning from a set of Structure-from-Motion (SfM) points, each Gaussian point is represented as follows G(x) = e−1 2 (x−µ)T Σ−1(x−µ), (1) where x is the coordinates in the 3D scene, µ and Σ are the mean position and covariance matrix of the Gaussian point, respectively. To ensure the positive semi-definite of Σ, Σ is represented as 3 x x Anchor Voxel Neural Gaussian Learnable offsets Masked Gaussian (opacity 𝛼< 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑) Decoding w/ bitstream Anchors at different levels Voxel Anchor forward (a) Anchor points with associated neural Gaussians. V"! = {𝐯! "#$, 𝐯% "#$} V"!"# = {𝐯! "#$, 𝐯$ "#$, 𝐯& "#$, 𝐯% "#$, 𝐯' "#$} 𝑀: 𝒫(V%!"#) →V%!"# Anchor forward Anchor division (From fine to coarse) V! = V#! = {𝐯" !#$, 𝐯% !#$} V!"# = V"!"# \V"! = {𝐯$ !#$, 𝐯& !#$, 𝐯' !#$} Deduplication of anchors (b) The proposed multi-level anchor division. Figure 2: (a): An illustration of the data structure we used following Scaffold-GS [19], where anchor points are used to extract common features of their associated neural Gaussians. (b): The proposed multi-level division of anchor points. The decoded anchors from higher (coarser) levels are directly forwarded to the lower (finer) level to avoid duplicate storage. Besides, taking decompression as an example, the already decoded anchors are used to predict anchors that are not decompressed yet, which greatly reduces the spatial redundancy among adjacent anchors. (Best zoom in for details.) Σ = RSST RT , where R and S are scaling and rotation matrixes, respectively. Besides, each neural Gaussian has the attributes of opacity α ∈R1 and view-dependent color c ∈R3 modeled by spherical harmonic [34]. All the attributes, i.e., [µ, R, S, α, c], in neural Gaussian points are learnable and optimized by the reconstruction loss of images rendered by the tile-based rasterization. Scaffold-GS [20]. While representing scenes with neural Gaussians greatly accelerates the rendering speed, the large amount of 3D Gaussians leads to significant storage overhead. To reduce the redundancy among adjunct 3D Gaussians, the most recent work, Scaffold-GS [19], proposes to introduce anchor points to capture common attributes of local 3D Gaussians as shown in Fig. 2 (a). Specifically, the anchor points are initialized from neural Gaussians by voxelizing the 3D scenes. Each anchor point has a context feature f ∈R32, a location x ∈R3, a scaling factor l ∈R3 and k learnable offset O ∈Rk×3. Given a camera at xc, anchor points are used to predict the view-dependent neural Gaussians in their corresponding voxels as follows, {ci, ri, si, αi}k i=0 = F(f, σc, ⃗dc) (2) where σc = \|\|x −xc\|\|2, ⃗dc = x−xc \|\|x−xc\|\|2 , the superscript i represents the index of neural Gaussian in the voxel, si, ci ∈R3 are the scaling and color respectively, and ri ∈R4 is the quaternion for rotation. The positions of neural Gaussians are then calculated as {µ0, ..., µk−1} = x + {O0, ..., Ok−1} · l, (3) where x is the learnable positions of the anchor and l is the base scaling of its associated neural Gaussians. After decoding the properties of neural Gaussians from anchor points, other processes are the same as the 3DGS [15]. By predicting the properties of neural Gaussians from the anchor features and saving the properties of anchor points only, Scaffold-GS [19] greatly eliminates the redundancy among 3D neural Gaussians and decreases the storage demand. 4 Methodology The overall framework is shown in Fig. 3. We first introduce how to divide anchors into levels with traceable mapping relationships among adjacent levels in Sec 4.1. Based on that, we present the entropy coding in an autoregressive way in Sec 4.2, and the overall training objective in Sec 4.3. 4.1 Anchor partitioning strategy We attempt to divide N anchors V = {vi}N i=0 = {(xi, fi, li, Oi)}N i=0 into K disjoint levels, i.e., V = V0 ∪V1 ∪... ∪VK−1 and Vi ∩Vj = ∅, ∀i ̸= j. For each anchor set Vi, it is expected to spawn over the whole scene, and be a relatively uniform downsampling of a finer set Vi−1. Assume that we encode/decode the scene using the order from level K −1 to level 0 where level K −1 is 4 Feature (x, y, z) Scaling Position Offset Context feature (from level 1) Distribution estimation Adaptive quantization 𝐟$ = 𝐟 𝑄𝐟 ⋅𝑄" Entropy estimation (Training) 𝐿#$%&'()(𝐟) Codec 010···101 (Testing) Quantized properties Learnable anchor attributes Distance, direction Hyperprior Entropy model 𝑝𝐳+\|-(𝐳-\|Θ) Codec 010···101 (Testing) 𝐿#$%&'()(𝐳-) (Training) Level dependent neural Gaussians Rendering Rendering result Reference 𝐿!, 𝐿""#$ (c) Neural Gaussian Splatting & 𝞪-blending 𝑖= 0 𝑁−1 A (a) Hyperprior Coding Anchor Feature Coding （Level 1） (b) Anchor Feature Coding A Anchor Feature Coding （Level 2） Hyperprior feature … 𝜎 𝜇 𝑄 Figure 3: The overall framework of the proposed method includes three levels, i.e., K = 3, to encode the anchors. The decoded anchors from a coarser level i + 1 are used to encode the anchors in level i. Besides, hyperprior features are used to predict the properties of anchors at all levels. For training, after finishing the coding of all levels, the anchor features after adaptive quantization are used to predict properties of neural Gaussians. The rendering loss is calculated and optimized together with the entropy coding loss Lentropy. For testing, after we decode anchor features from the bit stream, the rendering is exactly the same with Scaffold-GS [20] without introducing overhead. the coarsest level, we expect the mapping from Vi to Vi−1 is traceable and easy to obtain. In other words, given an anchor vk i = (xk i , f k i , lk i , Ok i ) from a coarser level k, we expect to quickly locate vk i ’s adjacent anchor set {vk−1 i } Nk−1 i i=0 in level k −1, where N k−1 i is the number of adjacent anchors, i.e., the anchors in the same voxel of level k −1 with vk i . As such, after decoding anchors at a coarser level, we can scatter the already decoded features to to-be-processed anchors as the prior for better coding efficiency. To achieve the requirements above, we propose to utilize a simple yet effective way to divide anchors into levels using a “bottom-up” method. As shown in Fig. 2 (b), given a set of anchors of the scene, we partition them into different fine-grained sets based on different voxel sizes ϵi as follows ˆVk = n M({vk−1 i : ˆxk i = ˆx}) : ˆx ∈ n ˆxk i : i = 1, 2, ..., \| ˆVk−1\| oo , (4) where \| · \| is the counting operation, ˆxk i is the anchor position after quantization using the voxel size of level k, and M : P( ˆVk−1) →ˆVk−1 (where P is the power set) is a mapping function that selects a representative anchor to level k −1 from a set of anchors {vk−1 i : ˆxk i = ˆx} that has the same position after quantization. The definition of ˆxk i and M are as follows ˆxk i = $ xk−1 i ϵk ' ϵk, xk−1 i ∈ˆVk−1 M({vk−1 i : ˆxk i = ˆx}) = vk−1 j s.t. j = min{i : ˆxk i = ˆx}, (5) where ˆV0 is initialized as the whole anchor set V. We select the anchor with the minimum index min{i : ˆxk i = ˆx} among a set of anchors in level k −1 that are in the same voxel. Besides, we filter out the repeated anchors among different levels as follows V2 = ˆV2, V1 = ˆV1\ ˆV2, V0 = ˆV0\ ˆV1 (6) where \ is the set difference operation. We keep the voxel size ϵ0 of the finest level (level 0) the same as the initial value ϵ, and set the voxel size of level i to ϵi = κi · ϵ. Since different scenes have different initial voxel sizes and anchor point distributions, using a fixed set of voxel scaling parameters {κi}K i=1 leads to a suboptimal performance. To avoid finetuning hyper-parameters for each scene, we propose to conduct a one-time parameter search after initializing anchors. Instead of 5 directly setting the scale κi, we set a target ratio between level i and i + 1 and expect \|Vi+1\| \|Vi\| ≈τ. Since \|Vi\| decreases monotonically with κi, we can easily and efficiently determine the values of {κi}K i=1 using a binary search. We empirically find that the performance of models among different senses is relatively robust to the selection of τ (refer to Fig. 6). 4.2 Coding with entropy models After dividing the anchors into multi-levels, in this section, we discuss how we use the already decoded anchors to predict ones that are not decompressed yet and how to encode attributes of anchors to improve the coding efficiency. Context modeling in anchor levels. To encode an anchor point vi = (xi, fi, li, Oi) into bitstreams efficiently using entropy coding, we need to estimate its distributions accurately. The core idea of the proposed method is to predict the properties of anchors additionally conditioned on already decompressed anchors. Taking the modeling of anchor feature f k i from the anchor vk i as an example, the details are as follows pf k(f k i \|ψk i ) = (N(µk i , σk i ) ∗U(−1 2∆k i , 1 2∆k i ))(f k i ), µk i , σk i , ∆k i = F k fi(ψk i ), (7) where F k fi is a MLP belonging to the level k, and ψk i is the prior of the anchor vk i as follows ψk i = [xk i ] k = K −1 [f k+1 j ; lk+1 j ; xk i ], k < K −1 , (8) where [·] is the concatenation operation among the channel dimension, f k+1 j , lk+1 j are the feature and scaling from the adjacent anchor vk+1 j in level k + 1 that are already decoded as shown in Fig. 2 (b). Hyperprior feature for anchor. While introducing the position xk i contributes to predicting the distribution of anchor features, it still lacks enough freedom to eliminate spatial redundancy. Therefore, we introduce a learnable hyperprior vector zi ∈R50//hc for each anchor vi where hc is a hyper-parameter to control the length of the hyperprior features. The hyperprior zi is modeled using the non-parametric, fully factorized density model [1] as follows: p˜z\|Θ(˜zi\|Θ) = 50//hc−1 Y j=0 pzj i \|Θ(j)(Θ(j)) ∗U −1 2, 1 2 (˜zj i ), (9) where ˜zi represents zi with quantization noise, j is the channel index and Θ is the network parameters for modeling the hyperprior. Since the hyperprior feature zi is quantized into integers ˆz and jointly optimized to reduce the size of the bitstream, it only occupies a small portion of storage as shown in Table 4. The final prior for coding features of the anchor vk i is ˆψk i = [ˆzk i ; ψk i ]. 4.3 Training objective The training objective of the proposed method is to jointly optimize the bitrate of coded anchor features and rendering loss measured by SSIM and L1 loss. The final training loss is L = Lscaffold + λeLentropy + λmLm (10) where Lscaffold is the training loss of [19], Lm is the masking loss from [18] to regularize the masking loss of offsets of neural Gaussians Ov, and Lentropy is the overall entropy loss that measures the cost of storing anchor properties defined as follows, Lentropy = E[−log2 p˜z\|Θ(˜zi\|Θ)] + K−1 X i=0  −log2   Y f∈{f k,lk,Ok} pf(fi\| ˆψk i )     (11) where the first term measures the cost of coding the hyperprior feature while the second term is the cost of coding features of anchor points in all the levels, and ˆψk i is the context feature that includes both the hyperprior feature zk i and the feature from already coded nearby anchor in level k + 1 as illustrated in Eq. 8. 6 Table 1: The quantitative results obtained from the proposed method ContextGS and other competitors. Baseline methods, namely 3DGS [15] and Scaffold-GS [20], are included for reference. The intermediary approaches are specifically designed for 3DGS compression. Our methodology showcases two results representing varying size and fidelity tradeoffs, achieved through adjustment of λe. Highlighted in red and yellow cells are the best and second-best results, respectively. Size measurements are expressed in megabytes (MB). Datasets Mip-NeRF360 [3] Tank&Temples [16] DeepBlending [14] BungeeNeRF [32] methods psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓ 3DGS [15] (SIGGRAPH’23) 27.49 0.813 0.222 744.7 23.69 0.844 0.178 431.0 29.42 0.899 0.247 663.9 24.87 0.841 0.205 1616 Scaffold-GS [20] (CVPR’24) 27.50 0.806 0.252 253.9 23.96 0.853 0.177 86.50 30.21 0.906 0.254 66.00 26.62 0.865 0.241 183.0 EAGLES [11] 27.15 0.808 0.238 68.89 23.41 0.840 0.200 34.00 29.91 0.910 0.250 62.00 25.24 0.843 0.221 117.1 LightGaussian [7] 27.00 0.799 0.249 44.54 22.83 0.822 0.242 22.43 27.01 0.872 0.308 33.94 24.52 0.825 0.255 87.28 Compact3DGS [18] (CVPR’24) 27.08 0.798 0.247 48.80 23.32 0.831 0.201 39.43 29.79 0.901 0.258 43.21 23.36 0.788 0.251 82.60 Compressed3D [26] (CVPR’24) 26.98 0.801 0.238 28.80 23.32 0.832 0.194 17.28 29.38 0.898 0.253 25.30 24.13 0.802 0.245 55.79 Morgenstern et al. [23] 26.01 0.772 0.259 23.90 22.78 0.817 0.211 13.05 28.92 0.891 0.276 8.40 − − − − Navaneet et al. [25] 27.16 0.808 0.228 50.30 23.47 0.840 0.188 27.97 29.75 0.903 0.247 42.77 24.63 0.823 0.239 104.3 HAC [5] 27.53 0.807 0.238 15.26 24.04 0.846 0.187 8.10 29.98 0.902 0.269 4.35 26.48 0.845 0.250 18.49 Ours (low-rate) 27.62 0.808 0.237 12.68 24.20 0.852 0.184 7.05 30.11 0.907 0.265 3.45 26.90 0.866 0.222 14.00 Ours (high-rate) 27.75 0.811 0.231 18.41 24.29 0.855 0.176 11.80 30.39 0.909 0.258 6.60 27.15 0.875 0.205 21.80 101 102 DeepBlending 29.00 29.25 29.50 29.75 30.00 30.25 PSNR (dB) 3DGS Scaffold-GS EAGLES Compact3DGS Compressed3D Morgenstern et al. Navaneet et al. HAC Ours 102 103 BungeeNeRF size(MB) 25.0 25.5 26.0 26.5 27.0 3DGS Scaffold-GS EAGLES Compact3DGS Compressed3D Morgenstern et al. Navaneet et al. HAC Ours 102 Mip-NeRF360 26.0 26.5 27.0 27.5 3DGS Scaffold-GS EAGLES Compact3DGS Compressed3D Morgenstern et al. Navaneet et al. HAC Ours Figure 4: The Rate-Distortion (RD) curves for quantitative comparison between our method with most recent SOTA competitors. It is worth noting that the x-axis is in log scale for better visualization. 5 Experiments 5.1 Implementation details We build our method based on Scaffold-GS [20]. The number of levels is set to 3 for all experiments and the target ratio among two adjacent iterations is 0.2. hc is set to 4, i.e., the dimension of the hyper-prior feature is a fourth of the anchor feature dimension. For a fair comparison, the dimension of the anchor feature f is set to 50 following [5] and we set the same λm = 5e −4. The setting of λe is discussed in Appendix A.3 since different values are used to evaluate different rate-distortion tradeoffs. For a fair comparison, we use the same training iterations with Scaffold-GS [20] and HAC [5], i.e., 30000 iterations. Besides, we use the same hyperparameters for anchor growing as Scaffold-GS [20] so that the final model has a similar or even smaller number of anchors, leading to faster rendering speed. More implementation details are in the supplementary materials. 5.2 Comparison with baselines Baseline, metric, and benchmark. We compare our method with 3DGS [15], Scaffold-GS [20] and some representative 3DGS compression works, including Compact3DGS [18], Compressed3D [26], EAGLES [11], LightGaussian [7], Morgenstern et al. [23], Navaneet et al. [25], and HAC [5]. The baseline methods include existing mainstream techniques, e.g., pruning [7, 18], codebooks [7, 18, 25, 26], and entropy coding [11, 5, 18, 23], and includes the most recent works. We utilize PSNR, SSIM, and LPIPS [35] to evaluate the rendering qualities of different methods and report the storage size measured in MB. We evaluate the performance of the models on several real-scene datasets, including BungeeNeRF [32], DeepBlending [14], Mip-NeRF360 [3], and Tanks&Temples [16]. To more comprehensively evaluate the performance of our method, following the previous prototype [5], we use all 9 scenes in Mip-NeRF360 [3]. The detailed results of each scene are reported in the Appendix A.3. To further evaluate the performance models among a wide range of compression ratios, we use Rate-Dsitoration (RD) curves as an additional metric. 7 (a) Reference image (The scene train from Tank&Temples dataset) (b) Compact3DGS (PSNR: 21.70 Size: 18 MB) (c) ScaffoldGS (PSNR: 22.47 Size: 60MB) (d) HAC (PSNR: 22.15 Size: 7.0MB) (e) Ours (PSNR: 22.42 Size: 6.4MB) (a) Reference image (The scene train from Tank&Temples dataset) (b) Compact3DGS (PSNR: 25.33 Size: 45.4MB) (c) ScaffoldGS (PSNR: 28.04 Size: 171MB) (d) HAC (PSNR: 27.60 Size: 17.3MB) (e) Ours (PSNR: 28.08 Size: 13.2MB) Figure 5: Visual comparisons between our method and baselines including Scaffold-GS [20], HAC [5], and Compact3DGS [18] on Bungeenerf [32] and Tank&Temples [16]. We report the PSNR (dB) of the image and the size of the 3D scene. (Best zoom in for details.) 0.2 0.4 0.6 0.8 Ratio 16 18 20 22 24 26 Size rome size (MB) rome PSNR (db) amsterdam size (MB) amsterdam PSNR (db) Figure 6: The ablation of different target ratio τ among different scenes. The PSNR remains relatively stable while the size of the scenes keeps increasing when increasing the τ, which demonstrates the effectiveness. Table 2: The ablation study of each component we proposed measured on BungeeNerf [32] dataset. “HP” and “CM” represent the hyperprior and anchor level context model respectively. Ours w/o HP w/o CM can be roughly regarded as a Scaffold-GS [20] model with entropy coding and masking loss [18]. Size (MB) PSNR SSIM LPIPS Scaffold-GS [20] 183.0 26.62 0.865 0.241 Ours w/o HP w/o CM 18.67 26.93 0.867 0.222 Ours w/o CM 15.03 26.91 0.866 0.223 Ours w/o HP 15.41 26.92 0.867 0.221 Ours 14.00 26.90 0.866 0.222 Results. As shown in Table 1, the proposed method achieves significant improvement compared to our backbone method Scaffold-GS [20] in terms of the size of the model, with a size reduction of 15× in average. Besides, the proposed method also achieves higher storage efficiency compared to the most recent competitors for the 3DGS compression, e.g., HAC [5], Compressed3D [26], and Compact3DGS [18]. It is worth noting that the proposed method also significantly improves rendering quality, even compared with the backbone model we use, i.e., Scaffold-GS [20]. This further verifies the observation from previous works that appropriate constraints on neural Gaussians can contribute to the rendering quality, e.g., entropy constraints [5] and pruning [33]. Visual comparisons between the proposed method and other competitors are shown in Fig. 5. As shown in the figure, the proposed method achieves better rending quality with greatly reduced size compared with most recent 3D compression works and also the backbone model. Besides, a comparison of the RD curves among the proposed method and most recent competitors is shown in Fig. 4, where the proposed method achieves better performance in a wide range of compression ratios. 5.3 Ablation studies and discussions Evaluation of target ratio τ among adjacent levels. To evaluate the performance of the proposed strategy that encodes anchors in a progressive way, we evaluate the performance of models trained under different ratios among adjacent levels. We disable the hyperprior feature to better explore the effect of different target ratios τ. As shown in Fig. 6, the PSNR remains relatively stable and the size gets relatively converged at the low ratio area. We select τ = 0.2 for all experiments. Ablation of each component. We verify the effectiveness of two main components in our methods, i.e., anchor level context model and hyperprior features, and the results are shown in Table 2. We build all the models on Scaffold-GS [20] and set the model with the entropy constraint and the 8 Table 3: The ablation study of our method w/ and w/o reusing anchors from coarser levels, i.e., anchor forward technique, measured on BungeeNerf [32] dataset. Size (MB) PSNR SSIM LPIPS Ours w/o dividing into levels 14.73 26.91 0.867 0.222 Ours w/o reusing anchors 15.54 26.91 0.862 0.230 Ours 13.80 26.89 0.867 0.222 Table 4: The storage cost of each component and rending qualities of our method and baselines evaluated on the scene rome in BungeeNeRF [32] dataset. “w/ APC” represents using anchor position coding, i.e., using the hyperprior features to code the anchor positions. (The encoding/decoding time is measured on an RTX3090.) Number of Anchors (K) Storage Cost (MB) Fidelity Speed (s) Hyper Position Feature Scaling Offset Mask MLPs Total PSNR SSIM Encode Decode Scaffold-GS [20] 61.9 N/A 7.08 75.16 14.18 70.88 N/A 0.047 184.4 26.25 0.872 N/A N/A Ours (w/ APC) 52.3 1.026 1.954 5.708 1.603 2.556 0.452 0.320 13.62 26.38 0.871 41.33 51.58 Ours 52.5 0.778 2.543 5.808 1.586 2.563 0.452 0.316 14.06 26.38 0.871 20.40 17.85 masking loss [18] as our baseline model, i.e., “Ours w/o HP w/o CM”. It is worth noting that our baseline model significantly improves the storage efficiency compared with Scaffold-GS [20] and even the latest SOTA methods. Both the proposed anchor-level context model and hyperprior feature for anchors significantly improve the compression rate compared with our strong baseline model, reducing the file size by 21% and 10.17%, respectively. Besides, using them together can further boost the performance with storage savings of 26.1% and 92.5% compared with the baseline we introduced above and Scaffold-GS [20] respectively. Ablation of anchor forward. A main difference between the proposed method and existing works for Level-of-Detail (LOD) techniques [28] is that the proposed method can reuse the anchors of different levels, i.e., anchor forward in Fig. 2 (b). For example, the anchors from different levels in [28] are stored separately. In contrast, the anchors from coarser levels are used in the final level (level 0) in our method, i.e., in an autoregressive manner. To verify the effectiveness of the proposed method that reuses the anchors in coarser levels, we do an ablation study in Table 3. As shown in the table, the model w/o reusing anchors of coarser levels to the finest level leads to serious redundancy, even slightly worse than the model w/o dividing anchors into different levels. This demonstrates the effectiveness of the proposed anchor forward technique for the anchor-level context model. Discussion on compressing anchor positions. One can utilize the hyperprior feature z to predict the distribution of anchor positions and the anchor position can thereby be compressed using entropy coding. However, we find that the precision of the anchor position is essential to the performance of the model and an adaptive quantization strategy leads to serious performance degradation. While a fixed quantization width is feasible and can retain the fidelity performance while effectively compressing the size of anchors, it leads to a greatly increased number of symbols that greatly decreases the coding speed. Since the anchor position only occupies a small portion of bitstreams as shown in Table 4, we do not encode anchors into bitstreams in all the experiments. Analysis of inference and decoding time. The rendering speed after decompression is the same as or even faster than Scaffold-GS [20] when the number of anchors is the same since we use the same data structure. However, as shown in Table 4, we can achieve higher rendering quality with fewer anchors due to the use of masking loss [18] therefore achieving faster rendering speed. For the decoding time, while the proposed method involves autoregressive coding, which is usually very slow in image compression tasks [21] due to its serial characteristics, it adds neglectable overhead to our method in both training and decompression compared to other entropy-coding-based 3DGS compression methods, such as HAC [5]. This is because, unlike autoregressive coding in image compression that predicts pixels/latent features one by one, introducing a loop of at least thousands of operations, we perform autoregressive coding group by group, introducing only a loop of 3 iterations. Additionally, there is no overlap of anchors among the coding of different levels, so the overall number of anchors to be processed remains the same as without dividing into levels. 9 6 Conclusion In this work, we introduce a pioneer study into utilizing the anchor-level context model in the compression of 3D Gaussian splatting models. We divide anchors into different levels and the anchors from coarser levels are first coded and then used to predict anchors that are not coded yet. Additionally, a hyperprior feature is used for each anchor that further reduces the channel-wised redundancy. Besides, we demonstrate that utilizing the proposed anchor forward technique, i.e., directly reusing the anchors from coarse levels to the final level, can achieve better performance than just using anchors of coarse levels as a prior. Extensive experiments demonstrate that the proposed methods achieve better performance than SOTA and concurrent works. 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Jiang, T. Lu, M. Yu, L. Xu, Z. Ni, and B. Dai. Octree-gs: Towards consistent real-time rendering with lod-structured 3d gaussians. arXiv preprint arXiv:2403.17898, 2024. [29] D. Rho, B. Lee, S. Nam, J. C. Lee, J. H. Ko, and E. Park. Masked wavelet representation for compact neural radiance fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20680–20690, 2023. [30] T. Takikawa, A. Evans, J. Tremblay, T. Müller, M. McGuire, A. Jacobson, and S. Fidler. Variable bitrate neural fields. In ACM SIGGRAPH 2022 Conference Proceedings, pages 1–9, 2022. [31] A. Van den Oord, N. Kalchbrenner, L. Espeholt, O. Vinyals, A. Graves, et al. Conditional image generation with pixelcnn decoders. Advances in neural information processing systems, 29, 2016. [32] Y. Xiangli, L. Xu, X. Pan, N. Zhao, A. Rao, C. Theobalt, B. Dai, and D. Lin. Bungeenerf: Progressive neural radiance field for extreme multi-scale scene rendering. In European conference on computer vision, pages 106–122. Springer, 2022. [33] R. Yang, Z. Zhu, Z. Jiang, B. Ye, X. Chen, Y. Zhang, Y. Chen, J. Zhao, and H. Zhao. Spectrally pruned gaussian fields with neural compensation. arXiv preprint arXiv:2405.00676, 2024. [34] Q. Zhang, S.-H. Baek, S. Rusinkiewicz, and F. Heide. Differentiable point-based radiance fields for efficient view synthesis. In SIGGRAPH Asia 2022 Conference Papers, pages 1–12, 2022. [35] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018. 11 A Appendix / supplemental material A.1 Broader impacts The social impacts of our work are mainly in three folds: 1. Accessibility: This work contributes to making advanced 3DGS [15, 20] models more accessible to a wider audience. The reduced size of the 3DGS representation means that it can be more easily stored, transmitted, and processed on various devices, potentially democratizing access to high-quality rendering capabilities. 2. Applications: Improved compression techniques for 3DGS could enhance various applications across industries, including virtual reality, gaming, medical imaging, and architectural visualization. These advancements may lead to more immersive experiences, better medical diagnostics, and more efficient design workflows. 3. Open Access: The commitment to releasing the code fosters transparency and collaboration within the research community. Open access to the code allows other researchers and practitioners to build upon this work, accelerating innovation in the field of view synthesis and compression. We do not find serious negative impacts since our work is only for compression. There is no concerns regarding the misuse of our models or generating fake data. A.2 Limitations A main and inevitable limitation of the proposed method is that the entropy coding process introduces extra computational costs to estimate the entropy of the anchor features during training encoding/decoding when saving/loading the 3D scene. For example, it requires extra time to decode the data of 3D scenes from the bitstream, making it challenging to start rendering at once when clicking the file. A.3 More experimental details and results We report the detailed comparison of our method w/ and w/o using the coding of anchor position in Table 5. To evaluate the performance among different rate-distortion (RD) tradeoffs, we utilize different λe, i.e., a larger λe leads to a smaller model size. For a fair comparison, we use the same normalization of the hyper-parameter λe as [5] by additionally utilizing the dimension of anchor features as the divisor so that the same λe can lead to the similar bias towards the RD tradeoffs. The detailed results of each scene with different λe are reported in Table 6, Table 8, Table 7, and Table 9. Table 5: Quantitative results of the effect of anchor positon coding on our method. For our approach, we give two results of different size and fidelity tradeoffs by adjusting λe. A smaller λe results in a larger size but improved fidelity, and vice versa. Datasets Mip-NeRF360 [3] Tank&Temples [16] DeepBlending [14] BungeeNeRF [32] methods psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓psnr↑ssim↑lpips↓size↓ w/ encoding anchors Ours (low-rate) 27.61 0.809 0.236 11.32 24.16 0.851 0.185 6.91 30.11 0.907 0.269 3.31 26.89 0.867 0.222 13.80 Ours (high-rate) 27.86 0.813 0.230 21.07 24.29 0.855 0.178 11.47 30.42 0.910 0.261 6.40 27.15 0.876 0.202 25.23 w/o encoding anchors Ours (low-rate) 27.62 0.808 0.237 12.68 24.20 0.852 0.184 7.05 30.11 0.907 0.265 3.45 26.90 0.866 0.222 14.00 Ours (high-rate) 27.75 0.811 0.231 18.41 24.29 0.855 0.176 11.80 30.39 0.909 0.258 6.60 27.15 0.875 0.205 21.80 12 Table 6: Per-scene results of our method on BungeeNerf [32] dataset. Scene Size PSNR SSIM LPIPS low-rate (λ = 0.004) rome 13.9631 25.9751 0.8669 0.2143 quebec 11.5909 30.1941 0.9337 0.1669 pompidou 15.2258 25.4676 0.8469 0.2446 hollywood 13.5183 24.5154 0.7772 0.3164 bilbao 13.1537 28.0842 0.8877 0.1932 amsterdam 16.5549 27.1694 0.8842 0.1956 Average 14.0011 26.9010 0.8661 0.2218 high-rate (λ = 0.001) rome 21.7033 26.6297 0.8825 0.1929 quebec 18.1517 30.5271 0.9400 0.1510 pompidou 23.4748 25.6218 0.8546 0.2330 hollywood 20.7082 24.7170 0.7876 0.3002 bilbao 20.7686 27.9842 0.8928 0.1770 amsterdam 26.0079 27.3979 0.8949 0.1754 Average 21.8024 27.1463 0.8754 0.2049 Table 7: Per-scene results of our method on DeepBlending [14] dataset. Scene Size PSNR SSIM LPIPS low-rate (λ = 0.004) drjohnson 3.94 29.68 0.906 0.261 playroom 2.96 30.53 0.907 0.269 Average 3.45 30.11 0.907 0.265 high-rate (λ = 0.0005) drjohnson 7.80 29.86 0.909 0.249 playroom 5.41 30.93 0.910 0.268 Average 6.60 30.39 0.909 0.258 Table 8: Per-scene results of our method on Tank&Template [16] dataset. Scene Size PSNR SSIM LPIPS low-rate (λ = 0.004) train 6.39 22.40 0.818 0.217 truck 7.72 26.00 0.885 0.150 Average 7.05 24.20 0.852 0.184 high-rate (λ = 0.0005) train 10.55 22.53 0.823 0.208 truck 13.06 26.05 0.888 0.143 Average 11.80 24.29 0.855 0.176 13 Table 9: Per-scene results of our method on the Mip-NeRF360 [3] dataset. Scene Size PSNR SSIM LPIPS low-rate (λ = 0.004) bicycle 21.82 25.08 0.738 0.271 bonsai 7.17 32.67 0.946 0.186 counter 6.30 29.38 0.912 0.197 flowers 16.71 21.31 0.576 0.377 garden 18.78 27.32 0.845 0.148 kitchen 7.00 31.28 0.925 0.131 room 4.50 31.71 0.923 0.207 stump 14.86 26.58 0.762 0.268 treehill 17.00 23.29 0.647 0.349 Average 12.68 27.62 0.808 0.237 high-rate (λ = 0.0005) bicycle 38.09 24.97 0.740 0.263 bonsai 12.43 32.93 0.951 0.182 counter 10.67 29.60 0.916 0.190 flowers 28.27 21.18 0.571 0.378 garden 31.26 27.39 0.851 0.136 kitchen 12.44 31.69 0.931 0.123 room 8.09 32.03 0.928 0.196 stump 24.22 26.56 0.761 0.263 treehill 28.78 23.09 0.645 0.346 Average 21.58 27.72 0.811 0.231 14 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We clearly summarized the contributions in both abstract and introduction. 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4,448	Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity Haoxuan Chen∗ ICME Stanford University haoxuanc@stanford.edu Yinuo Ren∗† ICME Stanford University yinuoren@stanford.edu Lexing Ying Department of Mathematics and ICME Stanford University lexing@stanford.edu Grant M. Rotskoff Department of Chemistry and ICME Stanford University rotskoff@stanford.edu Abstract Diffusion models have become a leading method for generative modeling of both image and scientiﬁc data. As these models are costly to train and evaluate, reducing the inference cost for diffusion models remains a major goal. Inspired by the recent empirical success in accelerating diffusion models via the parallel sampling technique [1], we propose to divide the sampling process into O(1) blocks with parallelizable Picard iterations within each block. Rigorous theoretical analysis reveals that our algorithm achieves e O(poly log d) overall time complexity, marking the ﬁrst implementation with provable sub-linear complexity w.r.t. the data dimension d. Our analysis is based on a generalized version of Girsanov’s theorem and is compatible with both the SDE and probability ﬂow ODE implementations. Our results shed light on the potential of fast and efﬁcient sampling of high-dimensional data on fast-evolving modern large-memory GPU clusters. 1 Introduction Diffusion and probability ﬂow based models [2–11] are now state-of-the-art in many ﬁelds, such as computer vision and image generation [12–22], natural language processing [23, 24], audio and video generation [25–29], optimization [30, 31], sampling and learning of ﬁxed classes of distributions [32–41], solving high-dimensional partial differential equations [42–46], and more recently several applications in physical, chemical and biological ﬁelds [47–63]. For a more comprehensive list of related work, one may refer to the following review papers [64–66]. While there are already many variants, such as denoising diffusion probabilistic models (DDPMs) [7], score-based generative models (SGMs) [9], diffusion schrödinger bridges [67], stochastic interpolants and ﬂow matching [2–4], etc., the recurring idea is to design a stochastic process that interpolates between the data distribution and some simple distribution, along which score functions or alike are learned by neural network-based estimators, and then perform inference guided by the learned score functions. Due to the sequential nature of the sampling process, the inference of high-quality samples from diffusion models often requires a large number of iterations and, thus, evaluations of the neural network-based score function, which can be computationally expensive [68]. Efforts have been ∗Equal contribution, alphabetical order. †Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Work Implementation Measure Approx. Time Complexity [100, Theorem 2] SDE TV(p0, bqT )2 e O(dδ−1) [104, Theorem 2] SDE DKL(pη∥bqT −η) e O(d2δ−2) [107, Corollary 1] SDE DKL(pη∥bqT −η) e O(dδ−2) [111, Theorem 3] ODE w/UMLC correction TV(pη, bqT −η)2 e O( √ dδ−1) Theorem 3.3 SDE w/parallel sampling DKL(pη∥bqT −η) e O(poly log(dδ−2)) Theorem 3.5 ODE w/parallel sampling TV(pη, bqT −η)2 e O(poly log(dδ−2)) Table 1: Comparison of the approximate time complexity (cf. Deﬁnition 2.1) of different implementations of diffusion models. η is a small parameter that controls the smooth approximation of the data distribution (cf. Section 3.1.1). made to accelerate this process by resorting to higher-order or randomized numerical schemes [69– 79], augmented dynamics [80], adaptive step sizes [81], operator learning [82], restart sampling [83], self-consistency [84–87] and knowledge distillation [88–90]. Recently, several empirical works [1, 91–94] leverage the Picard iteration and triangular Anderson acceleration to parallelize the sampling procedure of diffusion models and achieve empirical success in large-scale image generation tasks. Some other recent work [95, 96] also combine the parallel sampling technique with the randomized midpoint method [97] to accelerate the inference of diffusion models. This efﬁciency issue is closely related to the problem of bounding the required number of steps and evaluations of score functions to approximate an arbitrary data distribution on Rd to δ-accuracy, which has been analyzed extensively in the literature [98–115]. In terms of the dependency on the dimension d, the current state-of-the-art result for the SDE implementation of diffusion models is e O(d) [107], improved from the previous e O(d2) bound [104]. [111] gives a e O( √ d) bound for the probability ﬂow ODE implementation by considering a predictor-corrector scheme with the underdamped Langevin Monte Carlo (UMLC) algorithm. In this work, we aim to provide parallelization strategies, rigorous analysis, and theoretical guarantees for accelerating the inference process of diffusion models. The time complexity of previous implementations of diffusion models has been largely hindered by the discretization error, which requires the step size to scale with e O(1/d) for the SDE implementation and e O(1/ √ d) for the probability ﬂow ODE implementation. We show that the inference process can be ﬁrst divided into O(1) blocks with parallelizable evaluations of the score function within each, and thus reduce the overall time complexity to e O(poly log d). We provide the ﬁrst implementation of diffusion models with poly-logarithmic complexity, a signiﬁcant improvement over the current state-of-the-art polynomial results that sheds light on the potential fast and efﬁcient sampling of high-dimensional distributions with diffusion models on fast-developing memory-efﬁcient modern GPU clusters. 1.1 Contributions • We propose parallelized inference algorithms for diffusion models in both the SDE and probability ﬂow ODE implementations (PIADM-SDE/ODE) with exponential integrators, a shrinking step size scheme towards the data end, and the early stopping technique; • We provide a rigorous convergence analysis of PIADM-SDE, showing that our parallelization strategy yields a diffusion model with e O(poly log d) approximate time complexity; • We show that our strategy is also compatible with the probability ﬂow ODE implementation, and PIADM-ODE could improve the space complexity from e O(d2) to e O(d3/2) while maintaining the poly-logarithmic time complexity. 2 Preliminaries In this section, we brieﬂy recapitulate the framework of score-based diffusion models, deﬁne notations, and discuss related work. 2 2.1 Diffusion Models In score-based diffusion models, one considers a diffusion process (xs)s≥0 in Rd governed by the following stochastic differential equation (SDE): dxs = βs(xs)ds + σsdws, with x0 ∼p0, (2.1) where (ws)s≥0 is a standard Brownian motion, and p0 is the target distribution that we would like to sample from. The distribution of xs is denoted by ps. Once the drift βs(·), the diffusion coefﬁcient σs, and a sufﬁciently large time horizon T are speciﬁed, (2.1) also corresponds to a backward process ( ⃗ xt)0≤t≤T for another arbitrary diffusion coefﬁcient (υs)s≥0 [116]: d ⃗ xt = " − ⃗ βt( ⃗ xt) + ⃗ σt ⃗ σ⊤ t + ⃗ υt ⃗ υ⊤ t 2 ∇log ⃗ pt( ⃗ xt) # dt + ⃗ υtdwt, (2.2) where ⃗ ∗t denotes ∗T −t, with ⃗ p0 = pT and ⃗ pT = p0. For notational simplicity, we adopt a simple choice of the drift and the diffusion coefﬁcients in what follows: βt(x) = −1 2x, σt = Id, and υ = υId, under which (2.1) is an Ornstein-Uhlenbeck (OU) process converging exponentially to its stationary distribution, i.e. pT ≈bpT := N(0, Id), and (2.1) and (2.2) reduce to the following form: dxs = −1 2xsds + dws, and d ⃗ xt = 1 2 ⃗ xt + 1 + υ2 2 ∇log ⃗ pt( ⃗ xt) dt + υdwt. (2.3) In practice, the score function ∇ ⃗ pt( ⃗ xt) is often estimated by a neural network (NN) sθ t (xt), where θ represents its parameters, by minimizing the denoising score-matching loss [117, 118]: L(θ) := Ext∼pt h ∇log pt(xt) −sθ t (xt) 2i = Ex0∼p0 " Ext∼pt\|0(xt\|x0) " xt −x0e−t/2 1 −e−t −sθ t (xt) 2## , (2.4) and the backward process in (2.3) is approximated by the following SDE thereafter: dyt = 1 2yt + 1 + υ2 2 sθ t (yt) dt + υdwt, with y0 ∼N(0, Id). (2.5) Implementations. Diffusion models admit multiple implementations depending on the choice of the parameter υ in the backward process (2.2). The SDE implementation with υ = 1 is widely used in the literature for its simplicity and efﬁciency [10], while recent studies [111] claim that the probability ﬂow ODE implementation with υ = 0 may exhibit better time complexity. We refer to [111, 119] for theoretical and [120, 121] for empirical comparisons of different implementations. 2.2 Parallel Sampling Parallel sampling algorithms have been actively explored in the literature, including the parallel tempering method [122–124] and several recent studies [125–127]. For diffusion models, the idea of parallel sampling is based on the Picard iteration [128, 129] for solving nonlinear ODEs. Suppose we have an ODE dxt = ft(xt)dt and we would like to solve it for t ∈[0, T], then the Picard iteration is deﬁned as follows: x(0) t ≡x0, and x(k+1) t := x0 + Z t 0 fs(x(k) s )ds, for k ∈[0 : K −1]. (2.6) Under assumptions on the Lipschitz continuity of ft, the Picard iteration converges to the true solution exponentially fast, in the sense that ∥∥x(k) t −xt∥∥L∞([0,T ]) ≤δ with K = O(log δ−1) iterations. Unlike high-order ODE solvers, the Picard iteration is intrinsically parallelizable: for any t ∈[0, T], the computation of x(k+1) t relies merely on the values of the most recent iteration x(k) t . With sufﬁcient computational sources parallelizing the evaluations of f, the computational cost of solving the ODE no longer scales with T but with the number of iterations K. 3 Recently, this idea has been applied to both the Langevin Monte Carlo (LMC) and the underdamped Langevin Monte Carlo (UMLC) contexts [130]. Roughly speaking, it is proposed to simulate the Langevin diffusion process dxt = −∇V (xt)dt+dwt with the following iteration resembling (2.6): x(0) t ≡x0, and x(k+1) t := x0 − Z t 0 ∇V (x(k) t )ds + wt, for k ∈[0 : K −1], (2.7) where all iterations share a common Wiener process (wt)t≥0. It is shown that for well-conditioned log-concave distributions, parallelized LMC would achieve an iteration depth of K = e O(poly log d) that matches the indispensable time horizon T = e O(poly log d) to achieve exponential ergodicity (cf. [130, Theorem 13]). This promises a significant speedup in sampling high-dimensional distributions from the standard LMC of T = e O(d) iterations, hindered by the o(1/d) step size as imposed by the discretization error and now evaded by the parallelization. 2.3 Approximate Time Complexity A similar situation is expected in diffusion models, where the application bottleneck is largely the inference process with sequential iterations and expensive evaluations of the learned score function sθ t (·), which is often parametrized by large-scale NNs. Despite several unavoidable costs involving pre- and post-processing, data storage and retrieval, and arithmetic operations, we deﬁne the following notion of the approximate time complexity of the inference process of diffusion models: Deﬁnition 2.1 (Approximate time complexity). For a speciﬁc implementation of diffusion models (2.5), we deﬁne the approximate time complexity of the sampling process as the number of unparallelizable evaluations of the learned NN-based score function sθ t (·). This deﬁnition coincides with the notion of the number of steps required to reach a certain accuracy in [104, 100], iteration complexity in [107, 111], etc. in the previous theoretical studies of diffusion models. We have adopted this notion in Table 1 for a comparison of the current state-of-the-art results and our bounds in this work. We will use the notion of space complexity likewise to denote the approximate required storage during the inference. Trivially, the space complexity of the sequential implementation is O(d). Should no confusion occur, we omit the dependency of the complexities above on the accuracy threshold δ, etc., during our discussion, as we focus on applications of diffusion models to high-dimensional data distributions, following the standard practice in the literature. 3 Main Results Inspired by the acceleration achieved by the parallel sampling technique in LMC and ULMC, we aim to accommodate parallel sampling into the theoretical analysis framework of diffusion models. The beneﬁt of the parallel sampling technique in this scenario has been recently conﬁrmed by up to 14× acceleration achieved by the ParaDiGMS algorithm [1] and ParaTAA [92], where several practical compromises are made to mitigate GPU memory constraints and theoretical guarantees are still lacking. In this section, we will propose Parallelized Inference Algorithms for Diffusion Models with both the SDE and probability ﬂow ODE implementations, namely the PIADM-SDE (Algorithm 1) and PIADM-ODE (Algorithm 2), and present theoretical guarantees of our algorithms, including the approximate time complexity and space complexity, for both implementations in Section 3.1 and Section 3.2, respectively. Due to the large number of notations used in the presentation, we give an overview of notations in Appendix A.1 for readers’ convenience. 3.1 SDE Implementation We ﬁrst focus on the approximation, parallelization strategies, and error analysis of diffusion models with the SDE implementation, i.e. the forward and backward process (2.3) and its approximatation (2.5) with υ = 1. We will show that PIADM-SDE achieves an e O(poly log d) approximate time complexity with e O(d2) space complexity. 4 Outer Iterations n: N = O(log d) blocks bq0 ≈N(0, Id) bqtN ≈pdata h0 h1 hn−1 hN−1 η O(1) k = 0 k = 1 k = K e O(d−1) or e O(d−1/2) ... ... Mn = e O(d) or e O( √ d) parallalizable steps bqtn bqtn+1 Inner Iterations k: K = e O(log d) depth ϵn,0 ϵn,1 ϵn,M−1 Figure 1: Illustration of PIADM-SDE/ODE. The outer iterations are divided into O(log d) blocks of O(1) length. Within each block, the inner iterations are parallelized with e O(d) steps for SDE (cf. Theorem 3.3), or e O( √ d) for probability ﬂow ODE implementation (cf. Theorem 3.5). The overall approximate time complexity is KN = e O(poly log d). brown, green, blue, and red curves represent the computation graph at t = tn + τn,m for m = 1, 2, Mn −1, Mn. 3.1.1 Algorithm PIADM-SDE is summarized in Algorithm 1 and illustrated in Figure 1. The main idea behind our algorithm is the fact that (2.5) can be efﬁciently solved by the Picard iteration within a period of O(1) length, transferring e O(d) sequential computations to a parallelizable iteration of depth e O(log d). In the following, we introduce the numerical discretization scheme of our algorithm and the implementation of the Picard iteration in detail. Step Size Scheme. In our algorithm, the time horizon T is ﬁrst segmented into N blocks of length (hn)N−1 n=0 , with each hn ≤h := T/N = Ω(1), forming a grid (tn)N n=0 with tn = Pn j=1 hj. For any n ∈[0 : N −1], the n-th block is further discretized into a grid (τn,m)Mn m=0 with τn,0 = 0 and τn,Mn = hn. We denote the step size of the m-th step in the n-th block as ϵn,m = τn,m+1 −τn,m, and the total number of steps in the n-th block as Mn. For the ﬁrst N −1 blocks, we simply use the unique discretization, i.e. hn = h, ϵn,m = ϵ, and Mn = M := h/ϵ, for n ∈[0 : N −2] and m ∈[0 : M −1]. Following [104, 107], to curb the potential blow-up of the score function as t →T, which is shown by [107] for 0 ≤s < t < T to be of the order E Z t s ∥∇log ⃗ pτ( ⃗ xτ) −∇log ⃗ ps( ⃗ xs)∥2dτ ≲d t −s T −t 2 , we apply early stopping at time tN = T −η, where η is chosen in a way such that the O(√η) 2-Wasserstein distance between ⃗ pT and its smoothed version ⃗ pT −η that we aim to sample from alternatively, is tolerable for the downstream tasks. We also impose the exponential decay of the step size towards the data end in the last block. To be speciﬁc, we let hN−1 = h −δ, and discretize the interval [tN−1, tN] = [(N −1)h, T −η] into a grid (τN−1,m)MN−1 m=0 with step sizes (ϵN−1,m)MN−1 m=0 satisfying ϵN−1,m ≤ϵ ∧ϵ (h −τN−1,m+1) . (3.1) As shown in Lemma B.7, this exponential decaying step size scheme towards the data end is crucial to bound the discretization error in the last block. For the simplicity of notations, we introduce the following indexing function: for τ ∈[tn, tn+1], we deﬁne In(τ) to be the unique integer such that PIn(τ) j=1 ϵn,j ≤τ < PIn(τ)+1 j=1 ϵn,j. We also deﬁne 5 Algorithm 1: PIADM-SDE Input: by0 ∼bq0 = N(0, Id), a discretization scheme (T, (hn)N n=1 and (τn,m)n∈[1:N],m∈[0:M]) satisfying (3.1), the depth of iteration K, the learned NN-based score function sθ t (·). Output: A sample bytN ∼bqtN ≈ ⃗ pT . 1 for n = 0 to N −1 do 2 by(0) tn,τn,m ←bytn, ξm ∼N(0, Id) for m ∈[0 : Mn] in parallel; 3 for k = 0 to K −1 do 4 by(k) tn,0 ←bytn; 5 for m = 0 to Mn in parallel do 6 by(k+1) tn,τn,m ←e τn,m 2 by(k) tn,0 + Pm−1 j=0 e τn,m−τn,j+1 2 h 2 (eϵn,j −1) sθ tn+τn,j(by(k) tn,τn,j) + √eϵn,j −1ξj i ; (3.4) 7 end 8 end 9 bytn+1 ←by(K) tn,τn,Mn ; 10 end a piecewise function g such that gn(τ) = PIn(τ) j=1 ϵn,j. It is easy to check that under the uniform discretization for n ∈[1 : N −1], we have In(τ) = ⌊τ/ϵ⌋and gn(τ) = ⌊τ/ϵ⌋ϵ. Exponential Integrator. For each step τ ∈[tn + τn,m, tn + τn,m+1], we use the following exponential integrator scheme [77], as the numerical discretization of the SDE (2.5): bytn,τn,m+1 = eϵn,m/2 bytn,τn,m + 2 eϵn,m/2 −1 sθ tn+τn,m(bytn+τn,m) + √ eϵn,m −1ξ, where ξ ∼N(0, Id). Lemma B.3 shows its equivalence to approximating (2.5) as dbytn,τ = 1 2 bytn,τ + sθ tn+τn,m(bytn,τn,m) dτ + dwtn+τ, for τ ∈[τn,m, τn,m+1]. (3.2) Remark 3.1. One could also implement a straightforward Euler-Maruyama scheme instead of the exponential integrator (3.4), where an additional high-order discretization error term would emerge [104, Theorem 1], which we believe would not affect the overall e O(poly log d) time complexity with parallel sampling. Picard Iteration. Within each block, we apply Picard iteration of depth K. As shown by Lemma B.3, the discretized scheme (3.4) implements the following iteration for k ∈[0 : K −1]: dby(k+1) tn,τ = " 1 2 by(k+1) tn,τ + sθ tn+gn(τ) by(k) tn,gn(τ) # dτ + dwtn+τ, for τ ∈[0, hn]. (3.3) We denote the distribution of by(K) tn,τ by bqtn+τ. As proved in Lemma B.6, the iteration above would converge to (3.2) in each block exponentially fast, which given a sufﬁciently accurate learned score estimation sθ t should be close to the true backward SDE (2.3). One should also notice that the Gaussians ξm are only sampled once and used for all iterations. The parallelization for (3.4) in Algorithm 1 should be understood as that for any k ∈[0 : K −1], each sθ tn+τn,j(by(k) tn,τn,j) for j ∈[0 : Mn] is evaluated in parallel, with subsequent ﬂoating-point operations comparably negligible, resulting in the overall O(NK) approximate time complexity. 3.1.2 Assumptions Our theoretical analysis will be built on the following mild assumptions on the regularity of the data distribution and the numerical properties of the neural networks: 6 Assumption 3.1 (L2([0, tN]) δ-accurate learned score). The learned NN-based score sθ t is δ2accurate in the sense of E ⃗ p "N−1 X n=0 Mn−1 X m=0 ϵn,m sθ tn+τn,m ⃗ xtn+τn,m −∇log ⃗ ptn+τn,m ⃗ xtn+τn,m 2 # ≤δ2 2. (3.5) Assumption 3.2 (Regular and normalized data distribution). The data distribution p0 has ﬁnite second moments and is normalized such that covp0(x0) = Id. Assumption 3.3 (Bounded and Lipschitz learned NN-based score). The learned NN-based score function sθ t has bounded C1 norm, i.e. ∥∥sθ t (·)∥∥L∞([0,T ]) ≤Ms with Lipschitz constant Ls. Remark 3.2. Assumption 3.1 and the ﬁnite moment assumption in Assumption 3.2 are standard assumptions across previous theoretical works on diffusion models [100, 104, 111], while we adopt the normalization Assumption 3.2 from [107] to simplify true score function-related computations (cf. Lemma A.8). Assumption 3.3 can be easily satisﬁed by truncation, ensuring computational stability. Notice that the exponential integrator, one actually applies Picard iteration to e−t/2sθ t , a relaxation of Assumption 3.1 might be possible, which is left for future work. 3.1.3 Theoretical Guarantees The following theorem summarizes our theoretical analysis for PIADM-SDE (Algorithm 1): Theorem 3.3 (Theoretical Guarantees for PIADM-SDE). Under Assumptions 3.1, 3.2, and 3.3, given the following choices of the order of the parameters T = O(log(dδ−2)), h = Θ(1), N = O log(dδ−2) , ϵ = Θ d−1δ2 log−1(dδ−2) , M = O dδ−2 log(dδ−2) , K = e O(log(dδ−2)), and let L2 shne 7 2 hn ≪1, δ2 ≲δ, T ≲log η−1, the distribution bqtN that PIADM-SDE (Algorithm 1) generates samples from satisﬁes the following error bound: DKL(pη∥bqtN ) ≲de−T + dϵT + δ2 2 + dTe−K ≲δ2, with a total of KN = e O log2(dδ−2) approximate time complexity and dM = e O d2δ−2 space complexity for parallalizable δ-accurate score function computations. Remark 3.4. We would like to make the following remarks on the result above: • The acceleration from e O(d) to e O(poly log d) is at the cost of a trade-off with extra memory cost of M = e O(d) for computing and updating {sθ tn+τn,j(by(k) tn,τn,m)}m∈[0:Mn] simultaneously during each Picard iteration; • Compared with log-concave sampling [130], M being of order e O(d) instead of e O( √ d) therein is partly due to the time independence of the score function ∇log p(·) in general sampling tasks. Besides, the scaling M = e O(d) agrees with the current state-of-the-art dependency [107] for the SDE implementation of diffusion models; • As mentioned above, the scale of the step size ϵ within one block is still conﬁned to Θ(1/M) = eΘ (1/d). The block length h, despite being required to be small compared to 1/Ls, is of order Θ(1), resulting in only Θ(log d) blocks and thus e O(poly log d) total iterations. 3.1.4 Proof Sketch The detailed proof of Theorem 3.3 is deferred to Section B. The pipeline of the proof is to (a) ﬁrst decompose the error DKL( ⃗ ptN ∥bqtN ) into blockwise errors using the chain rule of KL divergence; (b) bound the error in each block by invoking Girsanov’s theorem; (c) sum up the errors in all blocks. The key technical challenge lies in Step (b). Different from all previous theoretical works [100, 104, 111], the Picard iteration in our algorithm generates K paths recursively in each block using the learned score sθ t . And therefore the ﬁnal path (by(K) tn,τ)τ∈[0,hn] depends on all previous paths (by(k) tn,τ)τ∈[0,hn] for k ∈[0 : K −1], ruling out a direct change of measure argument from the 7 naïve application of Girsanov’s theorem. To this end, we need a more sophisticated mathematical framework of stochastic processes, as given in Appendix A.2. We deﬁne the measurable space (Ω, F) with ﬁltrations (Ft)t≥0 to specify the probability measures on (Ω, F) of each Wiener process, and resort to one of the most general forms of Girsanov’s theorem ( [131, Theorem 8.6.6]). For example, in the n-th block, we apply the following change of measure procedure: 1. Let q\|Ftn be the measure where wt(ω) is the shared Wiener process in the Picard iteration (3.3) for any k ∈[0 : K −1]; 2. Deﬁne another process d e wtn+τ(ω) = dwtn+τ(ω) + δtn(τ, ω)dτ, where δtn(τ, ω) := sθ tn+gn(τ)(by(K−1) tn,gn(τ)(ω)) −∇log ⃗ ptn+τ(by(K) tn+τ(ω)); 3. Invoke Girsanov’s theorem, which yields that the Radon-Nikodym derivative of the measure ⃗ p\|Ftn with respect to q\|Ftn satisﬁes log d ⃗ p\|Ftn dq\|Ftn (ω) = − Z hn 0 δtn(τ, ω)⊤dwtn+τ(ω) −1 2 Z hn 0 ∥δtn(τ, ω)∥2dτ; 4. Conclude that ( e wtn+τ)τ≥0 is a Wiener process under the measure ⃗ p\|Ftn and thus (3.3) at iteration K satisﬁes the following SDE: dby(K) tn,τ(ω) = 1 2 by(K) tn,τ(ω) + ∇log ⃗ ptn+τ by(K) tn,τ(ω) dτ + d e wtn+τ(ω), i.e. the true backward SDE (2.3) with the true score function for τ ∈[tn, tn+1]. One should notice that this change of measure argument will cause an additional term in the bound of the discrepancy between the ﬁrst iteration by(1) tn,τ and the initial condition by(0) tn,τ in Lemma B.5. However, due to the exponential convergence of the Picard iteration, this term does not affect the overall error bound. 3.2 Probability Flow ODE Implementation In this section, we will show that our parallelization strategy is also compatible with the probability ODE implementation of diffusion models, i.e. the forward and backward process (2.3) and its approximatation (2.5) with υ = 0. We will demonstrate that PIADM-ODE (Algorithm 2) further improves the space complexity from e O(d2) to e O(d3/2) while maintaining the same e O(poly log d) approximate time complexity. 3.2.1 Algorithm Due to the space limit, we refer the readers to Section C.1 and Algorithm 2 for the details of our parallelization of the probability ﬂow ODE formulation of diffusion models. PIADM-ODE keeps the discretization scheme detailed in Section 3.1.1 that divides the time horizon T into N blocks and uses exponential integrators for all updating rules. Notably, PIADM-ODE has the following distinctions compared with PIADM-SDE (Algorithm 1): • Instead of applying Picard iteration to the backward SDE as in (3.2), we apply Picard iteration to the probability ﬂow ODE as in (C.3) within each block, which does not require sampling i.i.d. Gaussians to simulate a Wiener process; • The most signiﬁcant difference is the adoption of an additional corrector step [111] after running the probability ﬂow ODE with Picard iteration within one block. During the corrector step, one augments the state space with a Gaussian that represents the initial momentum and then simulates an underdamped Langevin dynamics for O(1) time with the learned NN-based score function at the time of the block end; • We then further parallelize the underdamped Langevin dynamics in the corrector step so that it can also be accomplished with O(log d) approximate time complexity, as a naïve implementation would result in e O( √ d) [130], which is incompatible with our desired poly-logarithmic guarantee. 8 3.2.2 Assumptions Due to technicalities speciﬁc to this implementation, we need ﬁrst to modify Assumption 3.1 and add assumption on the Lipschitzness of the true score functions ∇log pt, which is a common practice in related literature [104, 111]. Recent work on the probability ﬂow ODE implementation [112, 114] also adopts stronger assumptions compared to the SDE implementation. Assumption 3.1’ (L∞([0, tN]) δ-accurate learned score). For any n ∈[0 : N −1] and m ∈[0 : Mn −1], the learned NN-based score sθ tn,τn,m is δ∞-accurate in the sense of E ⃗ ptn+τn,m sθ tn+τn,m ⃗ xtn+τn,m −∇log ⃗ ptn+τn,m ⃗ xtn+τn,m 2 ≤δ2 ∞. Assumption 3.4 (Bounded and Lipschitz true score). The true score function ∇log pt has bounded C1 norm, i.e. ∥∥∇log pt(·)∥∥L∞([0,T ]) ≤Mp with Lipschitz constant Lp. Further relaxations on Assumption 3.4 to time-dependent assumptions accommodating the blow-up to the data end (e.g. [103, Assumption 1.5]) are left for further work. 3.2.3 Theoretical Guarantees Our results for PIADM-ODE are summarized in the following theorem: Theorem 3.5 (Theoretical Guarantees for PIADM-ODE). Under Assumptions 3.1’, 3.2, 3.3, and 3.4, given the following choices of the order of the parameters T = O(log(dδ−2)), h = Θ(1), N = O(log(dδ−2)), ϵ = Θ d−1/2δ log−1(d−1/2δ−1) , M = O(d1/2δ−1 log(d1/2δ−1)), K = e O(log(dδ−2)), for the outer iteration and T † = O(1) ≲L−1/2 p ∧L−1/2 s , h† = Θ(1), N † = O(1), ϵ† = Θ(d−1/2δ), M † = O(d1/2δ−1), K† = O(log(dδ−2)), for the inner iteration during the corrector step, and let L2 sh2eh ∨L2 sh†2eh†/γ ≪1, δ∞≲ δ log−1(dδ−2), and γ ≳L1/2 p , then the distribution bqtN that PIADM-ODE (Algorithm 2) generates samples from satisﬁes the following error bound: TV(pη, bqtN )2 ≲de−T + dϵ2T 2 + (T 2 + N 2)δ2 ∞+ dN 2e−K ≲δ2, with a total of (K + K†N †)N = e O log2(dδ−2) approximate time complexity and d(M ∨M †) = eΘ d3/2δ−1 space complexity for parallalizable δ-accurate score function computations. The reduction of space complexity by the probability ﬂow ODE implementation is intuitively owing to the fact that the probability ﬂow ODE process is a deterministic process in time rather than a stochastic process as in the SDE implementation, getting rid of the O(ϵ) term derived by Itô’s symmetry. This allows the discretization error to be bounded with O(ϵ2) instead (cf. Lemma B.7 and C.5). 3.2.4 Proof Sketch The details of the proof of Theorem 3.5 are provided in Section C. Along with the complexity beneﬁts the deterministic nature of the probability ﬂow ODE may bring, the analysis is technically more involved than that of Theorem 3.3 and requires an intricate interplay between statistical distances. Several major challenges and our corresponding solutions are summarized below: • The error of the parallelized algorithm within each block may now only be bounded by 2Wasserstein distance (cf. Theorem C.7) instead of any f-divergence that admits data processing inequality as in the SDE case by Girsanov’s theorem. The additional corrector step exactly handles this issue and would intuitively translate 2-Wasserstein proximity to TV distance proximity (cf. Lemma C.18), allowing the decomposition of the overall error into each block; 9 • For the corrector step, the underdamped Langevin dynamics as a second-order dynamics requires only O( √ d) steps to converge, instead of O(d) steps in its overdamped counterpart. We then adapt the parallelization technique mentioned in Section 2.2 to conclude that it can be accomplished with O(log d) approximate time complexity (cf. Theorem C.17). The error caused by the approximation to the true score and numerical discretization within this step is bounded in KL divergence by invoking Girsanov’s theorem(Theorem A.4) as in the proof of Theorem 3.3; • Different from the SDE case, where the chain rule of KL divergence can easily decouple the initial distribution and the subsequent dynamics, we need several interpolating processes between the implementation and the true backward process in this case. The ﬁnal guarantee is in TV distance as it connects with the KL divergence via Pinsker’s inequality and admits data processing inequality. We refer the readers to Figure 2 for an overview of the proof pipeline, as well as the notations and intuitions of the auxiliary and interpolating processes appearing in the proof. 4 Discussion and Conclusion In this work, we have proposed novel parallelization strategies for the inference of diffusion models in both the SDE and probability ﬂow ODE implementations. Our algorithms, namely PIADMSDE and PIADM-ODE, are meticulously designed and rigorously proved to achieve e O(poly log d) approximate time complexity and e O(d2) and e O(d3/2) space complexity, respectively, marking the ﬁrst inference algorithm of diffusion and probability ﬂow based models with sub-linear approximate time complexity. Our algorithm intuitively divides the time horizon into several O(1) blocks and applies Picard iteration within each block in parallel, transferring the time complexity into space complexity. Our analysis is built on a sophisticated mathematical framework of stochastic processes and provides deeper insights into the mathematical theory of diffusion models. Our ﬁndings echo and corroborate the recent empirical work [1, 91–94] that parallel sampling techniques signiﬁcantly accelerate the inference process of diffusion models. Theoretical exploration of the adaptive block window scheme therein presents an interesting future research potential. Possible future work also includes the investigation of how to apply our parallelization framework to other variants of diffusion models, such as the discrete [23, 132–142] and multi-marginal [143] formulations. Although we anticipate implementing diffusion models in parallel may introduce engineering challenges, e.g. scalability, hardware compatibility, memory bandwidth, etc., we believe that our theoretical contributions lay a solid foundation that not only supports but also motivates the empirical development of parallel inference algorithms for diffusion models since advancements continue in GPU power and memory efﬁciency. Acknowledgments and Disclosure of Funding LY acknowledges the support of the National Science Foundation under Grant No. DMS-2011699 and DMS-2208163. GMR is supported by a Google Research Scholar Award. References [1] Andy Shih, Suneel Belkhale, Stefano Ermon, Dorsa Sadigh, and Nima Anari. Parallel sampling of diffusion models. 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A.1 Notations We adopt the following notations throughout the paper: Notation Description [a : b] The set {a, a + 1, . . . , b} Id Identity matrix in Rd×d ⃗ ∗t ∗T −t b∗ Used to denote quantities produced by the algorithm e∗ Used to denote quantities related to the auxiliary processes ∗† Used to denote quantities related to the corrector step ∥· ∥ The Euclidean norm of a vector ≲or ≳ The inequality holds up to a constant factor ≪ Absolute continuity (for measures)/ much less than (for quantities) (xt)t≥0 The forward process of the diffusion model (2.3) ( ⃗ xt)t∈[0,T ] The backward process of the diffusion model (2.3) (yt)t∈[0,T ] The approximate backward process of the diffusion model (2.5) by(k) tn,τn,m The approximate value of the approximate process yt at time tn+τn,m after k iterations in the (n + 1)-th block bytn The value of the approximate process yt at time tn bqtn The distribution of bytn zt1:t2 The path (zt)t∈[t1,t2] of the process zt Df(· ∥·) The f-divergence between two distributions DKL(· ∥·) The KL divergence between two distributions TV (· , ·) The total variation distance between two distributions W2(·, ·) The 2-Wasserstein distance between two distributions Table 2: Summary of notations A.2 Preliminaries Theorem A.1 (Properties of f-divergence). Suppose p and q are two probability measures on a common measurable space (Ω, F) with p ≪q. The f-divergence between p and q is deﬁned as Df(p∥q) = EX∼q f dp dq , (A.1) where dp dq is the Radon-Nikodym derivative of p with respect to q, and f : R+ →R is a convex function. In particular, Df(· ∥·) coincides with the KullbackLeibler (KL) divergence when f(x) = x log x and Df(· ∥·) = TV coincides with the total variation (TV) distance when f(x) = 1 2\|x −1\|. For the f-divergence deﬁned above, we have the following properties: 1. (Data-processing inequality). Suppose H is a sub-σ-algebra of F, the following inequality holds Df (p\|H ∥q\|H) ≤Df(p ∥q); for any f-divergence Df(·∥·). 2. (Chain rule). Suppose X is a random variable generating a sub-σ-algebra FX of F, and p(·\|X) ≪q(·\|X) holds for any value of X, then DKL(p∥q) = DKL(p\|FX∥q\|FX) + EFX[DKL(p(·\|X)∥q(·\|X))]. 19 In this paper, we consider a probability space (Ω, F, p) on which (wt(ω))t≥0 is a Wiener process in Rd. The Wiener process (wt(ω))t≥0 generates the ﬁltration {Ft}t≥0 on the measurable space (Ω, F). For an Itô process zt(ω) with the following governing SDE: dzt(ω) = α(t, ω)dt + Σ(t, ω)dwt(ω), for any time t, we denote the marginal distribution of zt by pt, i.e. pt := p z−1 t (·) , where zt : Ω→Rm, ω 7→zt(ω), as well as the path measure of the process zt in the sense of pt1:t2 := p z−1 t1:t2(·) , where zt1:t2 : Ω→C([t1, t2], Rm), ω 7→(zt(ω))t∈[t1,t2]. For the sake of simplicity, we deﬁne the following class of functions: Deﬁnition A.2. For any 0 ≤t1 < t2, we deﬁne V(t1, t2) as the class of functions f(t, ω) : [0, +∞) × Ω→R such that 1. f(t, ω) is B × F-measurable, where B is the Borel σ-algebra on Rd; 2. f(t, ω) is Ft-adapted for all t ≥0; 3. The following Novikov condition holds E exp Z t2 t1 f 2(t, ω)dt < +∞, and V = ∩t>0V(0, t). For vectors and matrices, we say it belongs to Vn(t, ω) or Vm×n(t, ω) if each component of the vector or each entry of the matrix belongs to V(t, ω). Remark A.3. Novikov’s condition appeared in the third requirement is often relaxed to the squared integrability condition in the general deﬁnition of Itô processes, which requires E Z t2 t1 f 2(t, ω)dt < +∞. Here, we adopt the more restricted condition in the spirit of its necessity for Girsanov’s theorem to hold, as we shall see later. Similar to previous work [111], here we can avoid checking Novikov’s condition throughout our proofs below by using the approximation argument presented in [100]. A review of Girsanov can be found in textbooks like in [131, 144]. We will present the following generalized version of Girsanov’s theorem: Theorem A.4 (Girsanov’s Theorem [131, Theorem 8.6.6]). Let α(t, ω) ∈Vm, Σ(t, ω) ∈Vm×n, and (wt(ω))t≥0 be a Wiener process on the probability space (Ω, F, q). For t ∈[0, T], suppose zt(ω) is an Itô process with the following SDE: dzt(ω) = α(t, ω)dt + Σ(t, ω)dwt(ω), (A.2) and there exist processes δ(t, ω) ∈Vn and β(t, ω) ∈Vm such that 1. Σ(t, ω)δ(t, ω) = α(t, ω) −β(t, ω); 2. The process Mt(ω) as deﬁned below is a martingale with respect to the ﬁltration {Ft}t≥0 and probability measure q: Mt(ω) = exp − Z t 0 δ(s, ω)⊤dws(ω) −1 2 Z t 0 ∥δ(s, ω)∥2ds , then there exists another probability measure p on (Ω, F) such that 1. p ≪q with the Radon-Nikodym derivative dp dq (ω) = MT (ω), 20 2. The process e wt(ω) as deﬁned below is a Wiener process on (Ω, F, p): e wt(ω) = wt(ω) + Z t 0 δ(s, ω)ds, 3. Any continuous path in C([t1, t2], Rm) generated by the process zt satisﬁes the following SDE under the probability measure p: dezt(ω) = β(t, ω)dt + Σ(t, ω)d e wt(ω). (A.3) Corollary A.5. Suppose the conditions in Theorem A.4 hold, then for any t1, t2 ∈[0, T] with t1 < t2, the path measure of the SDE (A.3) under the probability measure p in the sense of pt1:t2 = p z−1 t1:t2(·) is absolutely continuous with respect to the path measure of the SDE (A.2) in the sense of qt1:t2 = q z−1 t1:t2(·) . Moreover, the KL divergence between the two path measures is given by DKL(pt1:t2∥qt1:t2) = DKL(pt1∥qt1) + Eω∼p\|Ft1 1 2 Z t2 t1 ∥δ(t, ω)∥2dt (A.4) Proof. First, by Theorem A.1, we have DKL(pt1:t2∥qt1:t2) = DKL(p\|Ft1 ∥q\|Ft1 )+Ez∼p\|Ft1 DKL p(ez−1 t1:t2(·))\|ezt1 = ez)∥q(ez−1 t1:t2(·))\|ezt1 = ez) . From Girsanov’s theorem (Theorem A.4), we have that the measure p\|Ft1 is absolutely continuous with respect to q\|Ft1 , which allows us to compute the second term above as follows: DKL p(ez−1 t1:t2(·)\|ezt1 = ez)∥q(ez−1 t1:t2(·)\|ezt1 = ez) = Eezt1:t2 " log dp(ez−1 t1:t2(·)\|ezt1 = z) dq(ez−1 t1:t2(·)\|ezt1 = z) # = Eω∼p\|Ft1 " log dp\|Ft1 dq\|Ft1 # = Eω∼p\|Ft1 − Z t2 t1 δ(t, ω)⊤dwt(ω) −1 2 Z t2 t1 ∥δ(t, ω)∥2dt = Eω∼p\|Ft1 − Z t2 t1 δ(t, ω)⊤(d e wt(ω) −δ(t, ω)dt) −1 2 Z t2 t1 ∥δ(t, ω)∥2dt = Eω∼p\|Ft1 1 2 Z t2 t1 ∥δ(t, ω)∥2dt , and therefore DKL(pt1:t2∥qt1:t2) = DKL(pt1∥qt1) + Eω∼p\|Ft1 1 2 Z t2 t1 ∥δ(t, ω)∥2dt , which completes the proof. A.3 Helper Lemmas Lemma A.6 ([107, Lemma 2]). For the backward process (2.3), we have for 0 ≤s < t < T, d dt E ∥∇log ⃗ pt( ⃗ xt) −∇log ⃗ ps( ⃗ xs)∥2 ≤1 2E ∥∇log ⃗ ps( ⃗ xs)∥2 + E ∥∇2 log ⃗ pt( ⃗ xt)∥2 F . Lemma A.7 ([107, Lemma 3]). For the forward process (2.3), we have for 0 ≤t < T, E [∇log pt(xt)] ≤dσ−2 t , and E ∥∇2 log pt(xt)∥2 F ≤dσ−4 t + 2 d dt σ−4 t E [tr Σt] , where the posterior covariance matrix Σt := covp0\|t(x0) and σ2 t = 1−e−t. Moreover, the posterior covariance matrix Σt satisﬁes E [tr Σt] ≲d ∧dσ2 t . 21 Lemma A.8. For any n ∈[0 : N −1] and τ ∈[0, hn], under the assumption covp0(x0) = Id, we have E ∥ ⃗ xtn∥2 ≤2d, (A.5) and E h ∥ ⃗ xtn − ⃗ xtn+τ∥2i ≤3d. (A.6) Proof. Conditioned on x0, we have that ⃗ xtn = xT −tn ∼N e−1 2 (T −tn)x0, (1 −e−(T −tn))Id , and ⃗ xtn+τ = xT −tn−τ ∼N e−1 2 (T −tn−τ)x0, (1 −e−(T −tn−τ))Id for any τ ∈[0, hn]. Therefore, we have E ∥ ⃗ xtn∥2 = E E ∥xT −tn∥2x0 ≤E h E h ∥xT −tn −e−1 2 (T −tn)x0∥2x0 i + ∥e−1 2 (T −tn)x0∥2i ≤d(1 −e−(T −tn)) + e−(T −tn)E[∥x0∥2] ≤2d. Taking the difference between them then implies that for any τ ∈[0, hn], E h ∥ ⃗ xtn − ⃗ xtn+τ∥2i = E h E h ∥xT −tn −xT −tn−τ∥2 \|x0 ii ≤d(2 −e−(T −tn) −e−(T −tn−τ)) + e−1 2 (T −tn) −e−1 2 (T −tn−τ)2 E[∥x0∥2] ≤2d + e−(T −tn−τ)(1 −e−1 2 τ)2E[∥x0∥2] ≤3d. Lemma A.9 (Lemma 9 in [104]). For bq0 ∼N(0, Id) and ⃗ p0 = pT is the distribution of the solution to the forward process (2.3), we have TV( ⃗ p0, bq0)2 ≤DKL( ⃗ p0∥bq0) ≲de−T . B Details of SDE Implementation In this section, we will present the missing proofs for Theorem 3.3. For readers’ convenience, we reiterate the backward process (2.3) d ⃗ xt = 1 2 ⃗ xt + ∇log ⃗ pt( ⃗ xt) dt + dwt, with ⃗ x0 ∼pT , (B.1) and its approximate version (2.5) with the learned score function dyt = 1 2yt + sθ t (yt) dt + dwt, with y0 ∼N(0, Id). The ﬁltration Ft refers to the ﬁltration of the SDE (B.1) up to time t. 22 B.1 Auxiliary Process We would like ﬁrst to consider the errors that Algorithm 1 may cause within one block of update. To this end, we consider the following auxiliary process for τ ∈[0, hn] conditioned on the ﬁltration Ftn at time tn: Deﬁnition B.1 (Auxiliary Process). For any n ∈[0 : N −1], we deﬁne the auxiliary process (by(k) tn,τ)τ∈[0,hn] as the solution to the following SDE recursively for k ∈[0 : K −1]: dby(k+1) tn,τ (ω) = " 1 2 by(k+1) tn,τ (ω) + sθ tn+gn(τ) by(k) tn,gn(τ)(ω) # dτ + dwtn+τ(ω), (B.2) with the initial condition by(0) tn,τ(ω) ≡bytn(ω) for τ ∈[0, hn], and by(k) tn,0(ω) ≡bytn(ω) for k ∈[1 : K] (B.3) where bytn(ω) = by(K) tn−1,τn−1,Mn−1 (ω) if n ∈[1 : N −1] and byt0(ω) ∼N(0, Id). The iteration should be perceived as a deterministic procedure to each event ω ∈Ω, i.e. each realization of the Wiener process (wt)t≥0. The following lemma clariﬁes this fact and proves the well-deﬁnedness and parallelability of the iteration in (B.2). Lemma B.2. The auxiliary process (by(k) tn,τ(ω))τ∈[0,hn] is Ftn+τ-adapted for any k ∈[0 : K] and n ∈[0 : N −1]. Proof. Since the initialization by(0) tn,τ(ω) ≡bytn(ω) for τ ∈[0, hn], where bytn(ω) is Ftn-adapted, it is obvious that by(0) tn,τ(ω) is Ftn+τ-adapted. Now suppose that (by(k) tn,τ(ω))τ∈[0,hn] is Ftn+τ-adapted, since gn(τ) ≤τ, we have the following Itô integral well-deﬁned and Ftn+τ-adapted: Z τ 0 sθ tn+gn(τ ′) by(k) tn,gn(τ ′)(ω) dτ ′, and therefore (B.2) has a unique strong solution (by(k+1) tn,τ (ω))τ∈[0,hn] that is also Ftn+τ-adapted. The lemma follows by induction. Lemma B.3 (Equivalence between (3.4) and (B.2)). For any n ∈[0 : N −1], the update rule (3.4) in Algorithm 1 is equivalent to the exact solution of the auxiliary process (B.2) for any k ∈[0 : K −1] and τ ∈[0, hn]. Proof. The dependency on ω will be omitted in the proof below. Rewriting (B.2) and multiplying e−τ 2 on both sides yield d h e−τ 2 by(k+1) tn,τ i = e−τ 2 dby(k+1) tn,τ −1 2 by(k+1) tn,τ dτ = e−τ 2 h sθ tn+gn(τ) by(k) tn,gn(τ) dτ + dwtn+τ i . Integrating on both sides from 0 to τ implies e−τ 2 by(k+1) tn,τ −by(k+1) tn,0 = Z τ 0 e−τ′ 2 sθ tn+gn(τ ′) by(k) tn,gn(τ ′) dτ ′ + dwtn+τ ′ ! = Mn X m=0 Z τ∧τn,m+1 τ∧tn,m e−τ′ 2 sθ tn+τn,m by(k) tn,τn,m dτ ′ + Z τ 0 e−τ′ 2 dwtn+τ ′ = Mn X m=0 2 e− τ∧τn,m 2 −e− τ∧τn,m+1 2 sθ tn+τn,j by(k) tn,τn,m + Z τ 0 e−τ′ 2 dwtn+τ ′, and then multiplying e τ 2 on both sides above yields by(k+1) tn,τ = e τ 2 by(k+1) tn,0 + Mn X m=0 2 e τ∧τn,m+1−τ∧τn,m 2 −1 e 0∨(τ−τn,m+1) 2 sθ tn+τn,m by(k) tn,τn,m + Mn X m=0 Z τ∧τn,m+1 τ∧τn,m e τ−τ′ 2 dwtn+τ ′, 23 where, by Itô isometry, we have Z τ∧τn,m+1 τ∧τn,m e τ−τ′ 2 dwtn+τ ′ ∼N 0, eτ∧τn,m+1−τ∧τn,m −1 e0∨(τ−τn,m+1)Id for τ > τn,m and equals to 0 otherwise. Plugging in τ = τj,m gives us (3.4), as desired. B.2 Errors within Block We shall invoke Girsanov’s theorem (Theorem A.4) in the procedure as detailed below: 1. Setting (A.2) in Theorem A.4 as the auxiliary process (B.2) at iteration K, where wt(ω) is a Wiener process under the measure q\|Ftn ; 2. Deﬁning another process e wtn+τ(ω) governed by the following SDE: d e wtn+τ(ω) = dwtn+τ(ω) + δtn(τ, ω)dτ, where δtn(τ, ω) := sθ tn+gn(τ)(by(K−1) tn,gn(τ)(ω)) −∇log ⃗ ptn+τ(by(K) tn+τ(ω)), (B.4) and computing the Radon-Nikodym derivative of the measure ⃗ p\|Ftn with respect to q\|Ftn as d ⃗ p\|Ftn dq\|Ftn (ω) := exp − Z hn 0 δtn(τ, ω)⊤dwtn+τ(ω) −1 2 Z hn 0 ∥δtn(τ, ω)∥2dτ ! , 3. Concluding that (B.2) at iteration K under the measure q\|Ftn satisﬁes the following SDE: dby(K) tn,τ(ω) = 1 2 by(K) tn,τ(ω) + ∇log ⃗ ptn+τ by(K) tn,τ(ω) dτ + d e wtn+τ(ω), with ( e wtn+τ)τ≥0 being a Wiener process under the measure ⃗ p\|Ftn . If we replace by(K) tn,gn(τ)(ω) by ⃗ xtn+τ(ω), one should notice (B.5) is immediately the original backward SDE (2.3) with the true score function on t ∈[tn, tn+1]: d ⃗ xtn+τ(ω) = 1 2 ⃗ xtn+τ(ω) + ∇log ⃗ ptn+τ( ⃗ xtn+τ(ω)) dτ + d e wtn+τ(ω). (B.5) Remark B.4. The applicability of Girsanov’s theorem here relies on the Fτ-adaptivity of sθ tn+gn(τ) by(K−1) tn,gn(τ)(ω) established by Lemma B.2. One should notice the change of measure procedure above depends on the number of iterations K, and different K would lead to different transform (B.4). Then Corollary A.5 provides the following computation DKL( ⃗ ptn+1∥bqtn+1) ≤DKL( ⃗ ptn:tn+1∥bqtn:tn+1) = DKL( ⃗ ptn∥bqtn) + Eω∼q\|Ftn " 1 2 Z hn 0 ∥δtn(τ, ω)∥2dτ # , (B.6) 24 where the ﬁrst inequality is by the data-processing inequality (Theorem A.1). Now, the problem remaining is to bound the discrepancy quantiﬁed by Z hn 0 ∥δtn(τ, ω)∥2dτ = Z hn 0 sθ tn+gn(τ)(by(K−1) tn,gn(τ)(ω)) −∇log ⃗ ptn+τ(by(K) tn,τ(ω)) 2 dτ ≤3 Z hn 0 ∇log ⃗ ptn+gn(τ) by(K) tn,gn(τ)(ω) −∇log ⃗ ptn+τ by(K) tn,τ(ω) 2 dτ \| {z } :=Atn(ω) + Z hn 0 sθ tn+gn(τ) by(K) tn,gn(τ)(ω) −∇log ⃗ ptn+gn(τ) by(K) tn,gn(τ)(ω) 2 dτ \| {z } :=Btn(ω) + Z hn 0 sθ tn+gn(τ) by(K) tn,gn(τ)(ω) −sθ tn+gn(τ) by(K−1) tn,gn(τ)(ω) 2 dτ ! . (B.7) Before we continue our proof, we would like ﬁrst to provide the following lemma bounding the behavior of the auxiliary process (B.2) when k = 0 for τ ∈[0, hn]. Lemma B.5. For any n ∈[0 : N −1], suppose the initialization bytn in (B.3) of the auxiliary process (B.2) follows the distribution of ⃗ xtn ∼ ⃗ ptn, then the following estimate holds sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn h ∥by(1) tn,τ(ω) −by(0) tn,τ(ω)∥2i ≤hne 7 2 hn M 2 s + 2d + 3e 7 2 hnEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + 3e 7 2 hnhnL2 s sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(K) tn,τ(ω) −by(K−1) tn,τ (ω) 2 . (B.8) Proof. Let ztn,τ = by(1) tn,τ −by(0) tn,τ. For k = 0, we can rewrite (B.2) as dztn,τ = " 1 2 ztn,τ + by(0) tn,τ + sθ tn+gn(τ) by(0) tn,gn(τ) # dτ + dwtn+τ, By applying Itô’s lemma and plugging in the expression of wtn+τ given by Theorem A.4, we have d∥ztn,τ∥2 = " ∥ztn,τ∥2 + z⊤ tn,τ by(0) tn,τ + 2z⊤ tn,τsθ tn+gn(τ) by(0) tn,gn(τ) + d # dτ + 2z⊤ tn,τ d e wtn+τ(ω) −δtn(τ, ω)dτ , (B.9) By integrating from 0 to τ and taking the expectation on both sides of (B.9), we obtain that Eω∼ ⃗ p\|Ftn ∥ztn,τ∥2 = Eω∼ ⃗ p\|Ftn "Z τ 0 ∥ztn,τ ′∥2 + z⊤ tn,τ ′ by(0) tn,τ ′ + 2z⊤ tn,τ ′sθ tn+gn(τ ′) by(0) tn,gn(τ ′) + d ! dτ ′ # + 2Eω∼ ⃗ p\|Ftn Z τ 0 z⊤ tn,τ ′ d e wtn+τ ′(ω) −δtn(τ ′, ω)dτ ′ , and by AM-GM, we further have Eω∼ ⃗ p\|Ftn ∥ztn,τ∥2 ≤Eω∼ ⃗ p\|Ftn "Z τ 0 " 7 2∥ztn,τ ′∥2 + 1 2 by(0) tn,τ ′ 2 + sθ tn+gn(τ ′) by(0) tn,gn(τ ′) 2 + d + ∥δtn(τ, ω)∥2 # dτ ′ # ≤ Z τ 0 Eω∼ ⃗ p\|Ftn 7 2∥ztn,τ ′∥2 + ∥δtn(τ, ω)∥2 dτ ′ + 1 2E by(0) tn,τ 2 + M 2 s + d τ, 25 where δtn(τ, ω) is deﬁned in (B.4). Similar to (B.7), we may use triangle inequality to upper bound ∥δtn(τ, ω)∥2, which implies that for any τ ∈[0, hn] Eω∼ ⃗ p\|Ftn ∥ztn,τ∥2 ≤7 2 Z τ 0 Eω∼ ⃗ p\|Ftn ∥ztn,τ ′∥2 dτ ′ + 1 2E by(0) tn,τ 2 + M 2 s + d τ + 3Eω∼ ⃗ p\|Ftn Z τ 0 sθ tn+gn(τ)(by(K−1) tn,gn(τ)(ω)) −sθ tn+gn(τ)(by(K) tn,gn(τ)(ω)) 2 dτ ′ + 3Eω∼ ⃗ p\|Ftn Z τ 0 sθ tn+gn(τ)(by(K) tn,gn(τ)(ω)) −∇log ⃗ ptn,gn(τ)(by(K) tn,gn(τ)(ω)) 2 dτ ′ + 3Eω∼ ⃗ p\|Ftn Z τ 0 ∇log ⃗ ptn+gn(τ)(by(K) tn,gn(τ)(ω)) −∇log ⃗ ptn+τ(by(K) tn+τ(ω)) 2 dτ ′ ≤7 2 Z τ 0 Eω∼ ⃗ p\|Ftn ∥ztn,τ ′∥2 dτ ′ + 1 2E by(0) tn,τ 2 + M 2 s + d τ + 3L2 s Z τ 0 Eω∼ ⃗ p\|Ftn by(K) tn,gn(τ ′)(ω) −by(K−1) tn,gn(τ ′)(ω) 2 dτ ′ + 3Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] , where in the second inequality above, we have used the fact that sθ t (·) is Ls-Lipschitz for any t. By Grönwall’s inequality, we have that for any τ ∈[0, hn] Eω∼ ⃗ p\|Ftn ∥ztn,τ∥2 ≤e 7 2 τ 1 2E by(0) tn,τ 2 + M 2 s + d τ + 3e 7 2 τEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + 3e 7 2 τL2 s Z τ 0 Eω∼ ⃗ p\|Ftn by(K) tn,gn(τ ′)(ω) −by(K−1) tn,gn(τ ′)(ω) 2 dτ ′. (B.10) By assumption, by(0) tn,τ = bytn follows the distribution of ⃗ xtn ∼ ⃗ ptn, which allows us to bound the second moment of bytn for any n ∈[0 : N] by Lemma A.8: E ∥bytn∥2 = E ∥ ⃗ xtn∥2 ≤2d. Substituting (A.5) into (B.10) then yields that for any τ ∈[0, hn] Eω∼q\|Ftn ∥ztn,τ∥2 ≤τe 7 2 τ M 2 s + 2d + 3e 7 2 τEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + 3τe 7 2 τL2 s sup τ ′∈[0,hn] Eω∼ ⃗ p\|Ftn by(K) tn,τ ′(ω) −by(K−1) tn,τ ′ (ω) 2 . Taking supremum with respect to τ ∈[0, hn] on both sides above completes our proof. As utilized in the proof of the existence of solutions of SDEs, the following lemma demonstrates the exponential convergence of the iteration deﬁned in (B.2). Lemma B.6 (Exponential convergence of Picard iteration in PIADM-SDE). For any n ∈[0, N], suppose the initialization bytn in (B.3) of the auxiliary process (B.2) follows the distribution of ⃗ xtn ∼ ⃗ ptn, then the two ending terms by(K) tn,τ and by(K−1) tn,τ of the sequence {by(k) tn,τ}k∈[0:K−1] satisfy the following exponential convergence rate sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(K) tn,τ(ω) −by(K−1) tn,τ (ω) 2 2 ≤ L2 shne2hnK−1 he 7 2 hn M 2 s + 2d 1 −3 (L2shne2hn)K−1 e 7 2 hnhnL2s + 3 L2 shne2hnK−1 e 7 2 hnEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] 1 −3 (L2shne2hn)K−1 e 7 2 hnhnL2s . Proof. For each ω ∈Ωconditioned on the ﬁltration Ftn, subtracting (B.2) from the process as deﬁned by dby(k) tn,τ(ω) = 1 2 by(k) tn,τ(ω) + sθ tn+gn(τ) by(k−1) tn,gn(τ)(ω) dτ + dwtn+τ(ω), (B.11) 26 we have d by(k+1) tn,τ (ω) −by(k) tn,τ(ω) = 1 2 by(k+1) tn,τ (ω) −by(k) tn,τ(ω) + sθ tn+gn(τ) by(k) tn,gn(τ)(ω) −sθ tn+gn(τ) by(k−1) tn,gn(τ)(ω) dτ, where the diffusion term dwtn+τ cancels each other out. Now we may use the formula above to compute derivative d dτ ′ by(k+1) tn,τ ′ (ω) −by(k) tn,τ ′(ω) 2 explicitly, integrate it from τ ′ = 0 to τ, and obtain the following inequality by(k+1) tn,τ (ω) −by(k) tn,τ(ω) 2 = Z τ 0 2 by(k+1) tn,τ ′ (ω) −by(k) tn,τ ′(ω) ⊤ sθ tn+gn(τ ′) by(k) tn,gn(τ ′)(ω) −sθ tn+gn(τ ′) by(k−1) tn,gn(τ ′)(ω) dτ ′ + Z τ 0 by(k+1) tn,τ ′ (ω) −by(k) tn,τ ′(ω) 2 dτ ′ ≤2 Z τ 0 by(k+1) tn,τ ′ (ω) −by(k) tn,τ ′(ω) 2 dτ ′ + Z τ 0 sθ tn+gn(τ ′) by(k) tn,gn(τ ′)(ω) −sθ tn+gn(τ ′) by(k−1) tn,gn(τ ′)(ω) 2 dτ ′ ≤2 Z τ 0 by(k+1) tn,τ ′ (ω) −by(k) tn,τ ′(ω) 2 dτ ′ + L2 s Z τ 0 by(k) tn,gn(τ ′)(ω) −by(k−1) tn,gn(τ ′)(ω) 2 dτ ′. By Grönwall’s inequality, we have by(k+1) tn,τ (ω) −by(k) tn,τ(ω) 2 ≤L2 se2τ Z τ 0 by(k) tn,gn(τ ′)(ω) −by(k−1) tn,gn(τ ′)(ω) 2 dτ ′. (B.12) Taking expectation on both sides above further implies that for any τ ∈[0, hn], Eω∼ ⃗ p\|Ftn by(k+1) tn,τ (ω) −by(k) tn,τ(ω) 2 ≤L2 se2τ Z τ 0 Eω∼ ⃗ p\|Ftn by(k) tn,gn(τ ′)(ω) −by(k−1) tn,gn(τ ′)(ω) 2 dτ ′ ≤L2 sτe2τ sup τ ′∈[0,τ] Eω∼ ⃗ p\|Ftn by(k) tn,τ ′(ω) −by(k−1) tn,τ ′ (ω) 2 . (B.13) Furthermore, we take supremum over τ ∈[0, hn] on both sides above and iterate (B.12) over k ∈N, which indicates sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(k+1) tn,τ (ω) −by(k) tn,τ(ω) 2 ≤L2 shne2hn sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(k) tn,τ(ω) −by(k−1) tn,τ ′ (ω) 2 ≤ L2 shne2hnk sup τ∈[0,hn] E by(1) tn,τ(ω) −by(0) tn,τ(ω) 2 ≤ L2 shne2hnk he 7 2 hn M 2 s + 2d + 3 L2 shne2hnk e 7 2 hnEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + 3 L2 shne2hnk e 7 2 hnhnL2 s sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(K) tn,τ(ω) −by(K−1) tn,τ (ω) 2 , (B.14) 27 where the last inequality follows from Lemma B.5. By rearranging the inequality above, setting k = K −1 and using the assumption that L2 shne2hn ≪1, we obtain sup τ∈[0,hn] Eω∼ ⃗ p\|Ftn by(K) tn,τ(ω) −by(K−1) tn,τ (ω) 2 ≤ L2 shne2hnK−1 he 7 2 hn M 2 s + 2d + 3 L2 shne2hnK−1 e 7 2 hnEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] 1 −3 (L2shne2hn)K−1 e 7 2 hnhnL2s , (B.15) as desired. The following lemma from [107] bounds the expectation of the term Atn(ω) in (B.7): Lemma B.7 ([107, Section 3.1]). We have Eω∼ ⃗ p\|Ftn [Atn(ω)] ≲ϵdhn, for n ∈[0 : N −2], and Eω∼ ⃗ p\|Ftn AtN−1(ω) ≲ϵd log η−1, where η is the parameter for early stopping. Proof. Notice that Eω∼ ⃗ p\|Ftn [Atn(ω)] =Eω∼ ⃗ p\|Ftn "Z hn 0 ∇log ⃗ ptn+gn(τ) by(K) tn,gn(τ)(ω) −∇log ⃗ ptn+τ by(K) tn,gn(τ)(ω) 2 dτ # =Eω∼ ⃗ p\|Ftn " Mn X m=0 Z τn,m+1 τn,m ∇log ⃗ ptn+τn,m by(K) tn,τn,m(ω) −∇log ⃗ ptn+τ by(K) tn,τ(ω) 2 dτ # , = Mn X m=0 Z τn,m+1 τn,m Eω∼ ⃗ p\|Ftn ∇log ⃗ ptn+τn,m ⃗ xtn+τ(ω) −∇log ⃗ ptn+τ ⃗ xtn+τ(ω) 2 dτ, where for the last equality, we use the fact that the process by(K) tn,τ(ω) follows the backward SDE with the true score function under the measure ⃗ p. In the following, we drop the superscript ω ∼ ⃗ p\|Ftn of the expectation for simplicity. By Lemma A.6 and A.7, we have E ∇log ⃗ ptn+τn,m ⃗ xtn+τ(ω) −∇log ⃗ ptn+τ ⃗ xtn+τ(ω) 2 ≤ Z τ 0 1 2E h ∥∇log ⃗ ptn+τn,m ⃗ xtn+τn,m(ω) ∥2i + E ∥∇2 log ⃗ ptn+τ ′ ⃗ xtn+τ ′(ω) ∥2 F dτ ′ ≤ Z τ 0 1 2d ⃗ σ−2 τ ′ + d ⃗ σ−4 τ ′ dτ ′ + ⃗ σ−4 tn+τn,mE h tr ⃗ Σtn+τn,m i − ⃗ σ−4 tn+τE h tr ⃗ Σtn+τ i , Now noticing that ⃗ σ2 t = σ2 T −t ≲T −t, we further have Z τn,m+1 τn,m E ∇log ⃗ ptn+τn,m ⃗ xtn+τ(ω) −∇log ⃗ ptn+τ ⃗ xtn+τ(ω) 2 dτ ≲ Z τn,m+1 τn,m Z τ ′ 0 d (T −tn −τn,m+1)2 dτ ′dτ + ϵn,m E h tr ⃗ Σtn+τn,m i −E h tr ⃗ Σtn+τn,m+1 i (T −tn −τn,m)2 ≲d ϵ2 n,m (T −tn −τn,m+1)2 + ϵ E h tr ⃗ Σtn+τn,m i −E h tr ⃗ Σtn+τn,m+1 i T −tn −τn,m , 28 and thus Mn X m=0 Z τn,m+1 τn,m E ∇log ⃗ ptn+τn,m ⃗ xtn+τ(ω) −∇log ⃗ ptn+τ ⃗ xtn+τ(ω) 2 dτ ≲d Mn X m=0 ϵ2 n,m (T −tn −τn,m+1)2 + Mn X m=0 ϵ T −tn −τn,m E h tr ⃗ Σtn+τn,m i −E h tr ⃗ Σtn+τn,m+1 i ≤dϵ2Mn + ϵE h tr ⃗ Σtn+τn,0 i T −tn −τn,0 + Mn X m=0 ϵϵn,mE h tr ⃗ Σtn+τn,m i (T −tn −τn,m+1)(T −tn −τn,m) ≤dϵ2Mn + ϵd + dϵ2Mn ≲dϵ2Mn. For n ∈[0, N −2], we have Mnϵ = hn and thus Eω∼ ⃗ p\|Ftn [Atn(ω)] ≲ϵdhn, and for n = N −1, we have MN ≲ Z h η 1 ϵτ dτ = log η−1ϵ−1 and thus Eω∼ ⃗ p\|Ftn AtN−1(ω) ≲ϵ2dMn ≲ϵd log η−1. B.3 Overall Error Bound Proof of Theorem 3.3. We ﬁrst continue the computation in (B.6) and (B.7): DKL( ⃗ ptn+1∥bqtn+1) ≤DKL( ⃗ ptn∥bqtn) + Eω∼ ⃗ p\|Ftn " 1 2 Z hn 0 ∥δtn(τ, ω)∥2dτ # ≤DKL( ⃗ ptn∥bqtn) + 3Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] +3Eω∼ ⃗ p\|Ftn "Z hn 0 sθ tn+gn(τ) by(K) tn,gn(τ)(ω) −sθ tn+gn(τ) by(K−1) tn,gn(τ)(ω) 2 dτ # ≤DKL( ⃗ ptn∥bqtn) + 3Eω∼ ⃗ p\|Ftn " Atn(ω) + Btn(ω) + L2 s Z hn 0 by(K) tn,gn(τ)(ω) −by(K−1) tn,gn(τ)(ω) 2 dτ # ≤DKL( ⃗ ptn∥bqtn) + 3Eω∼ ⃗ p\|Ftn " Atn(ω) + Btn(ω) + hnL2 s sup τ∈[0,hn] by(K) tn,τ(ω) −by(K−1) tn,τ (ω) 2 # . Then plugging in the result of Lemma B.6, we have DKL( ⃗ ptn+1∥bqtn+1) ≤DKL( ⃗ ptn∥bqtn) + 3Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + 3hnL2 s L2 shne2hnK−1 he 7 2 hn M 2 s + 2d 1 −3 (L2shne2hn)K−1 e 7 2 hnhnL2s +hnL2 s 9 L2 shne2hnK−1 e 7 2 hnEω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] 1 −3 (L2shne2hn)K−1 e 7 2 hnhnL2s ≲DKL( ⃗ ptn∥bqtn) + 1 + e−Khnehn 1 −e−Khnehn Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + e−Kh2 nehnd ≲DKL( ⃗ ptn∥bqtn) + Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + e−Kh2 nehnd, where we used the assumption that L2 shne 7 2 hn ≪1. 29 The term PN−1 n=0 Eω∼ ⃗ p\|Ftn [Btn(ω)] is bounded by Assumption 3.1 as N−1 X n=0 Eω∼ ⃗ p\|Ftn [Btn(ω)] ≤Eω∼ ⃗ p\|Ftn "N−1 X n=0 Z hn 0 sθ tn+gn(τ) by(K) tn,gn(τ)(ω) −∇log ⃗ ptn+gn(τ) by(K) tn,gn(τ)(ω) 2 dτ # =Eω∼ ⃗ p\|Ftn "N−1 X n=0 Mn−1 X m=0 ϵn,m sθ tn+τn,m by(K) tn,τn,m(ω) −∇log ⃗ ptn+τn,m by(K) tn,τn,m(ω) 2 # =Eω∼ ⃗ p\|Ftn "N−1 X n=0 Mn−1 X m=0 ϵn,m sθ tn+τn,m ⃗ xtn+τ(ω) −∇log ⃗ ptn+τn,m ⃗ xtn+τ(ω) 2 # ≤δ2 2, where the last equality is because the process by(K) tn,τ(ω) under measure ⃗ p follows the backward SDE (B.5). Thus, by Theorem A.1 and plugging in the iteration relations above DKL(pη∥bqtN ) = DKL( ⃗ ptN ∥bqtN ) ≤DKL( ⃗ p0∥bq0) + N−1 X n=0 Eω∼ ⃗ p\|Ftn [Atn(ω) + Btn(ω)] + e−Kh2 nehnd ≤DKL( ⃗ p0∥bq0) + N−2 X n=0 ϵdhn + ϵd log η−1 + N−1 X n=0 Eω∼ ⃗ p\|Ftn [Btn(ω)] + e−Kh2 nehndN ≤de−T + ϵd(T + log η−1) + δ2 2 + e−KdT ≤de−T + ϵdT + δ2 + e−KdT, as T ≳log η−1, hn ≲1, and δ2 ≲δ, and then it is straightforward to see that the following choices of parameters T = O(log(dδ−2)), h = Θ(1), N = O log(dδ−2) , ϵ = Θ d−1δ2 log−1(dδ−2) , M = O dδ−2 log(dδ−2) , K = e O(log(dδ−2)), would yield an overall error of O(δ2). C Details of Probability Flow ODE Implementation In this section, we provide the details of the parallelized algorithm for the probability ﬂow ODE formulation of diffusion models. We ﬁrst introduce the algorithm and deﬁne the necessary notations, then discuss the error analysis during the predictor and corrector steps, respectively, and ﬁnally provide the proof of Theorem 3.5. C.1 Algorithm In the parallelized inference algorithm for diffusion models in the probability ﬂow ODE formulation, we adopt the same discretization scheme as in Section 3.1.1 and the exponential integrator for all updating rules. For each block, we ﬁrst run a predictor step, which consists of running the probability ﬂow ODE in parallel. Then we run a corrector step, which runs an underdamped Langevin dynamics in parallel to correct the distribution of the samples. The algorithm is summarized In Algorithm 2. Parallelized Predictor Step The parallelization strategies in the predictor step are similar to those in the SDE algorithm (Algorithm 1). The only difference here is that instead of applying Picard iteration to the backward SDE as in (3.2), we apply Picard iteration to the probability ﬂow ODE as in (C.3), which does not require i.i.d. samples from standard Gaussian distribution. As shown in Lemma C.3, the update rule in the predictor step (C.1) in Algorithm 2 is equivalent to running 30 Algorithm 2: PIADM-ODE Input: by0 ∼bq0 = N(0, Id), a discretization scheme (T, (hn)N n=1 and (τn,m)n∈[1:N],m∈[0:Mn]) satisfying (3.1), parameters for the corrector step (T †,N †, h†, M †, ϵ†), the depth of iteration K and K†, the learned NN-based score sθ t (·). Output: A sample byT ∼bqT ≈ ⃗ pT . 1 for n = 0 to N −1 do 2 ▷Predictor Step (Section C.2) 3 by(0) tn,τn,m ←bytn for m ∈[0 : Mn]; 4 for k = 1 to K do 5 by(k) tn,0 ←bytn; 6 for m = 1 to Mn in parallel do 7 by(k) tn,τn,m ←1 2e τn,m 2 by(k−1) tn,0 + 1 2 Pm−1 j=0 e τn,m−τn,j+1 2 (eϵn,j −1) sθ tn+τn,j(by(k−1) tn,τn,j) (C.1) 8 end 9 end 10 ▷Corrector Step (Section C.3) 11 bu(0) tn,0 ←by(K) tn,hn and bv(0) tn,0 ∼N(0, Id); 12 for n† = 0 to N † −1 do 13 (bu(0) tn,n†h†,m†ϵ†, bv(0) tn,n†h†,m†ϵ†) ←(butn,n†h†, bvtn,n†h†) for m† ∈[0 : M †]; 14 for k† = 1 to K† do 15 (bu(k†) tn,n†h†,0, bv(k†) tn,n†h†,0) ←(butn,n†h†, bvtn,n†h†); 16 ξj† ∼N 0, 2γ(1 + γ−2)(1 −e−γϵ†)2e−2γ((M †−j†+1)ϵ†)Id for j† ∈[0 : M †]; 17 for m† = 1 to M † in parallel do 18  bu(k†) tn,n†h†,m†ϵ† bv(k†) tn,n†h†,m†ϵ†  ←G(m†ϵ†)  bu(k†−1) tn,n†h†,0 bv(k†−1) tn,n†h†,0   + m−1 X j†=0 G((m† −j† −1)ϵ†) Id −G(ϵ†) " 0 sθ tn+1(bu(k†−1) tn,n†h†,j†ϵ†) # + m−1 X j†=0 G((m† −j† −1)ϵ†) 0 ξj† ; (C.2) 19 end 20 end 21 (butn,(n†+1)h†, bvtn,(n†+1)h†) ←(bu(K†) tn,n†h†,h†, bv(K†) tn,n†h†,h†); 22 end 23 bytn+1 ←butn,T †; 24 end 31 bqtn = bqtn,0 bqtn,hn = bπ bu tn,0 bπ bu tn,T † = bqtn+1 by(k) tn,τ bu(k) tn,n†h†,τ †, bv(k) tn,n†h†,τ † ⃗ ptn eqtn,hn = πu tn,0 = eπ eu tn,0 eπ eu tn,T † πu tn,T † ey(k) tn,τ eu(k) tn,n†h†,τ †, ev(k) tn,n†h†,τ † utn,n†h†+τ †, vtn,n†h†+τ † ⃗ ptn+1 = π∗,u∗ tn,0 π∗,u∗ tn,T † = ⃗ ptn+1 ⃗ xtn+τ u∗ tn,n†h†+τ †, v∗ tn,n†h†+τ † W2(eqtn,hn, ⃗ ptn+1) (Theorem C.7) DKL(πtn,T †∥eπtn,T †) (Theorem C.17) TV(πtn,T †, ⃗ ptn+1) (Lemma C.18) TV(bπ bu tn,T †, eπ eu tn,T †) (Theorem A.1) Figure 2: Illustration of the proof pipeline of Theorem 3.5 for PIADM-ODE within the n-th block. the auxiliary predictor process (C.3). The auxiliary predictor process takes in the result from the previous corrector step (or the initialization if n = 0) and outputs by(K) tn,hn as the initialization for the next corrector step. Parallelized Corrector Step The parallelization of the underdamped Langevin dynamics is similar to that mentioned in Section 2.2. Given a sample resulting from the predictor step, we initialize the auxiliary corrector process (Deﬁnition C.8) which is an underdamped Langevin dynamics with the initialization butn,0 = y(K) tn,hn and the augmented variable bvtn,0 ∼N(0, Id) representing the momentum. We run the underdamped Langevin dynamics for time T †, which is set to be of order Ω(1) so that it is large enough to correct the distribution of the samples (cf. Lemma C.18) while being comparably short to ensure numerical stability (cf. Theorem C.17). Following a similar strategy as in Section 2.2 and in Algorithm 1, we further divide the time horizon T † into N † blocks with step size h†, and for each block the block length h† into M † steps with step size ϵ†. Within each block, we run the underdamped Langevin dynamics in parallel for K† iterations. As shown in Lemma C.9, the update rule in the corrector step (C.2) in Algorithm 2 is equivalent to running the auxiliary corrector process (C.11). In the following subsections, we proceed to provide theoretical guarantees for the algorithm. C.2 Parallelized Predictor Step Deﬁnition C.1 (Auxiliary Predictor Process). For any n ∈[0 : N −1], we deﬁne the auxiliary predictor process (by(k) tn,τ)τ∈[0,hn] as the solution to the following ODE recursively for k ∈[0 : K −1]: dby(k+1) tn,τ = " 1 2 by(k+1) tn,τ + 1 2sθ tn+gn(τ) by(k) tn,gn(τ) # dτ, (C.3) with the initial condition by(0) tn,τ ≡bytn for τ ∈[0, hn], and y(k) tn,0 ≡bytn for k ∈[1 : K] (C.4) where bytn = butn−1,N †h† if n ∈[1 : N −1] and byt0 ∼N(0, Id). We will also denote the probability distribution of by(K) tn,τ as bqtn,τ. 32 Deﬁnition C.2 (Interpolating Process). For any n ∈[0 : N −1], we deﬁne the interpolating process (ey(k) tn,τ)τ∈[0,hn] as the solution to the following ODE recursively for k ∈[0 : K −1]: dey(k+1) tn,τ = " 1 2 ey(k+1) tn,τ + 1 2sθ tn+gn(τ) ey(k) tn,gn(τ) # dτ, (C.5) with initial condition ey(0) tn,τ ≡ey(0) tn,0 for τ ∈[0, hn], and ey(k) tn,0 ≡ey(0) tn,0 for k ∈[1 : K], where ey(0) tn,0 ∼ ⃗ ptn. We will also denote the probability distribution of ey(K) tn,τ as eqtn,τ. Similar to the equivalence between (3.4) and (B.2), we have the following lemma: Lemma C.3 (Equivalence between (C.1) and (C.3)). For any n ∈[0 : N −1], the update rule (C.1) in Algorithm 2 is equivalent to the exact solution of (C.3) for any k ∈[0 : K −1] and τ ∈[0, hn]. Proof. Rewriting (C.3) and multiplying e−τ 2 on both sides yield d h e−τ 2 by(k+1) tn,τ i = e−τ 2 dby(k+1) tn,τ −1 2 by(k+1) tn,τ dτ = e−τ 2 2 sθ tn+gn(τ) by(k) tn,gn(τ) dτ Integrating on both sides from 0 to τ implies e−τ 2 by(k+1) tn,τ −by(k+1) tn,0 = Z τ 0 e−τ 2 2 sθ tn+gn(τ ′) by(k) tn,gn(τ ′) dτ ′ =1 2 Mn X m=0 Z τ∧τn,m+1 τ∧tn,m e−τ′ 2 sθ tn+τn,m y(k) tn,τn,m dτ ′ = Mn X m=0 e− τ∧τn,m 2 −e− τ∧τn,m+1 2 sθ tn+τn,j by(k) tn,τn,m , and then multiplying e τ 2 on both sides above yields by(k+1) tn,τ = e τ 2 by(k+1) tn,0 + Mn X m=0 e τ∧τn,m+1−τ∧τn,m 2 −1 e 0∨(τ−τn,m+1) 2 sθ tn+τn,m by(k) tn,τn,m . Plugging in τ = τn,m gives us (C.1), as desired. Lemma C.4 (Error between the interpolating process and the true process). Under the Picard iteration, we have that the ending process {by(K) tn,τ}τ∈[0,hn] satisﬁes the following exponential convergence rate sup τ∈[0,hn] E ey(K) tn,τ − ⃗ xtn+τ 2 ≤3d h2 nehn+ 3 2 L2 s 2 !K + ehn+ 3 2 hn/2 1 −h2nehn+ 3 2 L2s/2 hnδ2 ∞+ E[Dtn] , where Dtn := Z hn 0 sθ tn+gn(τ ′) ⃗ xtn+gn(τ ′) −sθ tn+τ ′( ⃗ xtn+τ ′) 2 dτ ′. Proof. Recall that the backward true process { ⃗ xtn+τ}τ∈[0,hn] satisﬁes the following backward SDE within one block d ⃗ xtn+τ = 1 2 ⃗ xtn+τ + 1 2∇log ⃗ ptn+τ( ⃗ xtn+τ) dτ. (C.6) By subtracting (C.6) from (C.5), we obtain that d dτ ey(k+1) tn,τ − ⃗ xtn+τ = 1 2 h ey(k+1) tn,τ − ⃗ xtn+τ i + 1 2 h sθ tn+gn(τ) ey(k) tn,gn(τ) −sθ tn+gn(τ)( ⃗ xtn+gn(τ)) i + 1 2 h sθ tn+gn(τ)( ⃗ xtn+gn(τ)) −∇log ⃗ ptn+gn(τ)( ⃗ xtn+gn(τ)) i + 1 2 h ∇log ⃗ ptn+gn(τ)( ⃗ xtn+gn(τ)) −∇log ⃗ ptn+τ( ⃗ xtn+τ) i . (C.7) 33 Then by d ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ 2 = 2 ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ ⊤ d ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ , and integrating for τ ′ ∈[0, hn], we have ey(k+1) tn,τ − ⃗ xtn+τ 2 = Z τ 0 ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ ⊤ sθ tn+gn(τ ′) ey(k) tn,gn(τ ′) −sθ tn+gn(τ ′)( ⃗ xtn+gn(τ ′)) dτ ′ + Z τ 0 ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ ⊤ sθ tn+gn(τ ′)( ⃗ xtn+gn(τ ′)) −∇log ⃗ ptn+gn(τ ′)( ⃗ xtn+gn(τ ′)) dτ ′ + Z τ 0 ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ ⊤ ∇log ⃗ ptn+gn(τ ′)( ⃗ xtn+gn(τ ′)) −∇log ⃗ ptn+τ ′( ⃗ xtn+τ ′) dτ ′ + Z τ 0 ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ 2 dτ ′. Using AM-GM inequality and taking expectations on both sides, we further upper bound the summation above as E ey(k+1) tn,τ − ⃗ xtn+τ 2 ≤ 1 + 3 2hn Z τ 0 E ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ 2 dτ ′ +hn 2 Z τ 0 E sθ tn+gn(τ ′) ey(k) tn,gn(τ ′) −sθ tn+gn(τ ′)( ⃗ xtn+gn(τ ′)) 2 dτ ′ +hn 2 Z τ 0 E sθ tn+gn(τ ′)( ⃗ xtn+gn(τ ′)) −∇log ⃗ ptn+gn(τ ′)( ⃗ xtn+gn(τ ′)) 2 dτ ′ +hn 2 E Z τ 0 ∇log ⃗ ptn+gn(τ ′)( ⃗ xtn+gn(τ ′)) −∇log ⃗ ptn+τ ′( ⃗ xtn+τ ′) 2 dτ ′ \| {z } ≤Dtn ≤ 1 + 3 2hn Z τ 0 E ey(k+1) tn,τ ′ − ⃗ xtn+τ ′ 2 dτ ′ + hn 2 τδ2 ∞+ E [Dtn] +L2 shn 2 Z τ 0 E ey(k) tn,gn(τ ′) − ⃗ xtn+gn(τ ′) 2 dτ ′, where the last equality is by Assumption 3.1’. Applying Grönwall’s inequality, we have E ey(k+1) tn,τ − ⃗ xtn+τ 2 ≤e(1+ 3 2hn )τL2 shn 2 Z τ 0 E ey(k) tn,gn(τ ′) − ⃗ xtn+gn(τ ′) 2 dτ ′ + e(1+ 3 2hn )τhn 2 τδ2 ∞+ E[Dtn] ≤τe(1+ 3 2hn )τL2 shn 2 sup τ ′∈[0,τ] E ey(k) tn,τ ′ − ⃗ xtn+τ ′ 2 + e(1+ 3 2hn )τhn 2 τδ2 ∞+ E[Dtn] , (C.8) and by taking supremum sup τ∈[0,hn] E ey(k+1) tn,τ − ⃗ xtn+τ 2 ≤h2 nehn+ 3 2 L2 s 2 sup τ ′∈[0,τ] E ey(k) tn,τ ′ − ⃗ xtn+τ ′ 2 + ehn+ 3 2 hn 2 hnδ2 ∞+ E[Dtn] (C.9) 34 Given that constant hn is sufﬁciently small, which ensures L2 shne 5 2 hn ≪1, iterating the above inequality for k ∈[0 : K −1] gives us that sup τ∈[0,hn] E ey(K) tn,τ − ⃗ xtn+τ 2 ≤ h2 nehn+ 3 2 L2 s 2 !K sup τ∈[0,hn] E ey(0) tn,τ − ⃗ xtn+τ 2 + ehn+ 3 2 hn/2 1 −h2nehn+ 3 2 L2s/2 hnδ2 ∞+ E[Dtn] , Notice that by Lemma A.8, we have E by(0) tn,τ − ⃗ xtn+τ 2 = E h ∥ ⃗ xtn − ⃗ xtn+τ∥2i ≤3d, substituting which into (C.9) then gives us that sup τ∈[0,hn] E ey(K) tn,τ − ⃗ xtn+τ 2 ≤3d h2 nehn+ 3 2 L2 s 2 !K + ehn+ 3 2 hn/2 1 −h2nehn+ 3 2 L2s/2 hnδ2 ∞+ E[Dtn] , as desired. Now it remains to bound Ctn and Dtn in Lemma C.4. We ﬁrst bound Dtn using the following lemma: Lemma C.5. For any n ∈[0 : N −1], we have that E [Dtn] ≲dϵ2hn. Proof. For any n ∈[0 : N −2], we have T −tn+1 ≳O(1) and thus by [111, Corollary 1] that E ∇log ⃗ ptN−1+τn,m( ⃗ xtN−1+τn,m) −∇log ⃗ ptN−1+τ ′( ⃗ xtN−1+τ ′) 2 ≲dϵ2 n,m, for any τ ′ ∈[τn,m, τn,m+1], and thus E [Dtn] = Z hn 0 E ∇log ⃗ ptn+gn(τ ′)( ⃗ xtn+gn(τ ′)) −∇log ⃗ ptn+τ ′( ⃗ xtn+τ ′) 2 dτ ′ = Mn X m=0 Z τn,m+1 τn,m E ∇log ⃗ ptn+τn,m( ⃗ xtn+τn,m) −∇log ⃗ ptn+τ ′( ⃗ xtn+τ ′) 2 dτ ′ ≲ Mn X m=0 dϵ2 n,mϵn,m ≤dϵ2hn. For n = N −1, notice that by the step size schedule (cf. Section 3.1.1) and suppose ϵ ≤1/2, we have T −τ 2 ≤T −gn(τ) ≤T −τ, and then again [111, Corollary 1] states E ∇log ⃗ ptn+ϵn,m( ⃗ xtn+ϵn,m) −∇log ⃗ ptn+τ ′( ⃗ xtn+τ ′) 2 ≲ dϵ2 n,m T −τn,m , and thus E DtN−1 = Z hN−1 0 E ∇log ⃗ ptN−1+gn(τ ′)( ⃗ xtN−1+gn(τ ′)) −∇log ⃗ ptN−1+τ ′( ⃗ xtN−1+τ ′) 2 dτ = MN−1 X m=0 Z τn,m+1 τn,m E ∇log ⃗ ptN−1+τn,m( ⃗ xtN−1+gn(τ ′)) −∇log ⃗ ptN−1+τ ′( ⃗ xtN−1+τ ′) 2 dτ ′ ≲ MN−1 X m=0 dϵ2 n,m T −τn,m ϵn,m ≤ MN−1 X m=0 dϵ2 n,mϵ ≲ Z T −tN−1 δ∞ dτdτ ≲dϵ2hN−1. 35 Remark C.6. The above lemma is able to achieve a better dependency on ϵ compared to Lemma B.7, because the backward process ( ⃗ xt)t∈[0,T ] is now a deterministic process in the probability ﬂow ODE formulation, instead of a stochastic process as in the SDE formulation as in Lemma B.7. Thus, intuitively applying Cauchy-Schwarz rather than Itô symmetry gives us a O(ϵ2)-dependency rather than O(ϵ)-dependency. Theorem C.7. Under Assumptions 3.1’, 3.2, 3.3, and 3.4, then the distribution eqtn,hn that the parallelized predictor step generates samples from satisﬁes the following error bound: W2(eqtn,hn, ⃗ ptn+1)2 ≲de−K + h2 nδ2 ∞+ dϵ2h2 n, for n ∈[0 : N −1]. Proof. By the deﬁnition of 2-Wasserstein distance, we have for any coupling of ey(K) tn,hn and ⃗ xtn+hn, W2(eqtn,hn, ⃗ ptn+1)2 ≤E ey(K) tn,hn − ⃗ xtn+hn 2 , and therefore W2(eqtn,hn, ⃗ ptn+1)2 ≤E ey(K) tn,hn − ⃗ xtn+hn 2 ≤ sup τ∈[0,hn] E ey(K) tn,τ − ⃗ xtn+τ 2 ≤3d h2 nehn+ 3 2 L2 s 2 !K + ehn+ 3 2 hn/2 1 −h2nehn+ 3 2 L2s/2 hnδ2 ∞+ E[Dtn] ≲de−K + h2 nδ2 ∞+ dϵ2h2 n, where for the second to last inequality we used Lemma C.4, the last inequality is due to Lemma C.5 and the assumption h2 nehnL2 s ≪1. C.3 Parallelized Corrector Step After each predictor step, we run the corrector step for O(1) time to reduce the error. Particularly, we apply the Parallelized underdamped Langevin dynamics algorithm [130] to the corrector step, which yields O(1) approximate time complexity compared to the ordinary implementation of the ULMC dynamics as in [111]. In the following, we will drop the dependency on ω for notational simplicity, and we refer readers to Appendix A.2 and B.2 to review the change of measure arguments and the application of Girsanov’s theorem A.4. We will also use a general notation ∗† to distinguish the time in the backward process and the inner time in the corrector step of the n-th block. We ﬁrst deﬁne the true underdamped Langevin dynamics (utn,t†, vtn,t†)t≥0: dutn,t† = vtn,t†dt† dvtn,t† = −γvtn,t†dt† −∇log ⃗ ptn+1(utn,t†)dt† + √2γdbtn,t†, (C.10) with initial condition utn,0 ≡ey(K†) tn,hn from the predictor step and vtn,0 ∼N(0, Id), where (btn,t†)t≥0 is a Wiener process. We may also write the system of SDEs above in the following matrix form: d utn,t† vtn,t† = 0 Id 0 −γId utn,t† vtn,t† − 0 ∇log ⃗ ptn+1(utn,t†) dt† + 0 0 0 √2γId d b′ tn,t† btn,t† . We run this underdamped Langevin dynamics until the pre-determined time horizon T †. We also deﬁne the joint probability distribution of (utn,t†, vtn,t†) at time t as πtn,t†(utn,t†, vtn,t†) and its marginal on utn,t† as πu tn,t†(utn,t†). Similar to the parallelizing strategy in Section 3.1.1, we discretize the time interval [0, T †] into N † blocks with length h† = T †/N †. Within the n-th block, we further divide the block [n†h†, (n+1)h†] into M † steps, each with step size ϵ† = h†/M †. 36 Deﬁnition C.8 (Auxiliary corrector process). For any n† ∈[0 : N † −1], we deﬁne the auxiliary corrector process (bu(k†) tn,n†h†,τ †)τ †∈[0,h†] as the solution to the following SDE recursively for k† ∈ [0 : K† −1]: ( dbu(k+1) tn,n†h†,τ † = bv(k+1) tn,n†h†,τ †dτ †, dbv(k+1) tn,n†h†,τ † = −γbv(k+1) tn,n†h†,τ †dτ † −stn+1 bu(k†) tn,n†h†,gn(τ †) dτ † + √2γdbtn,n†h†+τ † (C.11) with the initial condition ( bu(0) tn,n†h†,τ † ≡butn,n†h† bv(0) tn,n†h†,τ † = bvtn,n†h† for τ † ∈[0, h†], and    bu(k†) tn,n†h†,τ † ≡butn,n†h† bv(k†) tn,n†h†,0 ≡bvtn,n†h† for k ∈[1 : K†], (C.12) where butn,n†h† := bu(K†) tn,(n†−1)h†,h†, bvtn,n†h† := bv(K†) tn,(n†−1)h†,h† for n† ∈[1 : N † −1], and butn,0 = y(K) tn,hn, bvtn,0 ∼N(0, Id). We deﬁne the joint probability distribution of (butn,t†, bvtn,t†) at time t as bπtn,t†(butn,t†, bvtn,t†) and its marginal on butn,t† as bπ bu tn,t†(butn,t†). We will also denote the resulting probability distribution of bπ bu tn,T † as bqtn+1. Lemma C.9 (Equivalence between (C.2) and (C.11)). For any n† ∈[0 : N † −1], the update rule in Algorithm 2 is equivalent to the exact solution of the auxiliary process (C.11) for any k† ∈[0 : K† −1] and τ † ∈[0, h†]. Proof. Without loss of generality, we will prove the lemma for m† = M †. The proof for m† ∈[0 : M † −1] can be done similarly. We ﬁrst rewrite (C.2) into the matrix form: d  eu(k†) tn,n†h†,τ † ev(k†) tn,n†h†,τ †  =   0 Id 0 −γId  eu(k†) tn,n†h†,τ † ev(k†) tn,n†h†,τ †  − " 0 stn+1 eu(k†) tn,n†h†,gn(τ †) # dτ † + 0 0 0 √2γId d b′ tn,n†h†+τ † btn,n†h†+τ † . (C.13) Deﬁne the time-dependent matrix G(·) as G(t†) := " Id 1−e−γt† γ Id 0 e−γt†Id # = exp 0 Id 0 −γId t† , (C.14) satisfying that d dt† G(t†) = 0 Id 0 −γId G(t†) = G(t†) 0 Id 0 −γId . Now we multiply G(−τ †) on both sides of (C.13) to obtain: d  G(−τ †)  eu(k†) tn,n†h†,τ † ev(k†) tn,n†h†,τ †    = −G(−τ †) " 0 stn+1 eu(k†) tn,n†h†,gn(τ †) # dτ † + G(−τ †) 0 0 0 √2γId d b′ tn,n†h†+τ † btn,n†h†+τ † . 37 Integrating on both sides from 0 to h† and multiplying G(h†) on both sides, we have  eu(k†) tn,n†h†,τ ev(k†) tn,n†h†,τ  −G(h†)  eu(k†) tn,n†h†,0 ev(k†) tn,n†h†,0   = − Z h† 0 G(h† −τ †′) " 0 stn+1 eu(k†) tn,n†h†,g(τ †′) # dτ †′ + Z h† 0 G(h† −τ †′) 0 0 0 √2γId d b′ tn,n†h†+τ †′ btn,n†h†+τ †′ = − M †−1 X m†=0 Z (m†+1)ϵ† m†ϵ† G(h† −τ †′)dτ †′ " 0 stn+1 eu(k†) tn,n†h†,m†ϵ† # + M †−1 X m†=0 Z (m†+1)ϵ† m†ϵ† G(h† −τ †′) 0 0 0 √2γId d b′ tn,n†h†+τ †′ btn,n†h†+τ †′ = − M †−1 X m†=0 G(ϵ†) −Id G((M † −m† −1)ϵ†) " 0 stn+1 eu(k†) tn,n†h†,m†ϵ† # + M †−1 X m†=0 Z (m†+1)ϵ† m†ϵ† G(h† −τ †′) 0 0 0 √2γId d b′ tn,n†h†+τ †′ btn,n†h†+τ †′ . By Itô isometry, we have Z (m†+1)ϵ† m†ϵ† G(h† −τ †′) 0 0 0 √2γId d b′ tn,n†h†+τ †′ btn,n†h†+τ †′ ∼N 0, 0 0 0 √2γId G((M † −m† −1)ϵ†)⊤G(ϵ†) −Id ⊤ G(ϵ†) −Id G((M † −m† −1)ϵ†) 0 0 0 √2γId ! ∼ " 0 N 0, 2γ(1 + γ−2)(1 −e−γϵ†)2e−2γ(M †−m†+1)ϵ†)Id # , as desired Deﬁnition C.10 (Interpolating corrector process). For any n† ∈[0 : N † −1], we deﬁne the interpolating corrector process (bu(k†) tn,n†h†,τ †)τ †∈[0,h†] as the solution to the following SDE recursively for k† ∈[0 : K† −1]: ( deu(k+1) tn,n†h†,τ † = ev(k+1) tn,n†h†,τ †dτ †, dev(k+1) tn,n†h†,τ † = −γev(k+1) tn,n†h†,τ †dτ † −stn+1 eu(k†) tn,n†h†,gn(τ †) dτ † + √2γdbtn,n†h†+τ † (C.15) with the initial condition ( eu(0) tn,n†h†,τ † ≡eutn,n†h† ev(0) tn,n†h†,τ † = evtn,n†h† for τ † ∈[0, h†], and    eu(k†) tn,n†h†,τ † ≡eutn,n†h† ev(k†) tn,n†h†,0 ≡evtn,n†h† for k ∈[1 : K†], (C.16) where eutn,n†h† := eu(K†) (n†−1)h†,h†, evtn,n†h† := ev(K†) (n†−1)h†,h† for n† ∈[1 : N † −1], and eutn,0 = ey(K) tn,hn, evtn,0 ∼N(0, Id). 38 We deﬁne the joint probability distribution of (eutn,t†, evtn,t†) at time t as eπtn,t†(eutn,t†, evtn,t†) and its marginal on eutn,t† as eπ eu tn,t†(eutn,t†). We invoke Girsanov’s theorem (Theorem A.4) again by the following procedure 1. Setting (A.2) as the auxiliary process (C.15) at iteration K†, where btn,t†(ω) is a Wiener process under the measure Q; 2. Deﬁning another process ebtn,n†h†+τ † governed by the following SDE: debtn,n†h†+τ † = dbtn,n†h†+τ † −ϕtn,n†h†(τ †)dτ †, (C.17) where ϕtn,n†h†(τ †) = 1 √2γ stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) (C.18) and computing the Radon-Nikodym derivative of the measure P with respect to Q as dP dQ = exp Z h† 0 ϕtn,n†h†(τ †)⊤dbtn,n†h†+τ † −1 2 Z h† 0 ∥ϕnh(τ †)∥2dτ † ! ; (C.19) 3. Concluding that (C.15) at iteration K† under the measure Q satisﬁes the following SDE:    deu(K†) n†h†,τ † = ev(K†) n†h†,τ †dτ † dev(K†) tn,n†h†,τ † = −γev(K†) tn,n†h†,τ †dτ † −∇log ⃗ ptn+1(eu(K†) n†h†,τ †)dτ † + √2γdebtn,n†h†+τ †, (C.20) with (ebtn,n†h†+τ †)τ †≥0 being a Wiener process under the measure P. If we replace (eu(K†) n†h†,τ †, ev(K†) n†h†,τ †) by (utn,n†h†+τ †, vtn,n†h†+τ †), one should notice (C.20) is immediately the original backward SDE (C.10) with the true score function on t ∈[n†h†, (n + 1)h†]: ( dutn,n†h†+τ † = vtn,n†h†+τ †dτ † dvtn,n†h†+τ † = −γvtn,n†h†+τ †dτ † −∇log ⃗ ptn+1(utn,n†h†+τ †)dτ † + √2γdebtn,n†h†+τ †. (C.21) We further deﬁne the joint probability distribution of (utn,t†, vtn,t†) at time t as πtn,t†(utn,t†, vtn,t†) and its marginal on utn,t† as πu tn,t†(utn,t†). Remark C.11. The application of Girsanov’s theorem A.4 is by writing the system of SDEs in the matrix form. Deﬁnition C.12 (Stationary process). Under the P-measure that is deﬁned by the Radon-Nikodym derivative (C.19), we may deﬁne a stationary underdamped Langevin process for n† ∈[0 : N † −1] and τ † ∈[0, h†] as ( du∗ tn,n†h†+τ † = v∗ tn,n†h†+τ †dτ †, dv∗ tn,n†h†+τ † = −γv∗ tn,n†h†+τ †dτ † −∇log ⃗ ptn+1(u∗ n†h†+τ †)dτ † + √2γdebtn,n†h†+τ †, (C.22) with the initial condition u∗ tn,n†h† ∼ ⃗ ptn+1 and v∗ tn,n†h† ∼N(0, Id). We deﬁne the joint probability distribution of (u∗ tn,t†, v∗ tn,t†) at time t as π∗ tn,t†(u∗ tn,t†, v∗ tn,t†) and its marginal on u∗ tn,t† as π∗,u∗ tn,t†(u∗ tn,t†). Thus, from Corollary A.5, we have that DKL(πtn,n†h†∥eπtn,n†h†) ≤DKL(πtn,(n−1)h†∥eπtn,(n−1)h†) + N †−1 X n=0 DKL(πtn,n†h†:(n+1)h†∥eπtn,n†h†:(n+1)h†) ≤DKL(πtn,(n−1)h†∥eπtn,(n−1)h†) + 1 4γ EP "Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † # . (C.23) 39 By triangle inequality, we have Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † ≤5 Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −stn+1(eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) 2 dτ † +5 Z h† 0 stn+1(eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −stn+1(u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ†) 2 dτ † +5 Z h† 0 stn+1(u∗ tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ†) 2 dτ † +5 Z h† 0 ∇log ⃗ ptn+1(u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(u∗ tn,n†h†,τ †) 2 dτ † +5 Z h† 0 ∇log ⃗ ptn+1(u∗ tn,n†h†+τ †) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † ≤5L2 s Z h† 0 eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† 2 dτ † +5 L2 s Z h† 0 eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† 2 dτ † + L2 p Z h† 0 eu(K†) tn,n†h†,τ † −u∗ tn,n†h†+τ † 2 dτ † ! \| {z } :=Etn,n†h† +5h†δ2 ∞+ 5L2 p Z h† 0 u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+τ † 2 dτ † \| {z } :=Ftn,n†h† , (C.24) where we used the Lipschitz continuity of the learned score function (Assumption 3.3) and the true score function (Assumption 3.4), and the δ∞-accuracy of the learned score function at each time step (Assumption 3.1’). Now we proceed to bound the terms in the error decomposition (C.24). We ﬁrst bound the Ftn,n†h† term by the following lemma: Lemma C.13. For any n ∈[0 : N −1] and τ † ∈[0, h†], we have EP " u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+τ † 2# ≤dϵ†2, and therefore EP Ftn,n†h† ≤dh†ϵ†2. Proof. By the deﬁnition of (u∗ tn.n†h†+τ, v∗ tn,n†h†+τ) as the stationary underdamped Langevin dynamics (C.22), we have EP " u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+τ † 2# = EP   Z τ † ⌊τ† ϵ† ⌋ϵ† v∗ tn,n†h†+τ †′dτ †′ 2  ≤ϵ† Z τ † ⌊τ† ϵ† ⌋ϵ† EP v∗ tn,n†h†+τ †′ 2 dτ †′ ≤dϵ†2, where the ﬁrst inequality follows from Cauchy-Schwarz inequality and the last inequality is by the fact that v∗ tn,n†h†+τ †′ ∼N(0, Id), for any τ †′ ∈[0, h†]. 40 Consequently, we have EP Ftn,n†h† = Z h† 0 EP " u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+τ † 2# dτ † ≤dh†ϵ†2. The term Etn,n†h† can be bounded with the following lemma: Lemma C.14. For any n† ∈[0 : N † −1], suppose that γ ≲L−1/2 p and T † ≲L−1/2 p , then we have the following inequality for any τ † ∈[0, h†] EP eu(K†) tn,n†h†,τ † −u∗ tn,n†h†+τ † 2 ≲W 2 2 (eqtn,hn, ⃗ ptn+1), and therefore EP Etn,n†h† ≲h†(L2 s + L2 p)W 2 2 (eqtn,hn, ⃗ ptn+1). Proof. Recall that under the measure P, eu(K†) tn,n†h†,τ † follows the dynamics of utn,n†h†,τ † (C.21) for τ † ∈[0, h†], which coincides with that of u∗ tn,n†h†+τ †. As the only difference between the two processes utn,n†h†,τ † and u∗ tn,n†h†+τ † is the initial condition, we can invoke Lemma 10 proved in [111] to deduce that EP eu(K†) tn,n†h†,τ † −u∗ tn,n†h†+τ † 2 ≲W 2 2 (πtn,n†h†, ⃗ ptn+1), where the assumption that γ ≲L−1/2 p and T † ≲L−1/2 p is required. Now notice that u∗ tn,n†h†+τ † and utn,n†h†,τ † also follow the same dynamics with the true score function for τ † ∈[0, n†h†], for any coupling of u∗ tn,n†h† and utn,n†h†, we have W 2 2 (πtn,n†h†, ⃗ ptn+1) ≤E h ∥utn,n†h† −u∗ tn,n†h†∥2i ≤W 2 2 (πtn,0, ⃗ ptn+1) = W 2 2 (eqtn,hn, ⃗ ptn+1), where the last equality is again by [111, Lemma 10]. Therefore, we have EP Etn,n†h† = Z h† 0 EP " L2 s eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −u∗ tn,n†h†+⌊τ† ϵ† ⌋ϵ† 2 + L2 p eu(K†) tn,n†h†,τ † −u∗ tn,n†h†+τ † 2# dτ † ≤h†(L2 s + L2 p)W 2 2 (eqtn,hn, ⃗ ptn+1). Now, we provide lemmas that are used to bound the ﬁrst term in (C.24). Lemma C.15. For any n† ∈[0 : N † −1], we have the following estimate: sup τ †∈[0,h†] EP eu(1) tn,n†h†,τ † −eu(0) tn,n†h†,τ † 2 ≤5L2 sh†e(3+γ)h† 2γ sup τ †∈[0,h†] EP eu(K†−1) tn,n†h†,τ † −eu(K†) tn,n†h†,τ † 2 +5h†e(3+γ)h† 2γ EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† + h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d. 41 Proof. Let µtn,n†h†,τ † := eu(1) tn,n†h†,τ † −eu(0) tn,n†h†,τ † and νtn,n†h†,τ † := ev(1) tn,n†h†,τ † −ev(0) tn,n†h†,τ †. Then for k = 0, we may rewrite (C.15) as follows    dµtn,n†h†,τ † = νtn,n†h†,τ † + ev(0) tn,n†h†,τ † dτ † dνtn,n†h†,τ † = −γ(νtn,n†h†,τ † + ev(0) tn,n†h†,τ †)dτ † −stn+1(eu(0) tn,n†h†,τ †)dτ † + √2γdbtn,n†h†+τ † (C.25) On the one hand, by using the ﬁrst equation in (C.25), we may compute the derivative d dτ †′ µtn,n†h†,τ †′ 2 = 2µ⊤ tn,n†h†,τ †′ νtn,n†h†,τ †′ + ev(0) tn,n†h†,τ †′ and integrate it for τ †′ ∈[0, τ †], which yields µtn,n†h†,τ † 2 = 2 Z τ † 0 µ⊤ tn,n†h†,τ †′(νtn,n†h†,τ †′ + ev(0) tn,n†h†,τ †′)dτ †′ ≤2 Z τ † 0 µtn,n†h†,τ †′ 2 dτ †′ + Z τ † 0 νtn,n†h†,τ †′ 2 dτ †′ + Z τ † 0 ev(0) tn,n†h†,τ †′ 2 dτ †′. Applying Gronwall’s inequality, we have µtn,n†h†,τ † 2 ≤e2τ † Z τ † 0 νtn,n†h†,τ †′ 2 dτ †′ + Z τ † 0 ev(0) tn,n†h†,τ †′ 2 dτ †′ ! . We then take expectation with respect to the path measure P and then the supremum with respect to τ † ∈[0, h†], implying that sup τ †∈[0,h†] EP h µtn,n†h†,τ † 2i ≤ sup τ †∈[0,h†] e2τ † Z τ † 0 EP h νtn,n†h†,τ †′ 2i dτ †′ + e2τ † Z τ † 0 EP ev(0) tn,n†h†,τ †′ 2 dτ †′ ! ≤h†e2h† sup τ †∈[0,h†] EP h νtn,n†h†,τ †′ 2i + h†e2h†d. (C.26) On the other hand, by applying Itô’s lemma and plugging in the expression of btn,n†h†+τ † given by (C.17), we have d∥νtn,n†h†,τ †∥2 = − " 2γ∥νtn,n†h†,τ †∥2 + 2γν⊤ tn,n†h†,τ † ev(0) tn,n†h†,τ † + 2ν⊤ tn,n†h†,τ †stn+1 eu(0) tn,n†h†,τ † −2γd # dτ † +2ν⊤ tn,n†h†,τ † p 2γ debtn,n†h†+τ † + ϕtn,n†h†(τ †)dτ † , (C.27) Then similarly, we may compute the derivative of ∥νtn,n†h†,τ †∥2, integrate it for τ † ∈[0, h†], and take the supremum with respect to τ † to obtain EP ∥νtn,n†h†,τ †∥2 =EP " − Z τ † 0 2γ∥νtn,n†h†,τ †′∥2 + 2γν⊤ tn,n†h†,τ †′ ev(0) tn,n†h†,τ †′ −2γd ! dτ †′ # +EP " − Z τ † 0 2ν⊤ tn,n†h†,τ †′stn+1 eu(0) tn,n†h†,τ †′ dτ †′ # +2 p 2γEP "Z τ † 0 ν⊤ tn,n†h†,τ †′ debtn,n†h†+τ †′ + ϕtn,n†h†(τ †′)dτ †′# . 42 By Itô’s lemma, this equals to EP ∥νtn,n†h†,τ †∥2 =EP " − Z τ † 0 2γ∥νtn,n†h†,τ †′∥2 + 2γν⊤ tn,n†h†,τ †′ ev(0) tn,n†h†,τ †′ −2γd ! dτ †′ # +EP " − Z τ † 0 2ν⊤ tn,n†h†,τ †′stn+1 eu(0) tn,n†h†,τ †′ + 2 p 2γν⊤ tn,n†h†,τ †′ϕtn,n†h†(τ †′)dτ †′ # . Applying AM-GM gives EP ∥νtn,n†h†,τ †∥2 ≤ Z τ † 0 EP h (1 + γ)∥νtn,n†h†,τ †′∥2 + ∥ϕtn,n†h†(τ †′)∥2i dτ †′ + Z τ † 0 EP γ ev(0) tn,n†h†,τ †′ 2 + stn+1 eu(0) tn,n†h†,τ †′ 2 + 2γd dτ †′ ≤ Z τ † 0 EP h (1 + γ)∥νtn,n†h†,τ †′∥2 + ∥ϕtn,n†h†(τ †′)∥2i dτ †′ + γE ev(0) tn,n†h†,0 2 + M 2 s + 2γd τ † =(1 + γ) Z τ † 0 EP ∥νtn,n†h†,τ †′∥2 dτ †′ + Z τ † 0 EP h ∥ϕtn,n†h†(τ †′)∥2i dτ †′ + τ † 3γd + M 2 s , where in the last equality, we used the initialization of the auxiliary corrector process ev(0) tn,n†h†,0 ∼ N(0, Id). Again, we apply Gronwall’s inequality to the above inequality and take the supremum with respect to τ † ∈[0, h†] to obtain sup τ †∈[0,h†] EP ∥νtn,n†h†,τ †∥2 ≤e(1+γ)h† Z h† 0 EP ∥ϕtn,n†h†(τ †)∥2 dτ † + h†e(1+γ)h† 3γd + M 2 s ≤e(1+γ)h† 2γ EP "Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † # + h†e(1+γ)h† 3γd + M 2 s , (C.28) and for the difference term within the expectation, we decompose it again by the triangle inequality in (C.24), i.e. Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † ≤5L2 s Z h† 0 eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† 2 dτ † + 5Etn,n†h† + 5h†δ2 ∞+ 5L2 pFtn,n†h†, to obtain that sup τ †∈[0,h†] EP ∥νtn,n†h†,τ †∥2 ≤5L2 se(1+γ)h† 2γ EP "Z h† 0 eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† 2 dτ † # +5e(1+γ)h† 2γ EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† + h†e(1+γ)h† 3γd + M 2 s ≤5L2 se(1+γ)h† 2γ h† sup τ †∈[0,h†] EP " eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† 2# +5e(1+γ)h† 2γ EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† + h†e(1+γ)h† 3γd + M 2 s , 43 substituting which into (C.26) completes our proof of this Lemma. Lemma C.16 (Exponential convergence of Picard iteration in the corrector step of PIADM-ODE). For any n† ∈[0, N † −1], then the two ending terms eu(K†) n†h†,τ † and eu(K†) n†h†,τ † of the sequence {eu(k†) n†h†,τ †}k†∈[0:K†−1] satisfy the following exponential convergence rate sup τ †∈[0,h†] EP eu(K†) n†h†,τ † −eu(K†−1) n†h†,τ † 2 ≤CK† 5h†e(3+γ)h† 2γ EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† + h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d ! , (C.29) where the coefﬁcient CK† = L2 sh†2eh† 2γ !K†−1 ,  1 −5L2 sh†e(3+γ)h† 2γ L2 sh†2eh† 2γ !K†−1 . Proof. We subtract the dynamics of eu(k+1) n†h†,τ † and eu(k) n†h†,τ † in (C.15) to obtain d eu(k+1) n†h†,τ † −eu(k†) n†h†,τ † = ev(k+1) n†h†,τ † −ev(k†) n†h†,τ † dτ †. Then, we use the formula above to compute the derivative d dτ †′ eu(k+1) n†h†,τ †′ −eu(k†) n†h†,τ †′ 2 = 2 eu(k+1) n†h†,τ †′ −eu(k†) n†h†,τ †′ ⊤ ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ and integrate for τ †′ ∈[0, τ †] to obtain eu(k+1) n†h†,τ † −eu(k†) n†h†,τ † 2 =2 Z τ † 0 eu(k+1) n†h†,τ †′ −eu(k†) n†h†,τ †′ ⊤ ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ dτ †′ ≤ Z τ † 0 eu(k+1) n†h†,τ †′ −eu(k†) n†h†,τ †′ 2 dτ †′ + Z τ † 0 ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ 2 dτ †′ Applying Grönwall’s inequality gives us that eu(k+1) n†h†,τ † −eu(k†) n†h†,τ † 2 ≤eτ † Z τ † 0 ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ 2 dτ †′ and taking the supremum with respect to τ † ∈[0, h†] on both sides above implies sup τ †∈[0,h†] EP eu(k+1) n†h†,τ † −eu(k†) n†h†,τ † 2 ≤h†eh† sup τ †∈[0,h†] EP ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ 2 . (C.30) We then apply a similar argument for ev(k+1) n†h†,τ † −ev(k†) n†h†,τ † as well d ev(k+1) tn,n†h†,τ † −ev(k†) tn,n†h†,τ † = −γ ev(k+1) tn,n†h†,τ † −ev(k†) tn,n†h†,τ † dτ † − stn+1(eu(k†) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −stn+1(eu(k−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) dτ †, 44 integrate which for τ † ∈[0, τ †] to obtain ev(k+1) n†h†,τ † −ev(k†) n†h†,τ † 2 = − Z τ † 0 2γ ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ 2 dτ †′ −2 Z τ † 0 ev(k+1) n†h†,τ †′ −ev(k†) n†h†,τ †′ ⊤ stn+1(eu(k†) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ†) −stn+1(eu(k−1) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ†) dτ †′ ≤1 2γ Z τ † 0 stn+1(eu(k†) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ†) −stn+1(eu(k−1) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ†) 2 dτ †′ ≤L2 s 2γ Z τ † 0 eu(k†) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ† −eu(k−1) tn,n†h†,⌊τ†′ ϵ† ⌋ϵ† 2 dτ †′. And then taking the supremum with respect to τ † ∈[0, h†] on both sides above implies sup τ †∈[0,h†] EP ev(k+1) n†h†,τ † −ev(k†) n†h†,τ † 2 ≤h†L2 s 2γ sup τ †∈[0,h†] EP eu(k†) tn,n†h†,τ † −eu(k−1) tn,n†h†,τ † 2 (C.31) Substituting (C.31) into (C.30) and iterating over k ∈[1 : K† −1], we obtain that sup τ †∈[0,h†] EP eu(K†) n†h†,τ † −eu(K†−1) n†h†,τ † 2 ≤L2 sh†2eh† 2γ sup τ †∈[0,h†] EP eu(K†−1) tn,n†h†,τ † −eu(K†−2) tn,n†h†,τ † 2 ≤ L2 sh†2eh† 2γ !K†−1 sup τ †∈[0,h†] EP eu(1) tn,n†h†,τ † −eu(0) tn,n†h†,τ † 2 ≤ L2 sh†2eh† 2γ !K†−1 5h†e(3+γ)h† 2γ EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† + L2 sh†2eh† 2γ !K†−1 h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d + L2 sh†2eh† 2γ !K†−1 5L2 sh†e(3+γ)h† 2γ sup τ †∈[0,h†] EP eu(K†−1) tn,n†h†,τ † −eu(K†) tn,n†h†,τ † 2 , where we plug in the results from Lemma C.15 in the last inequality. Rearranging the inequality above completes our proof. Theorem C.17. Under Assumptions 3.1’, 3.2, 3.3, and 3.4, given the following choices of the order of the parameters T † = O(1), N † = O(1), h† = Θ(1) M † = Θ(d1/2δ−1), ϵ† = Θ(d−1/2δ), K† = O(log(dδ−2)) and let L2 sh†2eh† 2γ ≪1, γ ≲L−1/2 p , T † ≲L−1/2 p ∧L−1/2 s , δ∞≲δ then the distribution eπtn,T † satisﬁes the following error bound: DKL(πtn,T †∥eπtn,T †) ≲T †W 2 2 (eqtn,hn, ⃗ ptn+1) + T †δ2 ∞+ dT †ϵ†2 + e−K†T †h†d ≲W 2 2 (eqtn,hn, ⃗ ptn+1) + δ2, with a total of K†N † = O log(dδ−2) approximate time complexity and M = Θ d1/2δ−2 space complexity for parallalizable δ-accurate score function computations. 45 Proof. Now, we continue the computation by plugging the decomposition in (C.24) and all the error bounds derived above into the equation. First for the last term in (C.23) EP "Z h† 0 stn+1(eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ†) −∇log ⃗ ptn+1(eu(K†) tn,n†h†,τ †) 2 dτ † # ≤5L2 sh† sup τ †∈[0,h†] EP " eu(K†−1) tn,n†h†,⌊τ† ϵ† ⌋ϵ† −eu(K†) tn,n†h†,⌊τ† ϵ† ⌋ϵ† 2# + 5EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† ≤5 1 + L2 sh†CK† 5h†e(3+γ)h† 2γ ! EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† +5L2 sh†CK† h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d , where the last inequality is by Lemma C.16. We further substitute Lemma C.14 and C.13 into (C.23) to obtain DKL(πtn,n†h†∥eπtn,n†h†) ≤DKL(πtn,(n−1)h†∥eπtn,(n−1)h†) + 52γ + L2 sCK†5h†2e(3+γ)h† 4γ2 EP Etn,n†h† + h†δ2 ∞+ L2 pFtn,n†h† +5L2 sh†CK† 2γ h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d ≲DKL(πtn,(n−1)h†∥eπtn,(n−1)h†) +52γ + L2 sCK†5h†2e(3+γ)h† 4γ2 h†(L2 s + L2 p)W 2 2 (eqtn,hn, ⃗ ptn+1) + h†δ2 ∞+ dh†ϵ†2 +5L2 sh†CK† 2γ h†2e(3+γ)h† 3γd + M 2 s + h†e2h†d ≲DKL(πtn,(n−1)h†∥eπtn,(n−1)h†) + h†W 2 2 (eqtn,hn, ⃗ ptn+1) + h†δ2 ∞+ dh†ϵ†2 + e−K†h†2d, and then sum over n to obtain DKL(πtn,T †∥eπtn,T †) = DKL(πtn,N †h†∥eπtn,N †h†) ≲DKL(πtn,0∥eπtn,0) + N †h†W 2 2 (eqtn,hn, ⃗ ptn+1) + N †h†δ2 ∞+ dN †h†ϵ†2 + e−K†N †h†2d =T †W 2 2 (eqtn,hn, ⃗ ptn+1) + T †δ2 ∞+ dT †ϵ†2 + e−K†T †h†d. Then, it is straightforward to see that when the following order of the parameters holds T † = O(1), h† = Θ(1), N † = O(1), ϵ† = Θ(d−1/2δ), M † = O(d1/2δ−1), K† = O(log(dδ−2)) and δ∞≤δ, we have DKL(πtn,T †∥eπtn,T †) ≲W 2 2 (eqtn,hn, ⃗ ptn+1) + δ2. Lemma C.18. Suppose T † ≲L−1/2 p , then we have TV(πtn,T †, ⃗ ptn+1) ≤ q DKL(πtn,T †∥ ⃗ ptn+1) ≲ 1 L 1 4p (T †) 3 2 W2(πtn,0, ⃗ ptn+1) ≲W2(eqtn,hn, ⃗ ptn+1). Proof. A complete proof of the Lemma above is presented in [111, Lemma 9], which is derived based on [145, Corollary 4.7 (1) ]. 46 C.4 Overall Error Bound We are now ready to prove Theorem 3.5. Proof of Theorem 3.5. Notice that the interpolating corrector process (eutn,n†h†,τ †, evtn,n†h†,τ †) is constructed to follow the same dynamics as the auxiliary corrector process (butn,n†h†,τ †, bvtn,n†h†,τ †) in the corrector step. Therefore, we have by data processing inequality that TV(bπ bu tn,T †, eπ eu tn,T †) ≤TV(bπ bu tn,0, eπ eu tn,0) = TV(bqtn,hn, eqtn,hn), (C.32) and again, since the interpolating predictor process eytn,n†h† is constructed to follow the same dynamics as the auxiliary predictor process bytn,n†h† in the predictor step, we further have by data processing inequality that TV(bqtn,hn, eqtn,hn) ≤TV(bqtn,0, eqtn,0) = TV(bqtn, ⃗ ptn). (C.33) Furthermore, applying triangle inequality, Pinsker’s inequality along with Theorem C.17 and Theorem C.7 proved above, we may upper bound the second term above as follows TV(πtn,T †, eπtn,T †)2 ≲DKL(πtn,T †∥eπtn,T †) ≲W 2 2 (eqtn,hn, ⃗ ptn+1) + δ2 (C.34) Summarizing the above inequalities, we have TV(eπ eu tn,T †, ⃗ ptn+1)2 = TV(eπ eu tn,T †, π∗,u∗ tn,T †)2 ≤TV(eπ eu tn,T †, πu tn,T †)2 + TV(πu tn,T †, π∗,u∗ tn,T †)2 ≤TV(eπtn,T †, πtn,T †)2 + TV(πtn,T †, π∗ tn,T †)2 ≤TV(eπtn,T †, πtn,T †)2 + TV(πtn,T †, ⃗ ptn+1)2 ≲W 2 2 (eqtn,hn, ⃗ ptn+1) + δ2 + W 2 2 (eqtn,hn, ⃗ ptn+1) ≲de−K + h2 nδ2 ∞+ dϵ2h2 n + δ2, (C.35) where the second last inequality is deduced from Theorem C.17 and Lemma C.18 and the last inequality is derived via Theorem C.7. Therefore, for any n ∈[0 : N −1], applying triangle inequality along with data processing inequality (cf. Theorem A.1) yields TV(bqtn+1, ⃗ ptn+1) = TV(bπ bu tn,T †, ⃗ ptn+1) ≤TV(bπ bu tn,T †, eπ eu tn,T †) + TV(eπ eu tn,T †, ⃗ ptn+1) ≤TV(qtn, ⃗ ptn) + d1/2e−K/2 + hnδ∞+ d1/2ϵhn + δ. (C.36) where the last inequality is derived by plugging in (C.32), (C.33) and (C.36). Applying Lemma A.9 and summing the inequalities above further give us that TV(bqtN , pη) = TV(bqtN , ⃗ ptN ) ≲TV(bq0, ⃗ p0) + N−1 X n=0 d1/2e−K 2 + hnδ + d1/2ϵhn + δ ≲d1/2e−T/2 + Nd1/2e−K/2 + Tδ∞+ d1/2ϵT + δN. (C.37) By setting the parameters T = O(log(dδ−2)), h = Θ(1), N = O(log(dδ−2)), ϵ = Θ d−1/2δ log−1(d−1/2δ−1) , M = O(d1/2δ−1 log(d1/2δ−1)), K = e O(log(dδ−2)), and letting δ∞≲δT −1 ≲δ log−1(dδ−2), we ﬁnally obtained the upper bound TV(bqtN , pη)2 ≲de−T + N 2de−K + δ2 + dϵ2T 2 ≤δ2 as desired. 47 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope? 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4,449	Stochastic Kernel Regularisation Improves Generalisation in Deep Kernel Machines Edward Milsom School of Mathematics University of Bristol edward.milsom@bristol.ac.uk Ben Anson School of Mathematics University of Bristol ben.anson@bristol.ac.uk Laurence Aitchison School of Engineering Mathematics and Technology University of Bristol laurence.aitchison@gmail.com Abstract Recent work developed convolutional deep kernel machines, achieving 92.7% test accuracy on CIFAR-10 using a ResNet-inspired architecture, which is SOTA for kernel methods. However, this still lags behind neural networks, which easily achieve over 94% test accuracy with similar architectures. In this work we introduce several modifications to improve the convolutional deep kernel machine’s generalisation, including stochastic kernel regularisation, which adds noise to the learned Gram matrices during training. The resulting model achieves 94.5% test accuracy on CIFAR-10. This finding has important theoretical and practical implications, as it demonstrates that the ability to perform well on complex tasks like image classification is not unique to neural networks. Instead, other approaches including deep kernel methods can achieve excellent performance on such tasks, as long as they have the capacity to learn representations from data. 1 Introduction Neural network Gaussian Processes (NNGPs) (Lee et al., 2017) are a key theoretical tool in the study of neural networks. When a randomly initialised neural network is taken in the infinite-width limit, it becomes equivalent to a Gaussian process with the NNGP kernel. A large body of work has focused on improving the predictive accuracy of NNGPs (Novak et al., 2018; Garriga-Alonso et al., 2018; Arora et al., 2019; Lee et al., 2020; Li et al., 2019; Shankar et al., 2020; Adlam et al., 2023), but they still fall short of finite neural networks (NNs). One hypothesis is that this is due to the absence of representation learning; the NNGP kernel is a fixed and deterministic function of its inputs (MacKay, 1998; Aitchison, 2020), but finite neural networks fit their top-layer representations to the task; this aspect of learning is considered vital for the success of contemporary deep learning (Bengio et al., 2013; LeCun et al., 2015). Exploring this hypothesis through the development of NNGP-like models equipped with representation learning could deepen our understanding of neural networks. Although there are some theoretical frameworks for representation learning (Dyer & Gur-Ari, 2019; Hanin & Nica, 2019; Aitchison, 2020; Li & Sompolinsky, 2020; Antognini, 2019; Yaida, 2020; Naveh et al., 2020; Zavatone-Veth et al., 2021; Zavatone-Veth & Pehlevan, 2021; Roberts et al., 2021; Naveh & Ringel, 2021; Halverson et al., 2021; Seroussi et al., 2023), they are generally not scalable enough to handle important deep learning datasets like CIFAR-10. One recent promising proposal in this area is deep kernel machines (DKMs Yang et al., 2023; Milsom et al., 2024). DKMs are an entirely kernel-based method (i.e. there are no weights and 38th Conference on Neural Information Processing Systems (NeurIPS 2024). no features) that is nonetheless able to learn representations from data. DKMs narrow the gap to DNNs, achieving 92.7% performance on CIFAR-10 Milsom et al., 2024 vs 91.2% (Adlam et al., 2023) for traditional infinite-width networks. However, 92.7% is still far from standard DNNs like ResNets, which achieve around 95% performance on CIFAR-10. Here, we narrow this gap further, achieving 94.5% performance on CIFAR-10, by introducing two key modifications. First, we introduce a regularisation scheme inspired by dropout (Srivastava et al., 2014) where we randomly sample positive-definite matrices in place of our learned Gram matrices. We refer to this scheme as stochastic kernel regularisation (SKR). Second, we use single-precision floating point arithmetic to greatly accelerate training, allowing us to train for more epochs under a fixed computational budget. This presents challenges concerning numerical stability, and we develop a number of mitigations to address these. 2 Background: Convolutional Deep Kernel Machines Here, we provide a brief overview of convolutional deep kernel machines. For a more in-depth introduction see Appendix A, or Milsom et al. (2024). Deep kernel machines are a class of supervised learning algorithms that compute a kernel from inputs and transform this kernel through a series of learnable mappings over Gram matrices. We can then perform prediction with the top layer kernel representation using e.g. GP regression/classification. A deep kernel machine is similar in structure to a deep Gaussian process, with the key difference being that instead of features Fℓ∈RP ×Nℓat each layer, where P is the number of datapoints and Nℓis the number of features at layer ℓ, we work with Gram matrices Gℓ∈RP ×P . We define the Gram matrices as the (normalised) dot product between features: Gℓ= 1 Nℓ Fℓ(Fℓ)T . (1) Many common kernel functions K(·), such as the arccos (Cho & Saul, 2009) and squared-exponential, depend on features only through their pairwise dot products, and thus can be computed in this reparametrised model as K(Gℓ) instead of the usual K(Fℓ). To obtain a deep kernel machine, all layers are taken in the infinite-width limit, i.e. Nℓ→∞, and the likelihood function is scaled to retain representation learning. Under this limit the Gram matrices, {Gℓ}L ℓ=1, which were previously random variables whose distribution we might approximate through variational inference (see “deep kernel processes” Aitchison et al., 2021; Ober & Aitchison, 2021; Ober et al., 2023), become deterministic / point-distributed, and can therefore be treated as learnable parameters. We learn the Gram matrices by optimising the deep kernel machine objective (which is itself derived from the evidence lower-bound (ELBO) for variational inference in the infinite-width limit) using gradient ascent: L(G1, . . . , GL) = log P Y\|GL −PL ℓ=1νℓDKL N 0, Gℓ N 0, K(Gℓ−1) . (2) Since the computations involved are very similar to Gaussian processes, DKMs naturally scale like O(P 3) with the number of training points. Implementations of shallow Gaussian processes often compute the full kernel matrix using lazy evaluation (Novak et al., 2019; Gardner et al., 2018; Charlier et al., 2021) which avoids storing the entire kernel matrix in memory at once. This process is often very slow (Google’s neural tangents library takes 508 GPU hours to compute the Myrtle-10 kernel on CIFAR-10 Google, 2020; Novak et al., 2019), and therefore infeasible for our setting where the model is both deep, and requires thousands of iterations of gradient ascent. Instead, previous work on deep kernel machines utilises sparse variational inducing point schemes, inspired by similar work on deep Gaussian processes (Salimbeni & Deisenroth, 2017). These schemes replace the Pt training points with Pi pseudo-datapoints, where Pi ≪Pt, which leads to O(P 3 i + P 2 i Pt) computations which are much cheaper. In deep Gaussian processes, a separate set of P ℓ i inducing points is learned for each layer ℓ, approximating the input-output functions locally. Deep kernel machines typically use an analogous inducing point scheme, learning an inducing Gram matrix Gℓ ii ∈RP ℓ i ×P ℓ i at each layer (rather than the full Gram matrix for all training points, Gℓ∈RPt×Pt) by optimising a similar objective to Eq. (2) (see Eq. 27 in Appendix A for further details). Prediction of train/test points is described in Algorithm 1, but at a high level, it is similar to Gaussian process prediction, except instead of working with a feature vector partitioned into inducing points and train/test points, we have a Gram matrix partitioned into 4 blocks, 1 Nℓ Fℓ i Fℓ t Fℓ i Fℓ t T = 1 NℓFℓ i (Fℓ i )T 1 NℓFℓ i (Fℓ t )T 1 NℓFℓ t (Fℓ i )T 1 NℓFℓ t (Fℓ t )T = Gℓ ii (Gℓ ti)T Gℓ ti Gℓ tt . (3) 2 The goal in prediction is to compute the conditional expectation of Gℓ ti ∈RP ℓ t ×P ℓ i and Gℓ tt ∈RP ℓ t ×P ℓ t conditioned on the learned inducing block Gℓ ii ∈RP ℓ i ×P ℓ i . Convolutional deep kernel machines combine deep kernel machines with convolutional kernel functions from the infinite-width neural network literature / Gaussian process literature (van der Wilk et al., 2017; Dutordoir et al., 2020; Garriga-Alonso et al., 2018; Novak et al., 2018). In these settings, kernel matrices are of size PWH × PWH where P is the number of training images, and W, H are the width and height of the images, respectively. In particular, this implies that inducing Gram matrices Gℓ ii are of size P ℓ i WH × P ℓ i WH which is too expensive even for CIFAR-10 (e.g. 10 32 × 32 inducing CIFAR-10 images results in a 10, 240 × 10, 240 kernel matrix). To avoid this exorbitant computational cost, Milsom et al. (2024) proposed an inducing point scheme where the learned inducing blocks Gℓ ii have no “spatial” dimensions, i.e. they are of size P ℓ i × P ℓ i , whilst the predicted train / test blocks Gℓ ti ∈RPtW H×P ℓ i , Gℓ it ∈RP ℓ i ×PtW H, Gℓ tt ∈RPtW H×PtW H retain their spatial dimensions. They use linear maps Cℓ∈RDP ℓ i ×P ℓ−1 i to map the P ℓ−1 i non-spatial inducing points into P ℓ i patches with D pixels, which can be used by the convolutional kernel. The full prediction algorithm is given in Algorithm 1. In practice, the IID likelihood functions used at the output layer mean only the diagonal of Gℓ tt is needed, dramatically reducing the computation and storage requirements to a vector Gℓ t := diag(Gℓ tt) ∈RP ℓ t W H, and only minibatches of the data are required (Yang et al., 2023). The parameters are updated via gradient descent by computing the objective (Equation 27 in Appendix A) using the predictions, and backpropagating. 3 Methods We seek to improve the generalisation of the convolutional deep kernel machine via two main strategies. First, we seek to reduce overfitting of representations by introducing randomness to the learned inducing Gram matrices at train time, replacing them with samples from the Wishart distribution. We refer to this process as “stochastic kernel regularisation” to avoid ambiguity with terms like “random sampling”. Second, we improve the numerical stability of the algorithm enough to utilise lower-precision TF32 cores in modern NVIDIA GPUs, which are significantly faster but more prone to round-off errors. Using TF32 cores makes training roughly 5× faster than the implementation in Milsom et al. (2024), which used double-precision floating points, and therefore allows us to train for significantly more epochs. Numerical stability is improved through a combination of stochastic kernel regularisation and a Taylor approximation to the problematic log-determinant term in the objective function, which we show has no negative effect on predictive performance in our ablation experiments. We also observed that keeping the regularisation strength ν in the DKM objective (Equation 2) non-zero was crucial in preserving numerical stability. 3.1 Stochastic kernel regularisation Milsom et al. (2024) observed that convolutional deep kernel machines suffer from overfitting. Under the infinite-width limit, the distributions over Gram matrices become point distributions, and offer no stochastic regularising effect. Inspired by dropout in neural networks (Srivastava et al., 2014), we introduce random noise into the training process to reduce overfitting of representations. Specifically, we replace the inducing Gram matrices Gℓ ii at each layer with a sample ˜Gℓ ii from the Wishart distribution, ˜Gℓ ii ∼W(Gℓ ii/γ, γ), (4) which has expectation Gℓ ii and variance inversely proportional to γ. Strictly speaking, the Wishart distribution has support over positive-definite matrices only. This positive-definiteness constraint corresponds to the requirement γ ≥P ℓ i , which in turn upper-bounds the variance of the samples. In highly expressive models with many inducing points, it may be beneficial to have much higher variance samples than this, so we relax the constraint on γ, leading to potentially singular matrices, and then apply jitter to the samples, i.e. ˜Gℓ ii 7→˜Gℓ ii + λI, to ensure positive-definiteness. Random sampling is disabled at test-time, though we still add the same jitter to eliminate bias between train and test predictions. We refer to this process as stochastic kernel regularisation (SKR). 3 Algorithm 1 Convolutional deep kernel machine prediction. Changes from this paper are in red. Input: Batch of datapoint inputs: Xt ∈RPtW H×ν0 Output: Distribution over predictions Y∗ t Parameters: Inducing inputs Xi, inducing Gram matrices {Gℓ ii}L ℓ=1, inducing output GP approximate posterior parameters µ1, . . . , µνL+1, Σ (shared across classes), inducing “mix-up” parameters {Cℓ}L ℓ=1, where Cℓ∈RDP ℓ i ×P ℓ−1 i Initialise full Gram matrix G0 ii G0 it G0 ti G0 tt = 1 ν0 XiXT i XiXT t XtXT i XtXT t for ℓin (1, . . . , L) do Apply kernel non-linearity Φ(·) (e.g. arccos kernel) Φii ΦT ti Φti Φtt = Φ Gℓ−1 ii (Gℓ−1 ti )T Gℓ−1 ti Gℓ−1 tt Apply convolution and “mix up” inducing points (see Appendix A or Milsom et al. (2024)). Indexing d represents pixels within a patch, and (i, r) denotes “image / feature map i, spatial location r” Kii = 1 D P d Cℓ dΦii(Cℓ d)T Kti(i,r) = 1 D P d Φti(i,r+d)(Cℓ d)T i.e. Kti = conv2D(Φti, filters = Cℓ) Ktt(i,r),(j,s) = 1 D P d Φtt (i,r+d),(j,s+d) similar to the avg_pool2D operation Apply stochastic kernel regularisation to inducing Gram matrix ˜Gℓ ii ∼W(Gℓ ii/γ, γ) + λI Predict train/test components of Gram matrix conditioned on inducing component ˜Gℓ ii Ktt·i = Ktt −KtiK−1 ii KT ti . Gℓ ti = KtiK−1 ii ˜Gℓ ii Gℓ tt = KtiK−1 ii ˜Gℓ iiK−1 ii KT ti + Ktt·i end for Average over spatial dimension, forming an additive GP (van der Wilk et al., 2017). (r) and (s) index spatial locations in a feature map, and S is the total number of spatial locations. GFlat ii = ˜GL ii GFlat ti = 1 S P r Gti L (r) GFlat tt = 1 S2 P rs Gtt L (r),(s) Final prediction using standard Gaussian process expressions Kii KT ti Kti Ktt = Φ GFlat ii (GFlat ti )T GFlat ti GFlat tt Sample features f t;L+1 λ conditioned on K and inducing outputs Q(f i;L+1 λ ) ∼N(µλ, Σ) f t;L+1 λ ∼N(KtiK−1 ii µλ, Ktt −KtiK−1 ii Kit + KtiK−1 ii ΣK−1 ii Kit) Monte-Carlo average over softmax of features to obtain categorical distribution over classes Y∗ t = E softmax(f t;L+1 1 , . . . , f t;L+1 νL+1 ) Training: Compute DKM objective (Eq. 27) with Taylor approximation (Eq. 8) using true labels Yt and backpropagate to update parameters. 3.2 Enabling lower-precision floating point arithmetic Previous implementations of deep kernel machines (Yang et al., 2023; Milsom et al., 2024) used double-precision floating point arithmetic , which is very slow. Modern GPUs are highly optimised for lower-precision floating point operations. For example, the NVIDIA A100 GPU marketing material (NVIDIA, 2021) quotes 9.7 TFLOPS for FP64 operations, 19.5 TFLOPS for FP32 operations, and 156 TFLOPS for TensorFloat-32 (TF32) operations, a proprietary standard that has the 8-bit exponent (range) of FP32 but the 10-bit mantissa (precision) of FP16, for a total of 19 bits including the sign bit (Kharya, 2020). Therefore, switching from FP64 to TF32 numbers suggests potential speedups of up to 8×, though in reality speedups will be more modest as not all operations support TF32. Working with kernel methods in low precision arithmetic requires care to ensure numerical stability. For example, direct inversion of the kernel matrix Kii should be avoided, and instead all operations 4 of the form K−1 ii B should instead be computed by factorising Kii and solving the system KiiX = B (Trefethen & Bau, 1997). Unfortunately, the convolutional deep kernel machine as presented in Milsom et al. (2024) is highly unstable with low-precision arithmetic. Subroutines for the exact computation of Cholesky decompositions fail completely (halting any further progress in training) when large round-off errors accumulate. This problem is particularly acute when dealing with large kernel matrices, which are typically very ill-conditioned. The usual solution is to add jitter to the kernel matrices, but we found this was insufficient when using such low-precision arithmetic (Table 3). Instead, we hypothesised that the problem lay not in the kernel matrices K, but rather in the learned inducing Gram matrices Gℓ ii. In particular, we observed that the condition number of Gℓ ii tended to worsen over time (Fig. 1), suggesting that learning highly expressive representations led to ill-conditioned Gram matrices. Though the stochastic kernel regularisation scheme we proposed did result in improved condition numbers during training (Fig. 1), we still observed occasional failures in our large-scale experiments on CIFAR-10 (see ablations in Table 3). We suspected that the issue might be due to the regularisation / KL-divergence terms in Eq. (2). These KL-divergence terms can be written as DKL (N (0, G)∥N (0, K)) = Tr(K−1G) −logdet(K−1G) + const. (5) This should be understood as a function, with two arguments, G and K. To evaluate the objective (Eq. 2), we would set G = Gℓ, and K = K(Gℓ). The KL divergence is problematic in terms of stability for two reasons. Firstly, the log-determinant term is a highly unstable operation, particularly in the backward pass which involves inverting the kernel matrix (Petersen & Pedersen, 2012). Secondly, computing K−1G for the trace requires a forward and backward substitution using the cholesky of K, which is typically a very ill-conditioned kernel matrix. To reduce the number of unstable operations, we replaced the log-determinant and trace terms with their second-order Taylor expansions. Since we expect the Gram representations to be close to those of the NNGP, i.e. G−1K ≈I, our Taylor expansions are taken around λi = 1, where λi is the ith eigenvalue of G−1K. In particular, the log-determinant term can be approximated as, −logdet(K−1G) = logdet(G−1K) (6a) = X i log λi (6b) ≈ X i (λi −1) −1 2(λi −1)2 (6c) = Tr(G−1K −I) −1 2Tr (G−1K −I)2 . (6d) In the “≈” step we have taken the second order Taylor expansion of log(λi) around λi = 1, and in the final step we have used the fact that the trace of a matrix is equal to the sum of its eigenvalues. Similarly for the trace term we have, Tr(K−1G) = X i 1 λi (7a) ≈ X i 1 −(λi −1) + (λi −1)2 (7b) = −Tr(G−1K −I) + Tr (G−1K −I)2 + const. (7c) Putting these approximations together we obtain, DKL (N (0, G)∥N (0, K)) ≈1 2Tr (G−1K −I)2 + const = 1 2\|G−1K −I\|2 F + const, (8) where \| · \|F is the Frobenius norm. Computing G−1K should be more stable than K−1G since the inverse is only backpropagated to the learnt cholesky of G, rather than through K to earlier parts of the model, avoiding further compounding of round-off errors. 4 Experiments 4.1 Image classification experiments We evaluated our method on the CIFAR-10 dataset (Krizhevsky & Hinton, 2009), containing 60, 000 RGB images (50, 000 train, 10, 000 test) of size 32 × 32 divided into 10 classes. We use the same 5 Method Test Accuracy (%) Test Log-Likelihood Conv. Deep Kernel Machine (This Paper) 94.52 ± 0.0693 −0.3611 ± 0.0073 Conv. Deep Kernel Machine (Milsom et al., 2024) 92.69 ± 0.0600 −0.6502 ± 0.0125 Tuned Myrtle10 Kernel DA CG (Adlam et al., 2023) 91.2 NNGP-LAP-flip (Li et al., 2019) 88.92 Neural Network (Adam) 94.55 ± 0.0361 −1.3003 ± 0.0226 Neural Network (SGD + Weight Decay) 95.36 ± 0.0523 −0.2112 ± 0.0037 Table 1: Test metrics on CIFAR-10 using a DKM and a neural network with the same architecture. We report means and 1 standard error over 4 random seeds. architecture as in Milsom et al. (2024) for ease of comparison, i.e. a ResNet20-inspired architecture with an extra size-2 stride in the first block, so that the output feature map sizes of the 3 blocks are {16, 8, 4} respectively. Wherever the ResNet architecture contains a convolution layer, we use a convolutional deep kernel machine layer as described in the loop in Algorithm 1. That is, we apply a base kernel (in our case, the normalised Gaussian kernel described in Shankar et al. (2020), a more efficient / numerically stable alternative to arccos kernels), perform the (kernel) convolution, and then predict the train/test Gram matrix blocks conditioned on the kernel and the inducing Gram matrix block. In place of batch norm we use the batch kernel normalisation approach suggested by Milsom et al. (2024). Skip connections compute convex combinations of the kernel before and after pairs of layers. At the final layer, we average over the remaining spatial locations, forming an additive GP kernel akin to convolutional GPs (van der Wilk et al., 2017) that is used to make the final predictions. A categorical likelihood function is used in the top-layer GP. Since the number of inducing points can vary between layers, we use {512, 1024, 2048} inducing points in the three blocks of convolutions, respectively, giving more expressive power to the later layers (similar to how ResNets are wider in later layers). For the stochastic kernel regularisation, we used γ = P ℓ i /4 and a jitter size of λ = 0.1, and for the objective we used a regularisation strength of ν = 0.001. We train all parameters by optimising the sparse DKM objective function (Equation 27 with Taylor approximated terms from Section 3.2) using Adam (Kingma & Ba, 2017), with β1 = 0.8, β2 = 0.9 and with an initial learning rate of 0.01 which is divided by 10 at epochs 800 and 1100, for a total of 1200 epochs. The model is implemented1 in PyTorch (Paszke et al., 2019). We also train a neural network with the same architecture for comparison, using a modified version of the popular "pytorch-cifar" GitHub repository2. In the interest of fair comparison, we use network widths of {512, 1024, 2048} in the three blocks so that the model has a comparable number of parameters to the convolutional deep kernel machine. The neural network was trained for 1200 epochs, and we report results using two different optimisers. One model used Adam with an initial learning rate of 0.001 and the same learning rate scheduling as the convolutional deep kernel machine, and the other used SGD with a momentum term of 0.9, a weight decay strength of 0.0005, an initial learning rate of 0.1 and a cosine annealing learning rate scheduler. We ran all experiments with 4 random seeds, and report the results in Table 1 with 1 standard error of the mean, assuming normally distributed errors for statistical tests. On an NVIDIA A100 with TF32 matmuls and convolutions enabled, the Adam-trained neural network takes ~45s per epoch, whilst our model takes ~260s per epoch. We estimate (very roughly) a total time, including the ablations and CIFAR-100 experiments detailed later, of around 2000 GPU hours for all experiments in this paper, and around 2-3 times that number when including preliminary and failed experiments during the entire project. The deep kernel machine matches the Adam-trained neural network with a mean test accuracy of 94.52% compared to 94.55% for the neural network (two-tailed t-test with unequal variances gives a p-value of 0.7634, suggesting no significant difference). Furthermore, the deep kernel machine provides better uncertainty quantification as measured by (mean) log-likelihood on the test data (higher is better), with an average of −0.3611 compared to the Adam-trained neural network’s −1.3003. Our model also far surpasses the convolutional deep kernel machine presented in Milsom et al. (2024). However, all these models still lag behind the SGD-trained network, which achieves a higher test accuracy of 95.36% (p-value of 0.0001 when compared to our model) and higher test 1Code available at https://github.com/edwardmilsom/skr_cdkm 2https://github.com/kuangliu/pytorch-cifar 6 Method Test Accuracy (%) Test Log-Likelihood Conv. Deep Kernel Machine (This Paper) 75.31 ± 0.0814 −1.4652 ± 0.0183 Conv. Deep Kernel Machine (Milsom et al., 2024) 72.05 ± 0.2300 −2.0553 ± 0.0207 Neural Network (AdamW) 74.13 ± 0.0442 −1.9183 ± 0.0070 Neural Network (SGD + Weight Decay) 79.42 ± 0.0380 −0.8890 ± 0.0021 Table 2: Test metrics on CIFAR-100 using a DKM and a neural network with the same architecture. We report means and 1 standard error over 4 random seeds. Ablation Test Accuracy Test Log-Likelihood Failures No ablation/ Our full method 94.52 ± 0.0693 −0.3611 ± 0.0073 0/4 No Taylor approximation to objective 94.46 ± 0.0406 −0.3951 ± 0.0081 1/4 No SKR 93.71 ± 0.0150 −0.4512 ± 0.0168 2/4 No Taylor + No SKR 93.25 (1 run) −0.5113 (1 run) 3/4 No SKR but keep λ = 0.1 jitter 93.46 (1 run) −0.4762 (1 run) 3/4 νℓ= 0 (Eq. 2) Fail Fail 4/4 200 epochs 93.45 ± 0.0225 −0.2607 ± 0.0016 0/4 Table 3: Test metrics on CIFAR-10 with different ablations applied to our headline model (Table 1). We report means and 1 standard error over the random seeds that ran to completion. Failures indicates how many of the 4 random seed runs for each setting resulted in a numerical error. log-likelihood of −0.2112 (p-value 0.00002 when compared to our model). SGD is well known to train neural networks with better generalisation properties, and in particular for ResNets (Zhou et al., 2021; Keskar & Socher, 2017; Gupta et al., 2021), so this is perhaps not too surprising. We briefly experimented with using SGD to optimise the deep kernel machine but found it generally less stable than Adam. We hypothesise this is because the deep kernel machine has many different “types” of parameters to optimise, as seen in Algorithm 1, which may benefit from different optmisation strategies, whilst the neural network only has weights and a few batchnorm parameters to optimise. We further evaluated our method on the CIFAR-100 dataset (Krizhevsky & Hinton, 2009), with results being presented in Table 2. As in CIFAR-10, we found significant improvements over previous deep kernel machine work (Milsom et al., 2024), and we found our method is competitive with a ResNet trained with Adam, but still lags behind a ResNet trained with SGD, which is known to perform excellently on these tasks (Zhou et al., 2021; Keskar & Socher, 2017; Gupta et al., 2021). Note that we additionally had to use weight decay and a cosine annealing learning rate schedule with the Adam-trained ResNet to obtain acceptable performance on CIFAR-100. To further investigate the effects of our changes, we ran a series of ablation experiments that are presented in Table 3. We report test accuracies and test log-likelihoods, but also the number of times each ablation failed out of the 4 random seeds as a proxy for numerical stability. Our experiments verified that stochastic kernel regularisation (SKR) did yield a statistically significant improvement in test accuracy (p-value 0.0009). To verify that the improvement was in fact coming from the random sampling of matrices and not an implicit regularising effect of the large amount of jitter, we tested the model with SKR disabled but still applying the jitter λ. We found that performance was still far worse than with SKR enabled; only 1 seed ran to completion without a numerical error for this setting, so we cannot compute the standard deviation necessary for the t-test, but based on the other experiments it is very unlikely the variance would be high enough for this not to be statistically significantly lower than our headline number. Furthermore, our Taylor approximation in the objective function did not harm performance. In fact, on log-likelihoods we obtain a p-value of 0.02953, suggesting a statistically significant improvement when using our Taylor approximated objective, but we believe this would require further investigation to verify. We also tested training with only 200 epochs, scheduling down the learning rate at epochs 160 and 180, and found that training for a 1200 epochs did indeed give a substantial boost to test accuracy. We found no single trick was enough to ensure stability over all our training runs, but rather a combination of our proposed modifications was necessary. We provide some brief analysis of the stability of the learned Gram matrices in the next section. 7 1 1000 2000 Epoch 105 108 1011 1014 1017 1020 1023 Condition Number γ = ∞ γ = P ℓ i γ = P ℓ i /10 γ = P ℓ i /100 1 1000 2000 Epoch ν = 0 ν = 10−15 ν = 10−12 ν = 10−9 ν = 10−6 ν = 10−3 ν = 100 ν = 103 ν = 106 1 1000 2000 Epoch ν = 0 ν = 10−15 ν = 10−12 ν = 10−9 ν = 10−6 ν = 10−3 ν = 100 ν = 103 ν = 106 Figure 1: Effects of different regularisation methods on Gram matrix condition number, in the toy binary classification problem trained for 2000 epochs. The left plot shows the condition numbers when different amounts of stochastic kernel regularisation (γ) are applied. The middle and right plots show the condition numbers when the coefficient ν of the KL regularisation terms are varied, with and without a Taylor approximation, respectively. 4.2 Effects of regularisation on Gram matrix condition number To further investigate the numerical stability of our model, we ran a 1 layer deep kernel machine with the squared exponential kernel and 100 inducing points on a toy binary classification problem. We show the condition number of the learned Gram matrix at each epoch for various values of the SKR parameter γ (left, Fig. 1), the DKM objective regularisation strength ν when using the Taylor KL divergence terms (middle, Fig. 1), and ν when using the true KL divergence (right, Fig. 1). For the plot varying γ, we used ν = 0, so that the effect of the Taylor approximation to the KL terms is irrelevant. Note that γ = ∞refers to no SKR. We ran these experiments in double precision to avoid numerical errors, and used the Adam optimiser with learning rate fixed at 0.01. These experiments took about 1 CPU hour in total. We observe that more variance (smaller γ) in stochastic kernel regularisation slows down the rate at which the condition number of Gii worsens. This makes sense, as the noise we add to the Gram matrix makes it more difficult to learn the “optimal" Gram matrix for the data, which would likely be highly overfitted, leading to extreme eigenvalues. However, running the experiment for long enough eventually leads to the same condition number for all settings of γ (see Fig 2 in Appendix B). We may expect this behaviour since the expected value of the SKR samples matches the true Gram matrix, but it’s not clear how effectively the optimiser could achieve this outside of simple toy examples. It is clear that, when using our proposed Taylor approximation to the KL divergence terms in the objective, even tiny values for the strength ν of these terms result in learned Gram matrices with condition numbers orders of magnitude better than without, and this effect grows proportionally with ν. We also see an improvement in condition number when not using the Taylor approximation to the KL divergence, but only for large ν. Setting ν too large tends to harm generalisation (Milsom et al., 2024), so it is beneficial to use the Taylor approximation which reduces condition numbers even for small ν. We also observed that the minimum condition number achieved across all settings of ν was a few orders of magnitude lower when using the Taylor approximation vs. when using the true KL divergence. Furthermore, the behaviour in the plot using the true KL divergence is rather erratic. For example, the ν = 100 curve (right, Fig. 1) initially rises to very poor condition numbers, but after a few hundred epochs rapidly drops to a much smaller condition number. This leads us to believe that the difference between these two schemes can be explained by optimisation. Our Taylor approximated terms penalise the Frobenius norm \|GK −I\|2 F, a much simpler operation than the true KL divergence terms which penalise Tr G−1K −logdet G−1K . This complex penalty may result in a difficult optimisation landscape in practice. 8 5 Related Work There is a substantial body of literature attempting to push the performance of kernel methods to new heights. These methods can broadly be split into “kernel learning” and “fixed kernel” methods. Deep kernel machines, already extensively discussed in this paper, fall into the former category, as does the area of “deep kernel learning” (e.g. Wilson et al., 2016; Achituve et al., 2023; Ober et al., 2021, to name a few). In deep kernel learning, a neural network is used to produce rich features which are then passed as inputs into a traditional “shallow” kernel machine, aiming to give the best of both deep learning and kernel methods. Another “kernel learning” method is the “convolutional kernel machine” (Mairal et al., 2014; Mairal, 2016), which draw theoretical connections between kernel methods and neural networks, though the resulting model is fundamentally a neural-network-like architecture based on features, which distinguishes it from deep kernel machines. Song et al. (2017) also utilised deep neural networks to generate task-specific representations in a more complex model involving an ensemble of RKHS subspace projections. The main difference between deep kernel machines and these other methods is that deep kernel machines do not involve neural networks at any stage; the representations are learned directly as Gram matrices, not features. By contrast, “fixed kernel” methods do not perform any representation learning during training, instead fixing their feature space before data is seen via the choice of kernel function. Though this could cover practically the entire field of kernel methods, the best performing methods on image tasks typically utilise kernels derived from the infinite-width neural network literature (Lee et al., 2017; Jacot et al., 2018; Lee et al., 2020), sometimes called “neural kernels” (Shankar et al., 2020). In particular, Adlam et al. (2023) pushed the state of the art for CIFAR-10 test accuracy with “fixed kernels” to 91.2%, using Myrtle kernels (Shankar et al., 2020), a type of neural kernel, by massively scaling up their method with distributed preconditioned conjugate gradient methods. Apart from the obvious lack of representation learning in this work, another key difference from our work is that they focus on computing large full-rank kernel matrices and finding approximate solutions using iterative solvers, whilst we use sparse inducing point approximations resulting in smaller kernel matrices, which we then solve exactly. Deep kernel machines can be viewed as an infinite-width limit of deep kernel processes (Yang et al., 2023) or deep Gaussian processes (Damianou & Lawrence, 2013) with a modified likelihood function, which results in the Gram matrices having point distributions. This can lead to overfitting. In deep Gaussian processes and deep kernel processes, the representations (Gram matrices in the case of deep kernel processes) have continuous distributions with broad support, thereby offering a regularising effect. Our stochastic kernel regularisation scheme can be seen as analogous to sampling the inducing Gram matrix Gℓ ii in a deep kernel process. Unlike a deep kernel process, the other blocks Gℓ ti and Gℓ tt in our model remain deterministic, simplifying the model implementation. Other approaches to regularising kernel methods include “kernel dropout” proposed by Song et al. (2017), though in their context dropout refers to randomly removing latent representations from their ensemble during training. This is therefore very different to our setting. In the neural kernel literature, Lee et al. (2020) identified a correspondence between diagonal regularisation of kernels (jitter) and early stopping in neural networks, and found this usually improved generalisation. In this paper, we focused on regularising the learned intermediate representations / Gram matrices, rather than the final kernel, and found that diagonal regularisation had little effect on generalisation when applied to these matrices. Previous work has attempted to improve numerical stability in kernel methods, though using different approaches. For example, Maddox et al. (2022) developed strategies to ensure numerical stability when using conjugate gradient solvers for GPs with low-precision arithmetic, but we do not use numerical solvers in this paper. van der Wilk et al. (2020) circumvent the issue of computing inverses entirely using an approach based on a reparametrisation of the variational parameters, but applying such an approach to the deep kernel machine domain would be a substantial undertaking, which we leave to future work. 6 Limitations Though we have considerably advanced the state-of-the-art for kernel methods, from 92.7% (Milsom et al., 2024) to 94.5% in CIFAR-10, there still remains a gap to the best performing neural networks, both in terms of accuracy, and in terms of runtime. Nonetheless, given that we have shown that 9 representation learning in kernel methods has dramatically improved performance in kernel methods, from 91.2% (Adlam et al., 2023) to 94.5%, it is becoming increasingly likely that representation learning really is the key reason that NNGPs underperform DNNs. We leave further narrowing or even closing the remaining gap to DNNs for future work. Constraints on computational resources meant that we could only run a limited number of experiments, so we focused on providing concrete insights on a single dataset with a series of ablations, rather than performance metrics for multiple datasets with no further analysis. Nevertheless, we provide all the code necessary to run these experiments on other datasets. 7 Conclusion In this paper we have increased the kernel SOTA for CIFAR-10 to 94.5% test accuracy using deep kernel machines, considerably higher than the previous record of 92.7% (Milsom et al., 2024), and significantly higher than NNGP-based approaches, such as the 91.2% achieved by Adlam et al. (2023). We achieved this by developing a novel regularisation method, stochastic kernel regularisation, and by exploiting modern GPU hardware with lower-precision arithmetic, which required us to improve the numerical stability of the algorithm via a multi-faceted approach. We have highlighted the important role that representation learning plays in deep learning, which is unfortunately absent from NNGP-based theory. 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Asymptotics of representation learning in finite bayesian neural networks. Advances in neural information processing systems, 34: 24765–24777, 2021. Zhou, P., Feng, J., Ma, C., Xiong, C., Hoi, S., and E, W. Towards theoretically understanding why sgd generalizes better than adam in deep learning, 2021. A Introduction to (Convolutional) Deep Kernel Machines An introduction to both fully-connected and convolutional deep kernel machines can be found in (Milsom et al., 2024), but for completeness we provide an overview here. In particular, we show how sampling a large number of features at each layer of a deep Gaussian process gives rise to a deep kernel method. A.1 Fully-connected deep kernel machines A DKM can been seen as a wide deep Gaussian process (DGP), optimised using a tempered ELBO objective. To see why, we first define a DGP where subsequent layers of features are conditionally multivariate Gaussian. Assume we have P input data points X ∈RP ×N0, and corresponding label categories y ∈{1, . . . , C}P , where C is the number of categories. Then we can place the following DGP prior on the data, F0 = X, (9a) P(Fℓ\| Fℓ−1) = Nℓ Y λ=1 N(f ℓ λ; 0, Kfeatures(Fℓ−1)), (9b) P(y \| FL+1) = P Y i=1 Categorical(yi; softmax((FL+1)i,:)). (9c) Here Fℓ∈RP ×Nℓdenotes the Nℓfeatures at layer ℓ, and f ℓ λ is the λth feature at layer ℓ. Kfeatures(·) is a kernel function that takes in features (as kernel functions usually do), written with the subscript "features" to different it later from kernel functions that take Gram matrices as input. Notice that the 13 final layer features FL+1 are logits for a categorical distribution over labels, though the likelihood distribution can be easily changed for the regression setting. To derive an ELBO, we perform variational inference by defining the following approximate posterior over the features at intermediate and final layers, P(Fℓ\| X, Y) ≈Q(Fℓ) = Nℓ Y λ=1 Q(f ℓ λ) = Nℓ Y λ=1 N(f ℓ λ; 0, Gℓ), (10a) P(FL+1 \| X, Y) ≈Q(FL+1) = Nℓ Y λ=1 Q(f L+1 λ ) = NL+1 Y λ=1 N(f L+1 λ ; µλ, Σ). (10b) We learn the mean and covariance at the final layer, but only the covariances Gℓ∈RP ×P at the intermediate layers. This choice is justified by the fact that after taking the limit Nℓ→∞, this approximate posterior family (Eq. 10) contains the true posterior (see Yang et al. (2023) for more details). The ELBO of the DGP with respect to the variational parameters is, LELBO(G1, . . . , GL, µ1, . . . , µNL+1, Σ) = (11) EQ(FL+1) log P(y \| FL+1) − NL+1 X λ=1 DKL(Q(f L+1 λ ) \|\| P(f L+1 λ \| FL)) − L X ℓ=1 βNℓDKL(Q(f ℓ) \|\| P(f ℓ\| Fℓ−1)). (12) Here β is parameter that tempers the prior. Before proceeding, we make the following assumption that the kernel can be calculated using only the sample feature covariance ˆGℓ: Kfeatures(Fℓ) = K( ˆGℓ), (13a) where ˆGℓ= 1 NℓFℓ(Fℓ)T . (13b) Assumption 13 is actually not very restrictive — it is satisfied by common kernels (such as RBF, Matern), and indeed any isotropic kernel. It is also satisfied in the limit Nℓ→∞when Kfeature(X) = ReLU(X)ReLU(X)T by the arccosine kernel (Cho & Saul, 2009). We are now ready to recover a DKM. We set Nℓ= Nνℓfor each intermediate layer ℓ= 1, . . . , L, and temper with β = N −1. In the limit N →∞, Yang et al. (2023) showed that the limiting ELBO is, LELBO →L := EQ(FL+1) log P(y \| FL+1) − NL+1 X λ=1 DKL(N(µλ, Σ) \|\| N(0, K(GL)) − L X ℓ=1 νℓDKL(N(0, Gℓ) \|\| N(0, K(Gℓ−1))), (14) where K(·) : RP ×P →RP ×P a kernel function satisfying Assumption 13. We can evaluate the expected log-likelihood in the DKM objective (Eq. (14)) using the reparameterisation trick, and the KL divergence terms can be calculated in closed form. By optimizing L w.r.t. the variational parameters we ‘train’ the DKM. Optimisation is not possible in closed form in general (it is possible in the linear kernel case for regression problems, see Yang et al. (2023)), but we can train the parameters using gradient descent. The number of parameters is O(P 2) and the time complexity for evaluating the objective is O(P 3), therefore we can only optimise Eq. (14) directly for small datasets. We later discuss an inducing point method that enables linear scaling in the number of datapoints, but first we introduce DKMs with convolutions. A.2 Convolutional deep kernel machines Above we outlined how a fully-connected DKM is a DGP with wide intermediate layers trained by tempering the ELBO. We establish a DKM for the convolutional setting by introducing a convolution 14 into the DGP in Eq. (9), F0 = X, (15a) P(Hℓ\| Fℓ) = Mℓ Y λ=1 N(hℓ λ; 0, Kfeatures(Fℓ)), (15b) W ℓ dµλ ∼iid N(0, (\|D\| Mℓ−1)−1), (15c) F ℓ ir,λ = X d∈D Mℓ−1 X µ=1 Hℓ−1 i(r+d),µW ℓ dµλ, (15d) for ℓ= 1, . . . , L, as well as a spatial pooling layer before final classification, FL flat = SpatialPool(FL) (15e) P(FL+1 \| FL) = NL+1 Y λ=1 N(f L+1 λ ; 0, Kfeatures(FL flat)), (15f) P(y \| FL+1) = P Y i=1 Categorical(yi; softmax((FL+1)i,:)). (15g) Here, we consider datapoints to be spatial locations (a.k.a. patches) across a range of images, and we index these with i (image) and r (location). By concatenating all patches (i.e. every patch in every image) together, we can represent an entire dataset of images with a single feature matrix Fℓ∈RP \|S\|×Nℓ, where S is the set of patches in an image and P is the number of images. For us, y is a set of image-level labels rather than patch-level labels, so we also include a spatial pooling layer. The pooled features FL flat have size P × Nℓ. The convolution (Eq. 15d) uses convolutional weights Wℓ∈R\|D\|×Mℓ−1×Nℓ, where D is the set of spatial locations in the filter. In our context, we only consider 2-dimensional images, therefore D will contain \|D\| 2-dimensional locations of patches. We proceed by deriving a convolutional kernel. The conditional covariance of the features Fℓhas closed form, E[F ℓ ir,µF ℓ js,µ′ \| Hℓ−1] = E X dµ Hℓ−1 i(r+d)µW ℓ dµ,λ X d′µ′ Hℓ−1 j(s+d′)µ′W ℓ d′µ′,λ (16) = X dd′µµ′ Hℓ−1 i(r+d)µHℓ−1 j(s+d′)µ′E W ℓ dµ,λW ℓ d′µ′,λ (17) = 1 DMℓ X d∈D Mℓ X µ=1 Hℓ−1 i,(r+d)µHℓ−1 j,(s+d)µ′ (18) = 1 D X d∈D ˆΩℓ−1 i(r+d),j(s+d), (19) where ˆΩ ℓis the sample covariance of Hℓ. Under the layer-wise, infinite-width limit Mℓ→∞, the sample covariance becomes the true covariance, ˆΩ ℓ→Kfeatures(Fℓ). This means that we can compute the covariance of Fℓconditioned only on the previous layer, E[F ℓ ir,µF ℓ js,µ′ \| Fℓ−1] = 1 D X d∈D (Kfeatures(Fℓ−1))i(r+d),j(s+d). (20) We can view Eq. 20 as a kernel convolution operation, and we introduce the following notation for it: (Γ(K))ir,js = 1 D P d∈D Ki(r+d),j(s+d). Equipped with a convolutional kernel Γ, we can recover a convolutional deep kernel machine by taking the limit Nℓ→∞. We again use the approximate posterior defined in Eq. 10 and temper the 15 prior. This gives us the convolutional DKM objective, L := EQ(FL+1) log P(y \| FL+1) − NL+1 X λ=1 DKL(N(µλ, Σ) \|\| N(0, K(SpatialPool(GL))) − L X ℓ=1 νℓDKL(N(0, Gℓ) \|\| N(0, Γ(K(Gℓ−1)))), (21) where again we used Assumption 13. The spatial pool operation used in this paper is mean pooling (see Algorithm 1). This convolutional DKM objective implies a fixed convolutional kernel, but allows flexibility at intermediate layers via the variational parameters Gℓ. A.3 Inducing point approximations Due to O(P 2) parameters, and O(P 3) computational cost, it is not feasible to optimise the DKM objectives (Eq. 14 and Eq. 21) directly. To resolve this scaling problem, we appeal to inducing point methods. The idea is to approximate the full train/test dataset with a smaller set of datapoints. We demonstrate an inducing point convolutional DKM here. See Appendix M in Yang et al. (2023) for the fully-connected case. We define the train/test datapoint features Fℓ t ∈RPt×Nℓ, and the inducing datapoints Fℓ i ∈RP ℓ i ×Nℓ; Pt is the number of train/test datapoints and P ℓ i is the number of inducing points at layer ℓ. We take the inducing points and train/test points to be jointly distributed according to the deep Gaussian process in Eq. 15. In other words, the concatenation of features Fℓ= Fℓ i Fℓ t ∈R(P ℓ i +Pt)×Nℓ (22) satisfies the prior in Eq. 15, with the exception that the convolution for the inducing points is slightly modified so that, (F ℓ i )i,λ = X d∈D Mℓ X µ=1 W ℓ dµ,λ X i′ Cℓ di,i′(Hℓ−1 i )i′,µ. (23) Milsom et al. (2024) motivated the extra ‘mixup’ parameter Cℓ∈R\|D\|P ℓ i ×P ℓ−1 i as allowing informative correlations between the inducing points and the train/test points to be learned. We can view the matrix multiplication CℓHℓ i in Eq. 23 as having the effect of taking non-spatial inducing points Hℓ i and mapping to them to inducing patches. This allows them to correlate meaningfully with the train/test patches. The covariances among inducing and train/test features can then be calculated, E (F ℓ i )i,λ(F ℓ i )j,λ \| Hℓ−1 = 1 D X d∈D X i′ X j′ Cℓ di,i′Cℓ dj,j′(ˆΩℓ−1 ii )i′j′ := (Γℓ ii( ˆΩ ℓ−1))i,j, (24a) E (F ℓ i )i,λ(F ℓ t )js,λ \| Hℓ−1 = 1 D X d∈D X i′ Cℓ di,i′(ˆΩℓ−1 it )i′,j(s+d) := (Γℓ it( ˆΩ ℓ−1))i,js (24b) E (F ℓ t )is,λ(F ℓ t )jv,λ \| Hℓ−1 = 1 D X d∈D (ˆΩℓ−1 tt )i(s+d),j(v+d) := (Γℓ ti( ˆΩ ℓ−1))is,jv. (24c) Here, ˆΩ ℓ= 1 MℓHℓ(Hℓ)T is the sample covariance for combined inducing and train/test samples Hℓ= Hℓ i Hℓ t T . Suffices ii refer to inducing-inducing correlations, ti to train/test-inducing correlations, and tt refer to train/test-train/test correlations; though they may better be understood as referring to different blocks of a covariance matrix. Eq. 24 gives us a learnable convolutional kernel, which we call Γℓ(·). When we take the convolutional widths Mℓto be large, we have, P(Fℓ\| Fℓ−1) = Nℓ Y λ=1 N(f ℓ λ; 0, Γℓ(Kfeatures(Fℓ−1))). (25) 16 As in the non-inducing case, we will compute an ELBO. To do so, we place an approximate posterior on the inducing points, similar to Eq. 10, Q(Fℓ i ) = Nℓ Y λ=1 Q(f ℓ i;λ) = Nℓ Y λ=1 N(f ℓ i;λ0, Gℓ ii), (26a) Q(FL+1 i ) = Nℓ Y λ=1 Q(f L+1 i;λ ) = NL+1 Y λ=1 N(f L+1 i;λ ; µλ, Σ). (26b) Taking the layer-wise infinite-width limit Nℓ→∞(again tempering the prior as in Eq. 14), we recover the following limit of the ELBO, Linducing := EQ(FL+1) log P(y \| FL+1 t ) (27a) − NL+1 X λ=1 DKL(N(µλ, Σ) \|\| N(0, K(SpatialPool(GL ii))) − L X ℓ=1 νℓDKL(N(0, Gℓ ii) \|\| N(0, Γℓ(K(Gℓ−1 ii )))), (27b) However, it still remains to perform inference on the train/test points. To this end, we ‘connect‘ the train/test points to the inducing points by assuming the approximate posterior over all features decomposes like so, Q(Fℓ\| Fℓ−1) = P(Fℓ t \| Fℓ i , Fℓ−1)Q(Fℓ i ). (28) Due to the Gaussian structure of the DGP prior, the first term in Eq. 28 is Gaussian and can be written down using standard conditional Gaussian expressions, P(Fℓ t \| Fℓ i , Fℓ−1) = Nℓ Y λ=1 N(f ℓ t;λ; Γℓ ti(Γℓ ii)−1f ℓ i;λ, Γℓ tt −Γℓ ti(Γℓ ii)−1Γℓ it), (29) where Γℓis the result after applying the non-linearity kernel and then the convolutional kernel, i.e. Γℓ= Γ(Kfeatures(Fℓ)). In other words, we can write Fℓ t in terms of standard multivariate Gaussian noise Ξ ∈RPt×Nℓ, Fℓ t = Γℓ ti(Γℓ ii)−1Fℓ i + Γ1/2 tt·i Ξ, (30a) where Γtt·i = Γℓ tt −Γℓ ti(Γℓ ii)−1Γℓ it. (30b) In the infinite-width limit Nℓ→∞, the sample feature covariance must converge to the true covariance by the law of large numbers, 1 Nℓ Fℓ(Fℓ)T →E f ℓ(f ℓ)T = Gℓ= Gℓ ii Gℓ it Gℓ ti Gℓ tt . (31) We already know the true inducing point covariance matrices, Gℓ ii, because they are parameters in our approximate posterior. However we can write down the remaining blocks of the covariance using Eq. 30. We identify Gℓ ti and Gℓ tt as, Gℓ ti = lim Nℓ→∞ 1 NℓFℓ t(Fℓ i )T = Γℓ ti(Γℓ ii)−1Gℓ ii (32a) Gℓ tt = lim Nℓ→∞ 1 NℓFℓ t(Fℓ t)T = Γℓ ti(Γℓ ii)−1Gℓ ii(Γℓ ii)−1Γℓ it + Γtt·i. (32b) Equations 32 and 24, as seen in Algorithm 1, allow us to propagate train/test points through the model alongside learned inducing points. In the above, we treat datapoints as independent. This allows us to perform minibatch training, thus greatly improving the scaling of the method over the full-rank version. Alongside the reduction in learnable parameters from the inducing point scheme, we get linear scaling with the size of the dataset. 17 1 5000 10000 Epoch 105 108 1011 1014 1017 1020 1023 Condition Number γ = ∞ γ = P ℓ i γ = P ℓ i /10 γ = P ℓ i /100 Figure 2: Effects of stochastic kernel regularisation on Gram matrix condition number strength, in the toy binary classification problem trained for 10000 epochs. See Section 4.2. B Extra Figures C Licenses • ResNets from https://github.com/kuangliu/pytorch-cifar/ are MIT licensed. • CIFAR-10 is from https://www.cs.toronto.edu/~kriz/cifar.html and has no license evident. 18 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,450	Enhancing Semi-Supervised Learning via Representative and Diverse Sample Selection Qian Shao1,3∗, Jiangrui Kang2∗, Qiyuan Chen1,3∗, Zepeng Li4, Hongxia Xu1,3, Yiwen Cao2, Jiajuan Liang2†, and Jian Wu1† 1College of Computer Science & Technology and Liangzhu Laboratory, Zhejiang University 2BNU-HKBU United International College 3WeDoctor Cloud 4The State Key Laboratory of Blockchain and Data Security, Zhejiang University {qianshao, qiyuanchen, lizepeng, einstein, wujian2000}@zju.edu.cn {kangjiangrui, yiwencao, jiajuanliang}@uic.edu.cn Abstract Semi-Supervised Learning (SSL) has become a preferred paradigm in many deep learning tasks, which reduces the need for human labor. Previous studies primarily focus on effectively utilising the labelled and unlabeled data to improve performance. However, we observe that how to select samples for labelling also significantly impacts performance, particularly under extremely low-budget settings. The sample selection task in SSL has been under-explored for a long time. To fill in this gap, we propose a Representative and Diverse Sample Selection approach (RDSS). By adopting a modified Frank-Wolfe algorithm to minimise a novel criterion α-Maximum Mean Discrepancy (α-MMD), RDSS samples a representative and diverse subset for annotation from the unlabeled data. We demonstrate that minimizing α-MMD enhances the generalization ability of low-budget learning. Experimental results show that RDSS consistently improves the performance of several popular SSL frameworks and outperforms the state-of-the-art sample selection approaches used in Active Learning (AL) and Semi-Supervised Active Learning (SSAL), even with constrained annotation budgets. Our code is available at RDSS. 1 Introduction Semi-Supervised Learning (SSL) is a popular paradigm which reduces reliance on large amounts of labeled data in many deep learning tasks [40, 59]. Previous SSL research mainly focuses on effectively utilising labelled and unlabeled data. Specifically, labelled data directly supervise model learning, while unlabeled data help learn a desirable model that makes consistent and unambiguous predictions [53]. Besides, we also find that how to select samples for annotation will greatly affect model performance, particularly under extremely low-budget settings (see Section 7.2). The prevailing sample selection methods in SSL have many shortcomings. For example, random sampling may introduce imbalanced class distributions and inadequate coverage of the overall data distribution, resulting in poor performance. Stratified sampling randomly selects samples within each class, which is impractical in real-world scenarios where the label for each sample is unknown. Existing researchers also employ representativeness and diversity strategies to select appropriate samples for annotation. Representativeness [13] ensures that the selected subset distributes similarly with the entire dataset, and diversity [54] is designed to select informative samples by pushing away ∗These authors contributed equally to this work. †Corresponding authors. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). a) Representativeness only … b) Diversity only … c) 𝛼-MMD (ours) … Figure 1: Visualization of selected samples from a dog dataset. The red and grey circles respectively symbolize the selected and unselected samples. a) The selected samples often contain an excessive number of highly similar instances, leading to redundancy; b) The selected samples contain too many edge points, unable to cover the entire dataset; c) The selected samples represent the entire dataset comprehensively and accurately. them in feature space. And focusing on only one aspect presents significant limitations (Figure 1a and b). To address these issues, Xie et al. [57] and Wang et al. [50] employ a combination of the two strategies for sample selection. These methods set a fixed ratio for representativeness and diversity, restricting the ultimate performance through our empirical evidence (see Section 7.4). Fundamentally, they lack a theoretical basis to substantiate their effectiveness. We observe that Active Learning (AL) primarily focuses on selecting the right samples for annotation, and numerous studies transfer the sample selection methods of AL into SSL, giving rise to SemiSupervised Active Learning (SSAL) [51]. However, most of these approaches exhibit several limitations: (1) They require randomly selected samples to begin with, which expends a portion of the labelling budget, making it difficult to work effectively with a very limited budget (e.g., 1% or even lower) [6]; (2) They involve human annotators in iterative cycles of labelling and training, leading to substantial labelling overhead [57]; (3) They are coupled with the model training so that samples for annotation need to be re-selected every time a model is trained [50]. In summary, selecting the appropriate samples for annotation is challenging in SSL. To address these challenges, we propose a Representative and Diverse Sample Selection approach (RDSS) that requests annotations only once and operates independently of the downstream tasks. Specifically, inspired by the concept of Maximum Mean Discrepancy (MMD) [14], we design a novel criterion named α-MMD. It aims to strike a balance between representativeness and diversity via a trade-off parameter α (Figure 1c), for which we find an optimal interval adapt to different budgets. By using a modified Frank-Wolfe algorithm called Generalized Kernel Herding without Replacement (GKHR), we can get an efficient approximate solution to this minimization problem. We prove that under certain Reproducing Kernel Hilbert Space (RKHS) assumptions, α-MMD effectively bounds the difference between training with a constrained versus an unlimited labelling budget. This implies that our proposed method could significantly enhance the generalization ability of learning with limited labels. We also give a theoretical assessment of GKHR with some supplementary numerical experiments, showing that GKHR performs well in learning with limited labels. Furthermore, we evaluate our proposed RDSS across several popular SSL frameworks on the datasets CIFAR-10/100 [19], SVHN [30], STL-10 [9] and ImageNet [10]. Extensive experiments show that RDSS outperforms other sample selection methods widely used in SSL, AL or SSAL, especially with a constrained annotation budget. Besides, ablation experimental results demonstrate that RDSS outperforms methods using a fixed ratio. The main contributions of this article are as follows: • We propose RDSS, which selects representative and diverse samples for annotation to enhance SSL by minimizing a novel criterion α-MMD. Under low-budget settings, we develop a fast and efficient algorithm, GKHR, for optimization. 2 • We prove that our method benefits the generalizability of the trained model under certain assumptions and rigorously establish an optimal interval for the trade-off parameter α adapt to the different budgets. • We compare RDSS with sample selection strategies widely used in SSL, AL or SSAL, the results of which demonstrate superior sample efficiency compared to these strategies. In addition, we conduct ablation experiments to verify our method’s superiority over the fixed-ratio approach. 2 Related Work Semi-Supervised Learning Semi-Supervised Learning (SSL) effectively utilizes sparse labeled data and abundant unlabeled data for model training. Consistency Regularization [34, 20, 45], PseudoLabeling [21, 56] and their hybrid strategies [40, 63, 35] are commonly used in SSL. Consistency Regularization ensures the model’s output stays stable even when there’s noise or small changes in the input, usually from the data augmentation [55]. Pseudo-labelling integrates high-confidence data pseudo-labels directly into training, adhering to entropy minimization [23]. Moreover, an integrative approach that combines the aforementioned strategies can also achieve substantial results [53, 59]. Even though these approaches have been proven effective, they usually assume that labelled samples are randomly selected from each class (i.e., stratified sampling), which is not practical in real-world scenarios where the label for each sample is unknown. Active Learning Active learning (AL) aims to optimize the learning process by selecting the appropriate samples for labelling, reducing reliance on large labelled datasets. There are two different criteria for sample selection: uncertainty and representativeness. Uncertainty sampling selects samples about which the current model is most uncertain. Earlier studies utilized posterior probability [22, 49], entropy [18, 26], and classification margin [47] to estimate uncertainty. Recent research regards uncertainty as training loss [17, 60], influence on model performance [11, 24] or the prediction discrepancies between multiple classifiers [8]. However, uncertainty sampling methods may exhibit performance disparities across different models, leading researchers to focus on representativeness sampling, which aims to align the distribution of selected subset with that of the entire dataset [36, 39, 27]. Most AL approaches are difficult to perform well under extremely low-label settings. This may be because they usually require randomly selected samples to begin with and involve human annotators in iterative cycles of labelling and training, leading to substantial labelling overhead. Model-Free Subsampling Subsampling is a statistical approach which selects a subset with size m as a surrogate for the full dataset with size n ≫m. While model-based subsampling methods depend heavily on the model assumptions [1, 61], improper choice of the model could lead to bad performance of estimation and prediction. In that case, model-free subsampling is preferred in data-driven modelling tasks, as it does not depend on the model assumptions. There are mainly two kinds of popular model-free subsampling methods. The one is induced by minimizing statistical discrepancies, which forces the distribution of subset to be similar to that of full data, in other words, selects representative subsamples, such as Wasserstein distance [13], energy distance [28], uniform design [65], maximum mean discrepancy [7] and generalized empirical F-discrepancy [66]. The other tends to select a diverse subset containing as many informative samples as possible [54]. The above-mentioned methodologies either exclusively focus on representativeness or diversity, which are difficult to effectively apply to SSL. 3 Problem Setup Let X be the unlabeled data space, Y be the label space, Xn = {xi}i∈[n] ⊂X be the full unlabeled dataset containing pairwise different data, and Im = {i1, i2, · · · , im} ⊂[n](m < n) be an index set contained in [n], our goal is to find an index set I∗ m = {i∗ 1, i∗ 2, · · · , i∗ m} ⊂[n](m < n) such that the selected set of samples XI∗ m = {xi∗ 1, xi∗ 2, · · · , xi∗m} is the most informative. After that, we can get access to the true labels of selected samples and use the set of labelled data S = {(xi, yi)}i∈I∗ m and the rest of the unlabeled data to train a deep learning model. 3 Following the methodology of previous works, we use representativeness and diversity as criteria for evaluating the informativeness of selected samples. Representativeness ensures the selected samples distribute similarly to the full unlabeled dataset. Diversity is proposed to prevent an excessive concentration of selected samples in high-density areas of the full unlabeled dataset. Furthermore, the cluster assumption in SSL suggests that the data tend to form discrete clusters, in which boundary points are likely to be located in the low-density area. Therefore, under this assumption, selected samples with diversity contain more boundary points than the non-diversified ones, which is desired in training classifiers. As a result, our goal can be formulated by solving the following problem: max Im⊂[n] Rep(XIm, Xn) + λDiv(XIm, Xn), (1) where Rep(XIm, Xn) and Div(XIm, Xn) quantify the representativeness and diversity of selected samples respectively and λ is a hyperparameter to balance the trade-off representativeness and diversity. Besides, we propose another two fundamental settings which are beneficial to the implementation of the framework: (1) Low-budget learning. The budget for many of the real-world tasks which require sample selection procedures is relatively low compared to the size of unlabeled data. Therefore, we set m/n ≤0.2 in default in the following context, including the analysis of the sampling algorithm and the experiments; (2) Sampling without Replacement. Compared with the setting of sampling with replacement, sampling without replacement offers several benefits which better match our tasks, including bias and variance reduction, precision increase and representativeness enhancement [25, 46]. 4 Representative and Diversity Sample Selection The Representative and Diverse Sample Selection (RDSS) framework consists of two steps: (1) Quantification. We quantify the representativeness and diversity of selected samples by a novel concept called α-MMD (6), where λ is replaced by α as the trade-off hyperparameter; (2) Optimization. We optimize α-MMD by GKHR algorithm to obtain the optimally selected samples XI∗ m. 4.1 Quantification of Diversity and Representativeness In classical statistics and machine learning problems, the inner product of data points x, y ∈X, defined by ⟨x, y⟩, is employed to as a similarity measure between x, y. However, the application of linear functions can be very restrictive in real-world problems. In contrast, kernel methods use kernel functions k(x, y), including Gaussian kernels (RBF), Laplacian kernels and polynomial kernels, as non-linear similarity measures between x, y, which are actually inner products of the projections of k(x, y) in some high-dimensional feature space [29]. Let k(·, ·) be a kernel function on X × X, and we employ k(·, ·) to measure the similarity between any two points and the average similarity, denoted by Sk(XIm) = 1 m2 X i∈Im X j∈Im k (xi, xj) , (2) to measure the similarity between the selected samples. Obviously, S(XIm) can evaluate the diversity of XIm since larger similarity implies smaller diversity. As a statistical discrepancy which measures the distance between distributions, the maximum mean discrepancy (MMD) is introduced here to quantify the representativeness of XIm to Xn. Proposed by Gretton et al. [14], MMD is formally defined below: Definition 4.1 (Maximum Mean Discrepancy). Let P, Q be two Borel probability measures on X. Suppose f is sampled from the unit ball in a reproducing kernel Hilbert space (RKHS) H associated with its reproducing kernel k(·, ·), i.e., ∥f∥H ≤1, then the MMD between P and Q is defined by MMD2 k(P, Q) := sup ∥f∥H≤1 Z fdP − Z fdQ 2 = E [k (X, X′) + k (Y, Y ′) −2k(X, Y )] , (3) where X, X′ ∼P and Y, Y ′ ∼Q are independent copies. 4 We can next derive the empirical version for MMD that is able to measure the representativeness of XIm = {xi}i∈Im relative to Xn = {xi}n i=1 by replacing P, Q with the empirical distribution constructed by XIm, Xn in (3): MMD2 k(XIm, Xn) := 1 n2 n X i=1 n X j=1 k (xi, xj)+ 1 m2 X i∈Im X j∈Im k (xi, xj)−2 mn n X i=1 X j∈Im k (xi, xj) . (4) Optimization objective. Set Rep(·, ·) = −MMD2 k(·, ·) and Div(·) = −Sk(·) in (1), where k is a proper kernel function, our optimization objective becomes min Im⊂[n] MMD2 k(XIm, Xn) + λSk(XIm). (5) Set λ = 1−α αm , since Pn i=1 Pn j=1 k (xi, xj) is a constant, the objective function in (5) can be rewritten by α MMD2 k(XIm, Xn) + 1 −α m Sk(XIm) + α(α −1) n2 n X i=1 n X j=1 k (xi, xj) =α2 n2 n X i=1 n X j=1 k (xi, xj) + 1 m2 X i∈Im X j∈Im k (xi, xj) −2α mn n X i=1 X j∈Im k (xi, xj) = sup ∥f∥H≤1  1 m X i∈Im f(xi) −α n n X j=1 f(xj)   2 , (6) which defines a new concept called α-MMD, denoted by MMDk,α(XIm, Xn). This new concept distinguishes our method from those existing methods, which is essential for developing the sampling algorithms and theoretical analysis. Note that α-MMD degenerates to classical MMD when α = 1 and degenerates to average similarity when α = 0. As α decreases, λ increases, thereby encouraging the diversity for sample selection. Remark 1. In the following context, all the kernels are assumed to be characteristic and positive definite if not specified. The following illustrates the advantages of the two properties. Characteristics kernels. The MMD is generally a pseudo-metric on the space of all Borel probability distributions, implying that the MMD between two different distributions can be zero. Nevertheless, MMD becomes a proper metric when k is a characteristic kernel, i.e., P → R X k(·, x)dP for any Borel probability distribution P on X [29]. Therefore, MMD induced by characteristic kernels can be more appropriate for measuring representativeness. Positive definite kernels. Aronszajn [2] showed that for every positive definite kernel k(·, ·), i.e., its Gram matrix is always positive definite and symmetric, it uniquely determines an RKHS H and vice versa. This property is not only important for evaluating the property of MMD [43] but also required in optimizing MMD [32] by Frank-Wolfe algorithm. 4.2 Sampling Algorithm In the previous research [36, 27, 50, 38, 58], sample selection is usually modelled by a nonconvex combinatorial optimization problem. In contrast, following the idea of [4], we regard minIm∈[n] MMD2 k,α(XIm, Xn) as a convex optimization problem by exploiting the convexity of α-MMD, and then solve it by a fast iterative minimization procedure derived from Frank-Wolfe algorithm (see Appendix A for derivation details): xi∗ p+1 ∈arg min i∈[n] fI∗ p(xi), I∗ p+1 ←I∗ p ∪{i∗ p+1}, I0 = ∅, (7) where fIp(xi) = P j∈Ip k (xi, xj) −αp Pn l=1 k(xi, xl). As an extension of kernel herding [7], its corresponding algorithm (see Algorithm 2) is called Generalized Kernel Herding (GKH). Note that fIp(xi) is iteratively updated in Algorithm 2, which can save a lot of running time. However, GKH can select repeated samples that contradict the setting of sampling without replacement. To address this issue, we propose a modified iterating formula based on (7): xi∗ p+1 ∈arg min i∈[n]\I∗ p fI∗ p(xi), I∗ p+1 ←I∗ p ∪{i∗ p+1}, I∗ 0 = ∅, (8) 5 Algorithm 1 Generalized Kernel Herding without Replacement Require: Data set Xn = {x1, · · · , xn} ⊂X; the number of selected samples m < n; a positive definite, characteristic and radial kernel k(·, ·) on X × X; trade-off parameter α ≤1. Ensure: Selected samples XI∗ m = {xi∗ 1, · · · , xi∗m}. 1: For each xi ∈Xn calculate µ(xi) := Pn j=1 k(xj, xi)/n. 2: Set β1 = 1, S0 = 0, I = ∅. 3: for p ∈{1, · · · , m} do 4: i∗ p ∈arg mini∈[n]\I∗ p Sp−1(xi) −αµ(xi) 5: For all i ∈[n]\I∗ p, update Sp(xi) = (1 −βp)Sp−1(xi) + βpk(xi∗ p, xi) 6: I∗ p+1 ←I∗ p ∪{i∗ p}, p ←p + 1, set βp = 1/p. 7: end for which admits no repetitiveness in the selected samples. Its corresponding algorithm (see Algorithm 1) is thereby named as Generalized Kernel Herding without Replacement (GKHR), employed as the sampling algorithm for RDSS. Computational complexity. Despite the time cost for calculating kernel functions, the computational complexity of GKHR is O(mn), since in each iteration, the steps in lines 4 and 5 of Algorithm 2 respectively require O(n) computations. Note that GKH has the same order of computational complexity as GKHR. 5 Theoretical Analysis 5.1 Generalization Bounds Recall the core-set approach in [36], i.e., for any h ∈H, R(h) ≤bRS(h) + \|R(h) −bRT (h)\| + \| bRT (h) −bRS(h)\|, where T is the full labeled dataset and S ⊂T is the core set, R(h) is the expected risk of h, bRT (h), bRS(h) are empirical risk of h on T, S. The first term bRS(h) is unknown before we label the selected samples, and the second term \|R(h) −bRT (h)\| can be upper bounded by the so-called generalization bounds [3, 64] which do not depend on the choice of core set. Therefore, to control the upper bound of R(h), we only need to analyse the upper bound of the third term \| bRT (h) −bRS(h)\| called core-set loss, which requires several mild assumptions. Shalit, et al. [37] derived a MMDtype upper bound for \| bRT (h) −bRS(h)\| to estimate individual treatment effect, while our bound is generalized to a wider range of tasks. Let H1 = {h\|h : X →Y} be a hypothesis set in which we are going to select a predictor and suppose that the labelled data T = {(xi, yi)}n i=1 are i.i.d. sampled from a random vector (X, Y ) defined on X × Y. We firstly assume that H1 is an RKHS, which is mild in machine learning theory [3, 5]. Assumption 5.1. H1 is an RKHS associated with bounded positive definite kernel k1 where the norm of any h ∈H1 is bounded by Kh. We further make RKHS assumptions on the functional space of E(Y \|X) and Var(Y \|X) that are fundamental in the field of conditional distribution embedding [41, 43]. Assumption 5.2. There is an RKHS H2 associated with bounded positive definite kernel k2 such that E(Y \|X) ∈H2 and the norm of any E(Y \|X) is bounded by Km. Assumption 5.3. There is an RKHS H3 associated with bounded positive definite kernel k3 such that Var(Y \|X) ∈H3 and the norm of any Var(Y \|X) is bounded by Ks. We next give a α-MMD-type upper bound for the core-set loss by the following theorem: Theorem 5.4. Take k = k2 1 + k1k2 + k3, then under assumptions 1-3, for any selected samples S ⊂T, there exists a positive constant Kc such that the following inequality holds: \| bRT (h) −bRS(h)\| ≤Kc(MMDk,α(XS, XT ) + (1 −α) √ K)2, where 0 ≤α ≤1, 0 ≤maxx∈X k(x, x) = K and XS, XT are projections of S, T on X. 6 Therefore, minimizing α-MMD can optimize the generalization bound for R(h) and benefit the generalizability of the trained model (predictor). 5.2 Finite-Sample-Error-Bound for GKHR The concept of convergence does not apply to analyzing GKHR. With n fixed, GKHR iterates for at most n times and then returns XI∗ n = Xn. Consequently, we analyze the performance of GKHR by its finite-sample-error bound. Previous to that, we make an assumption on the mean of fI∗ p over the full unlabeled dataset. Assumption 5.5. For any I∗ p returned by GKHR, 1 ≤p ≤m −1, there exists p + 1 elements {xjl}p+1 l=1 in Xn such that fI∗ p(xj1) ≤· · · fI∗ p(xjp+1) ≤ Pn i=1 fI∗ p(xi) n . When m is not relatively small, this assumption is rather unrealistic. Nevertheless, under our lowbudget setting, especially when m ≪n, the assumption becomes an extension of the principle that "the minimum is never larger than the mean", which still probably makes sense. We can then show that the decaying rate for optimization error of GKHR can be upper bounded by O(log m/m): Theorem 5.6. Let XI∗ m be the samples selected by GKHR, under assumption 4, it holds that MMD2 k,α XI∗ m, Xn ≤C2 α + B 2 + log m m + 1 (9) where B = 2K, 0 ≤maxx∈X k(x, x) = K, C2 α = (1 −α)2K where K is defined in Lemma B.6. 6 Choice of Kernel and Hyperparameter Tuning In this section, we make some suggestions for choosing the kernel and tuning the hyperparameter α. Choice of kernel. Recall Remark 1 in Section 4.1, we only consider characteristic and positive definite kernels in RDSS. Since the Gaussian kernels are the most commonly used kernels in the field of machine learning and statistics [3, 15], we introduce Gaussian kernel as our choice, which is defined by k(x, y) = exp(−∥x −y∥2 2/σ2). The bandwidth parameter σ is set to be the median distance between samples in the aggregate dataset [15], i.e., σ = Median({∥x −y∥2\|x, y ∈Xn}), since the median is robust and also compromises between extreme cases. Tuning trade-off hyperparameter α. According to Theorem 5.6 and Lemma B.3, by straightforward deduction we have MMDk XI∗ m, Xn ≤Cα + O r log m m ! + (1 −α) √ K to upper bound the MMD between the selected samples and the full dataset under a low-budget setting. We can just set α ∈[1 − 1 √m, 1) so that the upper bound of the MMD would not be larger than the one of α-MMD in the perspective of the order of magnitude. 7 Experiments In this section, we first explain the implementation details of our method RDSS in Section 7.1. Next, we compare RDSS with other sampling methods by integrating them into two state-of-the-art (SOTA) SSL approaches (FlexMatch [63] and Freematch [53]) on five datasets (CIFAR-10/100, SVHN, STL-10 and ImageNet-1k) in Section 7.2. The details of the datasets, the visualization results and the computational complexity of different sampling methods are shown in Appendix D.2, D.3, and D.4, respectively. We also compare against various AL/SSAL approaches in Section 7.3. Lastly, we make quantitative analyses of the trade-off parameter α in Section 7.4. 7 7.1 Implementation Details of Our Method First, we leverage the pre-trained image feature extraction capabilities of CLIP [33], a vision transformer architecture, to extract features. Subsequently, the [CLS] token features produced by the model’s final output are employed for sample selection. During the sample selection phase, the Gaussian kernel function is chosen as the kernel method to compute the similarity of samples in an infinite-dimensional feature space. The value of σ for the Gaussian kernel function is set as explained in Section 6. To ensure diversity in the sampled data, we introduce a penalty factor given by α = 1 − 1 √m, where m denotes the number of selected samples. Concretely, we set m = {40, 250, 4000} for CIFAR-10, m = {400, 2500, 10000} for CIFAR-100, m = {250, 1000} for SVHN, m = {40, 250} for STL-10 and m = {100000} for ImageNet. Next, the selected samples are used for two SSL approaches, which are trained and evaluated on the datasets using the codebase Unified SSL Benchmark (USB) [52]. The optimizer for all experiments is standard stochastic gradient descent (SGD) with a momentum of 0.9 [44]. The initial learning rate is 0.03 with a learning rate decay of 0.0005. We use ResNet-50 [16] for the ImageNet experiment and Wide ResNet-28-2 [62] for other datasets. Finally, we evaluate the performance with the Top-1 classification accuracy metric on the test set. Experiments are run on 8NVIDIA Tesla A100 (40 GB) and 2Intel 6248R 24-Core Processor. We average our results over five independent runs. 7.2 Comparison with Other Sampling Methods Main results We apply RDSS on Flexmatch and Freematch to compare with the following three baselines and two SOTA methods in SSL under different annotation budget settings. The baselines conclude Stratified, Random and k-Means, while the two SOTA methods are USL [50] and ActiveFT [57]. The results are shown on Table 1 from which we have several observations: (1) Our proposed RDSS achieves the highest accuracy, outperforming other sampling methods, which underscores the effectiveness of our approach; (2) USL attains suboptimal results under most budget settings yet exhibits a significant gap compared to RDSS, particularly under severely constrained ones. For instance, FreeMatch achieves a 4.95% rise on the STL-10 with a budget of 40; (3) In most experiments, RDSS either approaches or surpasses the performance of stratified sampling, especially on SVHN and STL-10. However, the stratified sampling method is practically infeasible given that the category labels of the data are not known a priori. Results on ImageNet We also compare the second-best method USL with RDSS on ImageNet. Following the settings of FreeMatch [53], we select 100k samples for annotation. FreeMatch, using RDSS and USL as sampling methods, achieves 58.24% and 56.86% accuracy, respectively, demonstrating a substantial enhancement in the performance of our method over the USL approach. Table 1: Comparison with other sampling methods. Due to stratified sampling limitations, the results are marked in grey. Top and second-best performances are bolded and underlined, respectively, excluding stratified sampling. Metrics represent mean accuracy and standard deviation over five independent runs. Dataset CIFAR-10 CIFAR-100 SVHN STL-10 Budget 40 250 4000 400 2500 10000 250 1000 40 250 Applied to FlexMatch [63] Stratified 91.45±3.41 95.10±0.25 95.63±0.24 50.23±0.41 67.38±0.45 73.61±0.43 89.60±1.86 93.66±0.49 75.33±3.74 92.29±0.64 Random 87.30±4.61 93.95±0.91 95.17±0.59 45.58±0.97 66.48±0.98 72.61±0.83 87.67±1.16 94.06±1.14 65.81±1.21 90.70±0.79 k-Means 81.23±8.71 94.59±0.51 95.09±0.65 41.60±1.24 65.99±0.57 71.53±0.42 90.28±0.69 93.82±1.04 55.43±0.39 90.64±1.05 USL [50] 91.73±0.13 94.89±0.20 95.43±0.15 46.89±0.46 66.75±0.37 72.53±0.32 90.03±0.63 93.10±0.78 75.65±0.60 90.77±0.36 ActiveFT [57] 70.87±4.14 93.85±1.37 95.31±0.75 25.69±0.64 57.19±2.06 70.96±0.75 89.32±1.87 92.53±0.43 55.57±1.42 87.28±1.19 RDSS (Ours) 94.69±0.28 95.21±0.47 95.71±0.10 48.12±0.36 67.27±0.55 73.21±0.29 91.70±0.39 95.70±0.35 77.96±0.52 93.16±0.41 Applied to FreeMatch [53] Stratified 95.05±0.15 95.40±0.23 95.80±0.29 51.29±0.56 67.69±0.58 73.90±0.53 92.58±1.05 94.22±0.78 79.16±5.01 91.36±0.18 Random 93.41±1.24 93.98±0.91 95.56±0.17 47.16±1.25 66.09±1.08 72.09±0.99 91.62±1.88 94.40±1.28 76.66±2.43 90.72±0.97 k-Means 88.05±5.07 94.80±0.48 95.51±0.37 44.07±1.94 66.09±0.39 71.69±0.72 93.30±0.46 94.68±0.72 63.22±4.92 89.99±0.87 USL [50] 93.81±0.62 95.19±0.18 95.78±0.29 47.07±0.78 66.92±0.33 72.59±0.36 93.36±0.53 94.44±0.44 76.95±0.86 90.58±0.58 ActiveFT [57] 78.13±2.87 94.54±0.81 95.33±0.53 26.67±0.46 56.23±0.85 71.20±0.68 92.60±0.51 93.71±0.54 63.31±2.99 86.60±0.30 RDSS (Ours) 95.05±0.13 95.50±0.20 95.98±0.28 48.41±0.59 67.40±0.23 73.13±0.19 94.54±0.46 95.83±0.37 81.90±1.72 92.22±0.40 7.3 Comparison with AL/SSAL Approaches First, we compare RDSS against various traditional AL approaches on CIFAR-10/100. AL approaches conclude CoreSet [36], VAAL [39], LearnLoss [60] and MCDAL [8]. For a fair comparison, we 8 exclusively use samples selected by RDSS for supervised learning compared to other AL approaches, considering that AL relies solely on labelled samples for supervised learning. The implementation details are shown in Appendix D.5. The experimental results are presented in Table 2, from which we observe that RDSS achieves the highest accuracy under almost all budget settings when relying solely on labelled data for supervised learning, with notable improvements on CIFAR-100. Second, we compare RDSS with sampling methods used in SSAL when applied to the same SSL framework (i.e., FlexMatch or FreeMatch) on CIFAR-10. The sampling methods conclude CoreSetSSL [36], MMA [42], CBSSAL [12], and TOD-Semi [17]. In detail, we tune recent SSAL approaches with their public implementations and run experiments under an extremely low-budget setting, i.e., 40 samples in a 20-random-and-20-selected setting. Table 3 illustrates that the performance of most SSAL approaches falls below that of random sampling methods under extremely low-budget settings. This inefficiency stems from the dependency of sample selection on model performance within the SSAL framework, which struggles when the model is weak. Our model-free method, in contrast, selects samples before training, avoiding these pitfalls. Table 2: Comparison with AL approaches under Supervised Learning (SL) paradigm. The best performance is bold and the second best performance is underlined. Dataset CIFAR-10 CIFAR-100 Budget 7500 10000 7500 10000 CoreSet 85.46 87.56 47.17 53.06 VAAL 86.82 88.97 47.02 53.99 LearnLoss 85.49 87.06 47.81 54.02 MCDAL 87.24 89.40 49.34 54.14 SL+RDSS (Ours) 87.18 89.77 50.13 56.04 Whole Dataset 95.62 78.83 Table 3: Comparison with SSAL approaches. The green (red) arrow represents the improvement (decrease) compared to the random sampling method. Method FlexMatch FreeMatch Stratified 91.45 95.05 Random 87.30 93.41 CoreSetSSL 87.66 ↑0.36 91.24 ↓2.17 MMA 74.61 ↓12.69 87.37 ↓6.04 CBSSAL 86.58 ↓0.72 91.68 ↓1.73 TOD-Semi 86.21 ↓1.09 90.77 ↓2.64 RDSS (Ours) 94.69 ↑7.39 95.05 ↑1.64 Third, we directly compare RDSS with the above AL/SSAL approaches when applied to SSL, which may better reflect the paradigm differences. The experimental results and analysis are in the Appendix D.6. 7.4 Trade-off Parameter α We analyze the effect of different α with Freematch on CIFAR-10/100. The results are presented in Table 4, from which we have several observations: (1) Our proposed RDSS achieves the highest accuracy under all budget conditions, surpassing those that employ a fixed value; (2) The α that achieve the best or the second best performance are within the interval we set, which is in line with our theoretical derivation in Section 6; (3) The experimental outcomes exhibit varying degrees of reduction compared to our approach when the representativeness or diversity term is removed. Table 4: Effect of different α. The grey results indicate that the α is outside the interval we set in Section 6, i.e., α < 1 −1/√m, while the black results indicate that the α is within the interval we set, i.e., 1 −1/√m ≤α ≤1. Among them, α = 0 and α = 1 indicate the removal of the representativeness and diversity terms, respectively. The best performance is bold, and the secondbest performance is underlined. Dataset CIFAR-10 CIFAR-100 Budget (m) 40 250 4000 400 2500 10000 0 85.54±0.48 93.55±0.34 94.58±0.27 39.26±0.52 63.77±0.26 71.90±0.17 0.40 92.28±0.24 93.68±0.13 94.95±0.12 42.56±0.47 65.88±0.24 71.71±0.29 0.80 94.42±0.49 94.94±0.37 95.15±0.35 45.62±0.35 66.87±0.20 72.45±0.23 0.90 94.33±0.28 95.03±0.21 95.20±0.42 48.12±0.50 67.14±0.16 72.15±0.23 0.95 94.44±0.64 95.07±0.26 95.45±0.38 48.41±0.59 67.11±0.29 72.80±0.35 0.98 94.51±0.39 95.02±0.15 95.31±0.44 48.33±0.54 67.40±0.23 72.68±0.22 1 94.53±0.42 95.01±0.23 95.54±0.25 48.18±0.36 67.20±0.29 73.05±0.18 1 −1/√m (Ours) 95.05±0.13 95.50±0.20 95.98±0.28 48.41±0.59 67.40±0.23 73.13±0.19 9 8 Conclusion In this work, we propose a model-free sampling method, RDSS, to select a subset from unlabeled data for annotation in SSL. The primary innovation of our approach lies in the introduction of α-MMD, designed to evaluate the representativeness and diversity of selected samples. 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An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation. Journal of Computational and Graphical Statistics, 32(1):329–339, 2023. [66] M. Zhang, Y. Zhou, Z. Zhou, and A. Zhang. Model-free subsampling method based on uniform designs. IEEE Transactions on Knowledge and Data Engineering, 2023. 13 A Algorithms A.1 Derivation of Generalized Kernel Herding (GKH) Proof. The proof technique is borrowed from [32]. Let us firstly define a weighted modification of α-MMD. For any w ∈Rn such that w⊤1 = 1, the weighted α-MMD is defined by MMD2 k,α,Xn(w) = w⊤Kw −2αw⊤p + α2K, where K = [k(xi, xj)]1≤i,j≤n, K = 1⊤K1/n2, p = (e⊤ 1 K1/n, · · · , e⊤ n K1/n), {ei}n i=1 is the set of standard basis of Rn. It is obvious that for any Ip ⊂[n], MMD2 k,α,Xn(wp) = MMD2 k,α(XIp, Xn), where (wp)i = 1/p if i ∈Ip, and (wp)i = 0 if not. Therefore, weighted α-MMD is indeed a generalization of α-MMD. Let K∗= K −2αp1⊤+ α2K11⊤ we obtain the quadratic form expression of weighted α-MMD by MMD2 k,α,Xn(w) = w⊤K∗w, where K∗is strictly positive definite if the unlabeled data are pairwise different, w ̸= wn and k is a characteristic kernel according to [32]. Recall our low-budget setting (so w ̸= wn holds) and assumption for kernel, K∗is indeed a strictly positive definite matrix. Thus MMD2 k,α,Xn is a convex functional w.r.t. w, leading to the fact that minw⊤1=1 MMD2 k,α,Xn(w) can be solved by Frank-Wolfe algorithm. Then for 1 ≤p < n, sp ∈arg min s⊤1=1 s⊤(Kwp −αp) = arg min ei,i∈[n] e⊤ i (Kwp −αp). Let ei∗p = sp, under uniform step size in Frank-Wolfe algorithm, we have wp+1 = p p + 1 wp + 1 p + 1ei∗p, w0 = 0 as the update formula of Frank-Wolfe algorithm, which is equivalent to i∗ p ∈arg min i∈[n] X j∈Ip k(xi, xj) −αp n X l=1 k(xi, xl). then we can immediately derive the iterating formula in (7). A.2 Pseudo Codes Algorithm 2 Generalized Kernel Herding Require: Data set Xn = {x1, · · · , xn} ⊂X; the number of selected samples m < n; a positive definite, characteristic and radial kernel k(·, ·) on X × X; trade-off parameter α ≤1. Ensure: selected samples XI∗ m = {xi∗ 1, · · · , xi∗m}. 1: For each xi ∈Xn calculate µ(xi) := Pn j=1 k(xj, xi)/n. 2: Set β1 = 1, S0 = 0, I = ∅. 3: for p ∈{1, · · · , m} do 4: i∗ p ∈arg mini∈[n] Sp−1(xi) −αµ(xi) 5: For all i ∈[n], update Sp(xi) = (1 −βp)Sp−1(xi) + βpk(xi∗p, xi) 6: I∗ p+1 ←I∗ p ∪{i∗ p}, p ←p + 1, set βp = 1/p. 7: end for B Technical Lemmas Lemma B.1 (Lemma 2 [32]). Let (tk)k and (αk)k be two real positive sequences and A be a strictly positive real. If tk satisfies t1 ≤A and tk+1 ≤(1 −αk+1) tk + Aα2 k+1, k ≥1, with αk = 1/k for all k, then tk < A(2 + log k)/(k + 1) for all k > 1. 14 Lemma B.2. The selected samples XI∗ m generated by GKH (Algorithm 2) satisfies MMD2 k,α XI∗ m, Xn ≤M 2 α + B 2 + log m m + 1 (10) where B = 2K, 0 ≤maxx∈X k(x, x) ≤K, M 2 α is defined by M 2 α := min w⊤1=1,w≥0 MMD2 k,α,Xn (w) Proof. Following the notations in Appendix A, let pα = αp, we could straightly follow the proof for finite-sample-size error bound of kernel herding with predefined step sizes given by [32] to derive Lemma B.2, without any other technique. The detailed proof is omitted. Lemma B.3. Let H be an RKHS over X associated with positive definite kernel k, and 0 ≤ maxx∈X k(x, x) ≤K. Let Xm = {xi}m i=1, Yn = {yj}m j=1, xi, yj ∈X. Then for any α ≤1, \| MMDk,α(Xm, Yn) −MMDk(Xm, Yn)\| ≤(1 −α) √ K Proof. \|MMDk,α(Xm, Yn) −MMDk(Xm, Yn)\| = sup ∥f∥H≤1  1 m m X i=1 f (xi) −α n n X j=1 f (yj)  − sup ∥f∥H≤1  1 m m X i=1 f (xi) −1 n n X j=1 f (yj)   ≤ sup ∥f∥H≤1 1 −α n n X i=1 f (yi) = 1 −α n sup ∥f∥H≤1 n X i=1 f (yi) = 1 −α n sup ∥f∥H≤1 n X j=1 ⟨f, k(·, yj)⟩H ≤ 1 −α n sup ∥f∥H≤1 n X j=1 ⟨f, k(·, yj)⟩H ≤ 1 −α n sup ∥f∥H≤1 n X j=1 ∥f∥H∥k(·, yj)∥H ≤(1 −α) √ K. Lemma B.4 (Proposition 12.31 [48]). Suppose that H1 and H2 are reproducing kernel Hilbert spaces of real-valued functions with domains X1 and X2, and equipped with kernels k1 and k2, respectively. Then the tensor product space H = H1 ⊗H2 is an RKHS of real-valued functions with domain X1 × X2, and with kernel function k ((x1, x2) , (x′ 1, x′ 2)) = k1 (x1, x′ 1) k2 (x2, x′ 2) . Lemma B.5 (Theorem 5.7 [31]). Let f ∈H1 and g ∈H2, where H1, H2 be two RKHS containing real-valued functions on X, which is associated with positive definite kernel k1, k2 and canonical feature map ϕ1, ϕ2, then for any x ∈X, f(x) + g(x) = ⟨f, ϕ1(x)⟩H1 + ⟨g, ϕ2(x)⟩H2 = ⟨f + g, (ϕ1 + ϕ2)(x)⟩H1+H2 , where H1 + H2 = {f1 + f2\|fi ∈Hi} and ϕ1 + ϕ2 is the canonical feature map of H1 + H2. Furthermore, ∥f + g∥2 H1+H2 ≤∥f∥2 H1 + ∥g∥2 H2. Lemma B.6. For any unlabeled dataset Xn ⊂X and any subset XIm, MMD2 k,α(Xn, Xn) = (1 −α)2K, MMD2 k,α(XIm, Xn) ≤(1 + α2)K, where K = Pn i=1 Pn j=1 k(xi, xj)/n2, K = maxx∈X k(x, x). Lemma B.6 is directly derived from the definition of α-MMD. 15 C Proof of Theorems Proof for Theorem 5.4. The proof borrows the technique introduced in [37] for decomposing the expected risk of hypotheses. Firstly, let us denote that H4 = H1 ⊗H1 + H1 ⊗H2 + H3, with kernel k4 = k2 1 + k1k2 + k3 and canonical feature map ϕ4 = ϕ1 ⊗ϕ1 + ϕ1 ⊗ϕ2 + ϕ3. Under the assumptions in Theorem 5.4, according to Theorem 4 in [41], we have for any x ∈X, h(x) = ⟨h, ϕ1(x)⟩H1 , E[Y \|x] = ⟨E[Y \|X], ϕ2(x)⟩H2 , Var(Y \|x) = ⟨Var(Y \|X), ϕ3(x)⟩H3 where ϕ1, ϕ2, ϕ3 are canonical feature maps in H1, H2, H3. Denote that m = E[Y \|X] and s = Var(Y \|X). Now by definition, R(h) = E [ℓ(h(x), y)] = Z X Z Y ℓ(h(x), y)p(y\|x)p(x)dxdy = Z X f(x)p(x)dx where f(x) = Z Y (y −h(x))2p(y\|x)dy = Var(Y \|x) −2h(x)E[Y \|x] + h2(x) = ⟨s, ϕ3(x)⟩H3 −2 ⟨h, ϕ1(x)⟩H1 ⟨m, ϕ2(x)⟩H2 + ⟨h, ϕ1(x)⟩H1 ⟨h, ϕ1(x)⟩H1 = ⟨s, ϕ3(x)⟩H3 −⟨2h ⊗m, (ϕ1 ⊗ϕ2)(x)⟩H1⊗H2 + ⟨h ⊗h, (ϕ1 ⊗ϕ1)(x)⟩H1⊗H1 = ⟨s −2h ⊗m + h ⊗h, ϕ4(x)⟩H4 where the fourth equality holds by Lemma B.4 and the last equality holds by Lemma B.5, then f ∈H4, and ∥f∥H4 = ∥s −2h ⊗m + h ⊗h∥H4 ≤∥s∥H4 + ∥2h ⊗m∥H4 + ∥h ⊗h∥H4 ≤∥s∥H3 + 2∥m∥H2∥h∥H1 + ∥h ⊗h∥H1⊗H1 = ∥s∥H3 + 2∥m∥H2∥h∥H1 + ∥h∥2 H1 ≤K2 h + 2KhKm + Ks where the second inequality holds by Lemma B.5. Therefore, let β = 1/(K2 h + 2KhKm + Ks) we have ∥βf∥H4 = β∥f∥H4 ≤1. Then bRT (h) −bRS(h) = Z X f(x)dPT (x) − Z X f(x)dPS(x) =(K2 h + 2KhKm + Ks) Z X βf(x)dPT (x) − Z X βf(x)dPS(x) ≤(K2 h + 2KhKm + Ks) sup ∥f∥H4≤1 Z X f(x)dPT (x) − Z X f(x)dPS(x) =(K2 h + 2KhKm + Ks) MMDk4(XS, XT ) where PT denotes the empirical distribution constructed by XT , so does PS. Recall Lemma B.3, we have Theorem 5.4. Proof for Theorem 5.6. Following the notations in Appendix A, we further define w∗= 1/n, C2 α = MMD2 k,α,Xn(w∗) = (1 −α)2K (11) bw = arg min 1⊤w=1 MMD2 k,α,Xn(w) = α K−1 −K−111⊤K−1 1⊤K−11 p + K−11 1⊤K−11 16 Let pα = αp, we have (pα −Kbw) ∝1. Define ∆α(w) := MMD2 k,α,Xn(w) −C2 α = bg(w) −bg(w∗) where bg(w) = (w −bw)⊤K (w −bw). The related details for proving the equality are omitted, since they are completely given by the proof of alternative expression of MMD in Pronzato [32]. By the convexity of bg(·), for j = arg mini∈[n]\I∗ p fI∗ p(xi), bg (w∗) ≥bg (wp) + 2 (w∗−wp)⊤K (wp −bw) ≥bg (wp) + 2 min j∈[n]\I∗ p (ej −wp)⊤K (wp −bw) where the second inequality holds with the assumption in Theorem 5.6 (w∗−ej)⊤K (wp −bw) = (w∗−ej)⊤(Kwp −pα) = Pn i=1 fI∗ p(xi) n −fI∗ p(xjp+1) ≥ Pn i=1 fI∗ p(xi) n −fI∗ p(xj) ≥0 therefore, we have for B = 2K, ∆α(wp+1) =bg (wp) −bg (w∗) + 2 p + 1 (ej −wp)⊤K (wp −bw) + 1 (p + 1)2 (ej −wp)⊤K (ej −wp) = p p + 1(bg (wp) −bg (w∗)) + 1 (p + 1)2 B = p p + 1∆α(wp) + 1 (p + 1)2 B (12) where wp+1 = pwp/(p+1)+ej/(p+1), and obviously B upper bounds (ej −wp)⊤K (ej −wp). Since α ≤1, it holds from Lemma B.6 that ∆α(w1) ≤MMD2 k,α,Xn(w1) ≤(1 + α2)K ≤B therefore by Lemma B.1, we have MMD2 k,α(XI∗ m, Xn) = MMD2 k,α,Xn(wp) ≤C2 α + B 2 + log m m + 1 D Additional Experimental Details and Results D.1 Supplementary Numerical Experiments on GKHR Consider the fact that GKH is a convergent algorithm (Lemma B.2) and the finite-sample-size error bound (10) holds without any assumption on the data, we conduct some numerical experiments to empirically compare GKHR with GKH on datasets generated by four different distributions on R2. Firstly, we define four distributions on R2: 1. Gaussian mixture model 1 which consists of four Gaussian distributions G1, G2, G3, G4 with mixture weights [0.95, 0.01, 0.02, 0.02], 2. Gaussian mixture model 2 which consists of four Gaussian distributions G1, G2, G3, G4 with mixture weights [0.3, 0.2, 0.15, 0.35], 3. Uniform distribution 1 which consists of a uniform distribution defined in a circle with radius 0.5, and a uniform distribution defined in a annulus with inner radius 4 and outer radius 6, 4. Uniform distribution 2 defined on [−10, 10]2. where G1 = N 1 2 , 2 0 0 5 , G2 = N −3 −5 , 1 0 0 2 G3 = N −5 4 , 8 0 0 6 , G4 = N 15 10 , 4 0 0 9 17 (a) n=1000 (b) n=3000 (c) n=10000 (d) n=30000 Figure 2: The performance comparison between GKHR and GKH with different m, n over ten independent runs. The blue line is the mean value of D, the red dotted line over (under) the blue line is the mean value of D plus (minus) its standard deviation, and the pink area is the area between the upper and lower red dotted lines. 18 To consistently evaluate the performance gap between GKHR and GKH at the same order of magnitude, we propose the following criterion D = D1 −D2 D1 + D2 where D1 = MMD2 k,α(X(1) I∗ m, Xn), D2 = MMD2 k,α(X(2) I∗ m, Xn), X(1) Im is the selected samples from GKHR and X(2) Im is the selected samples from GKH. Positive value of D implies that GKH outperforms GKHR, and negative values of D implies that GKHR outperforms GKH. Large absolute value of D shows large performance gap. The experiments are conducted as follows. We generate 1000,3000,10000,30000 random samples from the four distributions separately, then use GKHR and GKH for sample selection under the low-budget setting, i.e., m/n ≤0.2. The α is set by m/n. We report the results over ten independent runs in Figure 2, which shows that although the performance gap tends to grow as m grows, when m is relatively small, the performance of GKHR is similar to that of GKH. Therefore, under the low-budget setting, GKHR and GKH have similar performance on minimizing α-MMD over various type of distributions, which convinces us that GKHR could work well in the sample selection task. D.2 Datasets For experiments, we choose five common datasets: CIFAR-10/100, SVHN, STL-10 and ImageNet. CIFAR-10 and CIFAR-100 contain 60,000 images with 10 and 100 categories, respectively, among which 50,000 images are for training, and 10,000 images are for testing; SVHN contains 73,257 images for training and 26,032 images for testing; STL-10 contains 5,000 images for training, 8,000 images for testing and 100,000 unlabeled images as extra training data. ImageNet spans 1,000 object classes and contains 1,281,167 training and 100,000 test images. The training sets of the above datasets are considered as the unlabeled dataset for sample selection. D.3 Visualization of Selected Samples To offer a more intuitive comparison between various sampling methods, we visualized samples chosen by stratified, random, k-Means, USL, ActiveFT and RDSS (ours). We generate 5000 samples from a Gaussian mixture model defined on R2 with 10 components and uniform mixture weights. One hundred samples are selected from the entire dataset using different sampling methods. The visualisation results in Figure 3 indicate that our selected samples distribute more similarly with the entire dataset than other counterparts. D.4 Computational Complexity and Running Time We compute the time complexity of various sampling methods and recorded the time required to select 400 samples on the CIFAR-100 dataset for each method. The results are presented in Table 5, where m represents the annotation budget, n denotes the total number of samples, and T indicates the number of iterations. The sampling time was obtained by averaging the duration of three independent runs of the sampling code on an idle server without any workload. As illustrated by the results, the sampling efficiency of our method surpasses that of all other methods except for random and stratified sampling. This discrepancy is likely because the execution time of other algorithms is affected by the number of iterations T. Table 5: Efficiency comparison with other sampling methods. Method Time complexity Time (s) Random O(n) ≈0 Stratified O(n) ≈0 k-means O(mnT) 579.97 USL O(mnT) 257.68 ActiveFT O(mnT) 224.35 RDSS (Ours) O(mn) 132.77 19 (a) Stratified (b) Random (f) RDSS (ours) (e) ActiveFT (d) USL (c) K-Means Figure 3: Visualization of selected samples using different sampling methods. Points of different colours represent samples from different classes, while black points indicate the selected samples. D.5 Implementation Details of Supervised Learning Experiments We use ResNet-18 [16] as the classification model for all AL approaches and our method. Specifically, We train the models for 300 epochs using SGD optimizer (initial learning rate=0.1, weight decay=5e− 4, momentum=0.9) with batch size 128. Finally, we evaluate the performance with the Top-1 classification accuracy metric on the test set. D.6 Direct Comparison with AL/SSAL The comparative results with AL/SSAL approaches are shown in Figure 4 and Figure 5, respectively. The specific values corresponding to the comparative results in the above two figures are shown in Table 6. And the above results are from [8], [12] and [17]. Table 6: Comparative results with AL/SSAL approaches. Dataset CIFAR-10 CIFAR-100 Budget 40 250 500 1000 2000 4000 5000 7500 10000 400 2500 5000 7500 10000 Active Learning (AL) CoreSet [36] 80.56 85.46 87.56 37.36 47.17 53.06 VAAL [39] 81.02 86.82 88.97 38.46 47.02 53.99 LearnLoss [60] 81.74 85.49 87.06 36.12 47.81 54.02 MCDAL [8] 81.01 87.24 89.40 38.90 49.34 54.14 Semi-Supervised Active Learning (SSAL) CoreSetSSL [36] 90.94 92.34 93.30 94.02 63.14 66.29 68.63 CBSSAL [12] 91.84 92.93 93.78 94.55 63.73 67.14 69.34 TOD-Semi [17] 79.54 87.82 90.3 36.97 52.87 58.64 Semi-Supervised Learning (SSL) with RDSS FlexMatch+RDSS (Ours) 94.69 95.21 95.71 48.12 67.27 73.21 FreeMatch+RDSS (Ours) 95.05 95.50 95.98 48.41 67.40 73.13 According to the results, we have several observations: (1) AL approaches often necessitate significantly larger labelling budgets, exceeding RDSS by 125 or more on CIFAR-10. This is primarily because AL paradigms are solely dependent on labelled samples not only for classification but also for feature learning. (2) SSAL and our methods leverage unlabeled samples, surpassing traditional AL approaches. However, this may not directly reflect the advantages of RDSS, as such performance enhancements could be inherently attributed to the SSL paradigm itself. Nonetheless, these experimental outcomes offer insightful implications: SSL may represent a more promising paradigm under scenarios with limited annotation budgets. 20 0 2000 4000 6000 8000 10000 Budget 80.0 82.5 85.0 87.5 90.0 92.5 95.0 Accuracy CoreSet VAAL LearnLoss MCDAL CoreSetSSL CBSSAL TOD-Semi FlexMatch+RDSS (Ours) FreeMatch+RDSS (Ours) Figure 4: Comparison with AL/SSAL approaches on CIFAR-10. 0 2000 4000 6000 8000 10000 Budget 35 40 45 50 55 60 65 70 75 Accuracy CoreSet VAAL LearnLoss MCDAL CoreSetSSL CBSSAL TOD-Semi FlexMatch+RDSS (Ours) FreeMatch+RDSS (Ours) Figure 5: Comparison with AL/SSAL approaches on CIFAR-100. E Limitation The choice of α depends on the number of full unlabeled data points, independent of the information on the shape of data distribution. This may lead to a loss of effectiveness of RDSS on those datasets with complicated distribution structures. However, it outperforms fixed-ratio approaches on the datasets under different budget settings. F Potential Societal Impact Positive societal impact. Our method ensures the representativeness and diversity of the selected samples and significantly improves the performance of SSL methods, especially under low-budget settings. This reduces the cost and time of data annotation and is particularly beneficial for resourceconstrained research and development environments, such as medical image analysis. Negative societal impact. When selecting representative data for analysis and annotation, the processing of sensitive data may be involved, increasing the risk of data leakage, especially in sensitive fields such as medical care and finance. It is worth noting that most algorithms applied in these sensitive areas are subject to this risk. 21 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We summarize our contributions in the last paragraph of Section 1. The RDSS framework, including the quantification method and sample algorithm, is illustrated in Section 4. Theoretical analysis on RDSS is presented in Section 5. Section 6 suggests the choice for kernel and tuning parameters. The comparison results with other methods are shown in Section 7.2 and Section 7.3. And we analyze the effect of different α in Section 7.4. 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Reviewers will be specifically instructed to not penalize honesty concerning limitations. 3. Theory Assumptions and Proofs 22 Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes] Justification: For the generalization bound in Section 5.1, the required assumptions are Assumption 5.1, 5.2 and 5.3, the proof is given by the Proof of Theorem 5.4 in Appendix C. For the finite-sample-error bound in Section 5.2, the required assumption is Assumption 5.5, the proof is given by the Proof of Theorem 5.6 in Appendix C. Other technical lemmas and their proofs or references are presented in Appendix B. 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4,451	DEX: Data Channel Extension for Efficient CNN Inference on Tiny AI Accelerators Taesik Gong12∗ Fahim Kawsar13 Chulhong Min1 1Nokia Bell Labs 2UNIST 3University of Glasgow taesik.gong@unist.ac.kr {fahim.kawsar, chulhong.min}@nokia-bell-labs.com Abstract Tiny machine learning (TinyML) aims to run ML models on small devices and is increasingly favored for its enhanced privacy, reduced latency, and low cost. Recently, the advent of tiny AI accelerators has revolutionized the TinyML field by significantly enhancing hardware processing power. These accelerators, equipped with multiple parallel processors and dedicated per-processor memory instances, offer substantial performance improvements over traditional microcontroller units (MCUs). However, their limited data memory often necessitates downsampling input images, resulting in accuracy degradation. To address this challenge, we propose Data channel EXtension (DEX), a novel approach for efficient CNN execution on tiny AI accelerators. DEX incorporates additional spatial information from original images into input images through patch-wise even sampling and channel-wise stacking, effectively extending data across input channels. By leveraging underutilized processors and data memory for channel extension, DEX facilitates parallel execution without increasing inference latency. Our evaluation with four models and four datasets on tiny AI accelerators demonstrates that this simple idea improves accuracy on average by 3.5%p while keeping the inference latency the same on the AI accelerator. The source code is available at https://github.com/Nokia-Bell-Labs/data-channel-extension. 1 Introduction Tiny machine learning (TinyML) is an active research field focused on developing and deploying machine learning models on extremely resource-constrained devices, such as microcontroller units (MCUs) and small IoT sensors. Compared to cloud-based AI, TinyML on devices offers benefits in privacy preservation, low latency, and low cost. While research efforts in TinyML, such as model compression techniques [15, 17, 25, 27, 31, 32], have successfully reduced the size of AI models to fit into memory-constrained MCUs, the fundamental limitation in the processing capability of MCUs leads to long inference latency. This limitation hinders the widespread adoption of on-device AI, especially for real-time applications. Recently, the advent of tiny AI accelerators like the Analog Devices MAX78000 [34] and Google Coral Micro [8] has revolutionized the TinyML field by dramatically boosting the model inference speed and leading a new phase of on-device AI. For instance, the MAX78000 AI accelerator [34] achieves 170× faster inference latency compared to an MCU processor (MAX32650 [33]). To enable such acceleration, these tiny AI accelerators introduce several hardware optimization techniques. They often feature multiple convolutional processors (e.g., 64 processors in MAX78000 [34]) ∗This work was done entirely while the author was affiliated with Nokia Bell Labs. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 64 Parallel Processors Pooling Engine Caching Conv Engine Data memory (512KB) Weight memory (432KB) Figure 1: The architecture of a tiny AI accelerator (MAX78000 [34]). MAX78000 MAX32650 STM32F7 0 500 1000 1500 2000 Latency (ms) 2 13 350 1760 123 754 KWS FaceID MAX78000 MAX32650 STM32F7 0 200 400 Energy (mJ) 0.14 0.4 83.7 42.1 47.5 464 KWS FaceID Figure 2: Comparison between an AI accelerator (MAX78000) and MCUs (MAX32650 and STM32F7). and parallelize per-channel CNN operations across these processors. For further optimization, the memory architecture allows each processor to have a dedicated memory instance, i.e., per-processor memory instance. This design enables simultaneous memory access to multiple channels from different processors. While these hardware-level optimizations bring significant performance improvements, we found that they also have several constraints at the expense of the optimizations. First, the per-processor memory architecture highly restricts the supported input image size because the data memory each processor can use for its input/output channels is limited to the capacity of its dedicated memory instance, which is a fraction of the total data memory divided by the number of processors. Consequently, most vision models for these accelerators are designed to support very small images, such as 32×32 pixels. Given that images captured by cameras are often generated with higher resolutions, downsampling is inevitable, leading to accuracy degradation due to information loss from the original image. Second, we found that processors and data memory are underutilized for the input layer due to the per-processor memory architecture; since input images typically have a low number of channels (e.g., RGB three channels), only a limited number of processors tied to memory instances are utilized while the remaining processors remain idle. For instance, on the MAX78000, 61 of 64 processors and per-processor memory instances remain unused in the first layer. In this work, we propose a novel approach, Data channel EXtension (DEX), to overcome these constraints while still benefiting from the acceleration power of tiny AI accelerators. The core idea is to boost accuracy by extending the data channels to incorporate additional image information into unused data memory instances and processors, instead of simple downsampling. Owing to the parallel processing and memory access capabilities of tiny AI accelerators, our method can achieve this accuracy improvement without compromising inference latency. Specifically, DEX involves two procedures: (1) pair-wise even sampling, where pixels from the original image are evenly sampled, and (2) channel-wise stacking, which arranges these samples across multiple channels. To measure the impact of DEX on accuracy and resource utilization, we conducted experiments on the MAX78000 [34] and MAX78002 [37] tiny AI accelerator platforms. DEX was evaluated on four models, SimpleNet [16], WideNet [16], EfficientNetV2 [48], and MobileNetV2 [45], using four vision datasets: ImageNette [18], Caltech101 [11], Caltech256 [14], and Food101 [2]. Our results show that DEX improves average accuracy by 3.5%p compared to the original model with downsampling and 3.6%p compared to the existing coordinate augmentation approach (CoordConv [29]), without increasing inference latency. Additionally, DEX maximizes data memory and processor utilization, demonstrating its effectiveness in enhancing model performance on resource-constrained devices. In summary, DEX can significantly enhance the performance of neural networks on tiny AI accelerators, leading to more efficient and effective deployment of AI on resource-constrained devices. 2 Preliminary: tiny AI accelerators The advent of tiny AI accelerators marks a pivotal shift towards on-device AI, greatly enhancing privacy and reducing latency. While a number of tiny-scale AI accelerators have emerged recently, such as Analog Devices MAX78000/MAX78002 [34, 37], Google Coral Micro [8], and GreenWaves GAP-8/GAP-9 [12], only a few are commercially available with access and control over their operations. In this paper, we focus on the MAX78000 [34] and MAX78002 [37] as our primary platforms since they are the most widely used tiny AI accelerator research platforms [1, 6, 13, 39, 40, 43] owing to the disclosed hardware details and open-source tools, enabling in-depth analysis and modification of their operations. 2 Conv 1 Conv 2 3 channels 32 channels 64 channels 𝑃𝑟! 𝑃𝑟"# 𝑃𝑟$ 𝑃𝑟% 𝑃𝑟%$ 𝑃𝑟%% 𝑃𝑟# 𝑃𝑟! 𝑃𝑟"# 𝑃𝑟$ 𝑃𝑟% 𝑃𝑟%$ 𝑃𝑟%% 𝑃𝑟# 𝑃𝑟! 𝑃𝑟"# 𝑃𝑟$ 𝑃𝑟% 𝑃𝑟%$ 𝑃𝑟%% 𝑃𝑟# Conv 3 Input in use Figure 3: Processor utilization with varying input channels on the AI accelerator. Architecture of tiny AI accelerators. The distinctive characteristic of tiny AI accelerators compared to conventional microcontroller units (MCUs) is parallel processors that parallelize per-channel CNN operations across these processors. Figure 1 depicts an abstracted architecture of the MAX78000; MAX78002 has a similar architecture to MAX78000 with increased memory (1.3 MB data and 2 MB weight memory). Further details are in Appendix A.1. It has 64 parallel convolutional processors, each capable of performing specific operations independently. To maximize performance, each processor has a dedicated memory instance, i.e., per-processor memory instance that optimizes data transfer with parallel access. For each CNN layer, operations on individual channels are assigned to separate convolutional processors and executed simultaneously, significantly reducing latency typically associated with convolutional algorithms. Each processor has a pooling engine, an input cache, and a convolution engine that can handle 3 by 3 kernels. The CNN accelerator includes 512 KB of data memory and 432 KB of weight storage memory. Within the 512 KB of data memory, an 8 KB per-processor memory instance is allocated to each of the 64 processors. Figure 3 shows the utilization of the processors (Pri) for executing CNNs with varying sizes of the input channels. Each processor communicates with a dedicated memory instance for each data channel. For example, given a three-channel image, three parallel processors are utilized in the first layer. Performance gain over MCUs. A recent benchmark study [35] demonstrates the remarkable performance gain of the MAX78000 in terms of latency and energy consumption. Figure 2 shows that the MAX78000 significantly outperforms widely-used MCUs (MAX32650 with a Cortex-M4 at 120 MHz [33], and a high-performance MCU, the STM32F7 with a Cortex-M7 at 216 MHz [47]) for face detection (FaceID) and keyword spotting (KWS). For KWS, latency is drastically reduced to only 2.0 ms, compared to 350 ms for the MAX32650 and 123 ms for the STM32F7. Accordingly, energy efficiency of the MAX78000 is also significant; it consumes only 0.40 mJ for FaceID, dramatically less than the 42.1 mJ and 464 mJ required by the MAX32650 and STM32F7, respectively. 3 DEX: Data channel extension for efficient CNN inference on AI accelerators 3.1 Constraints of per-processor memory instances in tiny AI accelerators for images As mentioned in §2, tiny AI accelerators leverage per-processor memory instances for faster data transfer with parallel access. However, we disclose that this causes several constraints at the expense of rapid data access: (1) low image resolution and (2) underutilized processors and data memory. Low image resolution due to limited per-processor memory size. MAX78000 [34] has 512 KB data memory which is divided into 64 segments of 8 KB memory instances per processor, each storing the data of each input channel. This memory architecture highly restricts the supported input resolution. For instance, an input image with a shape 3 × 224 × 224 (channel, height, and weight), which is a typical size of ImageNet [9], does not fit the MAX78000 even with Q7 format (one byte for each value), as memory limit for each channel is 8 KB (224 × 224 ∼50 KB > 8 KB). Thus, the current practice on tiny AI accelerators is to shrink the resolution of input images by downsampling and accordingly, to design small models to process lower-resolution images, e.g., 3 × 32 × 32. However, with this, it loses most of the information of the original image, which might lead to sub-optimal performance. Underutilized processors and data memory for the input layer. Although per-processor memory instances allow simultaneous memory access from different processors, it also brings inefficiency in data memory and processor utilization, especially in the input layer. Specifically, given an input image I with the number of channels CI, height HI, and width WI (e.g., 3 × 224 × 224) as shown in Figure 4(a), Figure 4(b) illustrates the downsampled image with the number of channels CI, height HO, and width WO (e.g., 3 × 32 × 32), and its data memory usage in the AI accelerator. 3 !! "! !" "" #! !" "" #" (b) Downsampling (a) Original image $ (c) DEX exceeds data memory limit #! Data memory Idle Data memory Processors Processors Different samples !"! !"& !"$ !"% !"# !"' !"" !"! !"& !"$ !"% !"# !"' !"" Figure 4: Comparison among different input data. (a) an original image that exceeds the data memory limit of the AI accelerator, (b) a downsampled image that fits the data memory but does not fully utilize parallel processors and data memory, and (c) a DEX-generated image that incorporates more information from original image by extending data across channels with full utilization of parallel processors and data memory instances. !! "! #! !" "" #" Patch &#$ !!! !!" !!# !!$ !"! '#$ "%!" #%!" &#$ $ = 0 $ = 1 $ = 2 $ = 3 Patch-wise even sampling (e.g., * = 4) '#$ (% !&!" , (% +, !&!" Channel-wise stacking (e.g., ,& = 12) Original image $ Output image ' Figure 5: Overview of DEX. DEX divides the original image I into multiple patches. DEX then evenly samples pixels from each patch Pij and constructs an output pixel Oij by stacking samples across channels. With three RGB channels, channel data are separately stored for each data memory instances for parallel execution. As there is N processors and corresponding data memory instances, it leaves the remaining N −3 processors and data memory instances idle. This provides an opportunity to utilize these idle data memory instances and parallel processors, which we detail in the following section. 3.2 DEX procedure As aforementioned, we note two key observations: (1) the input image needs to be downsampled due to the limited memory of tiny AI accelerators, which means most of the pixel information cannot be utilized, and (2) there exist idle data memory instances and processors that could process up to N channels in parallel. Although several recent studies have found efficient model architectures on tiny AI accelerators [39, 43], existing studies lack considerations on this inefficiency for input image processing in CNNs (further discussion on related work is in Appendix 5). Based on our observations, we propose Data channel EXtension (DEX) for efficient CNN execution on tiny AI accelerators. The key intuition behind DEX is that we can utilize the remaining data memory to incorporate additional information from the original image into neural networks by extending the input data across channels. By utilizing this additional memory and processors, we can incorporate extra sample information for feature learning without sacrificing latency. Figure 4(c) shows the input data reshaped via DEX, where each channel contains different pixel information from the original image. With DEX extending data across channels (from CI to CO), it can fully utilize the data memory and associated parallel processors. Figure 5 shows an overview of the procedure of DEX. Given an input image I with a number of channels CI, height HI, and width WI, DEX generates an output image O with an extended number of channels CO, height HO, and width WO (e.g., 64 × 32 × 32) via patch-wise even sampling and channel-wise stacking. Patch-wise even sampling. The purpose of patch-wise even sampling is to select samples evenly spaced across the original image while keeping the spatial relationship among pixels. We first define 4 a patch from the original image in which a corresponding output pixel is generated. We denote i-th row and j-th column of patch Pij in I as: Pij = I i · HI HO : (i + 1) · HI HO , j · WI WO : (j + 1) · WI WO , (1) where [:, :] refers to a 2-D array slicing operation, specifying the selection of rows and columns sequentially. The number of patches is determined by the resolution of the output image, i.e., HO × WO. For each patch Pij, we generate the corresponding output data Oij. This ensures that the spatial relationships among pixels in the input image are preserved in the output, maintaining spatial consistency throughout the process. The next step is to sample pixels within the patch considering the memory budget. Specifically, we define K = ⌈CO CI ⌉as the number of samples to be selected in each patch. Given the height and width of patch HPij = j (i + 1) · HI HO k − j i · HI HO k and WPij = j (i + 1) · WI WO k − j i · WI WO k , the i-th row and j-th column of output Oij can be represented by: Oij = Pij lk WPij , lk mod WPij \| lk = k · HPij · WPij −1 K −1 , for k = 0, 1, . . . , K −1 , (2) which means a collection of evenly distributed samples within each patch to encourage diverse information while minimizing the use of localized pixel information. With patch-wise even sampling, selected samples are evenly distributed both across patches and within each patch. Channel-wise stacking. Channel-wise stacking arranges sampled data across multiple channels and keeps this procedure for all pixels to maintain data integrity. Channel-wise stacking is beneficial as it maintains consistency within each channel, preserving the spatial and contextual relationships of the sampled data. Specifically, after patch-wise even sampling, the samples are stacked across the channel axis in ascending order of the index k, and this is repeated for each Oij. Note that lk = 0 when K = 1, and this is identical to traditional downsampling. If K > CO CI , it fills up the target channel with P’s data until the limit and discards the remaining channels. For instance, when using RGB channels (CI = 3) and if CO = 64 and K = 22, it takes only the red channel for i = 21 and discards the remaining green and blue channels that exceed the channel limit of 64. Algorithm 1 provides the pseudo-code that describes the procedure of DEX’s channel extension algorithm. Algorithm 1 DEX Channel Extension Algorithm Input: Source image I in a shape (CI, HI, WI) Output: Reshaped image O in a shape (CO, HO, WO) 1: O ←zeros(CO, HO, WO) 2: for i ←0 to HO −1 do 3: for j ←0 to WO −1 do 4: start_row, end_row ←floor(i · HI HO ), floor((i + 1) · HI HO ) 5: start_col, end_col ←floor(j · WI WO ), floor((j + 1) · WI WO ) 6: Pij ←I[:, start_row : end_row, start_col : end_col] ▷Patch Pij of I 7: K ←ceil( CO CI ) ▷Number of samples to be selected in Pij 8: for k ←0 to K −1 do ▷Get channels of Oij from Pij 9: HPij ←end_row −start_row 10: WPij ←end_col −start_col 11: lk = k · floor( HPij ·WPij −1 K−1 ) 12: O[k · CI : (k + 1) · CI, i, j] = Pij[:, floor( lk WPij ), lk mod WPij] 13: return O 3.3 Further analysis on DEX 5 -'_)+ , !" "" #" ⋆ Σ -%-./-0_1#2, #" ⋆ Σ ⋆ Σ -'_)*+ , Conv Image ! Kernels 1st layer output #! #! Added Added Figure 6: The initial CNN layer’s operation with DEX. Understanding how DEX leads to performance improvement. DEX’s ability to incorporate additional pixel information from the original image can improve the accuracy of CNNs. The extended channels provide further samples of adjacent areas in the original image, significantly broadening the receptive fields of features in the initial CNN layer. This expansion allows the model to detect more complex and subtle features early in the processing pipeline, which is critical for the nuanced understanding and interpretation of visual data. Specifically, Figure 6 visualizes how the first CNN layer operates with DEX, where L1 kernel_size and L1 c_out refer to the kernel size and the output channel size of the first layer, respectively. It illustrates the application of the convolution operation across each enhanced channel (CO as opposed to CI), where distinct kernel weights are applied to each channel. This ensures that the additional information is integrated into the output feature maps, thereby enriching the model’s feature extraction capabilities. The convolutional layer processes the increased channel input, which is reflected in weight sums that construct output channels. Impact of channel extension on the number of parameters. Given the first CNN layer’s kernel size L1 kernel_size and the first layer’s channel output size L1 c_out, the number of parameters required for the input layer can be calculated as OC · L1 kernel_size · L1 c_out. If OC is 3, it is the same as the traditional downsampling without our channel extension. Note that this channel extension does not incur additional inference latency on the AI accelerator. We found that the channel extension increases ∼3% of the total parameters as we show in our experiment §4.2. The rest of the layers remain the same. In addition to its simplicity, we have several reasons to change the first layer only, which we discuss further in §6. Utilization of the original image information. With traditional downsampling, the utilization of the original input is HO·WO HI·WI , but with DEX, this is extended to CO CI · HO·WO HI·WI . For instance, given a 3 × 256 × 256 input image, a downsampled image 3 × 32 × 32 utilizes only 1.6% of the original information, while with DEX and an output channel size CO = 64, it can utilize 33.3% of the original information. DEX can accommodate all the information when CO = CI · HI·WI HO·WO Maximum number of output channels. Increasing the number of output channels allows DEX to accommodate the original image information. The number of output channels denoted as OC, that can be extended without increasing latency on AI accelerators is limited by the number of data memory instances DN, i.e., OC < DN. For example, the MAX78000 has 64 data memory instances, allowing it to support up to OC = 64 output channels without affecting inference latency. 4 Evaluation 4.1 Experimental settings Here we explain experimental settings. Further details are in Appendix A. On-device testbed. We evaluated DEX on the off-the-shelf MAX78000 feather board [36] and MAX78002 Evaluation Kit [38], which are a development platform for the MAX78000 [34] and MAX78002 [37], respectively, as shown in Figure 9. In this paper, we select these accelerators because they provide open-source tools for thorough analysis and modification of their internal processes, making them the most widely used tiny AI accelerator research platforms [1, 6, 13, 39, 40, 43]. Model training and deployment. In our experiment, we use four models officially supported in the Analog Devices MAX78000/78002 Training framework [20]: SimpleNet [16], WideNet [16], 6 Table 1: Average classification accuracy (%) and corresponding standard deviations over three runs for each dataset and method. Bold type indicates those of the highest classification accuracy. Dataset Method SimpleNet WideNet EfficientNetV2 MobileNetV2 AVG (%) ImageNette Downsampling 57.8 ± 1.2 61.8 ± 0.2 51.3 ± 0.5 62.0 ± 0.7 58.2 CoordConv 58.0 ± 1.1 61.7 ± 0.2 51.9 ± 0.1 61.6 ± 0.3 58.3 CoordConv (r) 55.4 ± 1.5 61.4 ± 0.2 51.7 ± 1.0 61.2 ± 1.1 57.4 DEX (ours) 61.4 ± 0.6 65.6 ± 0.6 56.8 ± 0.5 64.4 ± 0.6 62.0 Caltech101 Downsampling 54.6 ± 2.1 55.8 ± 1.2 38.6 ± 0.9 51.4 ± 1.6 50.1 CoordConv 53.8 ± 1.6 56.5 ± 0.1 38.7 ± 0.2 49.8 ± 0.5 49.7 CoordConv (r) 52.7 ± 0.5 56.0 ± 1.7 38.2 ± 1.0 49.7 ± 1.2 49.1 DEX (ours) 56.9 ± 1.3 61.1 ± 1.4 45.9 ± 1.9 53.3 ± 1.7 54.3 Caltech256 Downsampling 19.8 ± 0.6 20.8 ± 0.5 14.7 ± 0.4 22.4 ± 1.0 19.4 CoordConv 19.8 ± 0.5 21.3 ± 0.8 14.8 ± 0.8 22.7 ± 0.8 19.6 CoordConv (r) 20.0 ± 1.6 20.9 ± 0.6 14.5 ± 0.3 22.7 ± 0.4 19.5 DEX (ours) 22.8 ± 0.5 22.9 ± 0.9 18.3 ± 0.9 26.3 ± 0.5 22.6 Food101 Downsampling 16.0 ± 0.4 17.7 ± 0.7 12.1 ± 0.2 22.4 ± 0.6 17.1 CoordConv 16.1 ± 0.8 17.7 ± 0.3 12.0 ± 0.1 21.7 ± 0.3 16.9 CoordConv (r) 16.3 ± 0.4 17.3 ± 0.6 12.0 ± 0.6 20.9 ± 0.3 16.6 DEX (ours) 18.4 ± 0.4 20.9 ± 0.4 16.4 ± 0.1 23.3 ± 1.1 19.8 EfficientNetV2 [48], and MobileNetV2 [45]. The supported models from the framework were trained via quantization-aware training with 8-bit integers in PyTorch [41]. We follow the official training configuration (details in Appendix A.2). The checkpoints are synthesized as embedded C codes for via the Analog Devices MAX78000/70002 Synthesis framework [19]. SimpleNet and WideNet are developed for MAX78000 while EfficientNetV2 and MobileNetV2 are for MAX78002 considering the size of the models. All models are originally designed to take 3 × 32 × 32 inputs, and DEX increases the number of the channels in the first layer to 64. Datasets. We evaluated on four common vision datasets: (1) ImageNette [18], a ten-class subset of ImageNet [9] with 9469/3925 train/test samples with the original image shape of 3 × 350 × 350, (2) Caltech101 [11] with 101 objects classes having 6941/1736 train/test samples with the original image shape of 3 × 300 × 300, (3) Caltech256 [14] with 256 objects classes having 23824/5956 train/test samples with the original image shape of 3 × 300 × 300, and (4) Food101 [2] with 101 food categories with 75750/25250 train/test samples with the original image shape of 3 × 512 × 512. Baselines. For baselines, we compare with the Downsampling method which is a straightforward way to reduce the size of the input under memory-constrained devices. It downsamples the input image to 3 × 32 × 32. In addition, we compare DEX with CoordConv [29] which pointed out the limitation of traditional CNNs that relied on RGB images for the coordinate transformation problem and introduced the augmentation of i and j coordinates, which improved object detection efficiency by using two extra channels. The authors of CoordConv also introduced the third channel for an r coordinate, where r = p (i −h/2)2 + (j −w/2)2, which they found effective in some experiments. 4.2 Result Overall accuracy. Table 1 shows the overall accuracy for four different datasets with the baselines and DEX. As shown, extending data channels to utilize additional input information improves accuracy in DEX. Specifically, DEX achieved 3.5%p higher accuracy compared to downsampling and 3.6% higher accuracy compared to CoordConv across datasets. CoordConv shows lower accuracy compared with downsampling (0.1%p degradation on average), showing they are not very effective solutions. This finding aligns with previous results indicating that CoordConv is useful for specific tasks such as object detection, where coordinate information is important [29]. We found CoordConv (r) has a similar pattern to CoordConv. Overall, DEX’s accuracy improvement shows the effectiveness of using extra information from the original image for feature learning. Resource usage. Table 2 compares the resource usage of the baseline and DEX. First, we found that, although DEX extends the number of channels in the first CNN layer to 64, its impact on the 7 Table 2: Model size (Size), utilization of the original image information (InfoRatio), accelerator’s processor utilization for the first layer (ProcUtil), and inference latency on the accelerator (Latency) for different models and methods averaged over three runs. Model Method InputChan Size (KB) InfoRatio (×) ProcUtil (%) Latency (µs) SimpleNet Downsampling 3 162.6 1.0 4.7 2592 ± 1 CoordConv 5 162.9 1.0 7.8 2592 ± 2 CoordConv (r) 6 163.0 1.0 9.4 2592 ± 2 DEX (ours) 64 171.2 21.3 100.0 2591 ± 1 WideNet Downsampling 3 306.4 1.0 4.7 3820 ± 1 CoordConv 5 306.9 1.0 7.8 3820 ± 0 CoordConv (r) 6 307.1 1.0 9.4 3819 ± 1 DEX (ours) 64 319.3 21.3 100.0 3818 ± 1 EfficientNetV2 Downsampling 3 742.4 1.0 4.7 11688 ± 2 CoordConv 5 743.0 1.0 7.8 11685 ± 3 CoordConv (r) 6 743.2 1.0 9.4 11689 ± 1 DEX (ours) 64 759.6 21.3 100.0 11690 ± 2 MobileNetV2 Downsampling 3 1317.8 1.0 4.7 3553 ± 4 CoordConv 5 1318.2 1.0 7.8 3554 ± 1 CoordConv (r) 6 1318.4 1.0 9.4 3554 ± 2 DEX (ours) 64 1330.7 21.3 100.0 3552 ± 3 3 6 18 36 64 Channel Size 55 60 65 Accuracy (%) ImageNette 3 6 18 36 64 Channel Size 40 45 50 55 60 Caltech101 3 6 18 36 64 Channel Size 15 20 25 Caltech256 3 6 18 36 64 Channel Size 15 20 25 Food101 SimpletNet WideNet EfficientNetV2 MobileNetV2 AVG Figure 7: Accuracy of DEX varying the channel size. The shaded areas are standard deviations. model size is negligible (an average increment of 3.2% compared to no channel extension). DEX utilizes 21.3× more image information compared to downsampling, which is the primary reason for the accuracy improvement. As expected, DEX does not increase on-device inference latency, even though it maximally utilizes the processors on the AI accelerators for information processing. This result is consistent across the four datasets, as all the models are designed to take the same input size in MAX78000 and MAX78002. Accuracy according to the channel size. We varied the size of the channels from 3 (downsampling) to 6, 18, 36, and 64 with DEX to understand the impact of the channel size in terms of accuracy. Figure 7 shows the accuracy variation according to the channel size across the four datasets. As shown, it seems that a higher number of channels increases accuracy in general. This means that selecting the highest channel size supported in AI accelerators might be an effective strategy in practice, considering that it does not incur the latency increase. Still, there are some cases where the accuracy of the highest channel size (64) is not the best among them. This means there might be an optimal number of channels tailored to a specific dataset and model architecture, which might be found in the model development process. Resource usage according to the channel size. We also measure resource usage varying the channel size. First, we measured the model size and inference latency as shown in Table 3. The model size increment is negligible and inference latency remains the same across different numbers of channels. The model size and inference latency are the same for the four datasets as all the models are designed to take the same input size in MAX78000 and MAX78002. Second, we measure the information utilization from the original image and processor utilization in the AI accelerators (Figure 8). 8 Table 3: Model size (Size) with relative increment (%) compared to the three channels and average inference latency on the accelerator (Latency) with standard deviations over three runs, varying the channel size. Model Chan = 3 Chan = 6 Chan = 18 Chan = 36 Chan = 64 SimpleNet 162.6 163.0 (+0.3%) 164.7 (+1.3%) 167.3 (+2.9%) 171.2 (+5.3%) WideNet 306.4 307.1 (+0.2%) 309.6 (+1.0%) 313.4 (+2.3%) 319.3 (+4.2%) EfficientNetV2 742.4 743.2 (+0.1%) 746.6 (+0.6%) 751.7 (+1.3%) 759.6 (+2.3%) Size (KB) MobileNetV2 1317.8 1318.4 (+0.0%) 1321.0 (+0.2%) 1324.8 (+0.5%) 1330.7 (+1.0%) SimpleNet 2592 ± 1 2592 ± 2 2591 ± 1 2590 ± 1 2591 ± 1 WideNet 3820 ± 1 3820 ± 2 3825 ± 1 3819 ± 3 3818 ± 1 EfficientNetV2 11688 ± 2 11691 ± 2 11692 ± 3 11691 ± 0 11690 ± 2 Latency (µs) MobileNetV2 3553 ± 4 3553 ± 1 3552 ± 1 3554 ± 0 3552 ± 3 36 18 36 64 Channel Size 0 10 20 Utilization (%) ImageNette CalTech101 CalTech256 Food101 (a) Information utilization. 36 18 36 64 Channel Size 0 25 50 75 100 Utilization (%) Proc Util (b) Processor utilization. Figure 8: Resource usage varying the channel size. The utilization of the original image information depends on the size of the original data size, which grows linearly according to the channel size. We found a correlation between information utilization rate and accuracy improvement. For example, Caltech101 and Caltech256 had utilization rates of 24.3%, improving accuracy by 4.2%p and 3.2%p, respectively, while Food101 had an 8.3% utilization rate with a 2.7%p accuracy improvement. The processor utilization linearly increases until 100% with 64 channels size, which is the number of parallel processing units in the evaluated platforms. Table 4: Comparison of data extension strategies. Method InputChan InfoRatio (×) Accuracy Downsampling 3 1.0 57.8 ± 1.2 Repetition 64 1.0 56.3 ± 0.8 Rotation 64 1.0 55.4 ± 0.7 Tile per channel 64 21.3 39.4 ± 0.7 Patch-wise seq. 64 21.3 61.0 ± 1.5 Patch-wise rand. 64 21.3 60.4 ± 1.0 DEX 64 21.3 61.4 ± 0.6 Comparison of alternative data extension strategies in DEX. To understand the effectiveness of our patch-wise even sampling and channel-wise stacking, we compared DEX with other possible data channel extension strategies. We compared with four strategies: repeating the same downsampled image across the channels (Repetition), generating slightly different images through rotation (Rotation), dividing the original image into multiple tiles and stacking those tiles across channels (Tile), patch-wise sequential sampling (Patch-wise seq.) that samples pixels sequentially within a patch, and patch-wise random sampling (Patch-wise rand.) that randomly samples within a patch. Further implementation details are in Appendix A.5. In this experiment, we used SimpleNet and evaluated it on ImageNette. Table 4 shows the results. Repetition does not improve accuracy over downsampling, indicating that merely increasing the number of kernels does not lead to performance gains. Rotation shows a slight decrease in accuracy compared to Repetition, which suggests that slight changes through rotation do not enhance performance. Interestingly, Tile shows low accuracy, demonstrating the importance of having a complete view of the original image in each channel, rather than focusing on specific regions. Both Patch-wise sequential and Patch-wise random samplings show lower accuracy than DEX’s patch-wise even sampling, highlighting the importance of even sampling for better performance. 5 Related work TinyML. Tiny Machine Learning (TinyML) is an emerging field that focuses on adapting machine learning techniques for highly resource-constrained devices, such as microcontroller units (MCUs). These devices often come with limited memory, typically hundreds of kilobytes of SRAM. Research in this area has mostly concentrated on reducing model size through various compression techniques, such as model pruning [15, 17, 25, 27, 31, 32], model quantization [7, 15, 42, 44, 49, 53, 54], and neural architecture search (NAS) [4, 5, 10, 24]. In addition, several studies have explored the 9 efficient utilization of memory resources (e.g., SRAM). Examples include optimizing on-device training processes [23, 28] and designing memory-efficient neural architectures [26, 52]. Unlike these approaches that primarily target MCUs, our research utilizes the distinctive architecture of tiny AI accelerators to enhance both memory efficiency and overall performance. Tiny AI accelerators. Several studies have leveraged tiny AI accelerators for small-scale on-device AI applications. For instance, TinyissimoYOLO [39] offers a quantized, memory-efficient, and ultra-lightweight object detection network, showcasing its effectiveness on the MAX78000 platform. Additionally, KP2DTiny [43] introduces a quantized neural keypoint detector and descriptor specifically optimized for MAX78000 and Coral AI accelerators. Moreover, Synergy represents a multi-device collaborative inference platform across wearables equipped with tiny AI accelerators [13]. Another line of studies utilized tiny AI accelerators in battery-free or intermittent computing scenarios [1, 6]. Traditionally, hardware accelerators on low-power AI platforms were capable of only one-shot atomic executions of a neural network inference without intermediate result backups. A study proposed a toolchain to address this that allows neural networks to execute intermittently on the MAX78000 platform [6]. To the best of our knowledge, there has been no work that manipulates data and models to efficiently utilize computing resources considering the unique architecture of tiny AI accelerators. Image channel extension in CNNs. Several studies have explored augmenting images with additional information to construct multi-channel inputs for Convolutional Neural Networks (CNNs). Liu et al. proposed a multi-modality image fusion approach, combining visible, mid-wave infrared, and motion images for enhanced object detection [30], while Wang et al. presented depth-aware CNN for image segmentation [50]. These approaches require extra sensing channels to acquire data, such as infrared cameras and depth cameras. Similarly, other research has incorporated location data to improve performance for segmentation [51] and object detection tasks [29]. For instance, CoordConv [29] pointed out the limitation of traditional CNNs that relied solely on RGB images for the coordinate transformation problem and introduced the augmentation of i and j coordinates, which improved object detection efficiency. However, these methodologies often necessitate additional sensor modalities or are tailored for specific applications such as object detection, which restricts their general use. Nevertheless, adapting findings from those studies within DEX could be an interesting future direction. 6 Discussion and conclusion We introduced DEX, a novel method to enhance CNN efficiency on tiny AI accelerators by augmenting input data across unused memory. Evaluations on four image datasets and models showed that DEX improves accuracy without increasing inference latency. This method maximizes the processing and memory capabilities of tiny AI accelerators, making it a promising solution for efficient AI model execution on resource-constrained devices. Limitations and potential societal impacts. We modified only the initial CNN layer due to simplicity, effectiveness, and memory constraints. The first layer, representing image data in three channels (RGB), has the most unused processors after initial data assignment. Extending channels at the first layer significantly increases data utilization with minimal impact on model size. This approach aligns with the design of weight memory in tiny AI accelerators, which maximizes model capacity by collective use across processors. We think DEX might be less effective in certain tasks where incorporating more pixel information is not beneficial. In those cases, alternative data extension strategies might be used instead of patch-wise even sampling to utilize the additional channel budget. While our focus was on small models supported by the MAX78000/MAX78002 platforms, evaluating larger models could be valuable, given rapid AI hardware advancements. 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All data storing and model inference is done only in the AI Accelerator part (MAX78000 and MAX78002). Table 5: Comparison of MAX78000 and MAX78002. Component MAX78000 [34] MAX78002 [37] MCU Processor Arm Cortex-M4 (100 MHz), RISC-V Arm Cortex-M4 (120 MHz), RISC-V Flash Memory 512 KB 2.5 MB SRAM 128 KB 384 KB CNN Processor 64 parallel CNN processors 64 parallel CNN processors Data Memory 512 KB 1.3 MB Weight Memory 432 KB 2 MB Bias Memory 2 KB 8 KB In this paper, we focus on the MAX78000 [34] and MAX78002 [37] as our primary platforms since they are the most popular research platforms [1, 6, 13, 39, 40, 43] owing to the disclosed hardware details and open-source tools, enabling in-depth analysis and modification of their operations. Figure 9 shows our testbed. Note that all operations required for the model inference are done under the AI accelerator part (highlighted with the red boxes), while the entire boards are bigger for development purposes. For on-device deployment and measurement, we used MAX78000 for SimpleNet and WideNet and MAX78002 [37] for EfficientNetV2 and MobileNetV2, following the memory requirement. For the sake of explanation, we assumed each processor is mapped to one memory instance in this paper, although MAX78000/MAX78002 group four data memory instances together to communicate with four processors in reality. Our experiments were conducted on the actual implementations. Table 5 compares MAX78000 and MAX78002. A.2 Model training 15 We followed the official model training code for MAX78000 and MAX78002 platforms [20]. Here, we detail the hyperparameters used in the official training code. We trained models with NVIDIA A40 GPUs. For all models, quantization-aware training is conducted with support for batch normalization after convolutional layers through batch normalization fusing [21]. This fusing operation integrates the effects of batch normalization directly into the parameters of the preceding convolutional layer by adjusting the weights and bias values. Consequently, after the fusing/folding process, the network no longer contains any batch normalization layers. Instead, the effects of batch normalization are reflected in the modified weights and biases of the preceding convolutional layers. SimpleNet. SimpleNet [16] was trained for 300 epochs using the Adam optimizer [22] with an initial learning rate of 0.001 and a batch size of 32. A multi-step learning rate scheduler was used with milestones set at epochs 100, 150, and 200, and a multiplicative factor of learning rate decay value of 0.25. Quantization-aware training (QAT) was introduced starting at epoch 240. During QAT, a shift quantile of 0.985 was applied to manage activation ranges. The weight precision was primarily set to 2 bits. However, exceptions were made for certain layers: the 1st convolutional layer utilized 8-bit weights, while the 2nd, 11th, 12th, 13th, and 14th convolutional layers used 4-bit weights. WideNet. WideNet [16] was trained for 300 epochs using the Adam optimizer [22] with an initial learning rate of 0.001 and a batch size of 100. A multi-step learning rate scheduler was used with milestones set at epochs 100, 150, and 200, and a multiplicative factor of learning rate decay value of 0.25. Quantization-aware training (QAT) was introduced starting at epoch 240. During QAT, a shift quantile of 0.985 was applied to manage activation ranges. The weight precision was primarily set to 2 bits. However, exceptions were made for certain layers: the 1st convolutional layer utilized 8-bit weights, while the 2nd, 11th, 12th, 13th, and 14th convolutional layers used 4-bit weights. EfficientNetV2. EfficientNetV2 [48] was trained for 300 epochs using the Adam optimizer [22] with an initial learning rate of 0.001 and a batch size of 100. A multi-step learning rate scheduler was used with milestones set at epochs 50, 100, 150, 200, and 250 and a multiplicative factor of learning rate decay value of 0.5. Quantization-aware training (QAT) was introduced starting at epoch 210. During QAT, a shift quantile of 0.995 was applied to manage activation ranges. The weight precision was primarily set to 8 bits. MobileNetV2. MobileNetV2 [45] was trained for 300 epochs using the stochastic gradient descent optimizer (SGD) [3] with an initial learning rate of 0.1 and a batch size of 128. A multi-step learning rate scheduler was used with milestones set at epochs 100, 150, 175, and 250 and a multiplicative factor of learning rate decay value of 0.235. Quantization-aware training (QAT) was introduced starting at epoch 200. During QAT, a shift quantile of 1.0 was applied to manage activation ranges. The weight precision was primarily set to 8 bits. A.3 Datasets ImageNette. Imagenette [18] is a smaller, more manageable subset of ImageNet [9], containing 10 classes. These classes include tench, English springer, cassette player, chain saw, church, French horn, garbage truck, gas pump, golf ball, and parachute. ImageNette has 9469/3925 train/test samples with the original image shape of 3 × 350 × 350. All images were normalized with the ImageNet mean (0.485, 0.456, 0.406) and standard deviations (0.229, 0.224, 0.225), and then converted to Q7 format (one byte per data) to support on-device inference with the tiny AI accelerator platforms (MAX78000 and MAX78002). Caltech101. Caltech101 [11] is a dataset composed of images representing objects from 101 different categories, in addition to a background clutter category. Each image features a single object and is labeled accordingly. The number of images per category ranges from approximately 40 to 800, resulting in a total of around 8677 images. Caltech101 has 6941/1736 train/test samples and the original image shape of 3 × 300 × 300. All images were normalized with the ImageNet mean (0.485, 0.456, 0.406) and standard deviations (0.229, 0.224, 0.225), and then converted to Q7 format (one byte per data) to support on-device inference with the tiny AI accelerator platforms (MAX78000 and MAX78002). 16 Caltech256. Caltech256 [14] built upon its previous version, Caltech101, offering enhancements such as larger category sizes, additional and more extensive clutter categories, and increased overall difficulty. The dataset contains 29780 images across 256 classes after removing the clutter class. Caltech256 has 23824/5956 train/test samples with the original image shape of 3 × 300 × 300. All images were normalized with the ImageNet mean (0.485, 0.456, 0.406) and standard deviations (0.229, 0.224, 0.225), and then converted to Q7 format (one byte per data) to support on-device inference with the tiny AI accelerator platforms (MAX78000 and MAX78002). Food101. Food101 [2] includes 101 food categories, each with 750 images for training and 250 images for testing, which is a total of 101000 images. The original images were rescaled to have a maximum side length of 512 pixels. This dataset has 75750/25250 train/test samples and the original image with the shape of 3 × 512 × 512. All images were normalized with the ImageNet mean (0.485, 0.456, 0.406) and standard deviations (0.229, 0.224, 0.225), and then converted to Q7 format (one byte per data) to support on-device inference with the tiny AI accelerator platforms (MAX78000 and MAX78002). A.4 Baseline details Downsampling. Downsampling is a straightforward method that collects samples evenly distributed across the original image. This approach is equivalent to the case when the number of channels is equal to three in DEX. CoordConv. CoordConv [29] pointed out the limitation of traditional CNNs that relied solely on RGB images for the coordinate transformation problem and introduced the augmentation of i and j coordinates, which improved object detection efficiency. We referred to the Pytorch implementation of CoordConv2 for implementing this baseline. CoordConv (with r). The authors of CoordConv also introduced the third channel for an r coordinate, where r = p (i −h/2)2 + (j −w/2)2, which they found effective in some experiments. Similar to CoordConv, we referred to the Pytorch implementation of CoordConv for implementing this baseline. A.5 Alternative data channel extension methods’ details (a) Repetition (b) Rotation −30°~30° (c) Tile 𝑃𝑖𝑗 𝑘= 0 𝑘= 1𝑘= 2𝑘= 3 Same samples (d) Patch-wise sequential sampling 𝑃𝑖𝑗 𝑘= 0 𝑘= 1 𝑘= 2 𝑘= 3 (e) Patch-wise random sampling Figure 10: Visulaization of four alternative data extension methods. Repetition. Repetition (Figure 10(a)) repeats the same downsampled image across the channels until it reaches the maximum possible number of input channels, which is the same as the number of data memory instances (64). Rotation. Rotation (Figure 10(b)) generates slightly different images through rotating images from the downsampled image. It makes rotated images until it reaches the maximum possible number of input channels. The angle of rotation ranges from -30 to 30 degrees. For instance, given a downsampled three-channel image input and target channel size of 64, it generates rotated images with an angle linearly spaced between -30 to 30 degrees. Tile. Tile (Figure 10(c)) divides the original image into multiple tiles and stacks those tiles across channels. Specifically, given the number of images take K = ⌈CO CI ⌉, it finds the nearest square 2https://github.com/walsvid/CoordConv 17 number S that is higher than or equal to K, (e.g., S = 52 > 22 when K = 22). The original image is then divided into equal-sized patches. Each patch is subsequently downsampled to the target size. The downsampled patches are collected and concatenated along the channel dimension, forming a new image with the desired number of channels. If the total number of patches exceeds the target number of channels, the excess patches are discarded. Patch-wise sequential sampling. Patch-wise sequential sampling (Figure 10(d)) is similar to DEX but it involves sequential sampling within a patch instead of even sampling. Specifically, it samples the first K samples for each patch and follows the same channel-wise stacking procedure in DEX. Patch-wise random sampling. Patch-wise random sampling (Figure 10(e)) is similar to DEX but it involves random sampling within a patch instead of even sampling. Specifically, it samples randomly-selected K samples for each patch and follows the same channel-wise stacking procedure in DEX. B Additional Experiments B.1 Overhead of the channel expansion on devices Channel expansion latency. The latency of the channel expansion process depends on the processor’s computational capability. During our evaluation, we pre-processed data on a powerful server, and thus data processing was negligible. We additionally conducted data processing on the ultra-low-power MCU processor on the board (Arm Cortex-M4) to understand the data processing overhead on less-capable devices. We measured the overhead of applying DEX to expand channels from a 3 × 224 × 224 image (a typical size for ImageNet) to 64 × 32 × 32 (the highest channel expansion used in our accelerators) on the MAX78002’s Arm Cortex-M4 (120MHz). This process took 2.2 ms on the Arm Cortex-M4. In terms of memory, this addition took the SRAM memory of 62KB (64 × 32 × 32 Bytes - 3 × 32 × 32 Bytes) on the processor. However, since DEX extends data to a size that the data memory in the AI accelerator can accommodate, this additional memory will not be an issue from the AI accelerator’s perspective. Impact on end-to-end inference performance. Note that the MCU processor and the AI accelerator are independent processing components that run in parallel. This means that if the inference latency on the accelerator is higher than the data processing latency, data can be pre-processed for the next inference during the current inference (and thus data processing latency can be hidden). For inference, the inference latency of EfficientNet (11.7ms) is higher than the data processing latency of 2.2ms, and thus the inference throughput remains the same under continuous inference. However, this depends on the scenario. The end-to-end impact of data processing latency depends on the processor’s computational capability, the dimension of the data, and the size of channel expansion. For instance, in scenarios where data processing is done and transferred in more capable machines (e.g., cloud servers, smartphones, etc.) than the MCU processor on the tiny AI accelerator, the impact of data processing can be even more negligible. B.2 Power consumption We measured the power consumption of the inference on MAX78000 by varying the size of the channel extension with a Monsoon Power Monitor. The result is shown in Table 6. As the number of channels increased, power consumption increased accordingly. This is because a higher number of channels uses more processors in the AI accelerator, leading to increased power consumption. Table 6: The power consumption of inference measured by varying the size of the channel extension with a Monsoon Power Monitor. All numbers are in milliwatts (mW). Model Chan = 3 Chan = 6 Chan = 18 Chan = 36 Chan = 64 SimpleNet 53.82 53.85 58.21 61.42 68.9 WideNet 60.74 61.37 63.76 67.92 77.14 18 C Example images generated from DEX Original image $ Downsampled (. = 0) Downsampled (. = 1) Downsampled (. = 2) Downsampled (. = 6) Downsampled (. = 5) Downsampled (. = 4) Downsampled (. = 3) Figure 11: Examples images generated from an original 3 × 350 × 350 image (ImageNette) to 3 × 32 × 32 downsampled image via DEX. k = 0 to k = 6 cases are shown only. Each generated image contains different pixel information, which collectively enhances feature learning in CNNs. Original image $ Downsampled (. = 0) Downsampled (. = 1) Downsampled (. = 2) Downsampled (. = 6) Downsampled (. = 5) Downsampled (. = 4) Downsampled (. = 3) Figure 12: Examples images generated from an original 3 × 350 × 350 image (ImageNette) to 3 × 32 × 32 downsampled image via DEX. k = 0 to k = 6 cases are shown only. Each generated image contains different pixel information, which collectively enhances feature learning in CNNs. D License of assets Datasets. ImageNette dataset (Apache-2.0 license), Caltech101 (CC BY 4.0), Caltech256 (CC BY 4.0), and Food101 dataset (MIT license). 19 Original image $ Downsampled (. = 0) Downsampled (. = 1) Downsampled (. = 2) Downsampled (. = 6) Downsampled (. = 5) Downsampled (. = 4) Downsampled (. = 3) Figure 13: Examples images generated from an original 3 × 350 × 350 image (ImageNette) to 3 × 32 × 32 downsampled image via DEX. k = 0 to k = 6 cases are shown only. Each generated image contains different pixel information, which collectively enhances feature learning in CNNs. Codes. 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4,452	Large Scale Transfer Learning for Tabular Data via Language Modeling Josh Gardner♮,∗ Juan C. Perdomo# Ludwig Schmidt♮,♭ ♮University of Washington, #Harvard University, ♭Stanford University ∗Corresponding author, jpgard@cs.washington.edu Abstract Tabular data – structured, heterogeneous, spreadsheet-style data with rows and columns – is widely used in practice across many domains. However, while recent foundation models have reduced the need for developing task-specific datasets and predictors in domains such as language modeling and computer vision, this transfer learning paradigm has not had similar impact in the tabular domain. In this work, we seek to narrow this gap and present TABULA-8B, a language model for tabular prediction. We define a process for extracting a large, high-quality training dataset from the TabLib corpus, proposing methods for tabular data filtering and quality control. Using the resulting dataset, which comprises over 2.1B rows from 4.2M unique tables, we fine-tune a Llama 3-8B large language model (LLM) for tabular data prediction (classification and binned regression) using a novel packing and attention scheme for tabular prediction. Through evaluation across a test suite of 329 datasets, we find that TABULA-8B has zero-shot accuracy on unseen tables that is over 15 percentage points (pp) higher than random guessing, a feat that is not possible with existing state-of-the-art tabular prediction models (e.g. XGBoost, TabPFN). In the few-shot setting (1-32 shots), without any fine-tuning on the target datasets, TABULA-8B is 5-15 pp more accurate than XGBoost and TabPFN models that are explicitly trained on equal, or even up to 16× more data. We release our model, code, and data along with the publication of this paper. 1 1 Introduction 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 191 32-Shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline Figure 1: TABULA-8B outperforms SOTA tabular baselines across 0 −32shot tasks from five tabular benchmarks. Transfer learning - the ability of a model to accurately solve prediction tasks on data it was not trained on - is one of the defining hallmarks of recent foundation models in domains such as vision [38] and language [6]. Among their many advantages, transferable models expand the scope of problems that can be tackled via machine learning by reducing the need for curated, task-specific models and datasets. Such models also can provide both absolute performance and sample-efficiency gains over task-specific models when applied to new tasks [38, 41, 53]. In this work, we introduce a new model and dataset for large-scale transfer learning on tabular data. Tabular, spreadsheet-style data underlies applications in healthcare, finance, government, and the natural sciences [4, 16, 50]. 1For links to all code, data, and model, see Section 7. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Yet, despite the potential impacts of transferable foundation models for tabular data [58], the core practices of machine learning on tabular data have remained largely unchanged. The prevailing paradigm is still to train single-task models (e.g., XGboost [7]) using a fixed schema on data from the same distribution on which the model will be deployed. Here, we aim to bridge this gap. We introduce TABULA-8B, a language model for tabular prediction which can flexibly solve classification tasks across unseen domains, including where data is scarce. Our methodology expands the scope what is possible in these settings, thereby democratizing access to prediction in low-resource contexts and providing state-of-the-art, training-free transfer learning on any tabular data. Since the model only requires a forward pass to perform inference on the target data, it also avoids the privacy or computational considerations that arise in other approaches that require fine-tuning on local, and potentially sensitive, datasets. In particular, given a small number of examples (shots), and without any fine-tuning on the task, TABULA-8B outperforms state-of-the-art gradient-boosted decision trees and tabular deep learning methods that are explicitly trained on the target data (see Figure 1). Furthermore, TABULA-8B is capable of zero-shot prediction, a behavior which is not possible using these prior methods. To enable these results, we build a new dataset for tabular prediction, The Tremendous TabLib Trawl (T4), that allows us to scale up training by several orders of magnitude (10,000× more data) relative to previous work. 1.1 Our Contributions This paper has the following three main contributions: TABULA-8B, a Tabular Prediction Model: We build TABULA-8B (Tabular Llama 3 - 8B), a model for prediction on tabular data. On an evaluation suite consisting of 329 tables drawn from five tabular benchmarks, TABULA-8B has zero-shot accuracy 17 percentage points (pp) above random guessing. In the few-shot setting (1-32 examples), TABULA-8B is 5-15 pp more accurate than state-of-the-art methods (XGboost, TabPFN, CatBoost) that are trained on equal number of shots, and these methods require 2-8× more data to achieve the performance of our model. TABULA-8B outperforms a variety of strong tabular baselines and even commercial LLMs such as Claude Instant and Claude 3 Sonnet. T4 - A Large Scale, High Quality Training Dataset: We build and release The Tremendous TabLib Trawl (T4), a filtered collection of 4.2M unique tables (consisting of over 2.1B rows, a total of 100B tokens) from TabLib [13]. We detail the recipe used to construct T4, including a suite of methods for filtering web-scale tabular data at several levels (table, row, column), removing unwanted information such as PII and code, and selecting unsupervised prediction targets from these tables. Open-Source Release: As part of our publication, we release all relevant infrastructure (code, models, and data) with the hopes that the community will build on our work. We provide high-quality, efficient implementations of data pre-processing and model training pipelines, including our new row-causal tabular masking (RCTM) attention and packing scheme for training on tabular data. We also share the code used to filter T4 from TabLib, enabling future work that extends our dataset construction methodology. 1.2 Preliminaries & Project Scope Our work is concerned with prediction models for tabular data. We define both below. Tabular Data: For our purposes, tabular data has three main properties. (i) Structured: It consists of elements with a “key-value” structure, often represented as a table with keys (or “headers”) representing column names, and rows that consist of values. (ii) Heterogeneous: The values are of mixed types, including numeric, boolean, categorical, ordinal, text, date/time, etc. Missing values may be present. (iii) Exchangeable: The ordering of rows and columns in the dataset is arbitrary. In particular, any permutation of the rows, or columns, still represents the same tabular dataset. Prediction Task Definition: The main focus of this work is prediction on tabular data. In tabular prediction, the goal is to predict the value y of a specific target column for a row in a dataset using the key-value pairs x from all other columns. More specifically, we focus on classification tasks where values y for the target column belong to a finite set C. Binned regression tasks, in which a real-valued y is discretized into a finite set of numeric values (as in [59]) also fit this definition. 2 2 Related Work Our work builds on a line of foundation modeling, tabular data prediction, and natural language processing research. Given space constraints, here we focus on the most closely related literature. Transfer Learning and Foundation Models: The idea of building general purpose models via autoregressive next-token prediction on large scale datasets was pioneered in a series of papers in natural language processing [6, 11, 36, 40, 57]. These results have since led to the development of foundation models capable of solving diverse tasks in other modalities including vision [38, 63], audio [39, 66], code [22, 42], time series [10, 18, 21], and graphs [62], as well as multi-modal models [53, 55]. Our work also build upon on the demonstrated capacity of transformers to perform few-shot or in context-learning [6, 17], which entails making predictions on examples from a previously unseen dataset, given only a few labeled examples from that task. Large-Scale Dataset Curation: The construction of large, high-quality datasets has emerged as one of the most critical, and challenging, issues in the development of transferable models. Several major milestones in this space [6, 40, 41, 53–56] stand out in their effort spent curating and cleaning web-scale datasets – often while using a model architecture and training recipe that only slightly differs from prior work. This has led to a number of modality-specific methods for large-scale dataset curation; for example, the use of heuristics [41] and model-based quality scoring to select high-quality text data [6, 9, 56]; methods for selecting aligned audio-transcript pairs for speech [39]; or the use of CLIP scores to filter for aligned image-text pairs [46]. However, to the best of our knowledge no prior work has developed analogous methods for tabular data. This lack of large-scale training data has been a critical bottleneck toward the development of tabular foundation models. Models for Tabular Prediction: Despite the fact that deep learning methods are now the norm in domains such as computer vision or NLP, methods based on gradient-boosted decision trees (GBDTs) [5, 7, 37] continue to be at or near state-of-the-art in tabular prediction [20]. Drawing upon recent breakthroughs in other modalities, the field has now developed deep learning-inspired approaches [19, 52] that are competitive with tree-based models in-distribution, where models are trained and evaluated on the same data, but the benefits of such approaches relative to GBDTs appears to be limited in practice [20, 33]. In particular, [25] introduces TabPFN, a transformer model for tabular data that outperforms XGBoost in certain regimes [33] and is capable of making predictions on unseen datasets (with some constraints on dataset size and label space; see D.2). A related recent work, CARTE [30], explores the use of graph-based architectures for tabular transfer, based on key-value encodings and pretrained on a large knowledge graph. Several recent works [12, 23, 59, 66] have explored fine-tuning LLMs on individual tables, or on small collections of tables (< 200). The main idea in this closely related line of work is to reduce classification to next-token prediction by first serializing a row as text (see Figure 2b for an illustration) and then training an LLM to predict the serialized labels. These studies demonstrate that this LLM approach is often competitive with trees or tabular deep learning methods in-distribution [12, 23, 59]. However, in cases where models were evaluated out-of-distribution, they were less accurate than SOTA methods trained on these held-out tables [65]. Our work builds on this promising line of work. Relative to these prior efforts, we specifically address the (1) lack of large-scale and training data; and (2) the inability of exiting methods to be competitive when evaluated out-of-distribution. 3 TABULA-8B - Model Design and Training Our overall approach is to fine-tune the pretrained Llama 3-8B language model [54] on tabular prediction tasks using a new web-scale corpus, T4. We use Llama 3-8B as our starting point since it is a high-quality, open-source model trained on over 15T tokens that demonstrates strong performance on a diverse set of downstream tasks [54], particularly at its relatively modest size (which makes fine-tuning, inference, and deployment more accessible). Serialization and Tabular Language Models: As discussed previously, our methodology extends ideas pioneered in previous work [12, 23, 59, 65] demonstrating how LLMs can be trained to perform tabular prediction tasks by serializing rows as text, converting the text to tokens, and then using the same loss function and optimization routines used in language modeling. Serialization refers to the procedure of converting a row of data into text, for instance by concatenating substrings of the form “the <key> is <value>”. Prior works investigated the impact of different serialization formats 3 Row 1 Row 2 Row 3 Source Token Index Target Token Index (a) Predict the value of weather: \|\|sun\|\|rain\|\|snow\|\| The date is 2015-03-22. The precipitation is 1.0. The temp_max is 11.699. What is the value of weather? \|\|sun\|\|rain\|\|snow\|\| <\|endinput\|> rain<\|endcompletion\|>Predict the value of weather: \|\|sun\|\|rain\|\| snow\|\| The date is 2015-09-19. The precipitation is 1.0. The temp_max is 14.722. What is the value of weather? \|\|sun\|\|rain\|\| snow\|\|<\|endinput\|>sun<\|endcompletion\|> date precipitation temp_max weather 2015-03-22 1.0 11.699 rain 2015-09-19 0.0 14.722 sun Serialization (b) Figure 2: 2a: Illustration of the row-causal tabular mask (RCTM) representing a batch during training. Each triangular block represents potentially many rows from a single table (detail shown at left). Shaded groups within this block represent tokens from one row in the table. This structure implicitly trains the model for few-shot learning by permitting it to attend to previous rows from the table, but not to rows in other tables. 2b: Serialization of tabular data into text. The model is trained to produce the tokens following the <\|endinput\|> token. [12, 23, 51], demonstrating that performance is largely insensitive to the exact mapping (e.g. using “{ <key>: <value> }”) and other strategies do not improve upon this “the <key> is <value>” structure. We adopt a similar serialization strategy, illustrated in Figure 2b. Given a row of data from a table, the corresponding serialization has three main parts: (i) a prefix containing a prompt (always “Predict the value of <target column name>”) followed by a list of possible label values (“val1 \|\| . . . \|\| valNumClasses \|\|), (ii) the example consisting of all key, value pairs for the columns used as features, and (iii) a suffix prompting the model with a question (“What is the value of <target column name>?”) again followed by the possible labels. For multiple-shot samples, we concatenate their serializations. We introduce three special tokens into the Llama 3 vocabulary to ensure these sequences are properly tokenized: \|\|, to delimit answer choices; <\|endinput\|> to denote the end of an input sequence (the last token before the targets or model generation begin); and <\|endcompletion\|> to indicate the end of a completion. Training Procedure: We train TABULA-8B using a standard language modeling setup where the model is trained to minimize the cross-entropy over the sequence of target tokens. We only compute loss over the subsequence of target tokens: the tokens starting after the <\|endinput\|> token, up to and including <\|endcompletion\|>. This objective focuses training on learning the desired target label, as in [12, 23, 59], rather than developing a broader generative model of tabular data as in [65]. Relative to prior studies on tabular prediction with LLMs, our work has one main methodological innovation. We introduce an efficient attention masking scheme, row-causal tabular masking (RCTM), tailored to few-shot tabular prediction whereby the model is allowed to attend to all previous samples from the same table in a batch, but not to samples from other tables (this is sometimes referred to as “cross-contamination” in the language modeling literature [31]). However, by appropriately masking out values, RCTM also enables packing examples into the same batch (as effectively zero padding is required during training despite the large variance in the size of each tokenized table or row), thereby increasing model throughput (see Figure 2a). Taken together, these have the effect of training the model to use multiple “shots” during training and mitigates the potential loss of few-shot learning capabilities that has been observed to occur during fine-tuning [29, 61, 64]. The RCTM masking structure is shown in Figure 2a. Lower-triangular blocks correspond to rows from the same table that are present in the batch. This is similar to the “in-context pretraining” method from [48], except that (i) our procedure encourages the model to aggregate information across multiple rows of a given table, rather than attending across documents, and (ii) our procedure only trains the model to predict the target tokens, not the input features. We show that RCTM has a drastic impact on few-shot performance through an ablation experiment (see Section F.1). Training Details: The final model is trained for 40k steps with a global batch size of 24 (with sample packing, this is roughly equivalent to a global batch size of 600 rows of tabular data). The model 4 Target Selection Filtering Figure 3: Sketch of dataset generation pipeline. 627M tables from TabLib [13] are filtered by applying rules at the table, row, and column level. Then, for each table, we identify valid and high-quality prediction targets in an unsupervised manner and use the results for training TABULA-8B. sees roughly 8B tokens during training; we note that is less than 10% of the 100Btokens in T4, and less than one one thousandth of TabLib itself. We fully fine-tune all model parameters, as opposed to parameter-efficient fine-tuning, since full-fine tuning consistently benefits from scale [24, 64]. Reproducibility details are given in Appendix B. 4 Dataset Construction: Building The Tremendous TabLib Trawl (T4) Beginning from a web-scale corpus of raw data (TabLib), we apply various filters to produce a high-quality subset, and transform the results into a set of prediction tasks for training. As the result of this procedure, we produce T4 (The Tremendous Tablib Trawl), which we release with this paper. Original Raw Data Source: TabLib [13] is a publicly-available dataset consisting of 627M tables extracted from two main sources: Common Crawl and Github (see [13] for more details). Due to its scale and diversity, TabLib presents a unique opportunity for training foundation-scale models on the tabular data. However, like other web-scale datasets, the vast majority of its contents are low quality and not suitable for training. For instance, TabLib contains numerous system logs with inscrutable statistics, tables of software documentation, and call sheets with personally identifiable information (PII). To the best of our knowledge, no previous work has addressed the task of filtering TabLib into a usable training set, and no publicly-available models have been trained on this corpus. Filtering Strategies: Filtering large collections of raw data to extract a higher-quality subset is an essential component in the development of foundation models [e.g. 41], yet to date, no previous work has addressed this core issue for tabular data. Filtering a web-scale dataset like TabLib into a usable subset of high-quality tables is critical to leverage its diversity and scale, but also raises unique challenges specific to tabular data, such as missing data, web “content” that is formatted as HTML tables that does not satisfy our definition of tabular data, and PII. To turn TabLib into a usable training set, we develop a set of filtering methods to identify high-quality tables for prediction. Conceptually, our filtering occurs at three levels, each applied sequentially: tables (entire tables are removed from the pool), columns (individual columns are removed from a table), and rows (rows are removed from a table). Similar to previous approaches [38, 40, 41, 54, 56], we use a mix of heuristics and rule-based methods to remove low-quality sources from our pool. We present the full list of our filtering rules in Section A. At a high level, our emphasis across all filtering strategies is to: (1) remove non-tabular data (such as text or PDFs incorrectly identified as tabular data during TabLib’s collection), (2) ensure the safety of chosen tables (e.g. by removing PII), and (3) find sources with high semantic content (e.g. by removing tables with too many missing values). As part of this filtering process we develop and apply simple methods for deduplication, English language filtering, filtering for missing data, PII removal, code removal, and more. Unsupervised Task Selection: As described in Section 1.2, we focus on methods for tabular prediction: predicting the value of a target column given the values of all other columns for an instance. Therefore, as part our data pipeline we develop new methods for selecting, in an unsupervised fashion, which column is the target column for each table in the corpus. Selecting targets of prediction for tabular data at scale is an under-explored problem. Prior work in this space operated on at most a few hundred tables and used either a combination of expensive queries to commercial LLMs or manual curation to identify tabular prediction targets [59, 65]. However, when operating on hundreds of millions of distinct tables with potentially no associated metadata, these strategies are not feasible. For each table, we select a prediction target programmatically by first identifying a subset of columns that are suitable for prediction according to various heuristics, and then choosing a specific column 5 at random from this set. The exact list of heuristics to arrive at this set is presented in Appendix A. Amongst others, these include excluding candidate columns if: the column name is numeric, it has only one unique value, or it has unique values for every row (excluding numeric columns). Final T4 Dataset Summary: Running this entire filtering process (from raw data to serialized examples ready for training) on all 70TB of TabLib using our open-sourced implementation takes about 4 hours on a CPU cluster. It yields a total of 4.2M tables, which equates to a table filtering rate of approximately 97.91%. Additional descriptive statistics for the dataset are given in Appendix A.3. The resulting dataset contains over 2.1B rows (approximately 100B Llama 3 tokens) for training of the downstream model, and occupies roughly 2TB compressed on disk. We note that 100B tokens is larger than the total number of tokens TABULA-8B sees during training. Therefore, the model sees each distinct table at most once during training, and our pipeline could be scaled up to support larger models or longer training runs. 5 Experimental Results 5.1 Evaluation Methodology We evaluate the transfer learning performance of TABULA-8B on a diverse set of established benchmarks previously considered in prior work (see Section 5.2 for a list). For each dataset, we use the predefined prediction target from the original benchmark. Due to computational constraints, we evaluate TABULA-8B on up to 128 test examples for each dataset and number of shots k. The term “few-shot” is unfortunately overloaded. It is used both to refer to models that make predictions on instances never seen during training, and to models that directly train on these examples before predicting on unseen samples. We do not fine-tune our model on test examples, in contrast to [23, 59]. Our methodology only requires performing forward passes through the network to generate predictions and avoids the need for computationally-expensive gradient updates. In zero-shot evaluations, given a row of a dataset along with the corresponding set of columns and possible labels, we first serialize the row into the same format used during training, and feed it into the model to generate a prediction following the <\|endinput\|> token. For few-shot evaluations, we perform the same procedure, except that we preprend the serialized “shots” as in Figure 2b. In contrast to methods like XGBoost that directly predict likelihoods of a set of labels, language models output likelihoods over a set of tokens (128k in the case of Llama 3). For each evaluation dataset, the values in the label set (e.g. “sun, rain, snow” in Figure 2b) can consist of a sequence of many individual tokens from this large vocabulary. Here, we use open-vocabulary (or “openended”) accuracy [1, 8] as the main evaluation metric for our model. In this setup, once the model is prompted with a serialized example, it is allowed to generate an arbitrary sequence of tokens. Once it produces the <\|endofcompletion\|> token, the generated text is then directly compared to the correct completion. Only an exact match, including the terminating <\|endofcompletion\|> token, is counted as accurate. This is more challenging than closed-vocabulary evaluation, where the model is only rated on assigning the highest probability to the correct completion from a predetermined set. 5.2 Evaluation Datasets We evaluate our model’s predictive performance across a collection of 329 publicly-available tabular datasets drawn from five tabular benchmarks (see Appendix H for a full list). These include: UniPredict Benchmark (169 datasets) [59]: We use the “supervised” subset of 169 datasets from the recently-introduced UniPredict benchmark. These are high-quality tabular datasets with generally informative column names and a mix of both categorical and continuous targets, drawn directly from Kaggle. While the model introduced in Wang et al. [59] was trained and tested on separate splits of these datasets, we only use them for testing. We make corrections to several datasets with targets erroneously treated as categorical in the original benchmark, described in Section 5.2. Grinsztajn Benchmark (45 datasets) [20]: The Grinsztajn benchmark is a curated suite of datasets consisting of numeric and categorical features. This dataset is notable in that the original study by Grinsztajn et al. found that gradient boosted decision trees (GBDTs) consistently outperformed deep learning-based methods on these tasks. 6 AutoML Multimodal Benchmark (AMLB) (8 datasets) [49] : A suite of tables which include one or more free-text fields (such as an Airbnb description, or a product review). The benchmark is considered challenging for tree-based methods due to the non-standard text-based features. However, it also poses a challenge for LLMs since some columns can contain highly variable lengths of text. OpenML CC-18 Benchmark (72 datasets) [2]: The OpenML Curated Classification Benchmark was created by applying filtering rules to extract a high-quality subset from the OpenML platform. The rules include: no artificial data sets, no subsets of larger data sets nor binarizations of other data sets, no data sets which are perfectly predictable by using a single feature or a simple decision tree. OpenML CTR-23 Benchmark (35 datasets) [14]: The OpenML Curated Tabular Regression (CTR) Benchmark is a curated set of tables for regression drawn from OpenML. The curation process is similar to that of OpenML-CC18, for regression tasks. We note that, being primarily intended for the evaluation of AutoML methods, the OpenML benchmarks are notable for lacking informative column names (i.e. names such as “Var1, Var2, .. . ” are common in OpenML benchmark datasets). We transform all regression tasks into a 4-class classification based on quartiles, as in [59] (see Appendix A.2 for details). Many datasets contain rows with missing data. We leave these as-is and do not remove any data. On a computational note, some datasets contain a large numbers of features and performing k-shot evaluations on these datasets at large k can exceed the model’s context window. Therefore, in few-shot evaluations, we always report results for the subset of datasets where k shots fit into the model’s context window, for the entire specified range of k. We provide more details on the evaluation datasets in Appendix D.1 and report per-dataset results in Appendix H. 5.3 Baselines When inspecting TABULA-8B’s performance, we compare against the following baselines: Llama 3-8B [54]: This is the base model from which TABULA-8B is fine-tuned. Comparing to the base model isolates the effects of the fine-tuning process. It also controls for any contamination of evaluation datasets that may be contained in pretraining data for Llama 3 (the exact training data for Llama 3 are not currently disclosed). We return to this point in Section 5.6. XGBoost [7]: XGBoost is a supervised learning gradient-boosted decision tree (GBDT) method. It is widely considered to be highly competitive in tabular prediction tasks [15, 20, 33]. TabPFN [25]: This a transformed-based hypernetwork pretrained to reflect a set of inductive biases germane to tabular data. TabPFN is thus considered especially effective for few-shot learning [25, 33]. Whenever possible, we perform hyperparameter tuning on XGBoost and TabPFN in order to maximize their performance. See Appendix D.2 for further details on baseline implementation and tuning. We also provide results comparing to additional supervised baseline models, and to commercial LLMs, in Section E.2. 5.4 Main Results: Assessing TABULA-8B’s Transfer Learning We present our main experiments evaluating the transfer learning ability of TABULA-8B in Figure 4. As a whole, TABULA-8B demonstrates strong transfer performance across the broad range of tasks. In the zero-shot regime (seen in the left-most point for each plot in Figure 4) – where the model is presented with no further information about the target dataset except for the serialized key-value pairs and set of possible labels for a single row –TABULA-8B is between 5 to 25 pp more accurate than a random baseline and 50 pp more accurate than the base Llama 3 model. This illustrates one of the key benefits of using language models for tabular prediction: after fine-tuning, TABULA-8B can leverage semantic information contained in the serialized data to make high-quality predictions. While XGboost and TabPFN are not capable of zero-shot prediction, this behavior has been observed in the original Llama 3 model [54, 57]. However, in our evaluations, Llama 3 performs below random guessing in the zero-shot setting. We hypothesize that the base Llama 3 model requires a small number of samples to understand the input-output format and task (as indicated by the large leap in Llama 3 performance from 0 →1 shot). In the few-shot setting, where each method additionally sees a small number of labeled examples, TABULA-8B’s performance steadily improves with the number of shots. In the regime of 1 to 32 7 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 120 UniPredict Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 45 OpenML CC18 Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 29 OpenML CTR23 Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 6 AMLB Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 44 Grinsztajn Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Impact of Potential Contamination TabuLa 8B, Decontaminated Subset XGBoost trained + tuned on k samples TabPFN on k samples TabuLa 8B, Possibly-Contaminated Subset XGBoost trained + tuned on k samples TabPFN on k samples Figure 4: Zero- and few-shot accuracy across five tabular benchmarks. For each benchmark, we evaluate on all tasks, but in the figures above we only display the subset of tasks where k shots fit into the 8192-token context window of TABULA-8B. Complete results are in Supplementary Section H. The final plot (lower right) shows curves separately over decontaminated vs. potentially-contaminated evaluation tasks (see Section 5.6); we find no impact on our overall findings due to contamination (and performance on tasks which may be in our training set is lower on average, across all models). shots, it outperforms state-of-the-art models (XGBoost and TabPFN) that are directly trained (and hyperparameter tuned) on each specific dataset by 5-20pp. Once we evaluate performance on 32, 64, or 128 shots (see Figure 7b), this gap begins to diminish, but the number of datasets that can fit > 32 shots into the 8192-token context window is both small and a relatively biased sample (due to their small number of features). TABULA-8B is consistently 10 to 20pp above the Llama 3-8B base model for the full range of shots, highlighting the benefits of our training procedure on T4. Improvements in Sample Efficiency: As discussed previously, the main benefit of transferable models is that they reduce the amount of data necessary to achieve good performance on new tasks. For instance, as seen in Figure 4, TABULA-8B only needs one shot to achieve 60% average accuracy on UniPredict tasks. However, both TabPFN and XGBoost only reaches 60% accuracy after 16 shots. Therefore, TABULA-8B reduces the amount of data necessary to achieve 60% accuracy by 16 fold relative to XGBoost annd TabPFN. We refer to this statistic as the relative sample efficiency (see D.3). TABULA-8B in general achieves higher accuracy than the benchmarks using less data. Hence, the relative sample efficiency is always > 1 (the exact ratio varies across benchmarks). Impact of Informative Column Headers: As shown in Figure 4, while TABULA-8B generally has higher accuracy than the baselines, this accuracy gap varies across benchmarks and the number of shots. For instance, for the UniPredict benchmark – which was specifically constructed to include datasets with semantically-meaningful column headers [59] – the gap to supervised baselines is much larger than in the OpenML benchmarks, which tend to have less semantically-meaningful column names. If meaningful column headers are absent, the model still performs well (matching or outperforming XGBoost at shots k ≤8), but its advantage over these strong baselines is lessened. We investigate this effect in further detail with a controlled experiment in Section F.2. 8 5.5 Further Robustness Evaluations and Ablation Experiments Robustness to Column Ordering. Apart from evaluating TABULA-8B’s transfer learning ability, we also investigate its robustness and the degree to which performance is affected by the order in which columns are presented (serialized); this order invariance is cited as a necessary attribute of tabular foundation models in [58]. We present these experiments in Appendix F.4. Our results demonstrate that changing column order does not alter performance in a statistically significant way, but that there may be a small (∼1pp) drop which we hypothesize is due to manual ordering of tabular columns in certain benchmark datasets which sometimes reflects a more “natural” ordering. Robustness to Feature Dropout. Language models may be uniquely susceptible to small changes in the downstream data; for example, the removal of specific features may affect language models’ prediction performance more than traditional supervised methods. We conduct an ablation study to assess the behavior of TABULA-8B as columns are removed from the test data. We assess removal both in order of descending and ascending importance. The results of these experiments, in Appendix F.3, demonstrate that TABULA-8B’s performance declines at a similar rate to an XGBoost model trained directly on the subset of features. Robustness to Column Header Removal. Another potential risk of tabular language models is that, while these models are able to utilize the semantic information in column names, the model may also be overly reliant on the presence of informative column names. In Appendix F.2 we conduct an ablation study to assess this. The results in Appendix F.2 demonstrate that there is a small decline in performance when column headers are removed (replaced with uninformative headers), but that TABULA-8B still outperforms baselines across all numbers of shots. We believe that this drop in performance is commensurate to the loss in information when column headers are eliminated. Importance of Row-Causal Tabular Mask. We evaluate the impact of the attention masking scheme introduced and described in Section 3. We conduct an ablation study, replacing this component of the model with a sample-wise causal attention (the same form of attention used during standard language model training, where attention across documents is prevented). Our results, detailed in Appendix F.1 and Figure 11, illustrate that this modification is central to the few-shot learning capabilities of TABULA-8B: when our mechanism is replaced with sample-wise attention the resulting model does not demonstrate few-shot learning capacity, and its performance degrades for k ≥16 (see Figure 11). Influence of the Base LLM. We conduct an ablation study of the base LLM to evaluate how TABULA-8B improves as the base LLM improves. In particular, we rerun our main training pipeline described in Section 3, but using LLama 1 and 2 as the initial language model instead of LLama 3. These results, provided in Section F.6, demonstrate that TABULA-8B improves along with the performance of the underlying base model. Taken together, these results highlight how the primary contribution of the paper is not the specific model we produce, as much as it is the methodology we present for generating tabular predictors from base language models. As LLMs continue to improve, so will the tabular models that are produced by applying our training methodology to new LLMs. 5.6 Assessing the Potential Impact of Data Contamination Given that T4 consists of 4.2Mtables sourced from public data sources (Common Crawl, Github) and that our evaluations are also comprised of public benchmarks, we investigate the extent and possible impact of data contamination – that is, training datasets that are part of the evaluation suite. In Section G, we explain our methodology to test for the potential presence of benchmark datasets in T4. Using a conservative identification strategy based on column matching (likely to include false positives). We find at most one-third of benchmark tables may occur at least once in the training set. When training large-scale models for transfer learning, it is not always clear a priori what the eventual application domains will be. Therefore, we believe that it is an important research question to understand the extent to which contamination may affect performance, as contamination may be difficult to prevent in some cases. Initial foundation modeling efforts in non-tabular domains adopted a similar approach, and found mixed or no impact from overlap [38, 40]. We evaluate the impact of contamination in our experimental setup by evaluating TABULA-8B separately on “potentially contaminated” vs. uncontaminated tables. Our results are shown in the bottom right plot of Figure 4, as well as in Figures 17 and 18. Summarizing, we find no clear evidence that contamination affects model performance on the test suite, or that transfer ability is affected by contamination. In fact, as seen in Figure 4, the gap between TABULA-8B and XGBoost is in fact 9 larger if we restrict evaluation to the benchmark tables which we verify are not in T4. In addition to verifying that our results continue to hold over a diverse set of tables which we know the model did not see in training, it also shows that having some amount of potential contamination did not upwardly bias our estimate of TABULA-8B’s transfer learning ability. We hypothesize that the observed gap in Figure 4 is due to our conservative duplication procedure being more likely to flag datasets with generic or common column names, which also leads to worse baseline performance on these tasks. We present more comprehensive investigation on the effects of contamination in Appendix G. 6 Discussion Limitations: TABULA-8B has several limitations. First, TABULA-8B has a limited context window of 8192 tokens. This restricts the number of examples that can be utilized for few-shot learning, as well as the additional information (such as text context or extended feature descriptions) that are available to the model. We expect that this limitation will be eased as the availability of longer-context models grows [e.g. 53]. Second, TABULA-8B has 8B parameters, which makes serving and inference expensive and limits the environments it may be deployed in. Lastly, given that it uses a pretrained LLM as a base-model and fine tunes on a web-scale corpora of historical datasets that likely contain various social biases, TABULA-8B introduces new potential fairness considerations that are not present when using preexisting supervised methods such as XGBoost. We hope that by open sourcing the model, data, and code, we might enable future research addressing these important directions. Future Work: Our work on transfer learning for tabular data is the first its kind at this scale, and there are several avenues for future research. These can be coarsely categorized as either improvements (of the existing dataset and model) or extensions (deeper investigations into the model and data itself). On the improvements side, we see several promising directions. These include improvements in tabular data filtering (this has been the main axis of improvement in recent generations of language models); scaling the model + data + compute; exploring the use of inference-time strategies to improve prediction (such as self-consistency [60], prompt ensembling [1, 38], or in-context example selection [43]); and introducing extra information during both training and inference, such as contextual information or samples from different, but related, tables. On the extensions side, we hope that our work opens avenues toward deeper understanding of tabular foundation models including: understanding potential biases or unwanted behavior with respect to sensitive features (features such as race, age, and gender are common in tabular datasets); using tabular foundation models to address small-sample problems which might be aided by a high-quality pretrained model (such as in the Fragile Families Challenge [44]); and extending this approach to new tasks beyond prediction, such as data generation, explanation, data wrangling, and more. 7 Accessing Open-Source Code, Data, and Model Weights TabLib Preprocessing Code: A Python module for filtering TabLib, tabliblib, along with scripts and configurations used to perform the filtering, are available at https://github.com/ mlfoundations/tabliblib. Model Training and Inference Code: We provide rtfm, a Python module used to train TABULA-8B, perform inference and evaluation, and process data, at https://github.com/mlfoundations/ rtfm. T4 Dataset: The T4 dataset is available via public credentialized access on Hugging Face datasets at https://huggingface.co/datasets/mlfoundations/t4-full. Because the dataset is derived from TabLib, users must first obtain permission to access TabLib at https://huggingface. co/datasets/approximatelabs/tablib-v1-full. Evaluation Datasets: The full evaluation suite used to evaluate TABULA-8B is available via Hugging Face Datasets at https://huggingface.co/datasets/mlfoundations/ tabula-8b-eval-suite. Each dataset includes: a CSV file containing the raw data; a TableShift [16] FeatureList JSON object; and a YAML file with associated metadata. Model Weights: TABULA-8B weights are available via Hugging Face at https://huggingface. co/mlfoundations/tabula-8b. 10 Acknowledgments and Disclosure of Funding JG was supported by a Microsoft Grant for Customer Experience Innovation. This work was also in part supported by the NSF AI Institute for Foundations of Machine Learning (IFML, CCF-2019844), Google, Open Philanthropy, and the Allen Institute for AI. JCP was supported by the Harvard Center for Research on Computation and Society. We are grateful to Foundry2 for providing the compute infrastructure used to train TABULA-8B. Our research also utilized computational resources and services provided by the Hyak computing cluster at the University of Washington, and from Stability AI. The authors also gratefully acknowledge the Gauss Centre for Supercomputing3 for funding this project by providing computing time on the GCS Supercomputer JUWELS[28] at Julich Supercomputing Centre (JSC). We are particularly appreciative of support from Jenia Jitsev at JSC. We also acknowledge Approximate Labs4 and express our appreciation for their development and release of TabLib, along with their communication and support as we utilized the dataset. We are grateful to Jeffrey Li, Mike Merrill, Jonathan Hayase, and Nilesh Tripuraneni for feedback on a early versions of this paper. We also benefited greatly from advice from Matt Jordan, Alex Fang, and Jeffrey Li on large-scale data preprocessing. References [1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in neural information processing systems, 35:23716–23736, 2022. [2] Bernd Bischl, Giuseppe Casalicchio, Matthias Feurer, Frank Hutter, Michel Lang, Rafael G. Mantovani, Jan N. van Rijn, and Joaquin Vanschoren. 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A precise implementation may be found as a part of our software release associated with this paper. Our pipeline involves a language identification step; for language detection, we make use of the fasttext library [27].5 List of Table Filtering Rules Level Name Description Motivation / Hypothesis Table English Filtering Drop where a language ID model score is below a fixed threshold All downstream benchmark datasets are in English Table Schema Heterogeneity Drop tables where every cell is of the same type Encourages understanding of mixed data types Table Row Count Drop table with fewer than k rows Anecdotally, many “very small” tables in TabLib are general web-text tables not useful/suitable for ML. Table Column Count Drop tables with fewer than k columns after column filters are applied Exclude tables that lack a reasonable amount of features Table Parse Error Drop tables where the headers suggest there was a parsing error. These tables are likely the result of bad table detection, and they almost definitely contain low-quality headers. Table Drop PII Drop table where > x% of the cells match a regex for phone number or email Don’t want to train on PII for privacy reasons. Also not likely to be present in downstream tasks. Table Drop Code Drop table with any cell that has probability > p of containing code. Lots of the data in TabLib is from GitHub and other technical documentation. Code is common. Code also confuses the model a lot, apparently due to special characters and whitespace. This code can be unevenly broken/spread across cells due to the tablib parser. Table Too many unnamed columns Drop table if the fraction of “Unnamed: ” columns is greater than a threshold. Discard low-quality data; unnamed columns tend to be of significantly lower quality based on manual data inspection. 5https://pypi.org/project/fasttext-langdetect/ 16 List of Row and Column Filtering Rules Level Name Description Motivation / Hypothesis Column Drop FreeText Drop columns with long headers (> 256 characters) The TabLib process used to scrape the tables can result in tables with “headers” that are actually just rows of data. One indicator of this is very long headers (e.g. a text column that ends up as a header). Column Drop Numeric Drop columns with names that are numeric. TabLib’s parsing removed tables with all-numeric headers “for most file formats”. This means that (a) some formats were missed, and (b) tables with many numeric headers and even one non-numeric header were still included. Column Drop Missing Drop any column with > x% values that are None, NaN, whitespace or empty string values Columns that are mostly missing will waste compute processing headers; empty cells usually won’t be informative (although sometimes a header alone can be useful). Column English Filtering Drop any text columns where average probability of English over rows is less than p Some tables contain English headers and non-English data. All of our downstream data is English. It’s hard to assess quality of non-English data. Column Drop Constant Drop columns where all values are the same. Constant features are not useful for prediction Row Drop Missing Drop any row with too many values that are None, NaN, whitespace or empty string values Rows with mostly missing data are likely to be uninformative. Row Drop Duplicates Drop duplicate rows This is non-standard in downstream tasks Row Drop PII (regexbased) Drop any row where PII is detected (phone number, email) Tables with small numbers of rows containing PII can still pass through the table-level PII filtering. Row Drop Code (regexbased) Drop any row where code is detected. Tables with small numbers of rows containing code can still pass through the table-level code filtering Row Drop ⌊ Drop any row where any of the values contain ⌊ symbol This is exclusively used as an indicator of hierarchy (again, common in technical documentation such as that found on Github). This is a sign that the row of the table isn’t self-contained and therefore probably not a candidate for a meaningful prediction task. 17 List of Target Column Selection Rules Name Description Motivation / Hypothesis Drop All Unique Drop non-numeric columns if all values all distinct This is probably not a classification target (most likely a unique identifier, a date, a number, etc.) Drop “Unnamed:” Drop any column whose name starts with “Unnamed:” “Unnamed:” is a special prefix used for unnamed columns in an Arrow table; we avoid making predictions on columns where there is no clear semantic information about the prediction target. Drop dates Drop any column with any date or time data type Not useful as classification targets in most cases Drop Too Long Drop any column which, when serialized, is greater than 256 characters This helps avoid choosing free-text columns as targets. Drop dates Drop any column with any date or time data type Not useful as classification targets in most cases A.2 Target Column Selection Our procedure for target column selection is based on the rules described in the above tables. Given a table, we consider all columns to be potentially valid targets unless they do not satisfy one of the listed criteria. Once target candidates are identified for a table, we choose a single candidate at random and use this as the prediction target. When there are both continuous and categorical candidates present, we choose a categorical candidate with probability p = 0.9 and a continuous candidate with probability 1 −p. This decision reflects our qualitative observation that our selection method tends to produce higher-quality categorical columns than numeric columns, and has the effect of showing the model classification tasks more often than (binned) regression tasks during training. In the case where we select a continuous target, we discretize it into a discrete set by selecting a number of quantiles uniformly at random over the interval [3, 8], and then discretizing the target value into these columns. We serialize the resulting quantiles as “less than 1,” “between 1 and 2.5”, “greater than 2.5” etc. Even after filtering, the tables in our pool contain columns which may not be meaningful prediction targets (for example, timestamps or UIDs). For a given table, we propose and apply a series of heuristics for identifying suitable candidates for target columns. These heuristics include excluding columns if: the name is numeric; only one unique value; has unique values for every row (excluding numeric columns); any row has a value longer than 256 characters; the column is of a date or timestamp type. We provide a detailed list of the exact target selection rules in Section A.2, and an implementation in our code. We do not drop such columns from the table; columns not meeting these criteria are simply kept as predictors but will never be used as prediction targets. Once target candidates are identified for a table, we choose a single candidate at random and use this as the prediction target. When there are both continuous and categorical candidates present, we choose a categorical candidate with probability p = 0.9 and a continuous candidate with probability 1 −p. This decision reflects our qualitative observation that our selection method tends to produce higher-quality categorical columns than numeric columns, and has the effect of showing the model classification tasks more often than (binned) regression tasks during training. A.3 Descriptive Statistics This section provides some basic descriptive statistics of the final dataset, shown in Figures 5 and 6. Figure 5a shows that T4 represents a wide variety of data types across its tables, but that data are primarily represented as float, int, and object (string/categorical) data types. Figure 5b shows 18 float64 int64 object bool datetime64[ns] uint64 datetime64[ns, UTC] Data Type of Column 101 102 103 104 105 106 107 Count (or %) of Columns in T4 41.6% 36.5% 21.3% 0.5% 0.1% 0.0% 0.0% Column Data Types Distribution Over T4 Tables (a) 0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Count 1e6 Proportion of Missing Values Distribution 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Distribution Function Histogram (Count) CDF (b) Figure 5: Summary metrics for the T4 dataset. 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Count 1e6 Row Count Distribution 0 100 200 300 400 500 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Count 1e6 Column Count Distribution 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Distribution Function Histogram (Count) CDF 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Distribution Function Histogram (Count) CDF Figure 6: Distribution of counts of rows per table and columns per table in T4. that, while most tables in T4 have little to no missing data, tables can have as much as 10% of data missing (a maximum threshold enforced by heuristics described above). Figure 6 provides a sense of the “shape” of tables in T4, showing that roughly 30% of tables have 64 columns, and roughly 30% have 1, 000 columns, the minimum and maximum number of rows set by our heuristic filters. In contrast, nearly all tables have fewer than 50 columns, with only a very small fraction (< 0.01%) having 500 or more columns (in practice, rows from tables with 500 columns would almost never fit inside the context window of T4 after serialization and tokenization). A.4 Implementation Details Our data processing implementation uses Ray Datasets6 to process tables in parallel. Our pipeline utilizes TabLib’s existing content_hash feature to read only a set of unique tables; this avoids performing an expensive deduplication step during online processing. The shards of TabLib are processed in chunks to avoid extra overhead due to very large collections in Ray. Our pipeline includes certain additional optimizations: for example, in TabLib, data is stored as serialized Arrow bytes (and the deserialized bytes cannot easily be passed between stages of the Ray pipeline); as a result, our pipeline avoids repeatedly deserializing these bytes and only does so at a single filtering step, and at the write step. 6https://docs.ray.io/en/latest/data/overview.html 19 Our preprocessing implementation is provided as a separate library, tabliblib, along with the release of this paper and in the supplementary material. B Training Details TABULA-8B is trained for 40 thousand steps with a global batch size of 24. We use a peak learning rate of 1e−5 warmed up over 10% of the total training to the peak learning rate, and then use a cosine decay schedule to zero. We do not use weight decay. We found that our model was quite sensitive to the selection of the learning rate, and that evaluation loss during training correlated strongly with performance on downstream tasks. C Description of Compute Resources Used Our final training run for TABULA-8B took approximately 6 days on a single node of 8 NVIDIA 80GB A100 GPUs on a commercial cloud provider. For our TabLib filtering and XGboost experiments, we used an academic CPU cluster. Our evaluations were distributed across two academic GPU clusters consisting of NVIDIA 40GB A40 GPUs and NVIDIA A100 GPUs. As a rule of thumb, evaluating the model on a single dataset over a grid of 8 values for k (number of shots) consumes around 4 GPU-hours. We estimate that the total number of GPU hours used across training, evaluation, and development is approximately 5, 000 −10, 000. D Evaluation Details In this section we provide more detail on the benchmarks datasets in our evaluation suite. We do not preprocess the datasets (no normalization, one-hot encoding, etc), as many datasets have been filtered for specific properties, and some of the downstream methods (e.g. TabPFN) perform best when preprocessing is not applied [25]. D.1 Evaluation Datasets D.1.1 UniPredict Benchmark We use the “supervised” subset of 169 datasets from the recently-introduced UniPredict [59] benchmark, obtained through correspondence with the authors. These are high-quality tabular datasets with generally informative column names and a mix of both categorical and continuous targets, drawn directly from Kaggle. The datasets are prostprocessed in [59] using a commercial LLM; however, we do not have access to the results of this postprocessing and instead use the datasets exactly as they are obtained from the Kaggle API. Note that while the model introduced in [59] was trained and tested on separate splits of these datasets, we only use them for testing. We obtain the complete set of tasks, and the corresponding target columns for each task, for UniPredict, via correspondence with the authors of UniPredict. However, we found that several tasks in the benchmark were incorrectly labeled as categorical when, in fact, these tasks were continuous (this is likely due to the use of an LLM to determine the target columns and their attributes in [59]). We manually verify the correct target type (continuous vs. categorical) for each of the 169 datasets, and apply corrections to TODO datasets in the benchmark. The exact set of benchmarks, target columns, and target type (continuous vs. non-continuous) used in our paper are provided in the supplementary material. We note that our results are not directly comparable to the original results in UniPredict. This is for at least two reasons: (1) due to the modifications described above, where there are errors present in the original categorization of the targets (continuous vs. non-continuous) and (2) because some of the Kaggle datasets in UniPredict are continuously updated (e.g. stock datasets) and the data or access dates are not reported in [59]. We do provide our full evaluation suite, including all UniPredict datasets, in the supplementary material in order for future comparisons to our work. 20 D.1.2 Grinsztajn Benchmark The Grinsztajn benchmark [20] is a curated suite of 45 datasets consisting of numeric and categorical features. This dataset is notable in that the original study by Grinsztajn et al. found that gradient boosted decision trees (GBDTs) consistently outperformed deep learning-based methods on these tasks. The benchmark is comprised of a mix of classification and regression tasks; for all regression tasks, we apply the discretization method used in UniPredict [59] and discretize the targets into quartiles. D.1.3 AutoML Multimodal Benchmark The AutoML Multimodal Benchmark [49]7 is a suite of tables which include one or more free-text fields (such as an Airbnb description, or a product review). The benchmark is considered challenging for tree-based methods due to the non-standard text-based features. However, it also poses a challenge for LLMs since some columns can contain highly variable lengths of text. D.1.4 OpenML CC-18 Benchmark The OpenML Curated Classification Benchmark [2] was created by applying filtering rules to extract a high-quality subset from the OpenML platform. The rules include: no artificial data sets, no subsets of larger data sets nor binarizations of other data sets, no data sets which are perfectly predictable by using a single feature or a simple decision tree. D.1.5 OpenML CTR-23 Benchmark The OpenML Curated Tabular Regression (CTR) Benchmark [14] is a curated set of tables for regression drawn from OpenML. The curation process is similar to that of OpenML-CC18, for regression tasks. As in [59], we convert the continuous regression targets into a finite set of discrete labels. We remove the solar_flare task from the benchmark, as 82% of the observations have the same regression target value (0) and thus we cannot apply our quartile-transformation method. D.2 Baselines For the supervised learning baselines (XGBoost, TabPFN, CatBoost, Logistic Regression), we conduct 10 independent trials, drawing separate training sets of size k shots, for each value of k. We use the full remaining dataset as the test set. Hyperparameter tuning: For each of the 10 independent trials, we tune the hyperparameters of the model. For XGBoost, we conduct 10 iterations of hyperparameter tuning using the HyperOpt hyperparameter optimization library and the hyperparameter grid defined in [16]. For TabPFN and L2-regularized Logistic Regression, we conduct a full grid search (since there is only a single hyperparameter). D.2.1 Llama 3 Base Model We use the pretrained Llama 3 model available on Hugging Face. For this model, we do not modify the tokenizer (i.e. by adding special tokens that are used by TABULA-8B), but the serialized data format is identical to the format seen by our model during training (Figure 2b). D.2.2 XGBoost For every dataset and number of shots, we tune the hyperparameters according to the grid from [16]. We use 10 iterations of the adaptive hyperparameter tuning method HyperOpt on this grid with 3-fold cross-validation, whenever the number of samples is greater than or equal to 3; when the number of samples is less than 3, we use the default settings. 7https://github.com/sxjscience/automl_multimodal_benchmark/tree/main 21 D.2.3 TabPFN TabPFN [25] is a hypernetwork that predicts the parameters of a neural network that can be used to classify the data. TabPFN is a pretrained Transformer model that takes a training dataset as input, and produces the parameters of a network as output; that network is then used to classify the test data. TabPFN is widely considered to be among the state-of-the-art methods for prediction on tabular data [25, 33], and has been shown to be especially effective for few-shot learning. We use the official TabPFN implementation8. TabPFN has one tunable hyperparameter, the number of model predictions that are ensembled with feature and class rotations (N_ensemble_configurations in the TabPFN codebase). We sweep over all values in the range [3, 2 · d] where d is the number of features. As noted in the package documentation, when N_ensemble_configurations > 2 · d for a binary classification task, no further averaging is applied. The official implementation of TabPFN has three limitations relevant to our experiments. First, TabPFN is limited to 100 features.9 As a result, when the number of features is greater than 100, we use TabPFN’s feature subsampling, which randomly selects 100 features. Second, TabPFN cannot be trained on datasets that have more than 10 input classes. We do not report the results of experiments where TabPFN cannot be trained on at least one of the 10 random iterates of each value of k. Third, TabPFN cannot be trained on fewer than \|C\| examples, where C is the set of potential classes. D.3 Relative Sample Efficiency For two classifiers f, g, let ND(f, α) and ND(g, α) denote the number of samples required for the classifiers to reach a performance level α on data D. The relative efficiency of f relative to g on dataset D at level α is equal to ND(f, α)/ND(g, α). (1) D.4 Generation Procedure We use the default generation settings of the Llama 3 Hugging Face model.10. This includes: temperature of 0.6, top-p 0.9. We do not tune these generation settings. E Detailed Results E.1 Results Beyond 32 Shots In this section, we provide additional results for larger numbers of shots not provided in the main text. Figure 7 provides extended results for both the baseline models, and for TABULA-8B. As shown in Figure 7, all models tend to improve with more shots. However, on the subset of datasets that can be used for 64- and 128-shot learning, we observe a narrower gap between TABULA-8B and baselines. We hypothesize that this is due to the fact that there is a selection bias: only datasets with small numbers of features and short column headers are candidates for 128-shot learning (due to the limited context window size of TABULA-8B). As a result, this biases those evaluations away from semantically-rich datasets where we expect TABULA-8B to excel. E.2 Additional Baseline Comparisons In this section, we provide comparisons to additional baselines not included in the main text. These include supervised baselines (Logistic Regression, CatBoost), and commercial LLMs (variants of Claude11). 8https://github.com/automl/TabPFN 9https://github.com/automl/TabPFN/blob/main/tabpfn/scripts/transformer_ prediction_interface.py#L105 10https://huggingface.co/meta-llama/Meta-Llama-3-8B 11https://www.anthropic.com/claude 22 0 20 40 60 80 100 120 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 191 32-Shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (a) 0 10 20 30 40 50 60 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 62 64-Shot Benchmark Tasks TabuLa 8B XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (b) 0 20 40 60 80 100 120 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 8 128-Shot Benchmark Tasks TabuLa 8B XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (c) Figure 7: 7a: Results for 32-shot tasks, with extended curves for baselines. 7b, 7c: Results for 64and 128-shot tasks. All models continue to improve as k increases, but the gap between methods may narrow. However, the nature of the datasets that can be used for 64-shot learning with TABULA-8B are considerably different – fewer features, with shorter, less semantically-meaningful column names – which may downwardly bias the observed performance of TABULA-8B with large k. For the Logistic Regression and Catboost baselines, we follow the same hyperparameter tuning and evaluation procedure described in the main text. These results, along with our main results, are presented in Figure 8. 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 272 16-shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Logistic Regression on k Samples CatBoost trained on k samples Random baseline (a) 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 187 32-shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Logistic Regression on k Samples CatBoost trained on k samples Random baseline (b) Figure 8: Few-shot curves for TABULA-8B along with additional supervised baselines (Logistic Regression, CatBoost). We conduct additional comparisons to strong commercial LLMs. In particular, we compare TABULA8B to two variances of Anthropic’s Claude models: (1) Claude Instant, a fast, relatively small model likely to be on the same order of magnitude of compute as TABULA-8B (the exact parameter counts of all Claude models are not publicly disclosed); and (2) Claude 3 Sonnet, a highly performant instruction-tuned LLM likely to be larger than TABULA-8B both in terms of parameter count and total training compute. Due to the cost of accessing these commercial models, we conduct this evaluation on a subset of our evaluation suite. Figure 9 shows that TABULA-8B significantly outperforms both models. Figure 9 shows two particularly interesting results: first, both Claude models show little to no improvement with increasing number of shots. This is also consistent with the behavior of the base Llama 3 model. We hypothesize that this behavior demonstrates the value of explicit training for few-shot learning, and likely 23 highlights the gap betweenn more generalized, task-agnostic training of these models relative to the task-specific training of TABULA-8B. Second, the commercial models perform worse than any other baseline, on average, beyond 3 shots (below which most baselines perform similarly). We hypothesize that this is due, at least in part, to the instruction-following training of the Claude models; no other set of models in our comparison undergo this second post-training phase likely to be utilized in the Claude training pipeline. This instruction-following may improve models alignment with user intentions but could decrease their ability to explicitly learn from data. 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Number of Shots 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 74 16-shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Logistic Regression on k Samples CatBoost trained on k samples Claude 3 Sonnet on k samples Claude Instant on k samples Random baseline 0 5 10 15 20 25 30 Number of Shots 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 46 32-shot Benchmark Tasks TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Logistic Regression on k Samples CatBoost trained on k samples Claude 3 Sonnet on k samples Claude Instant on k samples Random baseline Figure 9: Comparison of few-shot performance with commercial models, Claude Instant and Claude 3 Sonnet, along with other baselines. E.3 Performance on Numeric Tasks One concern with tabular LLMs is their perceived inability to represent numeric data: since language models represent all data as tokens in a learned embedding space, it may be more challenging for language models to learn the complex relationships between numeric features which are required for many classification tasks. In order to provide evidence of TABULA-8B’s performance on numeric data, in this section we report two different slices of our main results. First, we present results on the subset of our evaluation tasks that contain at least one numeric column (int or float data type). This excludes tables that are strictly non-numeric. Second, we present results on only the subset of our evaluation tasks that contain entirely numeric data. We note that both of these subsets exclude tables with purely textual data – precisely the datasets where we might expect language models to perform strongly. The results on these two subsets of our evaluation suite are shown in Figure 10. Figure 10a shows that, on the evaluation subset where all tables contain at least one numeric column, TABULA-8B still outperforms baselines across all numbers of shots. Figure 10a demonstrates that the mere presence of numeric features does not erode the performance of TABULA-8B relative to existing SOTA baseline methods. Figure 10b shows that, on the evaluation subset where all tables contain only numeric columns, TABULA-8B performs on par with existing SOTA baselines but generally only matches their performance. This result is perhaps unsurprising, as numeric-only data is the most advantageous setting for GBDT models and TabPFN (as GBDTs can directly learn splits over numeric values, and TabPFN is exclusively trained on numeric features). However, the ability of TABULA-8B to match these strong baselines, while exceeding them on non-numeric data and possessing capabilities no other baseline possesses (such as zero-shot prediction and transfer), is an indication of its strength and potential utility as a general tabular classifier. 24 0 2 4 6 8 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 287 8-shot Tasks With 1 or More Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 266 16-shot Tasks With 1 or More Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 177 32-shot Tasks With 1 or More Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (a) 0 2 4 6 8 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 59 8-shot Tasks With 100% Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 51 16-shot Tasks With 100% Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 36 32-shot Tasks With 100% Numeric Columns TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (b) Figure 10: Few-shot curves for TABULA-8B on the subset of evaluation tasks that contain at least one numeric column (10a) and on the subset of evaluation tasks that contain entirely numeric columns (10b). F Robustness and Ablation Studies We conduct a series of additional ablation studies to investigate robustness to column ordering, robustness to feature dropout, robustness to header removal, the impact of our causal masking procedure, and the impact of our data filtering procedure. We note that other aspects of our pipeline, such as the target selection procedure and the individual parameters of several of our TabLib processing heuristics, also affect the quality of our resulting model, but a comprehensive evaluation of these individual decisions is left to future work. We describe each ablation study below. F.1 Ablation Study: Row-Causal Table Masking (RCTM) In this section we conduct an ablation study of the row-causal table masking (RCTM) procedure used to train our model. To summarize the masking procedure (also described in Section 3 and Figure 2a): we explicitly allow the model to attend across samples within the same table in a batch. We hypothesize that this structure will encourage few-shot learning and will mitigate catastrophic forgetting which could cause the base model’s few-shot capabilities to deteriorate during fine-tuning. To do this, we design a controlled experiment. Both arms of the experiment use 10% of the compute of TABULA-8B (trained for 4k steps) but are otherwise identical. In one run, we remove the RCTM strategy described in Figure 2a, replacing it with a per-sample causal attention mask (the model is not allowed to attend to any samples besides the target sample, regardless of which table they are derived from). We evaluate both models over the full test suite (all benchmark tasks). The results of this study are shown in Figure 11. Our proposed masking scheme improves the models’ ability to attend across samples, while removing this masking causes the model not to learn from additional shots (for k ≤8) and to deteriorate as the number of shots grows. This figure 25 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Row-Causal Tabular Masking TabuLa 8B Compute-Matched Baseline XGBoost trained + tuned on k samples TabPFN on k samples Random baseline TabuLa 8B, No Tabular Masking Figure 11: Results of an ablation study comparing a model trained without our novel row-causal tabular masking (RCTM) scheme (described in Section 3 and illustrated in Figure 2a) vs. a baseline compute-matched version of TABULA-8B. RCTM improves the models’ ability to attend across samples, while removing RCTM causes the model not to learn from additional shots (for k ≤8) and to deteriorate as the number of shots grows. This result demonstrates the potential loss of few-shot capabilities during fine-tuning if it is not explicitly encouraged by the fine-tuning task. also demonstrates the potential loss of few-shot capabilities during fine-tuning if it is not explicitly encouraged by the fine-tuning task – but also that these capabilities can be maintained or improved over the base model if they are a part of the fine-tuning task. F.2 Robustness Evaluation: Informative Header Removal A potential disadvantage of a language modeling approach to tabular data prediction is that language models may be particularly reliant on semantically-informative column “headers” (column names, the keys in the tabular key-value structure), in contrast to traditional supervised learning methods which do not utilize the headers at all. This risk has been noted in previous work; for example, UniPredict [59] suggests an approach that uses a commercial LLM to “rewrite” the headers of a table to make them more informative. In this work, however, we seek to avoid expensive preprocessing and use only the provided headers from our training data. To understand this sensitivity to the semantic quality of the headers, we conduct a controlled experiment to test the effect of removing informative column headers from a dataset. Specifically, we do the following: starting from a benchmark with high-quality headers (UniPredict), we replace the original headers with “X1”, “X2”, . . . and the target with “Y”. Then, we evaluate TABULA-8B on the data. We do not alter the data itself; only the feature names are replaced. The results of this study are shown in Figure 12. We highlight a few key findings from these results. First, for small number of shots, the semantically-meaningful headers provide a performance benefit: for instance, in the 16-shot subset of Figure 12, semantically-meaningful headers provide a consistent accuracy gain of 3-5pp. Second, TABULA-8B can still outperform supervised baselines even without these headers: for example, Figure 12 shows that TABULA-8B still outperforms all baselines on the benchmark even without semantically-meaningful headers. This finding is further supported by our results on the OpenML benchmarks (CC18, CTR23) in Figure 4; these datasets also tend to have uninformative headers. Third, we observe that the utility of semantically-meaningful headers decreases as the number of shots increase. For example, at 32 shots, the performance with and without the original headers is effectively identical. We hypothesize that, as the number of shots grows, the model is increasingly utilizing the values provided in the shots (and their distribution) and is less reliant on the keys for providing information about the task. Collectively, the results of this ablation study suggest that TABULA-8B is robust to the semantic content of the headers, and that TABULA-8B is capable of providing effective tabular data predictions even in the absence of rich column headers. 26 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Column Headers (UniPredict 16-shot Subset) TabuLa 8B XGBoost trained + tuned on k samples TabPFN on k samples Random baseline TabuLa 8B (informative headers removed) 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Column Headers (UniPredict 32-shot Subset) Figure 12: Results of column header ablation study described in Section F.2. For low numbers of shots, there tends to be a positive effect from informative headers. However, as the number of shots increases, the utility of the headers decreases. F.3 Robustness Evaluation: Feature Dropout A potential risk of using language models for tabular data prediction may be their brittleness: language models, particularly in the few-shot setting, are known to be sensitive to details which should be irrelevant to the task difficulty, including order of examples [3, 32], whitespace [47], and prompt formatting [26, 45, 47]. Here, we are particularly interested in probing robustness to the removal of features. Some supervised models, including XGBoost, can be trained in a way that allows them to handle missing features at inference time, at a cost of slightly decreased predictive accuracy due to the loss of information. However, whether language models possess similar characteristics is unknown. We design an experiment to test how TABULA-8B’s zero- and few-shot performance degrades as features are removed from the evaluation datasets. First, on the training split of each dataset, we train a single XGBoost model using the same hyperparameter tuning procedure used for our baselines, and we extract the feature importance for all features in the dataset according to this model. Next, we evaluate TABULA-8B on each dataset, removing the top k features for k ∈[1, 5]. These features are removed in either descending or ascending order of importance (“important first” and “important last”, respectively). For comparison, we also train and evaluate XGBoost models. For these XGBoost models, we set 1/k fraction of the data to missing, uniformly at random. Then we train a hyperparameter-tuned model on this data and evaluate it on clean test data where the top-k features are set to missing (no other data is set to missing in the test data); this allows us to naturally leverage XGBoost’s robustness to missing data. This experiment is conducted on the same random sample of 32 datasets described in Section F.4. The results of this study are shown in Figure 13. Across 0-,4-,16-,and 32-shot evaluations, TABULA-8B shows a similar or favorable rate of decline in performance, relative to XGBoost, as the number of removed features increases (as indicated by the similar slope of the lines). We note that the XGBoost models have better absolute performance because these are full-shot models trained on the full training split; we use these models to compare the rate of decrease in accuracy as dropout increases, not to compare the accuracy itself. The results in Figure 13 also provide further evidence of when TABULA-8B may be favorable to standard supervised learning methods: namely, when the amount of missing data at test time is large. Finally, Figure 13 suggests that TABULA-8B is not brittle with respect to the features present in our evaluation datasets; removing these features causes only the expected drop in performance (and removing unimportant features is even associated with an increase in performance relative to the full feature set when the number of shots is larger than 8). 27 0 1 2 3 4 5 Features Removed (Important First) 0.3 0.4 0.5 0.6 0.7 0.8 Open-Vocabulary Accuracy Robustness Study - Feature Dropout XGBoost Full-Shot TabuLa 0-shot TabuLa 4-shot TabuLa 16-shot TabuLa 32-shot 0 1 2 3 4 5 Features Removed (Important Last) 0.3 0.4 0.5 0.6 0.7 0.8 Open-Vocabulary Accuracy Robustness Study - Feature Dropout Figure 13: Feature dropout ablation study results. We compare the few-shot performance of TABULA8B to the performance of an XGBoost model trained on the full training split of each dataset, progressively removing features at evaluation time. Features are removed in order of decreasing (“important first”) or increasing (“important last”) variable importance (see Section F.3 for details). TABULA-8B’s performance decreases at a rate consistent with XGBoost. F.4 Robustness Evaluation: Column Order Invariance Recent work has suggested that invariance to column ordering is an important property of tabular foundation models [58]. In this subsection, we conduct a controlled experiment to assess the sensitivity of our model to the ordering of the columns in the dataset. To assess this, we conduct the following experiment. We first choose a random subset of 32 tasks from our evaluation suite (we exclude the tasks from AMLB due to their relatively irregular structure as discussed above since our goal is to compare performance on relatively standard tabular data). For each task, we evaluate the model on a permuted version of the original data: that is, we randomly permute the columns, but otherwise leave the data unchanged. Due to computational constraints, we conduct this evaluation only at a coarse grid of (0, 8, 16, 32) shots. We compare the accuracy on these tasks to the accuracy on the original data. We use the same TABULA-8B model for both the permuted and non-permuted evaluations. The results of this study are shown in Figure 14. We observe a small drop (with Clopper-Pearson intervals overlapping at all points) after permuting the columns, but the general shape and rate of increase of the model under both cases is quite similar. We hypothesize that this drop is due to the fact that many tabular datasets, including those from our evaluation benchmark (which are drawn from manually-curated sources such as Kaggle, OpenML, and UCI Machine Learning Repository), have manually-selected column orderings that slightly improve prediction performance. This might include, for example, a "date" preceding the rest of the columns, or a "high" and "low" column located near each other in a stock dataset. These feature relationships, we hypothesize, can make it easier for models to pool information between related features and may contribute to the small drop observed on permuting the columns. We note that this sensitivity to order has been observed for language models in other contexts [32, 34]. Our results broadly show that TABULA-8B maintains consistent performance above baselines even under feature permutation. However, they also suggest that the sensitivity to prompting and formatting that affects LLMs in other contexts [32, 34] could possibly affect tabular LLMs. We believe further research into this issue is necessary, and may indeed point toward future, more effective methods for leveraging feature ordering to improve model performance. F.5 Ablation Study: Data Filtering We conduct a controlled experiment to assess the impact of our filtering strategies. In particular, we compare a dataset filtered according to the strategies described in Section 4 to a “minimally-filtered” 28 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Permuted Features (16-Shot Subset) TabuLa 8B TabuLa 8B (Permuted Features) XGBoost Trained + Tuned on k Samples TabPFN on k Samples Random Baseline 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Permuted Features (32-Shot Subset) Figure 14: Results of column permutation study described in F.4. We randomly permute the columns for a randomly-selected subset of 32 tasks, and evaluate TABULA-8B on the permuted data. We hypothesize that the slight drop in model performance is due to manually-crafted and semantically meaningful feature orderings in the data. However, our results broadly show that TABULA-8B maintains consistent performance above baselines even under feature permutation. TabLib dataset. We use a “minimally-filtered” dataset rather than an unfiltered dataset because applying no filtering could potentially result in a small number of very large tables dominating training. For the minimally-filtered baseline, we only apply the max-row count filter, max column filter, and max header length filter; the latter two help ensure that the resulting serializations are not too long so that the model is still able to perform few-shot training. The results of this study are shown in Figure 15. While our filtering strategy has a positive impact at larger numbers of shots (k ≥16), there is no impact evident at k < 16, with the minimally-filtered baseline performing similarly. We hypothesize that this is a reflection of the relatively limited additional filtering performed by the rest of our pipeline relative to the “minimal” baseline (which consists mainly of language filtering, PII and code filtering, and heuristics to remove excessive amounts of missing data). Additionally, given the clear impact of improvements in data quality for language model pretraining (e.g. [54]), we hypothesize that further refinements of our filtering pipeline would be likely to achieve further gains over minimal filtering. Finally, we emphasize that some of our filtering strategies (in particular, the PII detection) were designed to improve the safety of the model, not the quality, and we believe that some form of safety filtering should still be used regardless of its downstream effects. F.6 Ablation Study: Base Language Model In this section, we evaluate the improvements of TABULA-8B due to improvements in the underlying base language model. In order to do so, we train variants of TABULA-8B which are identical to the final version, except we use Llama 1 and Llama 2 as the base language models. This allows us to investigate the improvements in TABULA-8B as the underlying language model improves. We compare the Llama 1 and Llama 2 variants to a compute-matched Llama 3 variant (we use only 10% of the compute relative to our final model, as in our other ablation studies, but also compare to the 100%-compute TABULA-8B for reference). We note that, due to the smaller context sizes of the Llama 1 and Llama 2 models, we ensure the comparisons are example-matched; that is, we train Llama 2 (context size 4096) for 2x the update steps relative to Llama 3 (context size 8192), and train Llama 1 (context size 2048) for 4x the update steps. This ensures that all models see roughly the same number of tokens or examples during fine-tuning. The results of this study are shown in Figure 16. We only report the results up to 8 shots to allow for fair comparisons across all models, as Llama 1’s context size of 2048 is 4x smaller than TABULA-8B and can only fit up to 8 shots for many tasks. Figure 16 shows the clear improvement from better base models: as the underlying base models improve (Llama 1 →Llama 2 →Llama 3), the fine-tuned 29 0 5 10 15 20 25 30 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Ablation Study - Full vs. Minimal Data Filtering UniPredict Benchmark Tasks TabuLa 8B compute-matched baseline XGBoost Trained + Tuned on k Samples TabPFN on k Samples Random baseline TabuLa 8B (minimally-filtered TabLib) Random baseline Figure 15: Compute-matched comparison of TABULA-8B with our full filtering vs. an identical model trained on minimally-filtered TabLib (both models are trained for 10% of the number of steps as the final model, with identical hyperparameters). model also improves. These results provide hope that further improvements in language modeling could also lead to gains in tabular models. 0 1 2 3 4 5 6 7 8 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy Zero- and Few-Shot Performance Over 104 8-shot Benchmark Tasks TabuLa 8B TabuLa 8B Compute-Matched Baseline Llama 2 7B (example-matched) Llama 1 7B (example-matched) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline Figure 16: Results of base model ablation study. G Benchmark Contamination Study G.1 Identifying Potential Contamination Identifying contamination in tabular data is complex. As mentioned above, tabular data is invariant to permutations of rows and columns; as a result, the “same” tabular dataset could appear in a number of different permutations of its respective rows or columns. As a result, so-called “exact” deduplication methods are likely to be imperfect for tabular data (we also perform exact deduplication in our filtering step, so at most each individual dataset should only appear once in T4). When assessing the impact of duplication on our downstream evaluations, we are particularly concerned about the possibility of a model overfitting to the overall features and distribution of a dataset, not individual points in that distirbution – a model could learn unwanted information about a 30 Benchmark Potentially Leaked Tables Count (% of Benchmark Tables) Fuzzy Deduplication Strict Deduplication OpenML-CC18 23 (31.9%) 13 (18.1%) OpenML-CTR23 12 (34.2%) 6 (17.1%) AMLB 1 (12.5%) 0 UniPredict 50 (29.5%) 39 (23%) Grinsztajn 16 (35.5%) 4 (8.9%) Table 1: Counts of potentially contaminated tables according to “fuzzy” and “strict” procedures described. Note that “fuzzy” decontamination is stricter and is more likely to generate both true and false positives when checking for contamination. test set, for example, by observing points strictly from the training set of an i.i.d. split. As a result, we focus on eliminating datasets which have the same schema as a proxy for a dataset; we do not search for individual data points within that schema. We propose two levels of searching for contamination; we refer to these methods as “fuzzy” and “strict”. In fuzzy deduplication, we search for whether every column name in an evaluation dataset is present in the training dataset. In strict deduplication, we further add the condition that the number of columns must be identical. Note that we do not compute a matching directly between the columns of the two datasets, as checking for membership is much more efficient than checking for a compatible mapping between columns and our datasets can be up to 512 columns. Note that fuzzy deduplication will potentially exclude more datasets at the risk of potentially more false positives, since strict deduplication only adds conditions to the fuzzy deduplication. Fuzzy is therefore a more conservative deduplication mechanism than strict. We search over all 1.55M tables in T4 and apply both “fuzzy” and “strict” checks. Some descriptive metrics for this search are shown in Table 1. It is perhaps expected that, in most cases, benchmark tables pass our T4 filtering procedure and make it into T4 (one notable exception is AMLB, where most tables likely fail to pass our rules which filter for cells with large numbers of characters). Indeed, these are public benchmarks designed for learning on tabular data, and TabLib includes a significant component of datasets sourced from GitHub; many users likely work with these datasets and have uploaded them to GitHub. G.2 Impact of Contamination In this section we investigate the impact of potential contamination in our training data. In particular, we investigate whether the number of repetitions of a dataset in the training data is correlated with the downstream performance on that task, and we assess performance on “potentially contaminated” vs. “decontaminated” tasks. Figure 18 shows the relationship between the number of potential contamination instances for each dataset, and the TABULA-8B accuracy on that dataset. We find no clear relationship between this potential contamination and downstream performance, and believe that this reflects, at least in part, the conservative nature of our contamination test, which is likely to have may false positives for datasets with generic column names that may occur frequently in TabLib (for example, columns “Date”, “Open”, “Close” are common among stock datasets; columns “v1” “v2” are common generic variable names). Additionally, we believe that other considerations likely explain the lack of correlation between (potential) contamination in the training data and downstream benchmark berformance. For example: • TABULA-8B does not train on the full corpus; datasets that appear a small number of times may in fact not be seen during training. • Prior work has suggested mixed impact of contamination [e.g. 6, 40]; contamination does not always improve performance and can sometimes reduce performance. • We are only fine-tuning the model which can be thought of as an alignment step. It is possible that contamination in pretraining affects this more and that fine-tuning is less susceptible to memorization. 31 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy All Tasks (Potentially Contaminated Subset) TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (a) 0 2 4 6 8 10 12 14 16 Number of Shots 0.2 0.3 0.4 0.5 0.6 0.7 Open-Vocabulary Accuracy All Tasks (Decontaminated Subset) TabuLa 8B Llama 3 8B (no fine-tuning) XGBoost trained + tuned on k samples TabPFN on k samples Random baseline (b) Figure 17: 17a: Results curves on the subset of tasks identified as potentially contaminated according to our fuzzy decontamination procedure. 17b: Results curves on the subset of tasks not identified as potentially contaminated according to our fuzzy decontamination procedure. 0 200 400 600 800 1000 Duplicates Count (Fuzzy) 0.0 0.2 0.4 0.6 0.8 1.0 Test Accuracy of TabuLa-8B 0 Shots Benchmark grinsztajn_clf openml_cc18 unipredict openml_ctr23 amlb grinsztajn_reg 0 200 400 600 800 1000 Duplicates Count (Fuzzy) 0.0 0.2 0.4 0.6 0.8 1.0 Test Accuracy of TabuLa-8B 4 Shots 0 200 400 600 800 1000 Duplicates Count (Fuzzy) 0.2 0.4 0.6 0.8 1.0 Test Accuracy of TabuLa-8B 16 Shots 0 200 400 600 800 1000 Duplicates Count (Fuzzy) 0.2 0.4 0.6 0.8 1.0 Test Accuracy of TabuLa-8B 32 Shots Figure 18: Contamination rates vs. accuracy across varying numbers of shots. We find no clear relationship between contamination in the training set and downstream task performance. • Due to our use of deduplication, if a dataset does recur, it is not identical – so the model never really sees an identical table more than once unless we make multiple passes over the training set or sample with replacement. • Due to our use of row-level deduplication and random shuffling, the exact evaluation datasets in the same order are unlikely to ever be seen by the model even if provided during training; this may be a guard against memorization. We also separately report our results on the “possibly contaminated” vs. “decontaminated” subset of our evaluation suite, in Figure 17. Figure 17 shows that our model (and all baseline models) tend to perform better on the decontaminated subset. We hypothesize that this reflects a few factors. In particular, datasets in the “decontaminated” subset are likely to have unique column names. This “uniqueness” likely correlates positively with semantic quality; our model will tend to perform better on such datasets. Furthermore, we hypothesize that this reflects the strictness of our fuzzy decontamination check: it is likely that many of the tables flagged as “potentially contaminated” are false positives (our manual inspection confirmed this in many cases, although it is difficult to verify that two shuffled tabular datasets are identical; we leave such an analysis to future work). 32 H List of Evaluation Datasets and Per-Task Results We provide results for TABULA-8B on each individual task, along with the accuracy of a randomclass predictor (which is equivalent to 1 / number of classes) in this section. For the complete results for all values of shots and all baselines, see the supplementary material. Note that ‘NA’ results for TABULA-8B indicate that the serialized data with a given value of shots k exceeds the model’s context window size. TabuLa-8B XGBoost Random Task 0 4 32 4 32 amlb/data_scientist_salary 0.328 0.398 0.391 0.202 0.283 0.167 amlb/imdb_genre_prediction 0.844 0.742 0.828 0.528 0.554 0.500 amlb/jigsaw_unintended_bias100K 0.938 0.922 NA 0.942 NA 0.500 amlb/kick_starter_funding 0.594 0.680 0.609 0.644 0.649 0.500 amlb/melbourne_airbnb 0.727 NA NA NA NA 0.100 amlb/news_channel 0.297 0.227 NA 0.213 NA 0.167 amlb/product_sentiment_machine_hack 0.406 0.438 0.531 0.542 0.762 0.250 amlb/wine_reviews 0.648 0.711 NA 0.080 NA 0.033 grinsztajn/cat_clf/albert 0.445 0.500 NA 0.499 NA 0.500 grinsztajn/cat_clf/compas-two-years 0.547 0.539 0.602 0.499 0.582 0.500 grinsztajn/cat_clf/covertype 0.422 0.570 NA 0.500 NA 0.500 grinsztajn/cat_clf/default-of-credit-card-clients 0.508 0.531 0.539 0.500 0.618 0.500 grinsztajn/cat_clf/electricity 0.461 0.641 0.664 0.503 0.644 0.500 grinsztajn/cat_clf/eye_movements 0.508 0.430 NA 0.504 NA 0.500 grinsztajn/cat_clf/road-safety 0.508 0.477 NA 0.502 NA 0.500 grinsztajn/cat_reg/Bike_Sharing_Demand 0.352 0.344 0.508 0.250 0.424 0.250 grinsztajn/cat_reg/Brazilian_houses 0.344 0.547 0.797 0.256 0.769 0.250 grinsztajn/cat_reg/Mercedes_Benz_Greener_Manufa... 0.266 NA NA NA NA 0.250 grinsztajn/cat_reg/OnlineNewsPopularity 0.289 0.273 NA 0.253 NA 0.250 grinsztajn/cat_reg/SGEMM_GPU_kernel_performance 0.094 0.602 0.898 0.249 0.891 0.250 grinsztajn/cat_reg/analcatdata_supreme 0.461 0.453 0.914 0.624 0.980 0.333 grinsztajn/cat_reg/black_friday 0.203 0.234 0.297 0.250 0.331 0.250 grinsztajn/cat_reg/diamonds 0.344 0.547 0.688 0.248 0.736 0.250 grinsztajn/cat_reg/house_sales 0.258 0.453 NA 0.249 NA 0.250 grinsztajn/cat_reg/nyc-taxi-green-dec-2016 0.242 0.445 NA 0.249 NA 0.250 grinsztajn/cat_reg/particulate-matter-ukair-2017 0.305 0.375 NA 0.248 NA 0.250 grinsztajn/cat_reg/visualizing_soil 0.281 0.383 0.688 0.248 0.789 0.250 grinsztajn/cat_reg/yprop_4_1 0.203 0.219 NA 0.251 NA 0.250 grinsztajn/num_clf/Diabetes130US 0.531 0.508 0.484 0.500 0.523 0.500 grinsztajn/num_clf/MagicTelescope 0.555 0.703 0.695 0.494 0.670 0.500 grinsztajn/num_clf/MiniBooNE 0.461 0.539 NA 0.505 NA 0.500 grinsztajn/num_clf/bank-marketing 0.500 0.578 0.656 0.502 0.695 0.500 grinsztajn/num_clf/california 0.453 0.578 0.742 0.500 0.720 0.500 grinsztajn/num_clf/covertype 0.500 0.484 0.477 0.500 0.553 0.500 grinsztajn/num_clf/credit 0.562 0.578 0.656 0.505 0.659 0.500 grinsztajn/num_clf/default-of-credit-card-clients 0.516 0.477 0.523 0.500 0.601 0.500 grinsztajn/num_clf/electricity 0.547 0.609 0.602 0.503 0.655 0.500 grinsztajn/num_clf/eye_movements 0.469 0.547 NA 0.504 NA 0.500 grinsztajn/num_clf/heloc 0.500 0.508 NA 0.494 NA 0.500 grinsztajn/num_clf/house_16H 0.477 0.609 NA 0.503 NA 0.500 grinsztajn/num_clf/jannis 0.500 0.594 NA 0.500 NA 0.500 grinsztajn/num_clf/pol 0.523 0.484 0.773 0.512 0.810 0.500 grinsztajn/num_reg/Ailerons 0.266 0.289 NA 0.278 NA 0.250 grinsztajn/num_reg/Bike_Sharing_Demand 0.234 0.414 0.398 0.250 0.424 0.250 grinsztajn/num_reg/Brazilian_houses 0.250 0.570 0.812 0.256 0.769 0.250 grinsztajn/num_reg/MiamiHousing2016 0.250 0.352 NA 0.250 NA 0.250 grinsztajn/num_reg/california 0.289 0.406 NA 0.247 NA 0.250 grinsztajn/num_reg/cpu_act 0.359 0.320 NA 0.272 NA 0.250 33 grinsztajn/num_reg/diamonds 0.391 0.602 0.688 0.248 0.742 0.250 grinsztajn/num_reg/elevators 0.188 0.266 NA 0.266 NA 0.250 grinsztajn/num_reg/fifa 0.250 0.406 0.469 0.248 0.434 0.250 grinsztajn/num_reg/house_16H 0.336 0.320 NA 0.249 NA 0.250 grinsztajn/num_reg/house_sales 0.328 0.492 NA 0.249 NA 0.250 grinsztajn/num_reg/houses 0.258 0.312 0.445 0.247 0.418 0.250 grinsztajn/num_reg/isolet 0.262 NA NA NA NA 0.250 grinsztajn/num_reg/medical_charges 0.328 0.555 0.727 0.249 0.801 0.250 grinsztajn/num_reg/nyc-taxi-green-dec-2016 0.273 0.406 NA 0.249 NA 0.250 grinsztajn/num_reg/pol 0.602 0.617 0.867 0.683 0.815 0.500 grinsztajn/num_reg/sulfur 0.258 0.227 0.383 0.253 0.441 0.250 grinsztajn/num_reg/superconduct 0.242 0.438 NA 0.247 NA 0.250 grinsztajn/num_reg/wine_quality 0.578 0.508 0.516 0.538 0.622 0.333 grinsztajn/num_reg/year 0.227 0.289 NA 0.262 NA 0.250 openml_cc18/Fashion-MNIST 0.070 NA NA NA NA 0.100 openml_cc18/GesturePhaseSegmentationProcessed 0.266 0.312 NA 0.270 NA 0.200 openml_cc18/MiceProtein 0.125 0.211 NA 0.118 NA 0.125 openml_cc18/PhishingWebsites 0.516 0.656 NA 0.500 NA 0.500 openml_cc18/adult 0.727 0.812 0.828 0.763 0.764 0.500 openml_cc18/analcatdata_authorship 0.234 0.438 NA 0.365 NA 0.250 openml_cc18/analcatdata_dmft 0.125 0.156 0.125 0.182 0.188 0.167 openml_cc18/bank-marketing 0.508 0.758 0.844 0.889 0.880 0.500 openml_cc18/banknote-authentication 0.477 0.680 0.922 0.509 0.862 0.500 openml_cc18/blood-transfusion-service-center 0.516 0.664 0.586 0.787 0.749 0.500 openml_cc18/breast-w 0.938 0.961 0.984 0.569 0.900 0.500 openml_cc18/car 0.789 0.898 0.898 0.763 0.755 0.250 openml_cc18/churn 0.703 0.758 0.914 0.870 0.863 0.500 openml_cc18/cmc 0.320 0.383 0.352 0.330 0.450 0.333 openml_cc18/cnae-9 0.172 NA NA NA NA 0.111 openml_cc18/connect-4 0.297 0.414 NA 0.522 NA 0.333 openml_cc18/credit-approval 0.484 0.555 0.742 0.428 0.764 0.500 openml_cc18/credit-g 0.508 0.523 0.703 0.500 0.725 0.500 openml_cc18/diabetes 0.617 0.656 0.680 0.661 0.722 0.500 openml_cc18/dna 0.312 0.430 NA 0.467 NA 0.333 openml_cc18/electricity 0.438 0.633 0.711 0.544 0.695 0.500 openml_cc18/eucalyptus 0.273 0.352 0.359 0.211 0.407 0.200 openml_cc18/first-order-theorem-proving 0.211 0.195 NA 0.284 NA 0.167 openml_cc18/har 0.125 NA NA NA NA 0.167 openml_cc18/isolet 0.022 NA NA NA NA 0.038 openml_cc18/jm1 0.812 0.750 0.805 0.746 0.784 0.500 openml_cc18/jungle_chess_2pcs_raw_endgame_complete 0.477 0.555 0.516 0.409 0.589 0.333 openml_cc18/kc1 0.852 0.836 0.820 0.775 0.843 0.500 openml_cc18/kr-vs-kp 0.469 0.508 NA 0.490 NA 0.500 openml_cc18/letter 0.039 0.070 0.266 0.038 0.200 0.038 openml_cc18/madelon 0.531 NA NA NA NA 0.500 openml_cc18/mfeat-factors 0.078 0.172 NA 0.092 NA 0.100 openml_cc18/mfeat-fourier 0.086 0.172 NA 0.092 NA 0.100 openml_cc18/mfeat-karhunen 0.078 0.141 NA 0.092 NA 0.100 openml_cc18/mfeat-morphological 0.141 0.188 0.555 0.092 0.493 0.100 openml_cc18/mfeat-pixel 0.109 NA NA NA NA 0.100 openml_cc18/mfeat-zernike 0.141 0.141 NA 0.092 NA 0.100 openml_cc18/mnist_784 0.148 NA NA NA NA 0.100 openml_cc18/nomao 0.398 0.742 NA 0.542 NA 0.500 openml_cc18/numerai28.6 0.539 0.469 NA 0.495 NA 0.500 openml_cc18/optdigits 0.070 0.094 NA 0.092 NA 0.100 openml_cc18/ozone-level-8hr 0.445 0.852 NA 0.969 NA 0.500 openml_cc18/pc1 0.766 0.945 0.898 0.964 0.964 0.500 openml_cc18/pc3 0.820 0.875 NA 0.834 NA 0.500 openml_cc18/pc4 0.609 0.914 NA 0.890 NA 0.500 34 openml_cc18/pendigits 0.039 0.242 0.523 0.099 0.534 0.100 openml_cc18/phoneme 0.453 0.641 0.734 0.624 0.723 0.500 openml_cc18/qsar-biodeg 0.508 0.523 NA 0.579 NA 0.500 openml_cc18/satimage 0.180 0.320 NA 0.186 NA 0.167 openml_cc18/segment 0.195 0.438 NA 0.144 NA 0.143 openml_cc18/semeion 0.148 NA NA NA NA 0.100 openml_cc18/sick 0.938 0.914 NA 0.950 NA 0.500 openml_cc18/spambase 0.555 0.680 NA 0.560 NA 0.500 openml_cc18/splice 0.336 0.328 NA 0.378 NA 0.333 openml_cc18/steel-plates-fault 0.141 0.273 NA 0.201 NA 0.143 openml_cc18/texture 0.070 0.180 NA 0.090 NA 0.091 openml_cc18/tic-tac-toe 0.383 0.609 0.523 0.454 0.702 0.500 openml_cc18/vehicle 0.219 0.258 0.484 0.239 0.495 0.250 openml_cc18/vowel 0.055 0.125 0.172 0.106 0.358 0.091 openml_cc18/wall-robot-navigation 0.250 0.305 NA 0.398 NA 0.250 openml_cc18/wilt 0.594 0.812 0.969 0.959 0.957 0.500 openml_ctr23/Moneyball 0.477 0.375 0.609 0.243 0.595 0.250 openml_ctr23/QSAR_fish_toxicity 0.266 0.305 0.484 0.245 0.462 0.250 openml_ctr23/abalone 0.273 0.430 0.406 0.296 0.483 0.250 openml_ctr23/airfoil_self_noise 0.258 0.305 0.383 0.242 0.379 0.250 openml_ctr23/auction_verification 0.266 0.344 0.398 0.252 0.652 0.250 openml_ctr23/brazilian_houses 0.438 0.578 0.633 0.256 0.551 0.250 openml_ctr23/california_housing 0.320 0.375 0.461 0.247 0.419 0.250 openml_ctr23/cars 0.359 0.336 0.656 0.228 0.637 0.250 openml_ctr23/concrete_compressive_strength 0.328 0.484 0.562 0.246 0.446 0.250 openml_ctr23/cps88wages 0.391 0.367 0.422 0.255 0.332 0.250 openml_ctr23/cpu_activity 0.250 0.375 NA 0.272 NA 0.250 openml_ctr23/diamonds 0.438 0.680 0.742 0.247 0.751 0.250 openml_ctr23/energy_efficiency 0.391 0.461 0.719 0.238 0.749 0.250 openml_ctr23/fifa 0.352 0.359 NA 0.243 NA 0.250 openml_ctr23/fps_benchmark 0.242 0.367 NA 0.250 NA 0.250 openml_ctr23/geographical_origin_of_music 0.281 NA NA NA NA 0.250 openml_ctr23/grid_stability 0.242 0.297 NA 0.239 NA 0.250 openml_ctr23/health_insurance 0.406 0.500 0.633 0.482 0.591 0.333 openml_ctr23/kin8nm 0.312 0.242 0.297 0.251 0.342 0.250 openml_ctr23/kings_county 0.305 0.422 NA 0.249 NA 0.250 openml_ctr23/miami_housing 0.273 0.391 NA 0.250 NA 0.250 openml_ctr23/naval_propulsion_plant 0.211 0.227 NA 0.249 NA 0.250 openml_ctr23/physiochemical_protein 0.242 0.289 0.281 0.250 0.334 0.250 openml_ctr23/pumadyn32nh 0.219 0.273 NA 0.251 NA 0.250 openml_ctr23/red_wine 0.523 0.508 0.656 0.476 0.619 0.333 openml_ctr23/sarcos 0.227 0.203 NA 0.254 NA 0.250 openml_ctr23/socmob 0.391 0.523 0.695 0.248 0.598 0.250 openml_ctr23/space_ga 0.203 0.219 0.305 0.261 0.412 0.250 openml_ctr23/student_performance_por 0.375 0.258 NA 0.274 NA 0.250 openml_ctr23/superconductivity 0.250 0.352 NA 0.247 NA 0.250 openml_ctr23/video_transcoding 0.250 0.336 NA 0.249 NA 0.250 openml_ctr23/wave_energy 0.234 0.273 NA 0.256 NA 0.250 openml_ctr23/white_wine 0.484 0.594 0.594 0.651 0.660 0.333 unipredict/aakashjoshi123/exercise-and-fitness-... 0.383 0.641 0.766 0.253 0.761 0.250 unipredict/aakashjoshi123/spotify-top-hits-data 0.555 0.797 0.781 0.750 0.694 0.077 unipredict/abcsds/pokemon 0.977 0.984 0.992 0.060 0.128 0.059 unipredict/adityakadiwal/water-potability 0.562 0.469 0.516 0.524 0.591 0.500 unipredict/agirlcoding/all-space-missions-from-... 0.891 0.875 0.867 0.910 0.874 0.333 unipredict/ahsan81/food-ordering-and-delivery-a... 0.273 0.266 0.242 0.296 0.323 0.250 unipredict/ahsan81/superstore-marketing-campaig... 0.516 0.727 0.766 0.848 0.836 0.500 unipredict/akshaydattatraykhare/diabetes-dataset 0.586 0.742 0.672 0.661 0.704 0.500 unipredict/alexisbcook/pakistan-intellectual-ca... 0.609 0.656 NA 0.592 NA 0.083 unipredict/alirezachahardoli/bank-personal-loan-1 0.758 0.773 0.914 0.920 0.913 0.500 35 unipredict/altruistdelhite04/gold-price-data 0.562 0.680 0.867 0.241 0.603 0.250 unipredict/amirhosseinmirzaie/countries-life-ex... 0.766 0.719 NA 0.244 NA 0.250 unipredict/amirhosseinmirzaie/pistachio-types-d... 0.547 0.570 NA 0.523 NA 0.500 unipredict/ananthr1/weather-prediction 0.703 0.828 0.812 0.369 0.787 0.200 unipredict/andrewmvd/fetal-health-classification 0.211 0.734 NA 0.736 NA 0.333 unipredict/andrewmvd/udemy-courses 1.000 1.000 0.992 0.311 0.433 0.250 unipredict/arashnic/time-series-forecasting-wit... 0.992 1.000 0.984 0.251 0.903 0.250 unipredict/arnabchaki/data-science-salaries-2023 0.898 0.961 0.969 0.256 0.898 0.250 unipredict/arnabchaki/indian-restaurants-2023 0.281 0.289 0.242 0.257 0.308 0.250 unipredict/arnavsmayan/netflix-userbase-dataset 0.344 0.305 0.352 0.362 0.438 0.333 unipredict/arnavsmayan/vehicle-manufacturing-da... 0.055 0.047 0.062 0.077 0.094 0.059 unipredict/arslanr369/bitcoin-price-2014-2023 1.000 0.992 0.984 0.247 0.899 0.250 unipredict/ashishkumarjayswal/diabetes-dataset 0.516 0.680 0.742 0.661 0.712 0.500 unipredict/ashishkumarjayswal/movies-updated-data 0.656 0.555 0.641 0.158 0.276 0.091 unipredict/atharvaingle/crop-recommendation-dat... 0.055 0.320 0.695 0.049 0.438 0.045 unipredict/awaiskaggler/insurance-csv 0.398 0.547 0.695 0.245 0.519 0.250 unipredict/azminetoushikwasi/-lionel-messi-all-... 0.219 0.414 0.469 0.634 0.594 0.111 unipredict/barun2104/telecom-churn 0.844 0.836 0.836 0.717 0.857 0.500 unipredict/bhanupratapbiswas/bollywood-actress-... 0.258 0.430 0.656 0.567 0.440 0.125 unipredict/bhanupratapbiswas/fashion-products 0.359 0.367 0.406 0.328 0.349 0.333 unipredict/bhanupratapbiswas/uber-data-analysis 0.617 0.820 0.922 0.931 0.928 0.500 unipredict/bhanupratapbiswas/world-top-billiona... 0.383 0.531 NA 0.254 NA 0.250 unipredict/bharath011/heart-disease-classificat... 0.531 0.641 0.844 0.564 0.906 0.500 unipredict/bhavkaur/hotel-guests-dataset 0.430 0.594 0.867 0.865 0.836 0.333 unipredict/bhavkaur/simplified-titanic-dataset 0.555 0.523 0.734 0.647 0.735 0.500 unipredict/blastchar/telco-customer-churn 0.711 0.719 0.711 0.696 0.748 0.500 unipredict/bretmathyer/telemedicine-used 0.500 0.555 NA 0.494 NA 0.500 unipredict/buntyshah/auto-insurance-claims-data 0.602 0.594 NA 0.700 NA 0.500 unipredict/carolzhangdc/imdb-5000-movie-dataset 0.484 0.516 NA 0.247 NA 0.250 unipredict/chirin/africa-economic-banking-and-s... 0.953 0.953 0.977 0.877 0.969 0.500 unipredict/christinestevens/cstevens-peloton-data 0.984 0.992 1.000 0.151 0.164 0.143 unipredict/cpluzshrijayan/milkquality 0.406 0.406 0.711 0.361 0.836 0.333 unipredict/crxxom/manhwa-dataset 0.695 0.859 NA 0.589 NA 0.250 unipredict/dansbecker/aer-credit-card-data 0.609 0.742 0.930 0.545 0.967 0.500 unipredict/deependraverma13/diabetes-healthcare... 0.602 0.680 0.734 0.661 0.706 0.500 unipredict/desalegngeb/german-fintech-companies 0.758 0.789 NA 0.280 NA 0.250 unipredict/dileep070/heart-disease-prediction-u... 0.789 0.750 0.805 0.768 0.827 0.500 unipredict/dsfelix/us-stores-sales 0.547 0.648 NA 0.258 NA 0.250 unipredict/elakiricoder/gender-classification-d... 0.594 0.719 0.883 0.502 0.935 0.500 unipredict/fedesoriano/stroke-prediction-dataset 0.945 0.977 0.945 0.951 0.950 0.500 unipredict/gabrielsantello/cars-purchase-decisi... 0.383 0.609 0.820 0.580 0.843 0.500 unipredict/gauravduttakiit/resume-dataset 0.906 0.969 NA 0.043 NA 0.040 unipredict/geomack/spotifyclassification 0.422 0.641 0.984 0.536 0.981 0.500 unipredict/gyanprakashkushwaha/laptop-price-pre... 0.328 0.562 0.570 0.251 0.550 0.250 unipredict/hansrobertson/american-companies-pro... 0.250 0.297 0.242 0.250 0.314 0.250 unipredict/harishkumardatalab/medical-insurance... 0.336 0.586 0.750 0.235 0.681 0.250 unipredict/harshitshankhdhar/imdb-dataset-of-to... 0.430 0.445 NA 0.310 NA 0.250 unipredict/hashemi221022/bank-loans 0.773 0.844 0.867 0.920 0.887 0.500 unipredict/hashemi221022/diabetes 0.508 0.750 0.656 0.661 0.721 0.500 unipredict/hawkingcr/airbnb-for-boston-with-fra... 0.789 0.719 0.797 0.780 0.796 0.500 unipredict/hemanthhari/psycological-effects-of-... 0.414 0.391 0.570 0.245 0.615 0.143 unipredict/hesh97/titanicdataset-traincsv 0.797 0.734 0.703 0.587 0.706 0.500 unipredict/iamsumat/spotify-top-2000s-mega-dataset 0.305 0.406 0.516 0.258 0.276 0.250 unipredict/iqmansingh/company-employee-dataset 0.641 0.539 0.586 0.073 0.199 0.050 unipredict/ishadss/productivity-prediction-of-g... 0.391 0.430 NA 0.233 NA 0.250 unipredict/jainilcoder/netflix-stock-price-pred... 1.000 0.992 0.992 0.261 0.884 0.250 unipredict/jillanisofttech/brain-stroke-dataset 0.969 0.961 0.953 0.948 0.948 0.500 unipredict/kabure/german-credit-data-with-risk 0.594 0.586 0.695 0.500 0.720 0.500 unipredict/kandij/diabetes-dataset 0.562 0.758 0.711 0.661 0.717 0.500 36 unipredict/kanths028/usa-housing 0.195 0.281 NA 0.242 NA 0.250 unipredict/kingabzpro/cosmetics-datasets 0.859 0.836 NA 0.164 NA 0.167 unipredict/kreeshrajani/human-stress-prediction 0.547 0.633 0.664 0.500 0.496 0.500 unipredict/kumargh/pimaindiansdiabetescsv 0.117 0.172 0.141 0.156 0.171 0.077 unipredict/larsen0966/student-performance-data-set 0.297 0.266 NA 0.088 NA 0.077 unipredict/lightonkalumba/us-womens-labor-force... 0.891 0.992 1.000 0.508 0.986 0.500 unipredict/mahnazarjmand/bank-personal-loan 0.758 0.828 0.922 0.920 0.918 0.500 unipredict/maryalebron/life-expectancy-data 0.023 0.023 NA 0.030 NA 0.028 unipredict/maryammanoochehry/bank-personal-loan 0.844 0.891 0.883 0.920 0.914 0.500 unipredict/mathchi/diabetes-data-set 0.500 0.695 0.680 0.661 0.704 0.500 unipredict/mayankpatel14/second-hand-used-cars-... 0.234 0.203 0.297 0.226 0.659 0.250 unipredict/mayurdalvi/simple-linear-regression-... 0.453 0.516 0.594 0.548 0.543 0.500 unipredict/mayuriawati/bangalore-chain-restaura... 0.891 0.875 NA 0.086 NA 0.045 unipredict/mazlumi/ielts-writing-scored-essays-... 0.109 0.227 NA 0.122 NA 0.083 unipredict/mfaisalqureshi/spam-email 0.703 0.891 0.984 0.846 0.846 0.500 unipredict/mirichoi0218/insurance 0.250 0.633 0.719 0.258 0.708 0.250 unipredict/nancyalaswad90/review 0.617 0.641 0.680 0.661 0.703 0.500 unipredict/naveenkumar20bps1137/predict-student... 0.039 0.086 NA 0.061 NA 0.059 unipredict/nikhil1e9/netflix-stock-price 0.984 0.977 0.984 0.246 0.898 0.250 unipredict/noordeen/insurance-premium-prediction 0.461 0.531 0.734 0.258 0.677 0.250 unipredict/oles04/bundesliga-seasons 1.000 1.000 NA 0.529 NA 0.500 unipredict/oles04/top-leagues-player 0.484 0.641 0.609 0.264 0.271 0.250 unipredict/patelprashant/employee-attrition 0.805 0.820 NA 0.837 NA 0.500 unipredict/pavansubhasht/ibm-hr-analytics-attri... 0.820 0.883 NA 0.837 NA 0.500 unipredict/phangud/spamcsv 0.664 0.883 0.969 0.846 0.846 0.500 unipredict/prevek18/ames-housing-dataset 0.445 0.625 NA 0.252 NA 0.250 unipredict/primaryobjects/voicegender 0.445 0.516 NA 0.500 NA 0.500 unipredict/prkhrawsthi/bitcoin-usd-daily-price-... 0.961 0.969 0.984 0.248 0.899 0.250 unipredict/rajyellow46/wine-quality 0.352 0.453 0.438 0.372 0.426 0.143 unipredict/ravibarnawal/mutual-funds-india-deta... 0.203 0.227 0.219 0.228 0.310 0.167 unipredict/receplyasolu/6k-weather-labeled-spot... 0.141 0.172 0.273 0.124 0.211 0.125 unipredict/redwankarimsony/heart-disease-data 0.273 0.484 0.555 0.430 0.528 0.200 unipredict/reihanenamdari/breast-cancer 0.344 0.289 0.297 0.250 0.295 0.250 unipredict/rishikeshkonapure/hr-analytics-predi... 0.820 0.836 NA 0.837 NA 0.500 unipredict/rkiattisak/student-performance-in-ma... 0.453 0.477 0.547 0.250 0.494 0.250 unipredict/rounakbanik/pokemon 0.977 0.969 NA 0.963 NA 0.500 unipredict/rpaguirre/tesla-stock-price 0.977 0.992 0.984 0.247 0.882 0.250 unipredict/rtatman/chocolate-bar-ratings 0.109 0.141 0.227 0.138 0.149 0.100 unipredict/ruchi798/student-feedback-survey-res... 0.117 0.062 0.070 0.086 0.099 0.100 unipredict/ruchi798/tv-shows-on-netflix-prime-v... 0.250 0.312 0.477 0.283 0.457 0.167 unipredict/sabasaeed1953/stock-prices-of-2023 0.984 0.969 0.977 0.274 0.854 0.250 unipredict/saloni1712/chatgpt-app-reviews 0.625 0.602 0.633 0.365 0.519 0.200 unipredict/sanjanchaudhari/bankloan 0.656 0.648 0.594 0.628 0.669 0.500 unipredict/sanjanchaudhari/netflix-dataset 0.602 0.500 0.500 0.155 0.293 0.100 unipredict/sanjanchaudhari/user-behavior-on-ins... 0.469 0.547 0.766 0.500 0.796 0.500 unipredict/saunakghosh/nba-players-dataset 0.828 0.719 0.836 0.472 0.763 0.125 unipredict/saurabh00007/diabetescsv 0.578 0.695 0.703 0.661 0.703 0.500 unipredict/sbhatti/financial-sentiment-analysis 0.594 0.602 0.680 0.471 0.527 0.333 unipredict/shashankshukla123123/marketing-campaign 0.266 0.812 NA 0.888 NA 0.500 unipredict/shivamb/disney-movies-and-tv-shows 0.883 0.969 0.984 0.752 0.899 0.500 unipredict/shivamb/hm-stores-dataset 0.211 0.547 NA 0.458 NA 0.250 unipredict/shreyanshverma27/imdb-horror-chillin... 0.398 0.484 0.500 0.257 0.301 0.250 unipredict/shreyapurohit/anime-data 0.266 0.789 0.906 0.246 0.889 0.250 unipredict/shroukgomaa/babies-food-ingredients 0.289 0.320 NA 0.274 NA 0.250 unipredict/shubhamgupta012/titanic-dataset 0.742 0.703 0.734 0.697 0.682 0.500 unipredict/siddharthss/crop-recommendation-dataset 0.102 0.227 0.625 0.049 0.438 0.045 unipredict/sidhus/crab-age-prediction 0.047 0.148 0.195 0.088 0.192 0.053 unipredict/suraj520/dairy-goods-sales-dataset 0.617 0.508 NA 0.268 NA 0.250 unipredict/surajjha101/stores-area-and-sales-data 0.250 0.234 0.250 0.253 0.244 0.250 37 unipredict/surajjha101/top-youtube-channels-data 0.508 0.508 0.469 0.142 0.183 0.077 unipredict/tahzeer/indian-startups-by-state 0.062 0.141 NA 0.227 NA 0.012 unipredict/tarkkaanko/amazon 0.469 0.766 NA 0.750 NA 0.200 unipredict/team-ai/spam-text-message-classifica... 0.750 0.867 0.961 0.846 0.846 0.500 unipredict/teertha/ushealthinsurancedataset 0.312 0.617 0.711 0.258 0.708 0.250 unipredict/tejashvi14/employee-future-prediction 0.516 0.531 0.477 0.608 0.659 0.500 unipredict/tejashvi14/engineering-placements-pr... 0.617 0.594 0.773 0.487 0.724 0.500 unipredict/thedevastator/cancer-patients-and-ai... 0.039 0.109 NA 0.010 NA 0.029 unipredict/thedevastator/employee-attrition-and... 0.805 0.812 NA 0.837 NA 0.500 unipredict/thedevastator/higher-education-predi... 0.672 0.867 NA 0.654 NA 0.500 unipredict/therealsampat/predict-movie-success-... 0.617 0.883 NA 0.762 NA 0.500 unipredict/timoboz/tesla-stock-data-from-2010-t... 0.992 0.984 1.000 0.244 0.907 0.250 unipredict/uciml/mushroom-classification 0.555 0.734 0.961 0.502 0.952 0.500 unipredict/uciml/pima-indians-diabetes-database 0.648 0.664 0.703 0.661 0.705 0.500 unipredict/uciml/red-wine-quality-cortez-et-al-... 0.344 0.344 0.461 0.385 0.479 0.200 unipredict/varpit94/tesla-stock-data-updated-ti... 0.984 1.000 1.000 0.255 0.924 0.250 unipredict/vedavyasv/usa-housing 0.242 0.305 NA 0.242 NA 0.250 unipredict/vijayvvenkitesh/microsoft-stock-time... 0.984 0.977 0.953 0.260 0.859 0.250 unipredict/vikramamin/customer-churn-decision-t... 0.711 0.711 0.664 0.696 0.746 0.500 unipredict/vikramamin/time-series-forecasting-u... 0.211 0.359 0.578 0.256 0.247 0.250 unipredict/vstacknocopyright/blood-transfusion-... 0.484 0.570 0.672 0.787 0.749 0.500 unipredict/warcoder/earthquake-dataset 0.820 0.820 NA 0.259 NA 0.250 unipredict/whenamancodes/predict-diabities 0.672 0.695 0.781 0.661 0.718 0.500 unipredict/whenamancodes/students-performance-i... 0.500 0.453 0.500 0.252 0.481 0.250 unipredict/yasserh/titanic-dataset 0.711 0.727 0.805 0.587 0.721 0.500 unipredict/yasserh/wine-quality-dataset 0.391 0.359 0.500 0.346 0.511 0.200 I Model Card We provide a Model Card for TABULA-8B, as outlined in [35]. I.1 Model Details Person or organization developing model: This model was developed by the authors of this paper. Organizations providing computational support are listed in the Acknowledgements, but this model is not officially developed as part of any organization. The author affiliations are listed on the first page of this paper. Model date: This paper describes the May 2024 version of TABULA-8B. Model version: This paper describes version 1.0 of TABULA-8B. Model type: TABULA-8B is an autoregressive language model, identical in architecture to Llama 3 [54]. Information about training algorithms, parameters, fairness constraints or other applied approaches, and features: Our training procedure is described in Section 3. Our procedure for dataset construction, which includes methods for removing sensitive PII, is described in Sections 4 and A. Paper or other resource for more information: This paper is the primary resource for information about TABULA-8B. Implementation details can also be found at the open-source code release associated with the project. Citation details: See the first page of this paper. License: The model uses the Meta Llama 3 license (see https://llama.meta.com/llama3/ license/). Where to send questions or comments about the model: Send questions or comments directly to the corresponding authors, or file issues on the project git repo. 38 I.2 Intended Use Primary intended uses: This is a research-only release. The primary intended use of this model is for research on tabular data modeling, or for research applications on tabular data. Primary intended users: The primary intended users are scientific researchers interested in understanding, training, and applying tabular foundation models. Out-of-scope use cases: Commercial use, use of the model to attempt to identify, harm, or violate the privacy of individuals represented in the training data, and any other behavior that violates the Meta Llama 3 license is out of scope. I.3 Factors Relevant factors: The original Model Cards paper [35] identifies factors as “groups, instrumentation, and environments” relevant to summaries of model performance. One group relevant to our models’ performance is the task type (classification vs. binned regression). We report performance on these tasks separately; our results are discussed in Section 5. Broadly, we find that TABULA-8B’s overall performance profile relative to baselines is similar for both classification and binned regression tasks. Similarly, the different benchmarks may be viewed as different environments, each testing a different type of dataset. For example, UniPredict tests performance on datasets with informative headers; OpenML-CC18 tests performance on datasets without such headers and where traditional supervised learning methods can be tuned to good performance; Grinsztajn tests performance on datasets where GBDTs tend to perform best; and AMLB tests performance on tasks including free-form text. Our main results show that TABULA-8B’s overall performance relative to baselines is similar across these tasks; we analyze the differences in detail in the paper. Evaluation factors: Evaluating language models (LMs) is different from evaluating standard supervised learning methods: while the latter directly output a score or probability over the set of target labels, LMs only output next-token probabilities over their vocabularies; as a result, predicted probabilities are not directly available (although these can be obtained through the use of various heuristics). In order to avoid introducing additional degrees of freedom into the evaluation process, we do not use score-based evaluation methods that rely on evaluating predicted probabilities; we only evaluate based on exact matching (as in several works both in the tabular literature [12, 23] and in the broader language modeling literature [1, 8]. As a consequence, our evaluation does not use metrics which are sometimes used to evaluate tabular classification models, such as Area Under the Receiver Operating Characteristic Curve (AUC). I.4 Metrics Model performance measures: Our primary evaluation measures are based on accuracy. We use exact-match accuracy for language model generations, and top-1 accuracy for supervised learning model predictions. Decision thresholds: We use top-1 accuracy for supervised learning model predictions, but do not apply a specific threshold. Variation approaches: N/A I.5 Evaluation Data Datasets: We use a suite of five previously-proposed tabular benchmarks, comprising a total of 329 tables. Our evaluation datasets are described in detail in Sections 5.2 and D.1. Motivation: Using preexisting benchmark datasets allows us to compare the performance of our models to prior work in the tabular prediction literature. Additionally, using high-quality, curated benchmarks ensures that we are able to make reliable conclusion about overall model quality and performance relative to baselines. Preprocessing: Our preprocessing is described in Sections 5.2 and D.1. We perform minimal preprocessing on the datasets (no one-hot encoding, standardization, etc.) except for the logistic regression baseline, which requires this for best performance. 39 I.6 Training Data Our training data is described in 4, with further details in the supplementary. I.7 Quantitative Analyses Unitary results: Our unitary results are summarized in Section 5. We provide detailed analysis in the supplementary section and give per-dataset results in Section H. Intersectional results: We do not explicitly investigate tasks which include sensitive attributes, and so do not consider intersectional analysis in this work. We call for future work understanding the fairness properties of tabular foundation models in the future work (and our first-of-its-kind model will enable such research). I.8 Ethical Considerations There are important ethical considerations of both the data and model presented in this work. We discuss these in our Impact Statement. I.9 Caveats and Recommendations This model is for research use only. We recommend that more thorough research on both the impact of tabular training datasets, and the downstream performance of fine-tuned language models, be conducted before the deployment of tabular foundation models for real-world decisionmaking deployments. J Datasheet J.1 Motivation For what purpose was the dataset created? The dataset was created for training tabular data foundation models, serving a purpose similar to C4 [41] or other large-scale corpuses in the natural language processing community. Who created the dataset (e.g., which team, research group) and on behalf of which entity (e.g., company, institution, organization)? The dataset was created by the authors of this paper in their roles at the institutions listed in the affiliations section of this paper. Who funded the creation of the dataset? No funding was provided with the explicit purpose of creating this dataset. However, JG was supported by a Microsoft Grant for Customer Success. JCP was supported by the Harvard Center for Research on Computation and Society. J.2 Composition What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)? The instances that comprise the dataset represent tables extracted from the web (or individual rows of tables, depending on the downstream use of the data). All tables are publicly available and extracted from the Internet; in particular, all tables are available in the TabLib dataset from which T4 is filtered. How many instances are there in total (of each type, if appropriate)? The dataset consists of 4.2M tables where each table has many rows. The total number of rows across tables is 2.1B. Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set? 40 The dataset consists of a deterministically filtered subset of the TabLib dataset. What data does each instance consist of? The data consists of tables drawn from Github and CommonCrawl, as initially captured by the TabLib authors. Is there a label or target associated with each instance? Not by default. As part of our work, we select a target at random from filtered subset of the columns for each dataset. See A.2. Is any information missing from individual instances? Yes, many tables contain missing values for certain rows and columns. Are relationships between individual instances made explicit (e.g., users’ movie ratings, social network links)? In some cases, the tables have informative column headers describing the relationships between features for individual rows. Are there recommended data splits (e.g., training, develop- ment/validation, testing)? Not by default. We implement these as part of our training. Are there any errors, sources of noise, or redundancies in the dataset? We deduplicate the T4 dataset so that each table appears at most once. However, as is the case with most internet scale datasets, many of the tables contain noisy values whose correctness we do not manually inspect. Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)? It is self-contained. Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor–patient confidentiality, data that includes the content of individuals’ non- public communications)? We do our best to remove any kind of data that might be considered confidential or personally identifying. If someone finds tables with confidential information that still remain, we would appreciate if they contact us so we might remove them. Does the dataset contain data that, if viewed directly, might be offen- sive, insulting, threatening, or might otherwise cause anxiety? It is possible that there are tables with information that might be anxiety inducing. We do not explicitly filter for this type of information, but believe it is not common in our dataset. Does the dataset identify any subpopulations (e.g., by age, gender)? The instances (table rows) in the data represent a variety of entities, and the majority of these do not represent persons. However, for the subset of tables where each row does represent an individual person, it is possible that the dataset does identify subpopulations. Is it possible to identify individuals (i.e., one or more natural persons), either directly or indirectly (i.e., in combination with other data) from the dataset? While we aim to reduce obviously personally identifying data, we do not use techniques like differential privacy to formally defend against reidentification attacks. Does the dataset contain data that might be considered sensitive in any way (e.g., data that reveals race or ethnic origins, sexual orienta- tions, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic data; forms of government identification, such as social security numbers; criminal history)? We aim to remove this kind of information. Any other comments? 41 J.3 Collection Process How was the data associated with each instance acquired? The data was filtered from the original TabLib dataset [13]. What mechanisms or procedures were used to collect the data (e.g., hardware apparatuses or sensors, manual human curation, software programs, software APIs)? The data was filtered programatically according to hand picked heuristics as described in 4. If the dataset is a sample from a larger set, what was the sam- pling strategy (e.g., deterministic, probabilistic with specific sam- pling probabilities)? It was deterministically chosen according to hand coded filtering rules. Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)? The authors performed the data collection process. No external crowdsourcing or contractors were employed. Over what timeframe was the data collected? The original TabLib dataset was collected in 2023 by the original authors and contains tables published from a wide range of years. Our filtering was conducted during the spring of 2024. Were any ethical review processes conducted (e.g., by an institu- tional review board)? No. Did you collect the data from the individuals in question directly, or obtain it via third parties or other sources (e.g., websites)? Data was filtered from TabLib and hence not collected directly. Were the individuals in question notified about the data collection? We notified the TabLib authors of our effort, but not the owners of the publicly available tables in the original corpus. Did the individuals in question consent to the collection and use of their data? To the best of our knowledge, the data scraped in TabLib did not have a consent procedure. If consent was obtained, were the consenting individuals provided with a mechanism to revoke their consent in the future or for certain uses? NA Has an analysis of the potential impact of the dataset and its use on data subjects (e.g., a data protection impact analysis) been conducted? No. J.4 Preprocessing/cleaning/labeling Was any preprocessing/cleaning/labeling of the data done (e.g., discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? Yes. Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data (e.g., to support unanticipated future uses)? The raw data is available as part of the TabLib release [13]. Is the software that was used to preprocess/clean/label the data available? Yes, all the code used to filter the original corpus is available as part of our open source release. 42 J.5 Uses Has the dataset been used for any tasks already? We are not aware of any other uses of T4 apart from the training of TABULA-8B. Is there a repository that links to any or all papers or systems that use the dataset? Models trained on the dataset can be found using the Hugging Face dataset page. What (other) tasks could the dataset be used for? The dataset could be used to train generative models, LLM data science assistants, amongst others. Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses? Our filtering choices were optimized to train a tabular prediction (classification) model and may lead to suboptimal behavior for other tasks. Are there tasks for which the dataset should not be used? The dataset should not be used to try and identify private individuals. J.6 Distribution Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) on behalf of which the dataset was created? The dataset will be made publicly available on HuggingFace. How will the dataset will be distributed (e.g., tarball on website, API, GitHub)? It will be published at HuggingFace. When will the dataset be distributed? June 2024 Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)? The dataset is subject to the same usage and copyright restrictions as the original TabLib release. Have any third parties imposed IP-based or other restrictions on the data associated with the instances? Yes, the original TabLib authors place usage restrictions. See [] for more details. Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? The dataset is subject to the same usage and copyright restrictions as the original TabLib release. J.7 Maintenance Who will be supporting/hosting/maintaining the dataset? The authors of the paper. How can the owner/curator/manager of the dataset be contacted (e.g., email address)? Please contact jpgard@cs.washington.edu. Is there an erratum? The current manuscript, as published on the arxiv server, will serve as the main source of documenting errors. Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)? We do not foresee any updates. 43 If the dataset relates to people, are there applicable limits on the re- tention of the data associated with the instances (e.g., were the individuals in question told that their data would be retained for a fixed period of time and then deleted)? NA Will older versions of the dataset continue to be supported/hosted/maintained? NA If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so? Others are free to build on the dataset as long as they adhere to the original terms of use put forth by [13]. NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist: • You should answer [Yes] , [No] , or [NA] . • [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available. • Please provide a short (1–2 sentence) justification right after your answer (even for NA). The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper. The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes] " is generally preferable to "[No] ", it is perfectly acceptable to answer "[No] " provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] " or "[NA] " is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found. 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Guidelines: • The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper. • The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach. 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Guidelines: • The answer NA means that the paper does not include theoretical results. • All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems. 45 • The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 4. Experimental Result Reproducibility Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: We provide a detailed description of our methodology in Section 5 and release all the relevant code and datasets to reproduce both our training run and evals. Detailed description of how we created TabLib is included in Section 4 and Appendix A. 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Experimental Setting/Details Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: Yes, we pay careful attention to describing the data splits and hyperparameter tuning strategies in Section 5. Further detail is included as part of our open source release. Guidelines: • The answer NA means that the paper does not include experiments. • The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 7. Experiment Statistical Significance Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: We compute Clopper-Pearson confidence intervals whenever we evaluate empirical accuracy on a test set of examples. In all our figures, we report the number of datasets averaged over, with corresponding error bars for statistical significance. 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Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [Yes] Justification: We assess that fine-tuning of the Llama 3 model on publicly-available data poses no larger risk than the Llama 3 model itself, which is already open sourced and likely pretrained on data from the same sources as TabLib (Common Crawl, GitHub). Our model release is also required to adhere to the terms of use and Acceptable Use Policy12 of the Llama 3 model, which includes prohibiting the use of the model for violating the law or others’ rights, engaging in or facilitating harassment, processing, disclosing, generating, or inferring health, demographic, or other sensitive personal or private information about individuals without rights and consents required by applicable laws, and other harmful uses. We note that both the TabLib dataset [13] (from which our dataset is extracted), and also the original data used to create TabLib (extracted from public Common Crawl and Github data), are publicly available. As noted in the original release of TabLib [13], TabLib does appear to contain some personally identifying information (such as names, email addresses, and phone numbers), although at least some of this data may be synthetic. We take steps to aggressively remove tables containing PII from our released dataset (described in Section A), removing any table where we detect PII. As a result, our released subset of TabLib, which we refer to as T4, may indeed be safer than the original TabLib and may improve the safety of downstream models trained on it (relative to training on TabLib). Furthermore, the release of our data processing code will both enable transparency into the dataset creation process, but will also enable future work improving the safety of tabular data derived from TabLib and other similar sources. Guidelines: • The answer NA means that the paper poses no such risks. • Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets 12https://llama.meta.com/llama3/use-policy/ 49 Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: Yes, we explicitly acknowledge Meta and the Llama team for the Llama 3 model that we use as a starting point for TABULA-8B. Furthermore, we credit the creators of all benchmark suites we use for evaluation in Section 5 and the TabLib [13] authors for the initial work in compiling the data corpus. All relevant data sources are publicly available and our use is in line with previous research that has used them for model training or evaluation. 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4,453	WaveAttack: Asymmetric Frequency Obfuscation-based Backdoor Attacks Against Deep Neural Networks Jun Xia1,†, Zhihao Yue1,†, Yingbo Zhou1, Zhiwei Ling1, Yiyu Shi2, Xian Wei1, Mingsong Chen1∗ 1MoE Eng. Research Center of SW/HW Co-design Tech. and App., East China Normal University 2Department of Computer Science and Engineering, University of Notre Dame {jxia, 51215902034, 52215902009, 51215902044}@stu.ecnu.edu.cn, yshi4@nd.edu, {xwei, mschen}@sei.ecnu.edu.cn Abstract Due to the increasing popularity of Artificial Intelligence (AI), more and more backdoor attacks are designed to mislead Deep Neural Network (DNN) predictions by manipulating training samples or processes. Although backdoor attacks have been investigated in various scenarios, they still suffer from the problems of both low fidelity of poisoned samples and non-negligible transfer in latent space, which make them easily identified by existing backdoor detection algorithms. To overcome this weakness, this paper proposes a novel frequency-based backdoor attack method named WaveAttack, which obtains high-frequency image features through Discrete Wavelet Transform (DWT) to generate highly stealthy backdoor triggers. By introducing an asymmetric frequency obfuscation method, our approach adds an adaptive residual to the training and inference stages to improve the impact of triggers, thus further enhancing the effectiveness of WaveAttack. Comprehensive experimental results show that, WaveAttack can not only achieve higher effectiveness than state-of-the-art backdoor attack methods, but also outperform them in the fidelity of images (i.e., by up to 28.27% improvement in PSNR, 1.61% improvement in SSIM, and 70.59% reduction in IS). Our code is available at https://github.com/BililiCode/WaveAttack. 1 Introduction Along with the prosperity of Artificial Intelligence (AI), Deep Neural Networks (DNNs) have become increasingly prevalent in numerous safety-critical domains for precise perception and real-time control, such as autonomous vehicles [1], medical diagnosis, and industrial automation [2]. However, the trustworthiness of DNNs faces significant threats due to various notorious adversarial and backdoor attacks. Typically, adversarial attacks [3, 4] manipulate input data during the inference stage to induce incorrect predictions by a trained DNN, while backdoor attacks [5] tamper with training samples or processes to embed concealed triggers during training, which can be exploited to generate malicious outputs. Although adversarial attacks on DNNs frequently appear in various scenarios, backdoor attacks have attracted more attention because of their stealthiness and effectiveness. Generally, the performance of backdoor attacks can be evaluated by the following three objectives of an adversary: i) efficacy that refers to the effectiveness of an attack in causing the target model to produce incorrect outputs or exhibit unintended behavior; ii) specificity that denotes the precision of the attack in targeting a specific class; and iii) fidelity that represents the degree to which adversarial examples or poisoned training samples are indistinguishable from their benign counterparts [6]. Note that efficacy and specificity represent the effectiveness of backdoor attacks, while fidelity denotes the stealthiness of backdoor attacks. ∗Mingsong Chen is the corresponding author. † Equal contribution. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). In order to achieve higher stealthiness and effectiveness, existing backdoor attack methods (e.g. IAD [7], WaNet [8], BppAttack [9], and FTrojan [10]) are built based on various optimizations, which can be mainly classified into two categories. The former is the sample minimal impact method that can optimize the size of the trigger and minimize its pixel value, making the backdoor trigger difficult to detect in training samples for the purpose of achieving the high stealthiness of a backdoor attacker. Although these methods are promising in backdoor attacks, due to the explicit trigger influence on training samples, they cannot fully evade existing backdoor detection methods based on training samples. The latter is the latent space obfuscation-based methods, which can be integrated into any existing backdoor attack methods. Using asymmetric samples, these methods can obfuscate the latent space between benign samples and poisoned samples [11]. Although these methods can bypass latent space detection techniques, they suffer greatly from low image quality, making them extremely difficult to apply in practice. Therefore, how to improve both the effectiveness and stealthiness of backdoor attacks while minimally impacting the quality of training samples is becoming a significant challenge in the development of backdoor attacks. According to the work in [12], wavelet transform techniques have been widely investigated in various image-processing tasks [13, 14, 15], where high-frequency features can be utilized to enhance the generalization ability of DNNs and remain imperceptible to humans. Inspired by this finding, this paper introduces a novel backdoor attack method named WaveAttack, which adopts Discrete Wavelet Transform (DWT) to extract high-frequency components for highly stealthy backdoor trigger generation. To improve the impact of triggers and further enhance the effectiveness of our approach, we employ asymmetric frequency obfuscation that utilizes an asymmetric coefficient of the trigger in the high-frequency domain during the training and inference stages. This paper makes the following three contributions: • We introduce a promising frequency-based backdoor trigger generation method, which can effectively generate the backdoor residuals for the high-frequency component based on DWT, thus ensuring the high fidelity of poisoned samples. • We propose a novel asymmetric frequency-based obfuscation backdoor attack method to enhance the stealthiness and effectiveness of WaveAttack, which can increase stealthiness in latent spaces and improve the Attack Success Rate in training samples. • We conduct comprehensive experiments on four public benchmarks to demonstrate that WaveAttack outperforms state-of-the-art (SOTA) backdoor attack methods from the perspectives of both stealthiness and effectiveness. 2 Related Work Backdoor Attack. Typically, backdoor attacks try to embed backdoors into DNNs by manipulating their input samples and training processes. In this way, adversaries can control DNN output through concealed triggers, which results in manipulated predictions [16]. Depending on whether the training process is manipulated, existing backdoor attacks can be categorized into two types, i.e., training-unmanipulated and training-manipulated attacks. Specifically, training-unmanipulated attacks only inject a visible or invisible trigger into the training samples of some DNN, leading to its recognition errors [5]. For example, Chen et al. [17] introduced a Blend attack that generates poisoned data by merging benign training samples with specific key visible triggers. Moreover, there exists a large number of invisible trigger-based backdoor attack methods, such as natural reflection [18], human imperceptible noise [19], and image perturbation [10], which exploit the changes induced by real-world physical environments. Although these training-unmanipulated attacks are promising, due to their substantial impacts on training sample quality, most of them still can be easily identified somehow. As an alternative, training-manipulated attacks [8, 9] assume that adversaries from some malicious third party can control the key steps of the training process, thus achieving a stealthier attack. Although the above two categories of backdoor attacks are promising, most of them struggle with coarse-grained optimization of effectiveness and stealthiness, complicating the acquisition of superior backdoor triggers. Due to the significant difference in latent space and low poisoned sample fidelity, they cannot evade the latest backdoor detection methods. Backdoor Defense. There are two major types of backdoor defense methods, i.e., the detectionbased defense and erasure-based defense. The detection-based defenses can be further classified into two categories, i.e., sample-based and latent space-based detection methods. Specifically, sample2 based detection methods can identify the differences in the distribution between poisoned samples and benign samples [20], while latent space-based detection methods aim to find the disparity between the latent spaces of poisoned samples and benign samples [21]. Unlike the detection strategies described above that aim to prevent the injection of backdoors into DNNs by identifying poisoned samples during the training stages, erasure-based defenses can eradicate the backdoors from DNNs. So far, the erasure-based defenses can be classified into three categories, i.e., poison suppressionbased, model reconstruction-based, and trigger generation-based defenses. The poison suppressionbased methods [22] utilize the differential learning speed between poisoned and benign samples during training to mitigate the influence of backdoor triggers on DNNs. The model reconstructionbased methods [23, 24] use a selected set of benign data to rebuild DNN models, aiming to mitigate the impact of backdoor triggers. The trigger generation-based methods [25, 26] reverse engineer backdoor triggers by capitalizing on the effects of backdoor attacks on training samples. To the best of our knowledge, WaveAttack is the first attempt to generate backdoor triggers for the high-frequency component obtained through DWT. Unlike existing backdoor attack methods, WaveAttack first considers both the fidelity of poisoned samples and latent space obfuscation simultaneously. By using asymmetric frequency obfuscation, WaveAttack can not only acquire backdoor attack effectiveness but also achieve high stealthiness regarding both image quality and latent space. 3 Our Method In this section, we first present the preliminaries for the problem notations, threat model, and adversarial goal. Then, we visualize our motivations for adding triggers to the high-frequency components. Finally, we celebrate the attack process of our method, WaveAttack. 3.1 Preliminaries Notations. We follow the training scheme of Adapt-Blend [11]. Let D = {(xi, yi)}N i=1 be a clean training dataset, where xi ∈X = {0, 1, ..., 255}C×W ×H is an image, and yi ∈Y = {1, 2, ..., K} is its corresponding label. Note that K represents the number of labels. For a given training dataset, we select a subset of D with a poisoning rate pa as the payload samples Da = {(x′i, yt)\|x′i = T(xi), xi ∈X}, where T(·) is a backdoor transformation function, and yt is an adversary-specified target label. We use a subset of D with poisoning rate pr as the regularization samples Dr = {(x′i, yi)\|x′i = T(xi), xi ∈X}. For a given dataset, a backdoor attack adversary tries to train a backdoored model f that predicts x as its corresponding label, where x ∈D ∪Da ∪Dr. Threat Model. Similar to existing backdoor attack methods [7, 8, 9], we assume that adversaries have complete control over the training datasets, and model implementation. They can embed backdoors into the DNNs by poisoning the given training dataset. Moreover, in the inference stage, we assume that adversaries can only query backdoored models using any samples. Adversarial Goal. Throughout the attack process, adversaries strive to achieve two core goals, i.e., effectiveness and stealthiness. Effectiveness indicates that adversaries try to train backdoored models with a high ASR while ensuring that the decrease in Benign Accuracy (BA) remains imperceptible. Stealthiness indicates that samples with triggers have high fidelity and that there is no latent separation between poisoned and clean samples in the latent space. 3.2 Motivation Unlike humans who are not sensitive to high-frequency features, DNNs can effectively learn highfrequency features of images [12], which can be used for the generation of backdoor triggers. In other words, the poisoned samples generated by high-frequency features can easily escape various examination methods by humans. Based on this observation, if we can design backdoor triggers on top of high-frequency features, the stealthiness of corresponding backdoored attacks can be ensured. To obtain high-frequency components from the training samples, we resort to Discrete Wavelet Transform (DWT) to capture characteristics from both the time and frequency domains [27], allowing the extraction of multiple frequency components from the training samples. The reason why we adopt DWT rather than Discrete Cosine Transform (DCT) is that DWT can better capture high-frequency features from training samples (i.e., edges and textures) and allows superior reverse operations during both encoding and decoding phases, thus minimizing the impact on the 3 (a) Original (b) LL with noises (c) LH with noises (d) HL with noises (e) HH with noises Figure 1: A motivating example for the backdoor trigger design on high-frequency components. fidelity of poisoned samples. In our approach, we adopt a classic and effective biorthogonal wavelet transform method (i.e., Haar wavelet [28]), which mainly contains four kernel operations, i.e., LLT , LHT , HLT , and HHT . Here L and H denote the low and high pass filters, respectively, where LT = 1 √ 2 [1 1] , HT = 1 √ 2 [−1 1]. Note that, based on the four operations, the Haar wavelet can decompose an image into four frequency components (i.e., LL, LH, HL, HH) using DWT, where HH only contains the high-frequency information of a sample. Meanwhile, the Haar wavelet can reconstruct the image from the four frequency components via the Inverse Discrete Wavelet Transform (IDWT). To verify the motivation of our approach, Figure 1 illustrates the impact of adding the same noises to different frequency components on an image, i.e., Figure 1(a). We can find that, compared to the other three poisoned images, i.e., Figure 1(b) to 1(d), it is much more difficult to determine the difference between the original image and the poisoned counterpart in HH, i.e., Figure 1(e). Therefore, it is more suitable to inject triggers into the high-frequency component (i.e., HH) for backdoor attack purposes. 3.3 Implementation of WaveAttack In this subsection, we detail the design of our WaveAttack approach. As shown in Figure 2, we give an overview of our attack method WaveAttack. To be concrete, we first make samples poisoned into payload and regularization samples using our trigger design, which is implemented with frequency transformation. Then, we use benign samples, payload samples, and regularization samples to train a classifier to achieve the core goals of WaveAttack. 𝑯𝑯 × 𝜶 𝑯𝑯′ Residual Inputs 𝑯𝑳 𝑳𝑯 𝑳𝑳 IDWT DWT E D Payload Samples Benign Samples C ℒ𝒄 DWT: Discrete Wavelet Transform IDWT: Inverse DWT E: Encoder D: Decoder C: Classifier ℒ𝒓 Regularization Samples Random Split Figure 2: Overview of our attack method WaveAttack. Trigger Design. As mentioned above, our WaveAttack approach aims to achieve a stealthier backdoor attack, introducing triggers into the HH frequency component. Figure 2 contains the process of generating triggers using WaveAttack. First, we obtain the four components of the samples through DWT. Then, to generate imperceptible sample-specific triggers, we employ an encoderdecoder network as a generator g. These generated triggers are imperceptible additive residuals. Next, to achieve asymmetric frequency obfuscation, we multiply the residuals by a coefficient α, and generate the poisoned HH′ component with the triggers as follows: HH′ = HH + α · g(HH; ωg), (1) where ωg is the generator parameters. Finally, we can utilize IDWT to reconstruct four frequency components of poisoned samples. Specifically, we use a U-Net-like [29] generator to obtain residuals, although other methods (e.g., VAE [30]) can also be used by the adversary. This is because the skip connections of U-Net can effectively preserve the features of inputs with minimal impacts [29]. Optimization Objective. Our WaveAttack method has two networks to optimize. We aim to optimize a generator g to generate small residuals with minimal impact on the samples. Furthermore, our objective is to optimize a backdoored classifier c, enabling the effectiveness and stealthiness of WaveAttack. For the first optimization objective, we use the L∞norm to optimize small residuals. The optimization objective is defined as follows: Lr = \|\|g(HH; ωg)\|\|∞. (2) 4 For the second optimization objective, we train the classifier using the cross-entropy loss function in D, Da, and Dr dataset. The optimization objective is defined as follows: Lc = L(xp, yt; ωf) + L(xr, y; ωc) + L(xb, y; ωc), (3) where L(·) is the cross-entropy loss function, ωf is the classifier parameters, xb ∈D, xp ∈Da, and xr ∈Dr. The total loss function is as follows: Ltotal = Lc + Lr. (4) Algorithm 1 Training of WaveAttack Require: i) D, benign training dataset. ii) ωg, randomly initialized generator parameters. iii) ωc, randomly initialized classifier parameters. iv) pa, payload sample rate. v) pr, rate of regularization samples. vii) yt, target label. vi) E, # of epochs in training process. Ensure: i) ωg, well-trained generator model. ii) ωˆc, well-trained classifier model. 1: for e = 1, . . . , E do 2: for (x, y) in D do 3: b ←x.shape[0] 4: nm ←(pa + pr) × b 5: na ←pa × b 6: nr ←pr × b 7: xm ←x[:nm] 8: (LL, LH, HL, HH) ←DWT(xm) 9: resdiual ←α · g(HH; ωg) 10: HH′ ←HH + resdiual 11: xm ←IDWT(LL, LH, HL, HH′) 12: L1 ←L(xm[na:],yt;ωc) 13: L2 ←L(xm[:nr],y[na:nr];ωc) 14: L3 ←L(x[nm:],y[nm:];ωc) 15: L ←L1 + L2 + L3 + \|\|resdiual\|\|∞ 16: L.backward() 17: update(ωg, ωc) 18: end for 19: end for 20: Return ωg, ωˆc Algorithm Description. Algorithm 1 details the training process of our WaveAttack approach. At the beginning of WaveAttack training (Line 2), the adversary randomly selects a minibatch data (x, y) from D, which has b training samples. Lines 4-6 calculate the number of poisoned samples, payload samples, and regulation samples, respectively. Lines 7-11 denote the process of modifying samples by injecting triggers into the high-frequency component. After acquiring the modified samples in Line 7, Line 8 decomposes the samples into four frequency components (i.e., LL, LH, HL and HH) by DWT. Then, in Lines 9-10, we add the residual to the frequency component HH by Equation (1) and obtain the frequency component HH′. Line 11 reconstructs the samples from the four frequency components via IDWT. Lines 12-15 compute the optimization object using Equations (2) to (4). In Lines 16-17, we can use an optimizer (e.g., SGD optimizer) to update the parameters of the generator model and classifier model. Line 20 returns the welltrained generator model parameters ωg and the classifier model parameters ωˆc. Asymmetric Frequency Obfuscation. According to [11], regularization samples Dr can make DNNs learn the semantic feature of each class and the trigger feature, which can make the backdoor attack stealthy in the latent space. However, using the same trigger in samples during the inference process may diminish the fidelity of poisoned samples. Hence, it is crucial to devise an asymmetric frequency obfuscation method to enhance the effectiveness of backdoor attack methods. In our approach, we employ a coefficient α with a small value (i.e., α=1.0) to improve the stealthiness of triggers during the training process, while a larger value (i.e., α=100.0) is used to enhance the impact of triggers and further improve the effectiveness of WaveAttack. This method ensures that, during the inference process, the backdoored samples have sufficient “power” to activate the DNN backdoor, thus achieving a high ASR. 4 Experiments To demonstrate the effectiveness and stealthiness of our approach, we implemented WaveAttack using Pytorch and compared its performance with seven existing backdoor attack methods. We conducted all experiments on a workstation with a 3.6GHz Intel i9 CPU, 32GB of memory, an NVIDIA GeForce RTX3090 GPU, and a Ubuntu operating system. We designed comprehensive experiments to address the following three research questions: RQ1 (Effectiveness of WaveAttack): Can WaveAttack successfully inject backdoors into DNNs? RQ2 (Stealthiness of WaveAttack): How stealthy are the poisoned samples generated by WaveAttack compared to those generated by SOTA backdoor attack methods? RQ3 (Resistance to Existing Defenses): Can WaveAttack resist existing defense methods? 5 4.1 Experimental Settings Datasets and DNNs. We evaluated all the attack methods on four well-known benchmark datasets, i.e., CIFAR-10 [31], CIFAR-100 [31], GTSRB [32] and a subset of ImageNet (with the first 20 categories) [33]. The statistics of the datasets adopted in the experiments are presented in Table 6 (see Appendix 7.1). We used ResNet18 [34] as the base DNN for the effectiveness and stealthiness evaluation. In addition, we used VGG16 [35], SENet18 [36], ResNeXt29 [37], and DenseNet121 [38] to evaluate the generalizability of WaveAttack. Attack Configurations. To compare the performance of WaveAttack with SOTA attack methods, we considered nine SOTA backdoor attacks, i.e., BadNets [5], Blend [17], IAD [7], WaNet [8], BppAttack [9], Adapt-Blend [11], FTrojan [10], LIRA [39], and Fiba [40]. Note that, similar to our work, Adapt-Blend has asymmetric triggers, and FTrojan and Fiba are also frequency domain-based attack methods. We performed the attack methods using the default hyperparameters described in their original papers. Specifically, the poisoning rate is set to 10% with a target label of 0 to ensure a fair comparison. See the Appendix for more details on both data and attack settings. Evaluation Metrics. Similar to the existing work in [10], we evaluated the effectiveness of all attack methods using two metrics, i.e., Attack Success Rate (ASR) and Benign Accuracy (BA). To evaluate the stealthiness of all attack methods, we used three metrics, i.e., Peak Signal-to-Noise Ratio (PSNR) [41], Structure Similarity Index Measure (SSIM) [42], and Inception Score (IS) [43]. 4.2 Effectiveness Evaluation (RQ1) Effectiveness Comparison with SOTA Attack Methods. To evaluate the effectiveness of WaveAttack, we compared the ASR and BA of WaveAttack with nine SOTA attack methods. Since IAD [7] cannot attack the ImageNet dataset based on its open-source code, we do not provide its comparison result. Table 1 shows the attack performance of different attack methods. From this table, we can find that WaveAttack can acquire a high ASR without obviously degrading the BA. Especially for the datasets CIFAR-10 and GTSRB, our WaveAttack achieves the best ASR and BA compared to other SOTA attack methods. Compared to frequency domain-based attack methods (i.e., FTrojan and Fiba), WaveAttack outperforms FTrojan and Fiba in BA for CIFAR-10, CIFAR-100, GTSRB, and ImageNet datasets. Moreover, compared to the asymmetric-based method Adapt-Blend, WaveAttack can also obtain superior performance in terms of ASR and BA for all datasets. Table 1: Attack performance comparison between WaveAttack and seven SOTA attack methods. The best and the second-best results are highlighted and underlined, respectively. Method CIFAR-10 CIFAR-100 GTSRB ImageNet BA ↑ ASR ↑ BA ↑ ASR ↑ BA ↑ ASR ↑ BA ↑ ASR ↑ No attack 94.59 75.55 99.00 87.00 BadNets [5] 94.36 100 74.90 100 98.97 100 85.80 100 Blend [17] 94.51 99.91 75.10 99.84 98.26 100 86.40 100 IAD [7] 94.32 99.12 75.14 99.28 99.26 98.37 WaNet [8] 94.23 99.57 73.18 98.52 99.21 99.58 86.60 89.20 BppAttack [9] 94.10 100 74.68 100 98.93 99.91 85.90 99.50 Adapt-Blend [11] 94.31 71.57 74.53 81.66 98.76 60.25 86.40 90.10 FTrojan [10] 94.29 100 75.37 100 98.83 100 85.10 100 LIRA [39] 93.57 99.96 73.09 99.98 10.74 99.03 Fiba [40] 93.80 75.40 74.87 80.36 99.12 85.18 WaveAttack (Ours) 94.55 100 75.41 100 99.30 100 86.60 100 Table 2: Attack performance on different DNNs. Network No Attack WaveAttack BA ↑ BA ↑ ASR ↑ VGG16 [35] 93.62 93.70 99.76 SENet18 [36] 94.51 94.63 100 ResNeXt29 [37] 94.79 95.08 100 DenseNet121 [38] 95.29 95.10 99.78 Effectiveness on Different Networks. To evaluate the effectiveness of WaveAttack on various networks, we conducted experiments on CIFAR-10 using different networks (i.e., VGG16 [35], SENet18 [36], ResNeXt29 [37], and DenseNet121 [38]). Table 2 shows the attack performance of WaveAttack on these networks. From this table, we can find that our WaveAttack approach can successfully embed the backdoor into different networks. WaveAttack can not only cause malicious impacts of backdoor 6 attacks, but also maintain a classification performance with high BA, demonstrating the generalizability of WaveAttack on different network architectures. Table 3: Attack performance with different DWTs. Wavelet Dataset IS ↓ PSNR ↑ SSIM ↑ BA ↑ ASR ↑ Haar CIFAR-10 0.011 47.49 0.9979 94.55 100 CIFAR-100 0.005 50.12 0.9992 75.41 100 GTSRB 0.058 40.67 0.9877 99.30 100 DB CIFAR-10 0.007 47.53 0.9989 94.77 95.60 CIFAR-100 0.005 50.32 0.9994 76.64 80.43 GTSRB 0.022 41.95 0.9881 98.21 99.50 Effectiveness of WaveAttack with Different Discrete Wavelet Transforms. Due to simplicity and computational efficiency, we adopted the most common Haar wavelet in our wavelet transformation procedure. Since different wavelets are applicable to Discrete Wavelet Transform (DWT) in our method, we conducted experiments to incorporate the Daubechies (DB) wavelet, which has stronger orthogonality. Table 3 summarizes the experimental results of WaveAttack with different wavelets. From the table, we can find that the influence of different wavelets on the performance of our method is limited, indicating that WaveAttack maintains its effectiveness and stealthiness among different wavelet transformations. 4.3 Stealthiness Evaluation (RQ2) To evaluate the stealthiness of WaveAttack, we compared the images with the triggers generated by WaveAttack with the ones of SOTA attack methods. In addition, we used t-SNE [44] to visualize latent spaces for poisoned samples and benign samples from the target label. Stealthiness Results from The Perspective of Images. To show the stealthiness of triggers generated by WaveAttack, Figure 3 compares WaveAttack and SOTA attack methods using poisoned samples and their magnified residuals (×5) counterparts. From this figure, we can see that the residual generated by WaveAttack is the smallest and only leaves a few subtle artifacts. The trigger injected by WaveAttack is almost invisible to humans. Input Residual BadNets Blend IAD WaNet BppAttack FTrojan WaveAttack Adapt-Blend Figure 3: Comparison of examples generated by seven backdoor attacks. For each attack, we show the poisoned sample (top) and the magnified (×5) residual (bottom). We used three metrics (i.e., PSNR, SSIM, and IS) to evaluate the stealthiness of triggers generated by our WaveAttack. Table 4 shows the results of the stealthiness comparison between WaveAttack and nine SOTA attack methods. From this table, we can see that WaveAttack achieves the best stealthiness in the CIFAR-10 and ImageNet datasets. Note that although our WaveAttack only achieves the third-best SSIM score on the GTSRB dataset, it outperforms BadNets by up to 60.56% in PSNR and 67.5% in IS. Similarly, although our WaveAttack achieves the second-best SSIM score on the CIFAR-100 dataset, it is much better than LIRA in PSNR and IS. (a) BadNets (b) Blend (c) WaNet (d) Adapt-Blend (e) FTrojan (f) WaveAttack Figure 4: The t-SNE of feature vectors in the latent space under different attacks on CIFAR-10. We use red and blue points to denote poisoned and benign samples, respectively, where each point in the plots corresponds to a training sample from the target label. 7 Stealthiness Results from The Perspective of Latent Space. There are so many backdoor defense methods [45, 21] based on the assumption that there is a latent separation between poisoned and benign samples in latent space. Therefore, ensuring the stealthiness of the attack method from the perspective of latent space becomes necessary. We obtained feature vectors of the test result from the feature extractor (the DNN without the last classifier layer) and used t-SNE [44] for visualization. Figure 4 visualizes the distributions of feature representations of the poisoned samples and the benign samples from the target label under the six attacks. From Figure 4(a) to 4(c) and 4(e), we can observe that there are two distinct clusters, which can be used to detect poisoned samples or backdoor models [11]. However, as shown in 4(d) and 4(f), we can find that the feature representations of poisoned samples are intermingled with those of benign samples for Adapt-Blend and WaveAttack, i.e., there is only one cluster. Adapt-Blend and WaveAttack can achieve the best stealthiness from the perspective of latent space and break the latent separation assumption to evade backdoor defenses. Although Adapt-Blend exhibits a degree of stealthiness, Table 4 reveals that WaveAttack surpasses Adapt-Blend in image quality, suggesting that WaveAttack can achieve superior stealthiness. Table 4: Stealthiness comparison with existing attacks. Larger PSNR, SSIM, and smaller IS indicate better performance. The best and the second-best results are highlighted and underlined, respectively. Attack Method CIFAR-10 CIFAR-100 GTSRB ImageNet PSNR ↑ SSIM ↑ IS ↓ PSNR ↑ SSIM ↑ IS ↓ PSNR ↑ SSIM ↑ IS ↓ PSNR ↑ SSIM ↑ IS ↓ No Attack INF 1.0000 0.000 INF 1.0000 0.000 INF 1.0000 0.000 INF 1.0000 0.000 BadNets [5] 25.77 0.9942 0.136 25.48 0.9943 0.137 25.33 0.9935 0.180 21.88 0.9678 0.025 Blend [17] 20.40 0.8181 1.823 20.37 0.8031 1.600 18.58 0.6840 2.118 13.72 0.1871 2.252 IAD [7] 24.35 0.9180 0.472 23.98 0.9138 0.490 23.84 0.9404 0.309 WaNet [8] 30.91 0.9724 0.326 31.62 0.9762 0.237 33.26 0.9659 0.170 35.18 0.9756 0.029 BppAttack [9] 27.79 0.9285 0.895 27.93 0.9207 0.779 27.79 0.8462 0.714 27.34 0.8009 0.273 Adapt-Blend [11] 25.97 0.9231 0.519 26.00 0.9133 0.495 24.14 0.8103 1.136 18.96 0.6065 1.150 FTrojan [10] 44.07 0.9976 0.019 44.09 0.9972 0.017 40.23 0.9813 0.065 35.55 0.9440 0.013 LIRA [39] 46.77 0.9979 0.019 47.77 0.9995 0.018 40.44 0.9879 0.089 Fiba [40] 26.08 0.9734 0.061 26.24 0.9688 0.055 23.41 0.9130 0.079 WaveAttack (Ours) 47.49 0.9979 0.011 50.12 0.9992 0.005 40.67 0.9877 0.058 45.60 0.9913 0.007 4.4 Resistance to Existing Defenses (RQ3) To evaluate the robustness of WaveAttack against existing backdoor defenses, we implemented representative backdoor defenses (i.e., GradCAM [46], STRIP [47], Fine-Pruning [23], ANP [48] and Neural Cleanse [25]) and evaluated the resistance to them. We also show the robustness of WaveAttack against Spectral Signature [45] and other frequency detection methods [49] in the appendix. 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Entropy 0 500 1000 1500 2000 2500 3000 3500 Number of Inputs Clean Poison (a) CIFAR-10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Normalized Entropy 0 250 500 750 1000 1250 1500 1750 2000 Number of Inputs Clean Poison (b) CIFAR-100 0.0 0.5 1.0 1.5 2.0 Normalized Entropy 0 500 1000 1500 2000 2500 Number of Inputs Clean Poison (c) GTSRB Figure 5: STRIP normalized entropy of WaveAttack. Resistance to STRIP. STRIP [47] is a representative sample-based defense method. When entering a potentially poisoned sample into a model, STRIP will perturb it through a random set of clean samples and monitor the entropy of the prediction output. If the entropy of an input sample is low, STRIP will consider it poisoned. Figure 5 shows the entropies of benign and poisoned samples. From this figure, we can see that the entropies of the poisoned samples are larger than those of the benign samples, and STRIP fails to detect the poisoned samples generated by WaveAttack. Resistance to GradCAM. As an effective visualization mechanism, GradCAM [46] has been used to visualize intermediate feature maps of DNN, interpreting the predictions of DNN. 8 Figure 6: GradCAM visualization results for both clean and backdoored models. Existing defense methods [50, 51] exploit GradCAM to analyze the heatmap of input samples. Specifically, a clean model correctly predicts the class label, whereas a backdoored model predicts the target label. Based on this phenomenon, the backdoored model can induce an abnormal GradCAM heatmap compared to the clean model. If the heatmaps of poisoned samples are similar to those of benign sample counterparts, the attack method is robust and can withstand defense methods based on GradCAM. Figure 6 shows the visualization heatmaps of a clean model and a backdoored model attacked by WaveAttack. Please note that here “clean” denotes a clean model trained using benign training datasets. From this figure, we can find that the heatmaps of these models are similar and that WaveAttack can resist defense methods based on GradCAM. Resistance to Fine-Pruning. As a representative model reconstruction defense method, FinePruning (FP) [23] is based on the assumption that the backdoor can activate a few dormant neurons in DNNs. Therefore, pruning these dormant neurons can eliminate the backdoors in DNNs. To evaluate the resistance to FP, we gradually pruned the neurons of the last convolutional and fully connected layers. Figure 7 shows the performance comparison between WaveAttack and seven SOTA attack methods on CIFAR-10 by resisting FP. We find that along with more neurons being pruned, WaveAttack can acquire superior performance than other SOTA attack methods in terms of both ASR and BA. In other words, Fine-Pruning cannot eliminate the backdoor generated by WaveAttack. Note that, though the ASR and BA of WaveAttack are similar to those of Adapt-Blend at the final stage of pruning, the initial ASR (i.e., 71.57%) of Adapt-Blend is much lower than that (i.e., 100%) of WaveAttack. 0 100 200 300 400 Number of Filters Pruned (#) 0 20 40 60 80 100 Attack Success Rate (%) 0 100 200 300 400 Number of Filters Pruned (#) 80 82 84 86 88 90 92 94 96 Benign Accuracy (%) BadNets Blend IAD WaNet BppAttack Adapt-Blend FTrojan WaveAttack Figure 7: ASR comparison against Fine-Pruning. 0.0 0.2 0.4 0.6 0.8 Threshold 0 20 40 60 80 100 Attack Success Rate (%) 0.0 0.2 0.4 0.6 0.8 Threshold 65 70 75 80 85 90 95 Benign Accuracy (%) BadNets Blend IAD WaNet BppAttack Adapt-Blend FTrojan WaveAttack Figure 8: Attack performance comparison against ANP. Resistance to ANP. Figure 8 compares the attack performance between WaveAttack and SOTA attack methods on the dataset CIFAR-10 against the defense method, i.e., ANP [48], where we use the threshold to denote the pruning rate of neurons. We find that as more neurons are pruned, WaveAttack consistently outperforms the other SOTA attack methods in ASR and BA. Figure 9: Defense performance against NC. Resistance to Neural Cleanse. As a representative defense method for trigger generation, Neural Cleanse (NC) [25] assumes that the trigger designed by the adversary is small. Initially, NC optimizes a trigger pattern for each class label via an optimization process. Then, NC uses the Anomaly Index (i.e., Median Absolute Deviation [52]) to detect whether a DNN is backdoored. Similar to the work [25], we think the DNN is backdoored if the anomaly index is larger than 2. To evaluate the resistance to NC, we conducted experiments to evaluate our WaveAttack approach by resisting NC. Figure 9 shows the defense results against NC. Please note that here, “clean” denotes clean models trained by using benign training datasets, and “backdoored” denotes backdoored models by WaveAttack that are from the Subsection 4.2. From this figure, we 9 can see that the abnormal index of WaveAttack is smaller than 2 for all datasets, and WaveAttack can bypass NC detection. Resistance to Different Frequency Filtering Methods. From Table 10, we find that WaveAttack outperforms FTrojan in both BA and ASR under two frequency filtering methods. This is mainly because FTrojan only swaps the values of two random pixels of the samples after DCT transformation, while the quality (i.e., PSNR, SSIM, and IS) of training samples after attacks is neglected. Figure 10: Performance comparison considering different frequency filtering methods. Dataset CIFAR-10 CIFAR-100 Methods FTtrojan WaveAttack FTtrojan WaveAttack Metrics BA ↑ ASR ↑ BA ↑ ASR ↑ BA ↑ ASR ↑ BA ↑ ASR ↑ Gaussian 69.41 10.07 72.94 16.72 44.65 3.28 47.61 7.92 Wiener 66.59 12.13 69.58 77.08 41.90 6.42 42.19 76.00 Resistance to Frequency Detection Methods. Table 5 compares performance between different attack methods against the same defense method, i.e., the frequency detection method [49]. From this table, we can find that our method achieves a lower BDR than FTrojan, BppAttack, IAD, BadNets, and Blend. Note that, as studied in the experiment section, WaNet, and Adapt-Blend can be more easily detected by the latent space-based and sample-based detection methods, respectively. Table 5: Backdoor Detection Rate (BDR) comparison against the frequency detection method. Method BadNets Blend IAD WaNet BppAttack Adapt-Blend FTrojan WaveAttack BDR (%) 100 97.91 96.18 0.12 96.32 1.25 78.11 5.71 Resistance to Spectral Signature Spectral Signature [45] is a representative latent space-based detection defense method. Given a set of benign and poisoned samples, Spectral Signature first collects their latent features and computes the top singular value of the covariance matrix. Then, for each sample, the correlation score is calculated between its features and the top singular value used as the outlier score. If the samples have high outlier scores, they will be evaluated as poisoned. We randomly selected 9000 benign samples and 1000 poisoned samples. Figure 11 shows the histograms of the correlations between latent features of the samples and the top right singular vector of the covariance matrix. From this figure, we can find that the histograms of the poisoned data are similar to those of the benign data. Therefore, Spectral Signature fails to detect the poisoned data generated by WaveAttack. 8 6 4 2 0 2 4 6 Correlation with Top Right Singular Vector 0 200 400 600 800 1000 Number of Inputs Clean Poison (a) CIFAR-10 15 10 5 0 5 10 15 Correlation with Top Right Singular Vector 0 200 400 600 800 1000 Number of Inputs Clean Poison (b) CIFAR-100 20 15 10 5 0 5 10 15 Correlation with Top Right Singular Vector 0 200 400 600 800 1000 1200 1400 Number of Inputs Clean Poison (c) GTSRB Figure 11: The correlation with top right singular vector on different datasets. 5 Conclusion Although backdoor attacks on DNNs have attracted increasing attention from adversaries, few of them simultaneously consider both the fidelity of poisoned samples and latent space to enhance the stealthiness of their attack methods. To establish an effective and stealthy backdoor attack against various backdoor detection techniques, this paper proposed a novel frequency-based method called WaveAttack, which employs DWT to extract high-frequency features from samples to generate stealthier backdoor triggers. Furthermore, we introduced an asymmetric frequency obfuscation method to improve the impact of triggers and further enhance the effectiveness of WaveAttack. Comprehensive experimental results show that, compared with various SOTA backdoor attack methods, WaveAttack not only can achieve higher stealthiness and effectiveness but also can minimize the impact of image quality on well-known datasets. 10 6 Acknowledgements This work was supported by the Natural Science Foundation of China (62272170), “Digital Silk Road” Shanghai International Joint Lab of Trustworthy Intelligent Software (22510750100), and Shanghai Trusted Industry Internet Software Collaborative Innovation Center. 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Journal of the American Statistical Association, 69(346):383–393, 1974. 14 7 Appendix 7.1 Implementation Details for Experiments Settings of Datasets. Table 6 presents the setting of datasets used in our experiments. Table 6: Datasets Settings. Dataset Input Size Classes Training Images Test Images CIFAR-10 3×32×32 10 50000 10000 CIFAR-100 3×32×32 100 50000 10000 GTSRB 3×32×32 43 26640 12569 ImageNet subset 3×224× 224 20 26000 1000 Settings of Attacks. For a fair comparison, the settings of WaveAttack are consistent with those of the other seven SOTA attack methods. We used the SGD optimizer for training a classifier with a learning rate of 0.01, and the Adam optimizer for training a generator with a learning rate of 0.001. We decreased this learning rate by a factor of 10 after every 100 epochs. We considered various data augmentations, i.e., random crop and random horizontal flipping. For BadNets, we used a grid trigger placed in the bottom right corner of the image. For Blend, we applied a “Hello Kitty” trigger on CIFAR-10, CIFAR-100, and GTSRB datasets and used random noises on the ImageNet dataset. For other attack methods, we used the default settings in their respective papers. 7.2 Broader Impacts and Limitations Broader Impacts. In this work, we introduce a new effective and stealthy backdoor attack method named WaveAttack, which can stealthily compromise security-critical systems. If used improperly, the proposed attack method may pose a security risk to the existing DNN applications. Nevertheless, we hope that by emphasizing the potential harm of this malicious threat model, our work will stimulate the development of stronger defenses and promote greater attention from experts in the field. As a result, this knowledge promotes the creation of more secure and dependable DNN models and robust defensive measures. We would like to emphasize that our paper mainly focuses on introducing and evaluating the attack method. This paper aims to develop more powerful detection and defence mechanisms against such advanced backdoor attacks by proposing more advanced backdoor attack methods and addressing the weaknesses of state-of-the-art defence methods in future works. Limitations. Although our work shows exciting results for backdoor attacks, it requires more computing resources and runtime overhead than most existing backdoor attack methods due to the necessity of training a generator g to generate residuals of the various high-frequency components. Moreover, we do not consider a threat model, in which the adversary can only control the training dataset. In this threat model, we used our pre-trained generator to modify some benign samples in the training dataset. However, this limitation also appears in [11]. 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4,454	Provable Benefit of Cutout and CutMix for Feature Learning Junsoo Oh KAIST AI junsoo.oh@kaist.ac.kr Chulhee Yun KAIST AI chulhee.yun@kaist.ac.kr Abstract Patch-level data augmentation techniques such as Cutout and CutMix have demonstrated significant efficacy in enhancing the performance of vision tasks. However, a comprehensive theoretical understanding of these methods remains elusive. In this paper, we study two-layer neural networks trained using three distinct methods: vanilla training without augmentation, Cutout training, and CutMix training. Our analysis focuses on a feature-noise data model, which consists of several label-dependent features of varying rarity and label-independent noises of differing strengths. Our theorems demonstrate that Cutout training can learn low-frequency features that vanilla training cannot, while CutMix training can learn even rarer features that Cutout cannot capture. From this, we establish that CutMix yields the highest test accuracy among the three. Our novel analysis reveals that CutMix training makes the network learn all features and noise vectors “evenly” regardless of the rarity and strength, which provides an interesting insight into understanding patch-level augmentation. 1 Introduction Data augmentation is a crucial technique in deep learning, particularly in the image domain. It involves creating additional training examples by applying various transformations to the original data, thereby enhancing the generalization performance and robustness of deep learning models. Traditional data augmentation techniques typically focus on geometric transformations such as random rotations, horizontal and vertical flips, and cropping (Krizhevsky et al., 2012), or color-based adjustments such as color jittering (Simonyan and Zisserman, 2014). In recent years, several new data augmentation techniques have appeared. Among them, patch-level data augmentation techniques like Cutout (DeVries and Taylor, 2017) and CutMix (Yun et al., 2019) have received considerable attention for their effectiveness in improving generalization. Cutout is a straightforward method where random rectangular regions of an image are removed during training. In comparison, CutMix adopts a more complex strategy by cutting and pasting sections from different images and using mixed labels, encouraging the model to learn from blended contexts. The success of Cutout and CutMix has triggered the development of numerous variants including Random Erasing (Zhong et al., 2020), GridMask (Chen et al., 2020a), CutBlur (Yoo et al., 2020), Puzzle Mix (Kim et al., 2020), and Co-Mixup (Kim et al., 2021). However, despite the empirical success of these patch-level data augmentation techniques in various image-related tasks, a lack of comprehensive theoretical understanding persists: why and how do they work? In this paper, we aim to address this gap by offering a theoretical analysis of two important patch-level data augmentation techniques: Cutout and CutMix. Our theoretical framework draws inspiration from a study by Shen et al. (2022), which explores a data model comprising multiple label-dependent feature vectors and label-independent noises of varying frequencies and intensities. The key idea of this work is that learning features with low frequency can be challenging due to strong noises (i.e., low signal-to-noise ratio). We focus on how Cutout and CutMix can aid in learning such rare features. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1.1 Our Contributions In this paper, we consider a patch-wise data model consisting of features and noises, and use two-layer convolutional neural networks as learner networks. We focus on three different training methods: vanilla training without any augmentation, Cutout training, and CutMix training. We refer to these training methods in our problem setting as ERM, Cutout, and CutMix. We investigate how these methods affect the network’s ability to learn features. We summarize our contributions below: • We analyze ERM, Cutout, and CutMix, revealing that Cutout outperforms ERM since it enables the learning of rarer features compared to ERM (Theorem 3.1 and Theorem 3.2). Furthermore, CutMix demonstrates almost perfect performance (Theorem 3.3) by learning all features. • Our main intuition behind the negative result for ERM is that ERM learns to classify training samples by memorizing noise vectors instead of learning meaningful features if the features do not appear frequently enough. Hence, ERM suffers low test accuracy because it cannot learn rare features. However, Cutout alleviates this challenge by removing some of the strong noise patches, allowing it to learn rare features to some extent. • We prove the near-perfect performance of CutMix based on a novel technique that views the non-convex loss as a composition of a convex function and reparameterization. This enables us to characterize the global minimum of the loss and show that CutMix forces the model to activate almost uniformly across every patch of inputs, allowing it to learn all features. 1.2 Related Works Feature Learning Theory. Our work aligns with a recent line of studies investigating how training methods and neural network architectures influence feature learning. These studies focus on a specific data distribution composed of two components: label-dependent features and label-independent noise. The key contribution of this body of work is the exploration of which training methods or neural networks are most effective at learning meaningful features and achieving good generalization performance. Allen-Zhu and Li (2020) demonstrate that an ensemble model can achieve near-perfect performance by learning diverse features, while a single model tends to learn only certain parts of the feature space, leading to lower test accuracy. In other works, Cao et al. (2022); Kou et al. (2023a) explore the phenomenon of benign overfitting when training a two-layer convolutional neural network. The authors identify the specific conditions under which benign overfitting occurs, providing valuable insights into how these networks behave during training. Several other studies seek to understand various aspects of deep learning through the lens of feature learning (Zou et al., 2021; Jelassi and Li, 2022; Chen et al., 2022, 2023; Li and Li, 2023; Huang et al., 2023a,b). Theoretical Analysis of Data Augmentation. Several works aim to analyze traditional data augmentation from different perspectives, including kernel theory (Dao et al., 2019), margin-based approach (Rajput et al., 2019), regularization effects (Wu et al., 2020), group invariance (Chen et al., 2020b), and impact on optimization (Hanin and Sun, 2021). Moreover, many papers have explored various aspects of a recent technique called Mixup (Zhang et al., 2017). For example, studies have explored its regularization effects (Carratino et al., 2020; Zhang et al., 2020), its role in improving calibration (Zhang et al., 2022), its ability to find optimal decision boundaries (Oh and Yun, 2023) and its potential negative effects (Chidambaram et al., 2021; Chidambaram and Ge, 2024). Some works investigate the broader framework of Mixup, including CutMix, which aligns with the scope of our work. Park et al. (2022) study the regularization effect of mixed-sample data augmentation within a unified framework that contains both Mixup and CutMix. In Oh and Yun (2023), the authors analyze masking-based Mixup, which is a class of Mixup variants that also includes CutMix. In their context, they show that masking-based Mixup can deviate from the Bayes optimal classifier but require less training sample complexity. However, neither work provides a rigorous explanation for why CutMix has been successful. The studies most closely related to our work include Shen et al. (2022); Chidambaram et al. (2023); Zou et al. (2023). Shen et al. (2022) regard traditional data augmentation as a form of feature manipulation and investigate its advantages from a feature learning perspective. Both Chidambaram et al. (2023) and Zou et al. (2023) analyze Mixup within a feature learning framework. However, patch-level data augmentation such as Cutout and CutMix, which are the focus of our work, have not yet been explored within this context. 2 2 Problem Setting In this section, we introduce the data distribution and neural network architecture, and formally describe the three training methods considered in this paper. 2.1 Data Distribution We consider a binary classification problem on structured data, consisting of patches of labeldependent vectors (referred to as features) and label-independent vectors (referred to as noise). Definition 2.1 (Feature Noise Patch Data). We define a data distribution D on Rd×P × {−1, 1} such that (X, y) ∼D where X = x(1), . . . , x(P ) ∈Rd×P and y ∈{±1} is constructed as follows. 1. Choose the label y ∈{±1} uniformly at random. 2. Let {vs,k}s∈{±1},k∈[K] ⊂Rd be a set of orthonormal feature vectors. Choose the feature vector v ∈Rd for data point X as v = vy,k with probability ρk from {vy,k}k∈[K] ⊂Rd, where ρ1 + · · · + ρK = 1 and ρ1 ≥· · · ≥ρK. In our setting, there are three types of features with significantly different frequencies: common features, rare features, and extremely rare features, ordered from most to least frequent. The indices of these features partition [K] into (KC, KR, KE). 3. We construct P patches of X as follows. • Feature Patch: Choose p∗uniformly from [P] and we set x(p∗) = v. • Dominant Noise Patch: Choose ˜p uniformly from [P]\{p∗}. We construct x(˜p) = αu+ξ(˜p) where αu is feature noise drawn uniformly from {αv1,1, αv−1,1} with 0 < α < 1 and ξ(˜p) is Gaussian dominant noise drawn from N(0, σ2 dΛ). • Background Noise Patch: The remaining patches p ∈[P] \ {p∗, ˜p} consist of Gaussian background noise, i.e., we set x(p) = ξ(p) where ξ(p) ∼N(0, σ2 bΛ). Here, the noise covariance matrix is defined as Λ := I −P s,k vs,kv⊤ s,k which ensures that Gaussian noises are orthogonal to all features. We assume that the dominant noise is stronger than the background noise, i.e., σb < σd. Our data distribution captures characteristics of image data, where the input consists of several patches. Some patches contain information relevant to the image labels, such as cat faces for the label “cat,” while other patches contain information irrelevant to the labels, such as the background. Intuitively, there are two ways to fit the given data: learning features or memorizing noise. If a model fits the data by learning features, it can correctly classify test data having the same features. However, if a model fits the data by memorizing noise, it cannot generalize to unseen data because noise patches are not relevant to labels. Thus, learning more features is crucial for achieving better generalization. In real-world scenarios, different features may appear with varying frequencies. For instance, the occurrences of cat’s faces and cat’s tails in a dataset might differ significantly, although both are relevant to the “cat” label. Our data distribution reflects these characteristics by considering features with varying frequencies. To emphasize the distinctions between the three training methods we analyze, we categorize features into three groups: common, rare, and extremely rare. We refer to data points containing these features as common data, rare data, and extremely rare data, respectively. We emphasize that these terminologies are chosen merely to distinguish the three different levels of rarity, and even “extremely rare” features appear in a nontrivial fraction of the training data with high probability (see our assumptions in Section 2.4). Comparison to Previous Work. Our data distribution is similar to those considered in Shen et al. (2022) and Zou et al. (2023), which investigate the benefits of standard data augmentation methods and Mixup by comparing them to vanilla training without any augmentation. These results consider two types of features—common and rare—with different levels of rarity, along with two types of noise: feature noise and Gaussian noise. However, we consider three types of features: common, rare, and extremely rare, and three types of noise: feature noise, dominant noise, and background noise. This distinction allows us to compare three distinct methods and demonstrate the differences between them, whereas Shen et al. (2022) and Zou et al. (2023) compared only two methods. 3 2.2 Neural Network Architecture For the prediction model, we focus on the following two-layer convolutional neural network where the weights in the second layer are fixed at 1 and −1, with only the first layer being trainable. Several works including Shen et al. (2022) and Zou et al. (2023) also focus on similar two-layer convolutional neural networks. Definition 2.2 (2-Layer CNN). We define 2-layer CNN fW : Rd×P →R parameterized by W = {w1, w−1} ∈Rd×2. For each input X = x(1), . . . , x(P ) ∈Rd×P , we define fW (X) := X p∈[P ] ϕ D w1, x(p)E − X p∈[P ] ϕ D w−1, x(p)E , where ϕ(·) is a smoothed version of leaky ReLU activation, defined as follows. ϕ(z) :=      z −(1−β)r 2 z ≥r 1−β 2r z2 + βz 0 ≤z ≤r βz z ≤0 , where 0 < β ≤1 and r > 0. Previous works on the theory of feature learning often consider neural networks with (smoothed) ReLU or polynomial activation functions. However, we adopt a smoothed leaky ReLU activation, which always has a positive slope, to exclude the possibility of neurons “dying” during the complex optimization trajectory. Using smoothed leaky ReLU to analyze the learning dynamics of neural networks is not entirely new; there is a body of work that studies phenomena such as benign overfitting (Frei et al., 2022a) and implicit bias (Frei et al., 2022b; Kou et al., 2023b) by analyzing neural networks with (smoothed) leaky ReLU activation. A key difference between ReLU and leaky ReLU lies in the possibility of ReLU neurons “dying” in the negative region, where some negatively initialized neurons remain unchanged throughout training. As a result, using ReLU activation requires multiple neurons to ensure the survival of neurons at initialization, which becomes increasingly probable as the number of neurons increases. In contrast, the derivative of leaky ReLU is always positive, ensuring that a single neuron is often sufficient. Therefore, for mathematical simplicity, we consider the case where the network has a single neuron for each positive and negative output. We believe that our analysis can be extended to the multi-neuron case as we validate numerically in Appendix A.2. 2.3 Training Methods Using a training set sampled from the distribution D, we would like to train our network fW to learn to correctly classify unseen data points from D. We consider three learning methods: vanilla training without any augmentation, Cutout, and CutMix. We first introduce necessary notation for our data and parameters, and then formalize training methods within our framework. Training Data. We consider a training set Z = {(Xi, yi)}i∈[n] comprising n data points, each independently drawn from D. For each i ∈[n], we denote Xi = (x(1) i , . . . , x(P ) i ). Initialization. We initialize the model parameters in our neural network using random initialization. Specifically, we initialize the model parameter W (0) = {w(0) 1 , w(0) −1}, where w(0) 1 , w(0) −1 i.i.d. ∼ N(0, σ2 0Id). Let us denote updated model parameters at iteration t as W (t) = {w(t) 1 , w(t) −1}. 2.3.1 Vanilla Training The vanilla approach to training a model fW is solving the empirical risk minimization problem using gradient descent. We refer to this method as ERM. Then, ERM updates parameters W (t) of a model using the following rule. W (t+1) = W (t) −η∇W LERM W (t) , 4 where η is a learning rate and LERM(·) is the ERM training loss defined as LERM(W ) := 1 n X i∈[n] ℓ(yifW (Xi)), (1) where ℓ(·) is the logistic loss ℓ(z) = log(1 + e−z). 2.3.2 Cutout Training. Cutout (DeVries and Taylor, 2017) is a data augmentation technique that randomly cuts out rectangular regions of image inputs. In our patch-wise data, we regard Cutout training as using inputs with masked patches from the original data. For each subset C of [P] and i ∈[n], we define augmented data Xi,C ∈Rd×P as a data point generated by cutting the patches with indices in C out of Xi. We can represent Xi,C as Xi,C = x(1) i,C, . . . , x(P ) i,C , where x(p) i,C = ( x(p) i if p /∈C, 0 otherwise. Note that the output of the model fW (·) on this augmented data point Xi,C is fW (Xi,C) = X p/∈C ϕ D w1, x(p) i E − X p/∈C ϕ D w−1, x(p) i E . Then, the objective function for Cutout training can be defined as LCutout(W ) := 1 n X i∈[n] EC∼DC[ℓ(yifW (Xi,C))], where DC is a uniform distribution on the collection of subsets of [P] with cardinality C, where C is a hyperparameter satisfying 1 ≤C < P 2 .1 We refer to the process of training our model using gradient descent on Cutout loss LCutout(W ) as Cutout, and its update rule is W (t+1) = W (t) −η∇W LCutout W (t) , (2) where η is a learning rate. 2.3.3 CutMix Training. CutMix (Yun et al., 2019) involves not only cutting parts of images, but also pasting them into different images as well as assigning them mixed labels. For each subset S of [P] and i, j ∈[n], we define the augmented data point Xi,j,S ∈Rd×P as the data obtained by cutting patches with indices in S from data Xi and pasting them into Xj at the same indices S. We can write Xi,j,S as Xi,j,S = x(1) i,j,S, . . . , x(P ) i,j,S , where x(p) i,j,S = ( x(p) i if p ∈S, x(p) j otherwise. The one-hot encoding of the labels yi and yj are also mixed with proportions \|S\| P and 1 −\|S\| P , respectively. This mixed label results in the loss of the form \|S\| P ℓ(yifW (Xi,j,S)) + 1 −\|S\| P ℓ(yjfW (Xi,j,S)). From this, the CutMix training loss LCutMix(W ) can be defined as LCutMix(W ) := 1 n2 X i,j∈[n] ES∼DS " \|S\| P ℓ(yifW (Xi,j,S)) + 1 −\|S\| P ℓ(yjfW (Xi,j,S)) # , where DS is a probability distribution on the set of subsets of [P] which samples S ∼DS as follows.2 1DeVries and Taylor (2017) also employ a moderate size of cutting, such as cutting 16 × 16 pixels on CIFAR-10 data, which originally has 32 × 32 pixels. 2Other types of distributions, such as those considered in Yun et al. (2019), make the same conclusion. We adopt this distribution to make presentation simpler. 5 1. Choose the cardinality s of S uniformly at random from {0, 1, . . . , P}, and 2. Choose S uniformly at random from the collection of subsets of [P] with cardinality s. We refer to the process of training our network using gradient descent on CutMix loss LCutMix(W ) as CutMix, and its update rule is W (t+1) = W (t) −η∇W LCutMix W (t) , (3) where η is a learning rate. 2.4 Assumptions on the Choice of Problem Parameters To control the quantities that appear in the analysis of training dynamics, we make assumptions on several quantities in our problem setting. For simplicity, we use choices of problem parameters as a function of the dimension of patches d and consider sufficiently large d. We use the standard asymptotic notation O(·), Ω(·), Θ(·), o(·), ω(·) to express the dependency on d. We also use e O(·), eΩ(·), eΘ(·) to hide logarithmic factors of d. Additionally, poly(d) (or polylog(d)) represents quantities that increase faster than dc1(or (log d)c1) and slower than dc2 (or (log d)c2) for some constant 0 < c1 < c2. Similarly, o(1/poly(d)) (or o(1/polylog(d))) denotes some quantities that decrease faster than 1/dc (or 1/(log d)c) for any constant c. Finally, we use f(d) = o(g(d)/polylog(d)) when f(d)/g(d) = o(1/polylog(d)) for some function f and g of d. Assumptions. We assume that P = Θ(1) and P ≥8 for simplicity. Additionally, we consider a high-dimensional regime where the number of data points is much smaller than the dimension d, which is expressed as n = o αβσ−1 d σbd 1 2 /polylog(d) . We also assume that ρkn = ω n 1 2 log d for all k ∈[K], which ensures the sufficiency of data points with each feature. In addition, as we will describe in Section 4, the relative scales between the frequencies of features and the strengths of noises play crucial roles in our analysis, as they serve as a proxy for the “learning speed” in the initial phase. For common features k ∈KC, we assume ρk = Θ(1) and the learning speed of common features is much faster than that of dominant noise, which translates into the assumption σ2 dd = o(βn). For rare features k ∈KR, we assume ρk = Θ(ρR) for some ρR, and we consider the case where the learning speed of rare features is much slower than that of dominant noise but faster than background noise, which is expressed as ρRn = o α2σ2 dd/polylog(d) and σ2 bd = o(βρRn). Finally, for extremely rare features k ∈KE, we say ρk = Θ(ρE) for some ρE and their learning is even slower than that of background noises, which can be expressed as ρEn = o α2σ2 bd/polylog(d) . Lastly, we assume the strength of feature noise satisfies α = o n−1βσ2 dd/polylog(d) , and r, σ0, η > 0 are sufficiently small so that σ0, r = o (α/polylog(d)), η = o rσ−2 d d−1/polylog(d) . We list our assumptions in Assumption B.1 and there are many choices of parameters satisfying the set of assumptions, including: P = 8, C = 2, n = Θ d0.4 , α = Θ d−0.02 , β = 1 polylog(d), σ0 = Θ(d−0.2), r = Θ(d−0.2), σd = Θ d−0.305 , σb = Θ d−0.375 , ρR = Θ d−0.1 , ρE = Θ d−0.195 , η = Θ(d−1). 3 Main Results In this section, we provide a characterization of the high probability guarantees for the behavior of models trained using three distinct methods we have introduced. We denote by T ∗the maximum admissible training iterates and we assume T ∗= poly(d) η with a sufficiently large polynomial in d. In all of our theorem statements, the randomness is over the sampling of training data and the initialization of models and all results hold under the condition that d is sufficiently large. The following theorem characterizes training accuracy and test accuracy achieved by ERM. 6 Theorem 3.1. Let W (t) be iterates of ERM. Then with probability at least 1 −o 1 poly(d) , there exists TERM such that any T ∈[TERM, T ∗] satisfies the following: • (Perfectly fits training set): For all i ∈[n], yifW (T )(Xi) > 0. • (Random on (extremely) rare data): P(X,y)∼D [yfW (T )(X)>0]=1−1 2 P k∈KR∪KE ρk±o 1 poly(d) . The proof is provided in Appendix C.2. Theorem 3.1 demonstrates that ERM achieves perfect training accuracy; however, it performs almost like random guessing on unseen data points with rare and extremely rare features. This is because ERM can only learn common features and overfit rare or extremely rare data in the training set by memorizing noises to achieve perfect training accuracy. In comparison, we show that Cutout can perfectly fit both augmented training data and original training data, and it can also learn rare features that ERM cannot. However, Cutout still makes random guesses on test data with extremely rare features. We state these in the following theorem with the proof provided in Appendix D.2: Theorem 3.2. Let W (t) be iterates of Cutout training. Then with probability at least 1−o 1 poly(d) , there exists TCutout such that any T ∈[TCutout, T ∗] satisfies the following: • (Perfectly fits augmented data): For all i ∈[n] and C ⊂[P] with \|C\| = C, yifW (T )(Xi,C) > 0. • (Perfectly fits original training data): For all i ∈[n], yifW (T )(Xi) > 0. • (Random on extremely rare data): P(X,y)∼D [yfW (T )(X) > 0] = 1 −1 2 P k∈KE ρk ± o 1 poly(d) . In the case of CutMix, it is challenging to discuss training accuracy directly because the augmented data have soft labels generated by mixing pairs of labels. Instead, we prove that CutMix achieves a sufficiently small gradient of the loss, and the training accuracy on the original training data is perfect. We also demonstrate that CutMix achieves almost perfect test accuracy, as it learns all types of features regardless of rarity. Theorem 3.3. Let W (t) be iterates of CutMix training. Then with probability at least 1−o 1 poly(d) , there exists some TCutMix ∈[0, T ∗] that satisfies the following: • (Finds a near stationary point): ∇W LCutMix W (TCutMix) = 1 poly(d). • (Perfectly fits original training data): For all i ∈[n], yifW (TCutMix)(Xi) > 0. • (Almost perfectly classifies test data): P(X,y)∼D h yfW (TCutMix)(X) > 0 i = 1 −o 1 poly(d) . To prove Theorem 3.3, we characterize the global minimum of objective loss of CutMix. Surprisingly, at the global minimum, the model has the same outputs for all patches of the input data. In other words, the contributions of all feature vectors and noise vectors to the final outcome of the network are identical, regardless of their frequency and strength (see Section 4.2 for more details). Moreover, this uniform “contribution” is large enough, which allows the model to learn all types of features by reaching the global minimum. We provide the detailed proof in Appendix E.2. Our three main theorems elucidate the benefits of Cutout and CutMix. Cutout enables a model to learn rarer features than ERM, while CutMix can even outperform Cutout. These advantages in learning rarer features lead to improvements in generalization performance. 4 Overview of Analysis In this section, we discuss key proof ideas and the main challenges in our analysis. For ease of presentation, we consider the case α = 0. Although our assumptions do not allow the choice α = 0, the choice of nonzero α is to show guarantees on the test accuracy and does not significantly affect the feature learning aspect. 7 To provide the proof overview, let us introduce some additional notation. For each i ∈[n], recall that the corresponding input point can be written as Xi = (x(1) i , . . . , x(P ) i ). We use p∗ i and ˜pi to denote the indices of its feature patch and dominant noise patch, respectively. For each feature vector vs,k, where s ∈{±1} and k ∈[K], let Vs,k ⊂[n] represent the set of indices of data points having the feature vector vs,k, and Vs = SK k=1 Vs,k denotes the set of indices of data with label s. For each data point i ∈[n] and dominant or background noise patch p ∈[P] \ {p∗ i }, we refer to the Gaussian noise inside x(p) i as ξ(p) i . 4.1 Vanilla Training and Cutout Training We now explain why ERM fails to learn (extremely) rare features, while Cutout can learn rare features but not extremely rare features. Let us consider ERM. From (1), for s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }, the component of ws in the feature vector vs′,k’s direction is updated as D w(t+1) s , vs′,k E = D w(t) s , vs′,k E −ss′η n X j∈Vs′,k ℓ′(yjfW (t)(Xj))ϕ′ D w(t) s , vs′,k E , (4) and similarly, the “update” of inner product of ws with a noise patch ξ(p) i can be written as D w(t+1) s , ξ(p) i E ≈ D w(t) s , ξ(p) i E −syiη n ℓ′(yifW (t)(Xi))ϕ′ D w(t) s , ξ(p) i E ξ(p) i 2 , (5) where the approximation is due to the near-orthogonality of Gaussian random vectors in the highdimensional regime. This approximation shows that ⟨w(t+1) s , vs′,k⟩’s and ⟨w(t) s , ξ(p) i ⟩’s are almost monotonically increasing or decreasing. We address the approximation errors using a variant of the technique introduced by Cao et al. (2022), as detailed in Appendix B.3. From (4) and (5), we can observe that in the early phase of training satisfying −ℓ′(yifW (t)(Xi)) = Θ(1), the main factor for the speed of learning features and noises are the number of feature occurrence \|Vs′,k\| and the strength of noise ∥ξ(p) i ∥2. From our assumptions introduced in Section 2.4, if we compare the learning speed of different components, we have common features ≫dominant noises ≫rare features ≫background noises ≫extremely rare features, in terms of “learning speed.” Based on this observation, we conduct a three-phase analysis for ERM. • Phase 1: Learning common features quickly. • Phase 2: Fitting (extremely) rare data by memorizing dominant noises instead of learning features. • Phase 3: A model cannot learn (extremely) rare features since gradients of all data are small. The main intuition behind why ERM cannot learn (extremely) rare features is that the gradients of all data containing these features become small after quickly memorizing dominant noise patches. In contrast, since Cutout randomly cuts some patches out, there exist augmented data points that do not contain dominant noises and have only features and background noises. This allows Cutout to learn rare features, thanks to these augmented data. However, extremely rare features cannot be learned since the learning speed of background noise is much faster and there are too many background noise patches to cut them all out. Remark 4.1. Shen et al. (2022) conduct analysis on vanilla training and training using standard data augmentation, sharing the same intuition in similar but different data models and neural networks. Also, we emphasize that we proved the model cannot learn (extremely) rare features even if we run poly(d) η iterations of GD, whereas Shen et al. (2022) only consider the first iteration that achieves perfect training accuracy. Practical Insights. In practice, images contain features and noise across several patches. A larger cutting size can be more effective in removing noise but may also remove important features that the model needs to learn. Thus, there is a trade-off in choosing the optimal cutting size, a trend also observed in DeVries and Taylor (2017). One limitation of Cutout is that it may not effectively remove dominant noise. Thus, dominant noise can persist in the augmented data, leading to potential noise memorization. We believe that developing strategies that can more precisely detect and remove these noise components from the image input could enhance the effectiveness of these methods. 8 4.2 CutMix Training In learning dynamics of ERM and Cutout, inner products between weight and data patches evolve (approximately) monotonically, which makes the analysis much more feasible. However, analyzing the learning dynamics of CutMix involves non-monotone change of inner products, which is inevitable since CutMix uses mixed labels; this is also demonstrated in our experimental results (Section 5,especially the leftmost plot in Figure 1). Non-monotonicity and non-convexity of the problem necessitates novel proof strategies. Let us define Z := {zs,k}s∈{±1},k∈[K] ∪{z(p) i }i∈[n],p∈[P ]\{p∗ i } as a function of W as follows, z(p) i := ϕ D w1, ξ(p) i E −ϕ D w−1, ξ(p) i E , zs,k := ϕ(⟨w1, vs,k⟩) −ϕ(⟨w−1, vs,k⟩). Then, Z represents the contribution of each noise patch and feature vector to the neural network output, and the nonconvex function LCutMix(W ) can be viewed as the composition of Z(W ) and a convex function h(Z). By using the convexity of h(Z), we can characterize the global minimum of LCutMix(W ). Surprisingly, we show that any global minimizer W ∗= {w∗ 1, w∗ −1} satisfies ϕ D w∗ s, x(p) i E −ϕ D w∗ −s, x(p) i E = Cs, for all s ∈{±1}, i ∈Vs, and p ∈[P], with some constants C1, C−1 = Θ(1). In other words, at the global minimum, the output of model on each patch of the training data is uniform across the set of data with the same labels. We also prove that CutMix can achieve a point close to the global minimum within poly(d) η iterations. As a result, the model trained by CutMix can learn all features including extremely rare features. The complete proof of Theorem 3.3 appears in Appendix E.2. Remark 4.2. Zou et al. (2023) investigate Mixup in a similar feature-noise model and show that Mixup can learn rarer features than vanilla training, with its benefits emerging from the early dynamics of training. However, our characterization of the global minimum of LCutMix(W ) and experimental results in our setting (Section 5, Figure 1) suggest that the benefits of CutMix, especially for learning extremely rare features, arise from the later stages of training. This suggests that Mixup and CutMix have different underlying mechanisms for promoting feature learning. Practical Insights. The main underlying mechanism of CutMix is that it learns information almost uniformly from all patches in the training data. However, this approach also involves memorizing noise, which can potentially degrade performance in real-world scenarios. We believe that a more sophisticated strategy such as considering the positional information of patches as used in Puzzle Mix (Kim et al., 2020) or Co-Mixup (Kim et al., 2021) could improve the ability to learn more from patches containing features and reduce the impact of noise. 5 Experiments We conduct experiments both in our setting and real-world data CIFAR-10 to support our theoretical findings and intuition. We defer CIFAR-10 experiment results to Appendix A.1. For the numerical experiments on our setting, we set the number of patches P = 3, dimension d = 2000, number of data points n = 300, dominant noise strength σd = 0.25, background noise strength σb = 0.15, and feature noise strength α = 0.005. The feature vectors are given as the standard basis e1, e2, e3, e4, e5, e6 ∈Rd, where e1, e2, e3 are features for the positive label y = 1 and e4, e5, e6 are features for the negative label y = −1. We categorize e1 and e4 as common features with a frequency of 0.8, e2 and e5 as rare features with a frequency of 0.15, and lastly, e3 and e6 as extremely rare features with a frequency of 0.05. For the learner network, we set the slope of negative regime β = 0.1 and the length of the smoothed interval r = 1. We train models using three methods: ERM, Cutout, and CutMix with a learning rate η = 1. For Cutout, we cut a single patch of data (C = 1). We apply full-batch gradient descent for all methods; for Cutout and CutMix, we utilize all possible augmented data points.3 We note that this choice of problem parameters does not exactly match the technical assumptions in Section 2.4. However, we empirically observe the same conclusions, which suggests that our analysis could be extended beyond our assumptions. 3For CutMix, this may induce different choices of DS from those assumed in our analysis, but we mention that other general choices of DS do not alter the conclusions in our analysis. 9 For each feature vector v of the positive label, we plot the output of the learned filters for the feature vector ϕ(⟨w(t) 1 , v⟩) −ϕ(⟨w(t) −1, v⟩) throughout training in Figure 1. Our numerical findings confirm that ERM can only learn common features, Cutout can learn common and rare features but cannot learn extremely rare features, and CutMix can learn all types of features. Especially, CutMix learn common features, rare features, and extremely rare features almost evenly. Also, we observed nonmonotone behavior of the output in the case of CutMix, which motivated our novel proof technique. The same trends are observed with different architectures, such as a smoothed (leaky) ReLU network with multiple neurons, as detailed in Appendix A.2. 0 2000 4000 6000 8000 10000 Iterations 0 2 4 6 8 10 Common Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0.0 0.2 0.4 0.6 0.8 Rare Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0.0 0.2 0.4 0.6 0.8 Extremely Rare Feature Output ERM Cutout CutMix Figure 1: Numerical results on our problem setting. We validate our findings on the trends of ERM, Cutout, and CutMix in learning common feature (Left), rare feature (Center), and extremely rare feature (Right). The output of the common feature trained by CutMix shows non-monotone behavior. 6 Conclusion We studied how Cutout and CutMix influence the ability to learn features in a patch-wise featurenoise data model learning with two-layer convolutional neural networks by comparing them with vanilla training. We showed that Cutout enables the learning of rare features that cannot be learned through vanilla training by mitigating the problem of memorizing label-independent noises instead of learning label-dependent features. Surprisingly, we further proved that CutMix can learn extremely rare features that Cutout cannot learn. We also present our theoretical insights on the underlying mechanism of these methods and provide experimental support. Limitation and Future Work. Our work has some limitations related to the neural network architecture, specifically, the use of a 2-layer two-neuron smoothed leaky ReLU network. Extending our results to neural networks with deeper, wider, and more general activation functions is a direction for future work. Another future direction is to develop patch-level data augmentation based on our theoretical findings. Also, it would be interesting to perform theoretical analysis on state-of-the-art patch-level data augmentation such as Puzzle Mix (Kim et al., 2020) or Co-Mixup (Kim et al., 2021). These methods utilize patch location information, thus it may require the development of a theoretical framework capturing more complex characteristics of image data. Acknowledgement This work was supported by three Institute of Information & communications Technology Planning & Evaluation (IITP) grants (No. RS-2019-II190075, Artificial Intelligence Graduate School Program (KAIST); No. RS-2022-II220184, Development and Study of AI Technologies to Inexpensively Conform to Evolving Policy on Ethics; No. RS-2024-00457882, AI Research Hub Project) funded by the Korean government (MSIT), and a National Research Foundation of Korea (NRF) grant (No. RS-2019-NR040050) funded by the Korean government (MSIT). 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PMLR, 2023. 13 Contents 1 Introduction 1 1.1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Problem Setting 3 2.1 Data Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Neural Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Training Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Assumptions on the Choice of Problem Parameters . . . . . . . . . . . . . . . . . 6 3 Main Results 6 4 Overview of Analysis 7 4.1 Vanilla Training and Cutout Training . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 CutMix Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Experiments 9 6 Conclusion 10 A Additional Experimental Results 15 A.1 Experiments on CIFAR-10 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.2 Additional Experimental Results on Our Data Distribution . . . . . . . . . . . . . 19 B Proof Preliminaries 20 B.1 Properties of the Choice of Problem Parameters . . . . . . . . . . . . . . . . . . . 20 B.2 Quantities at the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 B.3 Feature Noise Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 C Proof for ERM 29 C.1 Proof of Lemma B.3 for ERM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 C.2 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 D Proof for Cutout 45 D.1 Proof of Lemma B.3 for Cutout . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 D.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 E Proof for CutMix 62 E.1 Proof of Lemma B.3 for CutMix . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 E.2 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 F Technical Lemmas 75 14 A Additional Experimental Results For all experiments described in this section and in Section 5, we use NVIDIA RTX A6000 GPUs. A.1 Experiments on CIFAR-10 Dataset A.1.1 Experimental Support for Our Intuition We compare three methods, ERM training, Cutout training, and CutMix training on CIFAR-10 classification. For ERM training, we apply only random cropping and random horizontal flipping on train dataset. In comparison, for Cutout training and CutMix training, we additionally apply Cutout and CutMix, respectively, on training data. For Cutout training, we randomly cut 16 × 16 pixels of input images, and for CutMix training, we sample the mixing ratio from a beta distribution Beta(0.5, 0.5). We train ResNet-18 (He et al., 2016) for 200 epochs with a batch size of 128 using SGD with a learning rate 0.1, momentum 0.9, and weight decay 5 × 10−4. Trained models using ERM, Cutout, and CutMix achieve test accuracy 95.16%, 96.05%, and 96.29%, respectively. We randomly generate augmented data using CutMix from pairs of cat images and dog images in CIFAR-10 with varying mixing ratios λ = 1, 0.8, 0.6 (Dog:Cat = λ : 1 −λ). We randomly make 5, 000 (cat, dot)-pairs in CIFAR-10 training set and apply CutMix randomly 10 times. By repeating this procedure 10 times, we generate total 5, 000 × 10 × 10 = 500, 000 augmented samples for each mixing ratio λ. We plot a histogram of dog prediction output subtracted by cat prediction output (before applying the softmax function), evaluated on 500, 000 augmented data in Figure 2. 15 10 5 0 5 10 15 20 Dog Output - Cat Output 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Frequency ERM Cutout CutMix 15 10 5 0 5 10 15 20 Dog Output - Cat Output 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Frequency ERM Cutout CutMix 15 10 5 0 5 10 15 20 Dog Output - Cat Output 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Frequency ERM Cutout CutMix Figure 2: Histogram of dog prediction output subtracted by cat prediction output evaluated on data points augmented by CutMix data using cat data and dog data with varying mixing ratio λ (Dog : Cat = λ : 1 −λ) (Left) λ = 1 , (Center) λ = 0.8, (Right) λ = 0.6 The leftmost plot represents the evaluation results for original dog images, as it uses a mixing ratio of λ = 1. We can observe that the output of the model trained using Cutout is skewed toward higher values compared to the output of the model trained using other methods. We believe this aligns with the theoretical intuition that Cutout learns more information from the original image using augmented data. The remaining two plots show the output for randomly augmented data using CutMix. We observe that the models trained with CutMix exhibit a shorter tail, supporting our intuition from the CutMix analysis that the models learn uniformly across all patches. A.1.2 Experimental Support for Our Findings We train ResNet-18 using ERM training, Cutout training, and CutMix training following the same experimental details described in Appendix A.1.1, except using only 10% of the training set. This data-hungry setting is intended to highlight the benefits of Cutout and CutMix. We then evaluated the trained models on the remaining 90% of the CIFAR-10 training dataset. The reason for evaluating the remaining training dataset is to analyze the misclassified data using C-score (Jiang et al., 2021), which is publicly available only for the training dataset. 15 C-score measures the structural regularity of data, with lower values indicating examples that are more difficult to classify correctly. In our framework, data with harder-to-learn features (corresponding to rarer features) would likely have lower C-scores. Since directly extracting and quantitatively evaluating features learned by the models is challenging, we use the C-score as a proxy to evaluate the misclassified data across models trained by ERM, Cutout, and CutMix. Table 1 illustrates that Cutout tends to misclassify data with lower C-scores compared to ERM, indicating that Cutout learns more hard-to-learn features than vanilla training. Furthermore, the data misclassified by CutMix has even lower C-scores than those misclassified by Cutout, suggesting that CutMix is effective at learning features that are the most challenging to classify. This observation aligns with our theoretical findings, demonstrating that CutMix captures even more difficult features compared to both ERM and Cutout. Table 1: Mean and quantiles of the C-score on misclassified data across models trained with ERM, Cutout, and CutMix. The results indicate that Cutout tends to misclassify data with lower C-scores compared to ERM, while CutMix exhibits even lower C-scores. Method Mean Q1 Q2 Q3 ERM 0.687 0.615 0.782 0.841 Cutout 0.679 0.599 0.775 0.837 CutMix 0.670 0.575 0.767 0.835 Since directly visualizing features learned by a model is challenging, we present data that were misclassified by the model trained with ERM but correctly classified by the model trained with Cutout instead. In Figure 3, we show 7 samples per class with the lowest C-scores, which are considered to have rare features. Similarly, we also visualize data misclassified by the model trained with Cutout but correctly classified by the model trained with CutMix to represent extremely rare data in Figure 4. This approach allows us to interpret some (extremely) rare features in CIFAR-10, such as frogs with unusual colors. 16 C-score: 0.024 C-score: 0.028 C-score: 0.029 C-score: 0.032 C-score: 0.032 C-score: 0.033 C-score: 0.037 C-score: 0.037 C-score: 0.038 C-score: 0.043 C-score: 0.045 C-score: 0.047 C-score: 0.049 C-score: 0.053 C-score: 0.056 C-score: 0.056 C-score: 0.057 C-score: 0.059 C-score: 0.060 C-score: 0.060 C-score: 0.065 C-score: 0.068 C-score: 0.072 C-score: 0.072 C-score: 0.077 C-score: 0.080 C-score: 0.081 C-score: 0.081 C-score: 0.083 C-score: 0.085 C-score: 0.085 C-score: 0.087 C-score: 0.088 C-score: 0.090 C-score: 0.091 C-score: 0.091 C-score: 0.094 C-score: 0.099 C-score: 0.102 C-score: 0.103 C-score: 0.105 C-score: 0.105 C-score: 0.108 C-score: 0.109 C-score: 0.111 C-score: 0.122 C-score: 0.124 C-score: 0.135 C-score: 0.143 C-score: 0.146 C-score: 0.150 C-score: 0.168 C-score: 0.180 C-score: 0.187 C-score: 0.187 C-score: 0.188 C-score: 0.204 C-score: 0.212 C-score: 0.213 C-score: 0.215 C-score: 0.222 C-score: 0.235 C-score: 0.255 C-score: 0.264 C-score: 0.308 C-score: 0.327 C-score: 0.344 C-score: 0.360 C-score: 0.432 C-score: 0.440 Figure 3: Examples of rare data in CIFAR-10 17 C-score: 0.017 C-score: 0.017 C-score: 0.019 C-score: 0.021 C-score: 0.022 C-score: 0.029 C-score: 0.033 C-score: 0.033 C-score: 0.035 C-score: 0.037 C-score: 0.037 C-score: 0.039 C-score: 0.041 C-score: 0.042 C-score: 0.045 C-score: 0.048 C-score: 0.049 C-score: 0.051 C-score: 0.052 C-score: 0.052 C-score: 0.054 C-score: 0.054 C-score: 0.055 C-score: 0.056 C-score: 0.059 C-score: 0.059 C-score: 0.062 C-score: 0.064 C-score: 0.065 C-score: 0.069 C-score: 0.069 C-score: 0.071 C-score: 0.074 C-score: 0.074 C-score: 0.082 C-score: 0.091 C-score: 0.091 C-score: 0.096 C-score: 0.101 C-score: 0.103 C-score: 0.104 C-score: 0.105 C-score: 0.110 C-score: 0.111 C-score: 0.113 C-score: 0.114 C-score: 0.118 C-score: 0.124 C-score: 0.126 C-score: 0.128 C-score: 0.132 C-score: 0.134 C-score: 0.136 C-score: 0.139 C-score: 0.148 C-score: 0.148 C-score: 0.154 C-score: 0.168 C-score: 0.171 C-score: 0.174 C-score: 0.178 C-score: 0.196 C-score: 0.206 C-score: 0.212 C-score: 0.216 C-score: 0.247 C-score: 0.247 C-score: 0.250 C-score: 0.276 C-score: 0.283 Figure 4: Examples of extreme data in CIFAR-10 18 A.2 Additional Experimental Results on Our Data Distribution In addition to the results described in Section 5, we further conducted numerical experiments on our data distribution by applying two variations to our architecture: increasing the number of neurons, and increasing the number of neurons with a smoothed ReLU activation (instead of smoothed leaky ReLU). We observed the same trends as predicted by our theoretical findings and shown in Figure 1. Let us describe the setting of our experiments in detail. In both cases, We set the number of patches P = 3, dimension d = 2000, and the number of data n = 300. The feature vectors are given by the standard basis e1, e2, e3, e4, e5, e6 ∈Rd, where e1, e2, e3 are features for the positive label y = 1 and e4, e5, e6 are features for the negative label y = −1. We categorize e1 and e4 as common features, e2 and e5 as rare features, and lastly, e3 and e6 as extremely rare features. We apply full-batch gradient descent with learning rate η = 1 and for Cutout and CutMix, we utilize all possible augmented data. For the multi-neuron with smoothed Leaky ReLU case (Figure 5), we use 10 neurons for each positive/negative output with the slope of negative regime β = 0.1 and the length of polynomial regime r = 1. We set the strength of dominant noise σd = 0.25, the strength of background noise σb = 0.12, and the strength of feature noise α = 0.05. In addition, frequencies of common features, rare features, and extremely rare features are set to 0.72, 0.15, and 0.03, respectively. For the multi-neuron with smoothed ReLU case i.e., β = 0 (Figure 6), we set the length of the polynomial regime as r = 1, and we use 10 neurons for each positive/negative output. We set the remaining problem parameters as follows: the strength of dominant noise σd = 0.25, the strength of background noise σb = 0.12, and the strength of feature noise α = 0.05. In addition, frequencies of common features, rare features, and extremely rare features are set to 0.75, 0.2, and 0.05, respectively. 0 2000 4000 6000 8000 10000 Iterations 0 2 4 6 8 Common Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0 1 2 3 4 Rare Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0.0 0.2 0.4 0.6 0.8 Extremely Rare Feature Output ERM Cutout CutMix Figure 5: Multi-neuron with a smoothed leaky ReLU actiation 0 2000 4000 6000 8000 10000 Iterations 0 2 4 6 8 10 Common Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0.0 0.5 1.0 1.5 2.0 Rare Feature Output ERM Cutout CutMix 0 2000 4000 6000 8000 10000 Iterations 0.0 0.2 0.4 0.6 0.8 Extremely Rare Feature Output ERM Cutout CutMix Figure 6: Multi-neuron with a smoothed ReLU 19 B Proof Preliminaries B.1 Properties of the Choice of Problem Parameters In our analysis, we consider the choice of problem parameters as a function of the dimension of patches d and consider sufficiently large d. Let us summarize the assumptions on the parameters for the problem setting and assume they hold. Assumption B.1. The following conditions hold. A1. (The number of patches) P = Θ(1) and P ≥8. A2. (Overparameterized regime): n = o αβσ−1 d σbd 1 2 /polylog(d) . A3. (Sufficient feature data): For all k ∈[K], ρkn = ω n 1 2 log d . A4. (Common feature vs dominant noise): For all k ∈KC, ρk = Θ(1) and σ2 dd = o(βn). A5. (Rare feature vs noise): For all k ∈KR, ρk = Θ(ρR) with ρRn = o α2σ2 dd/polylog(d) and σ2 bd = o(βρRn). A6. (Extremely rare feature vs background noise) For all k ∈KE, ρk = Θ(ρE) with ρEn = o α2σ2 bd/polylog(d) . A7. (Strength of feature noise) α = o n−1βσ2 dd/polylog(d) . A8. σ0σ2 dd, r = o (α/polylog(d)) , η = o rσ−2 d d−1/polylog(d) We now present some properties derived from Assumption B.1, which are frequently used throughout our proof. From (A3), for all k ∈[K], we have the following inequality: n ≥ρ1n ≥ρ2 kn = ω log2 d (6) From (A1) and (A2), and given that β < 1, σb < σd, we have d > (βσ−1 d σbd 1 2 )2 > n2P > nP. (7) From (A2), (A3), and (A6), and given that α, β < 1, we have σ2 dd > σ2 bd = ω(ρEn) = ω(1). (8) From (A1), (A2) and the fact that 0 < α < 1, we have nPβ−1σdσ−1 b d−1 2 = o α polylog(d) = o 1 polylog(d) (9) From (A7) and (A4), we have αβ−1 < αβ−2 = o n−1β−1σ2 dd polylog(d) = o 1 polylog(d) (10) From (8) and (A8), η = o(1) and then we have η ≤log(ηT ∗) 2 . (11) From (A2), (A3), (A4), and (A5) we have α−2 = o σ2 dd ρRn = o ρ−1 R = o n 1 2 = o d 1 4 . (12) 20 B.2 Quantities at the Beginning We characterize some quantities at the beginning of training. Lemma B.2. Let Einit the event such that all the following holds: • 25 52n ≤\|V1\|, \|V−1\| ≤27 52n • For each s ∈{±1} and k ∈[K], ρkn 4 ≤\|Vs,k\| ≤3ρkn 4 • ∪i∈V1,1{p∗ i } = [P] • For any s, s′ ∈{±1} and k ∈[K], D w(0) s , vs′,k E ≤σ0 log d. • For any s ∈{±1} and i ∈[n], D w(0) s , ξ(˜pi) i E ≤σ0σdd 1 2 log d. • For any s ∈{±1}, i ∈[n] and p ∈[P] \ {p∗ i , ˜pi}, D w(0) s , ξ(p) i E ≤σ0σbd 1 2 log d. • For any i, j ∈[n] with i ̸= j, 1 2σ2 dd ≤ ξ(˜pi) i 2 ≤3 2σ2 dd and D ξ(˜pi) i , ξ(˜pj) j E ≤σ2 dd 1 2 log d. • For any i, j ∈[n] and p ∈[P] \ {p∗ j, ˜pj}, D ξ(˜pi) i , ξ(p) j E ≤σdσbd 1 2 log d. • For any i, j ∈[n] and p ∈[P] \ {p∗ i , ˜pi}, q ∈[P] \ {p∗ j, ˜pj} with (i, p) ̸= (j, q), 1 2σ2 bd ≤ ξ(p) i 2 ≤3 2σ2 bd and D ξ(p) i , ξ(q) j E ≤σ2 bd 1 2 log d. • {vs,k}s∈{±1},k∈[K] ∪{x(p) i }i∈[n],p∈[P ]\{p∗ i } is linearly independent. Then, the event Einit occurs with probability at least 1 −o 1 poly(d) . Also, if ξ ∼N(0, σ2Λ) is independent of w(0) 1 , w(0) −1 and {(Xi, yi)}i∈[n], we have D w(0) 1 , ξ E , D w(0) −1, ξ E ≤σ0σd 1 2 log d, and D ξ, ξ(p) i E ≤σσdd 1 2 log d, for all i ∈[n] and p ∈[P] \ {p∗ i }, with probability at least 1 −o 1 poly(d) . Proof of Lemma B.2. Let us prove the first three points hold with probability at least 1 −o 1 poly(d) . By Höeffding’s inequality, P h\|V1\| −n 2 > n 52 i = P   X i∈[n] (1yi=1 −E[1yi=1]) > n 52   ≤2 exp −2 522 n = o 1 poly(d) , where the last equality is due to (6). In addition, for each s ∈{±1}, k ∈[K], by Höeffding’s inequality P h\|Vs,k\| −ρk 2 n > ρk 4 n i = P   X i∈[n] 1i∈Vs,k −E[1i∈Vs,k] > ρk 4 n   ≤2 exp −ρ2 k 8 n = o 1 poly(d) , where the last equality is due to (6). Also, for each i ∈[n] and p ∈[P], P[{i ∈V1,1} ∩{p∗ i = p}] = ρ1 P . 21 Hence, P ∪i∈V1,1{p∗ i } ̸= [P] ≤ X p∈[P ] P h ∩i∈[n] ({i ∈V1,1} ∩{p∗ i = p})∁i = P 1 −ρ1 P n ≤P exp −ρ1 P n = o 1 poly(d) . Next, we will prove the remaining. Let us refer to the standard deviation of the Gaussian noise vector in p-th patch of i-th data as σi,p. In other words, for each i ∈[n] and p ∈[P] \ {p∗ i }, σi,p = σd if p = ˜pi, σb otherwise. For each s, s′ ∈{±1} and k ∈[K], D w(0) s , vs′,k E ∼N(0, σ0). Hence, by Höeffding’s inequality, we have P h D w(0) s , vs′,k E > σ0 log d i ≤2 exp −(σ0 log d)2 2σ2 0 ! = o 1 poly(d) . Let {ul}l∈[d−2K] be an orthonormal basis of the orthogonal complement of Span({vs,k}s∈{±1},k∈[K]). Note that for each s ∈{±1}, i ∈[n] and p ∈[P] \ {p∗ i }, we can write ξ(p) i and ξ as ws(0) = σ0 X l∈[d−2K] zs,lul, ξ(p) i = σi,p X l∈[d−2K] z(p) i,l ul, ξ = σ X l∈[d−2K] zlul where zs,l, z(p) i,l , zl i.i.d. ∼N(0, 1). The sub-gaussian norm of standard normal distribution N(0, 1) is q 8 3. Then z(p) i,l 2 −1’s are mean zero sub-exponential with sub-exponential norm 8 3 (Lemma 2.7.6 in Vershynin (2018)). In addition, zs,lz(p) i,l ’s, z(p) i,l z(q) j,l ’s and z(p) i,l zl’s are mean zero sub-exponential with sub-exponential norm less than or equal to 8 3 (Lemma 2.7.7 in Vershynin (2018)). We use Bernstein’s inequality (Theorem 2.8.1 in Vershynin (2018)), with c being the absolute constant stated therein. We then have the following: 1 −P 1 2σ2 i,pd ≤ ξ(p) i 2 ≤3 2σ2 i,pd ≤P ξ(p) i 2 −σ2 i,p(d −2K) ≥σ2 i,pd 1 2 log d = P   X l∈[d−2K] z(p) i,l 2 −1 ≥d 1 2 log d   ≤2 exp −9cd log2 d 64(d −2K) ≤2 exp −9c log2 d 64 = o 1 poly(d) , in addition, P h D ξ(p) i , ξ(q) j E ≥σi,pσj,qd 1 2 log d i = P   X l∈[d−2K] z(p) i,l z(q) j,l ≥d 1 2 log d   ≤2 exp −9cd log2 d 64(d −2K) ≤2 exp −9c log2 d 64 = o 1 poly(d) . 22 Similarly, we have P h D w(0) s , ξ(p) i E ≥σ0σi,pd 1 2 log d i ≤2 exp −9c log2 d 64 = o 1 poly(d) . Lastly, the last result holds almost surely due to (7). Applying the union bound to all events, each of which is at most poly(d) due to (7), leads us to our first conclusion. In addition, for each s ∈{±1}, i ∈[n] and p ∈[P] \ {p∗ i }, P h D w(0) s , ξ E ≥σ0σd 1 2 log d i ≤2 exp −9c log2 d 64 = o 1 poly(d) , and P h D ξ(p) i , ξ E ≥σi,pσd 1 2 log d i ≤2 exp −9c log2 d 64 = o 1 poly(d) . Applying the union bound to all events, each of which is at most poly(d) due to (7), leads us to our second conclusion. B.3 Feature Noise Decomposition In our analysis, we use a technique that analyzes the coefficients of linear combinations of feature and noise vectors. A similar technique in a different data and network setting is introduced by Cao et al. (2022). Lemma B.3. If we run one of ERM, Cutout, and CutMix training to update parameters W (t) of a model fW (t), then there exist coefficients (corresponding to each method) γ(t) s (s′, k)’s and ρ(t) s (i, p)’s so that we can write W (t) = {w(t) 1 , w(t) −1} as w(t) s = w(0) s + X k∈[K] γ(t) s (s, k)vs,k − X k∈[K] γ(t) s (−s, k)v−s,k + X i∈Vs,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 − X i∈V−s,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiρ(t) s (i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρ(t) s (i, ˜pi) v−s,1 ξ(˜pi) i 2    where Fs denotes the set of indices of data with feature noise vs,1. Furthermore, if we run one of ERM and Cutout, the coefficients γ(t) s (s′, k)’s and ρ(t) s (i, p)’s are monotone increasing. We provide proof of Lemma B.3 for ERM in Appendix C.1, for Cutout in Appendix D.1 and for CutMix in Appendix E.1. Since Gaussian vectors in a high-dimensional regime are nearly orthogonal, we can use the coefficients to approximate the inner products or outputs of neurons. The following lemma quantifies the approximation error. Lemma B.4. Suppose the event Einit occurs and 0 ≤γ(t) s (s′, k), ρ(t) s (i, p) ≤e O(β−1) for all s, s′ ∈ {±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i } at iteration t. Then, for each s ∈{±1}, k ∈[K], i ∈[n], and p ∈[P] \ {p∗ i }, the following holds: • D w(t) s , vs,k E −γ(t) s (s, k) , ϕ D w(t) s , vs,k E −γ(t) s (s, k) = o 1 polylog(d) • D w(t) s , v−s,k E + γ(t) s (−s, k) , ϕ D w(t) s , v−s,k E + βγ(t) s (−s, k) = o 1 polylog(d) • D w(t) yi , ξ(p) i E −ρ(t) yi (i, p) , ϕ D w(t) yi , ξ(p) i E −ρ(t) yi (i, p) = o 1 polylog(d) • D w(t) −yi, ξ(p) i E + ρ(t) −yi(i, p) , ϕ D w(t) −yi, ξ(p) i E + βρ(t) −yi(i, p) = o 1 polylog(d) 23 • ϕ D w(t) yi , x(˜pi) i E −ρ(t) yi (i, ˜pi) , ϕ D w(t) −yi, x(˜pi) i E + βρ(t) −yi(i, ˜pi) = o 1 polylog(d) Proof of Lemma B.4. For each s ∈{±1}, k ∈[K] \ {1}, by (A8) and (8), we have D w(t) s , vs,k E −γ(t) s (s, k) = D w(0) s , vs,k E = e O(σ0) = o 1 polylog(d) . Similarly, by (A8) and (8), D w(t) s , v−s,k E + γ(t) s (−s, k) = D w(0) s , v−s,k E = e O(σ0) = o 1 polylog(d) , Next, we will consider the case of v1,1 and v−1,1. For each s ∈{±1}, we have D w(t) s , vs,1 E −γ(t) s (s, 1) ≤ D w(0) s , vs,1 E + α X i∈[n] ρ(t) s (i, ˜pi) ξ(˜pi) i −2 ≤e O(σ0) + e O αnβ−1σ−2 d d−1 = o 1 polylog(d) , where the last equality is due to (8) and (A7). Similarly, we have D w(t) s , v−s,1 E + γ(t) s (−s, 1) ≤ D w(0) s , v−s,1 E + α X i∈[n] ρ(t) s (i, ˜pi) ξ(˜pi) i −2 ≤e O(σ0) + e O αnβ−1σ−2 d d−1 = o 1 polylog(d) . Hence, from (A8) and the fact that \|ϕ(z) −z\| ≤(1−β)r 2 for any z ≥0, we have ϕ D w(t) s , vs,k E −γ(t) s (s, k) ≤ ϕ D w(t) s , vs,k E −ϕ γ(t) s (s, k) + ϕ γ(t) s (s, k) −γ(t) s (s, k) ≤ D w(t) s , vs,k E −γ(t) s (s, k) + (1 −β)r 2 = o 1 polylog(d) . and ϕ D w(t) s , v−s,k E + βγ(t) s (−s, k) = ϕ D w(t) s , v−s,k E −ϕ −γ(t) s (−s, k) ≤ D w(t) s , v−s,k E + γ(t) s (−s, k) = o 1 polylog(d) . For each i ∈[n], and p ∈[P] \ {p∗ i }, we have D w(t) yi , ξ(p) i E −ρ(t) yi (i, p) ≤ D w(0) yi , ξ(p) i E + X j∈[n],q∈[P ]\{p∗ i } (j,q)̸=(i,p) ρ(t) yi (j, q) D ξ(p) i , ξ(q) j E ξ(q) j 2 24 ≤e O σ0σdd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o 1 polylog(d) , where the last equality is due to (A8) and (9). By triangular inequality, (A8), and the fact that ϕ′ ≤1 and \|ϕ(z) −z\| ≤(1−β)r 2 for any z ≥0, we have ϕ D w(t) yi , ξ(p) i E −ρ(t) yi (i, p) ≤ ϕ D w(t) yi , ξ(p) i E −ϕ ρ(t) yi (i, p) + ϕ ρ(t) yi (i, p) −ρ(t) yi (i, p) ≤ D w(t) yi , ξ(p) i E −ρ(t) yi (i, p) + (1 −β)r 2 = o 1 polylog(d) . Also, if i ∈Fs for some s ∈{±1}, ϕ D w(t) yi , x(˜pi) i E −ρ(t) yi (i, ˜pi) ≤ ϕ D w(t) yi , ξ(˜pi) i E −ρ(t) yi (i, ˜pi) + ϕ D w(t) yi , x(˜pi) i E −ϕ D w(t) yi , ξ(˜pi) i E ≤ ϕ D w(t) yi , ξ(˜pi) i E −ρ(t) yi (i, ˜pi) + α D w(t) yi , vs,1 E ≤ ϕ D w(t) yi , ξ(˜pi) i E −ρ(t) yi (i, ˜pi) + αγ(t) yi (s, 1) + α · o 1 polylog(d) ≤e O αβ−1 + o 1 polylog(d) = o 1 polylog(d) , where we apply the triangular inequality, the fact that ϕ′ ≤1, the triangular inequality again, ρ(t) yi (s, 1) = e O(β−1) and (10) sequentially. Similarly, D w(t) −yi, ξ(p) i E + ρ(t) −yi(i, p) ≤ D w(0) −yi, ξ(p) i E + X j∈[n],q∈[P ]\{p∗ i } (j,q)̸=(i,p) ρ(t) −yi(j, q) D ξ(p) i , ξ(q) j E ξ(q) j 2 ≤e O(σ0σdd 1 2 ) + e O nPβ−1σdσ−1 b d−1 2 = o 1 polylog(d) , and ϕ D w(t) −yi, ξ(p) i E + βρ(t) −yi(i, p) = ϕ D w(t) −yi, ξ(p) i E −ϕ −ρ(t) −yi(i, p) ≤ D w(t) −yi, ξ(p) i E + ρ(t) −yi(i, p) = o 1 polylog(d) , Also, if i ∈Fs for some s ∈{±1}, ϕ D w(t) −yi, x(˜pi) i E + βρ(t) −yi(i, ˜pi) = ϕ D w(t) −yi, ξ(˜pi) i E + βρ(t) −yi(i, ˜pi) + ϕ D w(t) −yi, x(˜pi) i E −ϕ D w(t) −yi, ξ(˜pi) i E 25 ≤ ϕ D w(t) −yi, ξ(˜pi) i E + βρ(t) −yi(i, ˜pi) + α D w(t) −yi, vs,1 E ≤ ϕ D w(t) −yi, ξ(˜pi) i E + βρ(t) −yi(i, ˜pi) + αγ(t) −yi(s, 1) + α · o 1 polylog(d) ≤e O αβ−1 + o 1 polylog(d) = o 1 polylog(d) . We define the set W as the collection of W = {w1, w−1}, where w1 −w(0) 1 , w−1 −w(0) −1 are elements of the subspace spanned by {vs,k}s∈{±1},k∈[K] ∪ n x(p) i o i∈[n],p∈[P ]\{p∗ i }. The following lemma guarantees the unique expression of any W ∈W in the form of the feature noise decomposition. Lemma B.5. Suppose the event Einit occurs. For each element W = {w1, w−1} ∈W, there exist unique coefficients γs(s′, k)’s and ρs(i, p)’s such that ws = w(0) s + X k∈[K] γs(s, k)vs,k − X k∈[K] γs(−s, k)v−s,k + X i∈Vs p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i 2 − X i∈V−s p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiρs(i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρs(i, ˜pi) v−s,1 ξ(˜pi) i 2    for each s ∈{±1}. Using this fact, for each s∗∈{±1} and k∗∈[K], we can introduce a function Q(s∗,k∗) : W →Rd×2 such that for each W = {w1, w−1} ∈W, Q(s∗,k∗)(W ) = n Q(s∗,k∗) 1 (w1), Q(s∗,k∗) −1 (w−1) o is given by: Q(s∗,k∗) s (ws) = ss∗γs(s∗, k∗)vs∗,k∗+ ss∗ X i∈Vs∗,k∗,p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs∩Vs∗,k∗ ss∗ρs(i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s∩Vs∗,k∗ ss∗ρs(i, ˜pi) v−s,1 ξ(˜pi) i 2   . The function Q(s∗,k∗) plays a crucial role in Section C.2.4 and Section D.2.4. The key intuition behind our definition of Q(s∗,k∗) is that Q(s∗,k∗)(W (t)) represents the term updated by the data having the feature vector vs∗,k∗, where W (t) are the iterates of either ERM or Cutout. As expected from this intuition, if we sum all Q(s∗,k∗) 1 (w1) and Q(s∗,k∗) −1 (w−1) over all s∗∈{±1} and k∗∈[K], the result will be equal to w1 −w(0) 1 and w−1 −w(0) −1, respectively. Proof. From linear independency of {vs,k}s∈{±1},k∈[K] ∪ n x(p) i o i∈[n],p∈[P ]\{p∗ i }, we can express any element W = {w1, w−1} ∈W as ws = w(0) s + X k∈[K] ˜γs(s, k)vs,k − X k∈[K] ˜γs(−s, k)v−s,k 26 + X i∈Vs, p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i − X i∈V−s, p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i (13) with unique {˜γs(s, k), ˜γs(−s, k)}s∈{±1},k∈[K] and {ρs(i, p)}s∈{±1},i∈[n],p∈[P ]\{i∗}. If we define γs(s, k) and γs(−s, k) as γs(s, k) = ˜γs(s, k), γs(−s, k) = ˜γs(−s, k) for k ̸= 1, and γs(s, 1) = ˜γs(s, 1) −α X i∈Fs syiρs(i, ˜pi) ξ(˜pi) i −2 , γs(−s, 1) = ˜γs(−s, 1) + α X i∈F−s syiρs(i, ˜pi) ξ(˜pi) i −2 , then we have ws = w(0) s + X k∈[K] γs(s, k)vs,k − X k∈[K] γs(−s, k)v−s,k + X i∈Vs p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i 2 − X i∈V−s p∈[P ]\{p∗ i } ρs(i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiρs(i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρs(i, ˜pi) v−s,1 ξ(˜pi) i 2   . Next, we want to show the uniqueness part. Suppose {ˆγs(s, k), ˆγs(−s, k)}s∈{±1},k∈[K] and {ˆρs(i, p)}s∈{±1},i∈[n],p∈[P ]\{i∗} satisfies ws = w(0) s + X k∈[K] ˆγs(s, k)vs,k − X k∈[K] ˆγs(−s, k)v−s,k + X i∈Vs p∈[P ]\{p∗ i } ˆρs(i, p) ξ(p) i ξ(p) i 2 − X i∈V−s p∈[P ]\{p∗ i } ˆρs(i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiˆρs(i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiˆρs(i, ˜pi) v−s,1 ξ(˜pi) i 2   . We have ws = w(0) s + X k∈[K]\{1} ˆγs(s, k)vs,k − X k∈[K]\{1} ˆγs(−s, k)v−s,k + ˆγs(s, 1) + α X i∈Fs syiˆρs(i, ˜pi) ξ(˜pi) i −2 ! vs,1 −  ˆγs(−s, 1) −α X i∈F−s syiˆρs(i, ˜pi) ξ(˜pi) i −2  v−s,1 + X i∈Vs p∈[P ]\{p∗ i } ˆρs(i, p) ξ(p) i ξ(p) i 2 − X i∈V−s p∈[P ]\{p∗ i } ˆρs(i, p) ξ(p) i ξ(p) i 2 . From the uniqueness of (13), we have ˆγs(s, k) = ˜γs(s, k) = γs(s, k), ˆγs(−s, k) = ˜γs(−s, k) = γs(−s, k), for each s ∈{±1}, k ∈[K] \ {1}, and ˆρs(i, p) = ρs(i, p) for each i ∈[n], p ∈[P] \ {p∗ i }. Furthermore, ˆγs(s, 1) + α X i∈Fs syiˆρs(i, ˜pi) ξ(˜pi) i −2 = ˜γs(s, 1) = γs(s, 1) + α X i∈Fs syiρs(i, ˜pi) ξ(˜pi) i −2 , 27 and ˆγs(−s, 1)−α X i∈F−s syiˆρs(i, ˜pi) ξ(˜pi) i −2 = ˜γs(−s, 1) = γs(−s, 1)−α X i∈F−s syiρs(i, ˜pi) ξ(˜pi) i −2 . Hence, we obtain the uniqueness of the expression and Q(s∗,k∗) is well defined for each s∗∈{±1} and k∗∈[K]. 28 C Proof for ERM In this section, we use g(t) i := 1 1+exp(yifW (t)(Xi)) for each data i and iteration t, for simplicity. C.1 Proof of Lemma B.3 for ERM For s ∈{±1} and iterate t, w(t+1) s −w(t) s = −η∇wsLERM W (t) = η n X i∈[n] syig(t) i X p∈[P ] ϕ′ D w(t) s , x(p) i E x(p) i = η n  X i∈Vs g(t) i X p∈[P ] ϕ′ D w(t) s , x(p) i E x(p) i − X i∈V−s g(t) i X p∈[P ] ϕ′ D w(t) s , x(p) i E x(p) i  , and we have X i∈Vs g(t) i X p∈[P ] ϕ′ D w(t) s , x(p) i E x(p) i = X k∈[K] X i∈Vs,k g(t) i ϕ′ D w(t) s , vs,k E vs,k + X i∈Vs g(t) i X p∈[P ]\{p∗ i ,˜pi} ϕ′ D w(t) s , ξ(p) i E ξ(p) i + X i∈Vs∩Fs g(t) i ϕ′ D w(t) s , αvs,1 + ξ(˜pi) i E αvs,1 + ξ(˜pi) i + X i∈Vs∩F−s g(t) i ϕ′ D w(t) s , αv−s,1 + ξ(˜pi) i E αv−s,1 + ξ(˜pi) i , and X i∈V−s g(t) i X p∈[P ] ϕ′ D w(t) s , x(p) i E x(p) i = X k∈[K] X i∈V−s,k g(t) i ϕ′ D w(t) s , v−s,k E v−s,k + X i∈V−s g(t) i X p∈[P ]\{p∗ i ,˜pi} ϕ′ D w(t) s , ξ(p) i E ξ(p) i + X i∈V−s∩Fs g(t) i ϕ′ D w(t) s , αvs,1 + ξ(˜pi) i E αvs,1 + ξ(˜pi) i + X i∈V−s∩F−s g(t) i ϕ′ D w(t) s , αv−s,1 + ξ(˜pi) i E αv−s,1 + ξ(˜pi) i . Hence, if we define γ(t) s (s′, k)’s and ρ(t) s (i, p)’s recursively by using the rule γ(t+1) s (s′, k) = γ(t) s (s′, k) + η n X i∈Vs′,k g(t) i ϕ′ D w(t) s , vs′,k E , (14) ρ(t+1) s (i, p) = ρ(t) s (i, p) + η ng(t) i ϕ′ D w(t) s , x(p) i E ξ(p) i 2 , (15) starting from γ(0) s (s′, k) = ρ(0) s (i, p) = 0 for each s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P]\{p∗ i }, then we have w(t) s = w(0) s + X k∈[K] γ(t) s (s, k)vs,k − X k∈[K] γ(t) s (−s, k)v−s,k + X i∈Vs,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 − X i∈V−s,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 29 + α    X i∈Fs syiρ(t) s (i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρ(t) s (i, ˜pi) v−s,1 ξ(˜pi) i 2   , for each s ∈{±1}. Furthermore, γ(t) s (s′, k)’s and ρ(t) s (i, p)’s are monotone increasing. □ C.2 Proof of Theorem 3.1 To show Theorem 3.1, we present a structured proof comprising the following five steps: 1. Establish upper bounds on γ(t) s (s′, k)’s and ρ(t) s (i, p)’s to apply Lemma B.4 (Section C.2.1). 2. Demonstrate that the model learns common features quickly (Section C.2.2). 3. Show that the model overfits dominant noise in (extremely) rare data instead of learning its feature (Section C.2.3). 4. Confirm the persistence of this tendency until T ∗iterates (Section C.2.4). 5. Characterize train accuracy and test accuracy (Section C.2.5). C.2.1 Bounds on the Coefficients in Feature Noise Decomposition The following lemma provides upper bounds on Lemma B.3 during T ∗iterations. Lemma C.1. Suppose the event Einit occurs. For any t ∈[0, T ∗], we have 0 ≤γ(t) s (s, k) + βγ(t) −s(s, k) ≤4 log(ηT ∗), 0 ≤ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≤4 log (ηT ∗) , for all s ∈{±1}, k ∈[K], i ∈[n] and p ∈[P]\{p∗ i }. Consequently,γ(t) s (s′, k), ρ(t) s (i, p) = e O(β−1) for all s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }. Proof of Lemma C.1. The first argument implies the second argument since log(ηT ∗) = polylog(d) and γ(t) s (s′, k) ≤β−1 γ(t) s′ (s′, k) + βγ(t) s′ (s′, k) , ρ(t) s (i, p) ≤β−1 ρ(t) yi (i, p) + βρ(t) −yi(i, p) , for all s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }. We will prove this by using induction on t. The initial case t = 0 is trivial. Suppose the given statement holds at t = T and consider the case t = T + 1. Let ˜Ts,k ≤T denote the smallest iteration where γ( ˜Ts,k+1) s (s, k) + βγ( ˜Ts,k+1) −s (s, k) > 2 log(ηT ∗). We assume the existence of ˜Ts,k, as its absence would directly lead to our desired conclusion; to see why, note that the following holds, due to (14) and (11): γ(T +1) s (s, k) + βγ(T +1) −s (s, k) = γ(T ) s (s, k) + βγ(T ) −s (s, k) + η n X i∈Vs,k g(T ) i ϕ′ D w(T ) s , vs,k E + βϕ′ D w(T ) −s , vs,k E ≤2 log(ηT ∗) + 2η ≤4 log(ηT ∗) Now suppose there exists such ˜Ts,k ≤T. By (14), we have γ(T +1) s (s, k) + βγ(T +1) −s (s, k) = γ( ˜Ts,k) s (s, k) + βγ( ˜Ts,k) −s (s, k) + T X t= ˜Ts,k γ(t+1) s (s, k) + βγ(t+1) −s (s, k) −γ(t) s (s, k) −βγ(t) −s(s, k) ≤2 log(ηT ∗) + log(ηT ∗) + η n T X t= ˜Ts,k+1 X i∈Vs,k g(t) i ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E . 30 The inequality is due to γ( ˜Ts,k) s (s, k) + βγ( ˜Ts,k) −s (s, k) ≤2 log(ηT ∗) from our choice of ˜Ts,k and η n X i∈Vs,k g( ˜Ts,k) i ϕ′ D w( ˜Ts,k) s , vs,k E + βϕ′ D w( ˜Ts,k) −s , vs,k E ≤2η ≤log(ηT ∗), from (11). For each t = ˜Ts,k + 1, . . . T, and i ∈Vs,k, we have yifW (t)(Xi) = ϕ D w(t) s , vs,k E −ϕ D w(t) −s, vs,k E + X p∈[P ]\{p∗ i } ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≥γ(t) s (s, k) + βγ(t) −s(s, k) + X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −2P · o 1 polylog(d) ≥3 2 log(ηT ∗) The first inequality is due to Lemma B.4 and the second inequality holds due to (A7), (8), and our choice of t, γ(t) s (s, k) + βγ(t) −s(s, k) ≥2 log(ηT ∗). Hence, we obtain η n T X t= ˜Ts,k X i∈Vs,k g(t) i ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E ≤2η n T X t= ˜Ts,k X i∈Vs,k exp (−yifW (t)(Xi)) ≤2\|Vs,k\| n (ηT ∗) exp −3 2 log(ηT ∗) ≤ 2 √ηT ∗≤log(ηT ∗), where the last inequality holds for any reasonably large T ∗. Merging all inequalities together, we have γ(T +1) s (s, k) + βγ(T +1) −s (s, k) ≤4 log(ηT ∗). Next, we will follow similar arguments to show that ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) ≤4 log(ηT ∗) for each i ∈[n] and p ∈[P] \ {p∗ i }. Let ˜T (p) i ≤T be the smallest iteration such that ρ ( ˜T (p) i +1) yi (i, p) + βρ ( ˜T (p) i +1) −yi (i, p) > 2 log(ηT ∗). We assume the existence of ˜T (p) i , as its absence would directly lead to our desired conclusion; to see why, note that the following holds, due to (15) and (11): ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) = ρ(T ) yi (i, p) + βρ(T ) −yi(i, p) + η ng(T ) i ϕ′ D w(t) s , x(p) i E + βϕ′ D w(t) s , x(p) i E ξ(p) i 2 ≤2 log(ηT ∗) + 2η ≤4 log(ηT ∗), where the first inequality is due to ξ(p) i ≤3 2σ2 dd and (A4), and the last inequality is due to (11). Now suppose there exists such ˜T (p) i ≤T. By (15), we have ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) 31 = ρ ( ˜T (p) i ) yi (i, p) + βρ ( ˜T (p) i ) −yi (i, p) + T X t= ˜T (p) i ρ(t+1) yi (i, p) + βρ(t+1) −yi (i, p) −ρ(t) yi (i, p) −βρ(t) −yi(i, p) ≤2 log(ηT ∗) + log(ηT ∗) + η n T X t= ˜T (p) i +1 g(t) i ϕ′ D w(t) s , x(p) i E + βϕ′ D w(t) −s, x(p) i E ξ(p) i 2 The inequality is due to ρ ( ˜T (p) i ) yi (i, p) + βρ ( ˜T (p) i ) −yi (i, p) ≤2 log(ηT ∗) from our choice of ˜T (p) i and η ng ( ˜T (p) i ) i ϕ′ w ( ˜T (p) i ) s , x(p) i + βϕ′ w ( ˜T (p) i ) −s , x(p) i ξ(p) i 2 ≤2η ≤log(ηT ∗), from ξ(p) i 2 ≤3 2σ2 dd, (A4), and (11). For each t = ˜T (p) i + 1, . . . , T, if i ∈Vs,k, then we have yifW (t)(Xi) = ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E + X q∈[P ]\{p} ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E ≥ρ(t) yi (i, p) + βρ(t) −yi(i, p) + γ(t) yi (s, k) + βγ(t) −yi(s, k) + X q∈[P ]\{p,p∗ i } ρ(t) yi (i, q) + βρ(t) −yi(i, q) −2P · o 1 polylog(d) ≥3 2 log(ηT ∗). The first inequality is due to Lemma B.4 and the second inequality holds because from our choice of t, ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≥2 log(ηT ∗). Therefore, we have η n T X t= ˜T (p) i +1 g(t) i ϕ′ D w(t) yi , x(p) i E + βϕ′ D w(t) −yi, x(p) i E ξ(p) i 2 ≤η T X t= ˜T (p) i +1 exp (−yifW (t)(Xi)) ≤(ηT ∗) exp −3 2 log(ηT ∗) ≤ 1 √ηT ∗≤log(ηT ∗), where the first inequality is due to ξ(p) i 2 ≤ 3 2σ2 d, (A4) and the last inequality holds for any reasonably large T ∗. Merging all inequalities together, we conclude ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) ≤ 4 log(ηT ∗). C.2.2 Learning Common Features In the initial stages of training, the model quickly learns common features while exhibiting minimal overfitting to Gaussian noise. First, we establish lower bounds on the number of iterations ensuring that noise coefficients ρ(t) s (i, p) remain small, up to the order of 1 P . Lemma C.2. Suppose the event Einit occurs. There exists ˜T > n 6ηP σ2 dd such that ρ(t) s (i, p) ≤ 1 4P for all 0 ≤t < ˜T, s ∈{±1}, i ∈[n] and p ∈[P] \ {p∗ i }. 32 Proof of Lemma C.2. Let ˜T be the smallest iteration such that ρ( ˜T ) s (i, p) ≥ 1 4P for some s ∈ {±1}, i ∈[n] and p ∈[P] \ {p∗ i }. We assume the existence of ˜T, as its absence would directly lead to our conclusion. Then, for any 0 ≤t < ˜T, we have ρ(t+1) s (i, p) = ρ(t) s (i, p) + η ng(t) i ϕ′ D w(t) s , x(p) i E ξ(p) i 2 ≤ρ(t) s (i, p) + 3ησ2 dd 2n , where the inequality is due to g(t) i < 1, ϕ′ ≤1, and ξ(p) i 2 ≤3 2σ2 dd. Hence, we have 1 4P ≤ρ( ˜T ) s (i, p) = ˜T −1 X t=0 ρ(t+1) s (i, p) −ρ(t) s (i, p) < 3ησ2 dd 2n ˜T, and we conclude ˜T > n 6ηP σ2 dd which is the desired result. Next, we will show that the model learns common features in at least constant order within ˜T iterates. Lemma C.3. Suppose the event Einit occurs and ρk = ω σ2 dd βn for some k ∈[K]. Then, for each s ∈{±1}, there exists Ts,k ≤ 9n ηβ\|Vs,k\| such that γ(t) s (s, k) + βγ(t) −s(s, k) ≥1 for any t > Ts,k. Proof of Lemma C.3. Suppose γ(t) s (s, k) + βγ(t) −s(s, k) < 1 for all 0 ≤t ≤ n 6ηP σ2 dd. For each i ∈Vs,k, we have yifW (t)(Xi) = ϕ D w(t) s , vs,k E −ϕ D w(t) −s, vs,k E + X p∈[P ]\{p∗ i } ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≤γ(t) s (s, k) + βγ(t) −s(s, k) + X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) + 2P · o 1 polylog(d) ≤1 + 2P · 1 4P + 2P · o 1 polylog(d) ≤2. The first inequality is due to Lemma B.4, the second inequality holds since we can apply Lemma C.2, and the last inequality is due to (A1). Thus, g(t) i = 1 1+exp(yifW (t)(Xi)) > 1 9 and we have γ(t+1) s (s, k) + βγ(t+1) −s (s, k) = γ(t) s (s, k) + βγ(t) −s(s, k) + η n X i∈Vs,k g(t) i ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E ≥γ(t) s (s, k) + βγ(t) −s(s, k) + ηβ\|Vs,k\| 9n . Notice that \|Vs,k\| = ρkn. From the condition in the lemma statement, we have 9n ηβ\|Vs,k\| = o n 6ηP σ2 dd . If we choose t0 ∈ h 9n ηβ\|Vs,k\|, n 6ηP σ2 dd i , then 1 > γ(t0) s (s, k) + βγ(t0) −s (s, k) ≥ηβ\|Vs,k\| 9n t0 ≥1, and this is contradictory; therefore, it cannot hold that γ(t) s (s, k) + βγ(t) −s(s, k) < 1 for all 0 ≤t ≤ n 6ηP σ2 dd. Hence, there exists 0 ≤Ts,k < n 6ηP σ2 dd such that γ(Ts,k+1) s (s, k) + βγ(Ts,k+1) −s (s, k) ≥1 and choose the smallest one. Then we obtain 1 > γ(Ts,k) s (s, k) + βγ(Ts,k) −s (s, k) ≥ηβ\|Vs,k\| 9n Ts,k. Therefore, Ts,k < 9n ηβ\|Vs,k\| and this is what we desired. 33 What We Have So Far. For any common feature vs,k with s ∈{±1} and k ∈KC, it satisfies ρk = w σ2 dd βn due to (A4). By Lemma C.3, at any iterate t ∈ ¯T1, T ∗ with ¯T1 := maxs∈{±1},k∈KC Ts,k, the following properties hold if the event Einit occurs: • (Learn common features): For any s ∈{±1} and k ∈KC, γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1). • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 . C.2.3 Overfitting (extremely) Rare Data In the previous step, we have shown that common data can be well-classified by learning common features. In this step, we will show that the model correctly classifies (extremely) rare data by overfitting dominant noise instead of learning its features. We first introduce lower bounds on the number of iterates such that feature coefficients γ(t) s (s′, k) remain small, up to the order of α2β−1. This lemma holds for any kind of features, but we will focus on (extremely) rare features. This does not contradict the results from Section C.2.2 for common features since the upper bound on the number of iterations in Lemma C.3 is larger than the lower bound on the number of iterations in this lemma. Lemma C.4. Suppose the event Einit occurs. For each s ∈{±1} and k ∈[K], there exists ˜Ts,k > nα2 ηβ\|Vs,k\| such that γ(t) s′ (s, k) ≤α2β−1 for any 0 ≤t < ˜Ts,k and s′ ∈{±1}. Proof of Lemma C.4. Let ˜Ts,k be the smallest iterate such that γ( ˜Ts,k) s′ (s, k) > α2β−1 for some s′ ∈{±1}. We assume the existence of ˜Ts,k, as its absence would directly lead to our conclusion. For any 0 ≤t < ˜Ts,k, γ(t+1) s′ (s, k) = γ(t) s′ (s, k) + η n X i∈Vs,k g(t) i ϕ′ D w(t) s′ , vs,k E ≤γ(t) s′ (s, k) + η\|Vs,k\| n , and we have α2β−1 < γ( ˜Ts,k) s′ (s, k) = ˜Ts,k−1 X t=0 γ(t+1) s′ (s, k) −γ(t) s′ (s, k) ≤η\|Vs,k\| n ˜Ts,k. We conclude ˜Ts,k > nα2 ηβ\|Vs,k\| which is the desired result. Next, we will show that the model overfits (extremely) rare data by memorizing dominant noise patches in at least constant order within ˜Ts,k iterates. Lemma C.5. Suppose the event Einit occurs and ρk = o α2σ2 dd n . Then, for each i ∈Vs,k, there exists Ti ∈ h ¯T1, 18n ηβσ2 dd i such that X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) ≥1, for any t > Ti. Proof of Lemma C.5. Suppose P p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) < 1 for 0 ≤t ≤ nα2 ηβ\|Vs,k\|. From Lemma B.4 and Lemma C.4, we have yifW (t)(Xi) 34 = ϕ D w(t) s , vs,k E −ϕ D w(t) −s, vs,k E + X p∈[P ]\{p∗ i } ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≤γ(t) s (s, k) + βγ(t) −s(s, k) + X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) + 2P · o 1 polylog(d) ≤(1 + β)α2β−1 + 1 + 2P · o 1 polylog(d) ≤2, where the last inequality is due to (10). Thus, we have g(t) i = 1 1+exp(yifW (t)(Xi)) ≥1 9. Also, ρ(t+1) s (i, ˜pi) + βρ(t+1) −s (i, ˜pi) = ρ(t) s (i, ˜pi) + βρ(t) −s(i, ˜pi) + η ng(t) i ϕ′ D w(t) s , x(˜pi) i E + βϕ′ D w(t) −s, x(˜pi) i E ξ(˜pi) i 2 ≥ρ(t) s (i, ˜pi) + βρ(t) −s(i, ˜pi) + ηβσ2 dd 18n , where the last inequality is due to ξ(˜pi) i 2 ≥1 2σ2 dd and ϕ′ ≥β. Notice that \|Vs,k\| = ρkn. From the given condition in the lemma statement, we have 18n ηβσ2 dd = o nα2 ηβ\|Vs,k\| . If we choose t0 ∈ h 18n ηβσ2 dd, nα2 ηβ\|Vs,k\| i , then we have 1 > X p∈[P ]\{p∗ i } ρ(t0) s (i, p) + βρ(t0) −s (i, p) ≥ρ(t0) s (i, ˜pi) + βρ(t0) −s (i, ˜pi) ≥ηβσ2 dd 18n t0 ≥1. This is a contradiction; therefore it cannot hold that P p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) < 1 for all 0 ≤t ≤ nα2 ηβ\|Vs,k\|. Hence, we can choose the smallest 0 ≤Ti < nα2 ηβ\|Vs,k\| such that P p∈[P ]\{p∗ i } ρ(Ti+1) s (i, p) + βρ(Ti+1) −s (i, p) ≥1. For any 0 ≤t < Ti, 1 ≥ X p∈[P ]\{p∗ i } ρ(Ti) s (i, p) + βρ(Ti) −s (i, p) ≥ρ(Ti) s (i, ˜pi) + βρ(Ti) −s (i, ˜pi) ≥ηβσ2 dd 18n Ti, and we conclude that Ti ≤ 18n ηβσ2 dd. Lastly, we move on to prove Ti > ¯T1. Combining Lemma C.2 and Lemma C.3 leads to X p∈[P ]\{p∗ i } ρ( ¯T1) s (i, p) + βρ( ¯T1) −s (i, p) ≤1 2. Thus, we have Ti > ¯T1 and this is what we desired. What We Have So Far. For any k ∈KR ∪KE, it satisfies ρk = o α2σ2 dd n due to (A5). By Lemma C.5 at iterate t ∈[TERM, T ∗] with TERM := max s∈{±1} k∈KR∪KE max i∈Vs,k Ti ∈ ¯T1, T ∗ the following properties hold if the event Einit occurs: • (Learn common features): For s ∈{±1} and k ∈KC, γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1), 35 • (Overfit (extremely) rare data): For any s ∈{±1}, k ∈KR ∪KE, and i ∈Vs,k, X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) = Ω(1), • (Do not learn (extremely) rare features at TERM): For any s, s′ ∈{±1} and k ∈KR ∪KE, γ(TERM) s′ (s, k) ≤α2β−1. • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 . C.2.4 ERM cannot Learn (extremely) Rare Features Within Polynomial Times In this step, we will show that ERM cannot learn (extremely) rare features within the maximum admissible iterations T ∗= poly(d) η . From now on, we fix any s∗∈{±1} and k∗∈KR ∪KE. Recall that we defined the set W and the function Q(s∗,k∗) : W →Rd×2 in Lemma B.5. Let us omit superscripts for simplicity. For each iteration t, Q(W (t)) represents the cumulative updates contributed by data points with feature vector vs∗,k∗until t-th iteration. We will sequentially introduce several technical lemmas and by combining these lemmas, quantify update by data with feature vector vs∗,k∗after TERM and derive our conclusion. Let us define W ∗= {w∗ 1, w∗ −1}, where w∗ s = w(TERM) s + M X i∈Vs∗,k∗ ξ(˜pi) i ξ(˜pi) i 2 , for each s ∈{±1} with M = 4β−1 log 2ηβ2T ∗ α2 . Note that (12), β < 1, and T ∗= poly(d) η together imply M = e O β−1 . Note that W (t), W ∗∈W for any t ≥0. Lemma C.6. Suppose the event Einit occurs. Then, Q W (TERM) −Q(W ∗) 2 ≤12M 2\|Vs∗,k∗\|σ−2 d d−1, where ∥·∥denotes the Frobenius norm. Proof of Lemma C.6. For each s ∈{±1}, ss∗ Qs (w∗ s) −Qs w(TERM) s = Qs  ss∗M X i∈Vs∗,k∗ ξ(˜pi) i ξ(˜pi) i   = M X i∈Vs∗,k∗ ξ(˜pi) i ξ(˜pi) i 2 + αM    X i∈Fs∩Vs∗,k∗ vs,1 ξ(˜pi) i 2 + X i∈F−s∩Vs∗,k∗ v−s,1 ξ(˜pi) i 2   , and we have Q W (TERM) −Q(W ∗) 2 = Q1(w∗ 1) −Q1 w(TERM) 1 2 + Q−1(w∗ −1) −Q−1 w(TERM) −1 2 ≤2M 2    X i∈Vs∗,k∗ ξ(˜pi) i −2 + X i,j∈Vs∗,k∗,i̸=j D ξ(˜pi) i , ξ(˜pj) j E ξ(˜pi) i 2 ξ(˜pj) j 2    36 + 2M 2   α2   X i∈Fs∩Vs∗,k∗ ξ(˜pi) i −2   2 + α2   X i∈F−s∩Vs∗,k∗ ξ(˜pi) i −2   2  . From the event Einit defined in Lemma B.2 and (A2), we have X i,j∈Vs∗,k∗,i̸=j D ξ(˜pi) i , ξ(˜pj) j E ξ(˜pi) i 2 ξ(˜pj) j 2 ≤ X i∈Vs∗,k∗ X j∈Vs∗,k∗ ξ(˜pi) i −2 e O d−1 2 ≤ X i∈Vs∗,k∗ ξ(˜pi) i −2 e O nd−1 2 ≤ X i∈Vs∗,k∗ ξ(˜pi) i −2 In addition, we have α2   X i∈Fs∩Vs∗,k∗ ξ(˜pi) i −2   2 + α2   X i∈F−s∩Vs∗,k∗ ξ(˜pi) i −2   2 ≤   X i∈Fs∩Vs∗,k∗ ξ(˜pi) i −2   2 +   X i∈F−s∩Vs∗,k∗ ξ(˜pi) i −2   2 ≤ X i∈Fs∩Vs∗,k∗ ξ(˜pi) i −2 + X i∈F−s∩Vs∗,k∗ ξ(˜pi) i −2 = X i∈Vs∗,k∗ ξ(˜pi) i −2 , where the first inequality is due to α < 1 and the second inequality is due to P i∈Vs∗,k∗ ξ(˜pi) i −2 ≤ 2\|Vs∗,k∗\|σ−2 d d−1 < 1 from (A5). Hence, from Einit, we obtain Q W (TERM) −Q(W ∗) 2 ≤6M 2 X i∈Vs∗,k∗ ξ(˜pi) i −2 ≤12M 2\|Vs∗,k∗\|σ−2 d d−1. Lemma C.7. Suppose the Einit occurs. For any t ≥TERM and i ∈Vs∗,k∗, it holds that ⟨yi∇W fW (t)(Xi), Q(W ∗)⟩≥Mβ 2 . Proof of Lemma C.7. We have ⟨yi∇W fW (t)(Xi), Q(W ∗)⟩ = X p∈[P ] ϕ′ D w(t) s∗, x(p) i E D Qs∗(w∗ s∗), x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗(w∗ −s∗), x(p) i E . For any s ∈{±1} and p ∈[P] \ {p∗ i , ˜pi}, ss∗D Qs(w∗ s), ξ(p) i E = ρ(TERM) s (i, p) + X j∈Vs∗,k∗,q∈[P ]\{p∗ j } (j,q)̸=(i,p) ρ(TERM) s (j, q) D ξ(p) i , ξ(q) j E ξ(q) j 2 + X j∈Vs∗,k∗ M D ξ(p) i , ξ(˜pj) j E ξ(˜pj) j 2 37 ≥−e O nPβ−1σdσ−1 b d−1 2 −e O nMσbσ−1 d d−1 2 = −o 1 polylog(d) , (16) where the last equality is due to (9) and M = e O β−1 . Also, for any s ∈ {±1}, ss∗⟨Qs(w∗ s), vs∗,k∗⟩= γ(TERM) s (s∗, k∗) ≥0. In addition, ss∗D Qs(w∗ s), x(˜pi) i E = ss∗D Qs(w∗ s), ξ(˜pi) i E + ss∗D Qs(w∗ s), x(˜pi) i −ξ(˜pi) i E ≥ss∗D Qs(w∗ s), ξ(˜pi) i E −e O α2β−1ρk∗nσ−2 d d−1 = M + ρ(TERM) s (i, ˜pi) + X j∈Vs∗,k∗,q∈[P ]\{p∗ i } (j,q)̸=(i,˜pi) ρ(TERM) s (j, q) D ξ(˜pi) i , ξ(q) j E ξ(q) j 2 + X j∈Vs∗,k∗\{i} M D ξ(˜pi) i , ξ(˜pj) j E ξ(˜pj) j 2 −e O α2β−1ρk∗nσ−2 d d−1 ≥M −e O nPβ−1σdσ−1 b d−1 2 −e O α2β−1ρk∗nσ−2 d d−1 = M −o 1 polylog(d) ≥M 2 , (17) where the first inequality is due to the definition of Q and the second-to-last line is due to (9) and (A7). Hence, applying (16) and (17) for s = s∗, −s∗and combining with ϕ′ ≥β, we have ⟨yi∇W fW (t)(Xi), Q(W ∗)⟩≥Mβ −o 1 polylog(d) ≥Mβ 2 . By combining Lemma C.6 and Lemma C.7, we can obtain the following result. Lemma C.8. Suppose the event Einit occurs. η n T ∗ X t=TERM X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) ≤ Q W (TERM) −Q(W ∗) 2 + 2ηT ∗e−Mβ 4 , where ∥·∥denotes the Frobenius norm. Proof of Lemma C.8. Note that for any TERM ≤t < T ∗, Q W (t+1) = Q W (t) −η n∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) , and thus Q W (t) −Q (W ∗) 2 − Q W (t+1) −Q (W ∗) 2 = 2η n * ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) , Q W (t) −Q (W ∗) + −η2 n2 ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 38 = 2η n * ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)), Q W (t)+ −2η n X i∈Vs∗,k∗ ℓ′(yifW (t)(Xi)) ⟨∇W yifW (t)(Xi), Q (W ∗)⟩−η2 n2 ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 ≥2η n * ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)), Q W (t)+ −Mβη n X i∈Vs∗,k∗ ℓ′(yifW (t)(Xi)) −η2 n2 ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 , where the last inequality is due to Lemma C.7. By the chain rule, we have * ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)), Q W (t)+ = X i∈Vs∗,k∗ " ℓ′(yifW (t)(Xi)) × X p∈[P ] ϕ′ D w(t) s∗, x(p) i E D Qs∗ w(t) s∗ , x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗ w(t) −s∗ , x(p) i E # . For each s ∈{±1}, i ∈Vs∗,k∗, and p ∈[P], D w(t) s , x(p) i E − D Qs w(t) s , x(p) i E = D w(t) s −Qs w(t) s , x(p) i E ≤ X j∈[n]\Vs∗,k∗,q∈[P ]\{p∗ i } * ρ(t) s (j, q) ξ(q) j ξ(q) j 2 , x(p) i + + α X j∈F1\Vs∗,k∗ ρ(t) s (j, ˜pj) ξ(˜pj) j −2 D v1,1, x(p) i E + α X j∈F−1\Vs∗,k∗ ρ(t) s (j, ˜pj) ξ(˜pj) j −2 D v−1,1, x(p) i E ≤e O nPβ−1σdσ−1 b d−1 2 + e O α2β−1nσ−2 d d−1 = o 1 polylog(d) , where the last inequality is due to Lemma C.1 and the event Einit. By Lemma F.1, X p∈[P ] ϕ′ D w(t) s∗, x(p) i E D Qs∗ w(t) s∗ , x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗ w(t) s∗ , x(p) i E ≤ X p∈[P ] ϕ D w(t) s∗, x(p) i E −ϕ D w(t) −s∗, x(p) i E + rP + o 1 polylog(d) = yifW (t)(Xi) + o 1 polylog(d) where the last equality is due to r = o 1 polylog(d) . Therefore, we have Q W (t) −Q (W ∗) 2 − Q W (t+1) −Q(W ∗) 2 39 ≥2η n X i∈Vs∗,k∗ ℓ′ (yifW (t)(Xi)) yifW (t)(Xi) + o 1 polylog(d) −Mβ 2 −η2 n2 ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 ≥2η n X i∈Vs∗,k∗ ℓ′(yifW (t)(Xi)) yifW (t)(Xi) −Mβ 4 −η2 n2 ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 . From the convexity of ℓ(·), X i∈Vs∗,k∗ ℓ′(yifW (t)(Xi)) yifW (t)(Xi) −Mβ 4 ≥ X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) −ℓ Mβ 4 ≥ X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) −ne−Mβ 4 . In addition, by Lemma F.2, η2 n2 ∇ X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) 2 ≤8η2P 2σ2 dd\|Vs∗,k∗\| n2 X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) ≤η n X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)), where the last inequality is due to (A8), and we have Q W (t) −Q(W ∗) 2 − Q W (t+1) −Q(W ∗) 2 ≥η n X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) −2ηe−Mβ 4 . From telescoping summation, we have η n T ∗ X t=TERM X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) ≤ Q W (TERM) −Q (W ∗) 2 + 2ηT ∗e−Mβ 4 . Finally, we can prove that the model cannot learn (extremely) rare features within T ∗iterations. Lemma C.9. Suppose the event Einit occurs. For any T ∈[TERM, T ∗], we have γ(T ) s (s∗, k∗) = e O α2β−2 for each s ∈{±1}. Proof of Lemma C.9. For any T ∈[TERM, T ∗], we have γ(T ) s (s, k) = γ(TERM) s (s∗, k∗) + η n T −1 X t=TERM X i∈Vs∗,k∗ g(t) i ϕ′ D w(t) s , vs∗,k∗ E ≤γ(TERM) s (s∗, k∗) + η n T −1 X t=TERM X i∈Vs∗,k∗ g(t) i 40 ≤γ(TERM) s (s∗, k∗) + η n T −1 X t=TERM X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) , where the first inequality is due to ϕ′ ≤1 and the second inequality is due to −ℓ′ ≤ℓ. From the result of Section C.2.3 we know γ(TERM) s (s∗, k∗) ≤α2β−1. Additionally, by Lemma C.8 and Lemma C.6, we have η n (T −1) X t=TERM X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) ≤η n (T ∗) X t=TERM X i∈Vs∗,k∗ ℓ(yifW (t)(Xi)) ≤ Q W (TERM) −Q(W ∗) 2 + 2ηT ∗e−Mβ 4 ≤12M 2\|Vs∗,k∗\|σ−2 d d−1 + 2ηT ∗e−Mβ 4 = e O α2β−2 . The last line is due to (A5) and our choice M = 4β−1 log 2ηβ2T ∗ α2 . Thus, we have our conclusion. What We Have So Far. Suppose the event Einit occurs. For any t ∈[TERM, T ∗], we have • (Learn common features): For each s ∈{±1} and k ∈KC, γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1). • (Overfit (extremely) rare data): For each s ∈{±1}, k ∈KR ∪KE and i ∈Vs,k, X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) = Ω(1). • (Cannot learn (extremely) rare features): For each s ∈{±1} and k ∈KR ∪KE, γ(t) s (s, k), γ(t) −s(s, k) = O α2β−2 . • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 , C.2.5 Train and Test Accuracy In this step, we will prove that the model trained by ERM has perfect training accuracy but has near-random guesses on (extremely) rare data. For any i ∈Vs,k with s ∈{±1} and k ∈KC, by Lemma B.4, we have yifW (t)(Xi) = X p∈[P ] ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≥γ(t) s (s, k) + βγ(t) −s(s, k) + X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −2P · o 1 polylog(d) ≥γ(t) s (s, k) + βγ(t) −s(s, k) −o 1 polylog(d) = Ω(1) −o 1 polylog(d) > 0, for any t ∈[TERM, T ∗]. In addition, for any i ∈Vs,k with s ∈{±1} and k ∈KR ∪KE, we have yifW (t)(Xi) 41 = X p∈[P ] ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E = γ(t) s (s, k) + βγ(t) −s(s, k) + X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −2P · o 1 polylog(d) ≥ X p∈[P ]\{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −o 1 polylog(d) = Ω(1) −o 1 polylog(d) > 0, for any t ∈[TERM, T ∗]. We can conclude that ERM with t ∈[TERM, T ∗] iterates achieve perfect training accuracy. Next, let us move on to the test accuracy part. Let (X, y) ∼D be a test data with X = x(1), . . . , x(P ) ∈Rd×P having feature patch index p∗, dominant noise patch index ˜p, and feature vector vy,k. We have x(p) ∼N(0, σ2 bΛ) for each p ∈[P] \ {p∗, ˜p} and x(˜p) −αvs,1 ∼N(0, σ2 dΛ) for some s ∈{±1}. Therefore, for all t ∈[TERM, T ∗] and p ∈[P] \ {p∗, ˜p}, ϕ D w(t) 1 , x(p)E −ϕ D w(t) −1, x(p)E ≤ D w(t) 1 −w(t) −1, x(p)E ≤ D w(0) 1 −w(0) −1, x(p)E + X i∈[n],q∈[P ]\{p∗ i } ρ(t) 1 (i, q) −ρ(t) −1(i, q) D ξ(q) i , x(p)E ξ(q) i 2 ≤e O σ0σbd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o α polylog(d) , (18) with probability at least 1 −o 1 poly(d) due to Lemma B.2, (A8), (8), and (9). In addition, for any s′ ∈{±1}, we have D w(t) s′ , x(˜p) −αvs,1 E ≤ D w(0) s′ , x(˜p) −αvs,1 E + X i∈[n],q∈[P ]\{p∗ i } ρ(t) s′ (i, q) D ξ(q) i , x(˜p) −αvs,1 E ξ(q) i 2 = e O σ0σdd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o α polylog(d) , (19) with probability at least 1 −o 1 poly(d) due to Lemma B.2, (A8), (8), and (9). Case 1: k ∈KC By Lemma B.2, (A8), and (10), ϕ D w(t) 1 , x(˜p)E −ϕ D w(t) −1, w(˜p)E ≤ D w(t) 1 −w(t) −1, x(˜p)E ≤α D w(t) 1 −w(t) −1, vs,1 E + D w(t) 1 −w(t) −1, x(p) −αvs,1 E 42 ≤α γ(t) 1 (s, 1) + γ(t) −1(s, 1) + α D w(0) 1 , vs,1 E + α D w(0) −1, vs,1 E + o 1 polylog(d) ≤e O αβ−1 + e O (ασ0) + o 1 polylog(d) = o 1 polylog(d) , (20) with probability at least 1 −o 1 poly(d) . Suppose (18) and (20) holds. By Lemma B.4, we have yfW (t)(X) = ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + X p∈[P ]\{p∗} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E = γ(t) y (y, k) + βγ(t) −y(y, k) −o 1 polylog(d) = Ω(1) −o 1 polylog(d) > 0. Therefore, we have P(X,y)∼D h yfW (t)(X) > 0 \| x(p∗) = vy,k, k ∈KC i ≥1 −o 1 poly(d) . (21) Case 2: k ∈KR ∪KE By triangular inequality and ϕ′ ≤1, we have ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E = ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E + ϕ D w(t) s , x(˜p)E −ϕ D w(t) s , αvs,1 E − ϕ D w(t) −s, x(˜p)E −ϕ D w(t) −s, αvs,1 E ≥ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E − D w(t) s , x(˜p) −αvs,1 E − D w(t) −s, x(˜p) −αvs,1 E . In addition, ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E = ϕ αγ(t) s (s, 1) −ϕ −αγ(t) −s(s, 1) + ϕ D w(t) s , αvs,1 E −ϕ αγ(t) s (s, 1) − ϕ D w(t) −s, αvs,1 E −ϕ −αγ(t) −s(s, 1) ≥ ϕ αγ(t) s (s, 1) −ϕ −αγ(t) −s(s, 1) −α D w(t) s , vs,1 E −γ(t) s (s, 1) −α D w(t) −s, vs,1 E + γ(t) −s(s, 1) = α γ(t) s (s, 1) + βγ(t) −s(s, 1) −α · o 1 polylog(d) 43 = Ω(α), where the second equality is due to Lemma B.4 and (A8). If (19) holds, we have ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E = Ω(α) −o α polylog(d) = Ω(α). (22) Note that yfW (t)(X) = ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + ϕ D w(t) y , x(˜p)E −ϕ D w(t) −y, x(˜p)E + X p∈[P ]\{p∗,˜p} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E , and ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + X p∈[P ]\{p∗,˜p} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E ≤ D w(t) y −w(t) −y, vy,k E + o α polylog(d) ≤γ(t) 1 (y, k) + γ(t) −1(y, k) + D w(0) y −w(0) −y, vy,k E + o α polylog(d) ≤O(α2β−2) + e O(σ0) + o α polylog(d) = o α polylog(d) < ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E , where the first inequality is due to (18), second-to-last line is due to (A8), (8) and (10), and the last inequality is due to (22). Therefore, we have yfW (t)(X) > 0 if y = s. Otherwise, yfW (t)(X) < 0. Therefore, we have P(X,y)∼D h yfW (t)(X) > 0 \| x(p∗) = vy,k, k ∈KR ∪KE i = 1 2 ± o 1 poly(d) . (23) Hence, combining (21) and (23) implies P(X,y)∼D [yfW (t)(X) > 0] = X k∈KC ρk + 1 2 1 − X k∈KC ρk ! ± o 1 poly(d) = 1 −1 2 X k∈KR∪KE ρk ± o 1 poly(d) . □ 44 D Proof for Cutout In this section, we use g(t) i,C := 1 1+exp(yifW (t)(Xi,C)) for each data i, C ⊂[P] with \|C\| = C and iteration t, for simplicity. D.1 Proof of Lemma B.3 for Cutout For s ∈{±1} and iterate t, w(t+1) s −w(t) s = −η∇wsLCutout W (t) = η n X i∈[n] syiEC∼DC  g(t) i,C X p/∈C ϕ′ D w(t) s , x(p) i E x(p) i   = η n  X i∈Vs EC∼DC  g(t) i,C X p/∈C ϕ′ D w(t) s , x(p) i E x(p) i   − X i∈V−s EC∼DC  g(t) i,C X p/∈C ϕ′ D w(t) s , x(p) i E x(p) i    , and we have X i∈Vs EC∼DC  g(t) i,C X p/∈C ϕ′ D w(t) s , x(p) i E x(p) i   = X k∈[K] X i∈Vs,k EC∼DC h g(t) i,Cϕ′ D w(t) s , vs,k E · 1p∗ i /∈C i vs,k + X i∈Vs X p∈[P ]\{p∗ i ,˜pi} EC∼DC h g(t) i,Cϕ′ D w(t) s , ξ(p) i E · 1p/∈C i ξ(p) i + X i∈Vs∩Fs EC∼DC h g(t) i,Cϕ′ D w(t) s , αvs,1 + ξ(˜pi) i E · 1˜pi /∈C i αvs,1 + ξ(˜pi) i + X i∈Vs∩F−s EC∼DC h g(t) i,Cϕ′ D w(t) s , αv−s,1 + ξ(˜pi) i E · 1˜pi /∈C i αv−s,1 + ξ(˜pi) i , and X i∈V−s EC∼DC  g(t) i,C X p/∈C ϕ′ D w(t) s , x(p) i E x(p) i   = X k∈[K] X i∈V−s,k EC∼DC h g(t) i,Cϕ′ D w(t) s , v−s,k E · 1p∗ i /∈C i v−s,k + X i∈V−s X p∈[P ]\{p∗ i ,˜pi} EC∼DC h g(t) i,Cϕ′ D w(t) s , ξ(p) i E · 1p/∈C i ξ(p) i + X i∈V−s∩Fs EC∼DC h g(t) i,Cϕ′ D w(t) s , αvs,1 + ξ(˜pi) i E · 1˜pi /∈C i αvs,1 + ξ(˜pi) i + X i∈V−s∩F−s EC∼DC h g(t) i,Cϕ′ D w(t) s , αv−s,1 + ξ(˜pi) i E · 1˜pi /∈C i αv−s,1 + ξ(˜pi) i . Hence, if we define γ(t) s (s′, k)’s and ρ(t) s (i, p)’s recursively by using the rule γ(t+1) s (s′, k) = γ(t) s (s′, k) + η n X i∈Vs′,k EC∼DC h g(t) i,Cϕ′ D w(t) s , vs′,k E · 1p∗ i /∈C i , (24) 45 ρ(t+1) s (i, p) = ρ(t) s (i, p) + η nEC∼DC h g(t) i,Cϕ′ D w(t) s , x(p) i E · 1p/∈C i ξ(p) i 2 , (25) starting from γ(0) s (s′, k) = ρ(0) s (i, p) = 0 for each s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P]\{p∗ i }, then we have w(t) s = w(0) s + X k∈[K] γ(t) s (s, k)vs,k − X k∈[K] γ(t) s (−s, k)v−s,k + X i∈Vs,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 − X i∈V−s,p∈[P ]\{p∗ i } ρ(t) s (i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiρ(t) s (i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρ(t) s (i, ˜pi) v−s,1 ξ(˜pi) i 2   , for each s ∈{±1}. Furthermore, γ(t) s (s′, k)’s and ρ(t) s (i, p)’s are monotone increasing. □ D.2 Proof of Theorem 3.2 To show Theorem 3.2, we present a structured proof comprising the following five steps: 1. Establish upper bounds on γ(t) s (s′, k)’s and ρ(t) s (i, p)’s to apply Lemma B.4 (Section D.2.1). 2. Demonstrate that the model quickly learns common and rare features (Section D.2.2). 3. Show that the model overfits augmented data if it does not contain common or rare features (Section D.2.3). 4. Confirm the persistence of this tendency until T ∗iterates (Section D.2.4). 5. Characterize train accuracy and test accuracy (Section D.2.5). D.2.1 Bounds on the Coefficients in Feature Noise Decomposition The following lemma provides upper bounds on Lemma B.3 during T ∗iterations. Lemma D.1. Suppose the event Einit occurs. For any 0 ≤t ≤T ∗, we have 0 ≤γ(t) s (s, k) + βγ(t) −s(s, k) ≤4 log(ηT ∗), 0 ≤ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≤4 log (ηT ∗) , for all s ∈{±1}, k ∈[K], i ∈[n] and p ∈[P]\{p∗ i }. Consequently, γ(t) s (s′, k), ρ(t) s (i, p) = e O(β−1) for all s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }. Proof of Lemma D.1. The first argument implies the second argument since log(ηT ∗) = polylog(d) and γ(t) s (s′, k) ≤β−1 γ(t) s′ (s′, k) + βγ(t) s′ (s′, k) , ρ(t) s (i, p) ≤β−1 ρ(t) yi (i, p) + βρ(t) −yi(i, p) , for all s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }. We will prove the first argument by using induction on t. The initial case t = 0 is trivial. Suppose the statement holds at t = T and consider the case t = T + 1. Let ˜Ts,k ≤T denote the smallest iteration where γ( ˜Ts,k+1) s (s, k) + βγ( ˜Ts,k+1) −s (s, k) > 2 log(ηT ∗). We assume the existence of ˜Ts,k, as its absence would directly lead to our desired conclusion; to see why, note that the following holds, due to (24) and (11): γ(T +1) s (s, k) + βγ(T +1) −s (s, k) = γ(T ) s (s, k) + βγ(T ) −s (s, k) + η n X i∈Vs,k EC∼DC h g(T ) i,C · 1p∗ i /∈C i ϕ′ D w(T ) s , vs,k E + βϕ′ D w(T ) −s , vs,k E 46 ≤2 log(ηT ∗) + 2η ≤4 log(ηT ∗) By (24), we have γ(T +1) s (s, k) + βγ(T +1) −s (s, k) = γ( ˜Ts,k) s (s, k) + βγ( ˜Ts,k) −s (s, k) + T X t= ˜Ts,k γ(t+1) s (s, k) + βγ(t+1) −s (s, k) −γ(t) s (s, k) −βγ(t) −s(s, k) ≤2 log(ηT ∗) + log(ηT ∗) + η n T X t= ˜Ts,k+1 X i∈Vs,k EC∼DC g(t) i,C ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E · 1p∗ i /∈C . The inequality is due to γ( ˜Ts,k) s (s, k) + βγ( ˜Ts,k) −s (s, k) ≤2 log(ηT ∗) and η n X i∈Vs,k EC∼DC g( ˜Ts,k) i,C ϕ′ D w( ˜Ts,k) s , vs,k E + βϕ′ D w( ˜Ts,k) −s , vs,k E · 1p∗ i /∈C ≤2η ≤log(ηT ∗), from our choice of ˜Ts,k and η. For each t = ˜Ts,k + 1, . . . T, i ∈Vs,k, and C ⊂[P] such that \|C\| = C and p∗ i /∈C, we have yifW (t)(Xi,C) = ϕ D w(t) s , vs,k E −ϕ D w(t) −s, vs,k E + X p/∈C∪{p∗ i } ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≥γ(t) s (s, k) + βγ(t) −s(s, k) + X p/∈C∪{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −2P · o 1 polylog(d) ≥3 2 log(ηT ∗) The first inequality is due to Lemma B.4 and the second inequality holds due to (A7), (8), and our choice of t, γ(t) s (s, k) + βγ(t) −s(s, k) ≥2 log(ηT ∗). Hence, we obtain η n T X t= ˜Ts,k X i∈Vs,k EC∼DC g(t) i,C ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E · 1p∗ i /∈C ≤2η n T X t= ˜Ts,k X i∈Vs,k EC∼DC exp (−yifW (t)(Xi,C)) · 1p∗ i /∈C ≤2\|Vs,k\| n (ηT ∗) exp −3 2 log(ηT ∗) ≤ 2 √ηT ∗≤log(ηT ∗), where the last inequality holds for any reasonably large T ∗. Merging all inequalities together, we have γ(T +1) s (s, k) + βγ(T +1) −s (s, k) ≤4 log(ηT ∗). Next, we will follow similar arguments to show that ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) ≤4 log(ηT ∗) for each i ∈[n] and p ∈[P] \ {p∗ i }. 47 Let ˜T (p) i ≤T be the smallest iteration such that ρ ( ˜T (p) i +1) yi (i, p) + βρ ( ˜T (p) i +1) −yi (i, p) > 2 log(ηT ∗). We assume the existence of ˜T (p) i , , as its absence would directly lead to our desired conclusion; to see why, note that the following holds, due to (25) and (11): ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) = ρ(T ) yi (i, p) + βρ(T ) −yi(i, p) + η nEC∼DC h g(T ) i,C · 1p/∈C i ϕ′ D w(t) s , x(p) i E + βϕ′ D w(t) s , x(p) i E ξ(p) i 2 ≤2 log(ηT ∗) + 2η ≤4 log(ηT ∗), where the first inequality is due to ξ(p) i ≤3 2σ2 dd and (A4), and the last inequality is due to (11). Now we suppose there exists such ˜Ti ≤T. By (25), we have ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) = ρ ( ˜T (p) i ) yi (i, p) + βρ ( ˜T (p) i ) −yi (i, p) + T X t= ˜T (p) i ρ(t+1) yi (i, p) + βρ(t+1) −yi (i, p) −ρ(t) yi (i, p) −βρ(t) −yi(i, p) ≤2 log(ηT ∗) + log(ηT ∗) + η n T X t= ˜T (p) i +1 EC∼DC h g(t) i,C · 1p/∈C i ϕ′ D w(t) s , x(p) i E + βϕ′ D w(t) −s, x(p) i E ξ(p) i 2 The inequality is due to ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≤2 log(ηT ∗) our choice of ˜T (p) i and η nEC∼DC g ( ˜T (p) i ) i,C · 1p/∈C ϕ′ w ( ˜T (p) i ) s , x(p) i + βϕ′ w ( ˜T (p) i ) −s , x(p) i ξ(p) i 2 ≤2η ≤log(ηT ∗), from ξ(p) i 2 ≤3 2σ2 dd, (A4), and (11). For each t = ˜T (p) i + 1, . . . , T, and C ⊂[P] such that \|C\| = C and p /∈C, we have yifW (t)(Xi,C) = ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E + X q /∈C∪{p} ϕ D w(t) yi , x(q) i E −ϕ D w(t) −yi, x(q) i E ≥ρ(t) yi (i, p) + βρ(t) −yi(i, p) −2P · o 1 polylog(d) ≥3 2 log(ηT ∗). The first inequality is due to Lemma B.4 and the second inequality holds since from our choice of t, ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≥2 log(ηT ∗). Therefore, we have η n T X t= ˜T (p) i +1 EC∼DC h g(t) i,C · 1p/∈C i ϕ′ D w(t) yi , x(p) i E + βϕ′ D w(t) −yi, x(p) i E ξ(p) i 2 ≤η T X t= ˜T (p) i +1 EC∼DC exp (−yifW (t)(Xi,C)) 1p/∈C ≤(ηT ∗) exp −3 2 log(ηT ∗) 48 ≤ 1 √ηT ∗≤log(ηT ∗), where the first inequality is due to ξ(p) i 2 ≤3 2σ2 dd and (A4). Hence, we conclude ρ(T +1) yi (i, p) + βρ(T +1) −yi (i, p) ≤4 log(ηT ∗). D.2.2 Learning Common Features and Rare Features In the initial stages of training, the model quickly learns common features while exhibiting minimal overfitting to Gaussian noise. First, we establish lower bounds on the number of iterations, ensuring that background noise coefficients ρ(t) s (i, p) for p ̸= p∗ i , ˜pi remain small, up to the order of 1 P . Lemma D.2. Suppose the event Einit occurs. There exists ˜T > n 6ηP σ2 bd such that ρ(t) s (i, p) ≤ 1 4P for all 0 ≤t < ˜T, s ∈{±1}, i ∈[n] and p ∈[P] \ {p∗ i , ˜pi}. Proof of Lemma D.2. Let ˜T be the smallest iteration such that ρ( ˜T ) s (i, p) ≥ 1 4P for some s ∈ {±1}, i ∈[n] and p ∈[P] \ {p∗ i }. We assume the existence of ˜T, as its absence would directly lead to our conclusion. Then, for any 0 ≤t < ˜T, we have ρ(t+1) s (i, p) = ρ(t) s (i, p) + η nEC∼DC h g(t) i,Cϕ′ D w(t) s , x(p) i E · 1p/∈C i ξ(p) i 2 < ρ(t) s (i, p) + 3ησ2 bd 2n , where the inequality is due to g(t) i,C < 1, ϕ′ ≤1, and ξ(p) i 2 ≤3 2σ2 bd. Hence, we have 1 4P ≤ρ( ˜T ) s (i, p) = ˜T −1 X t=0 ρ(t+1) s (i, p) −ρ(t) s (i, p) < 3ησ2 bd 2n ˜T, and we conclude ˜T > n 6ηP σ2 bd which is the desired result. Next, we will show that the model learns common features in at least constant order within ˜T iterates. Lemma D.3. Suppose the event Einit occurs and ρk = ω σ2 bd βn for some k ∈[K]. Then, for each s ∈{±1}. there exists Ts,k ≤ 9nP ηβ\|Vs,k\| such that γ(t) s (s, k) + βγ(t) −s(s, k) ≥1 for any t > Ts,k. Proof of Lemma D.3. Suppose γ(t) s (s, k) + βγ(t) −s(s, k) < 1 for all 0 ≤t ≤ n 6ηP σ2 bd. For each i ∈Vs,k and C ⊂[P] with \|C\| = C such that p∗ i /∈C and ˜pi ∈C, we have yifW (t)(Xi,C) = ϕ D w(t) s , vs,k E −ϕ D w(t) −s, vs,k E + X p/∈C∪{p∗ i } ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E ≤γ(t) s (s, k) + βγ(t) −s(s, k) + X p/∈C∪{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) + 2P · o 1 polylog(d) ≤1 + 2P · 1 4P + 2P · o 1 polylog(d) ≤2. The first inequality is due to Lemma B.4, the second inequality holds since we can apply Lemma D.2, and the last inequality is due to (A1). Thus, g(t) i,C = 1 1+exp(yifW (t)(Xi,C)) > 1 9 and we have γ(t+1) s (s, k) + βγ(t+1) −s (s, k) 49 = γ(t) s (s, k) + βγ(t) −s(s, k) + η n X i∈Vs,k EC∼DC g(t) i,C ϕ′ D w(t) s , vs,k E + βϕ′ D w(t) −s, vs,k E · 1p∗ i /∈C ≥γ(t) s (s, k) + βγ(t) −s(s, k) + ηβ 9n X i∈Vs,k EC∼DC[1p∗ i /∈C∧˜pi∈C] = γ(t) s (s, k) + βγ(t) −s(s, k) + ηβ\|Vs,k\|C(P −C) 9nP(P −1) ≥γ(t) s (s, k) + βγ(t) −s(s, k) + ηβ\|Vs,k\| 9nP . From the given condition in the lemma statement, we have 9nP ηβ\|Vs,k\| = o n 6ηP σ2 bd . If we choose t0 ∈ h 9nP ηβ\|Vs,k\|, n 6ηP σ2 bd i , then 1 > γ(t0) s (s, k) + βγ(t0) −s (s, k) ≥ηβ\|Vs,k\| 9nP t0 ≥1, and this is contradictory; therefore, it cannot hold that γ(t) s (s, k) + βγ(t) −s(s, k) < 1 for all 0 ≤t ≤ n 6ηP σ2 bd. Hence, there exists 0 ≤Ts,k < n 6ηP σ2 bd such that γ(Ts,k+1) s (s, k) + βγ(Ts,k+1) −s (s, k) ≥1 and choose the smallest one. Then we obtain 1 ≥γ(t) s (s, k) + βγ(t) −s(s, k) ≥ηβ\|Vs,k\| 9nP Ts,k. Therefore, Ts,k ≤ 9nP ηβ\|Vs,k\| and this is what we desired. What We Have So Far. For any common feature or rare feature vs,k with s ∈{±1} and k ∈ KC ∪KR, it satisfies ρk = ω σ2 bd βn due to (A5). By Lemma D.3, at any iterate t ∈ ¯T1, T ∗ with ¯T1 := maxs∈{±1},k∈C Ts,k, the following properties hold if the event Einit occurs: • (Learn common/rare features): For s ∈{±1} and k ∈KC ∪KR, γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1), • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 . D.2.3 Overfitting Augmented Data In the previous step, we have shown that data containing common or rare features can be wellclassified by learning common and rare features. In this step, we will show that the model correctly classifies the remaining training data by overfitting background noise instead of learning its features. We first introduce lower bounds on the number of iterates such that feature coefficients γ(t) s (s′, k) remain small, up to the order of α2β−1. This lemma holds to any kind of features, but we will focus on extremely rare features. This does not contradict the results from Section D.2.2 for common features and rare features since the upper bound on the number of iterations in Lemma D.3 is larger than the lower bound on the number of iterations in this lemma. Lemma D.4. Suppose the event Einit occurs. For each s ∈{±1} and k ∈[K], there exists ˜Ts,k ≥ nα2 ηβ\|Vs,k\| such that γ(t) s′ (s, k) ≤α2β−1 for any 0 ≤t < ˜Ts,k and s′ ∈{±1}. Proo of Lemma D.4. Let ˜Ts,k be the smallest iterate such that γ(t) s′ (s, k) > α2β−1 for some s′ ∈ {±1}. We assume the existence of ˜Ts,k, as its absence would directly lead to our conclusion. For any 0 ≤t < ˜Ts,k, γ(t+1) s′ (s, k) = γ(t) s′ (s, k)+ η n X i∈Vs,k EC∼DC h g(t) i,Cϕ′ D w(t) s′ , vs,k E · 1p∗ i /∈C i ≤γ(t) s′ (s, k)+ η\|Vs,k\| n , 50 and we have α2β−1 ≤γ( ˜Ts,k) s′ (s, k) = ˜Ts,k−1 X t=0 γ(t+1) s′ (s, k) −γ(t) s′ (s, k) ≤η\|Vs,k\| n ˜Ts,k. We conclude ˜Ts,k ≥ nα2 ηβ\|Vs,k\| which is the desired result. Next, we will show that the model overfits data augmented not containing common or rare features in at least constant order within ˜Ts,k iterates. Lemma D.5. Suppose the event Einit occurs and ρk = o α2σ2 bd n . For each i ∈[n] and C ⊂[P] with \|C\| = C, if (1) i ∈Vyi,k and p∗ i /∈C or (2) i ∈[n] and p∗ i ∈C, then there exists Ti,C ∈ ¯T1, 18n( P C) ηβσ2 bd such that X p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) ≥1, for any t > Ti,C. Proof of Lemma D.5. We can address both cases in the statement simultaneously. Suppose P p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) < 1 for all 0 ≤t ≤ nα2 ηβ\|Vyi,k\|. From Lemma B.4 and Lemma D.4, we have yifW (t)(Xi,C) = X p/∈C ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E ≤γ(t) yi (yi, k) + βγ(t) −yi(yi, k) + X p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) + 2P · o 1 polylog(d) ≤(1 + β)α2β−1 + 1 + 2P · o 1 polylog(d) ≤2, and g(t) i,C = 1 1+exp(yifW (t)(Xi,C)) ≥1 9. Also, for each p /∈C ∪{p∗ i }, we have ρ(t+1) s (i, p) + βρ(t+1) −s (i, p) ≥ρ(t) s (i, p) + βρ(t) −s(i, p) + η nPC′∼DC[C′ = C]g(t) i,C ϕ′ D w(t) s , x(p) i E + βϕ′ D w(t) −s, x(p) i E ξ(p) i 2 ≥ρ(t) s (i, p) + βρ(t) −s(i, p) + ηβσ2 bd 18n P C , where the last inequality is due to ξ(p) i 2 ≥1 2σ2 bd and ϕ′ ≥β. We also have X p/∈C∪{p∗ i } ρ(t+1) s (i, p) + βρ(t+1) −s (i, p) ≥ X p/∈C∪{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) + ηβσ2 bd 18n P C From the given condition in the lemma statement, we have 18n( P C) ηβσ2 bd = o nα2 ηβ\|Vs,k\| . If we choose t0 ∈ 18n( P C) ηβσ2 bd , nα2 ηβ\|Vs,k\| , then we have 1 > X p/∈C∪{p∗ i } ρ(t0) s (i, p) + βρ(t0) −s (i, p) ≥ηβσ2 bd 18n P C t0 ≥1, 51 and this is a contradiction; therefore, it cannot hold that P p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) < 1 for all 0 ≤ t ≤ nα2 ηβ\|Vyi,k\|. Thus, there exists 0 ≤ Ti,C < nα2 ηβ\|Vs,k\| satisfying P p/∈C∪{p∗ i } ρ(Ti,C+1) s (i, p) + βρ(Ti,C+1) −s (i, p) ≥1 and let us choose the smallest one. For any 0 ≤t < Ti,C, we have 1 ≥ X p/∈C∪{p∗ i } ρ(Ti,C) s (i, p) + βρ(Ti,C) −s (i, p) ≥ ησ2 bd 18n P C Ti, and we conclude that Ti,C ≤ 18n( P C) ηβσ2 bd . Lastly, we move on to prove Ti,C > ¯T1. Combining Lemma D.2 and Lemma D.3 leads to X p/∈C∪{p∗ i }\{p∗ i } ρ( ¯T1) s (i, p) + βρ( ¯T1) −s (i, p) ≤1 2. Thus, we have Ti,C > ¯T1 and this is what we desired. What We Have So Far. For any k ∈KE, it satisfies ρk = o α2n σ2 bd due to (A6). By Lemma D.5 at iterate t ∈[TCutout, T ∗] with TCutout := max max k∈KE,i∈Vyi,k,p∗ i /∈C Ti,C, max i∈[n],p∗ i ∈C Ti,C ∈ ¯T1, T ∗ the following properties hold if the event Einit occurs: • (Learn common/rare features): For any s ∈{±1} and k ∈KC ∪KR, γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1), • (Overfit augmented data with extremely rare features or no feature): For each i ∈[n], k ∈KE, C ⊂[P] with \|C\| = C such that (1) i ∈Vyi,k and p∗ i /∈C or (2) i ∈[n] and p∗ i ∈C X p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) = Ω(1). • (Do not learn extremely rare features at TCutout): For any s, s′ ∈{±1} and k ∈KE, γ(TCutout) s′ (s, k) ≤α2β−1. • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 . D.2.4 Cutout cannot Learn Extremely Rare Features Within Polynomial Times In this step, We will show that Cutout cannot learn extremely rare features within the maximum admissible iterate T ∗= poly(d) η . we fix any s∗∈{±1} and k∗∈KE. Recall the function Q(s∗,k∗) : W →Rd×2, defined in Lemma B.5 and omit superscripts for simplicity. For each iteration t, Q(W (t)) represents quantities updates by data with feature vector vs∗,k∗until t-th iteration. We will sequentially introduce several technical lemmas and by combining these lemmas, quantify update by data with feature vector vs∗,k∗ after TCutout and derive our conclusion. Let us define W ∗= {w∗ 1, w∗ −1}, where w∗ s = w(TCutout) s + M X i∈Vs∗,k∗ X p∈[P ]\{p∗ i ,˜pi} ξ(p) i ξ(p) i 2 , where M = 4β−1 log 2ηβ2T ∗ α2 . Note that (12), β < 1, and T ∗= poly(d) η together imply M = e O β−1 . Note that W (t), W ∗∈W for any t ≥0. 52 Lemma D.6. Suppose the event Einit occurs. Then, Q W (TCutout) −Q(W ∗) 2 ≤8M 2P\|Vs∗,k∗\|σ−2 b d−1. where ∥·∥denotes the Frobenius norm. Proof of Lemma D.6. For each s ∈{±1}, ss∗ Qs (w∗ s) −Qs w(TCutout) s = Qs  ss∗M X i∈Vs∗,k∗ X p∈[P ]\{p∗ i ,˜pi} ξ(p) i ξ(p) i   = M X i∈Vs∗,k∗ X p∈[P ]\{p∗ i ,˜pi} ξ(p) i ξ(p) i 2 , and we have Q W (TCutout) −Q(W ∗) 2 = Q1(w∗ 1) −Q1 w(TCutout) 1 2 + Q−1(w∗ −1) −Q−1 w(TCutout) −1 2 ≤2M 2        X i∈Vs∗,k∗,p∈[P ]\{p∗ i ,˜pi} ξ(p) i −2 + X i,j∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi},q∈[P ]\{p∗ j ,˜pj} (i,p)̸=(j,q) D ξ(p) i , ξ(q) j E ξ(p) i 2 ξ(q) j 2        . From Einit and (A2), we have X i,j∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi},q∈[P ]\{p∗ j ,˜pj} (i,p)̸=(j,q) D ξ(p) i , ξ(q) j E ξ(p) i 2 ξ(q) j 2 ≤ X i∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi} X j∈Vs∗,k∗ p∈[P ]\{p∗ j ,˜pj} ξ(˜p) i −2 e O d−1 2 ≤ X i∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi} ξ(˜p) i −2 e O nPd−1 2 ≤ X i∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi} ξ(p) i −2 From the event Einit defined in Lemma B.2, we have X i∈Vs∗,k∗ p∈[P ]\{p∗ i ,˜pi} ξ(p) i −2 ≤2P\|Vs∗,k∗\|σ−2 d d−1, and we obtain Q W (TCutout) −Q(W ∗) 2 ≤4M 2 X i∈Vs∗,k∗,p∈[P ]\{p∗ i } ξ(p) i −2 ≤8M 2P\|Vs∗,k∗\|σ−2 b d−1. Lemma D.7. Suppose the Einit occurs. For any t ≥TCutout, i ∈Vs∗,k∗and any C ⊂[P] with \|C\| = C, it holds that ⟨yi∇W fW (t)(Xi,C), Q(W ∗)⟩≥Mβ 2 . 53 Proof of Lemma D.7. We have ⟨yi∇W fW (t)(Xi,C), Q(W ∗)⟩ = X p/∈C ϕ′ D w(t) s∗, x(p) i E D Qs∗(w∗ s∗), x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗(w∗ −s∗), x(p) i E . For any s ∈{±1} and p ∈[P] \ {p∗ i , ˜pi}, ss∗D Qs(w∗ s), ξ(p) i E ≥M + ρ(TCutout) s (i, p) − X j∈[n],q∈[P ]\{p∗ j } (j,q)̸=(i,p) ρ(TCutout) s (j, q) D ξ(p) i , ξ(q) j E ξ(q) j 2 −M X j∈Vs∗,k∗,q∈[P ]\{p∗ j ,˜pj} (j,q)̸=(i,p) D ξ(p) i , ξ(q) j E ξ(q) j 2 ≥M −e O nPβ−1σdσ−1 b d−1 2 = M −o 1 polylog(d) ≥M 2 , (26) where the last equality is due to (9). Also, for any s ∈{±1}, ss∗⟨Qs(w∗ s), vs∗,k∗⟩= γ(TCutout) s (s∗, k∗) ≥0. In addition, ss∗D Qs(w∗ s), x(˜pi) i E = ss∗D Qs(w∗ s), ξ(˜pi) i E + ss∗D Qs(w∗ s), x(˜pi) i −ξ(˜pi) i E = ss∗D Qs(w∗ s), ξ(˜pi) i E −e O α2β−1ρk∗nσ−2 d d−1 = ρ(TCutout) s (i, ˜pi) + X j∈[n],q∈[P ]\{p∗ i } (j,q)̸=(i,˜pi) ρ(TCutout) s (j, q) D ξ(˜pi) i , ξ(q) j E ξ(q) j 2 −e O α2β−1ρk∗nσ−2 d d−1 ≥−e O nPβ−1σdσ−1 b d−1 2 −e O α2β−1ρk∗nσ−2 d d−1 = −o 1 polylog(d) , (27) where the last equality is due to (9) and (A7). For any C ⊂[P] with \|C\| = C, there exists p ∈[P] \ {p∗ i , ˜pi} such that p ̸= C since C < P 2 . By applying (26) and (27) for s = s∗, −s∗and combining with ϕ′ ≥β, we have ⟨yi∇W fW (t)(Xi,C), Q(W ∗)⟩ ≥ ϕ′ D w(t) s∗, x(p) i E D Qs∗(w∗ s∗), x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗(w∗ −s∗), x(p) i E + X q /∈C∪{p} ϕ′ D w(t) s∗, x(q) i E D Qs∗(w∗ s∗), x(q) i E −ϕ′ D w(t) −s∗, x(q) i E D Q−s∗(w∗ −s∗), x(q) i E ≥Mβ −o 1 polylog(d) ≥Mβ 2 . 54 By combining Lemma D.6 and Lemma D.7, we can obtain the following result. Lemma D.8. Suppose the event Einit occurs. η n T ∗ X t=TCutout X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] ≤ Q W (TCutout) −Q(W ∗) 2 + 2ηT ∗e−Mβ 4 , where ∥·∥denotes the Frobenius norm. Proof of Lemma D.8. Note that for any TCutout ≤t < T ∗, Q W (t+1) = Q W (t) −η n∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] . Therefore, we have Q W (t) −Q (W ∗) 2 − Q W (t+1) −Q (W ∗) 2 = 2η n * ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] , Q W (t) −Q (W ∗) + −η2 n2 ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] 2 = 2η n * ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] , Q W (t)+ −2η n X i∈Vs∗,k∗ ⟨EC∼DC [ℓ′(yifW (t)(Xi,C))∇W yifW (t)(Xi,C)] , Q (W ∗)⟩ −η2 n2 ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] 2 ≥2η n * ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] , Q W (t)+ −Mβη n X i∈Vs∗,k∗ EC∼DC [ℓ′(yifW (t)(Xi,C))] −η2 n2 ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] 2 , where the last inequality is due to Lemma D.7. By the chain rule, for each C ⊂[P] with \|C\| = C, we have * ∇W X i∈Vs∗,k∗ ℓ(yifW (t)(Xi,C)), Q W (t)+ = X i∈Vs∗,k∗ " ℓ′(yifW (t)(Xi,C)) × X p/∈C ϕ′ D w(t) s∗, x(p) i E D Qs∗ w(t) s∗ , x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗ w(t) −s∗ , x(p) i E # . For each s ∈{±1}, i ∈Vs∗,k∗, and p ∈[P], D w(t) s , x(p) i E − D Qs w(t) s , x(p) i E = D w(t) s −Qs w(t) s , x(p) i E 55 ≤ X j∈[n]\Vs∗,k∗,q∈[P ]\{p∗ i } * ρ(t) s (j, q) ξ(q) j ξ(q) j 2 , x(p) i + + α X j∈F1\Vs∗,k∗ ρ(t) s (j, ˜pj) ξ(˜pj) j −2 D v1,1, x(p) i E + α X j∈F−1\Vs∗,k∗ ρ(t) s (j, ˜pj) ξ(˜pj) j −2 D v−1,1, x(p) i E ≤e O nPβ−1σdσ−1 b d−1 2 + e O α2β−1nσ−2 d d−1 = o 1 polylog(d) , where the last inequality is due to Lemma D.1 and the event Einit. By Lemma F.1, X p/∈C ϕ′ D w(t) s∗, x(p) i E D Qs∗ w(t) s∗ , x(p) i E −ϕ′ D w(t) −s∗, x(p) i E D Q−s∗ w(t) s∗ , x(p) i E ≤ X p/∈C ϕ D w(t) s∗, x(p) i E −ϕ D w(t) −s∗, x(p) i E + rP + o 1 polylog(d) = yifW (t)(Xi,C) + o 1 polylog(d) , where the last equality is due to r = o 1 polylog(d) . Therefore, we have Q W (t) −Q (W ∗) 2 − Q W (t+1) −Q(W ∗) 2 ≥2η n X i∈Vs∗,k∗ EC∼DC ℓ′ (yifW (t)(Xi,C)) yifW (t)(Xi,C) + o 1 polylog(d) −Mβ 2 −η2 n2 ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] 2 ≥2η n X i∈Vs∗,k∗ EC∼DC ℓ′(yifW (t)(Xi,C)) yifW (t)(Xi,C) −Mβ 4 −η2 n2 ∇W X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] 2 . From the convexity of ℓ(·), X i∈Vs∗,k∗ EC∼DC ℓ′(yifW (t)(Xi,C)) yifW (t)(Xi,C) −Mβ 4 ≥ X i∈Vs∗,k∗ EC∼DC ℓ(yifW (t)(Xi,C)) −ℓ Mβ 4 ≥ X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] −ne−Mβ 4 . In addition, by Lemma F.3, η2 n2 ∇ X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] 2 56 ≤8η2P 2σ2 dd\|Vs∗,k∗\| n2 X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] ≤η n X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))], where the last inequality is due to (A8), and we have Q W (t) −Q(W ∗) 2 − Q W (t+1) −Q(W ∗) 2 ≥η n X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] −2ηe−Mβ 4 . From telescoping summation, we have η n T ∗ X t=TCutout X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] ≤ Q W (TCutout) −Q (W ∗) 2 + 2ηT ∗e−Mβ 4 . Finally, we can prove that the model cannot learn extremely rare features within T ∗iterations. Lemma D.9. Suppose the event Einit occurs. For any T ∈[TCutout, T ∗], we have γ(T ) s (s∗, k∗) = e O(α2β−2) for each s ∈{±1}. Proof of Lemma D.9. For any T ∈[TCutout, T ∗], we have γ(T ) s (s∗, k∗) = γ(TCutout) s (s∗, k∗) + η n T −1 X t=TCutout X i∈Vs∗,k∗ EC∼DC h g(t) i,C · 1p/∈C i ϕ′ D w(t) s , vs∗,k∗ E ≤γ(TCutout) s (s∗, k∗) + η n T −1 X t=TCutout X i∈Vs∗,k∗ EC∼DC h g(t) i,C i ≤γ(TCutout) s (s∗, k∗) + η n T −1 X t=TCutout X i∈Vs∗,k∗ EC∼DC [ℓ(yifW (t)(Xi,C))] , where the first inequality is due to ϕ′ ≤1 and the second inequality is due to −ℓ′ ≤ℓ. From the result of Section D.2.3, γ(TCutout) s (s∗, k∗) ≤α2β−1 and by Lemma D.8 and Lemma D.6, we have η n (T −1) X t=TCutout X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] ≤η n (T ∗) X t=TCutout X i∈Vs∗,k∗ EC∼DC[ℓ(yifW (t)(Xi,C))] ≤ Q W (TCutout) −Q(W ∗) 2 + 2ηT ∗e−Mβ 2 ≤8M 2P\|Vs∗,k∗\|σ−2 b d−1 + 2ηT ∗e−Mβ 4 = e O α2β−2 . The last line is due to (A6) and M = 4β−1 log 2ηβ2T ∗ α2 . This finishes the proof. What We Have So Far. Suppose the event Einit occurs. For any t ∈[TCutout, T ∗], we have • (Learn common/rare features): γ(t) s (s, k) + βγ(t) −s(s, k) = Ω(1) for each s ∈{±1} and k ∈ KC ∪KR • (Overfit augmented data with extremely rare features or no feature): For each i ∈[n], k ∈KE, C ⊂ [P] with \|C\| = C such that (1) i ∈Vyi,k and p∗ i /∈C or (2) i ∈[n] and p∗ i ∈C X p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) = Ω(1). 57 • (Cannot learn extreme features): γ(t) s (s, k), γ(t) −s(s, k) = O α2β−2 for each s ∈{±1} and k ∈KE. • For any s ∈{±1}, i ∈[n], and p ∈[P] \ {p∗ i }, ρ(t) s (i, p) = e O β−1 , D.2.5 Train and Test Accuracy In this step, we will prove that the model trained by Cutout has perfect training accuracy on both augmented data and original data but has near-random guesses on test data with extremely rare data. For any i ∈Vs,k with s ∈{±1}, k ∈KC ∪KR and C ⊂[P] with \|C\| = C and p∗ i /∈C, yifW (t)(Xi,C) = X p/∈C ϕ D w(t) s , x(p) i E −ϕ D w(t) −s, x(p) i E = γ(t) s (s, k) + βγ(t) −s(s, k) + X p/∈C∪{p∗ i } ρ(t) s (i, p) + βρ(t) −s(i, p) −2(P −C) · o 1 polylog(d) ≥γ(t) s (s, k) + βγ(t) −s(s, k) −2(P −C) · o 1 polylog(d) = Ω(1) −o 1 polylog(d) = Ω(1), for any t ∈[TCutout, T ∗]. In addition, for any i ∈[n] and C ⊂[P] with \|C\| = C that does not correspond to the case above, by Lemma D.5 and Lemma B.4, we have yifW (t)(Xi,C) = X p/∈C ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E ≥ X p/∈C∪{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) −2(P −C) · o 1 polylog(d) = Ω(1) −o 1 polylog(d) = Ω(1), for any t ∈[TCutout, T ∗]. We can conclude that Cutout with t ∈[TCutout, T ∗] iterates achieve perfect training accuracy on augmented data. Next, we will show that Cutout achieves perfect training accuracy on the original data. For any i ∈[n], let us choose C ⊂[P] with \|C\| = C such that p∗ i ∈C. Then, from the result above, we have yifW (t)(Xi) = yifW (t)(Xi,C) + X p∈C ϕ D w(t) yi , x(p) i E −ϕ D w(t) −yi, x(p) i E ≥yifW (t)(Xi,C) + X p∈C\{p∗ i } ρ(t) yi (i, p) + βρ(t) −yi(i, p) −C · o 1 polylog(d) ≥Ω(1), for any t ∈[TCutout, T ∗] and we conclude that Cutout with t ∈[TCutout, T ∗] iterates achieve perfect training accuracy on original data. Lastly, let us move on to the test accuracy part. Let (X, y) ∼D be a test data with X = x(1), . . . , x(P ) ∈Rd×P having feature patch p∗, dominant noise patch ˜p, and feature vector vy,k. We have x(p) ∼N(0, σ2 bΛ) for each p ∈[P] \ {p∗, ˜p} and x(˜p) −αvs,1 ∼N(0, σ2 dΛ) for some s ∈{±1}. Therefore, for all t ∈[TCutout, T ∗] and p ∈[P] \ {p∗, ˜p}, ϕ D w(t) 1 , x(p)E −ϕ D w(t) −1, x(p)E 58 ≤ D w(t) 1 −w(t) −1, x(p)E ≤ D w(0) 1 −w(0) −1, x(p)E + X i∈[n],q∈[P ]\{p∗ i } ρ(t) 1 (i, q) −ρ(t) −1(i, q) D ξ(q) i , x(p)E ξ(q) i 2 ≤e O σ0σbd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o α polylog(d) , (28) with probability at least 1 −o 1 poly(d) due to Lemma B.2, (A8), (8), and (9).. In addition, for any s′ ∈{±1}, we have D w(t) s′ , x(˜p) −αvs,1 E ≤ D w(0) s′ , x(˜p) −αvs,1 E + X i∈[n],q∈[P ]\{p∗ i } ρ(t) s′ (i, q) D ξ(q) i , x(˜p) −αvs,1 E ξ(q) i 2 = e O σ0σdd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o α polylog(d) , (29) with probability at least 1 −o 1 poly(d) due to Lemma B.2, (A8), (8), and (9). Case 1: k ∈KC ∪KR By Lemma B.2, (A7), and (10), ϕ D w(t) 1 , x(˜p)E −ϕ D w(t) −1, x(˜p)E ≤ D w(t) 1 −w(t) −1, x(˜p)E ≤α D w(t) 1 −w(t) −1, vs,1 E + D w(t) 1 −w(t) −1, x(p) −αvs,1 E ≤α γ(t) 1 (s, 1) + γ(t) −1(s, 1) + α D w(0) 1 , vs,1 E + α D w(0) −1, vs,1 E + o 1 polylog(d) ≤e O αβ−1 + e O (ασ0) + o 1 polylog(d) = o 1 polylog(d) , (30) with probability at least 1 −o 1 poly(d) . Suppose (28) and (30) holds. By Lemma B.4, we have yfW (t)(X) = ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + X p∈[P ]\{p∗} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E = γ(t) y (y, k) + βγ(t) −y(y, k) −o 1 polylog(d) = Ω(1) −o 1 polylog(d) 59 > 0. Therefore, we have P(X,y)∼D h yfW (t)(X) > 0 \| x(p∗) = vy,k, k ∈KC ∪KR i ≥1 −o 1 poly(d) . (31) Case 2: k ∈KE By triangular inequality and ϕ′ ≤1, we have ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E = ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E + ϕ D w(t) s , x(˜p)E −ϕ D w(t) s , αvs,1 E − ϕ D w(t) −s, x(˜p)E −ϕ D w(t) −s, αvs,1 E ≥ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E − D w(t) s , x(˜p) −αvs,1 E − D w(t) −s, x(˜p) −αvs,1 E . In addition, ϕ D w(t) s , αvs,1 E −ϕ D w(t) −s, αvs,1 E = ϕ αγ(t) s (s, 1) −ϕ −αγ(t) −s(s, 1) + ϕ D w(t) s , αvs,1 E −ϕ αγ(t) s (s, 1) − ϕ D w(t) −s, αvs,1 E −ϕ −αγ(t) −s(s, 1) ≥ ϕ αγ(t) s (s, 1) −ϕ −αγ(t) −s(s, 1) −α D w(t) s , vs,1 E −γ(t) s (s, 1) −α D w(t) −s, vs,1 E + γ(t) −s(s, 1) = α γ(t) s (s, 1) + βγ(t) −s(s, 1) −α · o 1 polylog(d) = Ω(α), where the second equality is due to Lemma B.4 and (A8). If (29) holds, we have ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E = Ω(α) −o α polylog(d) = Ω(α). (32) Note that yfW (t)(X) = ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + ϕ D w(t) y , x(˜p)E −ϕ D w(t) −y, x(˜p)E + X p∈[P ]\{p∗,˜p} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E , and ϕ D w(t) y , vy,k E −ϕ D w(t) −y, vy,k E + X p∈[P ]\{p∗,˜p} ϕ D w(t) y , x(p)E −ϕ D w(t) −y, x(p)E ≤ D w(t) y −w(t) −y, vy,k E + o α polylog(d) 60 ≤γ(t) 1 (y, k) + γ(t) −1(y, k) + D w(0) y −w(0) −y, vy,k E + o α polylog(d) ≤O(α2β−2) + e O(σ0) + o α polylog(d) = o α polylog(d) < ϕ D w(t) s , x(˜p)E −ϕ D w(t) −s, x(˜p)E , where the first inequality is due to (28), the second-to-last line is due to (A8), (8), and (10) , and the last inequality is due to (32). Therefore, we have yfW (t)(X) > 0 if y = s. Otherwise, yfW (t)(X) < 0. P(X,y)∼D h yfW (t)(X) > 0 \| x(p∗) = vy,k, k ∈KE i = 1 2 ± o 1 poly(d) . (33) Hence, combining (31) and (33) implies P(X,y)∼D [yfW (t)(X) > 0] = X k∈KC∪KR ρk + 1 2 1 − X k∈KC∪KR ρk ! ± o 1 poly(d) = 1 −1 2 X k∈KE ρk ± o 1 poly(d) . □ 61 E Proof for CutMix E.1 Proof of Lemma B.3 for CutMix For each i, j ∈[n] and S ⊂[P], let g(t) i,j,S := −\|S\| P yiℓ′yifW (t)(Xi,j,S) − 1 −\|S\| P yjℓ′yjfW (t)(Xi,j,S) . For s ∈{±1} and iterate t, w(t+1) s −w(t) s = −η∇wsLCutMix W (t) = η n2 X i,j∈[n] ES∼DS  sg(t) i,j,S  X p∈S ϕ′ D w(t) s , x(p) i E x(p) i + X p/∈S ϕ′ D w(t) s , x(p) i E x(p) j     = sη n2 X s′∈{±1},k∈[K] X i∈Vs′,k,j∈[n] ES∼DS h g(t) i,j,S1p∗ i ∈S + g(t) j,i,S1p∗ i /∈S i ϕ′ D w(t) s , vs′,k E vs′,k + sη n2 X i,j∈[n],p∈[P ]\{p∗ i } ES∼DS h g(t) i,j,S1p∈S + g(t) j,i,S1p/∈S i ϕ′ D w(t) s , x(p) i E x(p) i . Hence, if we define γ(t) s (s′, k)’s and ρ(t) s (i, p)’s recursively by using the rule γ(t+1) s (s′, k) = γ(t) s (s′, k) + ss′η n2 X i∈Vs′,k,j∈[n] ES∼DS h g(t) i,j,S1p∗ i ∈S + g(t) j,i,S1p∗ i /∈S i ϕ′ D w(t) s , vs′,k E , ρ(t+1) s (i, p) = ρ(t) s (i, p) + syiη n2 X j∈[n] ES∼DS h g(t) i,j,S1p∈S + g(t) j,i,S1p/∈S i ϕ′ D w(t) s , x(p) i E ξ(p) i 2 , starting from γ(0) s (s′k) = ρ(0) s (i, p) = 0 for each s, s′ ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }, then we have w(t) s = w(0) s + X k∈[K] γ(t) s (s, k)vs,k − X k∈[K] γ(t) s (−s, k)v−s,k + X i∈Vs p∈[P ]\{˜pi} ρ(t) s (i, p) ξ(p) i ξ(p) i 2 − X i∈V−s p∈[P ]\{˜pi} ρ(t) s (i, p) ξ(p) i ξ(p) i 2 + α    X i∈Fs syiρ(t) s (i, ˜pi) vs,1 ξ(˜pi) i 2 + X i∈F−s syiρ(t) s (i, ˜pi) v−s,1 ξ(˜pi) i 2   , for each s ∈{±1}. □ E.2 Proof of Theorem 3.3 We will prove that the conclusion of Theorem 3.3 holds when the event Einit occurs. The proof of Theorem 3.3 is structured into the following six steps: 1. Introduce a reparametrization of the CutMix loss LCutMix(W ) to a convex function h(Z) for ease of analysis (Section E.2.1). 2. Characterize a global minimum of h(Z) (Section E.2.2). 3. Evaluate strong convexity constant in the region near the global minimum of h(Z) (Section E.2.3). 4. Show that near stationary point of h(Z) is close to a global minimum (Section E.2.4). 5. Prove that gradient descent on the CutMix loss LCutMix(W ) achieves a near-stationary point of the reparametrized function h(Z) and perfect accuracy on original training data (Section E.2.5). 6. Evaluate the test accuracy of a model in near-stationary point (Section E.2.6). 62 E.2.1 Reparametrization of CutMix Loss It is complicated to characterize the stationary points of CutMix loss LCutMix(W ) due to its nonconvexity. We will overcome this problem by introducing reparameterization of the objective function. Let us define z(p) i := ϕ D w1, x(p) i E −ϕ D w−1, x(p) i E , for i ∈[n], p ∈[P] and zs,k := ϕ(⟨w1, vs,k⟩) −ϕ(⟨w−1, vs,k⟩), for each s ∈{±1}, k ∈[K]. We can rewrite CutMix loss LCutMix(W ) as a function h(Z) of the defined variables Z := {zs,k}s∈{±1},k∈[K] ∪{z(p) i }i∈[n],p∈[P ]\{p∗ i } as follows. h(Z) := 1 n2 X i,j∈[n] ES∼DS  \|S\| P ℓ  yi  X p∈S z(p) i + X p/∈S z(p) j     + 1 −\|S\| P ℓ  yj  X p∈S z(p) i + X p/∈S z(p) j      , where we write z(p∗ i ) i = zs,k if i ∈Vs,k. For notational simplicity, let us consider Z as vectors in R2K+n(P −1) with the standard orthonormal basis {es,k}s∈{±1},k∈[K] ∪ n e(p) i o i∈[n],p∈[P ]\{p∗ i } which means Z = {zs,k}s∈{±1},k∈[K] ∪ n z(p) i o i∈[n],p∈[P ]\{p∗ i } = X s∈{±1},k∈[K] zs,kes,k + X i∈[n],p∈[P ]\{p∗ i } z(p) i e(p) i . If there is no confusion, we will use e(p∗ i ) i to represent es,k, for i ∈Vs,k. By the chain rule, ∇W LCutMix(W ) = J(W )∇Zh(Z), where each column of Jacobian matrix J(W ) ∈R2d×(n(P −1)+2K) is ∇W zs,k = ϕ′(⟨w1, vs,k⟩)vs,k −ϕ′(⟨w−1, vs,k⟩)vs,k ∈R2d, ∇W z(p) i =  ϕ′ D w1, x(p) i E x(p) i −ϕ′ D w−1, x(p) i E x(p) i  ∈R2d. Let us characterize the smallest singular value σmin(J(W )) of the Jacobian matrix J(W ). For any unit vector c = {cs,k}s∈{±1},k∈[K] ∪ n c(p) i o i∈[n],p∈[P ]\{p∗ i } ∈R2K+n(P −1), we have ∥J(W )c∥2 = X s∈{±1},k∈[K] c2 s,k ∥∇W zs,k∥2 + X i∈[n],p∈[P ]\{p∗ i } c(p) i 2 ∇W z(p) i 2 + X s1,s2∈{±1},k1,k2∈[K] (s1,k1)̸=(s2,k2) cs1,k1cs2,k2⟨∇W zs1,k1, ∇W zs2,k2⟩ + 2 X s∈{±1},k∈[K] i∈[n],p∈[P ]\{p∗ i } cs,kc(p) i D ∇W zs,k, ∇W z(p) i E + X i∈[n],p∈[P ]\{p∗ i } j∈[n],q∈[P ]\{p∗ j } (i,p)̸=(j,q) c(p) i c(q) j D ∇W z(p) i , ∇W z(q) j E . 63 For each s1, s2 ∈{±1}, k1, k2 ∈[K] such that (s1, k1) ̸= (s2, k2), and i ∈[n], p ∈[P] \ {p∗ i , ˜pi}, ⟨∇W zs1,k1, ∇W zs2,k2⟩= D ∇W zs1,k1, ∇W z(p) i E = 0, and if k1 > 1 D ∇W zs1,k1, ∇W z(p) i E = D ∇W zs1,k1, ∇W z(˜pi) i E = 0, since ⟨vs1,k1, vs2,k2⟩= D vs1,k1, ξ(p) i E = D vs1,k1, ξ(˜pi) i E = 0. Also, for each s ∈{±1} and i ∈Fs, then 2 cs,1c(˜pi) i D ∇W zs,1, ∇W z(˜pi) i E = 2 cs,1c(˜pi) i ϕ′(⟨w1, vs,1⟩)ϕ′ D w1, x(˜pi) i E + ϕ′(⟨w−1, vs,1⟩)ϕ′ D w−1, x(˜pi) i E α ≤4c2 s,1 ϕ′(⟨w1, vs,1⟩)2 + ϕ′(⟨w−1, vs,1⟩)2 α2 x(˜pi) i 2 + 1 4 c(˜pi) i 2 ϕ′ D w1, x(˜pi) i E2 + ϕ′ D w−1, x(˜pi) i E2 x(˜pi) i 2 < 1 2nc2 s,1 ϕ′(⟨w1, vs,1⟩)2 + ϕ′(⟨w−1, vs,1⟩)2 + 1 4 c(˜pi) i 2 ϕ′ D w1, x(˜pi) i E2 + ϕ′ D w−1, x(˜pi) i E2 x(˜pi) i 2 , where the last inequality holds since x(˜pi) i 2 = α2 + ξ(˜pi) i 2 ≥1 2σ2 dd = ω(nα2), where we apply the fact from the event Einit defined in Lemma B.2 and (A7). Also, D ∇W z−s,1, ∇W z(˜pi) i E = 0. Furthermore, for each i, j ∈[n], p ∈[P] \ {p∗ i }, q ∈[P] \ {p∗ j} with (i, p) ̸= (j, q) satisfies c(p) i c(q) j D ∇W z(p) i , ∇W z(q) j E = c(p) i c(q) j ϕ′ D w1, x(p) i E ϕ′ D w1, x(q) j E + ϕ′ D w−1, x(p) i E ϕ′ D w−1, x(q) j E D x(p) i , x(q) j E ≤ 1 4P n c(p) i 2 ϕ′ D w1, x(p) i E2 + ϕ′ D w−1, x(p) i E2 x(p) i 2 + 1 4P n c(q) j 2 ϕ′ D w1, x(q) j E2 + ϕ′ D w−1, x(q) j E2 x(q) j 2 = 1 4P n c(p) i 2 ∇W z(p) i 2 + c(q) j 2 ∇W z(q) j 2 where the last inequality is due to AM-GM inequality and x(p) i · x(q) j ≥2nP D x(p) i , x(q) j E , which we show through a case analysis. For the case p = ˜pi and q = ˜pj, this inequality holds since x(p) i · x(q) j ≥ ξ(p) i · ξ(q) j ≥2nP D ξ(p) i , ξ(q) j E + α2 ≥2nP D x(p) i , x(q) j E , where the second inequality is due to 1 2 ξ(p) i · ξ(q) j ≥2nP D ξ(p) i , ξ(q) j E , 1 2 ξ(p) i · ξ(q) j ≥2nPα2. which is implied by the fact from the event Einit defined in Lemma B.2, (A1), (A2), and (A7). In the remaining case, x(p) i · x(q) j ≥ ξ(p) i · ξ(q) j ≥2nP D ξ(p) i , ξ(q) j E = 2nP D x(p) i , x(q) j E , 64 where the second inequality is due to the fact from event Einit defined in Lemma B.2, (A1), and (A2). For s ∈{±1}, k ∈[K] and i ∈[n], p ∈[P] \ {p∗ i }, ∥∇W zs,k∥2 = ϕ′(⟨w1, vs,k⟩)2 + ϕ′(⟨w−1, vs,k⟩)2 ≥2β2, and ∇W z(p) i 2 = ϕ′ D w1, x(p) i E2 + ϕ′ D w−1, x(p) i E2 x(p) i 2 ≥β2σ2 i,pd ≥β2, where the last inequality is due to (8). By merging all inequalities together, we have ∥J(W )c∥2 = X s∈{±1},k∈[K] c2 s,k∥∇W zs,k∥2 + X i∈[n],p∈[P ]\{p∗ i } c(p) i 2 ∇W z(p) i 2 + X s∈{±1},i∈Fs cs,1c(˜pi) i D ∇W zs,1, ∇W z(˜pi) i E + X i∈[n],p∈[P ]\{p∗ i } j∈[n],q∈[P ]\{p∗ j } (i,p)̸=(j,q) c(p) i c(q) j D ∇W z(p) i , ∇W z(q) j E ≥ X s∈{±1},k∈[K] c2 s,k∥∇W zs,k∥2 + X i∈[n],p∈[P ]\{p∗ i } c(p) i 2 ∇W z(p) i 2 − X s∈{±1},i∈Fs 1 2nc2 s,1∥∇W zs,1∥2 + 1 4 c(˜pi) i 2 ∇W z(˜pi) i 2 − 1 4Pn X i∈[n],p∈[P ]\{p∗ i } j∈[n],q∈[P ]\{p∗ j } (i,p)̸=(j,q) c(p) i 2 ∇W z(p) i 2 + c(q) j 2 ∇W z(q) j 2 > 1 4 X s∈{±1},k∈[K] c2 s,k∥∇W zs,k∥2 + 1 4 X i∈[n],p∈[P ]\{p∗ i } c(p) i 2 ∇W z(p) i 2 ≥β2 4 , and we conclude σmin(J(W )) ≥β 2 for any W . E.2.2 Characterization of a Global Minimum of CutMix Loss In this section, we will check that h(Z) is strictly convex and it has a global minimum. For each i, j ∈[n] and S ⊂[P] let us define ai,j,S ∈R2K+n(P −1) as ai,j,S = X p∈S e(p) i + X p/∈S e(p) j , and then h(Z) = 1 n2 X i,j∈[n] ES∼DS \|S\| P ℓ(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P ℓ(yj⟨ai,j,S, Z⟩) . Since ℓ(·) is convex, h(Z) is also convex. Note that ∇h(Z) = 1 n2 X i,j∈[n] ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) ai,j,S , and ∇2h(Z) 65 = 1 n2 X i,j∈[n] ES∼DS \|S\| P ℓ′′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P ℓ′′(yj⟨ai,j,S, Z⟩) ai,j,Sa⊤ i,j,S = 1 n2 X i,j∈[n] ES∼DS ℓ′′(⟨ai,j,S, Z⟩)ai,j,Sa⊤ i,j,S , where the last equality holds since ℓ′′(z) = ℓ′′(−z) for any z ∈R. From the equation above, it suffices to show that {ai,j,S}i,j∈[n],S⊂[P ] spans R2K+n(P −1) to show strict convexity of h(Z). We define a function I : [P] →[n] such that for each p ∈[P], p∗ I(p) = p with x(p) I(p) = v1,1, where the existence is guaranteed by Lemma B.2 (but not necessarily unique). Then for any i ∈[n] and p ∈[p], we have ai,i,∅+ X q∈[P ]\{p} aI(q),i,{q} −(P −1)aI(p),i,{p} = X p′∈[P ] e(p′) i + X q∈[P ]\{p}  e1,1 + X p′∈[P ]\{q} e(p′) i  −(P −1)  e1,1 + X p′∈[P ]\{p} e(p′) i   = X p′∈[P ] e(p′) i +  (P −1)e(p) i + (P −2) X p′∈[P ]\{p} e(p′) i  −(P −1) X p′∈[P ]\{p} e(p′) i = Pe(p) i . (34) Hence, {ai,j,S}i,j∈[n],S⊂[P ] spans R2K+n(P −1) and h(Z) is strictly convex. Thus, it can have at most one global minimum. We want to show the existence of the global minimum and characterize it. n2∇h(Z) = X i,j∈[n] ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) ai,j,S = 2 X i,j∈[n] p∈[P ] ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) 1p∈S e(p) i . We can simplify terms as X j∈[n] ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) 1p∈S = X j∈Vyi ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) 1p∈S + X j∈V−yi ES∼DS \|S\| P yiℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P yjℓ′(yj⟨ai,j,S, Z⟩) 1p∈S = yi X j∈Vyi ES∼DS[ℓ′(yi⟨ai,j,S, Z⟩)1p∈S] + yi X j∈V−yi ES∼DS ℓ′(yi⟨ai,j,S, Z⟩) + 1 −\|S\| P 1p∈S = yi\|V−yi\|ES∼DS 1 −\|S\| P 1p∈S + yi X j∈[n] ES∼DS[ℓ′(yi⟨ai,j,S, Z⟩)1p∈S], where the second equality holds since ℓ′(z) + ℓ′(−z) = −1. Also, for any p ∈[P], ES∼DS 1 −\|S\| P 1p∈S = 1 P X q∈[P ] ES∼DS 1 −\|S\| P 1q∈S 66 = 1 P ES∼DS   1 −\|S\| P X q∈S 1q∈S   = 1 P ES∼DS 1 −\|S\| P \|S\| = P −1 6P . Hence, if X j∈[n] ES∼DS[ℓ′(yi⟨ai,j,S, Z⟩)1p∈S] + P −1 6P \|V−yi\| = 0, for all i ∈[n] and p ∈[P], then we have ∇h(Z) = 0. Let us consider a specific Z parameterized by z1, z−1, of the form z(p) i = yizyi for all i ∈[n] and p ∈[P]. We will find a stationary point with this specific form and then it should be the unique global minimum in the entire domain. Then for each i ∈[n] and p ∈[P], we have X j∈[n] ES∼DS[ℓ′(yi⟨ai,j,S, Z⟩)1p∈S] = X j∈Vyi ES∼DS[ℓ′(yi ⟨ai,j,S, Z⟩)1p∈S] + X j∈V−yi ES∼DS[ℓ′(yi⟨ai,j,S, Z⟩)1p∈S] = \|Vyi\| · ES∼DS[ℓ′(Pzyi)1p∈S] + \|V−yi\| · ES∼DS[ℓ′(\|S\|zyi −(P −\|S\|)z−yi)1p∈S] = 1 P X q∈[P ] \|Vyi\| · ES∼DS[ℓ′(Pzyi)1q∈S] + \|V−yi\| · ES∼DS[ℓ′(\|S\|zyi −(P −\|S\|)z−yi)1q∈S] = 1 P  \|Vyi\| · ES∼DS  ℓ′(Pzyi) X q∈S 1q∈S   +\|V−yi\| · ES∼DS  ℓ′(\|S\|zyi −(P −\|S\|)z−yi) X q∈S 1q∈S     = 1 P \|Vyi\| · ES∼DS[\|S\|ℓ′(Pzyi)] + \|V−yi\| · ES∼DS[\|S\|ℓ′(\|S\|zyi −(P −\|S\|)z−yi)] = \|Vyi\| 2 ℓ′(Pzyi) + \|V−yi\| P ES∼DS[\|S\|ℓ′(\|S\|zyi −(P −\|S\|)z−yi)]. From Lemma F.4, there exists a unique minimizer ˆZ = {ˆzs,k}s∈{±1},k∈[K] ∪ n ˆz(p) i o i∈[n],p∈[P ]\{p∗ i } of h(Z) and it satisfies sˆzs,k = z∗ s = Θ(1) for all k ∈[K] and yiˆz(p) i = z∗ yi = Θ(1) for all i ∈[n] and p ∈[P] \ {p∗ i } due to (A1). E.2.3 Strong Convexity Near Global Minimum We will show that h(Z) is strongly convex in a set G containing a global minimum ˆZ where G is defined as follows. G := n Z ∈R2K+n(P −1) : ∥Z −ˆZ∥∞< ∥ˆZ∥∞ o , here ∥·∥∞is ℓ∞norm. For any Z ∈G and a unit vector c ∈R2K+n(P −1) with c = P s∈{±1},k∈[K] cs,kes,k + P i∈[n],p∈[P ]\{p∗ i } c(p) i e(p) i , we have c⊤∇2h(Z)c = 1 n2 X i,j∈[n] ES∼DS ℓ′′(⟨ai,j,S, Z⟩)⟨ai,j,S, c⟩2 ≥ℓ′′(2P∥ˆZ∥∞) n2 X i,j∈[n] ES∼DS[⟨ai,j,S, c⟩2]. 67 Note that for each i ∈[n], p ∈[P], from (34), we have c(p) i = D c, e(p) i E = 1 P ⟨c, ai,i,∅⟩+ 1 P X q∈[P ]\{p} ⟨c, aI(q),i,{q}⟩−P −1 P c, aI(p),i,{p} , where we use the notational convention c(p∗ i ) i = cs,k for s ∈{±1}, k ∈[K] and i ∈Vs,k. By Cauchy-Schwartz inequality and the fact that PS∼DS[S = ∅], PS∼DS[S = {q}] ≥ 1 P (P +1) for all q ∈[P], c(p) i 2 =  1 P ⟨c, ai,i,∅⟩+ 1 P X q∈[P ]\{p} c, aI(q),i,{q} −P −1 P c, aI(p),i,{p}   2 ≤ 1 P 2 + P −1 P 2 + −P −1 P 2!  ⟨c, ai,i,∅⟩2 + X q∈[P ]\{p} c, aI(q),i,{q} 2 + c, aI(p),i,{p} 2   ≤ 1 P 2 + P −1 P 2 + −P −1 P 2! P(P + 1) X i,j∈[n] ES∼DS[⟨c, ai,j,S⟩2] ≤2P 2 X i,j,∈[n] ES∼DS ⟨c, ai,j,S⟩2 . Hence, we have c⊤∇2h(Z)c ≥ ℓ′′(2P∥ˆZ∥∞) (4K + 2n(P −1))P 2n2 (4K + 2n(P −1))P 2 X i,j∈[n] ES∼DS h ⟨c, ai,j,S⟩2i ≥ ℓ′′(2P∥ˆZ∥∞) (4K + 2n(P −1))P 2n2   X s∈{±1},k∈[K] c2 s,k + X i∈[n],q∈[P ]\{p∗ i } c(q) i 2   = ℓ′′(2P∥ˆZ∥∞) (4K + 2n(P −1))P 2n2 , and we conclude h(Z) is µ-strongly convex in G where µ := ℓ′′(2P ∥ˆ Z∥∞) (4K+2n(P −1))P 2n2 . Due to (A1), (A2), and the fact that ∥ˆZ∥∞= Θ(1), we have µ ≥ 1 poly(d). E.2.4 Near Stationary Points are Close to Global Minimum In this step, we want to show that near stationary points of h(Z) are close to a global minimum ˆZ. Lemma E.1. Suppose Z ∈R2K+n(P −1) satisfies ∥∇h(Z)∥< µϵ with some 0 < ϵ < ∥ˆ Z∥∞ 2 . Then, we have Z −ˆZ < ϵ. Proof of Lemma E.1. If Z = ˆZ, we immediately have our conclusion. We may assume Z ̸= ˆZ. Let us define a function g : R →R as g(t) = h ˆZ + t(Z −ˆZ) . Then g is convex and g′(t) = D ∇h ˆZ + t(Z −ˆZ) , Z −ˆZ E , g′′(t) = Z −ˆZ ⊤ ∇2h ˆZ + t(Z −ˆZ) Z −ˆZ . Furthermore, for 0 ≤t ≤t0 where t0 := ∥ˆ Z∥∞ 2∥Z−ˆ Z∥∞ , ˆZ + t(Z −ˆZ) ∈G, ∴g′′(t) ≥µ Z −ˆZ 2 . 68 We can conclude g is µ Z −ˆZ 2 -strongly convex in [0, t0]. From strong convexity in [0, t0] and convexity in R, we have (g′(t0) −g′(0))t0 = g′(t0)t0 ≥µ Z −ˆZ 2 t2 0, (g′(1) −g′(t0))(1 −t0) ≥0. If t0 < 1, we have ∥∇h(Z)∥ Z −ˆZ ≥ D ∇h(Z), Z −ˆZ E = g′(1) ≥g′(t0) ≥µ Z −ˆZ 2 t0, and ∥∇h(Z)∥≥µ Z −ˆZ t0 = µ Z −ˆZ ˆZ ∞ 2 Z −ˆZ ∞ ≥ µ ˆZ ∞ 2 , this is contradictory. Thus, we have t0 ≥1 and Z ∈G. From the strong convexity of h(Z) in G, we have µ Z −ˆZ ≤ ∇h(Z) −∇h( ˆZ) = ∥∇h(Z)∥< µϵ, and we have our conclusion Z −ˆZ < ϵ. E.2.5 Gradient Descent Achieves a Near Stationary Point We will show that LCutMix(W ) is a smooth function. Lemma E.2. Suppose the event Einit occurs. CutMix Loss LCutMix(W ) is L-smooth with L = 9r−1Pσ2 dd. Proof of Lemma E.2. Note that ∇w1LCutMix(W ) = 1 n2 X i,j∈[n] ES∼DS " \|S\| P yiℓ′(yifW (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjfW (Xi,j,S)) ×  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/∈S ϕ′ D w1, x(p) j E x(p) j    . Let f W = { e w1, e w−1} and W = {w1, w−1} be any parameters of the neural network fW . For any i, j ∈[n] and S ⊂[P], \|S\| P yiℓ′(yif f W (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjf f W (Xi,j,S)) ×  X p∈S ϕ′ D e w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D e w1, x(p) j E x(p) j   − \|S\| P yiℓ′(yifW (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjfW (Xi,j,S)) ×  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D w1, x(p) j E x(p) j   = \|S\| P yiℓ′(yif f W (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjf f W (Xi,j,S)) ×  X p∈S ϕ′ D e w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D e w1, x(p) j E x(p) j   − \|S\| P yiℓ′(yif f W (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjf f W (Xi,j,S)) ×  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D w1, x(p) j E x(p) j   69 + \|S\| P yiℓ′(yif f W (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjf f W (Xi,j,S)) ×  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D w1, x(p) j E x(p) j   − \|S\| P yiℓ′(yifW (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjfW (Xi,j,S)) ×  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/ ∈S ϕ′ D w1, x(p) j E x(p) j  . Since \|ℓ′\| ≤1, \|S\| P yiℓ′ yiff W (Xi,j,S) + 1 −\|S\| P yjℓ′ yjff W (Xi,j,S) ≤1, and since \|ϕ′\| ≤1, X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/∈S ϕ′ D w1, x(p) j E x(p) j ≤P max i∈[n],p∈[P ] x(p) i . In addition, since ϕ is r−1-smooth,  X p∈S ϕ′ D e w1, x(p) i E x(p) i + X p/∈S ϕ′ D e w1, x(p) j E x(p) j   −  X p∈S ϕ′ D w1, x(p) i E x(p) i + X p/∈S ϕ′ D w1, x(p) j E x(p) j   ≤ X p∈S ϕ′ D e w1, x(p) i E −ϕ′ D w1, x(p) i E x(p) i + X p/∈S ϕ′ D e w1, x(p) j E −ϕ′ D w1, x(p) j E x(p) j ≤r−1 X p∈S D e w1 −w1, x(p) i E x(p) i + r−1 X p/∈S D e w1 −w1, x(p) j E x(p) j ≤r−1P max i∈[n],p∈[P ] x(p) i 2 ∥e w1 −w1∥, and since ℓ′ and ϕ are 1-Lipschitz, we have \|S\| P yiℓ′(yiff W (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjff W (Xi,j,S)) − \|S\| P yiℓ′(yifW (Xi,j,S)) + 1 −\|S\| P yjℓ′(yjfW (Xi,j,S)) ≤ ff W (Xi,j,S) −fW (Xi,j,S) ≤ X p∈S D e w1 −w1, x(p) i E + D e w−1 −w−1, x(p) i E + X p/∈S D e w1 −w1, x(p) j E + D e w−1 −w−1, x(p) j E ≤P max i∈[n],j∈[P ] x(p) i (∥e w1 −w1∥+ ∥e w−1 −w−1∥) ≤ √ 2P max i∈[n],j∈[P ] x(p) i f W −W . 70 Therefore, ∇w1LCutMix(f W ) −∇w1LCutMix(W ) ≤r−1P max i∈[n],p∈[P ] x(p) i 2 ∥e w1 −w1∥+ √ 2P 2 max i∈[n],p∈[P ] x(p) i 2 f W −W ≤2r−1P max i∈[n],p∈[P ] x(p) i 2 f W −W , where the last equality is due to (A1) and (A8). In the same way, we can obtain ∇w−1LCutMix(f W ) −∇w−1LCutMix(W ) ≤2r−1P max i∈[n],p∈[P ] x(p) i 2 f W −W , and ∇LCutMix(f W ) −∇LCutMix(W ) ≤4r−1P max i∈[n],p∈[P ] x(p) i 2 f W −W ≤9r−1Pσ2 dd f W −W , where the last inequality holds since ξ(p) i 2 < 3 2σ2 dd and α2 ≤ 3 4σ2 dd due to (A7). Hence, LCutMix(W ) is L-smooth with L := 9r−1Pσ2 dd. Since our objective function LCutMix(W ) is L-smooth and η ≤1 L due to (A8), descent lemma (see Lemma 3.4 in Bubeck et al. (2015)) implies LCutMix W (t+1) −LCutMix W (t) ≤−η 2 ∇LCutMix W (t) 2 , and by telescoping sum, we have 1 T T −1 X t=0 ∇LCutMix W (t) 2 ≤2LCutMix W (0) ηT = Θ(1) ηT , (35) for any T > 0. Choose ϵ = µβ∥ˆ Z∥∞ polylog(d). Then from (35), there exists TCutMix ≤poly(d) η such that ∇LCutMix W (TCutMix) ≤ϵ. From characterization of σmin(J(W )) in Section E.2.1, ϵ ≥ ∇LCutMix W (TCutMix) ≥σmin J W (TCutMix) ∇h Z(TCutMix) ≥β 2 ∇h Z(TCutMix) , and thus ∇h Z(TCutMix) ≤2β−1ϵ = µ · 2∥ˆZ∥∞ polylog(d). For sufficiently large d, the RHS becomes smaller than µ · ∥ˆ Z∥∞ 4 . Then, by Lemma E.1 we have seen in Section E.2.4, Z(TCutMix) −ˆZ ≤∥ˆZ∥∞ 4 , and thus ϕ D w(TCutMix) yi , x(p) i E −ϕ D w(TCutMix) −yi , x(p) i E = Θ(1), for all i ∈[n] and p ∈[P], and therefore it reaches perfect training accuracy. 71 E.2.6 Test Accuracy of Solution Found by Gradient Descent The final step is showing that W (TCutMix) reaches almost perfect test accuracy. From the results of Section E.2.5, we have ϕ D w(TCutMix) s , vs,k E −ϕ D w(TCutMix) −s , vs,k E = Θ(1), ϕ D w(TCutMix) yi , ξ(p) i E −ϕ D w(TCutMix) −yi , ξ(p) i E = Θ(1), for each s ∈{±1}, k ∈[K], i ∈[n] and p ∈[P] \ {p∗ i }. For any u > v, by the mean value theorem, we have β(u −v) ≤ϕ(u) −ϕ(v) = (u −v)ϕ(u) −ϕ(v) u −v ≤(u −v). Hence, we have ϕ D w(TCutMix) s , vs,k E −ϕ D w(TCutMix) −s , vs,k E ≤ D w(TCutMix) s −w(TCutMix) −s , vs,k E , D w(TCutMix) s −w(TCutMix) −s , vs,k E ≤β−1 ϕ D w(TCutMix) s , vs,k E −ϕ D w(TCutMix) −s , vs,k E , and Ω(1) ≤ D w(TCutMix) s −w(TCutMix) −s , vs,k E ≤O(β−1), for each s ∈{±1} and k ∈[K]. Similarly, for all i ∈[n] and p ∈[P] \ {p∗ i }, ϕ D w(TCutMix) yi , ξ(p) i E −ϕ D w(TCutMix) −yi , ξ(p) i E ≤ D w(TCutMix) yi −w(TCutMix) −yi , ξ(p) i E , D w(TCutMix) yi −w(TCutMix) −yi , ξ(p) i E ≤β−1 ϕ D w(TCutMix) yi , ξ(p) i E −ϕ D w(TCutMix) −yi , ξ(p) i E , and Ω(1) ≤ D w(TCutMix) yi −w(TCutMix) −yi , ξ(p) i E ≤O(β−1). By Lemma B.3, w(TCutMix) 1 −w(TCutMix) −1 = w(0) 1 −w(0) −1 + X s∈{±1},k∈[K] sγ(s, k)vs,k + X i∈[n],p∈[P ]\{p∗ i } yiρ(i, p) ξ(p) i ξ(p) i 2 , where for each s ∈{±1}, γ(s, 1) = γ(TCutMix) 1 (s, 1) + γ(TCutMix) −1 (s, 1) + α X i∈Fs yi ρ(TCutMix) 1 (i, ˜pi) + ρ(TCutMix) −1 (i, ˜pi) ξ(˜pi) i −2 , and γ(s, k) = γ(TCutMix) 1 (s, k) + γ(TCutMix) −1 (s, k), ρ(i, p) = ρ(TCutMix) 1 (i, p) + ρ(TCutMix) −1 (i, p), for each s ∈{±1}, k ∈[K] \ {1}, i ∈[n] and p ∈[P] \ {p∗ i }. If we choose j ∈[n], q ∈[P] \ {p∗ j} such that ρ(j, q) = maxi∈[n],p∈[P ]\{p∗ i } ρ(i, p), then we have D w(TCutMix) yj −w(TCutMix) −yj , ξ(q) j E = D w(0) yj −w(0) −yj, ξ(q) j E + ρ(j, q) + yj X i∈[n],p∈[P ]\{p∗ i } (i,p)̸=(j,q) yiρ(i, p) D ξ(p) i , ξ(q) j E ξ(p) i 2 . 72 From the event Einit defined in Lemma B.2, (A8), and (8), D w(0) yj −w(0) −yj, ξ(q) j E = o 1 polylog(d) ≤1 2 D w(TCutMix) yj −w(TCutMix) −yj , ξ(q) j E , where the inequality holds since D w(TCutMix) yj −w(TCutMix) −yj , ξ(q) j E = Ω(1). In addition, by triangular inequality, we have X i∈[n],p∈[P ]\{p∗ i } (i,p)̸=(j,q) yiρ(i, p) D ξ(p) i , ξ(q) j E ξ(p) i 2 ≤ X i∈[n],p∈[P ]\{p∗ i } (i,p)̸=(j,q) ρ(i, p) D ξ(p) i , ξ(q) j E ξ(p) i 2 ≤ρ(j, q) e O nPσdσ−1 b d−1 2 ≤ρ(j, q) 2 , where the last inequality is due to (9). Hence, 1 3ρ(j, q) ≤ D w(TCutMix) yj −w(TCutMix) −yj , ξ(q) j E ≤3ρ(j, q) and we have ρ(j, q) = e O(β−1). Let (X, y) ∼D be a test data with X = x(1), . . . , x(P ) ∈Rd×P having feature patch p∗, dominant noise patch ˜p, and feature vector vy,k. We have x(p) ∼N(0, σ2 bΛ) for each p ∈[P] \ {p∗, ˜p} and x(˜p) −αvs,1 ∼N(0, σ2 dΛ) for some s ∈{±1}. Therefore, for all p ∈[P] \ {p∗, ˜p} ϕ D w(TCutMix) 1 , x(p)E −ϕ D w(TCutMix) −1 , x(p)E ≤ D w(TCutMix) 1 −w(TCutMix) −1 , x(p)E = D w(0) 1 −w(0) −1, x(p)E + X i∈[n],q∈[P ]\{p∗ i } ρ(i, q) D ξ(q) i , x(p)E ξ(q) i 2 ≤e O σ0σbd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o 1 polylog(d) , (36) with probability at least 1 −o 1 poly(d) due to Lemma B.2. In addition, ϕ D w(TCutMix) 1 , x(˜p)E −ϕ D w(TCutMix) −1 , x(˜p)E ≤ D w(TCutMix) 1 −w(TCutMix) −1 , x(˜p)E ≤α D w(TCutMix) 1 −w(TCutMix) −1 , vs,1 E + D w(TCutMix) 1 −w(TCutMix) −1 , x(˜p) −αvs,1 E ≤αβ−1 ϕ D w(TCutMix) 1 , vs,1 E −ϕ D w(TCutMix) −1 , vs,1 E + D w(0) 1 −w(0) −1, x(˜p) −αvs,1 E + X i∈[n],q∈[P ]\{p∗ i } ρ(i, q) D ξ(q) i , x(˜p) −αvs,1 E ξ(q) i 2 ≤e O αβ−1 + e O σ0σdd 1 2 + e O nPβ−1σdσ−1 b d−1 2 = o 1 polylog(d) , (37) with probability at least 1 −o 1 poly(d) , where the last equality is due to (8), (9), (10), and (A8). 73 Suppose (36) and (37) holds. Then, yfW (TCutMix)(X) = ϕ D w(TCutMix) y , vy,k E −ϕ D w(TCutMix) −y , vy,k E + X p∈[P ]\{p∗} ϕ D w(TCutMix) y , x(p)E −ϕ D w(TCutMix) −y , x(p)E = Ω(1) −o 1 polylog(d) > 0. Hence, we have our conclusion. □ 74 F Technical Lemmas In this section, we introduce technical lemmas that are used for proving the main theorems. We present their proofs here for better readability. The following lemma is used in Section C.2.4 and Section D.2.4: Lemma F.1. For any z, δ ∈R, \|ϕ(z) −(z + δ)ϕ′(z)\| ≤r + \|δ\|. Proof of Lemma F.1. ϕ(z) −zϕ′(z) =      z −1−β 2 r −z = −1−β 2 r = −1−β 2 r if z ≥r 1−β 2r z2 + βz − 1−β r z + β z = 1−β 2r z2 if 0 ≤z ≤r βz −βz = 0 if z < 0 , and we obtain \|ϕ(z) −(z + δ)ϕ′(z)\| ≤\|ϕ(z) −zϕ′(z)\| + \|δ\|ϕ′(z) ≤1 −β 2 r + \|δ\| ≤r + \|δ\|. The following lemma is used in Section C.2.4. Lemma F.2. Suppose Einit occurs. Then, for any model parameter W = {w1, w−1}, we have ∇W X i∈Vs,k ℓ(yifW (Xi)) 2 ≤8P 2σ2 dd\|Vs,k\| X i∈Vs,k ℓ(yifW (Xi)), for each s ∈{±1} and k ∈[K]. Proof of Lemma F.2. For each s ∈{±1} and i ∈[n], we have ∥∇wsfW (Xi)∥= X p∈[P ] ϕ′ D ws, x(p) i E x(p) i ≤P max p∈[P ] x(p) i ≤2Pσdd 1 2 , where the inequality is due to the condition from the event Einit defined in Lemma B.2 and (A7).. Therefore, for each s ∈{±1}, we have ∇ws X i∈Vs,k ℓ(yifW (Xi)) 2 = X i∈Vs,k ℓ′ (yifW (Xi)) ∇wsfW (Xi) 2 ≤  X i∈Vs,k ℓ′ (yifW (Xi)) ∥∇wsfW (Xi)∥   2 ≤4P 2σ2 dd  X i∈Vs,k ℓ′ (yifW (Xi))   2 ≤4P 2σ2 dd\|Vs,k\| X i∈Vs,k (ℓ′ (yifW (Xi)))2 ≤4P 2σ2 dd\|Vs,k\| X i∈Vs,k ℓ(yifW (Xi)) . The first inequality is due to triangular inequality, the third inequality is due to Cauchy-Schwartz inequality and the last inequality is due to 0 ≤−ℓ′ ≤1, which can be used to show (ℓ′)2 ≤−ℓ′ ≤ℓ. As a result, we have our conclusion: ∇W X i∈Vs,k ℓ(yifW (Xi)) 2 = ∇w1 X i∈Vs,k ℓ(yifW (Xi)) 2 + ∇w−1 X i∈Vs,k ℓ(yifW (Xi)) 2 75 ≤8P 2σ2 dd\|Vs,k\| X i∈Vs,k ℓ(yifW (Xi)). The following lemma is used in Section D.2.4. Lemma F.3. Suppose Einit occurs. Then, for any model parameter W = {w1, w−1}, we have ∇ X i∈Vs,k EC∼DC[ℓ(yifW (t)(Xi,C))] 2 ≤8P 2σ2 dd\|Vs,k\| X i∈Vs,k EC∼DC[ℓ(yifW (t)(Xi,C))] for each s ∈{±1} and k ∈[K]. Proof of Lemma F.3. For each s ∈{±1}, i ∈[n] and C ⊂[P] with \|C\| = C, we have ∥∇wsfW (Xi,C)∥= X p/∈C ϕ′ D ws, x(p) i E x(p) i ≤P max p∈[P ] x(p) i ≤2Pσdd 1 2 , where the inequality is due to the condition from the event Einit defined in Lemma B.2 and (A7). Therefore, for any s ∈{±1}, we have ∇ws X i∈Vs,k EC∼DC[ℓ(yifW (Xi,C))] 2 = X i∈Vs,k EC∼DC[ℓ′ (yifW (Xi,C)) ∇wsfW (Xi,C)] 2 ≤  X i∈Vs,k EC∼DC [ℓ′ (yifW (Xi,C)) ∥∇wsfW (Xi,C)∥]   2 ≤4P 2σ2 dd  X i∈Vs,k EC∼DC [ℓ′ (yifW (Xi,C))]   2 ≤4P 2σ2 dd\|Vs,k\| X i∈Vs,k EC∼DC h (ℓ′ (yifW (Xi,C)))2i ≤4P 2σ2 dd\|Vs,k\| X i∈Vs,k EC∼DC[ℓ(yifW (Xi,C))]. The first inequality is due to triangular inequality, the third inequality is due to Cauchy-Schwartz inequality and the last inequality is due to 0 ≤−ℓ′ ≤1, which can be used to show (ℓ′)2 ≤−ℓ′ ≤ℓ. As a result, we have our conclusion: ∇W X i∈Vs,k EC∼DC [ℓ(yifW (Xi,C))] 2 = ∇w1 X i∈Vs,k EC∼DC [ℓ(yifW (Xi,C))] 2 + ∇w−1 X i∈Vs,k EC∼DC [ℓ(yifW (Xi,C))] 2 ≤8P 2σ2 dd\|Vs,k\| X i∈Vs,k EC∼DC [ℓ(yifW (Xi,C))] . 76 The following lemma guarantees the existence and characterizes the minimum of the CutMix loss in Section E.2.2. Lemma F.4. Suppose the event Einit occurs. Let g1, g−1 : R × R →R be defined as gs(z1, z−1) := \|Vs\| \|V−s\|ℓ′(Pzs) + 2 P ES∼DS [\|S\|ℓ′(\|S\|zs −(P −\|S\|)z−s)] + P −1 3P , for each s ∈{±1}. There exist unique z∗ 1, z∗ −1 > 0 such that g1(z∗ 1, z∗ −1) = g−1(z∗ 1, z∗ −1) = 0. Furthermore, we have z∗ 1, z∗ −1 = Θ(1). Proof of Lemma F.4. For each z1 > 0, g−1(z1, 0) = \|V−1\| \|V1\| + 1 · −1 2 + 2 P ES∼DS[\|S\|ℓ′(−(P −\|S\|)z1)] + P −1 3P < \|V−1\| \|V1\| + 1 · −1 2 + P −1 3P < 0, since ℓ′(z) ≤−1 2 for any z ≤0 and we use 25 52n ≤\|V1\|, \|V−1\| ≤27 52n from the event Einit defined in Lemma B.2. In addition, g−1(z1, Pz1 + log 9) = \|V−1\| \|V1\| ℓ′(P 2z1 + P log 9) + 2 P ES∼DS[\|S\|ℓ′(\|S\|Pz1 + \|S\| log 9 −(P −\|S\|)z1)] + P −1 3P ≥ \|V−1\| \|V1\| + 1 ℓ′(log 9) + P −1 3P > 0, where we use 25 52n ≤\|V1\|, \|V−1\| ≤27 52n from the event Einit defined in Lemma B.2 and (A1) for the last inequality. Since z 7→g−1(z1, z) is strictly increasing and by intermediate value theorem, there exists S : (0, ∞) →(0, ∞) such that z = S(z1) is a unique solution of g−1(z1, z) = 0 and S(z1) < Pz1 + log 9. Note that S is strictly increasing since g−1(z1, z−1) is strictly decreasing with respect to z1 and strictly increasing with respect to z−1. Also, if S(z) is bounded above, i.e., there exists some U > 0 such that S(z) ≤U for any z > 0, lim z→∞g−1(z, S(z)) = lim z→∞ \|V−1\| \|V1\| ℓ′ (PS(z)) + 2 P ES∼DS \|S\|ℓ′\|S\|S(z) −(P −\|S\|)z + P −1 3P ≤lim z→∞ \|V−1\| \|V1\| ℓ′ (PU) + 2 P ES∼DS \|S\|ℓ′\|S\|U −(P −\|S\|)z + P −1 3P ≤−2 P ES∼DS \|S\| · 1\|S\|̸=P + P −1 3P = −P −1 P + 1 + P −1 3P < 0, and it is contradictory. Hence, we have limz→∞S(z) = ∞. Let us choose z > 0 such that z = 1 P log   3P 1 + \|V1\| \|V−1\| P −1 −1  , and thus ℓ′(Pz) = − P −1 3P 1 + \|V1\| \|V−1\| . We have g1(z, S(z)) = \|V1\| \|V−1\|ℓ′(Pz) + 2 P ES∼DS \|S\|ℓ′\|S\|z −(P −\|S\|)S(z) + P −1 3P 77 ≤ \|V1\| \|V−1\| + 1 ℓ′(Pz) + P −1 3P = 0. Next, we will prove the existence of z∗> 0 such that g1(z∗, S(z∗)) > 0. Let us choose ϵ > 0 such that ϵ−1 = max ( 3P(P + 1)\|V−1\| (P −2)(P + 2)\|V1\| + 3(P −1) P −2 , 3 2 1 + P(P + 1)\|V−1\| (P −1)(P −2)\|V1\| , 12P P −7 1 + \|V−1\| \|V1\| , 12P(P + 1) (P −2)(P + 2) 1 + \|V1\| \|V−1\| ) , (38) and note that ϵ = Θ(1). Since limz→∞S(z) = ∞, we can choose z∗such that ℓ′ 1 2 min {z∗, S(z∗)} = −ϵ 2. Then, for any t ≥z∗ 2 , we have −ϵ < ℓ′(t) < 0 and −1 < ℓ′(−t) < −1 + ϵ. (39) From the definition of S and (39) with t = PS(z∗) > 1 2 min{z∗, S(z∗)}, we have ES∼DS \|S\|ℓ′\|S\|S(z∗) −(P −\|S\|)z∗ = −P 2 \|V−1\| \|V1\| ℓ′PS(z∗) + P −1 3P < −P −1 6 + P\|V−1\| 2\|V1\| ϵ. (40) If S(z∗) −(P −1)z∗≥0, then PS(z∗) > (P −1)S(z∗) −z∗> . . . > 2S(z∗) −(P −2)z∗ = z∗+ S(z∗) + S(z∗) −(P −1)z∗ ≥z∗+ S(z∗) ≥1 2 min{z∗, S(z∗)}, and we have −P −1 6 + P\|V−1\| 2\|V1\| ϵ > ES∼DS \|S\|ℓ′\|S\|S(z∗) −(P −\|S\|)z∗ = 1 P + 1 ℓ′S(z∗) −(P −1)z∗ + P X m=2 mℓ′mS(z∗) + (P −m)z∗ ! ≥ 1 P + 1 −1 2 − P(P + 1) 2 −1 ϵ , where the last inequality is due to (39). This is contradictory to (38), especially the first term inside the maximum, and we have S(z∗) −(P −1)z∗< 0. In addition, if (P −1)S(z∗) −z∗≤0, then Pz∗> (P −1)z∗−S(z∗) > . . . > 2z∗−(P −2)S(z∗) = z∗+ S(z∗) + z∗−(P −1)S(z∗) ≥z∗+ S(z∗) ≥1 2 min{z∗, S(z∗)}, and we have −P −1 6 −P\|V−1\| 2\|V1\| ϵ < ES∼DS \|S\|ℓ′\|S\|S(z∗) −(P −\|S\|)z∗ = 1 P + 1 Pℓ′PS(z∗) + (P −1)ℓ′(P −1)S(z∗) −z∗ + P −2 X m=0 mℓ′mS(z∗) −(P −m)z∗ ! < 1 P + 1 −(P −1)(P −2) 2 (1 −ϵ) −P −1 2 , 78 where the last inequality is due to (39). This is contradictory to (38), especially the second term inside the maximum, and we have (P −1)S(z∗) −z∗> 0. Note that we have −ϵ 2 = ℓ′ 1 2 min{z∗, S(z∗)} ≥ℓ′ z∗ 2P , and since ϵ = Θ(1) in (38), we have z∗≤2P log 2 ϵ −1 = O(1). Thus, we have S(z∗) −(P −1)z∗< 0 < (P −1)S(z∗) −z∗< PS(z∗). One can consider dividing the interval [S(z∗) −(P −1)z∗, PS(z∗)] into a grid of length z∗+ S(z∗). Then, the interval is equally divided into P −1 sub-intervals and 0 belongs to one of them. In other words, there exists k ∈[P −2] such that kS(z∗) −(P −k)z∗≤0 < (k + 1)S(z∗) −(P −k −1)z∗, and note that if P = 3, then k = 1. The rest of the proof is divided into two cases: (k + 1)S(z∗) − (P −k −1)z∗≥1 2(z∗+ S(z∗)) or (k + 1)S(z∗) −(P −k −1)z∗< 1 2(z∗+ S(z∗)). In both cases, we show that g1(z∗, S(z∗)) > 0. Case 1: (k + 1)S(z∗) −(P −k −1)z∗≥1 2(z∗+ S(z∗)) From (39), we have −1 < ℓ′(−Pz∗) < · · · < · · · < ℓ′(k −1)S(z∗) −(P −k + 1)z∗ < −1 + ϵ, and −ϵ < ℓ′(k + 1)S(z∗) −(P −k −1)z∗ < · · · < ℓ′PS(z∗) < 0. Thus, we have ES∼DS \|S\|ℓ′\|S\|S(z∗) −(P −\|S\|)z∗) \| > 1 P + 1 kℓ′kS(z∗) −(P −k)z∗ −k(k −1) 2 −P 2 ϵ. and we obtain k > P −1 2 since k(k + 1) 2 = k(k −1) 2 + k > −P(P + 1) 2 ϵ −(P + 1)ES∼DS[\|S\|ℓ′(\|S\|S(z∗) −(P −\|S\|)z∗)] + kℓ′(kS(z∗) −(P −k)z∗) + k > −P(P + 1) 2 1 + \|V−1\| \|V1\| ϵ + (P −1)(P + 1) 6 ≥ P −1 2 P −1 2 + 1 2 , where the second inequality is due to (40) and the fact that ℓ′ ≥−1, and the last inequality is due to (38), especially the third term inside the maximum. Note that since k ∈N, k ≥P 2 . Note that from (39), we have −1 < ℓ′−PS(z∗) < · · · < ℓ′(P −k −1)z∗−(k + 1)S(z∗) < −1 + ϵ, and −ϵ < ℓ′(P −k + 1)z∗−(k −1)z∗ < · · · < ℓ′(Pz∗) < 0. Hence, we obtain ES∼DS \|S\|ℓ′\|S\|z∗−(P −\|S\|)S(z∗) ≥ 1 P + 1 −(P −k −1)(P −k) 2 −1 2(P −k) −((P −k + 1) + · · · + P) ϵ 79 ≥−(P −k)2 2(P + 1) − 1 P + 1 · P(P + 1) 2 ϵ ≥− P 2 8(P + 1) −P 2 ϵ, where we use k ≥P 2 for the last inequality. Therefore, we have g1(z∗, S(z∗)) = \|V1\| \|V−1\|ℓ′(Pz∗) + 2 P ES∼DS[\|S\|ℓ′(\|S\|z∗−(P −\|S\|)S(z∗))] + P −1 3P ≥− \|V1\| \|V−1\| + 1 ϵ − P 4(P + 1) + P −1 3P > 0, where the last inequality is due to (38), especially the fourth term inside the maximum. Case 2: (k + 1)S(z∗) −(P −k −1)z∗< 1 2(z∗+ S(z∗)) In this case, we have kS(z∗) −(P −k)z∗≤−1 2(z∗+ S(z∗)). From (39), we have −1 < ℓ′(−Pz∗) < · · · < ℓ′kS(z∗) −(P −k)z∗ < −1 + ϵ, and −ϵ < ℓ′(k + 2)S(z∗) −(P −k −2)z∗ < · · · < ℓ′PS(z∗) < 0. Thus, we have ES∼DS[\|S\|ℓ′(\|S\|S(z∗) −(P −\|S\|)z∗))\|] > 1 P + 1 (k + 1)ℓ′(k + 1)S(z∗) −(P −k −1)z∗ −k(k + 1) 2 −P 2 ϵ, and we obtain k > P −1 2 since (k + 1)2 2 = k(k + 1) 2 + k + 1 2 > −P(P + 1) 2 ϵ −(P + 1)ES∼DS[\|S\|ℓ′(\|S\|S(z∗) −(P −\|S\|)z∗)] + (k + 1)ℓ′((k + 1)S(z∗) −(P −k −1)z∗) + k + 1 2 > −P(P + 1) 2 1 + \|V−1\| \|V1\| ϵ + (P −1)(P + 1) 6 > P −1 2 + 1 2 2 , where the second inequality is due to (40) and the fact that ℓ′(z) ≥−1 2 ∀z ≥0, and the last inequality is due to our (38), especially the third term inside the maximum. Note that since k ∈N, we have k ≥P 2 . Note that from (39), we have −1 < ℓ′−PS(z∗) < · · · < ℓ′(P −k −2)z∗−(k + 2)S(z∗) < −1 + ϵ, and −ϵ < ℓ′(P −k)z∗−kz∗ < · · · < ℓ′(Pz∗) < 0. Hence, we obtain ES∼DS[\|S\|ℓ′(\|S\|z∗−(P −\|S\|)S(z∗)] ≥ 1 P + 1 −(P −k −1)(P −k) 2 −((P −k) + · · · + P) ϵ 80 ≥−(P −k)(P −k −1) 2(P + 1) − 1 P + 1 · P(P + 1) 2 ϵ ≥−(P −k)2 2(P + 1) − 1 P + 1 · P(P + 1) 2 ϵ ≥− P 2 8(P + 1) −P 2 ϵ, where we use k ≥P 2 for the last inequality. Therefore, we have g1(z∗, S(z∗)) = \|V1\| \|V−1\|ℓ′(Pz∗) + 2 P ES∼DS[\|S\|ℓ′(\|S\|z∗−(P −\|S\|)S(z∗))] + P −1 3P ≥− \|V1\| \|V−1\| + 1 ϵ − P 4(P + 1) + P −1 3P > 0, where the last inequality is due to (38), especially the fourth term inside the maximum. In both cases, we have g1(z∗, S(z∗)) > 0. By intermediate value theorem, there exist unique z∗ 1, z∗ −1 > 0 such that g1(z∗ 1, z∗ −1) = g−1(z∗ 1, z∗ −1) = 0. In addition, z ≤z∗ 1 ≤z∗and we have z1 = Θ(1) since z = Ω(1) and z∗= O(1). By using a similar argument, we can show that z∗ −1 = Θ(1), and we have our conclusion. 81 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The abstract and introduction accurately reflect the paper’s contributions. 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4,455	Interpretable Mesomorphic Neural Networks For Tabular Data Arlind Kadra Department of Representation Learning University of Freiburg kadraa@cs.uni-freiburg.de Sebastian Pineda Arango Department of Representation Learning University of Freiburg pineda@cs.uni-freiburg.de Josif Grabocka Department of Machine Learning University of Technology Nuremberg josif.grabocka@utn.de Abstract Even though neural networks have been long deployed in applications involving tabular data, still existing neural architectures are not explainable by design. In this work, we propose a new class of interpretable neural networks for tabular data that are both deep and linear at the same time (i.e. mesomorphic). We optimize deep hypernetworks to generate explainable linear models on a per-instance basis. As a result, our models retain the accuracy of black-box deep networks while offering free lunch explainability for tabular data by design. Through extensive experiments, we demonstrate that our explainable deep networks have comparable performance to state-of-the-art classifiers on tabular data and outperform current existing methods that are explainable by design. 1 Introduction Tabular data are arguably the most widely spread traditional data modality arising in a plethora of real-world application domains (Bischl et al., 2021; Borisov et al., 2022). There exists a recent trend to deploy neural networks for predictive tasks on tabular data (Kadra et al., 2021; Gorishniy et al., 2021; Somepalli et al., 2022; Hollmann et al., 2023). In a series of such application realms, it is important to be able to explain the predictions of deep learning models to humans (Ras et al., 2022), especially when interacting with human decision-makers, such as in healthcare (Gulum et al., 2021; Tjoa & Guan, 2021), or the financial sector (Sadhwani et al., 2020). Heavily parametrized models such as deep neural networks can fit complex interactions in tabular datasets and achieve high predictive accuracy, however, they are not explainable. In that context, achieving both high predictive accuracy and explainability remains an open research question for the Machine Learning community. In this work, we introduce mesomorphic neural architectures1, a new class of deep models that are both deep and locally linear at the same time, therefore, offering interpretability by design. In a nutshell, we propose a new architecture that is simultaneously (i) deep and accurate, as well as (ii) linear and explainable on a per-instance basis. Technically speaking, we learn deep hypernetworks that generate linear models that are accurate concerning the data point we are interested in explaining. Our interpretable mesomorphic networks for tabular data (dubbed IMN) are classification or regression models that identify the relevant tabular features by design. It is important to highlight that this work 1The etymology of the term mesomorphic is inspired by Chemistry as "pertaining to an intermediate phase of matter". For instance, a liquid crystal qualifies as both solid and liquid. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). tackles explaining predictions for a single data point (Lundberg & Lee, 2017), instead of explaining a model globally for the whole dataset (Ras et al., 2022). Similarly to existing prior works (AlvarezMelis & Jaakkola, 2018; Chen et al., 2018), we train deep models that generate explainable local models for a data sample of interest. In contrast, we train hypernetworks that generate linear models in the original feature space through a purely supervised end-to-end optimization. We empirically show that the proposed explainable deep models are both as accurate as existing blackbox classifiers for tabular datasets and achieve better performance compared to explainable end-to-end prior methods. At the same time, IMN is as interpretable as explainer techniques. Throughout this work, explainers can be categorized into two groups: i) interpretable surrogate models that are trained to approximate black-box models (Lundberg & Lee, 2017), and ii) end-to-end explainable methods by design. Concretely, we show that our method achieves comparable accuracy to competitive black-box classifiers and manages to outperform current state-of-the-art end-to-end explainable methods on the tabular datasets of the popular AutoML benchmark (Gijsbers et al., 2019). In addition, we compare our technique against state-of-the-art predictive explainers on the recent XAI explainability benchmark for tabular data (Liu et al., 2021) and empirically demonstrate that our method offers competitive interpretability. As a result, our method represents a significant step forward in making deep learning explainable by design for tabular datasets. Overall, this work offers the following contributions: • We present a technique that makes deep learning explainable by design via training hypernetworks to generate instance-specific linear models. • We offer ample empirical evidence that our method is as accurate as black-box classifiers, with the benefit of being as interpretable as state-of-the-art prediction explainers. 2 Proposed Method 2.1 Shallow Interpretability through Deep Hypernetworks Let us denote a tabular dataset consisting of N instances of M-dimensional features as X ∈RN×M and the C-dimensional categorical target variable as Y ∈{1, . . . , C}N. A model with parameters w ∈W estimates the target variable as f : RM × W →RC and is optimized by minimizing the empirical risk arg minw∈W PN n=1 L (yn, f(xn; w)), where L : {1, . . . , C} × RC →R+ is a loss function. An explainable model f is one whose predictions ˆyn = f(xn; w) for a data point xn are interpretable by humans. For instance, linear models and decision trees are commonly accepted to be interpretable by Machine Learning practitioners (Ribeiro et al., 2016; Lundberg & Lee, 2017). In this work, we rethink shallow interpretable models f(xn; w) by defining their parameters w ∈W to be the output of deep non-interpretable hypernetworks w(xn; θ) : RM × Θ →W, where the parameters of the hypernetwork are θ ∈Θ. We remind the reader that a hypernetwork (a.k.a. metanetwork, or "network of networks") is a neural network that generates the parameters of another network (Ha et al., 2017). In this mechanism, we train deep non-interpretable hypernetworks to generate interpretable models f in an end-to-end manner as arg minθ∈Θ PN n=1 L (yn, f(xn; w(xn; θ))). 2.2 Interpretable Mesomorphic Networks (IMN) Our method trains deep Multi-Layer Perceptron (MLP) hypernetworks that generate the parameters of linear models. For the case of multi-class classification, we consider linear models with parameters w ∈RC×(M+1), denoting one set of weights and bias terms per class, as f (xn; w)c = ez(xn;w)c/ PC k=1 ez(xn;w)k, with z (xn; w)c = PM m=1 wc,mxn,m + wc,0 representing the logit predictions for the c-th class. For the case of regression the linear model is simply f (xn; w) = PM m=1 wmxn,m + w0 with w ∈RM+1. Let us present our method IMN by starting with the case of multi-class classification following the hypernetwork mechanism explained in Section 2.1. The hypernetwork w(xn; θ) : RM × Θ → RC×(M+1) with parameters θ ∈Θ is a function that given a data point xn ∈RM generates the predictions as: 2 f (xn; w(xn; θ))c = ez(xn;w(xn;θ))c PC k=1 ez(xn;w(xn;θ))k , z (xn; w(xn; θ))c = M X m=1 w (xn; θ)c,m xn,m + w (xn; θ)c,0 (1) Instead of training weights w as in a standard linear classification, we use the output of an MLP network as the linear weights w(xn; θ). We illustrate the architecture of our mesomorphic network in Figure 1. In the case of regression, our linear model with hypernetworks is f(xn; w(xn; θ)) = PM m=1 w (xn; θ)m xn,m + w (xn; θ)0. We highlight that our experimental protocol (Section 4) includes both classification and regression datasets. Ultimately, we train the optimal parameters of the hypernetwork to minimize the following loss in an end-to-end manner: arg min θ∈Θ PN n=1 L (yn, f (xn; w(xn; θ))) + λ\|\|w (xn; θ) \|\|1. Input Features Input Layer TabResNet Backbone TabResNet Block Interpretable Weights W Figure 1: The IMN architecture. Our hypernetworks generate interpretable models that are accurate concerning a data point of interest (e.g. "Explain why patient xn is estimated to have cancer f(xn; w(xn; θ)) > 0.5 by analyzing the impact of features using the generated linear weights."). We stress that our novel method IMN does not simply train one linear model per data point, contrary to prior work (Ribeiro et al., 2016). Instead, the hypernetwork learns to generate accurate linear models by a shared network across all data points. As a result, generating the linear weights demands a single forward pass through the hypernetwork, rather than a separate optimization procedure. Furthermore, our method intrinsically learns to generate similar linear hyperplanes for neighboring data instances. The outputted linear models are accurate both in correctly classifying the data point xn, but also for the other majority of training instances in the neighborhood (see proof-of-concept experiment below). The outcome is a linear model with parameters w(xn; θ) that both interprets the prediction, but also serves as an accurate local model for the neighborhood of points. 2.3 Explainability Through Feature Attribution The generated linear models w (xn; θ) can be used to explain predictions through feature attribution (i.e. feature importance) (Liu et al., 2021). It is important to re-emphasize that our method offers interpretable predictions for the estimated target f(xn; w (xn; θ)) of a particular data point xn. Concretely, we can analyse the linear coefficients {w(xn; θ)1, . . . , w(xn; θ)M} to distill the importances of {xn,1, . . . , xn,M} by measuring the residual impact on the target. The impact of the m-th feature xn,m in estimating the target variable, is proportional to the change in the estimated target if we remove the feature (Hooker et al., 2019). Considering our linear models, the impact of the m-th feature is proportional to the change of the predicted target if we set the m-th feature to zero. In terms of notation, we multiply the feature vector element-wise with a Kronecker delta vector δm i = 1m̸=i. f(xn; w (xn; θ)) −f(xn ⊙δm; w (xn; θ)) ∝w(xn; θ)m xn,m (2) As a result, our feature attribution strategy is that the m-th feature impacts the prediction of the target variable by a signed magnitude of w(xn; θ)m xn,m. In our experiments, all the features are normalized to the same mean and variance, therefore, the magnitude w(xn; θ)m xn,m can be directly used to explain the impact of the m-th feature. In cases where the unsigned importance is required, a practitioner can use the absolute impact \|w(xn; θ)m xn,m\| as the attribution. Furthermore, to measure the global importance of the m-th feature for the whole dataset, we can compute 1 N PN n=1 \|w(xn; θ)m xn,m\|. 3 0 2 x1 0 1 x2 Globally Accurate 0 2 x1 0 1 x2 x′ Locally Interpretable {x \| w (x)T x + w0 (x) = 0} {x \| w(x′)T x + w0(x′) = 0} x ′ Figure 2: Investigating the accuracy and interpretability of IMN. Left: The global decision boundary of our method that separates the classes correctly. Right: The local hyperplane pertaining to an example x′ which correctly classifies the local example and retains a good global classification for the neighboring points. 2.4 Proof-of-concept: Globally Accurate and Locally Interpretable Classifiers As a proof of concept, we run our method on the half-moon toy task that consists of a 2-dimensional tabular dataset in the form of two half-moons that are not linearly separable. Initially, we investigate the global accuracy of our method. As shown in Figure 2 (left), our method correctly classifies all the examples. Furthermore, our method learns an optimal non-linear decision boundary that separates the classes (plotted in green). To determine the decision boundary, we perform a fine-grid prediction on all possible combinations of x1 and x2. Subsequently, we identify the points that exhibit the minimal prediction distance to a probability prediction of 0.5. Lastly, in Figure 2 (right) we investigate the local interpretability of our method, by taking a point x′ and calculating the corresponding weights (w (x′) , w (x′)0) generated by our hypernetwork, where we omited the dependence on θ for simplicity. The black line shows all the points that reside on the hyperplane w(x′) as {x \| w (x′)T x + w0 (x′) = 0}. It is important to highlight that the local hyperplane does not only correctly classify the point x′, but also the neighboring points, retaining an accurate linear classifier for the neighborhood of points. Table 1: Accuracy of local hyperplanes for neighboring points. Number of Neighbors Accuracy 10 0.84 25 0.82 50 0.78 100 0.77 200 0.77 To validate our claim that the per-example (local) hyperplane correctly classifies neighboring points, we conduct the following analysis: For every datapoint xn we take a specific number of nearest neighbor examples from every class, and we evaluate the classification accuracy of the hyperplane generated for the datapoint xn on the set of all neighbors. We repeat the above procedure with varying neighborhood sizes and we present the results in Table 1. The results indicate that the mesomorphic neural network generates hyperplanes that are accurate in the neighborhood of the point whose prediction we are interested in explaining. 3 Related Work Interpretable Models by Design: There exist Machine Learning models that offer interpretability by default. A standard approach is to use linear models (Tibshirani, 1996; Efron et al., 2004; Berkson, 1953) that assign interpretable weights to each of the input features. On the other hand, decision trees (Loh, 2011; Craven & Shavlik, 1995) use splitting rules that build up leaves and intermediate nodes. Every leaf node is associated with a predicted label, making it possible to follow the rules that led to a specific prediction. Bayesian methods such as Naive Bayes (Murphy et al., 2006) or Bayesian Neural Networks (Friedman et al., 1997) provide a framework for reasoning on the interactions of prior beliefs with evidence, thus simplifying the interpretation of probabilistic outputs. Instance-based models allow experts to reason about predictions based on the similarity to the train samples. The prediction model aggregates the labels of the neighbors in the training set, using the average of the top-k most similar samples (Freitas, 2014; Kim et al., 2015), or decision functions extracted from prototypes Martens et al. (2007). Attention-based models like TabNet (Arik & Pfister, 2021) make 4 use of sequential attention to generate feature weights on a per-instance basis, while, DANet (Chen et al., 2022) generates global importance weights for both the raw input features and higher order concepts. Neural additive models (NAMs) (Agarwal et al., 2021) use a neural network per feature to model the additive function of individual features to the output. However, these models trade-off the performance for the sake of interpretability, therefore challenging their usage on applications that need high performance. A prior similar work also trains hyper-networks to generate local models by learning prototype instances through an encoder model Alvarez-Melis & Jaakkola (2018). In contrast, we directly generate interpretable linear models in the original feature space. Interpretable Model Distillation: Given the common understanding that complex models are not interpretable, prior works propose to learn simple surrogates for mimicking the input-output behavior of the complex models (Burkart & Huber, 2021). Such surrogate models are interpretable, such as linear regression or decision trees (Ribeiro et al., 2016). The local surrogates generate interpretations only valid in the neighborhood of the selected samples. Some approaches explain the output by computing the contribution of each attribute (Lundberg & Lee, 2017) to the prediction of the particular sample. An alternative strategy is to fit globally interpretable models, by relying on decision trees (Frosst & Hinton, 2017; Yang et al., 2018), or linear models (Ribeiro et al., 2016). Moreover, global explainers sometimes provide feature importances (Goldstein et al., 2015; Cortez & Embrechts, 2011), which can be used for auxiliary purposes such as feature engineering. Most of the surrogate models tackle the explainability task disjointly, by first training a black box model, then learning a surrogate in a second step. Interpretable Deep Learning via Visualization: Given the success of neural networks in realworld applications in computer vision, a series of prior works (Ras et al., 2022) introduce techniques aiming at explaining their predictions. A direct way to measure the feature importance is by evaluating the partial derivative of the network given the input (Simonyan et al., 2013). CAM upscales the output of the last convolutional layers after applying Global Average Pooling (GAP), obtaining a map of the class activations used for interpretability (Zhou et al., 2016). DeepLift calculates pixel-wise relevance scores by computing differences with respect to a reference image (Shrikumar et al., 2017). Integrated Gradients use a baseline image to compute the cumulative sensibility of a black-box model f to pixel-wise changes (Sundararajan et al., 2017). Other methods directly compute the pixel-wise relevance scores such that the network’s output equals the sum of scores computed via Taylor Approximations (Montavon et al., 2017). 4 Experimental Protocol 4.1 Predictive Accuracy Experiments Baselines: In terms of interpretable white-box classifiers, we compare against Logistic Regression and Decision Trees, based on their scikit-learn library implementations (Pedregosa et al., 2011). On the other hand, we compare against two strong classifiers on tabular datasets, Random Forest and CatBoost. We use the scikit-learn interface for Random Forest, while for CatBoost we use the official implementation provided by the authors (Prokhorenkova et al., 2018). Lastly, in terms of interpretable deep learning architectures, we compare against TabNet (Arik & Pfister, 2021), a transformer architecture that makes use of attention for instance-wise feature-selection and NAM (Agarwal et al., 2021), a neural additive model which learns an additive function for every feature. For TabNet we use a well-maintained public implementation 2, while, for NAM we use the official public implementation from the authors 3. Protocol: We run our predictive accuracy experiments on the AutoML benchmark that includes 35 diverse classification problems, containing between 690 and 539 383 data points, and between 5 and 7 201 features. For more details about the datasets included in our experiments, we point the reader to Appendix C. In our experiments, numerical features are standardized, while we transform categorical features through one-hot encoding. For binary classification datasets we use target encoding, where a category is encoded based on a shrunk estimate of the average target values for the data instances belonging to that category. In the case of missing values, we impute numerical features with zero 2https://github.com/dreamquark-ai/tabnet 3https://github.com/AmrMKayid/nam 5 and categorical features with a new category representing the missing value. For CatBoost and TabNet we do not encode categorical features since the algorithms natively handle them. For all the methods considered we tune the hyperparameters with Optuna (Akiba et al., 2019), a well-known hyperparameter optimization (HPO) library. We use the default HPO algorithm (TPE) from the library and we tune every method for 100 HPO trials or a wall-time limit of 1 day, whichever condition gets fulfilled first. The HPO search spaces of the different baselines were taken from prior work (Gorishniy et al., 2021; Hollmann et al., 2023). For a more detailed description, we kindly refer the reader to Appendix C. Additionally, we use the area under the ROC curve (AUROC) as the evaluation metric. Lastly, the methods that offer GPU support are run on a single NVIDIA RTX2080Ti, while, the rest of the methods are run on an AMD EPYC 7502 32-core processor. 4.2 Explainability Experiments Baselines: First, we compare against Random, a baseline that generates random importance weights. Furthermore, BreakDown decomposes predictions into parts that can be attributed to certain features (Staniak & Biecek, 2018). TabNet offers instance-wise feature importances by making use of attention. LIME is a local interpretability method (Ribeiro et al., 2016) that fits an explainable surrogate (local model) to single instance predictions of black-box models. On the other hand, L2X is a method that applies instance-wise feature selection via variational approximations of mutual information (Chen et al., 2018) by making use of a neural network to generate the weights of the explainer. MAPLE is a method that uses local linear modeling by exploring random forests as a feature selection method (Plumb et al., 2018). SHAP is an additive feature attribution method (Lundberg & Lee, 2017) that allows local interpretation of the data instances. Last but not least, Kernel SHAP offers a reformulation of the LIME constrains (Lundberg & Lee, 2017). Metrics and Benchmark: As explainability evaluation metrics we use faithfulness (Lundberg & Lee, 2017), monotonicity (Luss et al., 2021) (including the ROAR variants (Hooker et al., 2019)), infidelity (Yeh et al., 2019) and Shapley correlation (Lundberg & Lee, 2017). For a detailed description of the metrics, we refer the reader to XAI-Bench, a recent explainability benchmark (Liu et al., 2021). For our explainability-related experiments, we use all three datasets (Gaussian Linear, Gaussian NonLinear, and Gaussian Piecewise) available in the XAI-Bench (Liu et al., 2021). For the state-of-the-art explainability baselines, we use the Tabular ResNet (TabResNet) backbone as the model for which the predictions are to be interpreted (same as for IMN). We experiment with different versions of the datasets that feature diverse ρ values, where ρ corresponds to the amount of correlation among features. All datasets have a train/validation set ratio of 10 to 1. Implementation Details: We use PyTorch as the main library for our implementation. As a backbone, we use a TabResNet where the convolutional layers are replaced with fully-connected layers as suggested by recent work (Kadra et al., 2021). For the default hyperparameters of our method, we use 2 residual blocks and 128 units per layer combined with the GELU activation (Hendrycks & Gimpel, 2016). When training our network, we use snapshot ensembling (Huang et al., 2017) combined with cosine annealing with restarts (Loshchilov & Hutter, 2019). We use a learning rate and weight decay value of 0.01, where, the learning rate is warmed up to 0.01 for the first 5 epochs, a dropout value of 0.25, and an L1 penalty of 0.1 on the weights. Our network is trained for 500 epochs with a batch size of 64. We make our implementation publicly available4. 5 Experiments and Results Hypothesis 1: IMN outperforms interpretable white-box models in terms of predictive accuracy. We compare our method against decision trees and logistic regression, two white-box interpretable models. We run all aforementioned methods on the AutoML benchmark and we measure the predictive performance in terms of AUROC. Lastly, we measure the statistical significance of the results using the autorank package Herbold (2020) that runs a Friedman test with a Nemenyi post-hoc test, and a 0.05 significance level. Figure 3 presents the average rank across datasets based on the AUROC performance. As observed IMN achieves the best rank across the AutoML benchmark 4Source code at https://github.com/ArlindKadra/IMN 6 datasets. Furthermore, the difference is statistically significant against both decision trees and logistic regression. The detailed per-dataset results are presented in Appendix C. Hypothesis 2: The explainability of IMN does not have a statistically significant negative impact on predictive accuracy. Additionally, it achieves a comparable performance against state-of-the-art methods. 1 2 3 Decision Tree Logistic Regression IMN CD Average Rank Figure 3: The critical difference diagram for the white-box interpretable methods. A lower rank indicates a better performance over datasets. This experiment addresses a simple question: Is our explainable neural network as accurate as a black-box neural network counterpart, that has the same architecture and same capacity?. Since our hypernetwork is a slight modification of the TabResNet Kadra et al. (2021), we compare it against TabResNet as a classifier. For completeness, we also compare against four other strong baselines, Gradient-Boosted Decision Trees (CatBoost), Random Forest, TabNet, and NAMs. Since the official implementation of NAMs only supports binary classification and regression, we separate the results into: i) results over 18 binary classification datasets (Figure 4 Top), and ii) results over all datasets (Figure 4 Bottom). 1 2 3 4 5 6 NAM TabNet Random Forest IMN TabResNet CatBoost CD Average Rank 1 2 3 4 5 Random Forest TabNet IMN TabResNet CatBoost CD Average Rank Figure 4: Black-box methods comparison with critical difference diagrams. Top: The average rank for the binary datasets present in the benchmark. Bottom: The average rank for all datasets present in the benchmark. A lower rank indicates a better performance. Connected ranks via a bold bar indicate that performances are not significantly different (p > 0.05). The results of Figure 4 demonstrate that IMN achieves a comparable performance to state-ofthe-art tabular classification models, while significantly outperforming explainable methods by design. IMN achieves a comparable performance to TabResNet, while outperforming TabNet and NAMs, indicating that its explainability does not harm accuracy in a significant way. There is no statistical significance of the differences between IMN, TabResNet and CatBoost. However, the difference in performance between IMNs, Random Forest, TabNet and NAMs is statistically significant. Additionally, we investigate the runtime performance of the different baselines (NAM is excluded since it cannot be run on the full benchmark). We present the results in Table 2. As expected, deep learning methods take a longer time to train, however, both IMN and TabResNet are the most efficient during inference. We observe that TabResNet takes longer to converge compared to IMN5, however, both methods demand approximately the same inference time. As a result, the explainability of our method comes as a free-lunch benefit. Lastly, IMN is 64x faster in inference compared to TabNet, an end-to-end deeplearning interpretable method. Hypothesis 1 and 2 are valid even when default hyperparameters are used, for more details we kindly refer the reader to Appendix B. Table 2: Aggregated training and inference times for all methods. Method Name Median Training Time (sec) Median Inference Time (sec) IMN (GPU) 192 0.025 TabResNet (GPU) 252 0.020 TabNet (GPU) 237 1.60 CatBoost (GPU) 63.2 0.20 Random Forest 42.55 2.20 Logistic Regression 0.23 0.07 Decision Tree 0.4 0.06 Hypothesis 3: IMNs offer competitive levels of interpretability compared to state-of-the-art explainer techniques. We compare against 8 explainer baselines in terms of 5 explainability metrics in the 3 datasets of the XAI benchmark (Liu et al., 2021), following the protocol we detailed in Section 4.2. The results of Table 3 demonstrate that IMN is competitive against all explainers across the indicated interpretability metrics. We tie in per5The number of training epochs is a hyperparameter. 7 Table 3: Investigating the interpretability of IMNs against state-of-the-art interpretability methods. The results are generated from the XAI Benchmark (Liu et al., 2021) datasets (with ρ = 0). Metric Dataset Random Breakd. Maple LIME L2X SHAP K. SHAP TabNet IMN Faithfulness (↑) Gaussian Linear 0.004 0.645 0.980 0.882 0.010 0.974 0.981 0.138 0.987 Gaussian Non-Linear -0.079 -0.001 0.487 0.796 0.155 0.926 0.970 0.161 0.621 Gaussian Piecewise 0.091 0.634 0.967 0.929 0.016 0.981 0.990 0.058 0.841 Faithfulness (ROAR) (↑) Gaussian Linear -0.039 0.494 0.548 0.544 0.049 0.549 0.550 0.041 0.639 Gaussian Non-Linear 0.050 0.006 0.040 -0.040 -0.060 -0.010 -0.036 -0.001 0.027 Gaussian Piecewise -0.055 0.372 0.347 0.450 0.015 0.409 0.426 0.072 0.404 Infidelity (↓) Gaussian Linear 0.219 0.041 0.007 0.007 0.034 0.007 0.007 0.049 0.007 Gaussian Non-Linear 0.075 0.086 0.021 0.071 0.089 0.030 0.022 0.047 0.018 Gaussian Piecewise 0.132 0.047 0.014 0.019 0.070 0.016 0.019 0.046 0.008 Monotonicity (ROAR) (↑) Gaussian Linear 0.487 0.605 0.700 0.652 0.437 0.680 0.667 0.585 0.785 Gaussian Non-Linear 0.497 0.542 0.645 0.587 0.457 0.670 0.632 0.493 0.637 Gaussian Piecewise 0.485 0.665 0.787 0.427 0.442 0.717 0.797 0.542 0.682 Shapley Correlation (↑) Gaussian Linear -0.016 0.246 0.999 0.942 -0.214 0.993 0.999 0.095 0.999 Gaussian Non-Linear -0.069 -0.179 0.686 0.872 -0.095 0.974 0.999 0.125 0.741 Gaussian Piecewise -0.078 0.099 0.983 0.959 0.157 0.991 0.999 0.070 0.875 Total Wins 1 0 2 2 0 2 7 0 7 0 0.5 1 Correlation 0.0 0.5 G. Linear 0 0.5 1 Correlation 0.50 0.75 0 0.5 1 Correlation 0 1 0 0.5 1 Correlation 0.00 0.01 0.02 IMN K. SHAP SHAP LIME Maple L2X BreakD. TabNet Random Figure 5: Performance analysis of different interpretability methods over a varying degree of feature correlation ρ. We present the performance of all methods on faithfulness (ROAR), monotonicity (ROAR), faithfulness, and infidelity (ordered from left to right) on the Gaussian Linear dataset for ρ values ranging from [0, 1]. formance with the second-best method Kernel-SHAP (Lundberg & Lee, 2017) and perform strongly against the other explainers. It is worth highlighting that in comparison to all the explainer techniques, the interpretability of our method comes as a free-lunch. In contrast, all the rival methods except TabNet are surrogate interpretable models to black-box models. Moreover, IMN strongly outperforms TabNet, the other baseline that offers explainability by design, achieving both better interpretability (Table 3) and better accuracy (Figure 4). Table 4: Interpretable method inference times. All the methods are run on the GPU and the time is reported in seconds. Method Name Credit-g Adult Christine IMN 0.01 0.02 0.02 TabNet 0.11 1.30 0.43 SHAP (TabResNet) 17.69 565.11 228.31 SHAP (CatBoost) 4.55 66.89 4317.61 As a result, for all surrogate interpretable baselines we first need to train a black-box model. Then, for the prediction of every data point, we additionally train a local explainer around that point by predicting with the black-box model multiple times. In stark contrast, our method combines prediction models and explainers as an all-in-one neural network. To generate an explainable model for a data point xn, IMN does not need to train a per-point explainer. Instead, IMN requires only a forward pass through the trained hypernetwork to generate a linear explainer. To quantify the difference in runtime between our method and other interpretable methods we compare the runtimes on a few datasets from the benchmark with a varying number of instances/features such as Credit-g (1000/21), Adult (48842/15), and Christine (5418/1637). Table 4 presents the results, where, as observed IMN has the fastest inference times, being 11-65x faster compared to TabNet which employs attention, 1710-11400x faster compared to SHAP that uses the same (TabResNet) backbone, and 455-215850x faster compared to SHAP that uses CatBoost as a backbone. 8 Table 5: The feature rank importances for the Census dataset. A lower rank is associated with a higher feature importance. Feature SHAP Decision Tree TabNet CatBoost IMN Age 2 5 2 3 3 Capital Gain 9 4 3 1 1 Capital Loss 10 9 14 4 5 Demographic 1 2 9 10 6 Education 5 3 5 9 9 Education num. 4 12 6 6 2 Hours per week 6 7 7 7 4 Race 8 10 5 12 7 Occupation 3 6 8 8 10 Relationship 7 1 1 2 8 Lastly, we compare all interpretability methods on 4 out of 5 metrics in the presence of a varying ρ factor, which controls the correlation of features on the Gaussian Linear dataset. Figure 5 presents the comparison, where IMN behaves similarly to other interpretable methods and has a comparable performance with the top methods in the majority of metrics. The results agree with the findings of prior work (Liu et al., 2021), where the performance in the interpretability metrics drops in the presence of feature correlations. Although our work focuses on tabular data, in Appendix A we present an application of IMN in the vision domain. Hypothesis 4: IMN offers a global (dataset-wide) interpretability of feature importance. The purpose of this experiment is to showcase that IMN can be used to analyze the global interpretability of feature attributions, where the dataset-wide importance of the m-th feature is aggregated as 1 N PN n=1 \|w(xn; θ)m xn,m\|. Since we are not aware of a public benchmark offering ground-truth global interpretability of features, we experiment with the Adult Census Income (Kohavi et al., 1996), a very popular dataset, where the goal is to predict whether income exceeds $50K/yr based on census data. We consider Decision Trees, CatBoost, TabNet, and IMN as explainable methods. Additionally, we use SHAP to explain the predictions of the TabResNet backbone. 1 2 3 4 5 Without top-k −1 0 1 2 3 Percentual decrease in AUROC IMN CatBoost TabNet Decision Tree SHAP Figure 6: Investigating the decrease in AUROC when removing the k-th most important feature. We present the importance that the different methods assign to features in Table 5. To verify the feature rankings generated by the models, we analyze the top 5 features of every individual method by investigating the drop in model performance if we remove the feature. The more important a feature is, the more accuracy should drop when removing that feature. The results of Figure 6 show that IMNs have a higher relative drop in the model’s accuracy when the most important predicted feature is removed. This shows that the feature ranking generated by IMN is proportional to the predictive importance of the feature and monotonously decreasing. In contrast, in the case of CatBoost, TabNet, SHAP, and Decision Trees, the decrease in accuracy is not proportional to the order of the feature importance (e.g. the case of Top-1 for Decision Trees, TabNet, SHAP or Top-2 for CatBoost). 10 5 10 4 10 3 10 2 10 1 odor spore-print-color gill-color ring-type stalk-surface-above-ring stalk-surface-below-ring population stalk-color-above-ring gill-size habitat stalk-root ring-number stalk-color-below-ring bruises%3F cap-shape cap-surface stalk-shape gill-spacing veil-color gill-attachment cap-color Feature Impacts Figure 7: Feature impacts for the mushroomedibility task. We additionally consider the task of predicting mushroom edibility (Lincoff, 1997). The odor feature allows one to predict whether a mushroom is edible or not and basing the predictions only on odor would allow a model to achieve more than 98.5% accuracy (Arik & Pfister, 2021). We run IMNs on the mushroom edibility task and we achieve a perfect test AUROC of 1. Furthermore, in Figure 7 we investigate the impact of every feature as described in Section 2.3, where, as observed, our method correctly identifies odor as the feature with the highest impact in the output. Based on the results, we conclude that IMNs offer global interpretability. 9 6 Conclusion In this work, we propose explainable deep networks that are comparable in performance to their black-box counterparts but also as interpretable as state-of-the-art explanation techniques. With extensive experiments, we show that the explainable deep learning networks outperform traditional white-box models in terms of performance. Moreover, the experiments confirm that the explainable deep-learning architecture does not include a significant degradation in performance or an overhead on time compared to the plain black-box counterpart, achieving competitive results against stateof-the-art classifiers in tabular data. Our method matches competitive state-of-the-art explainability methods on a recent explainability benchmark in tabular data, offering explanations of predictions as a free lunch. 7 Limitations and Future Work One potential limitation of our method is that although interpretable, the per-instance models are linear. A potential future work can focus on generating other types of non-linear interpretable models, such as decision trees. More concretely, the hypernetwork can generate the parameters of the decision splits and the decision value at each node, as well as the leaf weights. Another potential strategy is to generate local Support Vector Machines, by expressing the prediction for a data point as a function of the similarity of the informative neighbors. Acknowledgements JG, AK and SBA would like to acknowledge the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant number 417962828 and grant INST 39/963-1 FUGG (bwForCluster NEMO). 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We take the pre-trained backbone ϕ(·) : RH×W ×K →RD from PyTorch and change the output layer to a fully-connected layer w : RD →RH×W ×K×C that generates the weights for multiplying the input image x ∈RH×W ×K with K channels, and finally obtain the logits zc for the class c. In this experiment, we use λ = 10−3 as the L1 penalty strength. We fine-tuned the ImageNet pre-trained ResNet50 models, both for the explainable (IMN-ResNet) and the black-box (ResNet) variants for 400 epochs on the CIFAR-10 dataset with a learning rate of 10−4. To test whether the explainable variant is as accurate as the black-box model, we evaluate the validation accuracy after 5 independent training runs. IMN-ResNet achieves an accuracy of 87.49 ± 1.73 and the ResNet 88.76 ± 1.50, with the difference being statistically insignificant. We compare our method to the following image explainability baselines: Saliency Maps (Gradients) (Simonyan et al., 2013), DeepLift (Shrikumar et al., 2017), Integrated Gradients (Ancona et al., 2017) with SmoothGrad. All of the baselines are available via the captum library6. We compare the rival explainers to IMN-ResNet by visually interpreting the pixel-wise weights of selected images in Figure 8. The results confirm that IMN-ResNet generates higher weights for pixel regions that include descriptive parts of the object. B Plots 1 2 3 Decision Tree Logistic Regression IMN CD 1 2 3 4 5 6 NAM TabNet IMN Random Forest TabResNet CatBoost CD 1 2 3 4 5 6 7 TabNet DANet HyperTab IMN Random Forest TabResNet CatBoost CD Average Rank (Default Hyperparameters) Figure 9: The critical difference diagrams that represent the average rank over all datasets based on the AUROC test performance for: left) The white-box methods and IMN, middle) The black-box methods and IMN for the binary classification datasets, right) The black-box methods and IMN for the entire benchmark of datasets. In Figure 9, we repeat the experiments from Hypotheses 1 and 2, however, without performing hyperparameter optimization. Moreover, we consider two additional baselines, DANet (Chen et al., 6https://github.com/pytorch/captum 14 2022) and HyperTab (Wydma´nski et al., 2023). As observed, our findings are consistent and both hypotheses are validated even when default hyperparameters are used for all the methods considered. IMN TabResNet CatBoost R. Forest TabNet NAM 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Gain Figure 10: The gain distribution of the state-of-theart models. The gain is calculated by dividing the test AUROC against the test AUROC of a decision tree. To further investigate the results on individual datasets, in Figure 10 we plot the distribution of the gains in performance of all methods over a single decision tree model (with default hyperparameters). The gain G of a method m run on a dataset D for a single run is calculated as shown in Equation 3. G (m, DTree, D) = AUROC(m, D) AUROC(DTree, D) (3) The results indicate that all methods except NAM achieve a comparable gain in performance across the AutoML benchmark datasets, while, the latter achieves a worse performance overall. We present detailed results in Appendix C. Additionally, in Figure 11, we present the performance of the different explainers for the different explainability metrics. We present results for the Gaussian Non-Linear Additive and Gaussian Piecewise Constant datasets over a varying presence of correlation ρ between the features. The results show that our method achieves competitive results against Kernel Shap (K. SHAP) and LIME, the strongest baselines. 0 0.5 1 Correlation −0.25 0.00 G. Non-Linear Faithfulness (R) 0 0.5 1 Correlation 0.4 0.6 Monotonicity (R) 0 0.5 1 Correlation 0 1 Faithfulness 0 0.5 1 Correlation 0.00 0.01 0.02 Inﬁdelity 0 0.5 1 Correlation 0.00 0.25 G. Piecewise Constant 0 0.5 1 Correlation 0.50 0.75 0 0.5 1 Correlation 0 1 0 0.5 1 Correlation 0.00 0.01 0.02 IMN K. SHAP SHAP LIME Maple L2X BreakD. TabNet Random Figure 11: Performance analysis of all explainable methods on faithfulness (ROAR), monotonicity (ROAR), faithfulness, and infidelity. The results are shown for the Gaussian Non-Linear Additive and Gaussian Piecewise datasets where, correlation (ρ) ranges from [0, 1]. Lastly, to investigate how sensitive IMN is to the controlling hyperparameter configuration, we compare IMN and CatBoost (a method known for being robust to its hyperparameters in the community). Specifically, for every task, we plot the distribution of the performance of all hyperparameter configurations for every method. We present the results in Figure 12, where, as observed, IMN has a comparable sensitivity to CatBoost with regard to the controlling hyperparameter configuration. Moreover, in the majority of cases, the IMN validation performance does not vary significantly. 15 41142 1067 41147 4135 1464 54 41143 40996 41146 41164 41161 41138 12 40685 1590 23512 41150 31 1461 41166 41159 40981 1111 41169 41163 23517 1169 40984 1489 1468 1486 411683 41027 Task ID 0.5 0.6 0.7 0.8 0.9 1.0 Validation AUROC Distribution of Task Performances Method CatBoost IMN Figure 12: The distribution of the validation performance of the different hyperparameter configurations per task for CatBoost and IMN. C Tables To describe the 35 datasets present in our accuracy-related experiments, we summarize the main descriptive statistics in Table 6. The statistics show that our datasets are diverse, covering both binary and multi-class classification problems with imbalanced and balanced datasets that contain a diverse number of features and examples. Additionally, we provide the per-dataset performances for the accuracy-related experiments of every method with the default configurations. Table 7 summarizes the performances on the train split, where, as observed Random Forest and Decision Trees overfit the training data excessively compared to the other methods. Moreover, Table 8 provides the performance of every method on the test split, where, IMN, TabResNet, and CatBoost achieve similar performances. We provide the same per-dataset performances of every method with the best-found hyperparameter configuration during HPO for the train split in Table 9 and test split in Table 10. Lastly, we provide the HPO search spaces of the different methods considered in our experiments in Table 11, 12, 13, 14, 15, 16. 16 Table 6: Statistics regarding the AutoML benchmark datasets. Dataset ID Dataset Name Number of Instances Number of Features Number of Classes Majority Class Percentage Minority Class Percentage 3 kr-vs-kp 3196 37 2 52.222 47.778 12 mfeat-factors 2000 217 10 10.000 10.000 31 credit-g 1000 21 2 70.000 30.000 54 vehicle 846 19 4 25.768 23.522 1067 kc1 2109 22 2 84.542 15.458 1111 KDDCup09 appetency 50000 231 2 98.220 1.780 1169 airlines 539383 8 2 55.456 44.544 1461 bank-marketing 45211 17 2 88.302 11.698 1464 blood-transfusion-service-center 748 5 2 76.203 23.797 1468 cnae-9 1080 857 9 11.111 11.111 1486 nomao 34465 119 2 71.438 28.562 1489 phoneme 5404 6 2 70.651 29.349 1590 adult 48842 15 2 76.072 23.928 4135 Amazon employee access 32769 10 2 94.211 5.789 23512 higgs 98050 29 2 52.858 47.142 23517 numerai28.6 96320 22 2 50.517 49.483 40685 shuttle 58000 10 7 78.597 0.017 40981 Australian 690 15 2 55.507 44.493 40984 segment 2310 20 7 14.286 14.286 40996 Fashion-MNIST 70000 785 10 10.000 10.000 41027 jungle chess 44819 7 3 51.456 9.672 41138 APSFailure 76000 171 2 98.191 1.809 41142 christine 5418 1637 2 50.000 50.000 41143 jasmine 2984 145 2 50.000 50.000 41146 sylvine 5124 21 2 50.000 50.000 41147 albert 425240 79 2 50.000 50.000 41150 MiniBooNE 130064 51 2 71.938 28.062 41159 guillermo 20000 4297 2 59.985 40.015 41161 riccardo 20000 4297 2 75.000 25.000 41163 dilbert 10000 2001 5 20.490 19.130 41164 fabert 8237 801 7 23.394 6.094 41165 robert 10000 7201 10 10.430 9.580 41166 volkert 58310 181 10 21.962 2.334 41168 jannis 83733 55 4 46.006 2.015 41169 helena 65196 28 100 6.143 0.170 Table 7: The per-dataset train AUROC performance for all methods in the accuracy experiments with default hyperparameter configurations. The train performance is the mean value from 10 runs with different seeds. A dashed line ’-’ represents a failure of running on that particular dataset. Dataset ID Decision Tree Logistic Regression Random Forest TabNet TabResNet CatBoost IMN 3 1.000 0.990 1.000 0.980 1.000 1.000 1.000 12 1.000 1.000 1.000 1.000 1.000 1.000 1.000 31 1.000 0.795 1.000 0.514 1.000 0.963 1.000 54 1.000 0.955 1.000 0.492 1.000 1.000 1.000 1067 0.998 0.818 0.997 0.825 0.928 0.971 0.920 1111 1.000 0.822 1.000 0.966 0.899 0.895 1169 0.994 0.680 0.994 0.705 0.697 0.733 0.697 1461 1.000 0.908 1.000 0.947 0.945 0.948 0.942 1464 0.983 0.757 0.978 0.490 0.830 0.934 0.834 1468 1.000 1.000 1.000 0.493 1.000 1.000 1.000 1486 1.000 0.988 1.000 0.995 0.999 0.997 0.998 1489 1.000 0.813 1.000 0.947 0.974 0.982 0.977 1590 1.000 0.903 1.000 0.920 0.920 0.935 0.920 1596 1.000 0.951 1.000 0.949 0.992 0.997 0.995 4135 1.000 0.839 0.998 0.890 0.981 0.877 23512 1.000 0.683 1.000 0.820 0.863 0.831 0.853 23517 1.000 0.533 1.000 0.529 0.587 0.703 0.546 40685 1.000 0.999 1.000 0.988 0.999 0.994 40981 1.000 0.932 1.000 0.472 1.000 0.996 1.000 40984 1.000 0.983 1.000 0.990 0.999 1.000 0.999 40996 1.000 0.989 1.000 0.997 1.000 0.999 1.000 41027 1.000 0.799 1.000 0.980 0.982 0.989 0.982 41138 1.000 0.992 1.000 0.999 0.999 1.000 0.996 41142 1.000 0.942 1.000 0.951 1.000 0.999 1.000 41143 1.000 0.868 1.000 0.874 1.000 0.992 1.000 41146 1.000 0.967 1.000 0.989 1.000 1.000 1.000 41147 1.000 0.746 1.000 0.769 0.827 0.763 41150 1.000 0.938 1.000 0.896 0.985 0.988 0.985 41159 1.000 0.826 1.000 0.840 1.000 0.977 1.000 41161 1.000 1.000 1.000 0.999 1.000 1.000 1.000 41163 1.000 1.000 1.000 1.000 1.000 1.000 1.000 41164 1.000 0.994 1.000 0.968 1.000 0.983 1.000 41165 1.000 1.000 1.000 0.876 1.000 1.000 1.000 41166 1.000 0.889 1.000 0.943 0.978 0.992 0.995 41168 1.000 0.804 1.000 0.911 0.915 0.971 0.921 41169 1.000 0.854 1.000 0.867 0.938 0.998 0.986 17 Table 8: The per-dataset test AUROC performance for all methods in the accuracy experiments with default hyperparameter configurations. The test performance is the mean value from 10 runs with different seeds. A dashed line ’-’ represents a failure of running on that particular dataset. Dataset ID Decision Tree Logistic Regression NAM Random Forest TabNet TabResNet CatBoost IMN 3 0.987 0.990 0.977 0.998 0.983 0.999 0.999 0.999 12 0.938 0.999 0.998 0.995 0.999 0.999 0.999 31 0.643 0.775 0.717 0.795 0.511 0.756 0.790 0.751 54 0.804 0.938 0.927 0.501 0.968 0.934 0.957 1067 0.620 0.802 0.659 0.801 0.789 0.808 0.800 0.805 1111 0.535 0.816 0.544 0.793 0.778 0.843 0.816 1169 0.592 0.679 0.588 0.692 0.699 0.695 0.718 0.695 1461 0.703 0.908 0.827 0.930 0.926 0.931 0.937 0.930 1464 0.599 0.749 0.738 0.666 0.516 0.740 0.709 0.742 1468 0.926 0.996 0.995 0.495 0.995 0.996 0.994 1486 0.935 0.987 0.934 0.993 0.991 0.994 0.994 0.993 1489 0.842 0.805 0.806 0.962 0.928 0.949 0.948 0.950 1590 0.752 0.903 0.874 0.917 0.908 0.915 0.930 0.915 1596 0.942 0.951 0.997 0.949 0.991 0.996 0.994 4135 0.639 0.853 0.838 0.846 0.855 0.883 0.858 23512 0.626 0.683 0.583 0.794 0.803 0.825 0.804 0.823 23517 0.501 0.530 0.505 0.515 0.522 0.529 0.526 0.530 40685 0.967 0.994 1.000 0.986 0.995 0.993 40981 0.817 0.930 0.918 0.945 0.463 0.919 0.935 0.908 40984 0.946 0.980 0.995 0.985 0.994 0.995 0.994 40996 0.886 0.984 0.991 0.989 0.994 0.993 0.992 41027 0.792 0.797 0.931 0.976 0.979 0.974 0.978 41138 0.861 0.974 0.558 0.989 0.970 0.972 0.992 0.980 41142 0.626 0.742 0.724 0.796 0.713 0.782 0.822 0.775 41143 0.749 0.850 0.831 0.880 0.823 0.860 0.870 0.865 41146 0.910 0.966 0.983 0.974 0.982 0.988 0.981 41147 0.606 0.748 0.675 0.762 0.765 0.779 0.762 41150 0.867 0.938 0.912 0.981 0.896 0.984 0.984 0.984 41159 0.730 0.712 0.618 0.892 0.754 0.871 0.897 0.841 41161 0.857 0.995 0.972 0.999 0.997 0.998 1.000 0.998 41163 0.873 0.994 0.999 0.998 1.000 1.000 1.000 41164 0.786 0.898 0.925 0.888 0.913 0.935 0.902 41165 0.579 0.748 0.835 0.788 0.838 0.895 0.817 41166 0.699 0.882 0.927 0.918 0.952 0.949 0.943 41168 0.633 0.798 0.831 0.813 0.868 0.862 0.856 41169 0.554 0.841 0.800 0.842 0.883 0.866 0.865 18 Table 9: The per-dataset train AUROC performance for all methods in the accuracy experiments parametrized with the best hyperparameter configuration found during HPO. A dashed line ’-’ represents a failure to run on that particular dataset. Dataset ID Decision Tree Logistic Regression Random Forest TabNet TabResNet CatBoost IMN 3 0.999 0.995 0.997 1.000 1.000 1.000 12 0.986 1.000 0.999 1.000 1.000 1.000 1.000 31 0.820 0.785 0.945 0.982 0.806 0.888 0.813 54 0.930 0.960 0.971 0.988 0.990 1.000 0.968 1067 0.831 0.807 0.819 0.891 0.813 0.875 0.810 1111 0.836 0.829 0.885 0.825 0.842 0.844 1169 0.682 0.680 0.690 0.714 0.696 0.754 0.683 1461 0.900 0.907 0.922 0.946 0.947 0.954 0.944 1464 0.796 0.765 0.867 0.835 0.811 0.947 0.763 1468 0.972 1.000 0.996 1.000 1.000 1.000 1.000 1486 0.981 0.988 0.987 0.997 0.998 0.999 0.997 1489 0.908 0.815 0.938 0.993 0.995 1.000 0.998 1590 0.904 0.903 0.911 0.920 0.943 0.926 1596 0.931 0.951 0.946 0.952 0.998 4135 0.833 0.826 0.865 0.843 0.995 0.850 23512 0.737 0.684 0.766 0.831 0.869 0.900 0.859 23517 0.529 0.532 0.562 0.523 0.531 0.587 0.535 40685 1.000 0.999 1.000 1.000 1.000 1.000 0.996 40981 0.931 0.932 0.978 0.949 0.943 0.947 40984 0.989 0.988 0.996 0.999 0.999 1.000 0.999 40996 0.956 0.988 0.976 0.998 1.000 0.994 41027 0.873 0.801 0.905 0.999 0.996 0.989 0.996 41138 0.979 0.989 0.989 0.996 0.992 1.000 0.991 41142 0.786 0.874 0.860 0.976 0.952 1.000 41143 0.849 0.872 0.934 0.877 0.977 0.997 0.941 41146 0.966 0.968 0.986 1.000 1.000 0.999 0.994 41147 0.727 0.745 0.746 0.768 0.865 0.750 41150 0.946 0.956 0.961 0.968 0.989 1.000 0.989 41159 0.775 0.823 0.878 0.552 1.000 1.000 1.000 41161 0.860 0.999 0.961 0.999 1.000 1.000 1.000 41163 0.901 1.000 0.976 1.000 1.000 1.000 1.000 41164 0.707 0.975 0.901 0.998 0.982 0.999 0.999 41165 0.750 0.936 0.840 0.960 0.995 0.988 41166 0.822 0.892 0.867 0.986 0.984 1.000 0.999 41167 0.941 0.997 1.000 41168 0.787 0.804 0.823 0.908 0.900 0.944 0.908 41169 0.791 0.853 0.846 0.879 0.926 0.976 0.961 19 Table 10: The per-dataset test AUROC performance for all methods in the accuracy experiments parametrized with the best hyperparameter configuration found during HPO. A dashed line ’-’ represents a failure to run on that particular dataset. Dataset ID Decision Tree Logistic Regression NAM Random Forest TabNet TabResNet CatBoost IMN 3 0.999 0.997 0.977 0.998 1.000 1.000 1.000 12 0.960 0.998 0.998 0.998 0.998 0.998 1.000 31 0.765 0.852 0.717 0.826 0.741 0.844 0.826 0.863 54 0.844 0.959 0.924 0.955 0.966 0.935 0.959 1067 0.777 0.804 0.659 0.813 0.784 0.795 0.834 0.805 1111 0.805 0.814 0.544 0.829 0.801 0.839 0.806 1169 0.679 0.678 0.588 0.686 0.702 0.693 0.724 0.680 1461 0.903 0.907 0.827 0.919 0.917 0.932 0.939 0.933 1464 0.647 0.729 0.738 0.679 0.668 0.697 0.678 0.710 1468 0.954 0.999 0.988 0.987 0.999 0.994 0.999 1486 0.977 0.986 0.934 0.985 0.989 0.994 0.996 0.992 1489 0.890 0.802 0.806 0.920 0.945 0.958 0.955 0.956 1590 0.901 0.901 0.874 0.908 0.913 0.930 0.913 1596 0.931 0.951 0.946 0.952 0.997 4135 0.845 0.854 0.838 0.870 0.861 0.909 0.865 23512 0.725 0.681 0.583 0.754 0.803 0.817 0.808 0.817 23517 0.522 0.530 0.505 0.529 0.525 0.527 0.532 0.531 40685 0.908 0.999 1.000 1.000 0.999 1.000 1.000 40981 0.905 0.916 0.918 0.933 0.916 0.920 0.909 40984 0.981 0.989 0.991 0.995 0.994 0.996 0.993 40996 0.954 0.985 0.976 0.989 0.994 0.991 41027 0.870 0.800 0.900 0.999 0.993 0.980 0.991 41138 0.974 0.986 0.558 0.985 0.987 0.986 0.986 0.986 41142 0.744 0.811 0.724 0.794 0.814 0.819 0.808 41143 0.843 0.839 0.831 0.871 0.846 0.860 0.871 0.856 41146 0.966 0.962 0.975 0.991 0.984 0.987 0.984 41147 0.728 0.748 0.675 0.748 0.767 0.786 0.754 41150 0.942 0.954 0.912 0.958 0.967 0.985 0.986 0.984 41159 0.753 0.725 0.618 0.847 0.543 0.871 0.912 0.868 41161 0.849 0.997 0.972 0.950 0.996 0.999 0.999 0.998 41163 0.885 0.996 0.969 0.997 1.000 1.000 1.000 41164 0.697 0.914 0.877 0.869 0.919 0.937 0.919 41165 0.706 0.822 0.804 0.777 0.870 0.875 41166 0.811 0.886 0.862 0.929 0.954 0.956 0.947 41168 0.768 0.797 0.809 0.808 0.866 0.871 0.856 41169 0.762 0.845 0.823 0.839 0.884 0.869 0.873 20 Table 11: The hyperparameter search space for IMN. TabResNet has the same search space without weight normalization. Hyperparameter Type Range Log scale nr_epochs Integer [10, 500] learning_rate Continuous [1e-5, 1e-1] ✓ batch_size Categorical {32, 64, 128, 256, 512} weight_decay Continuous [1e −5, 1e −1] ✓ weight_norm Continuous [1e −5, 1e −1] ✓ dropout_rate Continuous [0, 0.5] Table 12: The hyperparameter search space for logistic regression. Hyperparameter Type Range Log scale C Continuous [1e −5, 5] penalty Categorical {l2, none} max_iterations Integer [50, 500] fit_intercept Categorical {True, False} Table 13: The hyperparameter search space for a decision tree. Hyperparameter Type Range Log scale criterion Categorical {Gini, Entropy} max_depth Integer [1, 21] min_samples_split Integer [2, 11] max_leaf_nodes Integer [3, 26] splitter Categorical {Best, Random} 21 Table 14: The hyperparameter search space for CatBoost. Hyperparameter Type Range Log scale learning_rate Continuous [1e −5, 1] ✓ random_strength Integer [1, 20] l2_leaf_reg Continuous [1, 10] ✓ bagging_temperature Continuous [1e −6, 1] ✓ leaf_estimation_iterations Integer [1, 20] iterations Integer [100, 4000] Table 15: The hyperparameter search space for Random Forest. Hyperparameter Type Range Log scale criterion Categorical {Gini, Entropy} max_depth Integer [1, 21] min_samples_split Integer [2, 11] max_leaf_nodes Integer [3, 26] n_estimators Integer [100, 4000] Table 16: Hyperparameter search space for the TabNet model. Hyperparameter Type Range Log scale n_a Categorical {8, 16, 24, 32, 64, 128} learning_rate Categorical {0.005, 0.01, 0.02, 0.025} gamma Categorical {1.0, 1.2, 1.5, 2.0} n_steps Categorical {3, 4, 5, 6, 7, 8, 9, 10} lambda_sparse Categorical {0, 0.000001, 0.0001, 0.001, 0.01, 0.1} batch_size Categorical {256, 512, 1024, 2048, 4096, 8192, 16384, 32768} virtual_batch_size Categorical {256, 512, 1024, 2048, 4096} decay_rate Categorical {0.4, 0.8, 0.9, 0.95} decay_iterations Categorical {500, 2000, 8000, 10000, 20000} momentum Categorical {0.6, 0.7, 0.8, 0.9, 0.95, 0.98} 22 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The results in Section 2.4 and Section 5 validate our claims. 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4,456	Rethinking Imbalance in Image Super-Resolution for Efficient Inference Wei Yu1, Bowen Yang 1, Qinglin Liu 1, Jianing Li 2, Shengping Zhang 1,∗, Xiangyang Ji 2,∗ 1 School of Computer Science and Technology, Harbin Institute of Technology 2 School of Information Science and Technology, Tsinghua University 20b903014@stu.hit.edu.cn, 2022211119@stu.hit.edu.cn, qlliu@hit.edu.cn, lijianing@pku.edu.cn, s.zhang@hit.edu.cn, xyji@tsinghua.edu.cn Abstract Existing super-resolution (SR) methods optimize all model weights equally using L1 or L2 losses by uniformly sampling image patches without considering dataset imbalances or parameter redundancy, which limits their performance. To address this issue, we formulate the image SR task as an imbalanced distribution transfer learning problem from a statistical probability perspective and propose a plugand-play Weight-Balancing framework (WBSR) for image SR to achieve balanced model learning without changing the original model structure or training data. Specifically, we develop a Hierarchical Equalization Sampling (HES) strategy to address data distribution imbalances, enabling better feature representation from texture-rich samples. To tackle model optimization imbalances, we propose a Balanced Diversity Loss (BDLoss) function, focusing on learning texture regions while disregarding redundant computations in smooth regions. After joint training of HES and BDLoss to rectify these imbalances, we present a gradient projection dynamic inference strategy to facilitate accurate and efficient reconstruction during inference. Extensive experiments across various models, datasets, and scale factors demonstrate that our method achieves comparable or superior performance to existing approaches with approximately a 34% reduction in computational cost. The code is available at https://github.com/aipixel/WBSR. 1 Introduction Image super-resolution (SR) aims to reconstruct high-resolution (HR) images with more details from low-resolution (LR) images. Recently, deep learning-based image SR methods have made significant progress in reconstruction performance through deeper network models and large-scale training datasets, but these improvements place higher demands on both computing power and memory resources, thus requiring more efficient solutions. Therefore, various techniques such as pruning [36, 53, 55], quantization [34, 39, 5], knowledge distillation [18, 45], and designing lightweight architectures [17, 25, 40] have been widely researched to accelerate inference and meet the requirements of deployment inference on resource-constrained platforms. However, these methods rely on static networks to process all input samples fairly, ignoring the different requirements of diverse samples for network computational cost, which limits the representation ability of the model. In contrast, dynamic neural network based methods [4, 42, 54, 48] can dynamically adjust the network structure or parameters and reduce the average computational cost, becoming a mainstream research focus in recent years. These methods can adaptively allocate networks with suitable computational ∗Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). ~0.08% pixels error Easy Medium Hard 48.8% 34.6% 16.6% ~19% pixels error ~52% pixels error Error_map Easy Medium Hard Training Data 𝑝𝑡𝑟𝑎𝑖𝑛𝑦 ~ Resampled Data 𝑝𝑟𝑒𝑠𝑦 ~ 𝜃 𝜃 𝑝𝑝𝑟𝑒𝑑𝑦∣𝑥 𝑝𝑝𝑟𝑒𝑑𝑦∣𝑥 BD loss (a) Data Distribution Imbalance (b) Model Optimization Imbalance (c) Weight Balancing 𝑝𝑏𝑎𝑙𝑦∣𝑥 × √ Resample loss 𝓛𝟏/𝓛𝟐 Imbalanced Weight Balanced Weight PSNR Value Number Figure 1: Illustration of (a) the data distribution from the widely used DIV2k [1] training set, (b) the reconstruction results of RCAN [51] model, and (c) the proposed weight-balancing framework. costs according to the content of the input samples during inference. Despite the advancements in these dynamic network solutions, practical applications are still hindered by two prevalent limitations: Data Distribution Imbalance. Existing SR methods [51, 52, 4, 26] mostly use uniformly sampled LR-HR patch pairs instead of the entire image to train models due to the limitation of memory resources. However, they ignore the underlying fact that patch contents in images exhibit imbalanced distributions (i.e., the abundant easily reconstructed smooth flat patches and rare hardly reconstructed edge texture patches), resulting in inherent data bias. Figure 1 (a) shows that the number proportion of easy flat patches (48.8%) is much larger than that of hard textured patches (16.6%). Model Optimization Imbalance. Current SR methods [27, 6, 12, 33] typically employ L1 or L2 losses to treat all patch areas and optimize each weight equally, which lacks reasonable optimization for their model training. Since the details lost in low-resolution images mainly exist in edges and texture locations, fewer computational resources of the model are required for those flat patches. Therefore, existing SR methods involve redundant calculations in flat areas, which leads to imbalanced inference performance where the model overfits in simple areas and underfits in complex ones and results in uneven distribution of model computational resources as shown in Figure 1 (b). For the same image, the optimized RCAN [51] model exhibits overfitting in the smooth background area (green box, with error pixels accounting for only 0.08%), while it shows obvious underfitting in the textured foreground area (red box, with error pixels accounting for up to 52%). Overall, these prevalent imbalance problems of data distribution and model optimization in the real world limit the performance of current image SR algorithms. As motivated, although this imbalance is a well-known observation in the classification task [3, 46, 21], we formulate the image SR task as an imbalanced distribution transfer learning problem from a statistical probability perspective. To mitigate the gap, we propose a plug-and-play weight-balancing framework, dubbed WBSR, to achieve balanced model learning without additional computation costs, which improves the restoration effect and inference efficiency of models without changing the original model structure and training data, as shown in Figure 1 (c). Specifically, to address the imbalance problem of data distribution, we develop a Hierarchical Equalization Sampling (HES) strategy, enabling better feature representation from texture-rich samples to mitigate data biases. Then, to solve the imbalance problem of model optimization, we propose a Balanced Diversity Loss (BDLoss) function, focusing on learning texture areas while disregarding redundant computations in smooth areas. After joint training of HES and BDLoss within WBSR to rectify these imbalances, we present a gradient projection dynamic inference strategy to facilitate accurate and efficient inference. In summary, we make the following three key contributions: (1) This paper is the first attempt to explore the imbalance in the image super-resolution field and gives a reasonable analysis from a perspective of probability statistics, i.e., the imbalance of data distribution and model optimization limits the algorithm performance. (2) We propose a plug-and-play weight-balancing framework dubbed WBSR upon HES and BDLoss to achieve balance training without additional computation costs, which improves the restoration effect and inference efficiency of models without changing the original model structure and training data. (3) Extensive experiments across various models, 2 datasets, and scale factors demonstrate that our achieves comparable or superior performance to existing methods with less computational cost. 2 Related Work 2.1 Deep Imbalanced Learning Deep imbalanced learning has attracted widespread attention due to the imbalanced data distribution caused by the difficulty of data acquisition in practical applications [20, 47]. The data imbalance problem presents a significant challenge in deep learning, when some classes have fewer samples than others, resulting in poorer model prediction performance for the minority class. Previous imbalanced learning methods [43, 50] have mainly studied data resampling techniques to solve this problem. For example, over-sampling minority classes [30, 3] and under-sampling common classes [2, 44]. However, oversampling increases memory storage and training time, while under-sampling causes overfitting problems [10, 5, 29]. Recently, several works [41, 49] attempt to develop data resampling methods as data augmentation strategies for the image super-resolution task to compensate for the imbalance of training patches between different classes. Another category of class-imbalanced learning methods is reweighting techniques. Recent reweighting methods assign weights to different classes [14, 7, 28] and training examples [15, 13, 38], which aim to modify their gradients to make models balance. [19] process from a domain adaptation perspective and enhance classic class-balanced learning by explicitly estimating the differences between different class distributions using meta-learning methods. In contrast, these methods balance the data loss by reweighting each class instead of sampling to achieve a balanced data distribution. 2.2 Dynamic Network for Efficient Image Super-Resolution Recent researches address this problem with efficient dynamic network frameworks, which mostly adopt content-aware modules to dynamically send image patches to sub-networks with different complexities to accelerate model inference. ClassSR [23] combines classification and SR in a unified framework, which uses an additional class module to classify image patches into different classes, and then applies the subnets to perform SR on different classes. ARM [4] further adopts the validation set to build an Edge-to-PSNR lookup table by mapping edge scores of image patches to the performance of each sub-network to select appropriate subnets to further improve efficiency. PathRestore [48] introduces a pathfinder to implement a multi-path CNN, which can dynamically select appropriate routes for different image areas according to the difficulty of restoration. However, these techniques still have two key problems. One is the additional amount of parameters and calculations brought by the introduction of classifiers or selectors, and the other is the neglect of data and network imbalance that affects the performance of the model. 3 Theoretical Analysis Let x and y denote LR and HR patches and L1 loss as an example (Note that the theoretical applies to the L2), the optimization object of SR task can be written as min θ E(x,y)\|pdata\|\|y −ˆy\|\|1 (1) where ˆy = fθ(x) represents the SR result estimated from LR x with SR model fθ. θ denotes the model parameters. pdata indicates the data distribution space. It aims to minimize all absolute errors between predicted images and ground-truth images from the whole data. From the natural assumption that the distribution of the training set is imbalanced, whereas the independent testing set is balanced [11, 14, 10], so we set the training data and testing data are drawn from different joint data distributions, ptrain(x, y) and pbal(x, y), respectively. The conditional probability p(x\|y) is the same in both training and testing sets due to the fixed downsampling degradation in the SR task. From the probabilistic view, the prediction ˆy of the SR network is considered as the mean of a noisy prediction distribution, which can be modeled as a Gaussian distribution p(y\|x; θ) = N(y; ˆy, σ2 noiseI) (2) where σ2 noise indicates the variance of the independently distributed error term. The prediction ˆy can be treated as the mean of a noisy prediction distribution. Eq. 2 can be interpreted as the distribution 3 form of Eq. 1, corresponding to the maximized negative log-likelihood (NLL) loss in the regression of the prediction distribution. Consequently, the prediction model trained by L1 actually captures the mean value of the entire solution space, i.e., the distribution of the training set. Theorem 1 (Distribution Transformation). Considering the discordance between ptrain(y\|x) and pbal(y\|x) attributable to the distribution shift. Given the identical conditional probability p(x\|y) across both the training and testing sets, we leverage the Bayes rule p(y\|x) ∝p(x\|y) · p(y) to establish the relationship through variable substitution as follows ptrain(y\|x) = ptrain(y) · pbal(y\|x) pbal(y) · pbal(x) ptrain(x) (3) This theorem reveals that the existence of imbalance issues stems from the direct proportionality between ptrain(y\|x) and ptrain(y) with a ratio of pbal(x) ptrain(x). When a specific type of patch sample is infrequently present in the training set, i.e., when ptrain(y) is low, the value of ptrain(y\|x) decrease as well, which results in a decrease in the accuracy of predictions. As a consequence, the trained SR model tends to underestimate the occurrence of rare patches during prediction. Meanwhile, considering that the integral of ptrain(y\|x) equals 1, we can obtain ptrain(y\|x) = ptrain(y\|x) R Y ptrain(y′\|x)dy′ (4) where Y denotes the entire training sample space. Then, we substitute Eq. 3 into Eq. 4 to model the relationship between the two distributions through explicit distribution transformation ptrain(y\|x) = pbal(y\|x) · ptrain(y) R Y pbal(y′\|x) · ptrain(y′)dy′ (5) where y′ denotes the integral variable. Diverging from previous works that focus on modeling ptrain(y\|x), our objective is to estimate pbal(y\|x) for achieving balanced prediction on the testing set. The detailed proof is available in the supplementary materials. The aforementioned theory proves that the imbalanced model optimization caused by imbalanced data distribution and loss function is reasonable. Therefore, our approach aims to correct this imbalance without introducing additional datasets or computational costs. 4 Methodology 4.1 Weight-Balancing Training Framework Based on the observed phenomenon and analysis, the imbalanced model optimization of image SR undoubtedly limits the reconstruction performance of the model, especially for rare hard texture patches. We consider attaining a robust model representation with balanced weights from the perspective of two aspects: data sampling and optimization function. Figure 2 (a) illustrates the training process of the proposed framework, dubbed WBSR, which consists of two main components: Hierarchical Equalization Sampling (HES) and Balanced Diversity Loss (BDLoss). Given input LR patches from the training set, we employ HES to sample a batch of approximately balanced patches to optimize each subnet model with our BDLoss Lbd. The overall optimization objective is min θ E(x,y)\|ptrainLbd(y −Smθ(x)) (6) where Smθ represents the m-th subnet in the supernet with parameters θm. We employ a divide-andconquer optimization strategy to learn nearly balanced weights, minimizing the overall objective by ensuring that each individual subnet within the supernet is well-optimized. Each subnet with varying computational cost shares the weights of the supernet and is intended to handle image patches of different complexities, which does not introduce additional complexity that impedes the inference speed. In the following, we describe the details of our HES and BDLoss, respectively. 4.1.1 Hierarchical Equalization Sampling Without prior data classification, we propose a simple yet effective Hierarchical Equalization Sampling (HES) strategy, which utilizes inherent gradient information of patches to perform sample-level 4 (a) Training Stage Imbalanced LR Patches from training set Imbalanced Weight Balanced Weight Update LR Patches Hierarchical Equalization Sampling S1 S2 S3 Gradient Projection Dynamic Inference Balanced Diversity Loss HR Patches Proj Map Dynamic Network SR Patches SR Image (b) Testing Stage … … LR Image … Chop Grad Combine Figure 2: Illustration of the proposed weight-balancing framework. (a) The training stage combines hierarchical equalization sampling and balanced diversity loss to jointly train a supernet model with balanced weights. (b) The testing stage adopts the gradient projection dynamic inference with a gradient projection map and multiple dynamic subnets for efficient inference. sampling and class-level sampling of difficult and easy classes to achieve equalization between the abundant simple samples and rare difficult samples. Sample-Level Sampling refers to uniformly sampling patches from the training dataset. Each sample is sampled with equal probability during the training stage, whose probability is Pi = 1 N . i indicates the i-th samples. N denotes the total number of training patch samples. It ensures that the model learns stable initial weights early in training, capturing general features across different sample types. Class-Level Sampling aims to assign a higher sampling probability to rare difficult samples. Unlike the image classification task where the number of categories is determined, samples in image SR are unclassified and the number is unknown. To address it, we calculate the gradient vectors online consisting of the mean and standard deviation of the gradient magnitude of the input samples in the horizontal and vertical directions, which assess the reconstruction difficulty of samples and then classify them using vector thresholds t to obtain the sampling probability. The threshold for the k-th class is defined as follows tk = t[k · N K ], k ∈[1, K] (7) where K is the number of classes. t1 and tK represent the gradient threshold of the simplest and most difficult classes, respectively. The number of samples for the k-th class corresponds to the Nk samples whose gradient vectors fall within the range from tk−1 to tk. The sampling possibility Pk can be calculated by Pk = PK j=1 1 Nj Nk · δk (8) where δ ∈(0, 1) indicates the exponential factor to avoid overfitting simple data by reducing the number of samples. It enables the sampled batch training data containing samples from difficult classes, thereby achieving equalized data sampling. The core concept of the proposed hierarchical equalization sampling strategy is to reconcile the data bias caused by the inherent imbalance, i.e., difficult samples are visually more important than smooth samples. During training and testing, the gradient vectors of image patches can be quickly exported using existing gradient operator [32]. Therefore, our HES method does not impose any additional computational burden and effectively leverages dataset information to enhance the model’s feature representation capabilities for hard samples. 5 4.1.2 Balanced Diversity Loss The commonly used L1 and L2 losses of previous methods treat all patches equally and perform gradient updates on each weight parameter, which ignores parameter redundancy and leads to overfitting on simple patches and underfitting on rare hard patches. To achieve reasonable optimization of models for diversity patches, we propose a novel Balanced Diversity Loss (BDLoss) to learn approximate balanced model weights, which performs distribution transformation by exploiting the training distribution without additional data to achieve balanced predictions. In accordance with Theorem 1, we first estimate the desired pbal(y\|x) by minimizing the NLL loss pbal(y\|x; θ) = N(y; ˆy, σ2 noiseI) (9) Definition 1. To balance the uncertainty of model diversity predictions and avoid excessive optimization, our BDLoss is defined as the likelihood function Lbd = −log ptrain(y\|x; θ) + λ\|\|θ\|\|2 (10) where log ptrain(y\|x; θ) denotes the converted conditional probability aimed to obtain balanced model weights θ. \|\| · \|\|2 indicates the L2 regularization function to prevent model overfitting. λ represents a regularization coefficient. Next, we derive the implementation of Lbd based on Eq. 9 log ptrain(y\|x; θ) = log pbal(y\|x; θ) · ptrain(y) R Y pbal(y′\|x; θ) · ptrain(y′)dy′ = log N(y; ˆy, σ2 noiseI) + log ptrain(y) −log Z Y N(y′; ˆy, σ2 noiseI) · ptrain(y′)dy′ (11) where log ptrain(y) is constant term that can be omitted. The first remaining term is a probability form of L1 loss as Eq. 2. The last term of log R Y N(y′; ˆy, σ2 noiseI) · ptrain(y′)dy′ indicates the key diversity balancing term that obeys Gaussian distribution, which involves the integral operation and necessitates finding a closed-form expression. Building upon the design of previous classification tasks [15, 19, 35], we utilize the Gaussian Mixture Model (GMM) technique to represent the constant term ptrain(y) = L X i=1 ϕiN(y; µi, σi) (12) where L denotes the number of Gaussian components. ϕ, µ, σ indicate the weights, means and covariances of multi-dimensional GMM, respectively. As the multiplication of two Gaussian functions results in another unnormalized Gaussian, the diversity balancing term can be expressed as Z Y N(y; ˆy, σ2 noiseI) · L X i=1 ϕiN(y; µi, Σi)dy = L X i=1 ϕisi Z Y N(y; ˜µi, ˜Σi)dy (13) where si, ˜µ, and ˜Σ are the norms, means, and covariances of the resulting unnormalized Gaussian, respectively. Now, the integral of the balanced diversity term adheres to a Gaussian distribution and is solved straightforwardly, so the BDLoss of Eq. 10 can be derivable as follows Lbd = −log N(y; ˆy, σ2 noiseI) + log L X i=1 ϕi · N(ˆy; µi, Σi + σ2 noiseI) + λ\|\|θ\|\|2 (14) 4.2 Gradient Projection Dynamic Inference Figure 2 (b) illustrates the testing process of our WBSR framework, we propose a gradient projection dynamic inference strategy to achieve a dynamic balance of efficiency and performance. It adaptively allocates the subnet model without any increase in additional parameters by calculating the gradient projection map based on the input content. Gradient Projection. We observe that patches with complex (simple) structures exhibit high (low) image gradient magnitude and do not suffer more (less) score degradation with SR scale changes. Following the approach described in Section 4.1.1, we calculate gradient vectors to measure the 6 complexity of the patch contents and construct a gradient projection map online to project the gradient vector of an image patch to the selection of each subnet model. At inference time, each patch can select a suitable subnet upon its gradient vector. When low-resolution noise exists in image patches, the edge detection methods [37, 8] ignore the local complexity of the patch and result in missed detections, thereby erroneously categorizing the patch as a simple sample. We count the changes in gradient strength by calculating the standard deviation directly, when there is a large amount of noise or texture changes of varying intensity in the local area of the patch, it can still be correctly assigned as a difficult sample. As shown in Figure 3, yellow boxes represent areas of local texture change, such as the clouds in the previous row and the railings in the next row. It can be intuitively seen that our gradient projection method can accurately distinguish local smooth regions or textured regions and assign them to the corresponding small or large subnets. Dynamic Inference. To facilitate the deployment of the model across any hardware resources, our dynamic supernet contains multiple subnets by gradually shrinking the model calculation with structured iteration to dynamically adapt various computational and performance requirements. During inference, we adopt the dynamic supernet to individually distribute image patches of K classes to M subnets to obtain better computational performance trade-offs. Given a new LR patch, we first calculate its gradient vector and derive its class ˆk according to the threshold t. Then, the selected subnet for inference can be easily obtained by equally splitting the gradient vector interval into a total of M subintervals, which can be expressed as (b) Gradient (c) Projected Patches (a) Edge Figure 3: Visualizations of (a) the edge detection results, (b) the gradient magnitude results, and (c) the projected subnet selection. For ease of observation, we visualize three assigned subnets with small, medium, and large computational costs as green, yellow, and red, respectively. m = ⌈ ˆk · M K ⌉ (15) where m ∈[1, M] denotes the index of the selected subnet to reconstruct this LR patch. ⌈·⌉indicates the ceiling function that tends to select the larger subnet. However, the larger subnet selection leads to better performance with heavier computation, we further consider selecting the inference subnet under the limited computational resources Ct ˆm = arg min m \|α· ˆk · Cm K −Ct\| (16) where ˆm indicates the selected optimal subnet under resource constraints. Cm denotes the computational cost of the m-th subnet. α is a hyperparameter that is utilized to strike a balance between the computational cost and performance, where higher values prioritize improved performance, while lower values favor reduced computational overhead. Consequently, our WBSR framework can be flexibly adjusted to accommodate diverse application scenarios based on actual performance and hardware resource requirements. 5 Experiments 5.1 Experimental Details Datasets and Metrics. Following the previous works [23, 4], we apply DIV2K [1] as the training dataset widely used for image SR, which includes 800 high-quality images with diverse contents and texture details. To verify the model performance under different image content distributions, four datasets are employed for model testing, including B100 [31], Urban100 [16], Test2K, and Test4K. Test2K and Test4K are downsampled from DIV8K [9]. For metrics, we adopt peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) to quantitatively evaluate all methods. Additionally, the FLOPs are calculated as the average results of all patches in the entire test dataset images. Implementation Details. Our proposed WBSR can be easily incorporated into existing CNN-based SR networks to achieve efficient inference. SRResNet [24] and RCAN [51] are selected as two baselines in our experiments for a fair comparison, we conduct extensive experiments on four datasets of different SR scales to verify the effectiveness of our framework. During training, we set nine 7 Scale Method #Pramas (M) B100 [31] Urban100 [16] Test2K [9] Test4K [9] PSNR↑ #FLOPs (G) PSNR↑ #FLOPs (G) PSNR↑ #FLOPs (G) PSNR↑ #FLOPs (G) ×2 SRResNet [24] 1.52 32.19 20.78 (100%) 32.11 20.78 (100%) 30.39 20.78 (100%) 31.90 20.78 (100%) +ClassSR [23] 3.12 31.68 14.75 (71%) 31.15 16.21 (78%) 30.24 14.13 (68%) 31.89 13.51 (65%) +ARM [4] 1.52 31.69 16.21 (78%) 31.16 16.83 (81%) 30.26 15.59 (75%) 31.90 13.71 (66%) +WBSR (Ours) 1.52 32.15 12.26 (59%) 31.98 13.30 (64%) 30.41 12.05 (58%) 32.02 12.68 (61%) RCAN [51] 15.59 32.40 130.40 (100%) 32.33 130.40 (100%) 30.86 130.4 (100%) 32.26 130.40 (100%) +ClassSR [23] 30.10 31.88 91.28 (70%) 31.72 103.02 (79%) 30.79 83.46 (64%) 32.24 83.46 (64%) +ARM [4] 15.59 31.89 99.10 (76%) 31.74 109.54 (84%) 30.80 105.62 (81%) 32.24 97.80 (75%) +WBSR (Ours) 15.59 32.34 88.67 (68%) 32.31 96.50 (74%) 30.91 75.63 (58%) 32.37 77.65 (60%) ×4 SRResNet [24] 1.52 27.34 5.19 (100%) 25.30 5.19 (100%) 26.19 5.19 (100%) 27.65 5.19 (100%) +ClassSR [23] 3.12 26.53 3.83 (74%) 24.53 4.23 (81%) 26.20 3.62 (70%) 27.66 3.30 (63%) +ARM [4] 1.52 26.53 4.34 (83%) 24.54 4.48 (86%) 26.21 3.76 (72%) 27.66 3.33 (64%) +WBSR (Ours) 1.52 27.36 3.99 (77%) 25.32 4.36 (84%) 26.26 3.37 (65%) 27.73 3.22 (62%) RCAN [51] 15.59 27.76 32.60 (100%) 25.82 32.60 (100%) 26.39 32.60 (100%) 27.89 32.60 (100%) +ClassSR [23] 30.10 26.70 22.82 (70%) 25.13 26.08 (80%) 26.39 21.22 (65%) 27.88 19.49 (60%) +ARM [4] 15.59 26.74 25.75 (79%) 25.14 28.36 (87%) 26.39 26.70 (82%) 27.88 25.10 (77%) +WBSR (Ours) 15.59 27.75 25.10 (77%) 25.81 27.01 (83%) 26.45 18.52 (57%) 27.94 19.40( 59%) Table 1: Quantitative comparison results of our method and other SOTA methods on the GoPro and H2D datasets. The optimal and suboptimal results are highlighted. subnets (i=9) with different parameters θi in each supernet. For SRResNet, the widths and depths of the subnets are set as ([36, 52, 64]) and ([4, 8, 16]), respectively. As for RCAN, the widths and depths of the subnets are configured as ([36, 52, 64]) and ([5, 10, 20]), respectively. The compared width adaptation algorithms [23, 4] also follow such model width configuration to ensure a fair comparison. All methods are implemented using PyTorch and trained on an NVIDIA GeForce RTX 3090 for 100 epochs with 16 batch sizes, where the first 70 epochs are sample-level sampling and the rest are class-level sampling. The former aims to maintain the original data distribution of the entire dataset, ensuring a stable and comprehensive feature representation. The latter focuses on correcting the imbalance of dataset and enhancing the model’s ability to represent difficult texture samples. The training times of SRResNet and RCAN are 25 and 28 GPU hours using a single GPU, respectively The training patch size is set to 128 × 128 and augmented by horizontal and vertical flipping to enhance its robustness. We utilize our Lbd loss along with the Adam optimizer [22], setting β1 = 0.9 and β2 = 0.999. To adjust the learning rate, we apply a cosine annealing learning strategy, starting with an initial learning rate of 2 × 10−4 and decaying to 10−7. 5.2 Comparison results Table 1 shows the quantitative performance of our approach coupled with various SR baselines in terms of the metrics, parameter number, and computational cost. ClassSR [23] with an additional classifier module has more parameters, resulting in additional computational and parameter costs. ARM [4] adopts the validation set to build an Edge-to-PSNR lookup table, which generates additional inference time and parameter storage overhead. In comparison, our framework achieves superior performance with less computation (average 62%) than baselines, without incurring additional parameters or computational costs. When tested on unseen datasets such as B100 and Urban100, which lie outside the trained distribution, the compared methods exhibit performance degradation due to overfitting to specific features of the original training dataset and a lack of generalization ability to diverse data. In contrast, our method maintains comparable performance to the original model with lower computational costs (averaging 70%), benefiting from our balanced sampling and optimization strategies during training. Figure 4 shows visual comparisons across four testing datasets. The SR images produced by ClassSR and ARM exhibit structural blur and noise. In contrast, our method recovers more detailed information, resulting in better outcomes that are more faithful to the HR ground truth. 5.3 Ablation Studies To verify the effectiveness of our WBSR upon HES and BDLoss, we conduct ablation studies on Tesk2K and Test4K with sclae factor of ×4, as shown in Table 2. Effectiveness of HES. During model training, we replace the original uniform sampling in baseline with our HES strategy. As the method “+HES” shown in Table 2, HES achieves a 0.05 dB average improvement in terms of PSNR compared with the baseline, which benefits from enhanced feature representations in hard texture patches. 8 ClassSR ARM Ours HR ClassSR ARM Ours HR B100 LR Urban100 LR ClassSR ARM Ours HR Test2K LR ClassSR ARM Ours HR Test4K LR Figure 4: Qualitative comparison results of our method with other methods for × 4 SR on the four testing datasets. Please zoom in for details. Table 2: Ablation studies of our WBSR on two benchmarks of × 4 SR. † indicates using the whole network with 100% FLOPs for inference. The optimal and suboptimal results are highlighted. Method (×4) Test2K [9] Test4K [9] PSNR ↑ SSIM↑ #FLOPs (G) PSNR↑ SSIM↑ #FLOPs (G) SRResNet [24] 26.19 0.7624 5.19 (100%) 27.65 0.7966 5.19 (100%) +HES 26.24 0.7665 3.58 (69%) 27.71 0.7986 3.43 (66%) +Lbd 26.21 0.7658 3.58 (69%) 27.70 0.7984 3.43 (66%) +WBSR 26.26 0.7673 3.37 (65%) 27.73 0.7993 3.22 (62%) +WBSR† 26.38 0.7684 5.19 (100%) 27.80 0.8026 5.19 (100%) RCAN [51] 26.39 0.7706 32.60 (100%) 27.89 0.8058 32.60 (100%) +HES 26.43 0.7748 20.86 (64%) 27.92 0.8086 19.89 (61%) +Lbd 26.42 0.7746 20.86 (64%) 27.91 0.8077 19.89 (61%) +WBSR 26.45 0.7755 18.52 (57%) 27.94 0.8106 19.40 (59%) +WBSR† 26.51 0.7756 32.60 (100%) 28.10 0.8138 32.60 (100%) Furthermore, we conduct additional experiments to compare our HES with existing sampling works [41, 49, 29] in Table 3. It shows that our HES outperforms the previous best sampling method BSPA of an average of 0.1dB in terms of PSNR and demonstrates the superiority and generalization capabilities of our HES. In “+WBSR†”, we can achieve even greater performance gains of 0.18 dB by integrating our HES with our BDLoss. HES first performs more sample-level sampling to learn generalized feature representations followed by fewer selective class-level sampling to focus on texture-rich regions to correct sample bias with stable learning and prevent the model’s overfitting, which mitigates the model oscillation and addresses the overfitting problem. Furthermore, our HES achieves balanced stable training with our BDLoss for diverse samples in each training step, which solves the instability and training bias issues. Effectiveness of BDLoss. To demonstrate the effect of BDLoss, we train the SR model using uniform sampling and replace only the L1 loss with Lbd. As shown for method “+Lbd ” in Table 2, the PSNR 9 Table 3: Quantitative comparison results of our method with other sampling strategies. Method B100 [31] Urban100 [16] PSNR↑ SSIM↑ #FLOPs (G) PSNR↑ SSIM↑ #FLOPs (G) RCAN 27.40 0.7306 32.60 (100%) 25.54 0.7684 32.60 (100%) +BSPA [29] 27.54 0.7348 32.60 (100%) 26.02 0.7839 32.60 (100%) +SamplingAug [41] 27.47 0.7323 32.60 (100%) 25.80 0.7771 32.60 (100%) +DDA [49] 27.51 32.60 (100%) 25.89 32.60 (100%) +HES 27.73 0.7388 32.60 (100%) 26.04 0.7863 32.60 (100%) +WBSR† 27.81 0.7402 32.60 (100%) 26.10 0.7889 32.60 (100%) +WBSR 27.77 0.7391 26.41 (81%) 26.03 0.7850 29.67 (91%) improved by balanced training is 0.06 dB with an average 65% computing cost compared to the baseline model, which demonstrates the superiority of our BDLoss. Effectiveness of Joint Training. When applying joint training of HES and BDLoss within WBSR to the baseline network, we can further improve the PSNR and SSIM results by 0.13 dB and 0.0043, respectively, which achieve an overall performance improvement with average 66% calculation. Furthermore, to fully demonstrate the effectiveness of our WBSR, we adopt the weight-balancing framework to retrain the full baseline model instead of the dynamic supernet model. It can be seen from the “+WBSR†”method of Table 2, our WBSR with a computational cost comparable to baseline obtains average PSNR and SSIM gains of 0.33 dB and 0.0078, respectively. Models trained using our WBSR show consistent performance improvements that are not affected by the skewness of the training sample distribution. In Figure 5, the SR performance on rare samples obtains gains, while the performance on abundant samples remains the same or slightly decreases, which proves that our weight-balancing strategy not only enhances the learning of texture areas and reduces redundant computation of flat areas. Additional experimental results are placed in supplementary materials. 6 Conclusion 1 2 3 4 5 n ··· Naïve Weights 1 2 3 4 5 n ··· Weight Balancing 1 2 3 4 5 n ··· Balanced Weights Naïve Performance Gain w/ WBSR (66%) Categories PSNR (dB) Figure 5: Illustration of the gain of our weightbalancing framework relative to the baseline model and its weight rectification diagram. In this paper, we rethink the imbalance problem in image SR from a statistical probability perspective and propose a plug-and-play WeightBalancing framework (WBSR) to achieve balanced model learning without changing the original model structure and training data. Specifically, to tackle the imbalance problem of data distribution, we propose a Hierarchical Equalization Sampling strategy (HES) to enhance the model’s capability to extract features from difficult samples to mitigate inherent data biases. Then, to solve the imbalance problem of model optimization, we propose a Balanced Diversity Loss (BDLoss) function to focus on learning texture regions and ignore redundant computations in those smooth regions. After joint training of HES and BDLoss, our WBSR rectifies the imbalance to achieve accurate and efficient inference via a gradient projection dynamic inference strategy. Extensive qualitative and quantitative experiments across various models, datasets, and scaling factors demonstrate that our method achieves comparable or superior performance to existing approaches with less computational cost. 7 Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grants 62272134 and 62072141, in part by the National Science and Technology Major Project under Grant 2021ZD0110901. 10 References [1] Eirikur Agustsson and Radu Timofte. NTIRE 2017 challenge on single image super-resolution: Dataset and study. In CVPRW, pages 1122–1131, 2017. 2, 7 [2] Jonathon Byrd and Zachary Lipton. What is the effect of importance weighting in deep learning? In International conference on machine learning, pages 872–881. PMLR, 2019. 3 [3] Kaidi Cao, Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning imbalanced datasets with label-distribution-aware margin loss. 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Real-time image demoireing on mobile devices. arXiv preprint arXiv:2302.02184, 2023. 1 [55] Yang Zhou, Yuda Song, Hui Qian, and Xin Du. Classpruning: Speed up image restoration networks by dynamic n: M pruning. arXiv preprint arXiv:2211.05488, 2022. 1 14 A Derivations and Proofs In this supplementary material, we give a detailed derivation of Theorem 1 from Eq. 4 to Eq. 5 ptrain(y\|x) = ptrain(y\|x) R Y ptrain(y′\|x)dy′ = pbal(y\|x) · pbain(y) pbal(y) · pbal(x) pbain(x) R Y pbal(y′\|x) · pbain(y′) pbal(y′) · pbal(x) pbain(x)dy′ = pbal(y\|x) · pbain(y) pbal(y) R Y pbal(y′\|x) · pbain(y′) pbal(y′) dy′ = pbal(y\|x) · ptrain(y) R Y pbal(y′\|x) · ptrain(y′)dy′ (17) Then, we give a derivation from Eq. 13 to Eq. 14. Since si in Eq. 13 indicates the norm of the product of multi Gaussians, its inherent characteristic can also be delineated as a Gaussian distribution si = N(ˆy; µi, Σi + σ2 noiscI) (18) where N denotes the probability density function of a Gaussian distribution parameterized by a mean vector µi and a covariance matrix Σi + σ2 noiscI. I denotes the identity matrix, ensuring that the covariance matrix remains positive definite. Building upon this foundation, the weighted Gaussian terms can be expressed as L X i=1 ϕisi = L X i=1 ϕi Z Y N(ˆy; µi, Σi + σ2 noiscI)dy (19) where ϕi denotes the weight of individual Gaussian components. This weighting mechanism allows the model to flexibly learn balanced optimization of parameters based on the different data characteristics. Subsequently, we substitute Eq. 19 and Eq. 12 into Eq. 13 to derive the pivotal diversity balancing term, which integrates the principles of predictive uncertainty with the underlying training distribution log Z Y N(y′; ˆy, σ2 moiscI) · ptrain(y′)dy′ = log L X i=1 ϕi · N(ˆy; µi, Σi + σ2 noiseI) (20) Finally, we integrate Eq. 20 into the our Lbd formula from Eq. 11 to obtain the final loss function expression Lbd = −log N(y; ˆy, σ2 noiseI) + log L X i=1 ϕi · N(ˆy; µi, Σi + σ2 noiseI) + λ\|\|θ\|\|2 (21) where balance weights θ are rectified through the joint optimization of two key terms with the loss: the diversity balancing term to ensure balanced optimization, and the regularization term to prevent overfitting. B Ablation Studies Visualization comparison of ablation. To further verify the feature representation ability of our BDLoss on hard texture patches, we also visualize the error map of SR results with GT images in Figure 6. As can be seen from the yellow box area, the error of the model trained with Lbd on the texture area is significantly smaller than that of L1, indicating that our Lbd function improves the fitting accuracy in challenging texture regions. In addition, the effectiveness of our gradient projection strategy can also be proved according to the subnet allocation map, which reasonably allocates large subnets (i.e., green masks) to complex texture areas. This demonstrates that our gradient projection strategy can effectively enhance the model’s ability to handle complex textures. 15 M 3 6 9 12 16 W&D 3&1 3&2 3&3 4&3 4&4 PSNR↑ 26.28 26.26 26.26 26.13 26.12 SSIM↑ 0.7678 0.7675 0.7673 0.7657 0.7654 Table 4: Influence of different numbers M of subnets. W represents the number of subnet widths, and D represents the number of subnet depths, constituting subnets with varying parameters and computational costs. Error map w/o BDLoss Error map w/ BDLoss Subnet Allocation Map Figure 6: Visual comparison of error map from SR model trained by L1 and our Lbd. Analysis of the number M of subnets. We verify the impact of the number of subnets on network inference performance in Table 4. As the number of subnets increases, the inference effect will decrease to a certain extent because the fixed training epochs cause the extra subnets to not be fully trained. Increasing the number of training epochs will solve this problem to a certain extent. However, to achieve a trade-off between performance and efficiency, we adopt a total of nine subnet branches of three widths and three depths to process different image patches. Analysis of the class k of samples. To analyze the impact of the number of sample categories, we visualize the variation curves of computational cost GFLOPs and performance PSNR under different K classes in Figure 7. When the K = 5 value is small, the results of gradient vector projection of GFLOPs PSNR (dB) Figure 7: Effectiveness of different class values k of samples. different patches are too concentrated, which will lead to inaccurate network division. Increasing the value of K results in a more suitable subnet selection. Considering the memory limitation, we set the 16 number of batches to 16 in per training step, and the number K of class at each training step in all experiments is set to 10 to achieve a better trade-off of performance and efficiency. C More Qualitative Results In Figures 8 - 11, we present more visual comparison results of our WBSR and other SOTA methods [23, 4] on different testing datasets at different scales. It can be seen that our algorithm can accurately reconstruct more spatial structures and more texture details and other algorithms suffer from loss of detail on difficult texture patches, which demonstrates the superiority of our algorithm. ClassSR ARM Ours HR B100 LR Figure 8: Qualitative comparison results of ×4 on B100 [31] dataset between our method and other methods. Please zoom in for details. ClassSR ARM Ours HR Urban100 LR Figure 9: Qualitative comparison results of ×2 on Urban100 [16] dataset between our method and other methods. Please zoom in for details. 17 ClassSR ARM Ours HR Tesk2K LR Figure 10: Qualitative comparison results of ×4 on Test2K [9] dataset between our method and other methods. Please zoom in for details. ClassSR ARM Ours HR Tesk4K LR Figure 11: Qualitative comparison results of ×2 on Test4K [9] dataset between our method and other methods. Please zoom in for details. 18 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The abstract and introduction accurately reflect the paper’s contributions and scope. 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4,457	Perceiving Longer Sequences With Bi-Directional Cross-Attention Transformers Markus Hiller, Krista A. Ehinger, and Tom Drummond School of Computing and Information Systems The University of Melbourne, Australia m.hiller@unimelb.edu.au Abstract We present a novel bi-directional Transformer architecture (BiXT) which scales linearly with input size in terms of computational cost and memory consumption, but does not suffer the drop in performance or limitation to only one input modality seen with other efficient Transformer-based approaches. BiXT is inspired by the Perceiver architectures but replaces iterative attention with an efficient bidirectional cross-attention module in which input tokens and latent variables attend to each other simultaneously, leveraging a naturally emerging attention-symmetry between the two. This approach unlocks a key bottleneck experienced by Perceiverlike architectures and enables the processing and interpretation of both semantics (‘what’) and location (‘where’) to develop alongside each other over multiple layers – allowing its direct application to dense and instance-based tasks alike. By combining efficiency with the generality and performance of a full Transformer architecture, BiXT can process longer sequences like point clouds, text or images at higher feature resolutions and achieves competitive performance across a range of tasks like point cloud part segmentation, semantic image segmentation, image classification, hierarchical sequence modeling and document retrieval. Our experiments demonstrate that BiXT models outperform larger competitors by leveraging longer sequences more efficiently on vision tasks like classification and segmentation, and perform on par with full Transformer variants on sequence modeling and document retrieval – but require 28% fewer FLOPs and are up to 8.4× faster. 1 1 Introduction Much of the data we obtain when perceiving our environment can be interpreted via a division into ‘what’ and ‘where’. If we consider for example the image pictured in Figure 1 on the left, we can easily describe its content by ‘what’ we see – the building, sky and a flag. If we were to draw conclusions on a more fine-grained level though, we would likely include more specific descriptions like “lower left corner” referring to their positions within the image – the ‘where’. In other words, ‘where’ denotes the actual geometric location of the individual elements (e.g. pixels) and ‘what’ the semantic entities (e.g. objects) that collectively describe the data as a whole. Note that this similarly applies to many other modalities, like point clouds or even language where we form words via letters that together have a certain meaning. Thanks to the few structural constraints placed on the input data paired with high performance, Transformers [44] have shown great capabilities in extracting both ’what’ and ’where’ for a range of input modalities, giving rise to significant advances across various fields such as Natural Language Processing [9] and Computer Vision [10, 41, 42]. However, their success comes at the high cost of scaling quadratically in memory and time with the input length, practically prohibiting their use on 1Code and models are publicly available at https://github.com/mrkshllr/BiXT. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). (a) (b) Seq: lat →tok (c) Seq: lat ←tok (d) Ours: lat ←→tok Figure 1: Emerging patterns when attending both ways. (a) Input image. (b) depicts the areas of the image that 4 different latents attend to, while (c) inversely shows which image regions attend to these latents (transformed into the same coordinate system for ease of interpretation). (d) displays which areas & latents are symmetrically attended to using our proposed bi-directional cross-attention. larger input data like point clouds, long documents, or high-resolution images when computational resources are limited. Several approaches have since been proposed to increase their efficiency, either by changing how the computationally expensive self-attention operation is realized [37, 45] or by exploiting the domainspecific structure of their data input [17, 29, 34, 43]. However, all these either face a reduction in the Transformer’s performance or limit its application to only one specific type of input [11]. In an attempt to preserve the generality by not imposing additional constraints on the input data, Jaegle et al. [18] employ a small set of latent vectors as a bottleneck to extract the ‘what’ via one-sided (iterative) cross-attention – and require an additional decoder to draw conclusions about ‘where’ [19]. While achieving linear complexity w.r.t. the input length, these ‘Perceiver’ architectures require between 360 - 707 GFLOPs to achieve around 78% accuracy on ImageNet1K – results that recent Transformer variants like ViT [41, 42] are able to obtain at a fraction of the compute. One possible explanation for this discrepancy is that the effective working memory of Perceiver architectures is strictly limited to the latents which therefore need to compensate via increased computation, whereas conventional Transformers like ViTs leverage the (larger) number of tokens across several layers. This raises an important question: Are the appealing individual properties of these two methods mutually exclusive, or can we in fact have the best of both worlds? In this paper, we set out to affirm the latter. We demonstrate that a small set of latent vectors appropriately combined with layerwise simultaneous refinement of both input tokens and latents makes it possible to pair the high performance and architectural simplicity of Transformers with the linear scaling of Perceivers – outperforming both in settings where compute is limited. We start off by investigating a naïve approach: sequentially applying cross-attention to refine ‘what’ and ‘where’, one after the other. We discover that approximately symmetric attention patterns naturally emerge between latents and tokens even when both are provided with complete flexibility. In other words, for most latents (‘what’) that pay attention to particular tokens (‘where’), these tokens in turn pay attention to exactly these latents (see Figure 1 and Section 3.1). Not only does this intuitively make sense – objects need to know ‘where’ they are located in the image, and image locations need to know ‘what’ objects are located there – it more importantly offers us a unique opportunity to save FLOPs, memory and parameters. As we will demonstrate in Section 2, this approximate symmetry means we only need to compute the attention matrix once, reducing the involved parameters by ∼1/3 to facilitate an efficient bidirectional information exchange via our proposed bi-directional cross-attention. Integrated into our bi-directional cross-attention Transformer architecture (BiXT), this forms a flexible and highperforming yet efficient way to process different input modalities like images, point clouds or text on a variety of instance-based (e.g. classification) or dense tasks (e.g. segmentation) – all while scaling linearly w.r.t. the input length. In summary, our main contributions include the following: 1. We introduce a novel bi-directional cross-attention Transformer architecture (BiXT) that scales linearly with the input size in terms of computational cost and memory consumption, allowing us to process longer sequences like point clouds, text or images at higher resolution. 2 2. We propose bi-directional cross-attention as an efficient way to establish information exchange that requires computation of the attention matrix only once and reduces the involved parameters by ∼1/3, motivated by a naturally emerging symmetry in cross-attention and showing significant improvements over uni-directional iterative methods like Perceiver. 3. We analyze BiXT’s advantage of processing longer sequences across a number of tasks using different input modalities and output structures in settings with limited computational resources – with our tiny 15M parameter model achieving accuracies up to 83.1% for classification on ImageNet1K without any modality-specific internal components, performing competitively for semantic image and point cloud part segmentation even among modality-specific approaches, and being up to 28% more efficient and 8.4× faster on LRA. 4. We further provide insights into BiXT’s extendibility: Thanks to its simple and flexible design, modality-specific components can easily be incorporated in a plug-and-play fashion should the need arise – further improving results while trading off generality. 2 Perceiving via Bi-Directional Cross-Attention We start this section by briefly revisiting the concept of attention before moving on to presenting our proposed bi-directional cross-attention methodology, followed by its use within our BiXT architecture (Figure 2). Please note that we define the concepts using single-head attention for brevity instead of the actually employed multi-head attention (MHA), and all methods directly generalize to MHA. 2.1 Background: The Attention Mechanism While self-attention has recently gained great popularity through its use in the Transformer architecture [44], we will start from a slightly more general point of view: Given a source sequence S ∈RN×DS and a target sequence T ∈RM×DT, attention aims to refine T by exhaustively discovering pairwise correlations between all elements of both sequences and integrating information from the source components of interest into the target. Formally, S is linearly projected into two D-dimensional representations using learnable matrices – yielding a key KS ∈RN×D and value V S ∈RN×D – while T is projected into one D-dimensional representation to obtain the query QT ∈RM×D. These representations are then used to compute the attention-based target refinement as ∆attn T =attn (QT , KS, V S)=softmax QT KT S √ D ! · V S, (1) with the scaled dot product ¯AT,S = 1/ √ D (QT KT S) ∈RM×N representing the scaled pairwise similarity between target and source elements. This concept is commonly referred to as crossattention (CA) between target T and source S. If a representation itself is to be refined given the context within, i.e. source and target are identical (S =T ), Equation (1) reduces to the well-known self-attention where the triplet key, query and value are all generated as a function of the same sequence elements. Note that computing the similarity matrix ¯AT,S has computational complexity O(NM). For selfattention used in Transformers where T = S and hence M = N, this yields quadratic complexity O(N 2) w.r.t. the input sequence length N, prohibiting its use on longer sequences when computational resources are limited. On the other hand, if cross-attention is employed with a fixed sequence length M =const ≪N, the complexity becomes linear O(N). 2.2 Bi-Directional Cross-Attention Reducing the complexity of attention from quadratic to linear without impairing performance or adding constraints w.r.t. input modalities is one of the main aspects of this work. We build our approach on the previously introduced notion that most data can be interpreted as ‘what’ and ‘where’ – and both need to pay attention to the other for optimal information exchange. We represent the ‘what’ via a small set of M learnable latent vectors and the ‘where’ via an input-dependent sequence of N tokens, respectively denoted via the subscripts lat and tok in the following and with M ≪N. Naïvely, one could simply apply two individual cross-attention operations sequentially – first querying 3 (optional) Input Tokenizer Latent Self-Attention (optional) Token Refinement ×L Learnable latent vectors (tokenized) input data Images Point Clouds … ‘What?’ ‘Where?’ Bi-Directional Cross-Attention Bi-Directional CrossAttention Learnable latent vectors (tokenized) input data * … 𝑁 𝑀 + + Norm Norm MLP MLP + + Row-wise Softmax Column-wise Softmax Norm Norm * Figure 2: BiXT architecture. (left) Input data passing through one layer of our Bi-Directional CrossAttention Transformer. (right) Internal structure of proposed efficient bi-directional cross-attention. information from one side and then the other by creating two query-key-value triplets. However, our analyses in Section 3.1 show that symmetric tendencies in the attention patterns between latents and tokens naturally emerge during training, offering a chance to further reduce the computational requirements and to increase efficiency via our bi-directional cross-attention as follows. We start by creating reference-value pairs Rlat ∈RM×D, V lat ∈RM×D and Rtok ∈RN×D, V tok ∈ RN×D via learnable linear projection from the latent vectors and tokens, respectively. Leveraging symmetry to create bi-directional information exchange, pairwise similarities between latents and tokens are then computed via a scaled dot product as ¯Alat,tok = RlatRT tok √ D ! = ¯AT tok,lat, (2) which is in turn used to obtain the attention-based refinement for both, the latents and tokens, via ∆attn lat = softmax ¯Alat,tok · V tok and ∆attn tok = softmax ¯Atok,lat · V lat. (3) Note that in addition to providing linear scaling w.r.t. to the input length N, Equation (2) requires evaluating the most computationally-expensive operation, namely the similarity matrix (O(MN)), only once and allows simultaneous refinement of latents and tokens as defined in Equation (3). The implicit reuse of the references as both query and key further reduces the parameter count of the linear projection matrices by 1/3 compared to naïve sequential cross-attention. 2.3 BiXT – Bi-Directional Cross-Attention Transformers Figure 2 (left) illustrates the individual components that make up our BiXT architecture. BiXT is designed in a simple symmetric, ladder-like structure allowing ‘what’ (latent vectors) and ‘where’ (tokens) to simultaneously attend to and develop alongside each other – making it equally-well suited for instance-based tasks like classification and dense tasks like semantic segmentation on a variety of input modalities. We start this section with a brief overview, followed by more detailed descriptions of the individual components. General overview. The raw input data is first passed through a tokenization module which projects the data into an embedding sequence of length N and optionally adds positional encodings, depending on the input modality and data structure. These tokens together with a fixed set of M learnable latent vectors are then passed to the first layer’s bi-directional cross-attention module for efficient refinement (details depicted in Figure 2 (right) and explained below). The latents are then further refined via latent self-attention, while the tokens are either directly passed on to the next layer (default) or optionally refined by a token refinement module which could include modality-specific components. The simultaneous ladder-like refinement of ‘what’ and ‘where’ is repeated for L layers, before the result is passed to task-specific output head(s). For instance-based tasks like classification, we simply average the set of latent vectors and attach a classification head to the output, while for tasks like segmentation that require outputs resembling the input data structure, the refined tokens are used. Efficient bi-directional information exchange. We use bi-directional cross-attention introduced in Section 2.2 to enable M latents and N tokens to simultaneously attend to each other in a time and memory efficient way, provided M ≪N. The detailed internal structure of our module is depicted in 4 Figure 2 (right) and defined via Equations (2) and (3). Apart from the efficient bi-directional attention computation, it follows the common Transformer-style multi-head attention in terms of normalization, activations and processing via feed-forward networks (FFN) introduced by Vaswani et al. [44] and can thus be easily implemented in modern deep learning frameworks. Three aspects are particularly worth noting here: 1) While our bi-directional attention imposes a ‘hard’ structural constraint of symmetry on the pair-wise similarity matrix between tokens and latents as defined in Equation (2), the actual information exchange is less strict: applying the row-wise and column-wise softmax operations to obtain the actual attention maps offers a certain degree of flexibility, since adding a constant to each element in a row keeps the resulting (latent) attention map unchanged while modulating the column-wise (token) one, and vice versa. More specifically, bi-directional CA between M latents and N tokens has in total MN −1 degrees of freedom (dof), only (M −1)·(N −1) of which are shared – leaving M +N −2 dof that can be used by the network for the modulation of the (non-strictly-symmetric) information exchange (see Appendix A.3 for detailed discussion). 2) Even if the latents and tokens symmetrically attend to each other, the actual information that is transferred is created via individual value projection matrices and thus offers flexibility in terms of content. 3) While tokens cannot directly communicate with each other as is possible when using computationally expensive self-attention, this communication can still take place over two layers in our structure by using a latent vector as temporary storage in a token-latent-token sequence. Since the total number of latents is usually larger than the semantic concepts required to describe one data sample, we can expect this to be possible without impairing performance. Latent vector refinement. After gathering information from the tokens, we use one multi-head self-attention operation [44] to further refine the information stored in the latents and provide direct information exchange with a global receptive field across latents. Note that since the number of latents M is fixed and significantly smaller than the input sequence, this operation is input-length independent and not particularly resource intensive. This step is similar to Perceiver [18, 19], but we only use one instead of several self-attention operations at each layer. Optional token refinement. In the majority of experiments presented in this paper, we simply pass the tokens returned by the bi-directional cross-attention to the next layer. However, our architectural structure also allows to easily include additional (e.g. data-specific) modules for further refinement in a plug-n-play manner. We demonstrate examples of this in Section 3, where we add a local refinement component exploiting grid-shaped data for semantic segmentation and a data-specific hierarchical grouping module for point cloud shape classification. Positional encodings. We use additive sinusoidal positional encodings [44] to represent the structure of input data, which is more efficient than learnt encodings for variable input size. For simplicity, we follow previous works [11] and create the encodings in 32 dimensions per input axis followed by a linear projection into the model’s token dimension D. This method is applicable independent of the raw data’s dimensions and thus easily handles data ranging from 2D images to 3D or 6D point clouds. Input tokenization. Tokenization can be performed in various ways and is the only input modalityspecific component in our architecture, akin to Perceiver-IO’s input adapters [19]. For image-based experiments, we follow common practice and use simple linear projection as our default tokenizer to embed image patches. For point cloud data, we simply encode the 3D or 6D points directly into embedding space using our sinusoidal positional encoder. We adhere to the guidelines of Tay et al. [40] for text-based hierarchical sequence modelling and document retrieval experiments on LRA. 3 Experimental Evaluation The purpose of our investigations presented in the following is twofold: 1) To provide qualitative and quantitative insights into our proposed bi-directional cross-attention and the underlying intuition of symmetry, and 2) to demonstrate how BiXT’s ability to efficiently and effectively process longer sequences positively affects various tasks. We focus the majority of our experiments around efficient architectures in the low FLOP, memory and parameter regime, and unless otherwise stated, we use BiXT-Ti with 64 latent vectors, embedding dimension 192 and 6 heads for all attention modules. 3.1 Symmetric Tendencies Emerge when Attending Both Ways We start by investigating the intuition underlying our work: When describing data like an image by asking ‘what’ is in it and ‘where’ things are, it intuitively makes sense that these two components are tightly interconnected, and that they will inform aka pay attention to each other. To this end, we set 5 Table 1: Bi-directional vs. iterative attention. (a) Classification accuracy on ImageNet1K. All architectures use 64 latent vectors and have been trained for 120 epochs with hyperparameters individually optimized. Architectural configurations noted in brackets. †indicates sharing of all, ‡of all but the 1st layer’s cross-attention parameters. Results reported as mean and (unbiased) std-dev over 3 randomly seeded training runs (see appendix for complete results). (b) Point cloud shape classification on ModelNet40. BiXT without (naïve) and with modality-specific components. (a) ImageNet1K @ 120epochs. Attention Top-1 Acc. FLOPs Mem. #Param Perceiver-like Iterative‡ (sa5-d8) 58.26 ± 2.34 1.58G 7.17M 19.05M Iterative‡ (sa6-d7) 54.94 ± 5.96 1.59G 7.23M 19.94M Iterative† (sa6-d8) 60.61 ± 1.11 1.82G 8.25M 22.16M Iterative† (sa4-d12) 56.03 ± 1.02 1.99G 9.10M 22.16M Iterative† (sa1-d24) 55.92 ± 0.67 1.79G 8.39M 11.93M Cross-Attn. Sequential (2-way, d11) 73.10 ± 0.53 1.66G 8.44M 14.60M Bi-Directional (d12) 73.86 ± 0.39 1.68G 7.86M 15.12M Sequential (2-way, d12) 73.79 ± 0.32 1.81G 9.24M 15.94M Bi-Directional (d13) 74.10 ± 0.14 1.82G 8.54M 16.38M (b) ModelNet40. Method OA mAcc Naïve, point-based PointNet Qi et al. [32] 89.2 86.0 Perceiver Jaegle et al. [18] 85.7 – BiXT (naïve) 89.6 86.4 Hierarchical, point grouping, etc. PointNet++ Qi et al. [33] 90.7 – PointMLP Ma et al. [25] 94.1 91.3 BiXT (+ grouping) 92.5 89.7 BiXT (+ grouping & hierarchy) 93.1 90.6 up a naïve architecture where latent vectors first query the tokens via cross-attention (CA), followed by the tokens querying the latents (i.e. using independent query-key-value triplets), before a further refinement step of the latent information via one self-attention operation – repeated over multiple layers and trained on ImageNet1K [36]. When looking at the resulting attention patterns depicted in Figure 1, we discover that most latents pay attention to parts of the image representing one specific ‘entity’ like a building ((b), top-left), a flag ((b), top-right) or parts of the sky ((b), lower-right) – supporting the notion that latent vectors represent ‘things’. More interestingly however, we discover in (c) that most of these image regions (tokens) are in turn also paying attention to exactly these latent vectors – showing a roughly symmetric information exchange and providing a qualitative indication that our idea of leveraging symmetry via our bi-directional architecture might be well justified. We additionally visualize the attention patterns after replacing the naïve sequential CA through our efficient bi-directional one in (d), and the results look surprisingly similar – clearly indicating that our symmetrically constrained approach can achieve similar information exchange while being significantly more efficient. 3.2 Attention – Iterative, Sequential or Bi-directional? We aim to provide conclusive insights about the two major advantages of our proposed bi-directional attention compared to Perceiver’s iterative attention: 1) Higher performance for comparable numbers of FLOPs, and 2) Ability to optionally extend the architecture via modality-specific components. We therefore choose two tasks that have also been investigated in the Perceiver paper: Image classification on ImageNet1K [36] and point cloud shape classification on ModelNet40 [49]. ImageNet classification. To provide a fair basis for comparison, we create a range of architectural configurations with iterative attention based on the insights reported by Jaegle et al. [18]. Targeting a similar FLOP count as our BiXT tiny, we experiment with different numbers of layers, varying numbers of self-attention operations per block and with sharing all CA parameters as well as all but the first layer’s (for details, see Perceiver paper and our appendix) – yielding a total of 10 architectures based on Perceiver’s iterative attention. Having optimized the hyperparameters (learning rate and schedule) for each individually, we run 3 randomly seeded training runs for the best 5 configurations and report their results after training for 120 epochs in Table 1 (a) together with BiXT and the naïve sequential CA variant. It is apparent that removing the bottleneck of iterative attention significantly boosts the performance, with both BiXT and sequential CA outperforming all iterative variants by a significant margin at comparable FLOP counts. Interestingly, we find the configuration with 8 blocks and 6 self-attention layers per block (sa6-d8) to achieve best performance among the iterative variants, which aligns with the ‘best’ configuration reported by Jaegle et al. [18]. Contrasting the two CA-based approaches with identical numbers of layers (‘d12’) demonstrates the clear advantage of our proposed bi-directional CA, requiring ∼7% fewer FLOPs, ∼15% less memory and 5% fewer parameters to achieve similar results as the sequential variant. This allows BiXT to use one additional layer at matching FLOP count, consistently outperforming the naïve approach across all our experiments while being still 7–8% more memory efficient. 6 Point cloud shape classification. To gain further quantitative insights how bi-directional attention affects processing of other modalities, we evaluate our approach on the ModelNet40 dataset [49]. BiXT again clearly outperforms Perceiver in terms of overall accuracy (OA) and is even competitive to other point-based methods like the seminal PointNet [32] (Figure 2 (b)). In contrast to Perceiver’s iterative attention that gathers information exclusively in the latents, BiXT’s simultaneous refinement of latents and tokens allows us to easily integrate data-specific modules for token refinement. To gauge the effect, we add the ‘affine grouping’ module from PointMLP [25] without and with hierarchical structure (i.e. point reduction). While BiXT is still outperformed by point cloud specific PointMLP, these optional modules help to boost the accuracy by up to 3.9% while trading off generality. 3.3 Image Classification Table 2: Classification on ImageNet1K using ‘few-FLOP’ Transformers. Note that we focus here on efficient models in the low FLOP and/or parameter regime. Perceiver architectures are included as contrast to our bi-directional attention. All methods have been trained on input resolutions of 2242, and ↑384 further fine-tuned on 3842. Note that different models may have received a different optimization effort. ∗result reproduced as not reported in original work. ‘(conv)’ indicates the use of a convolutional tokenizer (see appendix for details). Architecture FLOPs #Param Acc. ‘Generalists’ – no tokenizer, no vision-specific internals Perceiver Jaegle et al. [18] 707G 45M 78.0 Perceiver v2 Jaegle et al. [19] 404G 42M 78.6 Perceiver-IO Jaegle et al. [19] 407G 48M 79.0 ‘Vanillas’ – tokenizer, but no vision-specific internals Perceiver v2 (conv) Jaegle et al. [19] 367G 42M 77.4 Perceiver-IO (conv) Jaegle et al. [19] 369G 49M 82.1 DeiT-Ti/16 Touvron et al. [41] 1.3G 6M 72.2 DeiT3-Ti/16∗Touvron et al. [42] 1.3G 6M 75.4 BiXT-Ti/16 1.7G 15M 80.1 BiXT-Ti/16 (conv) 1.7G 15M 81.0 Vision-specific derivatives, incl. multi-scale / pyramidal PiT-Ti Heo et al. [16] 0.7G 5M 73.0 PiT-XS Heo et al. [16] 1.4G 11M 78.1 ViL-Ti-APE Zhang et al. [55] 1.3G 7M 76.3 ViL-Ti-RPB Zhang et al. [55] 1.3G 7M 76.7 PVTv1-Ti Wang et al. [46] 1.9G 13M 75.1 PVTv2-B1 Wang et al. [47] 2.1G 13M 78.7 XCiT-T12 El-Nouby et al. [11] 1.2G 7M 77.1 XCiT-T24 El-Nouby et al. [11] 2.3G 12M 79.4 BiFormer Zhu et al. [57] 2.2G 13M 81.4 Going finer w/ BiXT – smaller patches, larger images BiXT-Ti/8 [seq-len: 784] 4.7G 15M 81.9 BiXT-Ti/4 [seq-len: 3,136] 16.8G 15M 82.7 BiXT-Ti/16 ↑384 [seq-len: 576] 3.6G 15M 81.8 BiXT-Ti/8 ↑384 [seq-len: 2,304] 12.5G 15M 82.8 BiXT-Ti/4 ↑384 [seq-len: 9,216] 48.1G 15M 83.1 Comparison to SOTA. Note that we focus here on efficient Transformer models in the low FLOP and/or parameter regime, with results reported in Table 2. BiXT performs favourably with default and convolutional tokenizer against the other ‘vanilla’ Transformers, outperforming both versions of DeiT by a significant margin (6.2 – 11.8%) while being ∼200× more efficient than Perceiver (IO). These results are highly competitive even when compared to specialized visiononly architectures that leverage complex pyramidal multi-scale techniques, with BiXT outperforming all but one very recent method (which however requires 29% more FLOPs than our BiXT). Increasing feature resolution and input size. We keep the patch size fixed to 162 while reducing the stride of our linear patch projector to increase feature resolution (see appendix for ablation on patch sizes vs. stride). Note that our BiXT/4 model can easily process 3,136 tokens per 2242 image thanks to linear scaling, boosting the top-1 accuracy to 82.7%. Linear scaling also lets us process larger input images more efficiently – which we investigate by fine-tuning on 3842 for 30 epochs to reduce the required computational resources. Increasing the input size further notably improves the accuracy across architectures by up to 2.1%, however at the expense of higher FLOP counts. Nevertheless, BiXT shows that it is possible to achieve 83.1% on ImageNet with only 15M parameters and no vision-specific internals. Longer sequence beats model size. Most importantly, BiXT is able to efficiently leverage longer sequences to outperform larger competitors at fewer FLOPs: The most-recent DeiT3-S achieves 81.4% (4.6G FLOPs, 22M param), while BiXT obtains 81.8% at only 3.6G FLOPs & 15M parameters – see Appendix B.1 for further details. 3.4 Dense Tasks – Semantic Image Segmentation & Point Cloud Part Segmentation Semantic Segmentation. We investigate the transferability of our methods onto semantic image segmentation on ADE20K [56]. We follow common practice and first integrate BiXT pretrained on ImageNet1K together with SemFPN [21] as decoder. Our vanilla BiXT performs competitively against other methods with similar FLOP counts, while the more vision-specific variant BiXT+LPI with local token refinement is on par with even the improved pyramidal PvTv2 and outperforms the other models of comparable complexity (Table 3). Please refer to Appendix C for more details. 7 Table 3: Semantic Segmentation on ADE20K. We again focus here on efficient models in the low FLOP and/or parameter regime. All methods trained on 5122 images, and FLOPs are computed on 5122 images as well. Backbone FLOPs #Param mIoU. Using the Semantic FPN decoder [21] PVTv2-B0 Wang et al. [47] 25.0G 8M 37.2 ResNet18 He et al. [15] 32.2G 16M 32.9 PVTv1-Ti Wang et al. [46] 33.2G 17M 35.7 PVTv2-B1 Wang et al. [47] 34.2G 18M 42.5 XCiT-T12 El-Nouby et al. [11] − 8M 38.1 BiXT-Ti/16 31.8G 19M 39.2 BiXT-Ti/16 (conv) 31.8G 19M 41.4 BiXT-Ti/16 (+LPI from XCiT) 32.4G 19M 42.4 Simple linear predictor BiXT-Ti/16 6.4G 15M 40.6 BiXT-Ti/16 (conv) 6.4G 15M 42.3 BiXT-Ti/8 23.2G 15M 42.1 BiXT-Ti/8 (conv) 23.2G 15M 43.2 However, decoders like SemFPN were originally introduced for multi-scale CNN-like architectures and take feature maps at multiple resolutions as input. Non-hierarchical Transformers like BiXT therefore need to down- and upsample their feature maps at various stages – raising the question how this affects performance and to what extent results are caused by backbone, decoder, and their compatibility. To provide insights unaffected by these potential influences, we take inspiration from DINOv2 [27] and simply use a linear layer to directly predict a segmentation map at feature resolution from the last layer’s tokens, which is then upsampled using bilinear interpolation. Interestingly, our naïve approach is on par with the SemFPN variants but requires 80% fewer FLOPs, and outperforms by ∼1.6% at higher resolution while still being 32% more efficient (Table 3) – indicating that more research into the suitability of such decoders with non-hierarchical architectures might be needed. Point Cloud Part Segmentation. Since BiXT provides a similar generality as Perceiver regarding its input data structure but additionally allows the use of the dense, local token information, we determine its suitability for the segmentation of parts of a point cloud on ShapeNetPart [52]. The naïve application of BiXT with a linear classifier directly applied to the last layer’s tokens achieves a competitive class mIoU of 83.5% and outperforms other ‘simple’ methods like seminal PointNet [32] (class mIoU of 80.4%), but lags slightly behind recent more complex encoder-decoder methods like PointMLP [25] (class mIoU of 84.6%). Including a modality-specific token-refinement module and decoder however closes the gap and lets BiXT obtain a highly competitive class mIoU of 84.7% – as always trading off performance and generality. Please refer to Appendix D for more detailed results. 3.5 Beyond Visual Perception: Hierarchical Sequence Modeling and Document Retrieval Table 4: Hierarchical Sequence Modeling and Document Retrieval using the LRA benchmark [40]. Samples per second indicate empirical throughput at inference time for varying specified batch sizes ‘bs’ (using one NVIDIA A100). Arch. Accuracy FLOPs samples / s samples / s (%) ↑ (×106) ↓ (bs=32) ↑ (bs=256) ↑ Hierarchical Sequence Modeling - Long ListOps (2k) Transf. 39.10±0.57 137 5175 5357 BiXT 39.42±0.24 103 (-25%) 16891 (3.3×) 23804 (4.4×) Byte-level Document Retrieval - AAN (4k-8k) Transf. 82.34±0.31 535 751 751 BiXT 82.46±0.41 384 (-28%) 5703 (7.6×) 6325 (8.4×) Up to this point, we have demonstrated BiXT’s advantages on perception tasks centered around visual and 3D-structural reasoning. We now go one step further and investigate whether our claim of ‘BiXT performing at the same level as a full Transformer while being more efficient’ holds on tasks that are proven to require modeling of and reasoning over very long and often complex sequences. We evaluate the two tasks from the LRA benchmark with the ’longest required attention span’ [40]: hierarchical sequence modeling using Long-ListOps [26], and byte-level document retrieval using AAN [35]. Long-ListOps tests the ability to reason hierarchically over complex sequences composed of numbers, mathematical operators and brackets – requiring models to access all tokens and model the logical structure of inputs. ‘Retrieval’ evaluates the ability to encode and compress sequences of 4k length for matching and retrieval, requiring reasoning over 8k tokens in total. To allow fair comparison, we follow the setup in [50], and train both a full Transformer model and our BiXT variant for 5 random seeds each. While both models are on par in terms of accuracy, BiXT requires up to 28% fewer FLOPs and is up to 8.4× faster – clearly supporting our claim of significantly improving the efficiency for processing long sequences (Table 4). For additional details, please refer to the discussion in Appendix E. 8 128 160 192 224 256 288 320 Embedding dimension 74 76 78 80 Validation Accuracy # latents 64 lat 1 2 3 4 GFLOPs 196 576 784 2,304 Input sequence length 78 79 80 81 82 Validation Accuracy # latents 32 lat 64 lat 128 lat 2 4 6 8 10 12 14 GFLOPs Figure 3: Scaling trends. Ablating the influence of embedding dimension, varying numbers of latents and sequence lengths for ImageNet1K classification. All models trained with shorter schedule (only 300 epochs) to save computational resources, and comparisons should therefore be performed relative to each other. Red star-markers correspond to BiXT-Ti/16 (Acc. 80.1) from Table 2. Validation accuracy represented through solid lines, while dashed lines indicate the computational resources. 3.6 Scaling Trends – Number of Latents & Dimensions The majority of this paper is concerned with tiny efficient models; however, it is interesting to see whether our models follow previous Transformers in terms of scaling behavior. BiXT offers an additional degree of freedom in the number of latents. We therefore provide some insights into BiXT’s ImageNet1K performance changes for 32, 64 and 128 latents as well as various embedding dimensions (Figure 3). As expected, accuracy increases with both larger embedding dimension and number of latents – and it is worth noting that increasing the number of latents scales quadratically in FLOPs due to the self-attention-based latent refinement while increasing the sequence length scales linearly. Note that we use shorter training schedules for this ablation, and results are intended to be interpreted relative to each other. While we chose not to run excessive hyperparameter optimization and refrain from translating to very large architectures due to the large computational requirements involved, we did not observe any signs why BiXT should not behave like other Transformer architectures in terms of scaling and performance. We therefore anticipate to see similar tendencies as reported for related attention-based architectures, but leave this to future work. 3.7 Limitations & Discussion Our results obtained from the investigation of iterative vs. bi-directional attention as well as our experiments across multiple tasks and modalities clearly show that bi-directional attention offers advantages in a number of settings, both in terms of performance and efficiency. However, it is worth noting that by simultaneously refining the tokens alongside the latents, BiXT does not decouple the model’s depth from the input, unlike Perceiver models [18]. Therefore, very deep BiXT variants might potentially face difficulties in settings of extremely long sequences paired with limited compute and memory. However, we suspect most such scenarios to benefit from some form of preprocessing via a modality-specific input tokenizer, similar to the input-adapter-based concept used in Perceiver-IO [19] – shifting most applications again into regions where BiXT performs effectively and efficiently. Given the current popularity of natural language processing tasks, we would further like to note that BiXT in its current form is an encoder-based architecture (similar to BERT-like models), and we expect it to perform well on tasks that require understanding and modeling of entire sequences – which is what our results obtained in Section 3.5 / Table 4 on the LRA tasks indicate. However, as BiXT circumvents the expensive token self-attention of Transformers via our proposed bi-directional cross-attention, causal masking as commonly used in decoder-only methods for generative language tasks is not directly applicable to BiXT’s current attention mechanism, as information from later tokens would be able to ‘leak’ to earlier ones via the latent refinement. One possibility to establish causality in this setup could be to assign groups of tokens to specific latents by masking the bidirectional cross-attention and latent refinement accordingly (while trading off some processing resolution at training time), but we expect there to be numerous potential ways and leave this as an interesting area for future follow-up research. 9 4 Related work The introduction of Transformers [44] has helped self-attention to significantly gain in popularity, despite its caveat of scaling quadratically in computational time and memory with input length. Their flexibility regarding input modality and success in Natural Language Processing (NLP) [9] and Computer Vision (CV) [10, 41, 42] prompted a series of works targeting more efficient versions. Approximating the attention matrix via low-rank factorization has been employed across NLP [20, 45, 39], CV [6, 58, 23] and others [7], essentially avoiding the explicit computation through associativity, estimating a set of bases or using sampling – usually at the expense of performance. Others proposed to use tensor formulations [24, 3] or exploit the input data structure [29, 17, 34, 11] under the umbrella of sparsity, however limiting their use to only one specific input modality. The line of work closest related to ours are ‘memory-based approaches’ which employ some form of global memory to allow indirect interaction between local tokens. [4] propose to compose various local windowed patterns (sliding, dilated) with global attention on few ‘pre-selected’ and task-specific input locations for NLP tasks, while its vision derivative [55] provides global memory as tokens within a vision-pyramid architecture and employs four different pairwise attention operations combined with several sets of global tokens that are discarded at certain stages, introducing rather high architectural complexity. [1] additionally investigate the encoding of structured NLP inputs, whereas [54] propose a hand-crafted mix of random, window and global attention to sparsify and thus reduce attention complexity. [57] route information between selected tokens in a directed graph to achieve sparsity and skip computation in regions deemed irrelevant, whereas [5] split the input sequence and introduce dedicated latents for each chunk. [51] in turn use cross-attention-based dual-blocks for efficiency but combine these with merging-blocks that cast attention over the entire concatenated token sequence, introducing a shared representation space and preventing linear scaling. While these ideas of indirect local token communication via a shared global memory align with ours, BiXT realizes this goal in a much simpler and modality-independent manner when compared to the mix of highly modalityspecific components, attention patterns and strategies involved in these works. Preserving generality w.r.t. the input, [22] use a set of learnable ‘inducing points’ via cross-attention to query input data, while the recent Perceiver architectures [18, 19] similarly use a fixed set of latents to query input data – yet none offers the efficient simultaneous refinement of latents and tokens realized in our BiXT. Please see Appendix A.5 for some further in-detail discussion and a wider scope of related work. 5 Conclusion In this paper, we presented a novel bi-directional cross-attention Transformer architecture (BiXT) for which computational cost and memory consumption scale linearly with input size, motivated by a naturally emerging symmetry in two-way cross-attention that aligns with common intuition and has been empirically demonstrated in this work. By allowing the ‘what’ (latent variables) and ‘where’ (input tokens) to attend to each other simultaneously and develop alongside throughout the architectural stages, BiXT combines Perceiver’s linear scaling with full Transformer architectures’ high performance in a best-of-both-worlds approach. 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Reducing the computational costs of Transformer models reduces their energy consumption and therefore their impact on the environment; however, these benefits may be offset if users take advantage of the increased efficiency of our approach to implement more or larger models. A BiXT – General Aspects and Insights A.1 Code and Reproducibility We implemented our models in PyTorch [30] using the timm library, and will release all code and pretrained models. We further made use of the mmsegmentation library [8] for the semantic segmentation experiments. Point cloud experiments were built on the publicly released code base from Ma et al. [25]. A.2 Complexity Analysis The complexity of BiXT is dominated by the bi-directional cross-attention, in particular by a) the matrix multiplication to compute the similarity matrix and b) the two matrix multiplications to compute the refined outputs. Using the previously specified embedding dimension D, N tokens and M latent vectors, multiplication a) involves matrices of shape M ×D, D×N with result M ×N, and the two multiplications b) involve matrices of shape M ×N, N ×D with result M ×D and N ×M, M ×D with result N ×D. The overall complexity per layer is thus O(MND) = O(N) and linear in the size of the input N. A.3 Bi-Directional Attention and Degrees of Freedom In this section, we discuss the degrees of freedom (dof) inherent to our bi-directional cross-attention and provide some further insights into why the information exchange between latents and tokens is less restricted than might at first be expected. It is worth noting that there might be cases where the approximate symmetry that motivates our approach does not clearly emerge when using a naïve sequential method. Even in these cases, we however found our method to still consistently provide a net benefit across all experiments. We conjecture that multiple aspects contribute to this effect, one of which is that even though a ‘hard’ structural symmetry constraint is imposed on the pairwise similarity matrix, the actual attention matrices obtained after row- and column-wise softmax have additional non-shared degrees of freedom which can be used to modulate the information exchange. We discuss this in the following in more detail. (Another helpful aspect could be that having an additional layer due to BiXT’s higher efficiency can compensate for additionally required non-symmetric processing, and information exchange can also be realized across multiple layers via e.g. a token-latent-token sequence.) TLDR: Bi-directional cross-attention between M latents and N tokens has in total MN −1 dof, only (M−1)·(N−1) of which are shared – leaving M+N−2 dof that can be used by the network for the modulation of the (non-strictly-symmetric) information exchange. Gentle introduction. For ease of understanding, we start from a vector ¯v ∈RN and apply the softmax operation to obtain v = softmax(¯v). Given that all entries vi of this vector have to sum to 1 due to the applied softmax operation, v has N −1 dof. This can also be interpreted as “adding a constant to all elements of ¯v doesn’t change v”. Uni-directional cross-attention. Let us now consider the pair-wise similarity matrix ¯AT,S between target T and source S as introduced in Section 2.1. Casting uni-directional attention between M latents and N tokens to refine the latents, we obtain Alat,tok = softmax( ¯Alat,tok) ∈RM×N with 14 softmax (a) uni-directional, rowwise softmax (b) uni-directional, columnwise shared softmax softmax (c) bi-directional, row- & columnwise Figure A1: Degrees of Freedom. (a) Row-wise softmax for uni-directional cross-attention, based on matrix ∈RM×N with M · (N −1) degrees of freedom. (b) Column-wise softmax for uni-directional cross-attention, based on matrix ∈RM×N with N·(M−1) degrees of freedom. (c) Row- and columnwise softmax for our proposed bi-directional cross-attention, using the same matrix ∈RM×N with MN −1 degrees of freedom. the softmax applied row-wise – resulting in M ·(N −1) dof as visualized in Figure A1 a). Likewise, computing the attention matrix Atok,lat ∈RN×M between tokens and latents using a different set of key and query vectors yields N·(M−1) dof, which is visualized in its transposed form in Figure A1 b). →Therefore, sequentially applying two uni-directional cross-attention operations on two individual pair-wise similarity matrices provides a total of 2MN −M −N dof. Bi-directional cross-attention. Unlike the sequential approach, our proposed bi-directional crossattention uses the same pair-wise similarity matrix and obtains the attention matrices via row- and column-wise softmax. This can be interpreted as overlaying both operations and their respective degrees of freedom, and is visualized in Figure A1 c). As demonstrated by the shaded area, both softmax operations ‘share’ a total of (M −1)·(N −1) dofs. With the row-wise softmax yielding M ·(N −1) dof and the column-wise softmax N ·(M −1) dof, this results in a total of MN −1 dof – where the ‘1’ can be interpreted as “adding the same constant to all elements pre-softmax doesn’t change the result”. Note however that while adding the same constant to all elements of a row (pre-softmax) does not affect the results after the row-wise softmax, it does change the column-wise one. Therefore, the non-overlapping areas in Figure A1 c) can be interpreted as the dof that are unique to the attention maps obtained via row- or column-wise softmax, and can be used to modulate the resulting information flow to better accommodate potential deviations from symmetry. →Bi-directional cross-attention uses the same pairwise similarity matrix to obtain both attention maps and therefore has a total of MN −1 dof, (M −1)·(N −1) of which are shared and M +N −2 are unique. A.4 Types of Attention – Additional Results, Visualizations and Further Explanations An extended list of the results stated in Section 3.2 are presented in Table A1. Note that we performed an individual sweep over a set of learning rates for each individual architecture – usually starting at 4e−3 and lowering until stable training occurred. We then used these results to pick the best 5 architectural variants and training schemes, and ran them for an additional 2 random seeds. Note that all architectural variants, including BiXT and the sequential one have only been run in this setup for a total of maximum 3 runs, and no cherry-picking of results occurred for any of the architectures. Note that we have also tried stepped schedulers with the schedule proposed in the original Perceiver paper [18], but resorted back to using the cosine since it showed equal or superior results. To contrast the sequential attention to our default BiXT with 12 layers (d12) on a matching FLOP level, the sequential version uses only 11 layers (d11) due to its higher complexity per layer. This is due to the fact that our bi-directional cross-attention only requires 4 instead of 6 projection matrices (2×[R, V ] vs. 2×[Q, K, V ]) and only computes the attention matrix once (instead of twice). The hereby saved FLOPs (as well as parameters and memory) can then be spent on additional layers, further improving results. Architectures with one more layer each show the same trend. In other words, by holding FLOP and/or memory requirements constant, we consistently observe a net benefit with our bi-directional attention in terms of accuracy throughout our experiments. We empirically found that it additionally improved robustness/consistency across different parameter 15 initializations (seeds), which can be seen by the slightly smaller standard deviations of the bidirectional variants. Table A1: Architectural variants using iterative attention & cross-attention parameter sharing. Classification accuracy on the ImageNet1K dataset for varying types of attention. All architectures use 64 latent vectors and have been trained for 120 epochs with hyperparameters individually optimized. Cross-attention parameter sharing schemes: †indicates sharing of all, ‡of all but the 1st layer’s crossattention parameters. Architectural configurations noted in brackets. Three randomly seeded runs were performed for the ‘best’ architectures (judged by their performance on seed = 42), and mean and (unbiased) standard deviation are reported. One randomly seeded run reported for all other architectures. Attention type Acc.@1 (%) Acc.@5 (%) FLOPs Mem. #Param Iterative† (sa5-d8) 57.51 80.61 1.58G 7.17M 18.61M Iterative† (sa6-d7) 58.86 81.53 1.59G 7.23M 19.50M Iterative† (sa6-d8) 60.61 ± 1.11 82.75 ± 0.68 1.82G 8.25M 22.16M Iterative† (sa4-d12) 56.03 ± 1.02 79.38 ± 0.80 1.99G 9.10M 22.16M Iterative† (sa1-d22) 56.09 79.36 1.64G 7.70M 11.04M Iterative† (sa1-d24) 55.92 ± 0.67 79.33 ± 0.52 1.79G 8.39M 11.93M Iterative‡ (sa5-d8) 58.26 ± 2.34 81.02 ± 1.76 1.58G 7.17M 19.05M Iterative‡ (sa6-d7) 54.94 ± 5.96 78.39 ± 4.69 1.59G 7.23M 19.94M Iterative‡ (sa6-d8) 58.23 80.95 1.82G 8.25M 22.61M Iterative‡ (sa4-d12) 56.35 79.64 1.99G 9.10M 22.61M Sequential (2-way, d11) 73.10 ± 0.53 91.05 ± 0.28 1.66G 8.44M 14.60M Sequential (2-way, d12) 73.79 ± 0.32 91.48 ± 0.15 1.81G 9.24M 15.94M Bi-Directional (d12) 73.86 ± 0.39 91.55 ± 0.14 1.68G 7.86M 15.12M Bi-Directional (d13) 74.10 ± 0.14 91.61 ± 0.12 1.82G 8.54M 16.38M Visualizing the three types of attention. To further ease understanding and provide a clearer overview of the differences between the various investigated types of attention, we visualize the conceptual changes in the architectural layout when transitioning from ‘iterative’ over ‘sequential’ to our proposed efficient ‘bi-directional’ attention and their respective differences in Figure A2. (optional) Input Tokenizer Latent Self-Attention ×L Learnable latent vectors (tokenized) input data Images Point Clouds … Iterative Cross-Attention ×B (a) iterative: A ∈RM×N (optional) Input Tokenizer Latent Self-Attention ×L Learnable latent vectors (tokenized) input data Images Point Clouds … Iterative Cross-Attention Iterative Cross-Attention (b) sequential: 2 × A ∈RM×N (optional) Input Tokenizer Latent Self-Attention (optional) Token Refinement ×L Learnable latent vectors (tokenized) input data Images Point Clouds … ‘What?’ ‘Where?’ Bi-Directional Cross-Attention (c) bi-directional: A ∈RM×N Figure A2: Transitioning from iterative to bi-directional attention. (a) Perceiver-like iterative attention, creating a bottleneck and small effective working memory; (b) Naïve sequential attention ‘unblocking’ the bottleneck and extending working memory, but still markedly less efficient than: (c) Bi-directional cross-attention used in BiXT, combining efficient linear scaling with competitive performance across various tasks. Note that iterative attention attends to the (unrefined) input at every layer, while sequential and bi-directional attend to variants of the input refined by the previous layer. The Perceiver-like setup additionally uses multiple self-attention layers to refine between each iterative cross-attention (×B) in each architectural layer, whereas sequential and bi-directional variants only use one self-attention operation per architectural layer. Architectures are then built by stacking L layers. 16 Iterative CrossAttention Learnable latent vectors (tokenized) input data * … 𝑁 𝑀 + Norm MLP + Softmax Norm Norm (a) iterative attention Bi-Directional CrossAttention Learnable latent vectors (tokenized) input data * … 𝑁 𝑀 + + Norm Norm MLP MLP + + Row-wise Softmax Column-wise Softmax Norm Norm * (b) bi-directional attention Figure A3: Detailed structure of attention blocks. (a) Perceiver-like iterative attention, creating a bottleneck and small effective working memory; (b) Bi-directional cross-attention used in BiXT. Figure A3 shows the internal difference between the Perceiver-like iterative attention and our proposed bi-directional cross-attention in more detail. A.5 More Detailed Discussion of Most-Recent Related Work In the following, we provide some additional and more in-depth discussion of methods we see related to our proposed BiXT architecture. We start by taking a look at three methods mainly targeting the image space, and follow up with a more general discussion of related methods across modalities that focus on the long-sequence aspect – including recently proposed Structured State Space Sequence Models. Methods mainly targeting the image domain. » DualViT [51]. DualViT’s dual block used in the early layers of their architecture does to some extent show similarity to the naïve solution of sequential cross-attention, but is distinctly different from our bi-directional approach as it does not leverage any symmetry. Importantly, their multi-stage pyramidal vision-only architecture uses a large number of ‘merging blocks/layers’ (between 9 - 24) which cast full self-attention over the concatenated sequence of latents and tokens. This prevents linear scaling and also introduces a shared embedding space of latent vectors and tokens through the use of the same key-query-value projection matrices – whereas our architecture keeps those separate (aligned with the presented ’what’ and ’where’ analogy and the level of information they represent) and scales linearly with respect to the input length. » BiFormer [57]. BiFormer follows the common trend for vision-only approaches and employs a pyramidal structure. In contrast to previous work, the authors reduce the computational complexity through routing information between selected tokens via a directed graph, thus achieving sparsity to skip computation of certain regions that are deemed ‘irrelevant’. While this is a very neat way of dynamically reducing complexity, it is distinctly different from our approach and does not achieve true linear scaling. » FiT [5]. FiT explicitly divides a token sequence into subgroups of tokens to cast quadratic local/windowed self-attention within, and assigns a small set of latents to each group. Exchange between these latents is accomplished via one-way cross-attention within each group, followed by global information routing via multiple self-attention operations cast across the latents of all groups. The exact architectural structure in terms of composition varies between architectural variants (number of local and global layers per block + number of blocks). Our BiXT in contrast achieves its entire information exchange via our proposed efficient bi-directional cross-attention between latents and tokens, followed by one self-attention operation among latents. This significantly simplifies the architecture in terms of complexity, only requires one set of latents that efficiently interacts with the entire sequence and does not require any manual grouping of the input sequence. While our approach markedly differs from FiT in various aspects, their experimental setups are quite interesting and it is great to see that the research community is following similar directions in terms of decomposing and routing information among global latents and local sequence tokens. 17 Beyond Transformers – Recent developments in recurrent methods. As we were focusing mainly on Transformer-based approaches and perception-based tasks in the main paper, we kept this as the primary focus of the literature review of the main manuscript. Here, we provide some additional recent methods relevant in the context of long sequence processing (especially beyond perception-based data) that warrant closer discussion. While Transformer-based architectures have steadily gained in popularity over the last years, recurrent methods have recently enjoyed increased attention and have been both revisited and further improved across several works – e.g. by ’reinventing RNNs for the Transformer era’ [31] with the goal of combining Transformer-style efficient training with the fast inference speed of RNNs. An alternative to the well-known recursive methods like RNNs are the recently introduced structured state-space sequence (S4) models [13], which are based on a new way of parameterizing SSMs that makes their application to long sequence modelling tasks computationally feasible and training much more efficient. Multiple works have since proposed simplifications to the S4 model ([12, 14, 38]) – while others have used the gained insights to further improve well-known models like RNNs [28]. B ImageNet1K Experiments – Further Details This section outlines further details and additional insights regarding our image classification experiments conducted on the ImageNet1K dataset [36]. B.1 Longer Sequences Help to Beat Larger Models – Further Discussion and Results As reported in the main paper in Section 3.3, BiXT’s ability to efficiently leverage longer sequences helps it to outperform larger models – and often at fewer FLOPs. In the following, we contrast BiXT to different ‘evolutions’ of the ViT/DeiT family [10, 41, 42] with approximately matching parameter and/or FLOP counts. We start with our tiny BiXT and contrast it with the next larger Vision Transformer models – DeiT-S & DeiT3-S – in addition to the results shown in Table 2. This allows a much closer comparison in terms of FLOPs and parameters. Both DeiT-3 with 79.8% and the most-recent DeiT3-S with 81.4% use 22M parameters & 4.6GFLOPs. This is surpassed by both of our closest BiXT variants with fewer or similar FLOP counts (Table A2): • BiXT-Ti/16 ↑384 achieves 81.8% accuracy with 15M param & 3.6GFLOPs, and • BiXT-Ti/8 achieves 81.9% accuracy with 15M param & 4.6GFLOPs Note that the use of longer sequences, either via 384×384 images or through a patch size of 8, cannot be efficiently leveraged by DeiT variants as it would significantly increase their FLOP count due to the inherent quadratic scaling of their attention (∼15.5GFLOPs for DeiT-S↑384). In addition to matching DeiT3-S’s performance via longer sequence length, we have run some additional experiments for BiXT with increased embedding dimension 256 (given limited available resources). This approximately matches DeiT-S in terms of parameters (BiXT-d256 27M vs. DeiT-S 22M), with results included in Table A2: • Our ‘small’ BiXT-d256/16 achieves 81.7% and already outperforms the original ViT-B (77.91%) and recent DeiT3-S (81.4%), and is on par with DeiT-B (81.8%) at a fraction of the FLOP count (2.9G vs. 17.5G). • Our longer-sequence model BiXT-d256/8↑384 is on par even with the newest (mostoptimized) DeiT3-B while showing much higher parameter efficiency (26.7M vs 86.6M, albeit requiring slightly more FLOPs). » A Note Regarding Larger Models and Actual Complexity of Training « While it would indeed be very interesting to analyze larger models, we would like to note that this requires a substantial number of additional large experiments. Even though such models might at first appear to require moderate compute, the actually required computational budget not only encompasses the training runs but also the hyperparameter search. The importance of well-chosen hyperparameters and augmentation strategies grows significantly with model size, as can be seen 18 Table A2: Matching FLOP and parameter counts of Transformer models. Comparing evolutions of ViTs to variants of BiXT for image classification on ImageNet1K [36]. Note that different models might have received different levels of optimization effort, especially the ViT/DeiT variants across their multiple evolutions. Architecture Accuracy #Param FLOPs DeiT-S [41] 79.8% 22M 4.6G DeiT3-S [42] 81.4% 22M 4.6G BiXT-Ti/16 ↑384 81.8% 15M 3.6G BiXT-Ti/8 81.9% 15M 4.7G BiXT-d256/16 81.7% 27M 2.9G ViT-B [10] 77.9% 87M 17.5G DeiT-B [41] 81.8% 87M 17.5G DeiT3-B [42] 83.8% 87M 17.5G BiXT-Ti/4 82.7% 15M 16.8G BiXT-Ti/8 ↑384 82.8% 15M 12.5G BiXT-Ti/4 ↑384 83.1% 15M 48.1G BiXT-d256/8 83.2% 27M 8.1G BiXT-d256/8 ↑384 83.9% 27M 21.6G in the literature (e.g. in the transition from ViT [10] →DeiT [41] →DeiT3 [42] or ResNet [15] →ResNet strikes back [48]). This makes an appropriate exploration of this vast search space essential but computationally very expensive, and we (have to) leave this as an opportunity for future work. B.2 Computational Complexity, Sequence Length and Empirical Throughput aka ‘Latency’ The benefit of modeling input data at a higher resolution (e.g. smaller patches and larger images in vision) has been demonstrated across most works like ViT/DeiT. For example, increasing the input image size from 224 to 384 for DeiT3-S yields a boost of 2% in accuracy, but requires 3× as many FLOPs due to quadratic scaling of the attention with input sequence length. Reducing the patch size from 16×16 to 4×4 incurs 15.5× as many operations (Table A3). One of the main advantages of our BiXTin contrast to vanilla Transformers is its linear scaling with the input sequence length while maintaining competitive performance. Increasing the input size from 224 to 384 only incurs 2.2× as many FLOPs, and patch-size reduction to 4×4 less than 10× – a decrease by 26% and 35%, respectively. This allows BiXT to essentially process and model longer sequences much more efficiently than naïve Transformer models, boost results (see main paper) and extend its processing capabilities to regions where Transformer-like methods with full self-attention become infeasible. In our image segmentation experiments for example, BiXT processes sequences of up to 16,384 tokens during training – and up to 65,536 at inference time for 512 × 2048 images. Note that this aligns well with our obtained insights that BiXT is able to efficiently leverage a longer sequence to outperform a ‘larger’ DeiT model at fewer FLOPs (Section 3.3), as well as with the results obtained on the LRA benchmark in Section 3.5. Table A3 shows common sequence lengths encountered during image processing (classification on ImageNet [36], semantic segmentation on ADE20K [56]) and demonstrates the scaling differences for ViT/DeiT variants [10, 41, 42] and BiXT. While latency is closely linked to the FLOP counts, we additionally provide empirical data on the throughput (img/s) in this section. Note that these numbers are obtained with a batch size of 256 on a single A100 GPU with float32 precision (no amp) – and that given its popularity and maturity, DeiT might have received more optimization effort than our BiXT. As can be seen in Table A4, while the tiny version of DeiT3 [42] in its default configuration (patch 16) is faster than BiXT, our method significantly outperforms DeiT3 methods across all higher sequence lengths (i.e. larger images, smaller patches) – e.g. with BiXT-Ti384/4 (160img/s) being 6.4× faster than DeiT3-Ti384/4 (25img/s). 19 Table A3: Scaling of computational complexity. Relative increase in FLOPs and Activations (memory) over sequence length (w.r.t. baseline 224 / p16). Config 224 / p16 384 / p16 224 / p8 512 / p16 384 / p8 224 / p4 512 / p8 384 / p4 512 / p4 Seq. Len. 196 576 784 1,024 2,304 3,136 4,096 9,216 16,384 Increase in compute, measured in FLOPs BiXT Incr 1x 2.2x 2.8x 3.5x 7.5x 10.0x 12.9x 28.6x 50.6x DeiT/ViT Incr 1x 3.0x 3.9x 5.2x 11.5x 15.5x 20.4x 45.6x 81.0x Increase in memory consumption (activations, per sample) BiXT Incr 1x 2.2x 2.8x 3.6x 7.5x 10.1x 13.1x 29.3x 51.3x DeiT/ViT Incr 1x 3.0x 4.0x 5.2x 11.7x 15.9x 20.8x 46.8x 83.2x Table A4: Throughput. Empirical latency for different variants of DeiT3 and BiXT. Arch. 224 / p16 224 / p8 224 / p4 384 / p16 384 / p8 384 / p4 Empirical throughput, measured in img/s BiXT-Ti 5775 1971 527 2521 702 160 BiXT-d256 4085 1408 385 1823 510 119 Deit3-Ti 10263 1861 190 2730 325 25 Deit3-S 4784 852 90 1253 153 12 Deit3-B 1833 344 42 505 69 6 B.3 Model Configurations and Training Details Hyperparameter choice for the default ImageNet experiments: BiXT with 64 latents, 12 layers, embedding dimension for latents and tokens 192 paired with 6 heads (head dimension 32) – learning rate 2.5e−3, weight decay 0.05 and lambc optimizer, as well as cosine learning rate scheduler with linear warmup; stochastic dropout on self-attention and cross-attention 0.1 for all tiny models. Apart from these, we directly apply the augmentation and training proposed by Touvron et al. [42]. Our models have been trained between 300 (ablations) and 800 epochs on one or several A100 GPUs. Note that we did not conduct an extensive hyperparameter search, and we expect results to potentially improve if done so. Finetuning on images of size 384×384 was performed for 30 epochs using a batch size of 512 and an initial learning rate of 2.5e−5 with cosine decline, starting from the model trained on 224×224 images. We found empirically that increasing the stochastic dropout during finetuning to 0.2 can help to improve the results, and we hence use this as default value for our finetuning experiments. B.4 Ablating Patch Size for Fixed Sequence Lengths in Image Classification In this section, we investigate whether lowering the patch size to increase the resolution of the resulting feature maps is actually the most-suited way – or whether simply reducing the stride and thus creating tokens that originate from overlapping patches yield better results. Our experiments on image classification using the ImageNet1k [36] dataset with models using varying patch sizes and strides to keep the sequence lengths fixed show that the originally introduced and commonly used patch size of 16 × 16 pixels seems to be a good fit when using no overlapping patches (Table A5). Interestingly, we find that even when we increase the feature resolution and thus choose smaller strides, a patch size of 16 × 16 still yields best results across our experiments. One potential reason is that patch boundaries are randomly chosen and objects in images do naturally not match these boundaries, so that information has to be exchanged – whereas slight overlaps might ease this to some extent. Another potential reason for this behaviour is that significantly decreasing the patch size reduces the input information per patch, with an 82 RGB patch having a total of 192 channels, exactly matching the tiny embedding dimension. Smaller patches however would create a significant null space, which might be an additional reason for better performance when using larger patches. 20 Table A5: Varying patch sizes for fixed sequence lengths. ImageNet1k classification results for varying patch sizes are presented for three fixed sequence lengths (realised via stride). All models have been trained for 300 epochs using the same (default) hyperparameters and input images of size 224 × 224. Best results for each sequence length is highlighted in bold. Seq. length 196 (14 × 14) 784 (28 × 28) 3136 (56 × 56) Patch size 32 × 32 16 × 16 32 × 32 16 × 16 8 × 8 16 × 16 8 × 8 4 × 4 Acc. (%) 77.50 78.13 79.90 79.92 79.36 80.95 80.75 79.56 FLOPs 1.77G 1.68G 5.05G 4.71G 4.62G 16.81G 16.46G 16.38G Mem 7.27M 7.23M 20.25M 20.25M 20.25M 72.18M 72.18M 72.18M #Param 15.56M 15.11M 15.56M 15.12M 15.01M 15.12M 15.01M 14.98M B.5 Convolutional Tokenizer In addition to our default linearly-projecting tokenizer, we report results using a convolutional tokenizer as BiXT-Ti/16 (conv) in Table 2. This tokenizer follows El-Nouby et al. [11] and consists of a stack of four {conv - Batch Norm - GeLU} groups, using 3 × 3 convs with stride 1 and sequentially encoding the input channels into the specified embedding dimension D (via D/8, D/4, D/2, D). B.6 Token Refinement via Local Patch Interaction (XCiT) We integrate a slightly modified version of the ‘LPI’ module from El-Nouby et al. [11] together with their convolutional tokenizer for our vision-specific image segmentation experiments. Our LPI module consists of two depth-wise convolutional layers (3x3) with Layer Normalization (instead of the original Batch Normalization) and a GELU non-linearity in between. For further details, please refer to the original paper. C Semantic Image Segmentation Experiments – Further Details We investigate the transferability of our methods onto semantic image segmentation on the ADE20K dataset [56]. We follow common practice and integrate BiXT pretrained on ImageNet1K together with SemanticFPN [21] as decoder, train for 80k iterations with learning rate 6e−5 and weight decay 0.01 following El-Nouby et al. [11] and others. We choose a batch size of 32 due to the efficiency of our model on the 5122 images, and train on a single A100 GPU. Our vanilla BiXT performs competitively against other methods with similar FLOP counts, while the more vision-specific version BiXT+LPI with local token refinement is on par with even the improved pyramidal PvTv2 and outperforms the others (Table A6). Criticism on decoders & a potential alternative. Decoders like SemFPN were originally introduced for CNN-like architectures and use feature maps at multiple resolutions. Non-hierarchical Transformer architectures like BiXT thus need to downsample and up-convolve their feature maps at various stages – raising the question how this affects performance and to which extent results are caused by backbone, decoder and the compatibility of the two. To provide insights unaffected by these potential influences, we take inspiration from the recently published DINOv2 [27] and simply use a linear layer to directly predict a segmentation map at feature resolution from the last layer’s tokens, which we then upsample using bilinear interpolation. Interestingly, our naive approach clearly outperforms our SemFPN variants with 80% fewer FLOPs (6.4G vs 31.8G). Increasing the sequence length via smaller stride improves results further, with BiXT-Ti/8 (conv) clearly outperforming other methods while still requiring ∼32% fewer FLOPs. These insights are somewhat surprising and clearly indicate that more research into the suitability of these decoders with non-hierarchical architectures might be needed. 21 Table A6: Semantic Segmentation on ADE20K. We again focus here on efficient models in the low FLOP and/or parameter regime. All methods trained on 5122 images, and FLOPs are computed on 5122 images as well. Backbone FLOPs #Param mIoU. Using the Semantic FPN decoder [21] PVTv2-B0 Wang et al. [47] 25.0G 8M 37.2 ResNet18 He et al. [15] 32.2G 16M 32.9 PVTv1-Ti Wang et al. [46] 33.2G 17M 35.7 PVTv2-B1 Wang et al. [47] 34.2G 18M 42.5 XCiT-T12 El-Nouby et al. [11] − 8M 38.1 BiXT-Ti/16 31.8G 19M 39.2 BiXT-Ti/16 (conv) 31.8G 19M 41.4 BiXT-Ti/16 (+LPI from XCiT) 32.4G 19M 42.4 Simple linear predictor BiXT-Ti/16 6.4G 15M 40.6 BiXT-Ti/16 (conv) 6.4G 15M 42.3 BiXT-Ti/8 23.2G 15M 42.1 BiXT-Ti/8 (conv) 23.2G 15M 43.2 D Point Cloud Experiments – Further Details D.1 Training and Evaluation Details Note that we do not use any voting strategy or other multi-scale augmentation and simply follow the training regime of PointMLP [25] for most of our experiments. We use a standard BiXT architecture for the ‘naïve’ point cloud experiments as well as the ones using simple grouping – and reduce our architecture to 4 layers when using the decoder for part segmentation and the hierarchical approach for shape classification – paired with 32 and 24 neighbours, respectively (which are the default values used in other works like PointMLP). We train our models using a single A100 GPU (80Gb). D.2 Detailed Results for Point Cloud Part Segmentation Since BiXT provides a similar generality as Perceiver regarding its input data structure but additionally allows the use of the dense, local token information, we run experiments to determine its suitability regarding the segmentation of sub-parts of a point cloud – commonly referred to as point cloud part segmentation – on the ShapeNetPart dataset [52]. The detailed results of our experiments are reported in the form of class intersection over union (IoU) and instance IoU in Table A7, together with the individual results for all object classes. The naïve application of BiXT with a linear classifier directly applied to the last layer’s tokens achieves a competitive class mIoU of 83.5% (instance mIoU of 85.2%) and outperforms other simple methods like seminal PointNet [32](class mIoU of 80.4%), but lacks slightly behind recent more complex encoder-decoder methods like PointMLP [25] (class mIoU of 84.6%). Note, however, that methods in this space are usually highly specialized encoder-decoder structures. Including a modality-specific token-refinement (’geometric affine grouping’) and passing the encoded information to PointMLP’s Table A7: Point cloud part segmentation on ShapeNetPart [52]. Reported are the class IoU and instance IoU for BiXT and PointMLP [25]. Note that we only compare here to PointMLP due to investigating the use of their grouping module and decoder within BiXT. Method Cls. Inst. aero- bag cap car chair earguitar knife lamp laptop motor- mug pistol rocket skate- table mIoU mIoU plane phone bike board PointNet 80.4 83.7 83.4 78.7 82.5 74.9 89.6 73.0 91.5 85.9 80.8 95.3 65.2 93.0 81.2 57.9 72.8 80.6 PointMLP 84.6 86.1 83.5 83.4 87.5 80.5 90.3 78.2 92.2 88.1 82.6 96.2 77.5 95.8 85.4 64.6 83.3 84.3 BiXT (naïve) 83.5 85.1 83.9 81.4 91.5 79.0 89.5 76.2 91.9 87.3 79.3 95.8 73.1 95.0 84.2 63.7 80.4 83.5 BiXT (EncDec) 84.7 86.0 84.4 82.7 86.3 80.9 90.2 80.1 92.1 87.8 82.3 95.9 78.1 95.9 84.9 67.0 82.4 83.9 22 decoder [25] however closes the gap and lets BiXT obtain a highly competitive class mIoU of 84.7% (instance mIoU 86.0%) – as always trading off performance and generality. E Hierarchical Sequence Modeling and Document Retrieval – Further Details As detailed in the main paper’s body in Section 3.5, we investigate BiXT’s capabilities in modeling long sequences by using the Long Range Arena (LRA) benchmark proposed by Tay et al. [40]. We provide more details in the following. E.1 Training and Evaluation Details For our experiments, we follow the setup proposed by Xiong et al. [50] and use models with 2 layers. The embedding dimension is set to 64, and we employ a hidden dimension of 128 (i.e. mlp-ratio of 2), as well as 2 attention heads. This applies to both the Transformer and our BiXT architecture. BiXT employs 32 latents for both experiments. For the hierarchical sequence modeling experiments on Long ListOps [26], we use a vocabulary size of 32, and train for 40 epochs using a batch size of 32, learning rate of 2.5e-4, path-dropout rate of 0.02, the lamb optimizer [53] and a cosine scheduler with 1 epoch linear warm-up. For the byte-level document retrieval task on AAN [35], we use a vocabulary size of 128, and train for 20 epochs using a batch size of 32, learning rate of 2.5e-5, the lamb optimizer [53] and a cosine scheduler with 1 epoch linear warm-up. Models for both tasks are trained using a single A100 GPU. E.2 Detailed Results and Additional Discussion To investigate our claim of ‘BiXT performing at the same level as a full Transformer while being more efficient’ in the context of tasks that are proven to require modeling of and reasoning over very long and often complex sequences, we evaluate the two tasks from the Long Range Arena (LRA) benchmark with the ’longest required attention span’ [40]: hierarchical sequence modeling using Long-ListOps [26], and byte-level document retrieval using AAN [35]. Note that the LRA benchmark has been specifically designed to evaluate the capabilities of Transformer-like models in very long-context scenarios in a systematic and unified manner [40]. Long-ListOps tests the ability to reason hierarchically over complex sequences (length 2048) composed of numbers, mathematical operators and delimiters (brackets). To successfully solve this task, models are required to access all tokens and model the logical structure of the inputs while handling long contexts in order to make a prediction – a task considered to be “considerably challenging” [40]. For more information, we refer the interested reader to the original ListOps work [26] and the LRA benchmark [40], both of which provide more detail including a visualization of a shortened example sequence. The ‘retrieval’ task on the other hand is designed to evaluate the ability of models to encode and compress sequences of 4k length into representations that are useful for matching and retrieval. With each individual document being 4k bytes/characters in length, this requires reasoning over 8k tokens in total. To allow fair comparison, we follow the setup in [50] as detailed above in terms of model size and most hyperparameters. We train a full Transformer model and our BiXT variant for 5 random seeds each. We pick the best model based on validation accuracy, and report the mean and (unbiased) standard deviation across these models evaluated on the withheld test set in Table A8. While both models are on par in terms of accuracy, BiXT requires up to 28% fewer FLOPs and is up to 8.4× faster – outlining BiXT’s advantage in efficiently modeling long sequences. E.3 Alternative Setups Found in Related Works on LRA Note that we follow the ‘classic’ 2-layer setup as related works like [50], and run our architecture in direct comparison to a full Transformer [44] under the same conditions for fair comparison. 23 Table A8: Hierarchical Sequence Modeling and Document Retrieval using the LRA benchmark. Samples per second indicate empirical throughput at inference time. Arch. Accuracy FLOPs samples / s samples / s samples / s (%) ↑ (×106) ↓ (bs=32) ↑ (bs=128) ↑ (bs=256) ↑ Hierarchical Sequence Modeling - Long ListOps Transf. 39.10±0.57 137 5175 5316 5357 BiXT 39.42±0.24 103 (-25%) 16891 (3.3×) 22522 (4.2×) 23804 (4.4×) Byte-level Document Retrieval - AAN Transf. 82.34±0.11 535 751 756 751 BiXT 82.46±0.41 384 (-28%) 5703 (7.6×) 6225 (8.2×) 6325 (8.4×) Some recent approaches by Gu et al. [12, 13], Gupta et al. [14], and others have moved to target the alternate ‘free-for-all’ setting of LRA with often extensive task-specific hyperparameter and model optimization, e.g. see Table 11 (appendix) in the work by Smith et al. [38], where a specific architecture (layers, blocks, dimensions, initialization) is created for each task, paired with its own unique optimization configuration – requiring extensive search across possible configurations. Given that our goal of evaluating BiXT on the LRA benchmark is to support our claim of ‘being as performant as a full Transformer while being significantly more efficient’, we deem it more appropriate instead provide the side-by-side evaluations as previously described to reduce compute requirements and allow faster training. Note that recent work by Amos et al. [2] sheds new light on the comparability of methods under this ‘free-for-all’ setting and outlines significant changes in performance depending on a variety of factors like model initialization – further supporting our side-by-side model comparison using the same setup (including initialization method). 24 F Visualization of Latent-Token Attention To provide some additional qualitative insights into the bi-directional attention that is cast within BiXT, we provide three sets of attention maps overlaid onto the input image: • Figure A4: The attention maps of the four latent vectors presented in Figure 1(d) for all layers throughout the BiXT tiny architecture (layer 1, top-left to layer 12, bottom-right). • Figure A5 The attention maps of all latent vectors (64 in this case) for the final layer of our BiXT tiny architecture. • Figure A6 The attention maps of all latent vectors (64 in this case) for the second-last layer of our BiXT tiny architecture. Figure A4: Attention across layers. Bi-directional attention maps for the four selected tokens presented in Figure 1(d) across all layers: Starting with first layer on top left, ending with last layer (layer 12) on the bottom right. Displayed are the mean attention maps averaged across the heads of BiXT tiny with 64 latents. 25 Figure A5: Attention maps of final layer. Bi-directional cross-attention maps of all 64 latent vectors of the final layer (layer 12) of our BiXT tiny architecture. 26 Figure A6: Attention maps of penultimate layer. Bi-directional cross-attention maps of all 64 latent vectors of the second-last layer (layer 11) of our BiXT tiny architecture. 27 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,458	Constrained Human-AI Cooperation: An Inclusive Embodied Social Intelligence Challenge Weihua Du∗1Z, Qiushi Lyu∗2\, Jiaming Shan3, Zhenting Qi4, Hongxin Zhang5, Sunli Chen5 Andi Peng6, Tianmin Shu7, Kwonjoon Lee8, Behzad Dariush8, Chuang Gan5ˇ “ 1 Carnegie Mellon University, 2 Peking University, 3 University of California, Santa Barbara 4 Harvard University, 5 University of Massachusetts Amherst, 6 MIT 7 Johns Hopkins University, 8 Honda Research Institute USA Z weihuad@cs.cmu.edu, \ lvqiushi@stu.pku.edu.cn, ˇ “ chuangg@umass.edu Abstract We introduce Constrained Human-AI Cooperation (CHAIC), an inclusive embodied social intelligence challenge designed to test social perception and cooperation in embodied agents. In CHAIC, the goal is for an embodied agent equipped with egocentric observations to assist a human who may be operating under physical constraints—e.g., unable to reach high places or confined to a wheelchair—in performing common household or outdoor tasks as efficiently as possible. To achieve this, a successful helper must: (1) infer the human’s intents and constraints by following the human and observing their behaviors (social perception), and (2) make a cooperative plan tailored to the human partner to solve the task as quickly as possible, working together as a team (cooperative planning). To benchmark this challenge, we create four new agents with real physical constraints and eight longhorizon tasks featuring both indoor and outdoor scenes with various constraints, emergency events, and potential risks. We benchmark planning- and learningbased baselines on the challenge and introduce a new method that leverages large language models and behavior modeling. Empirical evaluations demonstrate the effectiveness of our benchmark in enabling systematic assessment of key aspects of machine social intelligence. Our benchmark and code are publicly available at https://github.com/UMass-Foundation-Model/CHAIC. 1 Introduction Humans possess a remarkable ability to observe, infer, and help others, even when others have different mental models and physical constraints in the world from themselves (Warneken and Tomasello, 2006). From a young age, humans are able to watch other people attempt to perform a task, and if other people fail, they can develop plans of action that best assist them. In contrast, AI agents struggle to exhibit such basic social skills and fail to adjust their plans for the specific humans they wish to aid (Valmeekam et al., 2022; Ngo et al., 2022), rendering them poor personalized helpers. For AI agents to best assist human partners in performing tasks in the real world, they must possess two fundamental capabilities: (1) contextual perception, i.e., the ability to follow and observe human behavior and identify the specific goals and constraints faced by each human; and (2) cooperative planning, i.e., the ability to plan actions that are best tailored to helping each human with different goals and constraints. While there have been some embodied benchmarks and environments designed to test general multi-agent intelligence (Puig et al., 2021, 2023b; Gan et al., 2021), such efforts have largely excluded the unique accessibility challenges that real humans may possess in the world and ∗Equal Contribution 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. Figure 1: Constrained Human-AI Cooperation (CHAIC) for benchmarking embodied agents that socially perceive and assist human partners with physical constraints. Left: A human partner is confined to a wheelchair and struggles to move past an obstacle. The helper agent infers the human partner’s constraints and intents and assists him by removing the obstacle. Right: In a moving house scenario, after observing a human partner fail to lift heavy furniture, the helper agent understands her intents and constraints and assists her in carrying the furniture together. neglect the differences among individuals. Moreover, outdoor scenarios and emergencies are also prevalent in human life, but receive little attention in the embodied intelligence community (Deitke et al., 2022). This paper introduces the first large-scale embodied social intelligence challenge with accessibility explicitly in mind: Constrained Human-AI Cooperation (CHAIC). In this challenge, an embodied agent with egocentric visual observation must actively perceive and cooperate with a human partner possibly with physical constraints in a near photo- and physically realistic virtual environment to complete common household and outdoor tasks as efficiently as possible. This is motivated by the idea that people who need the most help from autonomous agents are those who are currently not explicitly accounted for in embodied intelligence frameworks. In CHAIC, a helper agent needs to follow and observe the human partner to infer their goals and constraints; then, the agent plans a user-tailored strategy for aiding the human in efficiently performing tasks together; moreover, with the existence of unexpected emergencies, the agent needs to be reactive and adjust its strategy accordingly. To create the challenge with accessibility in mind, we design and implement four new agents with real physical constraints that reflect the rich diversity of human partners in the real world. For example, a human partner confined to a wheelchair struggles to move past obstacles or a human partner struggles with heavy furniture when moving house in an outdoor scene, shown in Figure 1, and eight long-horizon tasks featuring both indoor and outdoor scenes on top of the ThreeDWorld (Gan et al., 2021), explicitly motivating the development of embodied agents that prioritize accessibility efforts when learning and planning and can thrive in rich scenarios. We benchmark several baseline models, including planning- and learning-based agents, especially those powered by foundation models. We also introduce a new method for building agents that combines the behavior modeling capabilities of video models with the reasoning ability of large language models. Our benchmark results suggest that current baselines have difficulty modeling partner behaviors from raw RGB images, and LLM-driven agents are competitive agents in decisionmaking. We hope this new challenge will advance the study of social intelligence in embodied agents in complex scenarios including diverse human partners with constraints and rich indoor and outdoor scenes. This initiative calls on the community to develop and evaluate embodied agents with a strong emphasis on accessibility and inclusivity. Our contributions include: • We design and implement four new agents with real physical constraints and eight long-horizon tasks featuring both indoor and outdoor scenes on top of ThreeDWorld (Gan et al., 2021), simulating rich human constraints and scenarios in the real world. 2 • We introduce a new embodied social intelligence challenge with accessibility explicitly in mind: Constrained Human-AI Cooperation (CHAIC), to test embodied agents’ ability to actively perceive human partners’ intents and constraints from egocentric visual observations and make user-tailored cooperative plans to help constrained human partners in rich scenarios. • We benchmark several baseline models, including those powered by foundation models, especially a new agent with behavior modeling introduced by us, and conduct comprehensive analyses to identify and discuss the persisting challenges related to inter-agent perception and cooperation within complex environments. 2 Related Work Embodied Multi-Agent Cooperation Challenges Our benchmark and environment build on a rich history of realistic 3D simulated environments (Zhou et al., 2024; Li et al., 2023; Padmakumar et al., 2022; Kolve et al., 2017; Shridhar et al., 2020; Misra et al., 2018; Zhu et al., 2017; Xia et al., 2018; Savva et al., 2019; Xiang et al., 2020). Various tasks and methods have been introduced for multi-agent cooperation (Lowe et al., 2017; Samvelyan et al., 2019; Carroll et al., 2019; Suarez et al., 2019; Jaderberg et al., 2019; Amato et al., 2019; Baker et al., 2020; Bard et al., 2020; Jain et al., 2020; Puig et al., 2023b; Wen et al., 2022; Szot et al., 2023; Zhang et al., 2023, 2024). Specifically, Puig et al. (2021, 2023a) explored the inter-agent perception of isomorphic agents during household tasks. However, these works did not address the explicit challenge of actively perceiving diverse human partners with physical constraints from visual observations and adapting the cooperation strategy accordingly. In contrast, our challenge is designed explicitly not only to study the social perception of the partner’s goals and constraints from visual observations but also to capture the nuances of human physical mobility constraints that might impair the successful completion of such tasks. A contemporary work (Cao et al., 2024) also studies assistive agents for vulnerable groups but focuses only on indoor scenarios with oracle symbolic observations. In contrast, our proposed CHAIC Challenge features both indoor and outdoor scenarios, an egocentric visual observation, newly created physically constrained agents, and unexpected events, enabling rich, physics-driven interactions on real-world assistive tasks. Accessibility in AI Design People with disabilities or physical impairments are a central focus area of study in robotics, including care for wheelchair users, elderly users, and users with aging-related ailments like dementia (de Saille et al., 2022; Sundaresan et al., 2022; Broadbent et al., 2009; Benda et al., 2020; Cooper et al., 2016; Lee et al., 2017). These works often study the best ways to design for inclusivity; in other words, how to best build assistive robots to handle the explicit physical needs of the users in question (Benda et al., 2020). We build on these design principles to create the first-ever large-scale embodied intelligence environment that explicitly models such impairments. 3 The Constrained Human-AI Cooperation (CHAIC) Challenge The Constrained Human-AI Cooperation (CHAIC) Challenge seeks to study how embodied agents perform in terms of social perceptions of human partners with diverse physical constraints and cooperative planning abilities within rich scenarios. Built on top of ThreeDWorld, a realistic 3D embodied AI platform, we design and implement four new agents with real physical constraints (Section 3.1) and eight tasks featuring both indoor and outdoor scenes, including emergencies (Section 3.2). For each task, there is a constrained agent mimicking a human partner with capability constraints, trying to find and transport some target objects to a specific goal location, and a helper agent trying to infer the constrained agent’s goal and capability constraints through active perception of its behaviors to assist the constrained agent better. The success of the helper agent is measured by the ratio of target objects successfully transported by both of them. Figure 2 provides an overview of the challenge, with further details in Section 3.3. 3.1 Constrained Agents To enable the testing of embodied social intelligence with a diverse set of potential human partners, we have created four new simulated agents that may face physical constraints such as limited height, strength, and movement speed, reflective of real humans. 3 Figure 2: Overview of CHAIC challenge: We present four agents with diverse capability constraints and eight tasks built around these constraints, featuring both indoor and outdoor scenarios. The tasks are named no constraint, high container, high goal location, high target object, obstacle, low target object, shopping, and moving house (four of them are shown on the left). In each task, there are objects, containers, and a goal location. A helper agent needs to infer the partner’s intents and constraints from its egocentric observations (shown on the right) and make a tailored plan to assist the partner in transporting the intended target objects to the goal location using containers as tools. Each agent possesses two properties: reaching range and strength. An agent can successfully interact with objects whose heights are within its reaching range and whose weights are lighter than its strength limit. When an agent attempts an action that exceeds its capabilities, the action does not fail immediately but instead has a success rate. This rate is calculated using the formula exp(−δ/α)/β, where δ represents the excess amount, and α and β are constants. If an action exceeds multiple capability thresholds, the probabilities of success are multiplied. We have developed the following constrained agents: • Child Agent: A small child with a height of 1.2 m that has a reaching range of [0, 1.5] m. • Wheelchair Agent: An agent confined to a wheelchair or limping that may be blocked by obstacles in the house (e.g., a couch). Its reaching range is [0.25, 1.5] m. • Bicycle Agent: An agent walking with a bike that moves slowly. It must first dock the bike when picking up an object. The child accompanying it may run away, causing an emergency. • Frail Agent: An agent that is less capable of lifting heavy objects (e.g., furniture) and has only 1/6 the strength of a normal agent. 3.2 Tasks with Constrained Agents We designed eight tasks featuring indoor and outdoor scenes, including emergencies, in our CHAIC benchmark, utilizing the various constrained agents introduced earlier. Information about each task is shown in Table 1. 3.3 Challenge Details In CHAIC, an embodied helper agent Ah is tasked to infer the goal G and the constraints of a constrained agent Am and assist Am in finding and transporting a set of target objects Ot from random locations to a goal location Lg. There are containers scattered in the environment, which the agents can use to transport more objects simultaneously. An agent could take two objects at a time without a container, and the capacity of a container is set to three. Formally, a task in the challenge is defined by the goal G of the constrained agent Am (i.e., a set of goal predicates describing the final desired state) and an initial environment E where the helper agent Ah is placed alongside the constrained agent Am to complete the task. The ground truth goals and constraints of the constrained agent are hidden from the helper agent Ah, thereby explicitly motivating the need for active perception for the agent to infer intents and constraints. 4 Table 1: Tasks with constrained agents, including both indoor and outdoor scenes and rich features. Task Name Scene Description Agent Type Features No constraint Indoor Main agent with no constraints Normal agent N/A Low target Indoor Target objects on the ground Wheelchair agent N/A Obstacle Indoor Obstacles between most rooms Wheelchair agent Existence of obstacles High target Indoor Target objects in high places Child agent Fragile high targets may break High goal location Indoor Goal locations in high places Child agent Fragile high targets may break High container Indoor Containers in high places Child agent Fragile high targets may break Shopping Outdoor Main agent walks a bike with his child while shopping Bicycle agent Emergency event: the child runs away Moving house Outdoor Main agent moves all the furniture onto the truck Frail agent Agents can cooperate to lift furniture together Observation Space In CHAIC, actions may take several frames to finish and are executed asynchronously between agents. The agent will receive the following observation after its action is finished or failed: • Egocentric RGB-D Image: The agent receives 512 × 512 egocentric color and depth images, as shown in Figure 2. • Self-State: The agent is provided with information relevant to itself, including its current location, orientation, and the objects in its possession. Action Space The action space consists of three low-level navigation actions (move forward, turn left, turn right), three basic interaction actions (pick up A, put A in B, put A on B), and one idle action (wait). 3.3.1 Task Generation Indoor Task To generate an indoor task, a floorplan configuration with six to eight interconnected rooms and a target task is initially sampled from predefined sets. For each scene, objects related to goals in the predefined set are placed on low surfaces such as tables, chairs, sofas, and floors for low objects, and higher surfaces like cabinets or refrigerators for high objects. However, only a subset of the objects is the target object set. The target object set is a set that includes all objects related to a specific type like food or fruit, one object randomly selected from a non-target set, and two additional fragile vases if the task is a high-target task. The number of targets is around ten. Then, a goal location and up to six containers are added to the scene based on available space and task constraints. We randomly initialize two agents (one constrained agent Am and one helper Ah), and each agent is placed in a free space at least 0.5 meters away from the nearest wall. This setup ensures sufficient initial distance between the agents and the walls, allowing unrestricted movement at the beginning of the task. Outdoor Task The generation of outdoor tasks is largely the same as the indoor task generation. For the shopping task, six shops are generated and spread out on both sides of the road, and each shop sells one specific category of items. The goal location of the shopping task is a fixed, predetermined place in front of the bicycle agent’s house. In the moving house task, the target objects include five pieces of furniture on the road in front of a house. The goal location is a truck parked nearby. The details of outdoor task generation can be found in Appendix G. 5 Figure 3: LLM+BM Helper Implementation Pipeline: An overview of the LLM+BM Helper with specific modules for Perception, Behavior Modeling, Decision, and Execution. (1) The perception module detects objects from raw RGB images; (2) the memory module builds the semantic map of the environment using depth images and records behaviors; (3) the behavior modeling module recognizes the action of the partner and localizes the object corresponding to the action; (4) the decision module decides plans for the next steps using foundation models; and (5) the execution module generates low-level actions. 3.3.2 Dataset Construction For each of the eight tasks, we create 12 episodes for training and 12 episodes for testing, resulting in approximately 200 episodes in total. We ensure that the environments of the test set are different from those of the training set. We randomly sample the initial starting states for each task. An episode terminates when all goal predicates of the task are satisfied or when the maximum time step horizon T = 3000 frames is reached (for the moving furniture task, the maximum time step horizon is T = 1500). 4 Language Agent Augmented with Behavior Modeling Module We also introduce a new agent framework combining the prowess of action recognition models and the reasoning ability of large language models (LLMs). Due to their simplicity and generalization ability, LLMs can also be implemented in other environments or the real world. We built a behavior modeling module, which models the behaviors of the constrained agent via an action recognition model and incorporated it into the CoELA framework (Zhang et al., 2023) with four other modules: (1) the perception module, which transforms the raw RGB-D observations into structured semantic maps via an object detection model; (2) the memory module, which saves all the history information in a structured manner; (3) the decision module, which generates high-level plans and is driven by large language models; and (4) the execution module, which turns the generated plans into low-level actions. More details regarding these modules can be found in Appendix B.1. Figure 3 shows an overview of the framework. 4.1 Behavior Modeling Module To infer the intents and inabilities of constrained agents, the behavior modeling module extracts constrained agents’ actions and status from a sequence of egocentric images (i.e., a video). The behavior modeling module contains two parts: action recognition and action grounding. Action Recognition We adopt an action recognition model to enable the helper agent to recognize the actions of the constrained agent. We select the TSN model (Wang et al., 2016) pretrained on Kinetics-400 (Kay et al., 2017) as the base video action recognition model. There are four types of actions: pick up, put on, put in, and walking (including move forward, turn left, and turn right). Each action may be successfully executed or fail (except for put in and walking, which are always 6 successful), so there are six classes in total. We collect data by having an agent follow the constrained agent to observe its behaviors while executing a task and store the action video clips in the training set. During testing, the helper agent utilizes this model to recognize the actions of the constrained agent when it is within observation. The training details can be found in Appendix D.2. Action Grounding After the helper recognizes the action of the constrained agent, it looks up the semantic map in the memory module for the predicate of the action. For example, when the action is pick up, the predicate will be the nearest object to the constrained agent. Finally, the behavior modeling module identifies the action, the predicate of the action, and the status of the action of the constrained agent. 5 Experiments 5.1 Setup 5.1.1 Constrained Agent Policy The constrained agent takes ground truth object segmentation as observation to mitigate the impact of imperfect visual perception on performance and chooses actions based on a rule-based high-level planner designed with handwritten rules by human experts. At the beginning of an episode, the constrained agent will explore the environment to find more target objects, containers, and the goal location. Whenever the agent finds target objects or containers, it will pick them up if it has free hands. If more than 50% of the time steps are left and it does not have a container in hand, its priority will be to pick up a container; otherwise, it will pick up a target. If the agent cannot carry more objects, it will put the object on the goal location. If less than 25% of the time steps is left (37.5% if it has not found the goal location yet) and the agent is carrying a target object, it will put the object on the goal location immediately since it is often a long walk to the goal location. When selecting possible targets, the agent will opt for the closest one if multiple options are available. Moreover, at any time, if the agent can put an object in a container, it will do so. 5.1.2 Evaluation Metrics To evaluate the success of helper agents, we measure the following three metrics: • Transport rate (TR): The percentage of target objects that the agents successfully transported. We also calculate the Efficiency Improvement (EI) of having the helper as ∆M/M0, where ∆M denotes the increase in the transport rate after adding the helper, and M0 denotes the larger of the transport rates of the team or the constrained agent alone, for numerical stability. • Goal Inference Accuracy (IA): The ratio of target objects successfully transported by the helper to the total number of objects transported by the helper. • Emergency Rate (ER): For the shopping task, we calculate the ratio of frames where the child agent is away from the constrained agent to measure the helper agent’s ability to handle emergencies. 5.2 Baselines We test four types of planning-based helpers: Random Helper, Rule-Based Hierarchical Plan Helper (RHP), LLM+BM Helper, and VLM Helper. All the helpers share the same Perception Module, Memory Module, and Execution Module as the language agent introduced in Section 4, but the critical differences lie in the high-level planner. Meanwhile, an Oracle Helper is tested to demonstrate the upper-bound performance. Below is the description of each type of helper: • No Helper (w/o): The constrained agent performs the task solely without assistance from a helper. • Random Helper: A naive helper randomly selects a plan from a list of valid plans. • Rule-Based Hierarchical Plan Helper (RHP): This helper uses prior knowledge of the task and relies on handcrafted rules by human experts to make plans to assist the constrained agent in completing the task. Further details on the rules can be found in Appendix B.2. 7 Table 2: Quantitative results on CHAIC benchmark. We report the average Transport Rate (TR), Efficiency Improvement (EI) and Goal Inference Accuracy (IA) here. w/o means the main agent does the task solely without a helper. The Emergency Rate (ER) metric is also reported for the shopping task. Indoor Helper Agent No Constraint High Target High Container High Goalplace TR(EI)↑ IA↑ TR(EI)↑ IA↑ TR(EI)↑ IA↑ TR(EI)↑ IA↑ w/o 0.53 / 0.30 / 0.37 / 0.28 / Random 0.52(-0.02) 0.24 0.27(-0.05) 0.29 0.36(0.00) 0.25 0.33(0.10) 0.14 RHP 0.64(0.15) 0.15 0.35(0.11) 0.29 0.45(0.19) 0.21 0.35(0.18) 0.21 VLM (GPT-4o) 0.63(0.14) 0.24 0.33(0.06) 0.32 0.43(0.12) 0.40 0.26(-0.20) 0.33 LLM (GPT-4) + BM 0.65(0.17) 0.25 0.38(0.19) 0.29 0.49(0.24) 0.30 0.36(0.23) 0.35 Oracle 0.77(0.31) 0.88 0.49(0.37) 0.91 0.69(0.47) 0.91 0.61(0.56) 0.90 Indoor Outdoor Helper Agent Low Target Obstacle Shopping Furniture TR(EI)↑ IA↑ TR(EI)↑ IA↑ TR(EI)↑ IA↑ ER↓ TR(EI)↑ w/o 0.51 / 0.07 / 0.37 / / 0.17 Random 0.50(-0.01) 0.31 0.21(0.56) 0.24 0.39(0.05) 0.34 0.32 0.48(0.68) RHP 0.66(0.23) 0.28 0.44(0.77) 0.17 0.49(0.22) 0.44 0.30 0.65(0.72) VLM (GPT-4o) 0.69(0.26) 0.46 0.40(0.86) 0.35 0.50(0.25) 0.72 0.39 0.70(0.78) LLM (GPT-4) + BM 0.70(0.27) 0.43 0.42(0.89) 0.47 0.58(0.33) 0.74 0.38 0.69(0.77) Oracle 0.82(0.38) 0.91 0.60(0.87) 0.82 0.61(0.39) 0.87 0.17 0.76(0.80) • LLM+BM Helper: A language agent augmented with a Behavior Modeling module introduced in Section 4. Example prompts can be found in Appendix C.1. We use GPT-4 as our decision-making LLM. • VLM Helper: A vision-language agent similar to the LLM+BM Helper. The last 10 frames of egocentric RGB-D observation are added as visual inputs to perceive the constrained agent. We use GPT-4o as our decision-making VLM.2 • Oracle Helper: An oracle helper that knows the ground truth goal, as the ground truth object segmentation, and the task progress. It behaves the same way as the RHP and is close to the upper-bound performance a helper could achieve. We also tested some learning-based methods like reinforcement learning and SmartHelp (Cao et al., 2024), whose results can be found in Appendix E.3. 5.3 Main Results We conducted an extensive evaluation by deploying four baseline models across eight distinct constraint settings and measured four specific metrics, as outlined in Section 5.1.2. The results are presented in Table 2. Overall, the LLM+BM Helper emerges as a strong baseline, achieving the highest transport rate (TR) in 6 out of 8 tasks, the most significant efficiency improvement (EI) in 7 out of 8 tasks, and the best goal inference accuracy (IA) in 4 out of 8 tasks. Behavior Modeling Analysis Our LLM+BM Helper achieves a reasonable IA metric compared with other helpers, which shows our behavior model successfully models the partner’s behaviors to some extent. However, compared with the Oracle Helper, all the other baseline agents perform poorly on the IA metric. The IA metric reflects whether the helper successfully determines the needs of the constrained agent, so the gap shows all our baselines do not work well in inferring the behavior of the constrained agent from the raw RGB-D image sequence. Nevertheless, our fine-tuned action recognition model achieves 86% accuracy on the validation set (See Appendix D.2 for the action recognition model details). Two reasons contribute to the discrepancy: (1) Due to blocking or distance, the action clip received by the helper may be incomplete or out-of-distribution from training data. (2) The current LLM-based decision module is insufficient to balance observing the partner’s behavior and acting independently. 2The main experiments are carried out between May 28 to Jun.5, 2024. 8 LLM Can Infer Goals Correctly and Perform Actions Properly In analyzing some of the chain-of-thought outputs of LLM, we observe that the LLM-based helper can accurately infer the target objects desired by the constrained agent and formulate appropriate plans to collect them. For instance, in an outdoor shopping scene, the bike agent named David seeks some fruit. Initially, the LLM helper assesses, “Since David hasn’t picked any object yet, it’s challenging to precisely determine his target objects.” It then realizes, “No matter what object David wants, the best first step would be to maximize the efficiency of carrying objects by using a container,” and subsequently proceeds to pick up a container. Upon observing the bike agent picking an apple, the LLM helper deduces, “Considering the constraints and the objects David has shown interest in (i.e., an apple), the best course of action from the provided list would be to ‘goto and pick up target <apple>’.” With a container and a target object in both hands, the LLM helper notes, “Considering I am currently holding two target objects (one directly and one in a container), the optimal next action is to put the object in your one hand to the container in your other hand. This action will free up one of my hands, allowing me to pick up more target objects and transport them efficiently to the goal.” Meanwhile, the LLM helper is capable of picking other fruits besides apples, demonstrating its accuracy in inferring the object category. After freeing up one hand, the LLM helper states, “Based on the observed actions and status of David, it’s clear that his target objects are fruits, specifically apples...so picking up more grapes aligns with the goal.” Finally, after collecting several fruits and having both hands full, the LLM helper concludes, “the best action to take next is to ‘transport object in hand to goal space’. This action involves taking the container filled with target objects, along with the additional grape in the other hand, to the specified goal location.” The detailed analysis of these chain-of-thought outputs is shown in Appendix F.1. Dealing with Emergencies In outdoor shopping tasks, the helper needs to handle unpredicted emergencies, requiring swift responses. The Emergency Rate (ER) metric shows that even if LLMand VLM-based helpers can achieve high scores in normal tasks, they cannot handle emergencies as efficiently as RHP. To improve, some rule-based control may be required in LLM- and VLM-based helpers to help them prioritize and respond more effectively in urgent situations. Failure Case Analysis During the experiment, we discovered some common failure situations leading to poor performance, which might be helpful for further helper design. • Spatial Information Analysis: The LLM-based agents do not understand spatial information very well when provided with text inputs of object locations. They often choose a distant object rather than a nearby one, even if they share the same name. Additionally, they often underestimate the cost of reaching the goal location and fail to transport due to time limits. • Acting without Cooperation: In the obstacle task, a reasonable solution for the helper is to remove obstacles first to free the constrained agent. However, LLM- and VLM-based helpers often transport objects alone without assisting the constrained agent, leading to relatively bad performance in this task. • VLM is Unable to Infer the Targets Needed by Constrained Agents: In certain tasks, the VLM Helper baseline performs worse than both the random baseline and the No Helper baseline. This is primarily because VLM cannot accurately infer the preferred objects of constrained agents when they observe them picking up items, leading it to consistently follow. Consequently, the VLM Helper fails to transport any objects, making it less effective than randomly transporting some objects, as done by the random helper. Additionally, frequently following the constrained agent can interfere with their actions—such as blocking their path—resulting in the VLM Helper baseline sometimes performing worse than having no helper. However, the LLM+BM Helper transports some objects even if it does not infer the goal correctly from the BM model, achieving a relatively higher score than the VLM Helper. 6 Conclusion In this work, we proposed an accessibility-centered embodied social intelligence challenge: the Constrained Human-AI Cooperation (CHAIC) Challenge. This challenge includes four new agents with physical constraints and eight long-horizon tasks featuring both indoor and outdoor scenes, designed to test the critical skills of social perception and cooperation in embodied agents. Our experimental 9 results benchmarking both planning- and learning-based baselines illustrate the systematic evaluation that such a benchmark can provide for future efforts. We further perform an in-depth analysis of failure cases and provide insights for the future development of embodied social intelligence. Limitations While we aimed to preserve as much realism as possible, there are undoubtedly aspects of human behavior, particularly in how physical constraints manifest in the world, that are challenging to simulate. Meanwhile, the rule-based control of constrained agents makes their behavior lack diversity. This may be solved by leveraging LLMs to control constrained agents. Moreover, while we believe our challenge takes a good first step forward in introducing accessibility challenges to embodied social intelligence benchmarking efforts, we emphasize that our challenge is not representative of all possible constraints that such users may face. Acknowledgement We thank Qinhong Zhou for his insightful feedback and help with paper writing, and Jeremy Schwartz and Esther Alter for setting up and updating the ThreeDWorld environments. 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Tenenbaum, and Chuang Gan. Hazard challenge: Embodied decision making in dynamically changing environments, 2024. Yuke Zhu, Daniel Gordon, Eric Kolve, Dieter Fox, Li Fei-Fei, Abhinav Gupta, Roozbeh Mottaghi, and Ali Farhadi. Visual semantic planning using deep successor representations. In Proceedings of the IEEE international conference on computer vision, pages 483–492, 2017. 13 A More Information about the CHAIC Challenge A.1 Benchmark Usage The CHAIC Challenge can be accessed on the CHAIC GitHub as well as on its official webpage. A.2 Comparison with Other Embodied Challenges We compare the differences between our proposed challenge and others in Table 3. Our CHAIC Challenge represents the first large-scale embodied social intelligence challenge focused on accessibility, incorporating outdoor scenes with emergent events, and requires goal inference from observations. Table 3: Comparison between Various Embodied Challenges. Social rearrangement assumes oracle position information of both agents and the target objects. Smart Help assumes perfect perceiving of the other agents’ actions and statuses. Challenge Name Accessibility Multi-Agent Goal Observation Outdoor Emergent Setting Support Inference Type Scenes Event Watch-and-Help (Puig et al., 2021) × ✓ × Symbolic × × NOPA (Puig et al., 2023a) × ✓ ✓ Symbolic × × Social Rearrangement (Szot et al., 2023) × ✓ × Visual × × TDW-MAT (Zhang et al., 2023) × ✓ × Visual × × Hazard (Zhou et al., 2024) × × × Visual ✓ × Smart Help (Cao et al., 2024) ✓ ✓ ✓ Symbolic** × × CHAIC (Ours) ✓ ✓ ✓ Visual ✓ ✓ B Baseline Details B.1 More Details on LLM+BM Helper We have tested several baselines in the benchmark, and the LLM+BM helper has shown the best performance. The LLM+BM helper consists of multiple modules. In addition to the behavior modeling module discussed in the main paper, the helper has several other modules, which we introduce here: Perception Module The perception module is used to extract useful information from raw RGB-D images. Following (Zhang et al., 2023), we have fine-tuned a Mask R-CNN (He et al., 2017) model on the images collected in the training dataset to obtain object-wise segmentation masks. The training set contains the same kinds of objects but with different layouts and scene backgrounds compared to the test set. The training details are in Appendix D.1. Memory Module The memory module is designed to understand the environment’s layout and the positions of objects by an occupancy map and a semantic map. Firstly, the agent continuously updates a 2D grid-based top-down occupancy map when executing actions. Initially, all areas on the occupancy map are marked as unknown, and the map is updated using depth images. Utilizing depth images and camera intrinsics, the memory module first maps each pixel from the depth image into 3D space and then projects it onto the occupancy map. The semantic map, which also adopts a top-down view and maintains the same grid size as the occupancy map, records the locations of all detected objects within the grid. Decision Module An LLM-based decision module is used to generate a subgoal without any prior knowledge or specific design. The prompt encompasses six components: task description, self-information, information about other agents, task progress, semantic map information, and available plans. The LLM’s output should be a plan from the list of valid plans, with specific object IDs included if the plan involves actions like "pick up" or "put on". Detailed information about the prompt is available in Appendix C.1. Execution Module The execution module is a low-level executor, which serves as a low-level executor, bridging the gap between high-level plans and low-level actions, including navigation and exploration. When the high-level plan directs the picking up of a previously seen object or traveling 14 to a goal space, the navigation module can recursively generate a path with low-level navigation commands using the occupancy map generated from the memory module. This map distinguishes between free, unknown, occupied, and wall spaces, assigning increasingly higher costs respectively. The navigation module employs the A* algorithm (Hart et al., 1968) to determine the most efficient route from the agent’s current location to the target object and generates the necessary actions to follow this path. The path is recalculated whenever the occupancy map is updated. Additionally, for exploration tasks, the navigation module utilizes the frontier exploration method to enhance the efficiency of discovering new areas by repeatedly moving toward unknown spaces adjacent to known ones. B.2 Detailed Rules for Rule-Based Hierarchical Plan Helper (RHP) The rule for Rule-Based Hierarchical Plan Helper (RHP) is similar to that of the constrained agent described in Section 5.1.1. The only difference is that since the helper does not know the exact goal of the constrained agent, the ruled-based agent will choose the target object randomly among all available objects. In detail, at the start of an episode, the rule-based agent explores the environment to locate objects, containers, and the goal location. It picks up objects or containers if its hands are free. If over 50% of the time steps remain and the agent does not obtain a container, it prioritizes acquiring one; otherwise, it focuses on objects. When unable to carry more, it deposits objects at the goal location. If less than 25% of the time steps is left (or 37.5% without a goal location identified), it immediately places objects in hands at the goal. The rule-based agent chooses the nearest object when multiple are available and puts objects in containers whenever possible. C Prompt Details C.1 Detailed Prompt of Decision Module of LLM+BM Helper The LLM+BM Helper uses an LLM-based decision module to determine the plan for the next step. The prompt of the LLM-based decision module contains six parts: task description, self-information, information about other agents, task progress, semantic map information, and available plans. The decision module needs to select a plan and fill in the plan with proper parameters, and then the execution module will execute the plan. Following are prompt descriptions and examples of each part: C.1.1 Task Description The Task Description includes a detailed description of the task’s basic rules but does not explicitly show the constraints of the constrained agent. A prompt example is listed in Figure 4. C.1.2 Self-Information The Self Information contains information about the helper agent himself, including the previous actions of the helper with the statuses of these actions, the current position of the helper, and the objects that the helper is currently holding. A prompt example is listed in Figure 5. C.1.3 Information about Other Agents Information about Other Agents contains information about other agents, including the actions of the constrained agent that the helper has seen, together with their statuses, the objects that the helper has seen the constrained agent holding, the position of the constrained agent when the helper last saw him/her. For shopping tasks, it also includes the position of the child when the helper last saw her. A prompt example is listed in Figure 6. C.1.4 Task Progress The Task Progress contains all the objects that have been transported to the goal location, and the number of frames passed. A prompt example is listed in Figure 7. 15 Task Description Prompt Example You are Bob. A constrained human David is walking a bike with his left hand while accompanying his child. Your goal is to infer the target objects David wants from his actions , and help him transport as many wanted target objects as possible to <b03_fire_hydrant > (8882855) with the help of containers . Note that: - There are six shops in the environment , and the target objects are distributed in these shops. - David is accompanied by a bike and the bike has a basket , and David can put at most three things into the bike basket , but David has to move the bike with two hands and has to stop at a shop to put things into the basket. David moves slow. - You can hold two things at a time , and they can be either objects or containers. You can put objects into the container (only after the container is grasped) to hold more objects. - All objects are identified by a unique name and ID , e.g. <table > (712). - Actions cost several steps , and the maximum number of steps you can take is 3000. It may be costly to walk for long distance , so you need to transport objects to the goal location as early as possible. - A container can contain at most three objects , and will be lost once it is transported to the goal location. - Help David to supervise the child by following the child if she runs away. - David is trying to get the same kind of things , so you should pick the things that are of the same kind of the things that David picked. Figure 4: Task Description Prompt Example Self Information Prompt Example Your previous actions and status are: [(’ moving at frame 0’, ’ActionStatus .success ’), (’pick up wood_basket <7157967 > at frame 62’, ’ActionStatus .success ’), (’pick up grape <13682679 > at frame 134’, ’ActionStatus .success ’), (’put the object in the container at frame 198’, ’ActionStatus .success ’), (’pick up apple <4455088 > at frame 290’, ’ActionStatus .success ’), (’put the object in the container at frame 357’, ’ActionStatus .success ’)]. Your current position is: (9.78 , 3.26). You ’re holding a container <wood_basket > (7157967) with target objects <grape > (13682679) , <apple > (4455088) in it. Figure 5: Self Information Prompt Example Information about Other Agents Prompt Example David ’s previous actions and status are: [(’pick up orange <8822607 > at frame 433’, ’ActionStatus .success ’)]. You have seen David holding these objects: a bike with nothing in it , a bike with target object <apple > (15360225) in it , a bike with target objects <apple > (15360225) , <orange > (11935439) in it , a bike with target objects <apple > (15360225) , <orange > (11935439) , <orange > (8822607) in it. The last time you saw , David was at (3.63 , 2.39) (meters). The last time you saw , David ’s child was at (6.23 , -0.2) (meters). David ’s target objects and constraints should be infered from his actions and status. Figure 6: Other Agents’ Information Prompt Example C.1.5 Semantic Map Information The Semantic map information contains all objects, containers, and the goal location information in the semantic map, with their position and height. A prompt example is listed in Figure 8. 16 Task Progress Prompt Example Your current progress is: You ’ve taken 1460/3000 steps. You have found the goal position <b03_fire_hydrant > (14887175) . You and David have already transported <banana > (8826121) , <banana > (11770901) to the <b03_fire_hydrant > (14887175) . Figure 7: Task Progress Prompt Example Semantic Map Information Prompt Example You ’ve seen these objects: <orange > (11878369) is located at (8.86 , -1.94) (meters) with a height of 0.46 meters , <apple > (15231642) is located at (9.26 , -1.98) (meters) with a height of 0.42 meters , <apple > (9952058) is located at (9.71 , -2.0) (meters) with a height of 0.72 meters , <orange > (10130190) is located at (10.25 , -1.97) (meters) with a height of 0.44 meters , <orange > (8108449) is located at (10.53 , -1.91) (meters) with a height of 1.08 meters , <apple > (14366820) is located at (10.7 , 3.51) (meters) with a height of 0.77 meters , <grape > (15240406) is located at (10.92 , 3.48) (meters) with a height of 1.03 meters , <grape > (604227) is located at (10.88 , 3.67) (meters) with a height of 1.08 meters , <croissant > (4072) is located at (13.88 , -1.97) (meters) with a height of 0.37 meters , <burger > (9082737) is located at (14.25 , -1.98) (meters) with a height of 0.39 meters , container <wood_basket > (5304525) is located at (14.21 , -1.8) (meters) with a height of 1.13 meters , container <plastic_basket > (10421069) is located at (14.09 , 3.4) (meters) with a height of 1.13 meters , <b03_fire_hydrant > (14887175) is located at (1.73 , 5.55) (meters) with a height of 0.4 meters. Figure 8: Semantic Map Information Prompt Example C.1.6 Available Plans The Available Plans contains all the available plans that the helper agent can take. A prompt example is listed in Figure 9. Available Plan Prompt Example Given your goal , previous actions , progress , and objects you see , please choose the best action from the following action list to achieve your goal as soon as possible: [’explore ’, ’follow child ’, ’turn around ’, ’transport object in hand to goal space ’, ’goto and pick up target <orange > (11878369) ’, ’goto and pick up target <apple > (15231642) ’, ’goto and pick up target <apple > (9952058) ’, ’goto and pick up target <orange > (10130190) ’, ’goto and pick up target <orange > (8108449) ’, ’goto and pick up target <apple > (14366820) ’, ’goto and pick up target <grape > (15240406) ’, ’goto and pick up target <grape > (604227) ’, ’goto and pick up target <croissant > (4072) ’, ’goto and pick up target <burger > (9082737) ’, ’follow David ’]. Please choose one option from the list. Figure 9: Available Plan Prompt Example C.2 Detailed Prompt of Decision Module of VLM Helper The prompt of the decision module of VLM Helper is similar to that of LLM+BM Helper. The only difference is the VLM-based decision module perceives the behavior of the partner through raw RGB images as an additional observation. In detail, an image sequence with the last ten images is added to the input of the VLM-based decision module. 17 D Perception Model Details D.1 Detection Model for Object Detection Since the helper receives raw RGB-D images from the environment, an object detection model is necessary to identify objects within these images. We fine-tuned an object detection model using our dataset collected from training scenes. Data Collection To collect training data in the environment, a helper roams randomly and solely within the scenes, collecting egocentric images combined with ground truth segmentation. The environment is split into training and validation, and we collected 61K images at a resolution of 512 × 512 in total. There are 53 types of objects related to the benchmark, so the detection model has the same number of labels. Training Details We utilized the open-source code provided by MMDetection (Chen et al., 2019) as our training framework and selected a Mask R-CNN (He et al., 2017) model pre-trained on the COCO dataset (Lin et al., 2014) with a ResNet50 (He et al., 2016) backbone. The model was fine-tuned for four epochs, incorporating a warm-up stage of 500 steps and a batch size of 16. The optimizer employed was SGD with lr = 0.01, momentum = 0.9, and weight_decay = 0.0001. This finetuning process was finished on an NVIDIA A10G GPU in approximately six hours. The fine-tuned model achieved a 94.4% mAP@50 (Segmentation Mean Average Precision at 50% intersection over union) on the validation set. D.2 Action Recognition Model for Behavior Modeling Recognizing the actions of the partner is a crucial ability for understanding its intentions, while current foundation models cannot directly discern actions in the wild. Therefore, an auxiliary action recognition model is necessary for our baseline agents. Similarly to the detection model, we fine-tuned an action recognition model. Data Collection Collecting behavior data is more challenging than gathering object detection data. To simulate the real situation, we created a follower whose sole action is to track the constrained agent. This follower has access to the action history, which when indicating that the constrained agent is acting, triggers the extraction of an RGB image sequence from the observer’s viewpoint. These images are then concatenated into a video clip, each containing 50 to 100 frames at a resolution of 512 × 512. Sometimes the constrained agent is obscured, preventing full visibility throughout some actions. Consequently, we discarded any action clip where the visibility of the constrained agent was less than 20%. The dataset comprises six behaviors for the constrained agent: successful pick-up, fail pick-up, put-in, successful put-on, fail put-on and moving. In total, the dataset contains 3,000 video clips. Training Details We utilized the open-source tools provided by MMDetection (Contributors, 2020) for training and employed the Temporal Segment Network (TSN) (Wang et al., 2016), pre-trained on the Kinetics-400 dataset (Kay et al., 2017), as our base model with a ResNet50 backbone (He et al., 2016). The sampling strategy was set to 16 × 1 × 1 (number of clips, clip length, clip interval). We fine-tuned the model 100 epochs using the same optimizer as object detection and selected the best checkpoint from the validation set. The fine-tuned model achieved 86.1% top-1 accuracy on the validation set. E More Results E.1 Additional Metrics Besides the transport rate (TR), Efficiency Improvement (EI), and Goal Inference Accuracy (IA), we also calculated some meaningful metrics to measure the performance of helpers: • Completion Ratio of Helper (CR): The proportion of tasks completed by the helper relative to the total number of completed tasks; 18 Figure 10: Main Results with Error Bars: The visualization of the transport rate with 1-sigma error bar of the standard error. Table 4: Additional results on CHAIC benchmark. Here we report two more useful metrics: Completion Ratio of Helper (CR) and Standard Error of Transport Rate (STDTR). Commonly, a higher CR means the helper could do more parts in the tasks. Indoor Helper Agent Normal High Target High Container High Goalplace CR↑ STD CR↑ STD CR↑ STD CR↑ STD w/o / 0.03 / 0.02 / 0.03 / 0.05 Random 0.09 0.04 0.10 0.03 0.12 0.03 0.06 0.04 RHP 0.15 0.02 0.43 0.04 0.29 0.03 0.39 0.05 VLM (GPT-4o) 0.13 0.03 0.08 0.02 0.34 0.04 0.18 0.05 LLM (GPT-4) + BM 0.22 0.03 0.30 0.03 0.30 0.03 0.35 0.04 Oracle 0.51 0.03 0.64 0.04 0.66 0.03 0.73 0.04 Indoor Outdoor Helper Agent Low Target Obstacle Shopping Furniture CR↑ STD CR↑ STD CR↑ STD CR↑ STD w/o / 0.03 / 0.04 / 0.02 / 0.04 Random 0.09 0.04 0.09 0.04 0.07 0.02 0.73 0.05 RHP 0.36 0.03 0.19 0.04 0.34 0.02 0.74 0.04 VLM (GPT-4o) 0.39 0.02 0.17 0.03 0.34 0.03 0.82 0.05 LLM (GPT-4) + BM 0.38 0.03 0.45 0.05 0.46 0.03 0.78 0.05 Oracle 0.59 0.03 0.38 0.03 0.45 0.03 0.77 0.04 19 Table 5: Quantitative results of learning-based agents on CHAIC benchmark. We report the average Transport Rate (TR) and Efficiency Improvement (EI) here. w/o means the main agent does the task solely without a helper. Indoor Helper Agent Normal High Target High Container High Goalplace TR(EI)↑ TR(EI)↑ TR(EI)↑ TR(EI)↑ w/o 0.53 0.30 0.38 0.27 RL 0.45(-0.19) 0.26(-0.16) 0.28(-0.25) 0.25(-0.22) Smart-Help 0.46(-0.12) 0.24(-0.17) 0.26(-0.28) 0.31(0.01) Indoor Outdoor Helper Agent Low Target Obstacle Shopping Furniture TR(EI)↑ TR(EI)↑ TR(EI)↑ TR(EI)↑ w/o 0.51 0.08 0.37 0.17 RL 0.43(-0.16) 0.11(0.07) 0.32(-0.13) 0.67(0.74) Smart-Help 0.49(-0.04) 0.13(0.11) 0.32(-0.13) 0.57(0.70) • Standard Error of Transport Rate (STDTR): The standard error of transport rate among the test sets. The results of these metrics are shown in Table 4. E.2 Error Bar of Transport Rate We also visualize the 1-sigma standard error of transport rate for baseline helpers described in Section 5.2, shown in Figure 10. E.3 Comparison with Learning-Based Baselines Although the baselines in the main paper are all non-training baselines, we also tested some learningbased baselines. The results are shown in Table 5. However, we observe that in most tasks, our learning-based baselines perform similarly to the main agent operating without a helper (except for the outdoor furniture task where two agents make a crucial difference because of their different strength capacities and the relatively easy task setting). This is primarily due to the following reasons: • The inherent difficulty of vision-based reinforcement learning. Even with depth and segmentation information which we encoded as a semantic map, the agent is still hard to learn non-trivial features from the observations. • The slow data collection process and the inability to parallelize in ThreeDWorld make it time-consuming to gather large-scale online data for training. In our experiments, collecting rollouts of size 104 for each task requires one day on an NVIDIA A10G GPU. RL Baseline An end-to-end reinforcement learning helper trained on each task separately. We use the Stable-Baselines3 (Raffin et al., 2021) codebase and PPO (Schulman et al., 2017) algorithm to wrap and train our RL helpers. We made minor modifications to the observation space to fit our training needs. Namely, we pack the RGBD image, semantic map, agent position/direction, and status of agent-holding objects as a customized RL observation. The reward is designed as a linear combination of transported objects and distance to the nearest target object. A penalty is also applied for each invalid action. The policy network extracts features from each observation class with either CNN for images or MLP for scalar information and concatenates them. The concatenated features then pass through a two-layer MLP to produce a vector of length 64. This part is shared by both the actor and the critic in PPO. During training, we use a batch size of 2 and update the policy every 2 rollout step due to slow data collection. We use default training parameters in Stable-Baselines3, including γ = 0.99, λ = 0.95, lr = 3 × 10−4, etc. For each task, we train the model for 104 steps. 20 SmartHelp Baseline A helping method proposed by Cao et al. (2024). For the opponent modeling module, we set the window size w of the state feature to 5. The input of the opponent modeling is the observation of the helper agent, which contains information on each object in the helper’s view, as well as the constrained agent’s information, if visible to the helper. The information on each object includes the type, weight, position, and height. The information on the constrained agent includes position and orientation, the last action with its status, and the objects currently held by the agent. The helper receives the ground truth actions and statuses of the constrained agent. We run simulations with a random helper and an oracle constrained agent for 12 episodes each to collect the dataset of constrained agent trajectories used for training the opponent model. We collect 8355 trajectories after balancing, each containing observations at five discrete time points, together with the constrained agent’s goal and constraint. The constraint of the constrained agent is a 5-dimension vector, which includes the maximum and minimum heights the agent can reach, the maximum weight the agent can hold, whether the agent is holding a bike and whether the agent is confined to a wheelchair, with each dimension scaled to [0, 1]. The goal contains the type of the goal and possible target index. There are six types of goal: “explore”, “wait”, “pick”, “puton”, “putin”, “unknown”. If the helper agent doesn’t see the constrained agent for all five observations, the ground truth goal is set to “unknown”. We balance the number of data for each goal in the trajectory dataset. For some goals like “pick”, there is a target index for the goal indicating the type of the object it picks, and there are 53 types of objects in total. We trained the opponent model using the cross-entropy loss for goals and the mean squared error loss for constraints. We use the Adam optimizer with a learning rate of 1 × 10−6 and a batch size of 32. Finally, we achieved the goal prediction accuracy of 93%, and the mean squared error loss for the constraint is 0.02. After training the opponent modeling module, we use the same setting of the RL Baseline to train a policy module for 104 steps on each task. F Additional Qualitative Analysis of LLM+BM and VLM Helper Behaviors F.1 LLM+BM Helper Behaviors We analyze the chain-of-thought outputs of LLM and the actions of LLM+BM Helper, and Figure 11 is an example of the shopping task. LLM+BM helper knows to use containers to increase transport efficiency As shown in Figure 11a, at the beginning of the task, the helper did not see David, the constrained agent, pick anything yet. But he could reason to pick up the container to improve transportation efficiency. He saw two containers when he was planning and chose the container closer to his current position. LLM+BM helper can infer objects that constrained agent needs correctly As shown in Figure 11b, after seeing David picking up an apple, the helper could reason out that David wanted fruits, so he picked up an apple, and he could also pick up other fruits like grapes afterward. LLM+BM helper knows to put things in container continuously As shown in Figure 11c, after picking up the apple, the helper reasoned to put the apple into the container, freeing up one hand to pick up more target objects. LLM+BM helper knows when to transport the objects into goal space As shown in Figure 11d, after putting three objects into the container and picking an object in the other hand, the helper decided to transport them to goal space, the fire hydrant in front of the house of David’s. The helper analyzed that he could not get objects anymore while transporting these objects with maximum efficiency, and David needed the objects he transported. F.2 VLM Helper Behaviors We also analyzed some behaviors of the VLM Helper, and Figure 12 is an example of the high container task. 21 LLM+BM Helper Given that David … that he moves slowly, it would be efficient to collect objects that are of the same kind that David might choose based on his initial actions and status. Since David hasn‘t picked any object yet, it’s challenging to precisely determine his target objects… The best first step would be to maximize the efficiency of carrying objects by using a container. Among the visible containers, the <wood_basket> (7157967) is closer to my current position. Therefore, the best action to take first would be to 'goto and pick up container <wood_basket> (7157967)'. Given the goal is to help David transport as many wanted target objects as possible to the <b03_fire_hydrant> (8882855) and considering the constraints and the objects David has shown interest in (i.e., an apple), the best course of action from the provided list would be to 'goto and pick up target <apple> (4455088)'. This aligns with the objective of collecting similar kinds of things that David has picked, which are fruits in this case. … I currently hold an apple in one hand and a wood basket in the other hand. The optimal next action from the given options to achieve the goal as soon as possible, considering I am currently holding two target objects (one directly and one in a container), is to: put the object in my one hand to the container in my other hand. This action will free up one of my hands, allowing me to pick up more target objects and transport them efficiently to the goal. …the best action to take next is to 'transport object in hand to goal space'. This choice is optimal because: 1. The container is full, meaning I’m efficiently utilizing the container's capacity to transport as many target objects as possible in one trip. 2. Transporting the objects to the goal space now will minimize unnecessary steps and back-and-forth movement, adhering to the constraint of a maximum of 3000 steps. 3. Since David is collecting fruits (apples and oranges noted in his bike basket), and I have collected similar kinds (grapes and an apple), transporting these items aligns with the goal of assisting David by handling analogous target objects. a. helper picking up a container b. helper picking up an apple c. helper putting the apple to the container d. helper transport object in hand to goal space LLM+BM Helper LLM+BM Helper LLM+BM Helper Figure 11: LLM+BM Helper’s behaviors in one episode of the shopping task, together with the chain-of-thought outputs of the LLM-based decision module (Some of the outputs are omitted for space reasons). VLM Helper knows to explore at first As shown in Figure 12a, at the beginning of the task, the helper could not see the constrained agent. Therefore, the helper chose to explore to find the constrained agent and objects. VLM Helper knows to follow the constrained agent As shown in Figure 12b, after seeing the constrained agent, the helper chose to follow her to get the information about the things she needed. Then, he saw the constrained agent pick up bread. However, sometimes the VLM helper is unable to infer the objects needed by the constrained agent, causing him to consistently follow the constrained agent. VLM Helper sometimes infer objects that constrained agent needs correctly As shown in Figure 12c and Figure 12d, after seeing the constrained agent picking bread, the helper could collect other objects of this kind, like loaf bread and hamburger. VLM Helper cannot transport objects efficiently In this episode, the VLM Helper only transported one thing at a time to the goal space, which is inefficient. First, the helper didn’t use the container. Second, the helper didn’t use both hands to carry the target objects. The most efficient way is to hold one container with three objects in one hand and hold one object in the other hand, thus transporting four objects at a time. G Details of Outdoor Task Generation G.1 Details of Shopping Task Generation For the shopping task, six shops are generated and spread out on both sides of the road. Each shop sells one specific category of items. There are three categories of items, each of which is sold in exactly two stores. The item categories and specific items for each category are listed below: 22 a. helper explores at first b. helper seeing constrained agent picking bread c. helper picking loaf bread d. helper picking hamburger Figure 12: VLM Helper’s behaviors in one episode of the high container task • Fruit: apple, orange, grape, and banana. • Baked food: loaf bread, croissant, burger, and donut. • Drink: cola, pepsi, sprite, and fanta. For each episode, we first randomly select one category of items. Then randomly select several objects in this category, together with another object randomly selected in the other two categories as the target objects. The number of target objects is between 11 and 13. The goal location is a fixed, pre-determined place in front of the bicycle agent’s house. Then we randomly select three shops and put a container on each of them. Finally, we randomly initialize the two agents. Their initial positions are guaranteed at least 0.5 meters away from the nearest shop. G.2 Details of Moving House Task Generation We choose twelve common pieces of furniture for the moving house task. For each episode, we randomly select five pieces of furniture and randomly put them in the area in front of the house. We ensure that the furniture does not overlap each other. Then we set the initial positions of the agents near the place area. The goal location is a truck parked around 10 meters away from the place area. The frail agent’s strength is 100, while the helper agent’s strength is 600. The weights of furniture range from 50 to 900. H Statement H.1 Broader Impacts By building the CHAIC benchmark, our work tries to simulate the disability of human beings in a simulated environment. The benchmark proposes several tasks and scenes that are common in the real world. After setting up the benchmark, we built a few helper agents to help the constrained agents fulfill their tasks with visual observation only. This kind of helper agent has a wide range of potential usage in real life but there is not much research on this in academia. Through this research, we hope to pave the way to more friendly and helpful AI agents that can be implemented in both simulated environments and the real world. Potential Negative Impacts While some baselines are driven by foundation models and achieving the best scores in our experiments, applying them may generate malicious content and potentially 23 lead to bad actions. It is important to set up mechanisms to detect such actions before these helpers can be put into real-world usage. H.2 Responsibility The authors declare that they bear full responsibility for any violations of rights associated with this dataset. H.3 LICENSE The CHAIC benchmark is licensed under the MIT license. Meanwhile, the benchmark is built upon several open-source projects, and we list their licenses here: • ThreeDWorld: BSD-2-Clause license • MMAction2: Apache-2.0 license • MMDetection: Apache-2.0 license H.4 Resubmission Discussion The paper is withdrawn from CVPR2024, where the reviewers posted two main points: • The reviewers thought it was not natural that the helper could know the whole action history of the constrained agent. We agree with it, and currently, the helper cannot get any text information about the action history of the constrained agent and needs to infer it from raw RGB-D observation. • The reviewers thought the objects and tasks were not rich enough. Therefore, we create both indoor and outdoor scenes with various tasks in this submission, and the number of task-relevant objects increases from about 20 to over 50. I Datasheets I.1 Motivation • For what purpose was the dataset created? Was there a specific task in mind? Was there a specific gap that needed to be filled? Please provide a description. The dataset was established to research human-AI cooperation in embodied and realistic settings. • Who funded the creation of the dataset? If there is an associated grant, please provide the name of the grantor and the grant name and number. N/A. • Any other comments? No. I.2 Composition • What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)? Are there multiple types of instances (e.g., movies, users, and ratings; people and interactions between them; nodes and edges)? Please provide a description. One instance is a sequence of commands and the ThreeDWorld Platform can read the commands in the instance and then create the scenes and tasks. • How many instances are there in total (of each type, if appropriate)? There are 192 instances in total. There are 8 tasks, and each task contains 12 training instances and 12 testing instances. • Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set? If the dataset is a sample, then what is the larger set? Is the sample representative of the larger set (e.g., geographic coverage)? If so, please describe 24 how this representativeness was validated/verified. If it is not representative of the larger set, please describe why not (e.g., to cover a more diverse range of instances, because instances were withheld or unavailable). The dataset is a sample. We have an instance generation pipeline that can generate infinite instances for each task. • What data does each instance consist of? “Raw” data (e.g., unprocessed text or images) or features? In either case, please provide a description. Each instance consists of a command sequence used for task initialization in the ThreeDWorld Platform. • Is there a label or target associated with each instance? If so, please provide a description. No. • Is any information missing from individual instances? If so, please provide a description, explaining why this information is missing (e.g., because it was unavailable). This does not include intentionally removed information, but might include, e.g., redacted text. No. • Are relationships between individual instances made explicit If so, please describe how these relationships are made explicit. No. • Are there recommended data splits (e.g., training, development/validation, testing)? If so, please provide a description of these splits, explaining the rationale behind them. The training-testing split in the dataset has a ratio of 1:1. • Are there any errors, sources of noise, or redundancies in the dataset? If so, please provide a description. No. • Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)? If it links to or relies on external resources, a) are there guarantees that they will exist, and remain constant, over time; b) are there official archival versions of the complete dataset (i.e., including the external resources as they existed at the time the dataset was created); c) are there any restrictions (e.g., licenses, fees) associated with any of the external resources that might apply to a future user? Please provide descriptions of all external resources and any restrictions associated with them, as well as links or other access points, as appropriate. The dataset is linked to the ThreeDWorld Platform, a) which has been maintained and will be maintained for a long time at TDW GitHub, and we use its stable version 1.12.27. b) the complete dataset can be found at CHAIC GitHub. c) The 3D models used in the dataset can be publicly downloaded via Google Drive. The dataset is subjected to an MIT License. • Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor-patient confidentiality, data that includes the content of individuals’ non-public communications)? If so, please provide a description. No. • Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, or might otherwise cause anxiety? If so, please describe why. No. • Does the dataset relate to people? If not, you may skip the remaining questions in this section. No. Only humanoid agents are used in the dataset. • Any other comments? No I.3 Collection Process • How was the data associated with each instance acquired? Was the data directly observable (e.g., raw text, movie ratings), reported by subjects (e.g., survey responses), or 25 indirectly inferred/derived from other data (e.g., part-of-speech tags, model-based guesses for age or language)? If data was reported by subjects or indirectly inferred/derived from other data, was the data validated/verified? If so, please describe how. The data is not directly observable or reported. Data is created from an automatic scene generation pipeline and visualized via the ThreeDWorld Platform. • What mechanisms or procedures were used to collect the data (e.g., hardware apparatus or sensor, manual human curation, software program, software API)? How were these mechanisms or procedures validated? We use an automatic scene generation pipeline for each task to generate data. • If the dataset is a sample from a larger set, what was the sampling strategy (e.g., deterministic, probabilistic with specific sampling probabilities)? Random Sampling. • Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)? Since we automatically generated data, only authors were involved in the data collection process. • Over what timeframe was the data collected? Does this timeframe match the creation timeframe of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please describe the timeframe in which the data associated with the instances was created. The data was collected between 15 May and 10 June 2024. • Were any ethical review processes conducted (e.g., by an institutional review board)? If so, please provide a description of these review processes, including the outcomes, as well as a link or other access point to any supporting documentation. No. • Does the dataset relate to people? If not, you may skip the remainder of the questions in this section. No. • Any other comments? No. I.4 Preprocess/cleaning/labeling • Was any preprocessing/cleaning/labeling of the data done (e.g., discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? If so, please provide a description. If not, you may skip the remainder of the questions in this section. No. • Any other comments? No I.5 Uses • Has the dataset been used for any tasks already? If so, please provide a description. No. • Is there a repository that links to any or all papers or systems that use the dataset? If so, please provide a link or other access point. No. • What (other) tasks could the dataset be used for? Embodied behavior recognition, embodied object detection. • Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses? For example, is there anything that a future user might need to know to avoid uses that could result in unfair 26 treatment of individuals or groups (e.g., stereotyping, quality of service issues) or other undesirable harms (e.g., financial harms, legal risks) If so, please provide a description. Is there anything a future user could do to mitigate these undesirable harms? No. • Are there tasks for which the dataset should not be used? If so, please provide a description. No. • Any other comments? No I.6 Distribution • Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) on behalf of which the dataset was created? If so, please provide a description. Yes, the dataset is public and everyone is welcome to use it. • How will the dataset will be distributed (e.g., tarball on website, API, GitHub)? Does the dataset have a digital object identifier (DOI)? The dataset will be distributed via GitHub. Currently, the dataset does not have a DOI. • When will the dataset be distributed? Before NeurIPS 2024. • Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)? If so, please describe this license and/or ToU, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, as well as any fees associated with these restrictions. This dataset is licensed under the MIT License. • Have any third parties imposed IP-based or other restrictions on the data associated with the instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions. No. • Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any supporting documentation. No. • Any other comments? No. I.7 Maintainance • Who will be supporting/hosting/maintaining the dataset? Our CHAIC Team will maintain the dataset. • How can the owner/curator/manager of the dataset be contacted (e.g., email address)? We will release the non-anonymous version after review. • Is there an erratum? If so, please provide a link or other access point. There is no erratum yet, and the future erratum will be released in the GitHub repo if any. • Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)? If so, please describe how often, by whom, and how updates will be communicated to dataset consumers (e.g., mailing list, GitHub). This will be updated to GitHub when we find errors or make improvements. • If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g., were the individuals in question told that their data 27 would be retained for a fixed period of time and then deleted)? If so, please describe these limits and explain how they will be enforced. N/A. • Will older versions of the dataset continue to be supported/hosted/maintained? If so, please describe how. If not, please describe how its obsolescence will be communicated to dataset consumers. Yes, the old version will be kept in GitHub. • If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so? If so, please provide a description. Will these contributions be validated/verified? If so, please describe how. If not, why not? Is there a process for communicating/distributing these contributions to dataset consumers? If so, please provide a description. Others can contact paper authors when they want to contribute. • Any other comments? No. 28	2024	1414
4,459	Prospective Representation Learning for Non-Exemplar Class-Incremental Learning Wuxuan Shi1, Mang Ye1,2∗ 1School of Computer Science, Wuhan University, Wuhan, China 2 Taikang Center for Life and Medical Sciences, Wuhan University, Wuhan, China {wuxuanshi, yemang}@whu.edu.cn https://github.com/ShiWuxuan/NeurIPS2024-PRL Abstract Non-exemplar class-incremental learning (NECIL) is a challenging task that requires recognizing both old and new classes without retaining any old class samples. Current works mainly deal with the conflicts between old and new classes retrospectively as a new task comes in. However, the lack of old task data makes balancing old and new classes difficult. Instead, we propose a Prospective Representation Learning (PRL) approach to prepare the model for handling conflicts in advance. In the base phase, we squeeze the embedding distribution of the current classes to reserve space for forward compatibility with future classes. In the incremental phase, we make the new class features away from the saved prototypes of old classes in a latent space while aligning the current embedding space with the latent space when updating the model. Thereby, the new class features are clustered in the reserved space to minimize the shock of the new classes on the former classes. Our approach can help existing NECIL baselines to balance old and new classes in a plug-and-play manner. Extensive experiments on several benchmarks demonstrate that our approach outperforms the state-of-the-art methods. 1 Introduction In recent years, deep neural networks (DNNs) have achieved great success in static scenarios. Research attention is increasingly turning to extending the learning capability of DNNs to open and dynamic environments. An important aspect is to enable the network to accumulate knowledge from new tasks as the input stream is updated (i.e., incremental learning [1; 2; 3]). Whenever a new task arrives, it is costly to retrain the model with current and old data. Not to mention that the old data is not fully available. A typical alternative is to fine-tune the network with new data directly. However, this can lead to drastic performance degradation on previously learned tasks, a phenomenon known as catastrophic forgetting [4; 5]. While storing exemplars of each class is a simple approach to mitigate forgetting, it relies on the quality of saved exemplars and faces challenges in storage and privacy, especially for sensitive domains such as medical imaging. Hence, this paper focuses on coping with catastrophic forgetting during incremental learning without storing any old samples, which is called non-exemplar class-incremental learning (NECIL) [6; 7]. In NECIL, a serious challenge is to discriminate between old and new classes without access to old data. Most methods usually start considering handling conflicts between old and new classes only when new tasks arrive. While some methods use stored prototypes to model the distribution of old classes [8; 6; 9; 10], others extend the network structure to accommodate new classes [7; 11]. However, in the base phase (i.e., training on the first task), traditional training allows different classes ∗Corresponding Author: Mang Ye 38th Conference on Neural Information Processing Systems (NeurIPS 2024). to divide up all the embedding space, causing trouble for subsequent conflict resolution. As shown in Fig. 1, in the incremental phase (i.e., training on tasks after the first one), with the influx of new classes, there are overlaps of the old and new classes in the embedding space that are difficult to discriminate. Moreover, due to the unavailability of old class samples, handling this conflict with only new task data is intractable. Instead, we suggest addressing this issue by learning prospectively at the feature level, which requires a two-pronged effort in both the base and incremental phases. Traditional Training Prospective Learning Base Phase Class 1 Class 2 Class 3 ··· ··· ··· Incremental Phase Class 4 ··· Class 5 ··· overlap Reserved Reserved Reserved Figure 1: The traditional training paradigm in NECIL considers conflicts between old and new classes only when new classes arrive and is prone to overlap. We suggest prospective learning to reduce conflicts: (1) reserve space for unknown classes; (2) make the newly coming class embedded in the reserved space. Firstly, the model should make room in advance for the incoming classes in the future. Thus, the space of past classes does not need to be drastically squeezed when expanding new classes. To this end, during the base phase, we construct a preemptive embedding squeezing constraint to enforce intra-class concentration and inter-class reserved separation. Specifically, we push instances from the same class closer together and instances from different classes farther apart in a mini-batch. It allows more space to be reserved in the initial embedding space, thus making the model ready for future classes. Secondly, the model should minimize the shock and impact of the new classes on the past classes, i.e., embed the new classes into the reserved space as much as possible. However, achieving the desired embedding of new classes when the old class data is fully unavailable is difficult. Inspired by previous works [6; 10], we try to accomplish this using prototypes (typically the class mean in the deep feature space) saved for each old class. During the incremental phase, we propose a prototype-guided representation update mechanism. Concretely, we use the network learned from previous tasks to extract features from new task samples and project these features and the saved prototypes into a latent space. In the latent space, the new class features are pushed away from the region hosting the old class and embedded as much as possible in the reserved space with the help of prototypes. We guide the update of the current model representation through the latent space to reduce the shock of the new classes on the former classes. In summary, combining the above two ideas, our Prospective Representation Learning (PRL) scheme makes the following main contributions: • We impose a preemptive embedding squeezing constraint to reserve space for future classes by reinforcing intra-class concentration and inter-class reserved separation. • We propose a prototype-guided representation update strategy that utilizes the saved prototypes to reduce the impact of expanding new classes on old ones. • Extensive experiments on four benchmarks suggest the superior performance of our approach over the state-of-the-art. We also provide a detailed analys of our method. 2 Related Work 2.1 Class-Incremental Learning Mainstream CIL methods can be roughly divided into three categories: rehearsal-based methods, regularization-based methods, and structure-based methods. 2 Rehearsal-based methods store a portion of seen data in a fixed-size memory buffer and replay it as new data arrives. Based on the stored data, some works use knowledge distillation techniques to protect existing knowledge [12; 13; 14; 15], while others regularize the gradient to make more efficient use of the stored samples [16; 17; 18]. Additionally, several works design new strategies for memory management instead of simple random sampling. [19; 20; 21; 22]. Although rehearsal-based methods effectively mitigate catastrophic forgetting, they are encumbered by privacy concerns and become impractical under stringent storage constraints. Regularization-based methods estimate the importance of different parameters for past tasks and then limit the updating of these important parameters when learning new tasks [23; 24; 25; 26]. In incremental learning, the storage of importance weights becomes essential. However, these methods are encumbered by constraints on model parameters, consequently impeding knowledge transfer and leading to suboptimal performance, particularly in long-sequence task streams. Structure-based methods accommodate knowledge from new tasks by dynamically modifying the network structure. Some works extend the network by assigning new parameters of different forms to new tasks [27; 28; 29; 30]. While this approach adeptly manages extended task sequences and sustains the performance of established classes, the linear growth of network parameters with the number of tasks and the necessity for reasoning across multiple forward propagations pose significant challenges. Parameter fusion [31] and selecting partial parameters for expansion [32] mitigates this problem to some extent. An alternative is to mask part of the parameters that are highly correlated with the previous task at the parameter level or unit level [33; 34; 35; 36]. Their performance is limited by the backbone obtained on the first task. 2.2 Non-Exemplar Class-Incremental Learning Recently, some works begin to focus on NECIL [8; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46], due to privacy and memory concerns [47; 48; 49], where the algorithms have no access to any past data. Li et al. [50] combine knowledge distillation with fine-tuning in a first attempt at incremental learning without a memory buffer. Zhu et al. [38] propose class augmentation and semantic augmentation to address the representation bias and classifier bias caused by the lack of old task data. Yin et al. [37] use model inversion technology to generate samples from previous tasks to alleviate forgetting. Based on [37], Gao et al. [39] introduce relation-guided knowledge distillation to address the distributional gap between generated data and real data. Zhu et al. [6] combat catastrophic forgetting for the first time by preserving prototypes and augmenting them. Yu et al. [8] address the problem of prototype outdating in the current representation space by estimating the semantic drift of past tasks and compensating for it. Furthermore, Toldo et al. [9] subdivide the drift into feature drift and semantic drift and compensate for both, thereby achieving better results. Shi et al. [10] inject information about the current feature distribution into the prototype to model the distribution of past tasks. Wang et al. [51] improve the prototype augmentation method based on density to make the model more focused on features of the old class with low density. Malepathirana et al. [52] use the domain information obtained from topological relations to optimize prototype augmentation to reduce inter-class overlap. However, previous works deal with conflicts between old and new classes only after the new data arrives and lack prospective consideration. 2.3 Embedding Space Regularization Embedding space regularization has been extensively studied in literature [53; 54; 55; 56; 57]. Chaudhry et al. [53] first propose learning tasks in different (low-rank) embedding subspaces that are kept orthogonal to each other. They learn an isometric mapping by formulating network training as an optimization problem on the Stiefel manifold. Another idea is to implement orthogonality in the gradient space. Saha et al. [58] analyze network representations after learning each task with Singular Value Decomposition (SVD) to find the basis of the subspaces and store them in memory. Moreover, several methods promote forward compatibility through regularization in the initial phase. Zhou et al. [59] assign virtual prototypes to compress embeddings of known classes and reserve space for new classes. Shi et al. [60] encourage initial CIL learners to generate representations that are similar to those of models trained jointly on all classes. Compared to previous works, we target CIL in exemplar-free scenarios (NECIL). We consider how to resolve conflicts between new and old classes during the incremental phases, in addition to reserving space in the initial phase. 3 3 Methodology 3.1 Problem Statement The goal of NECIL is to continually train a unified model over a series of tasks to recognize all classes learned so far. The data stream can be defined as D = {D0, D1, . . . DT }, where T is the number of incremental phases. At any phase i, the training set Di consists of the sample set Xi(0 ≤i ≤T) and the label set Yi. In particular, the classes of all phases are disjoint, i.e., Yi ∩Yj = ∅, ∀i ̸= j. It is notable that only Dt is available at current phase t. There are no old training sets (i.e., D0:t−1) to access or save in memory. To facilitate analysis, we represent the model with two components: a feature extractor F with parameters θ and a unified classifier G with parameters φ. For a comprehensive evaluation of the model, the test set at phase t includes classes from all the seen label sets Y0 ∪Y1 ∪. . . ∪Yt. At the time of testing, the model does not have access to the task ID, i.e., it does not know from which task the test sample come. 3.2 Baseline We adapt the paradigm of existing NECIL works [6; 7; 11; 10] as our baseline, which primarily uses knowledge distillation and prototype rehearsal. Specifically, at the base phase (i.e., t = 0), the classification model is optimized under full supervision: argmin θt,φt Lt = Lce(θt, φt; Dt) = − E (x,y)∼Dt[y · log (Gφt(Fθt(x)))], (1) where Lt represents the overall loss function, Lce is the cross-entropy loss. At the incremental phase (i.e., t > 0), standard fully supervised training seeks to minimize the following objective: Lt = Lce(θt, φt; D0:t−1) + Lce(θt, φt; Dt). (2) However, this is especially challenging since previous training sets D0:t−1 are assumed to be unavailable in the NECIL setting. The absence of the first term in eq. (2) leads to a bias in favor of current classes in the feature extractor Fθt and the classifier Gφt. To address this problem, existing methods [38; 6; 10] adopt knowledge distillation and prototype rehearsal to cope with the bias. Specifically, they take the frozen feature extractor Fθt−1 from the previous phase t −1 as a teacher and the current one Fθt as a student. A distillation term is introduced to encourage the model to mimic the previous representation: Lkd(θt; θt−1, Dt) = X x∈Xi ∥Fθt(x) −Fθt−1(x)∥2, (3) where ∥· ∥2 denotes Euclidean distance. Knowledge distillation helps maintain existing knowledge in Fθt−1, thus mitigating bias in the current feature extractor. For the bias in the classifier, we use class-representative prototypes [6] to balance the optimization. Specifically, after the training of t −1 phase, we compute a prototype pc for each class c: pc = E (x,y)∼Dt−1[Fθt−1(x) \| y = c]. (4) All prototypes of learned classes P 0:t−1 = {pc, c}c∈Y0:t−1 are stored in memory. In each training iteration of current phase t, existing works [6; 38; 10] augment the memorized prototypes P 0:t−1 to ˜P 0:t−1 and train the classifier jointly with the current data Dt. In particular, the prototypes are involved in the standard classification optimization with the following objective: Lpro(φt; ˜P 0:t−1) = − E (˜pc,c)∼˜P 0:t−1 [c · log (Gφt(˜pc))]. (5) Compared to exemplar rehearsal, prototype rehearsal is more memory efficient and privacy secure. In conclusion, the overall loss function of the baseline can be expressed as: Lt = Lce(θt, φt; Dt) + α1Lkd(θt; θt−1, Dt) + α2Lpro(φt; ˜P 0:t−1), (6) where α1 and α2 are the weights of the distillation loss and prototype loss, respectively. The specific implementation of prototype augmentation is not our focus. In this paper, we implement our approach based on the pipeline in PRAKA [10]. Our method can be incorporated with different augmentations and plugged into other baselines, such as PASS [6] and IL2A [38]. 4 align A. Base Phase 𝐷0 B. Incremental Phase PES CE Loss Inter-class reserved separation Reserved Space Intra-class concentration Constraints prototype feature 𝐷𝑡 current model frozen old model KD Projector PGRU current space latent space stored prototypes … Preemptive Embedding Squeezing Figure 2: Overview of our Prospective Representation Learning (PRL) for NECIL. (A) During the base phase, we impose a preemptive embedding squeezing (PES) constraint to squeeze the space of the current class in preparation for accepting future new classes. (B) During the incremental phase, a prototype-guided representation update (PGRU) strategy is proposed to keep new class features away from old class prototypes in the latent space, which guides the update of the current model to mitigate the confusion of new classes with old classes. 3.3 Prospective Representation Learning An overview of our Prospective Representation Learning scheme is shown in Fig. 2. It consists of a preemptive embedding squeezing constraint in the base phase and a prototype-guided representation update strategy in the incremental phase. The specific implementation of the two components is described in the following. Preemptive Embedding Squeezing. In the base phase (t = 0), a common training paradigm of NECIL is to optimize the empirical loss over the training set Dt as eq. (1). Without consideration of the future incremental learning, it overspreads the embedding space. As new classes come in, the embedding of old classes needs to be squeezed to make room for new ones. However, striking a balance in this process is challenging, especially without the old data. Therefore, we would like to be proactive and reserve space for future classes by squeezing the embedding of current classes in the base phase. Specifically, we impose a preemptive embedding squeezing (PES) constraint to cluster features of the same class and make features of different classes separate from each other. To reduce complexity, the PES loss is computed over a mini-batch data B = {xi, yi}n i=1 ∈Dt, which can be formulated as: s = X ∀xi,xj∈B yi=yj ⟨Fθt(xi), Fθt(xj)⟩, (7) d = X ∀xi,xk∈B yi̸=yk ⟨Fθt(xi), Fθt(xk)⟩, (8) LP ES(θt; Dt) = (1 −s) + λ ∗(1 + d), (9) where n is the batch size, ⟨·, ·⟩denotes the cosine similarity operator. As LP ES is minimized, the first term (1 −s) facilitates intra-class concentration, and the second term (1 + d) aims to reinforce inter-class reserved separation, as shown in Fig. 2 (A). Since s, d ∈[−1, 1), both terms are greater than zero. The hyper-parameter λ controls the priority ratio of intra-class constraints and inter-class constraints. Since our PES is implemented in a vectorized manner on the mini-batch, it does not incur excessive computational burden. With the preemptive embedding squeezing constraint, the optimization objective for the base phase training in eq. (1) can be rewritten as: Lt = Lce(θt, φt; Dt) + γ ∗LP ES(θt; Dt). (10) where γ is a hyperparameter controlling the weights of loss. 5 Algorithm 1 Proposed Method input: Data streams D, Model {Fθ, Gφ}, Factors λ and γ, Projector Pϕt 1: for all phases t ∈{0, 1, .., T} do 2: Get training set Dt 3: for minibatch B = {xi, yi}n i=1 ∈Dt do 4: if t = 0 then 5: Compute Lt = Lce + γ ∗LP ES 6: Update model {Fθt, Gφt} 7: else 8: Get prototypes set P 0:t−1 9: Compute Lt = Lce + α1Lkd + α2Lpro + α3LP GRU 10: Update model {Fθt, Gφt} and projector Pϕt 11: end if 12: end for 13: Compute pc = E (x,y)∼Dt[Fθt(x) \| y = c] 14: Update prototypes set P 0:t−1 15: end for 16: return Model {Fθt, Gφt} Prototype-Guided Representation Update. In the incremental phase (t > 0), we would like to embed the new class into the previously reserved space. The plain idea is to keep the new classes well clustered and distanced from the old ones. To this end, we propose a prototype-guided representation update (PGRU) strategy, as shown in Fig. 2 (B), which employs prototypes as proxies for past classes to guide the embedding of new classes into the appropriate space. However, it is not practical to establish a relationship directly between the saved prototypes and the new class features extracted by the current model due to the continual updating of the current embedding space. To mitigate the mismatch between the old class prototype and the new class features, on the one hand, we use the frozen model from the previous phase t −1 to extract the new class features, which has been implemented in the baseline as shown in eq. (3); on the other hand, the new classes features and the saved prototypes are projected into a unified latent space. Then, we construct orthogonal structures between the new class features and the old class prototypes in the latent space: Lort = X ∀xi∈B ∀pc∈P 0:t−1 \|⟨Pϕt(Fθt−1(xi)), Pϕt(pc)⟩\|, (11) where \| · \| denotes the absolute value operator, P is a projector with parameters ϕ. Similarly, Lort is also implemented in the mini-batch to reduce computational costs. Inspired by [61], we use a simple undercomplete autoencoder as the projector. It consists of a linear layer followed by ReLU activation that maps the features to a low-dimensional subspace and another linear layer followed by sigmoid activation that maps the features back to high dimensions. When minimizing Lort, it will promote orthogonality between the new class features and the old class prototypes. By the above operations, we would like to allow the new class of features to be embedded in appropriate positions and to keep clustering in the latent space. Our ultimate goal is to guide the update of the current model. Hence, we align the current embedding space with the latent space as: Lalign = X x∈Xi LMSE(Pϕt(Fθt−1(xi)), Fθt(xi)), (12) where LMSE is mean squared error (MSE) loss. In summary, the PGRU loss can be defined as: LP GRU = Lort(ϕt; Dt, P 0:t−1) + Lalign(θt, ϕt; Dt). (13) In the incremental phase, the optimization objective in eq. (6) can be rewritten as: Lt =Lce(θt, φt; Dt) + α1Lkd(θt; θt−1, Dt)+ α2Lpro(φt; ˜P 0:t−1) + α3LP GRU(θt, ϕt; Dt, P 0:t−1). (14) The main procedure is summarized in algorithm 1. 6 4 Experiment 4.1 Experimental Setting Dataset. We conduct comprehensive experiments on four public datasets: CIFAR-100 [62], TinyImageNet [63], ImageNet-Subset and ImageNet-1K [64]. CIFAR-100 consists of 100 classes, where each class contains 500 training images and 100 testing images with size 32×32. TinyImageNet has 200 classes in total, and the image size is 64×64. Each class in TinyImageNet contains 500 training images and 50 testing images. ImageNet-1K is a large-scale dataset comprising about 1.28 million images for training and 50,000 for validation with 500 images per class. ImageNet-Subset is a 100-class subset randomly chosen (random seed 1993) from the original ImageNet-1K. The image size of ImageNet-1K is much larger than the other two datasets, which poses a test of sensitivity to large-scale data. Protocol. Following the setting in [6; 7; 10], we divide around half the classes for the base phase, and the rest are divided equally into all the incremental phases. For CIFAR-100 and ImageNet-Subset: 1) 50 classes for base phase and 5 incremental phases of 10 classes; 2) 50 classes for base phase and 10 incremental phases of 5 classes; 3) 40 classes for base phase and 20 incremental phases of 3 classes. For TinyImageNet, we start by training the model with 100 classes in the base phase and distribute the remaining classes into three incremental settings: 1) 5 incremental phases of 20 classes; 2) 10 incremental phases of 100 classes; 3) 20 incremental phases of 5 classes. Implementation details. Our method is implemented with PyCIL [65]. For a fair comparison with [6], we adopt ResNet-18 [66] as the backbone network. The batch size is set to 64 for CIFAR-100 and TinyImageNet and 128 for ImageNet-Subset and ImageNet-1K. During training, the model is optimized by the Adam optimizer with β1 = 0.9, β2 = 0.999 and ϵ = 1e−8 (weight decay 2e-4). For ImageNet-1K, the learning rate starts at 0.0005 for all phases. The learning rate decays to 1/10 of the previous value every 70 epochs (160 epochs in total) in the base phase and every 45 epochs (100 epochs in total) in each incremental phase. For other datasets, the learning rate starts from 0.001 and decays to 1/10 of the previous value every 45 epochs (100 epochs in total) for all phases. We use λ = 0.5 and γ = 0.1 for all datasets. Regarding the loss weights, for comprehensive performance considerations and with reference to previous studies [6; 51], we set α1 = 10, α2 = 10, and α3 = 2 for training. We conduct our experiments on an RTX4090 GPU. Metric. We evaluate the methods in terms of average incremental accuracy. Average incremental accuracy AT is computed as the average of the accuracy of all phases (including the base phase) and is a fair metric to compare the overall incremental performance of different methods: AT = 1 T + 1 T X t=0 at, (15) where at is the average accuracy over all seen classes on phase t. 4.2 Comparison with SOTA We compare our method with the state-of-the-art (SOTA) methods of NECIL (EWC [23], LwF_MC [67], MUC [68], SDC [8], PASS [6], SSRE [7], SOPE [11], POLO [51], PRAKA [10] and NAPAVQ [52]). "Fine-tuning" refers to continuously fine-tuning the network on the new task with only cross-entropy loss. "Joint" means that when learning a new task, all data from past tasks are available to jointly train the model, which can be considered as an upper bound of the CIL model. The results reported for PASS are obtained with self-supervised learning. The quantitative comparisons of average incremental accuracy are reported in Tab. 1. In comparison with the SOTA, our method improves by 1.4% and 6.0% on CIFAR-100 and TinyImageNet datasets, respectively. To further investigate the behavior of different methods on larger data, we also evaluated their performance on ImageNet-Subset. Compared with suboptimal results, PRL achieves an average improvement of 3.6%. The outstanding performance on ImageNet-Subset demonstrates the reliability of our method. To provide a more nuanced view of the changes in performance of the different methods over the course of incremental learning, we show accuracy curves for CIFAR-100, TinyImageNet and ImageNet-Subset in Fig. 3. The accuracy of our method remains ahead as we continue to learn new tasks. By prospective learning, our approach demonstrates strengths early on that will be maintained and even enlarged over the course of continuously learning new tasks. 7 Table 1: Quantitative comparisons of the average incremental accuracy (%) with other methods on CIFAR-100, TinyImageNet and ImageNet-Subset. P represents the number of incremental phases. The best performance is shown in bold, and the sub-optimal performance is underlined. The relative improvement compared to the SOTA NECIL methods is shown in red. CIFAR-100 TinyImageNet ImageNet-Subset Methods P=5 P=10 P=20 P=5 P=10 P=20 P=5 P=10 P=20 Fine-tuning 23.15 12.96 7.93 18.64 10.68 5.75 23.43 13.12 7.96 Joint 76.72 76.72 76.72 63.08 63.08 63.08 78.94 78.94 78.94 EWC [23] 24.48 21.20 15.89 18.80 15.77 12.39 — 20.40 — LwF_MC [67] 45.93 27.43 20.07 29.12 23.10 17.43 — 31.18 — MUC [68] 49.42 30.19 21.27 32.58 26.61 21.95 — 35.07 — SDC [8] 56.77 57.00 58.90 — — — — 61.12 — PASS [6] 63.47 61.84 58.09 49.55 47.29 42.07 64.40 61.80 51.29 SSRE [7] 65.88 65.04 61.70 50.39 48.93 48.17 — 67.69 — SOPE [11] 66.64 65.84 61.83 53.69 52.88 51.94 — 69.22 — POLO [51] 68.95 68.02 65.71 54.90 53.38 49.93 70.81 69.11 — PRAKA [10] 70.02 68.86 65.86 53.32 52.61 49.83 69.81 68.98 63.95 NAPA [52] 70.44 69.04 67.42 52.77 51.78 49.51 69.15 68.83 63.09 PRL (Ours) 71.26 70.17 68.44 58.12 57.24 54.51 72.85 71.54 66.88 Improvement +0.82 +1.13 +1.02 +3.22 +3.86 +2.57 +2.04 +2.32 +2.93 40 50 60 70 80 50 60 70 80 90 100 top-1 Acc(%) number of classes ImageNet -Subset (10 phases) 40 50 60 70 80 40 52 64 76 88 100 top-1 Acc(%) number of classes CIFAR-100 (20 phases) 30 40 50 60 70 100 120 140 160 180 200 top-1 Acc(%) number of classes TinyImageNet (20 phases) 40 50 60 70 80 50 60 70 80 90 100 top-1 Acc(%) number of classes CIFAR-100 (10 phases) 40 50 60 70 80 50 60 70 80 90 100 top-1 Acc(%) number of classes CIFAR-100 (5 phases) 30 40 50 60 70 100 120 140 160 180 200 top-1 Acc(%) number of classes TinyImageNet (10 phases) 30 40 50 60 70 100 120 140 160 180 200 top-1 Acc(%) number of classes TinyImageNet (5 phases) Figure 3: Detailed accuracy curves showing the top-1 accuracy of each incremental phase on CIFAR100, TinyImageNet and ImageNet-Subset. 4.3 Ablation Study To analyze the impact of each component in our method, we perform several ablation studies on CIFAR-100 and TinyImageNet datasets. We use the prototype augmentation technique in [10] as eq. (5) in our baseline. As shown in Tab. 2, The first line shows the performance of our baseline model. Our baseline is strong due to using the prototype augmentation in [10]. Even on the strong baseline model, both preemptive embedding squeezing (PES) constraint and prototype-guided representation update (PGRU) strategy can bring considerable performance improvements. Furthermore, the table shows that PES plays a more central role than PGRU. This is reasonable since the space reserved by PES for future classes is the basis for the PGRU to guide new classes to embed in the representation space during the incremental phase. 4.4 Analysis Visualization. To analyze the impact of PRL on representation learning, we visualize the embedding space of 2D feature vectors on CIFAR-100 (5 phases) with t-SNE [69] in Fig. 4. Specifically, we (1) 8 Table 2: Ablation study (in average incremental accuracy) of our method on CIFAR-100 and TinyImageNet datasets. CIFAR-100 TinyImageNet Methods P=5 P=10 P=20 P=5 P=10 P=20 baseline 69.25 68.52 65.93 55.04 54.15 51.65 baseline w/ PES 70.57 69.64 67.58 57.08 55.84 53.58 baseline w/ PGRU 70.36 69.23 67.17 56.79 56.05 53.16 PRL 71.26 70.17 68.44 58.12 57.24 54.51 Baseline PRL Base phase (t=0) old class features new class features Last phase (t=T=5) Baseline PRL Figure 4: Visualization of the impact of PRL on the feature representations. Dashed circles and arrows highlight observable differences between baseline and PRL. PRL visually concentrates the distribution of features within classes, disperses the distribution of features between classes, and mitigates inter-class confusion. visualize the features of a randomly selected subset of classes from D0 (old class features) after the base phase, and (2) visualize the old class features along with a subset of classes from DT (new class features) after the last phase. As shown in the first row, once the training of the base phase (t = 0) is complete, the model integrated with PRL has more tightly clustered intra-class distributions (blue circles) and more dispersed inter-class distributions (ò). Thus, more space is reserved for learning new classes. The second row is visualized after the last phase (t = T = 5). It can be observed that the overlap (red circles) in the baseline model increases, causing confusion between the old and new classes. In contrast, PRL reduces the overlap between classes, making them easier to distinguish. Moreover, the new classes are farther away from the old ones (ò) compared to the baseline. Comparison of the confusion matrix. Figure 5 compares the confusion matrices obtained by finetuning, PASS [6], NAPA-VQ [52] and our PRL on CIFAR-100. The diagonal entries indicate correct classification, while the non-diagonal entries indicate misclassification. Due to the forgetting of old classes, fine-tuning produces predictions that are biased toward the most recent classes, showing a strong confusion on the last task. PASS clearly mitigates this confusion but still predicts more intensively on recent tasks. The predictions of NAPA-VQ are largely centered on the diagonal, but its predictions are more accurate for the initial classes that appear in the base phase (the red patches are more localized in the first half of the diagonal). In contrast, there are more red patches visible along the diagonal and more evenly distributed in the confusion matrix of PRL, which explains the higher average accuracy of our method compared to NAPA-VQ and the absence of a serious bias towards either new or old classes. (b) PASS (c) NAPA-VQ (d) PRL(Ours) (a) Fine-tuning Figure 5: The comparison of confusion matrix of fine-tuning, PASS, NAPA-VQ and our method on CIFAR-100 (10 phases). 9 40 50 60 70 80 5 10 20 Accuracy (%) Phase Current Task Accuracy 40 50 60 70 80 0-4 0-9 0-19 Accuracy (%) Phase Old Tasks Average Accuracy Figure 6: During incremental learning, our method shows less performance degradation on past tasks. Meanwhile, in contrast to other methods other methods whose performance on new tasks declines as the number of tasks increases or remains poor, our method shows good plasticity in the performance of new tasks. Plasticity and stability analysis. An incremental learner should acquire new knowledge of the current task for the sake of plasticity and also preserve knowledge from previous tasks for the sake of stability [70; 71]. We present an analysis of the plasticity and stability of the different methods in Fig. 6. First, we observe a gradual decline in average performance on past tasks during incremental learning. This is rational because experiencing more tasks also results in heavier catastrophic forgetting. Nonetheless, our method exhibits better stability due to less degradation and consistently superior average performance on old tasks. Then we turn our attention to the current task and also found a performance degradation as more and more tasks are learned. This corresponds to a gradual reduction in plasticity since tasks are sampled uniformly from the set of possible tasks, which is consistent with observations from previous studies [72; 73]. PRAKA [10] starts with good performance, but its plasticity degrades as more tasks are learned. NAPA-VQ consistently performs poorly on the current task, which is also in line with the results in Fig. 5. Remarkably, PRL maintains a good performance on the current task and has yet to show a visible decline. In general, our method achieves a better trade-off between stability and plasticity. 5 Conclusion and Limitation In this work, we consider the conflict between old and new classes in NECIL from a prospective view. In the base phase, we construct a preemptive embedding squeezing constraint to reserve space for future classes by enforcing intra-class concentration and inter-class reserved separation. In the incremental phase, we propose a prototype-guided representation update (PGRU) strategy, which reduces the impact on the old class during model update by keeping the new class embedding away from the old class prototype. In cases where exemplars cannot be saved, waiting until the conflict arrives could exacerbate the problem, and we offer a novel solution. Through extensive experiments on four public benchmarks, our method exhibits excellent average performance and can provide a good balance between stability and plasticity. However, since the number and distribution of unknown classes cannot be predicted, how to rationally allocate the space of base classes in prospective learning is open to further discussion. Acknowledgments This work is partially supported by National Natural Science Foundation of China under Grant (62176188, 62361166629, 62225113) and Key Research and Development Project of Hubei Province (2022BAD175). The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University. References [1] M. Masana, X. Liu, B. Twardowski, M. Menta, A. D. Bagdanov, and J. 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Bonin, “The stability-plasticity dilemma: Investigating the continuum from catastrophic forgetting to age-limited learning effects,” Frontiers in psychology, vol. 4, p. 504, 2013. 14 A Appendix / supplemental material A.1 Detailed Description of the Accuracy Curve To facilitate comparison of future work with our method, we provide detailed values of the accuracy curves in Tab. 3, Tab. 4 and Tab. 5, where ’A’ represents the CIFAR-100 dataset, ’B’ represents the TinyImageNet dataset and ’C’ represents the ImageNet-Subset dataset, respectively. Table 3: Detailed values of accuracy under the setting of 5 phases. Phase Dataset 0 1 2 3 4 5 A 82.80 75.65 72.10 68.26 65.52 63.44 B 66.58 60.58 59.04 57.14 54.10 52.13 C 84.52 77.90 72.32 69.72 67.16 65.44 Table 4: Detailed values of accuracy under the setting of 10 phases. Phase Datasets 0 1 2 3 4 5 6 7 8 9 10 A 82.80 78.76 74.90 73.18 70.71 69.53 67.35 65.36 64.90 63.24 61.71 B 66.58 62.75 61.02 58.83 58.57 56.73 56.34 54.79 53.18 51.64 50.25 C 84.52 80.69 76.37 73.57 71.89 70.51 68.6 67.13 65.53 63.68 64.10 Table 5: Detailed values of accuracy under the setting of 20 phases. Phase Datasets 0 1 2 3 4 5 6 7 8 9 A 83.45 79.81 78.85 76.80 76.06 74.64 72.64 70.52 68.27 67.84 B 66.38 63.26 62.22 61.19 60.07 58.85 57.68 57.46 56.45 55.31 C 84.75 80.84 77.91 78.08 75.27 74.55 73.31 69.38 67.75 65.85 Phase Datasets 10 11 12 13 14 15 16 17 18 19 20 A 66.80 66.36 66.37 64.06 62.84 61.46 61.61 60.77 58.98 58.94 57.74 B 54.34 53.61 53.51 52.41 51.69 50.23 49.06 48.05 47.58 47.63 45.58 C 63.66 62.36 62.18 60.86 60.22 60.85 57.73 57.91 57.30 57.32 56.52 Evaluation on Large Datasets and Robustness. To further demonstrate the effectiveness of our method, we evaluated it on a large-scale dataset — ImageNet-1K. For ImageNet-1K, we allocate 500 classes for the base phase and 50 classes for each of the 10 incremental phases. As shown in Table 6, our method shows an improvement of 1.9% compared to the suboptimal results. The results of our method are obtained by averaging three replicate experiments, and we set a different random seed for each run. To illustrate the stability of our method, we report the standard deviation of these three results. As shown in Tab. 6, the random seed has little impact on the results of our approach. Table 6: The number after ± in the last line represents the standard deviation of three different runs. CIFAR-100 TinyImageNet Methods P=5 P=10 P=20 P=5 P=10 P=20 SOPE [11] 66.64 65.84 61.83 53.69 52.88 51.94 POLO [51] 68.95 68.02 65.71 54.90 53.38 49.93 NAPA [52] 70.44 69.04 67.42 52.77 51.78 49.51 PRL (Ours) 71.26±0.19 70.17±0.31 68.44±0.24 58.12±0.48 57.24±0.41 54.51±0.36 ImageNet-Subset ImageNet-1K Methods P=5 P=10 P=20 P=10 SOPE [11] — 69.22 — 60.20 POLO [51] 70.81 69.11 — 61.53 NAPA [52] 69.15 68.83 63.09 54.21 PRL (Ours) 72.85±0.25 71.54±0.27 66.88±0.37 62.74±0.34 15 Table 7: We report the performance gain of average incremental accuracy by applying PRL to other NECIL baselines. Absolute gains are marked in (red). CIFAR-100 Methods P=5 P=10 P=20 IL2A [38] 67.35 61.03 60.67 +PRL 69.53 (+2.18) 62.49 (+1.46) 62.36(+1.69) PASS [6] 63.47 61.84 58.09 +PRL 66.22 (+2.75) 62.85 (+1.01) 58.85 (+0.76) Plug-and-play with other NECIL methods. Existing NECIL methods mainly focus on backwardlooking means of resolving conflicts between old and new classes, which does not contradict our prospective learning. Therefore, we integrate PRL into the existing NECIL methods. Tab. 7 illustrates the performance gains achieved by incorporating PRL in these methods. In the setting of the CIFAR dataset with three different lengths of task sequences, PRL improved accuracy by an average of 2.7% for IL2A [38] and 2.3% for PASS [6], which demonstrates the good compatibility of our method. A.2 Impact of the hyper-parameter To investigate the sensitivity of our method to the hyper-parameters λ and γ, we performed ablation experiments on three settings (5 phases, 10 phases and 20 phases) of the CIFAR-100 dataset. In Fig. 7 we show the impact of λ, which controls the priority ratio of intra-class constraints and inter-class constraints. A smaller λ means that the preemptive embedding squeezing (PES) is more concerned with intra-class concentration. Conversely, for a larger λ, more emphasis is placed on inter-class separation. When the value of λ is either too large or too small, the performance of our method degrades, indicating that there is a need to maintain a certain balance between the intra-class constraint and the inter-class constraint. The best performance is achieved when λ is equal to 0.5, suggesting that for prospective learning in NECIL, intra-class concentration could be more important than inter-class separation. We also provide an analysis of the impact of the hyper-parameters γ in Fig. 8. The performance of our method is stable on the 5 phase setting. The performance of the model gradually increases as γ increases on the 10-phase and 20-phase settings, peaking at γ = 0.1. However, continued increase in the values of γ leads to a decline in model performance. We argue that too large loss weights cause PES to interfere with the optimization of cross-entropy for classification performance. In addition, our method is more sensitive to the values of λ and γ when there are more tasks (20 phases) to learn. For the hyperparameter of loss weight, we set α1 = 10, α2 = 10, and α3 = 2 by default. When a sensitivity analysis is performed on one of the hyperparameters, default settings are used for the remaining hyperparameters. As shown in Fig. 9, the left column shows the effect of changing the value of each hyperparameter on the average incremental accuracy of our method, and the right 𝜆 71.01 71.08 71.26 71.17 71.32 71.18 70.66 70.10 69.99 70.17 70.03 69.67 69.52 68.83 67.14 67.93 68.44 67.72 67.48 67.52 67.36 64 66 68 70 72 0.1 0.25 0.5 0.75 1 1.5 2 average accuracy (%) value of 5 phases 10 phases 20 phases Figure 7: Impact of the hyper-parameter λ in our preemptive embedding squeezing, which controls the priority ratio of intra-class constraints and inter-class constraints. Larger values of λ represent a stronger inter-class separation. 𝛾 70.87 71.14 71.26 70.92 71.16 70.71 69.66 70.02 70.17 69.24 68.41 66.19 67.78 67.81 68.44 67.76 67.49 65.81 62 64 66 68 70 72 0.02 0.05 0.1 0.2 0.3 0.5 average accuracy (%) value of 5 phases 10 phases 20 phases Figure 8: Impact of the hyper-parameter γ, which controls for the weight of the PES loss. Larger values of λ represent that the PES loss exerts a greater influence in the base phase of training compared to the cross-entropy loss. 16 69.54 70.88 71.30 71.26 71.19 71.04 65.68 68.69 69.54 70.17 70.03 70.38 61.38 67.12 68.14 68.44 68.79 68.87 60 64 68 72 1 3 5 10 15 20 average incremental accuracy (%) value of P=5 P=10 P=20 71.23 71.17 71.24 71.26 70.96 70.55 69.93 70.09 70.21 70.17 70.00 69.81 68.05 68.08 68.20 68.44 68.15 67.45 64 66 68 70 72 1 3 5 10 15 20 average incremental accuracy (%) value of P=5 P=10 P=20 70.88 71.35 71.26 71.22 71.04 70.79 69.72 70.09 70.17 70.25 70.02 69.94 67.54 68.32 68.44 68.29 68.41 68.05 65 67 69 71 73 0.5 1 2 3 5 10 average incremental accuracy (%) value of P=5 P=10 P=20 (c) Impact of hyperparameters on overall performance (b) (a) 50 60 70 80 1 3 5 10 15 20 accuracy (%) value of new classes old classes 50 60 70 80 1 3 5 10 15 20 accuracy (%) value of new classes old classes 50 55 60 65 70 0.5 1 2 3 5 10 accuracy (%) value of new classes old classes 𝛼1 𝛼1 𝛼2 𝛼2 𝛼3 𝛼3 (f) (e) (d) Impact of hyperparameters on on the new and old tasks Figure 9: The figures in the left column show the effect of changing the value of each hyperparameter on the average incremental accuracy of our method on CIFAR-100 dataset, where ’P’ denotes the number of incremental phases. The figures in the right column show the effect of changing the value of each hyperparameter in the last phase on the accuracy on the new and old tasks on CIFAR-100 dataset (P=10), respectively. column shows the effect of changing the value of each hyperparameter in the last phase on the accuracy on the new and old tasks, respectively. Among the three hyperparameters in eq. (14), α1 and α2 are common in previous NECIL methods and represent the weights of distillation loss and prototype loss, respectively. The main role of these two loss functions is to maintain the pre-existing knowledge of the model. Therefore, as shown in Fig. 9 (d) and Fig. 9 (e), as α1 and α2 get larger, the optimization of the model will be biased towards maintaining stability at the expense of plasticity, resulting in the model performing better on the old task and worse on the new task. It can be seen that as the value of α1 and α2 increases to a certain level its performance improvement on old tasks slows down. Excessively large values of and will bring much less gain on the old task than they will hurt performance on the new task. For the consideration of comprehensive performance and with reference to previous works [6; 51], we set α1 = 10 and α2 = 10 for our method. Then α3 controls the loss of the Prototype-Guided Representation Update (PGRU) proposed in this paper. In Fig. 9 (c), as α3 increases PGRU comes into play. The effect of increasing α3 on the overall performance of the algorithm fluctuates, which may be caused by overly strict constraints on the learning of new class representations. Overall, our algorithm is robust to the hyperparameters. 17 NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. 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4,460	Depth Anywhere: Enhancing 360 Monocular Depth Estimation via Perspective Distillation and Unlabeled Data Augmentation Ning-Hsu Wang albert.nhwang@gmail.com Yu-Lun Liu Department of Computer Science National Yang Ming Chiao Tung University yulunliu@cs.nycu.edu.tw Abstract Accurately estimating depth in 360-degree imagery is crucial for virtual reality, autonomous navigation, and immersive media applications. Existing depth estimation methods designed for perspective-view imagery fail when applied to 360-degree images due to different camera projections and distortions, whereas 360-degree methods perform inferior due to the lack of labeled data pairs. We propose a new depth estimation framework that utilizes unlabeled 360-degree data effectively. Our approach uses state-of-the-art perspective depth estimation models as teacher models to generate pseudo labels through a six-face cube projection technique, enabling efficient labeling of depth in 360-degree images. This method leverages the increasing availability of large datasets. Our approach includes two main stages: offline mask generation for invalid regions and an online semi-supervised joint training regime. We tested our approach on benchmark datasets such as Matterport3D and Stanford2D3D, showing significant improvements in depth estimation accuracy, particularly in zero-shot scenarios. Our proposed training pipeline can enhance any 360 monocular depth estimator and demonstrates effective knowledge transfer across different camera projections and data types. See our project page for results: albert100121.github.io/Depth-Anywhere. 1 Introduction In recent years, the field of computer vision has seen a surge in research focused on addressing the challenges associated with processing 360-degree images. The widespread use of panoramic imagery across various domains, such as virtual reality, autonomous navigation, and immersive media, has underscored the need for accurate depth estimation techniques tailored specifically for 360-degree images. However, existing depth estimation methods developed for perspective-view images encounter significant difficulties when applied directly to 360-degree data due to differences in camera projection and distortion. While many methods aim to address depth estimation for this camera projection, they often struggle due to the limited availability of labeled datasets. To overcome these challenges, this paper presents a novel approach for training state-of-the-art (SOTA) depth estimation models on 360-degree imagery. With the recent significant increase in the amount of available data, the importance of both data quantity and quality has become evident. Research efforts on perspective perceptual models have increasingly focused on augmenting the volume of data and developing foundation models that generalize across various types of data. Our method leverages SOTA perspective depth estimation foundation models as teacher models and generates pseudo labels for unlabeled 360-degree images using a six-face cube projection approach. By doing so, we efficiently address the challenge of labeling depth in 360-degree imagery by leveraging perspective models and large amounts of unlabeled data. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Ground Truth BiFuse++ BiFuse++ (p) Error Map Error Map RGB Depth Figure 1: Our proposed training pipeline improves existing 360 monocular depth estimators. This figure demonstrated the improvement of our proposed training pipeline tested on the Stanford2D3D [2] dataset in a zero-shot setting. Our approach consists of two key stages: offline mask generation and online joint training. During the offline stage, we employ a combination of detection and segmentation models to generate masks for invalid regions, such as sky and watermarks in unlabeled data. Subsequently, in the online stage, we adopt a semi-supervised learning strategy, loading half of the batch with labeled data and the other half with pseudo-labeled data. Through joint training with both labeled and pseudo-labeled data, our method achieves robust depth estimation performance on 360-degree imagery. To validate the effectiveness of our approach, we conduct extensive experiments on benchmark datasets such as Matterport3D and Stanford2D3D. Our method demonstrates significant improvements in depth estimation accuracy, particularly in zero-shot scenarios where models are trained on one dataset and evaluated on another. Furthermore, we demonstrate the efficacy of our training techniques with different SOTA 360-degree depth models and various unlabeled datasets, showcasing the versatility and effectiveness of our approach in addressing the unique challenges posed by 360-degree imagery. Our contributions can be summarized as follows: • We propose a novel training technique for 360-degree imagery that harnesses the power of unlabeled data through the distillation of perspective foundation models. • We introduce an online data augmentation method that effectively bridges knowledge distillation across different camera projections. • Our proposed training techniques significantly benefit and inspire future research on 360degree imagery by showcasing the interchangeability of state-of-the-art (SOTA) 360 models, perspective teacher models, and unlabeled datasets. This enables better results even as new SOTA techniques emerge in the future. 2 Related Work 360 monocular depth. Depth estimation for 360-degree images presents unique challenges due to the equirectangular projection and inherent distortion. Various approaches have been explored to address these issues: • Directly Apply: Some methods directly apply monocular depth estimation techniques to 360-degree imagery. OmniDepth [69] leverages spherical geometry and incorporates SphConv [42] to improve depth prediction with distortion. [70, 49] use spherical coordinates to overcome distortion with extra information. [12, 17] leverage other ground truth supervisions to assist on depth estimation. SliceNet [31] and ACDNet [68] propose advanced network architectures tailored for omnidirectional images. EGFormer [64] and HiMODE [18] introduce a transformer-based model that captures global context efficiently, while [45, 44] 2 focuses on integrating geometric priors into the learning process. [13] proposed to generate large-scale datasets with SfM and MVS, then apply to test-time training. • Cube: Other approaches use cube map projections to mitigate distortion effects. 360SelfNet [46] is the first work to self-supervised 360 depth estimation leveraging cubepadding [9]. BiFuse [47] and its improved version BiFuse++ [48] are two-branch architectures that utilize cube maps and equirectangular projections. UniFuse [16] combines equirectangular and cube map projections and simplifies the architecture. [3] combines two-branch techniques with transformer network. • Tangent Image: Tangent image projections are also popular. [37, 23, 30] convert equirectangular images into a series of tangent images, which are then processed using conventional depth estimation networks. PanoFormer [40] employs a transformer-based architecture to handle tangent images, while SphereNet [11] and HRDFuse [1] enhance depth prediction by collaboratively learning from multiple projections. 360 other works Beyond depth estimation, 360-degree imagery has been applied to depth completion tasks as follows [26, 8, 32, 57, 58, 15]. Other methods, such as [25] and [43] focus on the projection between camera models, while the former projects pinhole camera model images into a large field of view, whereas the latter transforms convolution kernels. Unlabeled / Pseudo labeled data. Utilizing unlabeled or pseudo-labeled data has become a significant trend to mitigate the limitations of labeled data scarcity. Techniques like [22, 71, 41, 56] leverage large amounts of unlabeled data to improve model performance through semi-supervised learning. In the context of 360-degree depth estimation, our approach generates pseudo labels from pre-trained perspective models, which are then used to train 360-degree depth models effectively. Zero-shot methods. Zero-shot learning methods aim to generalize to new domains without additional training data. [7, 54] target this directly with increasing training data., MiDaS [35, 5, 34] and Depth Anything [59] are notable for their robust monocular depth estimation across diverse datasets leveraging affine-invariant loss. [61] takes a step further to investigate zero-shot on metric depth. Marigold [19] leverages diffusion models with image conditioning and up-to-scale relative depth denoising to generate detailed depth maps. ZoeDepth [4] further these advancements by incorporating scale awareness and domain adaptation. [14, 50] leverage camera model information to adapt cross-domain depth estimation. Foundation models. Foundation models have revolutionized various fields in AI, including natural language processing and image-text alignment. In computer vision, models like CLIP [33] demonstrate exceptional generalization capabilities. [28] proposed a foundation visual encoder for downstream tasks such as segmentation, detection, depth estimation, etc. [20] proposed a model that can cut out masks for any objects. Our work leverages a pre-trained perspective depth estimation foundation model [59] as a teacher model to generate pseudo labels for 360-degree images, enhancing depth estimation by utilizing the vast knowledge embedded in these foundation models. 3 Methods In this work, we propose a novel training approach for 360-degree monocular depth estimation models. Our method leverages a perspective depth estimation model as a teacher and generates pseudo labels for unlabeled 360-degree images using a 6-face cube projection. Figure 2 illustrates our training pipeline, incorporating the use of Segment Anything to mask out sky and watermark regions in unlabeled data during the offline stage. Subsequently, we conduct joint training using both labeled and unlabeled data, allocating half of the batch to each. The joint training avoids limiting performance by teacher model. The unlabeled data is supervised using pseudo labels generated by Depth Anything, a state-of-the-art perspective monocular depth foundation model. With the benefit of our teacher model, the 360-degree depth model demonstrates an observable improvement on the zero-shot dataset, as shown in Figure 1. 3.1 Unleashing the Power of Unlabel 360 data Dataset statistics. 360-degree data has become increasingly available in recent years. However, compared to perspective-view depth datasets, labeling depth ground truths for 360-degree data presents greater challenges. Consequently, the availability of labeled datasets for 360-degree data is considerably smaller than that of perspective datasets. 3 Labeled data Unlabeled data Weight sharing 360 model 360 model Depth prediction Loss on GT Ground truth Valid pixel mask Same random rotation SAM Depth Anything Same random rotation Equi. to cube Equi. to cube Equi. to cube Pseudo GT Loss on pseudo GT with valid masks Valid masks (c) Random rotation (a) Supervised training with labeled data (b) Distillation with unlabeled data Figure 2: Training Pipeline. Our proposed training pipeline involves joint training on both labeled 360 data with ground truth and unlabeled 360 data. (a) For labeled data, we train our 360 depth model with the loss between depth prediction and ground truth. (b) For unlabeled data, we propose to distill knowledge from a pre-trained perspective-view monocular depth estimator. In this paper, we use Depth Anything [59] to generate pseudo ground truth for training. However, more advanced techniques could be applied. These perspective-view monocular depth estimators fail to produce reasonable equirectangular depth as a domain gap exists. Therefore, we distill knowledge by inferring six perspective cube faces and passing them through perspective-view monocular depth estimators. To ensure stable and effective training, we propose generating a valid pixel mask with Segment Anything [20] while calculating loss. (c) Furthermore, we augment random rotation on RGB before passing it into Depth Anything, as well as on predictions from the 360 depth model. Table 1: 360 monocular depth estimation lacks a large amount of training data. The number of images in 360-degree monocular depth estimation datasets alongside perspective depth datasets from the Depth Anything methodology. Perspective Equirectangular Labeled 1.5M Unlabeled 62 M Labeled 34K Unlabeled 344K Table 1 presents the data quantities available in some of the most popular 360-degree datasets, including Matterport3D [6], Stanford2D3D [2], and Structured3D [65]. Additionally, we list a multi-modal dataset, SpatialAudioGen [29], which consists of unlabeled 360-degree data used in our experiments. Notably, the amount of labeled and unlabeled data used in the perspective foundation model, Depth Anything [59], is significantly larger, with 1.5 million labeled images [24, 52, 10, 60, 55, 51] and 62 million unlabeled images [38, 63, 53, 62, 39, 21, 66, 20], making the amount in 360-degree datasets approximately 170 times smaller. Data cleaning and valid pixel mask generation Unlabeled data often contains invalid pixels in regions such as the sky and watermark, leading to unstable training or undesired convergence. To address this issue, we applied the GroundingSAM [36] method to mask out the invalid regions. This approach utilizes Grounded DINOv2 [27] to detect problematic regions and applies the Segment Anything [20] model to mask out the invalid pixels by segmenting within the bounding box. While Depth Anything [59] also employs a pre-trained segmentation model, DINOv2, to select sky regions. Brand logos and watermarks frequently appear after fisheye camera stitching. Therefore, additional labels are applied to enhance the robustness of our training process. We also remove all images with less than 20 percent of valid pixels to stablize our training progress. Perspective foundation models (teacher models). To tackle the challenges posed by limited data and labeling difficulties in 360-degree datasets, we leverage a large amount of unlabeled data alongside state-of-the-art perspective depth foundation models. Due to significant differences in camera projection and distortion, directly applying perspective models to 360-degree data often yields inferior results. Previous works have explored various methods of projection for converting equirectangular to perspective depth, as stated in Section 2. Among these, cube projection and tangent 4 Figure 3: Valid Pixel Masking. We used Grounded-Segment-Anything [36] to mask out invalid pixels based on two text prompts: “sky” and “watermark.” These regions lack depth sensor ground truth labels in all previous datasets. Unlike Depth Anything [59], which sets sky regions as 0 disparity, we follow ground truth training to ignore these regions during training for two reasons: (1) segmentation may misclassify and set other regions as zero, leading to noisy labeling, and (2) watermarks are post-processing regions that lack geometrical meaning. projection are the most common techniques. We selected cube projection to ensure a larger field of view for each patch, enabling better observation of relative distances between pixels or objects during the inference of the perspective foundation model and enhancing knowledge distillation. The comparison table can be find in the supplementary material. In our approach, we apply projection to unlabeled 360-degree data and then run Depth Anything on these projected patches of perspective images to generate pseudo-labels. We explore two directions for pseudo-label supervision: projecting the patch to equirectangular and computing in the 360-degree domain or projecting the 360-degree depth output from the 360 model to patches and computing in the perspective domain. Since training is conducted in an up-to-scale relative depth manner, stitching the patch of perspective images back to equirectangular with an aligned scale will lead to failure in training Figure 4, which is an additional research topic that is worth investigation. We opt to compute the loss in the perspective domain, facilitating faster and easier training without the need for additional alignment optimization. 3.2 Random Rotation Processing Directly applying Depth Anything on cube-projected unlabeled data does not yield improvements due to ignorance of cross-cube-face relation, leading to cube artifacts (Figure 5). This issue arises from the separate estimation of perspective cube faces, where monocular depth is estimated based on semantic information, lacking a comprehensive understanding of the entire scene. To address this, we propose a random rotation preprocessing step in front of the perspective foundation model. As depicted in Figure 2, the rotation is applied to equirectangular projection RGB images using a random rotation matrix, followed by cube projection. This results in a more diverse set of cube faces, capturing relative distances between ceilings, walls, windows, and other objects more effectively. With the proposed random rotation technique, knowledge distillation becomes more comprehensive as the point of view is not static. The inference by the perspective foundation model is performed on the fly, with parameters frozen during the training of the 360 model. In order to perform random rotation, we apply a rotation matrix on the equirectangular coordinates, noted as (θ, ϕ), and rotation matrix as R. (ˆθ, ˆϕ) = R · (θ, ϕ). (1) For equirectangular to cube projection, the field-of-view (FoV) of each cube face is equal to 90 degrees; each face can be considered as a perspective camera whose focal length is w/2, and all faces share the same center point in the world coordinate. Since the six cube faces share the same center point, the extrinsic matrix of each camera can be defined by a rotation matrix Ri. p is then the pixel 5 (a) RGB (b) Pseudo GT (w/o Rotation) (c) Pred on Equi (w/o Rotation) (d) Pseudo GT (w/ Rotation) (e) Pred on Equi (w/ Rotation) (f) Ours Figure 4: Qualitative visualization of a model trained directly on pseudo equirectangular data without scale alignment. We propose calculating the loss with pseudo ground truth on cube faces due to scale misalignment between the six faces during the cube-to-equirectangular projection. We showcase the results of a model trained on pseudo equirectangular data without scale alignment as a simple baseline to demonstrate the importance of calculating loss separately on each of the six faces. The images are presented from top to bottom as follows: (a) RGB images. (b) Pseudo cube ground truth projected directly to equirectangular. (c) Prediction trained with row 2. (d) Pseudo cube ground truth with rotation projected directly to equirectangular. (e) Prediction trained with row 4. (f) Our model’s predictions are trained on cube faces separately with rotation. on the cube face p = K · RT i · q, (2) where, q = "qx qy qz # = "sin(θ) · cos(ϕ) sin(ϕ) cos θ · cos ϕ # , K = "w/2 0 w/2 0 w/2 w/2 0 0 1 # , (3) where θ and ϕ are longitude and latitude in equirectangular projection and q is the position in Euclidean space coordinates. 3.3 Loss Function The training process closely resembles that of MiDaS, Depth Anything, and other cross-dataset methods. Our goal is to provide depth estimation for any 360-degree images. Following previous approaches that trained on multiple datasets, our training objective is to estimate relative depth. The depth values are first transformed into disparity space using the formula 1/d and then normalized to the range [0, 1] for each disparity map. To adapt to cross-dataset training and pseudo ground truths from the foundation model, we employed the affine-invariant loss, consistent with prior cross-dataset methodologies. This loss function disregards absolute scale and shifts for each domain, allowing for effective adaptation across different 6 Figure 5: Cube Artifact. Shown in the center row of the figure, an undesired cube artifact appears when we apply joint training with pseudo ground truth from Depth Anything [59] directly. This issue arises from independent relative distances within each cube face caused by a static point of view. Ignoring cross-cube relationships results in poor knowledge distillation. To address this, as shown in Figure 2(c), we randomly rotate the RGB image before inputting it into Depth Anything. This enables better distillation of depth information from varying perspectives within the equirectangular image. datasets and models. L1 = 1 HW HW X i=1 ρ(d∗ i , di), (4) where d∗ i and di are the prediction and ground truth, respectively. ρ represents the affine-invariant mean absolute error loss: ρ(d∗ i , di) = \| ˆd∗ i −ˆdi\|. (5) Here, ˆdi and ˆd∗ i are the scaled and shifted versions of the prediction d∗ i and ground truth di: ˆdi = di −t(d) s(d) , (6) where t(d) and s(d) are used to align the prediction and ground truth to have zero translation and unit scale: t(d) = median(d), s(d) = 1 HW HW X i=1 \|di −t(d)\|. (7) 4 Experiments These notations apply for all tables: M: Matterport3D [6], SF: Stanford2D3D [2], ST: Structured3D [65], SP: Spatialaudiogen [29], -all indicates using the entire train, validation, and test sets of the specific dataset, and (p) denotes using pseudo ground truth generated by Depth Anything [59]. Due to space limits, we provide the experimental setup in the appendix, including implementation details and evaluation metrics. 4.1 Baselines Recent state-of-the-art methods [1, 64, 47, 48, 16, 31, 40] have emerged. We chose UniFuse and BiFuse++ as our baseline models for experiments, as many of the aforementioned methods did not fully release pre-trained models or provide training code and implementation details. It’s worth noting that PanoFormer [40] is not included due to incorrect evaluation code and results. Both selected models are re-implemented with affine-invariant loss on disparity for a fair comparison and to demonstrate improvement. We conduct experiments on the Matterport3D [6] benchmark to demonstrate performance gains within the same dataset/domain, and we perform zero-shot evaluation on the Stanford2D3D [2] test set to demonstrate the generalization capability of our proposed training technique. To further validate its robustness, we evaluate additional baseline models [45, 64] in zero-shot setting, showcasing the effectiveness of our approach for non-dual-projection models. 7 Table 2: Matterport3D Benchmark. The upper section lists 360 methods trained with metric depths in meters using BerHu loss. All numbers are sourced from their respective papers. The lower section includes selected methods retrained with relative depth (disparity) using affine-invariant loss. Method Loss Train Test Abs Rel ↓ δ1 ↑ δ2 ↑ δ3 ↑ BiFuse [47] BerHu M M 0.845 0.932 0.963 UniFuse [16] BerHu M M 0.106 0.890 0.962 0.983 SliceNet [31] BerHu M M 0.872 0.948 0.972 BiFuse++ [48] BerHu M M 0.879 0.952 0.977 HRDFuse [1] BerHu M M 0.097 0.916 0.967 0.984 UniFuse [16] Affine-Inv M M 0.102 0.893 0.970 0.989 UniFuse [16] Affine-Inv M, ST-all (p) M 0.089 0.911 0.975 0.991 BiFuse++ [48] Affine-Inv M M 0.094 0.914 0.974 0.989 BiFuse++ [48] Affine-Inv M, ST-all (p) M 0.085 0.917 0.976 0.991 4.2 Benchmarks Evaluation We conducted our in-domain improvement experiment on the widely used 360-degree depth benchmark, Matterport3D [6], to showcase the results of perspective foundation model distillation on the two selected baseline models, UniFuse [16] and BiFuse++[48]. In Table 2, we list the metric depth evaluation results from state-of-the-art methods on this benchmark. Subsequently, we present the re-trained baseline models using affine-invariant loss on disparity to ensure a fair comparison with their original depth metric training. Finally, we demonstrate the improvement achieved with results trained on the labeled Matterport3D training set and the entire Structured3D dataset with pseudo ground truth. 4.3 Zero-Shot Evaluation Our goal is to estimate depths for all 360-degree images, making zero-shot performance crucial. Following previous works [47, 16], we adopted their zero-shot comparison setting, where models trained on the entire Matterport3D [6] dataset are tested on the Stanford2D3D [2] test set. In Table 3, the upper section lists methods trained with metric depth ground truth, with numbers sourced from their respective papers. The lower section includes models trained with affine-invariant loss on disparity ground truth. As shown in Figure 6, [16, 48] demonstrate generalization improvements with a lower error on the Stanford2D3D dataset. Depth Anything [59] and Marigold [19] are state-of-the-art zero-shot depth models trained with perspective depths. As shown in Table 3, due to the domain gap and different camera projections, foundation models trained with perspective depth cannot be directly applied to 360-degree images. We demonstrated the zero-shot improvement on UniFuse [16], BiFuse++ [48] and non-dual-projection methods [45, 64] with models trained on the entire Matterport3D [6] dataset with ground truth and the entire Structured3D [65] or SpatialAudioGen [29] dataset with pseudo ground truth generated using Depth Anything [59]. As Structured3D provides ground truth labels for its dataset, we also evaluate our models on its test set to assess how well they perform with pseudo labels. Table 4 shows the improvements achieved on the Structured3D test set when using models trained with pseudo labels. It’s worth noting that even when the 360 model is trained on pseudo labels from SpatialAudioGen, it performs similarly well. This demonstrates the success of our distillation technique and the model’s ability to generalize across different datasets. 4.4 Qualtative Results in the Wild We demonstrated the qualitative results in Figure 8 and Figure 7 360-degree images that were either captured by us or downloaded from the internet1. These examples showcase the zero-shot capability of our model when applied to data outside the aforementioned 360-degree datasets. 1Stig Nygaard, https://www.flickr.com/photos/stignygaard/49659694937, CC BY 2.0 DEED Dominic Alves, https://www.flickr.com/photos/dominicspics/28296671029/, CC BY 2.0 DEED Luca Biada, https://www.flickr.com/photos/pedroscreamerovsky/6873256488/, CC BY 2.0 DEED Luca Biada, https://www.flickr.com/photos/pedroscreamerovsky/6798474782/, CC BY 2.0 DEED 8 Table 3: Zero-shot Evaluation on Stanford2D3D. We perform zero-shot evaluations with models trained on other datasets. Following the original training settings, we train the 360 models [48, 16, 45, 64] on the entire Matterport3D dataset and then test on Stanford3D’s test set. Method Loss train test Abs Rel ↓ δ1 ↑ δ2 ↑ δ3 ↑ BiFuse [47] BerHu M-all SF 0.120 0.862 UniFuse [16] BerHu M-all SF 0.094 0.913 BiFuse++ [48] BerHu M-all SF 0.107 0.914 0.975 0.989 Depth Anything [59] Affine-Inv Pers. SF 0.248 0.635 0.899 0.97 Marigold [19] Affine-Inv Pers. SF 0.195 0.692 0.942 0.982 UniFuse [16] Affine-Inv M-all SF 0.090 0.914 0.976 0.990 UniFuse [16] Affine-Inv M-all, ST-all (p) SF 0.086 0.924 0.977 0.990 UniFuse [16] Affine-Inv M-all, SP-all (p) SF 0.090 0.920 0.978 0.990 BiFuse++ [48] Affine-Inv M-all SF 0.090 0.921 0.976 0.990 BiFuse++ [48] Affine-Inv M-all, ST-all (p) SF 0.082 0.931 0.979 0.991 BiFuse++ [48] Affine-Inv M-all, SP-all (p) SF 0.086 0.926 0.979 0.991 HoHoNet [45] Affine-Inv M-all SF 0.095 0.906 0.975 0.991 HoHoNet [45] Affine-Inv M-all, ST-all (p) SF 0.088 0.920 0.979 0.992 EGFormer [64] Affine-Inv M-all SF 0.098 0.906 0.972 0.989 EGFormer [64] Affine-Inv M-all, ST-all (p) SF 0.086 0.923 0.976 0.990 UniFuse UniFuse (p) RGB Error Error GT BiFuse++ BiFuse++ (p) RGB Error Error GT Figure 6: Zero-shot qualitative with UniFuse [16] (left) and BiFuse++ [48] (right) tested on Stanford2D3D. Table 4: Structured3D Test Set. We demonstrate the improvement on the Structured3D test set using pseudo ground truth for training. The lower section shows enhancements with models trained on pseudo ground truth from Matterport3D and SpatialAudioGen, indicating similar improvements. This highlights the successful distillation of Depth Anything. Method Loss train test Abs Rel ↓ δ1 ↑ δ2 ↑ δ3 ↑ UniFuse [16] Affine-Inv M-all ST 0.202 0.759 0.932 0.970 UniFuse [16] Affine-Inv M-all, ST-all (p) ST 0.130 0.887 0.953 0.977 UniFuse [16] Affine-Inv M-all, SP-all (p) ST 0.152 0.864 0.946 0.972 9 Table 5: Metric depth fine-tuning. We fine-tune our model trained with Matterport3D [6] ground truth label and Structured3D [65] pseudo label on relative depth with Stanford2D3D [2]’s training set metric depths with a single epoch. Method MAE ↓ Abs Rel ↓ RMSE ↓ RMSElog ↓ δ1 ↑ δ2 ↑ δ3 ↑ UniFuse [16] 0.208 0.111 0.369 0.072 0.871 0.966 0.988 UniFuse [16] (Ours) 0.206 0.118 0.351 0.049 0.910 0.971 0.987 RGB UniFuse UniFuse (p) Figure 7: Generalization ability in the wild with depth map visualization. We showcase zero-shot qualitative results using a combination of images we captured and randomly sourced from the internet to assess the model’s generalization ability. For privacy reasons, we have obscured the cameraman in the images. Figure 8: Generalization ability in the wild with point cloud visualization. We showcase zero-shot qualitative results in the point cloud using a combination of images we captured and randomly sourced from the internet to assess the model’s generalization ability. 4.5 Fine-Tuned to Metric Depth Estimation We use our pre-trained model as an initial weight and fine-tune on Stanford2D3D [2] metric depth to demonstrate the effectiveness of our pre-trained relative depth model’s ability to adapt to metric depth with a single epoch in Table 5 5 Conclusion Our proposed method significantly advances 360-degree monocular depth estimation by leveraging perspective models for pseudo-label generation on unlabeled data. The use of cube projection with random rotation and affine-invariant loss ensures robust training and improved depth prediction accuracy while bridging the domain gap between perspective and equirectangular projection. By effectively addressing the challenges of limited labeled data with cross-domain distillation, our approach opens new possibilities for accurate depth estimation in 360 imagery. This work lays the groundwork for future research and applications, offering a promising direction for further advancements in 360-degree depth estimation. Limitations. Our work faces limitations due to its heavy reliance on the quality of unlabeled data and pseudo labels from perspective foundation models. The results are significantly impacted by data quality (Section 3.1). Without data cleaning, the training process resulted in NaN values. 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ArXiv, abs/2006.06882, 2020. URL https://api.semanticscholar.org/CorpusID:219635973. 15 A Appendix / Supplemental Material A.1 Experimental Setup Implementation details. Our work is divided into two stages: (1) offline mask generation and (2) online joint training. (1) In the first stage, we use Grounded-Segment-Anything [36], which combines state-of-the-art detection and segmentation models. We set the BOX_THRESHOLD and TEXT_THRESHOLD to 0.3 and 0.25, respectively, following the recommendations of the official code. We use “sky” and “watermark” as text prompts. All pixels with these labels are set to False to form our valid mask for the second stage of training. (2) In the second stage, each batch consists of an equal mix of labeled and unlabeled data. We follow the backbone model’s official settings for batch size, learning rate, optimizer, augmentation, and other hyperparameters, changing only the loss function to affine-invariant loss. Unlike Depth Anything, which sets invalid sky regions to zero disparity, we ignore these invalid pixels during loss calculation, consistent with ground truth training settings. We average the loss for ground truth and pseudo ground truth during updates. All our experiments are conducted on a single RTX 4090, both offline and online. However, if future 360-degree state-of-the-art methods or perspective foundation models require higher VRAM usage, the computational resource requirements may increase. Metrics. In line with previous cross-dataset works, all evaluation metrics are presented in percentage terms. The primary metric is Absolute Mean Relative Error (AbsRel), calculated as: 1 M PM i=1 \|ai −di\|/di, where M is the total number of pixels, ai is the predicted depth, and di is the ground truth depth. The second metric, δj accuracy, measures the proportion of pixels where max(ai/di, di/ai) is within 1.25j. During evaluations, we follow [16, 47, 48] to ignore areas where ground truth depth values are larger than 10 or equal to 0. Given the ambiguous scale of self-training results, we apply median alignment after converting disparity output to depth before evaluation, as per the method used in [67]: d′ = d · median( ˆd) median(d), (8) where d is the predicted depth from inverse disparity and ˆd is the ground truth depth. This ensures a fair comparison by aligning the median depth values of predictions and ground truths. A.2 More Qualitative We demonstrate additional zero-shot qualitative results in Figure 9 and in-the-wild results in Figure 12. In-domain results on the Matterport3D test sets are showcased in Figure 10 and Figure 11. A.3 Dataset Statistic As described in Sec.3.1 of the main paper, there is a significant difference in the number of images between the perspective and equirectangular datasets. Detailed statistics of the datasets are listed in Table 6. A.4 Ground Truth and Pseudo Label Ratio Ablation Unlike many previous knowledge distillation approaches that use a higher proportion of pseudo labels during model training, we opt for an equal ratio of ground truth to pseudo labels. Through an ablation study exploring the relationship between this data ratio and model performance, we observe a robust improvement starting from 1 : 1 in Table 7. A.5 Perspective Camera Projection Ablation There are various perspective camera projections for panoramic imagery, with cube and tangent image projections being the most common, both widely used in previous works. In Table 8, we compare these two projections and observe similar improvements when applying our proposed training pipeline, demonstrating the effectiveness of our method. For robust knowledge distillation in relative depth estimation, we select the cube projection due to its wider field of view coverage. 16 RGB GT UniFuse UniFuse (p) BiFuse++ BiFuse++ (p) Figure 9: More qualitative tested on Stanford2D3D with zero-shot setting. GT UniFUse UniFUse (p) RGB Figure 10: In-domain qualitative with UniFuse. 17 GT BiFuse++ BiFuse++ (p) RGB Figure 11: In-domain qualitative with BiFuse++. Table 6: 360 monocular depth estimation lacks a large amount of training data. This table lists datasets used in 360-degree monocular depth estimation alongside perspective depth datasets from the Depth Anything methodology. The volume of training data for 360-degree imagery (right column) is significantly smaller than that for perspective imagery (left column) by about 200 times. This highlights the need for using perspective distillation techniques to enhance the limited data available for 360-degree depth estimation. Ground truth (GT) labels are noted where applicable, showing the available resources for training in these domains. Perspective Equirectangular Dataset Venue # of images GT labels Dataset Venue # of images GT labels MegaDepth [24] CVPR 2018 128K ✓ Stanford2D3D [2] arXiv 2017 1.4K ✓ TartanAir [52] IROS 2020 306K ✓ Matterport3D [6] 3DV 2017 10.8K ✓ DIML [10] arXiv 2021 927K ✓ Structured3D [65] ECCV 2020 21.8K ✓ BlendedMVS [60] CVRP 2020 115 K ✓ SpatialAudioGen [29] NeurIPS 2018 344K HRWSI [55] CVPR 2020 20K ✓ IRS [51] ICME 2021 103K ✓ ImageNet-21K [38] IJCV 2015 13.1M BDD100K [63] CVPR 2020 8.2M Google Landmarks [53] CVPR 2020 4.1M LSUN [62] arXiv 2015 9.8M Objects365 [39] ICCV 2019 1.7M Open Images V7 [21] IJCV 2020 7.8M Places365 [66] TPAMI 2017 6.5M SA-1B [20] ICCV 2023 11.1M Table 7: Ratios of GT and Pseudo label during training. We conduct additional experiments with varying ratios, which shows our method is robust across different ratios starting from 1:1. Ratio train(GT) train(Pseudo) test Abs Rel ↓ δ1 ↑ δ2 ↑ δ3 ↑ 1:1 M-all ST-all(p) SF 0.086 0.924 0.977 0.990 1:2 M-all ST-all(p) SF 0.087 0.923 0.977 0.990 1:4 M-all ST-all(p) SF 0.085 0.923 0.977 0.990 18 Table 8: Comparison between Tangent Image and Cube Projection. We compare two of the most commonly used perspective camera projections for panorama images. As shown in the table, both projections yield similar quantitative results. However, we select the cube projection for knowledge distillation due to its broader field of view coverage. Projection Method train test Abs Rel ↓ δ1 ↑ δ2 ↑ δ3 ↑ Cube UniFuse M-all+ST-all(p) SF 0.086 0.924 0.977 0.990 Tangent UniFuse M-all+ST-all(p) SF 0.087 0.923 0.978 0.991 RGB UniFuse UniFuse (p) Figure 12: Additional in-the-wild results. We compare our proposed joint-training method (Matterport3D (GT) + SpatialAudioGen (Pseudo)) with a model trained only on the Matterport3D dataset, using data randomly downloaded from the internet. 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4,461	A Bayesian Approach for Personalized Federated Learning in Heterogeneous Settings Disha Makhija Electrical and Computer Engineering University of Texas at Austin Austin, TX 78705 disham@utexas.edu Joydeep Ghosh Electrical and Computer Engineering University of Texas at Austin Austin, TX 78705 jghosh@utexas.edu Nhat Ho Statistics and Data Science University of Texas at Austin Austin, TX 78705 minhnhat@utexas.edu Abstract Federated learning (FL), through its privacy-preserving collaborative learning approach, has significantly empowered decentralized devices. However, constraints in either data and/or computational resources among participating clients introduce several challenges in learning, including the inability to train large model architectures, heightened risks of overfitting, and more. In this work, we present a novel FL framework grounded in Bayesian learning to address these challenges. Our approach involves training personalized Bayesian models at each client tailored to the unique complexities of the clients’ datasets and efficiently collaborating across these clients. By leveraging Bayesian neural networks and their uncertainty quantification capabilities, our local training procedure robustly learns from small datasets. And the novel collaboration procedure utilizing priors in the functional (output) space of the networks facilitates collaboration across models of varying sizes, enabling the framework to adapt well in heterogeneous data and computational settings. Furthermore, we present a differentially private version of the algorithm, accompanied by formal differential privacy guarantees that apply without any assumptions on the learning algorithm. Through experiments on popular FL datasets, we demonstrate that our approach outperforms strong baselines in both homogeneous and heterogeneous settings, and under strict privacy constraints. 1 Introduction Federated Learning (FL) has emerged as a pivotal paradigm in various real-world applications, offering a decentralized approach that allows participating clients to contribute to a shared model without compromising the privacy of their raw data. However, implementing FL in practical scenarios poses challenges due to the significant variability among participating clients in terms of their local data and computational resources. Clients with restricted compute capacity may encounter difficulty in training large machine learning models identical to those of other clients, and those with minimal data may struggle to obtain reliable estimates of local model parameters. Due to their ability to generalize under limited data and provide uncertainty quantifications [69, 2, 3], we consider Bayesian based learning methods to construct improved local models. Employing Bayesian learning in FL, however, would involve the following steps - each client doing local 38th Conference on Neural Information Processing Systems (NeurIPS 2024). posterior inference to obtain a distribution over weight parameters and then communicating the local posteriors to the server, the server receiving the local posteriors from the clients and aggregating them to obtain a global posterior distribution, which is then broadcast to the clients for the next round of training. This entire learning procedure, as it turns out, is highly resource and communicationintensive. For solving an m-dimensional federated least squares estimation, this method will require O(m3) computation on all the clients and server sites [4] which is much more as opposed to the cost of standard FL (generally O(m)). How could we then utilize the strengths of Bayesian methods for FL settings without paying such high costs? Additionally, given substantial variations in locally available computing resources, how do we still enable efficient learning and collaboration on all clients? We address these questions by proposing a framework that allows all clients to train their own personal Bayesian models (with varying model complexities), and achieves collaboration across clients by distilling knowledge from the peer clients via a shared unlabelled public dataset and instilling that knowledge in the local models in the form of priors. The challenge of transferring knowledge across models of different architectures is addressed by using the functional (output) space to indirectly determine priors on weights of the local model parameters. Furthermore, to prevent the data leaks in FL setup [22, 68, 23] and formally guarantee the privacy of the local client data, we present a differentially-private version of the algorithm applying a formal well-known standard of differential privacy [19], along with privacy analysis and a bound on the privacy loss of the entire procedure. This work provides a novel integrated Federated Learning (FL) framework, FedBNN, designed to tackle challenges arising from both limited data and heterogeneous computational resources across clients. Additionally, our method offers valuable characterizations of model uncertainties, and is able to operate under strict data privacy constraints, thereby extending the applicability of FL to crucial domains such as healthcare and legal, where these considerations are paramount. To the best of our knowledge, no previous work has jointly addressed all these learning challenges in the FL context. Our promising results significantly broaden the potential of FL for critical real-world applications. Specifically, our key contributions can be summarized as : • We propose a new approach to personalized federated learning utilizing Bayesian principles for improved robustness and reliability, particularly in contexts where data is scarce. Despite its Bayesian framework, this method is designed to be both computationally and communication efficient. • A novel collaboration mechanism based on assigning prior distributions over the model parameters via the output space, instead of directly sharing model parameters’ distributions, which can be computationally expensive and raise privacy concerns, is proposed to enable clients having different computational resources to train models with varying complexity. This is important because in real-world FL applications, clients often have vastly different capabilities. • We provide a formal differential privacy guarantee for our method that applies to general settings irrespective of the client’s learning algorithm and show that the method is able to learn effectively even under strict privacy guarantees. • We evaluated our method on several datasets and show that it outperforms the baselines by a significant margin, particularly in heterogeneous data and model settings. This makes FedBNN particularly well-suited for real-world FL applications, which often exhibit high degrees of heterogeneity. 2 Related Work This section provides a brief overview of the most relevant prior work in the fields of federated learning, Bayesian FL, and Differential Privacy in FL. Federated Learning FL was introduced as the FedAvg algorithm in the seminal work in [45]. Since then many different modifications have been proposed that tackle specific challenges, including global FL solutions as well as personalized solutions. FedPD [74], FedSplit [54], and FedDyn [1] proposed methods for finding better fixed-point solutions to the FL optimization problem. [40, 73, 66, 58, 15] show that point-wise aggregate of the local client models does not produce a good global model and propose alternate aggregation mechanisms to achieve collaboration. Personalized FL has been 2 Table 1: Contrasting our method, FedBNN, against previous works. Method Addressed Challenges Limited Data Heterogeneous Compute Uncertainty Quantification Privacy FedProx ✗ ✗ ✗ ✓ pFedME ✓ ✗ ✓ ✗ FOLA ✓ ✗ ✓ ✗ pFedGP ✓ ✗ ✓ ✗ pFedBayes ✓ ✗ ✓ ✗ FedPop ✓ ✗ ✓ ✗ FedAUX ✗ ✓ ✗ ✓ FedBNN ✓ ✓ ✓ ✓ approached in many ways like meta-learning [20, 8, 31, 33], multi-task learning [59, 38, 60], by clustering the clients [56, 26] and others [17, 37, 72, 57, 67, 43], [16] uses Bayesian view based analysis to obtain a better trade-off between personal and global models. Knowledge distillation for personalized FL settings has also been used previously for training heterogeneous models in non-Bayesian settings [36, 40, 50]. Several other methods have proposed enhancements in FL learning and privacy by using an auxiliary dataset like [15, 55, 36, 52]. Most of these methods rely on the well-established knowledge distillation procedures. But since the transfer of knowledge or information between Bayesian models itself has remained inadequately addressed, these methods are not easily extensible to Bayesian settings. Our method on the other hand, utilizes a novel method that enables collaboration across client specific Bayesian models by transferring knowledge through a prior specification mechanism in the output space, which also enhances the field of Bayesian knowledge distillation. Bayesian Federated Learning Bayesian approaches for federated learning can also be broadly divided as methods using Bayesian inference to obtain a global model and personalized Bayesian learning methods. Amongst the methods that train a global model, some methods just use Bayesian mechanisms for achieving collaboration among non-Bayesian local models, like FedBE [15] which uses Bayesian mechanism to aggregate the locally trained neural networks to obtain a Bayesian ensemble at the server, [9] which suggests using an MCMC based method for obtaining a global model from the local models, and PFNM [73] and FedMA [66] which use a Beta-Bernoullli process to obtain the global models. Other methods that train local Bayesian models at the clients and at the server include FedPA [4] that uses Laplace approximations for an efficient way of computing local and global posteriors, [18] that suggests the use of Bayesian Optimization and Thompson Sampling to obtain the solution to the global optimization problem, recently, [49] did an empirical study on various ways of aggregation mechanisms for local variational Bayesian neural networks and their effects on the solution. These methods that focus on obtaining a global solution are less suited for the statistical heterogeneity present across clients [14], and therefore we focus more on the methods that build personalized Bayesian solutions for clients. Among such methods, pFedGP [3] is a Gaussian Process based estimation method that utilizes Deep Kernel Learning to collaboratively train a single deep neural network with FedAvg and then uses personalized GPs for prediction. FedLoc [70] also uses GP in FL but for regression tasks. pFedBayes [75] uses variational inference locally to optimize a loss at each client that is a combination of the data likelihood term and distance to the prior and iteratively determines the prior from the global posterior distribution. FOLA [41] proposed using Laplace Approximation for posterior inference at both the server side and the client side, PAC-FL [11] and [34, 65, 51] also proposed variants of methods that assume Bayesian models on local clients but for all of them main assumption is that the local model parameters are generated from a shared global distribution thus making them useful only in homogeneous settings. All the methods described above choose priors by assuming a distribution over values for each weight, and thus choosing an appropriate and meaningful prior becomes a challenge [14]. These issues led us to use functional space priors instead which have been explored in limited centralized settings [63, 61, 21] but not in FL. But most importantly, none of these methods are designed or could be easily extended to work with compute heterogeneous settings limiting the applicability of these solutions in several real-world scenarios. Table 1 compares our approach with the most closely related works. Differential Privacy in FL Since decentralized learning does not guarantee that the data will remain private, it is important that a formal rigorous guarantee be given on the data that is leaked by the algorithm. Seminal works in DP propose using a Gaussian noise mechanism by adding Gaussian noise to the intermediate results and achieving a bound on the algorithm by using composition 3 results [19, 46, 32]. For FL, [24] and [44] independently proposed DP-FedSGD and DP-FedAvg algorithms, which enhance FedAvg by adding Gaussian noise to the local client updates. Several other works focus on analyzing the privacy-utility trade-off in DP in FL setting [25, 27, 5, 64, 39]. Recently, [30] proposed a DP-based solution for personalized FL that works only for linear models. And then [47] improved it for general models and heterogeneous data in FL. These methods, however, mostly focus on privacy guarantees while solving the non-Bayesian FL optimization problem. 3 Methodology In this section, we first go over the problem setting and background, and then present our proposed framework, FedBNN, with details of all the key components. 3.1 Background Problem Description Consider an FL setting with N clients where each client i has local dataset X i of size ni drawn from the local data distribution Di. The goal of a personalized federated learning procedure is to obtain optimal weights for each client’s local model, W∗ i , given the entire data, X = SN j=1 X j through collaboration but without compromising client data privacy. However, the learning procedure faces challenges that are posed due to - system heterogeneity and statistical heterogeneity. System heterogeneity refers to the variable amount of data and compute resources across clients, meaning, i) the data resources on each client vary widely, i.e., nk >> nl for some clients k and l, and ii) the compute across clients is non-identical due to which it is not possible to train models of uniform architectures across clients, leading to non-identical weights, i.e., Wi ̸= Wj for different clients i and j. Statistical heterogeneity implies that the data distribution across clients is non-IID. Bayesian Learning Instead of obtaining the optimal values of the model parameters, W∗ i , Bayesian learning aims to learn posterior distributions (probability distributions over the values) for all the model parameters from the given data - IP(W\|X). Thus, in a personalized Bayesian FL procedure, the modified goal would be to learn distributions for local weights, IP(Wi\|X) from X = SN j=1 X j. However, the exact inference for obtaining the posterior distribution for each of the weight parameter in the network is intractable and several approximations have been studied to obtain approximate distributions. Variational inference [29] is an approximation method that tries to learn parameterized distribution q(w\|θ) from a family of distributions Q, typically of simpler form, by optimizing the parameters θ such that the new distribution q(w\|θ∗), obtained for the optimal value of θ, is close to the desired posterior distribution IP(W\|X). Precisely, θ∗is obtained by solving the following optimization problem and its expansion given below θ∗= arg min θ:q(W\|θ)∈Q KL[q(W\|θ)\|\|IP(W\|X)] (1) = arg min θ:q(W\|θ)∈Q KL[q(W\|θ)\|\|p(W; ψ)] −Eq(W\|θ)[logIP(X\|W)] (2) and then q(w\|θ∗) is used in place of IP(W\|X). The optimization objective minimizes the distance of q(W\|θ) to a prior distribution p(W; ψ), used to encode any prior information about the parameters, while also maximizing the likelihood of the observed data X under q(W\|θ). A more detailed discussion of Bayesian learning is included in Appendix A. Even though Bayesian approaches are more computationally expensive than their point-estimation counterparts, their superior capabilities for uncertainty quantification and performance in small data settings outweigh the extra compute costs in many critical applications. Moreover, recent innovations like Bayes by Backprop [10] which carefully uses the backpropagated gradients for learning the parameters of posterior distributions, drastically reducing the added computation costs. 3.2 FedBNN Methodology The FedBNN framework works iteratively in two steps - local optimization on the individual clients to obtain local posterior distributions over the model parameters, and a global collaboration step 4 where the output from each client is appropriately aggregated at the server and broadcast to all the clients for the next rounds of training. These two steps are further described below, and the detailed algorithm and the overview diagram are included in the Appendix B in Algorithm 1 and Figure 2 respectively. Local Setting Let each client in the network be training a personalized Bayesian NN, which for the client i is denoted by Φi and is parameterised by weights Wi. As commonly used in the literature, we assume that the individual weights of the BNN are Normally distributed and satisfy mean-field decomposition, i.e., wi,α ∼N(µi,α, σ2 i,α) for α ∈[1, . . . , \|Wi\|] where µi,α is the mean of the Gaussian distribution for the parameter α on the ith client and σ2 i,α is the variance of the Gaussian distribution for the same parameter. To guarantee that σi,α takes non-negative values for all clients i and all parameters α, we use a technique commonly used in inference procedures [10], and replace each σi,α by another parameter ρi,α during the training, with σi,α = log(1 + exp(ρi,α)). The individual weights of the local BNN, wi,α, are also assumed to each have a Gaussian prior distribution, p(wi,α; ψi,α), parameterized by ψi,α = (µp i,α, σp i,α). 3.2.1 Global Collaboration We attain collaboration amongst clients via an auxiliary dataset called the Alignment Dataset (AD). This is an unlabeled dataset typically small in size, and is used for providing peer supervision to the individual clients by helping clients distill knowledge from other peer clients without explicitly sharing a large number of locally learned parameter weight distributions. The experiments in Figure 5 and Table 3 show the effect of the varying size and distribution of AD in achieving effective collaboration. In heterogeneous settings, the use of non-identical architecture models (Wi ̸= Wj) means that there is no direct way of aggregating the distributions for prior specification. In fact, even in homogeneous settings, aggregating the weight distributions can be prone to errors due to reasons like insufficient understanding of the weight space, non-alignment of weights across models, etc. Thus, for the purpose of collaboration, we use the function-space of the networks rather than the weight space. Specifically, in each global communication round, the server shares the AD with all the clients. The clients do a forward pass on AD to obtain the local output Φi(AD), where the local output of the ith client is approximated by drawing m sets of weight samples, W(j) i : j ∈[1, m], from its local posterior distribution IP(Wi\|X) using Monte Carlo sampling and aggregating the outputs under each of these samples Φi(AD) = 1 m Pm j=1 Φi(AD; W(j) i ). The obtained output for AD on each client is then sent back to server which forms an aggregated representation, denoted by ¯Φ(AD), obtained via a weighted aggregation of all clients’ outputs, i.e., ¯Φ(X) = PN j=1 wjΦj(X). By default, all weights are considered the same, however the formulation provides flexibility, for example to accommodate situations where the aggregation weights could represent the relative strength of each client in terms of its data or compute resources, i.e., clients with high compute (or data) resources receive more weight as compared to clients with lower amount of resources. The obtained ¯Φ(AD) is then uploaded to all the clients for use in the next round of local training. More details about the Alignment Dataset (AD) along with the explanations and experiments on the size, distribution, availability etc. of the AD are included in the Appendix E. 3.2.2 Local Optimization on Clients Prior Specification Design The Bayesian framework provides a natural way of incorporating supervision in the form of priors. Conventional methods in Bayesian deep learning provide direct priors for model weights as distribution over values. However, the relationship between the values of the model weights/parameters and the outputs is complex and the priors in model’s weight-space do not directly capture the desired functional properties. Also, since the number of parameters in a neural network is large, most prior specifications tend to take a simplistic form like an isotropic Gaussian, to make inference feasible. Thus, learning by specifying prior distributions over weights does not always help translate prior knowledge in the learning process. In this work, we consider a way of specifying priors in the functional space by first optimising the Bayesian neural networks over the prior parameters for a fixed number of steps so that the BNN achieves a desired functional output. These intuitive priors help in explicitly instilling the external knowledge during the training of the 5 neural networks. Let p(Wi; ψ) represent the prior function over the weights Wi and is parameterized by ψ, with ψ = {(µp i,α, σp i,α), α ∈[1, . . . , \|Wi\|]}, the prior parameters that determine the prior distributions are learned by solving an optimization problem as below: ψ∗ i = arg min ψ d(Y, Φi(AD; Wi)), where d is a suitable distance function and Y represents the desired output, resulting in optimal priors p(Wi; ψ∗). Below we provide details of the prior specification for our method. Local Optimization For the local optimization, the individual clients learn IP(Wi\|X i) via variational inference. As described above, a variational learning algorithm tries to find optimal parameters θ∗of a parameterized distribution q(Wi\|θ) among a family of distributions denoted by Q. In our setting, we set the family of distributions, Q, to be containing distributions of the form wi,α ∼N(µi,α, σ2 i,α) for each parameter wi,α for α ∈[1, . . . , \|Wi\|]. For inference in Bayesian neural networks, we use Bayes by Backprop [10] method to solve the variational inference optimization problem. At the beginning of each local optimization procedure (in each global communication round a specific client is selected), we use the global information obtained from the server ¯Φ(AD) to intialize the prior for the BNN. Specifically, at the beginning of each local training round, the selected clients first tune their priors to minimize the distance between the local output, Φi(AD; Wi) and the aggregated output obtained from the server, ¯Φ(AD). Since the aggregated output represents the collective knowledge of all the clients and may not be strictly precise for the local model optimization, we consider this aggregated output as “noisy" and correct it before using for optimization. Specifically, we generate Φcorrected i as a convex combination of the global output and the local output for a tunable parameter γ. For the ith client, Φcorrected i = γ ¯Φ(AD) + (1 −γ)Φi(AD; Wi). (3) The prior optimization steps then optimize the distance between Φcorrected i and Φi(AD; Wi) to train the prior parameters ψ, with the aim of transferring the global knowledge encoded in Φcorrected i to the local model. Precisely, ψ∗ i = arg min ψ d(Φcorrected i , Φi(AD; Wi)). (4) When the outputs Φ(X; W) are logits, we use cross-entropy or the negative log-likelihood loss as the distance measure. The optimization involves training the client’s personal BNN Φi to only learn the parameters of the prior distribution denoted by ψ. This way of initializing the BNN prior enables translating the functional properties, as captured by Φi(AD; Wi), to weight-space distributions. The optimal prior parameters are then kept fixed while training the BNN over the local dataset. The local optimization procedure now works to find the best q(Wi\|θ) fixing the prior distribution through the following optimization problem : θ∗ i = arg min θ:q(Wi\|θ)∈Q KL[q(Wi\|θ)\|\|p(Wi; ψ∗ i )] −Eq(Wi\|θ)[logIP(X i\|Wi)], (5) which is similar to the optimization problem defined in Equation 1 except that now the prior parameters are optimized so that the obtained prior distributions capture the global knowledge and can guide the local learning process to make q(Wi\|θ) close to the global collective knowledge. 3.2.3 Achieving Differential Privacy In this variation, to control the release of information from the clients, we add a carefully designed Gaussian mechanism wherein we add Gaussian noise to the Φi(AD) that is being shared by each client. Specifically, each client i uploads Φi(AD)DP = Φi(AD) + N(0, σ2 g) to the server and then the server aggregates Φi(AD)DP across clients to obtain and broadcast ¯Φ(AD)DP which is used by the clients in their next round of local optimization. The variance of the noise depends on the required privacy guarantee. 6 4 Privacy Analysis Though our algorithm is inherently quite private as it refrains from explicitly sharing model weights, we can also provide a formal Differential Privacy based guarantee. Our analysis in this section focuses on providing record-level DP guarantee over the entire dataset X. This analysis quantifies the level of privacy achieved towards any third party and an honest-but-curious server. In this section we directly present the key result of our analysis. Due to the lack of space, additional definitions, results and the proof for the theorem are mentioned in Appendix C. Theorem 4.1 (Privacy Budget). The proposed algorithm is (ϵ, δ)-differentially private, if the total privacy budget per global communication round per query is set to ρ = ϵ2 4EKlog 1 δ for E number of global communication rounds and K number of queries to the algorithm per round. The parameter ρ is related to the Gaussian noise by ρ = ∆2 2σ2 . The detailed proof is included in Appendix C. Our analysis does not assume any specifics of how each client is trained and is therefore applicable in more general settings. Note that we present a pessimistic analysis by providing a worstcase analytical bound, wherein we assume that a change in single data point may entirely change the output of the algorithm, and also since the public dataset remains common throughout the rounds, the actual privacy loss due to querying on the public dataset does not typically add up linearly. Yet the above analysis shows that we have several knobs to control to achieve the desired privacy-utility trade off - balancing the number of global communication rounds with local epochs, reducing the number of queries, and the standard noise scale. By appropriately tuning these controls we are able to achieve good performance with a single digit ϵ (≈9.98) privacy guarantee and δ = 10−4. 5 Experiments In this section, we present an experimental evaluation of our method and compare it with different baselines under diverse homogeneous and heterogeneous client settings. Specifically, we experiment with three types of heterogeneity - i) heterogeneity in data resources (amount of data), ii) heterogeneity in compute resources, and iii) statistical heterogeneity (non-IID data distribution across clients). We also discuss the change in performance of our method when the degree and type of heterogeneity changes. Due to the space constraint, additional experiments on varying the size and distribution of the AD, privacy-utility trade-off and model calibration are included in the Appendix E, G and D respectively. 5.1 Experimental Details Datasets We choose three different datasets commonly used in prior federated learning works from the popular FL benchmark, LEAF [13] including MNIST, CIFAR-10 and CIFAR-100. MNIST contains 10 different classes corresponding to the 10 digits with 50,000 28×28 black and white train images and 10,000 images for validation. CIFAR-10 and CIFAR-100 contain 50,000 train and 10,000 test-colored images for 10 classes and 100 classes respectively. The choice of these datasets is primarily motivated by their use in the baseline methods. Simulation Details We simulate three different types of heterogeneous settings - corresponding to heterogeneity in compute resources, data resources and the statistical data distribution. Before starting the training process, we create N different clients with different compute resources by randomly selecting a fraction of clients that represent clients with smaller compute. Since these clients do not have large memory and compute capacity, we assume that these clients train smaller-size BNNs as opposed to the other high-capacity clients that train larger VGG-based models. In particular, the small BNNs were constructed to have either 2 or 3 convolution layers, each followed by a ReLU and 2 fully-connected layers at the end, and a VGG9-based architecture was used for larger BNNs. The number of parameters in smaller networks is around 50K and that in larger networks is around 3M. Since the baselines only operate with identical model architectures across clients, we use the larger VGG9-based models on the baselines for a fair comparison. We include the results of our 7 method in both homogeneous compute settings (similar to baselines) as well as in heterogeneous compute settings wherein we assume that 30% of the total clients have smaller compute and are training smaller-sized models. Next, we also vary the data resources across clients and test the methods under 3 different data settings - small, medium and full. The small setting corresponds to each client having only 50 training data instances per class, for the medium and full settings each client has 100 data instances and all available data instances per class respectively for training. We simulate statistical heterogeneity by creating non-IID data partitions across clients. We work in a rather strict non-IID setting by assuming clients have access to data of disjoint classes. For each client a fraction of instance classes is sampled and then instances corresponding to the selected classes are divided amongst the specific clients. For the included experiments, we set number of clients N = 20 and divide the instances on clients such that each client has access to only 5 of the 10 classes for MNIST and CIFAR-10, and 20 out of 100 classes for CIFAR-100. Table 2: Test accuracy comparsion with baselines in non-IID settings. Method MNIST CIFAR10 CIFAR100 (small) (medium) (full) (small) (medium) (full) (small) (medium) (full) (Non-Bayesian) Local Training 88.7 ± 1.2 90.1 ± 1.0 91.9 ± 1.1 53.9 ± 2.1 59.5 ± 1.8 70.8 ± 1.4 28.8 ± 1.8 32.7 ± 1.9 43.5 ± 1.6 FedAvg 88.2 ± 0.5 90.15 ± 1.2 92.23 ± 1.0 43.14 ± 1.2 56.27 ± 1.8 78.17 ± 1.2 27.3 ± 1.9 32.81 ± 1.6 36.3 ± 0.2 FedProx 86.9 ± 0.8 89.91 ± 0.7 93.1 ± 0.4 44.27 ± 1.2 58.93 ± 0.9 79.19 ± 0.6 28.6 ± 2.7 34.31 ± 1.4 37.8 ± 0.9 FedAUX 90.1 ± 1.6 92.8 ± 1.34 94.4 ± 1.21 60.01 ± 1.96 68.6 ± 0.73 77.0 ± 0.84 37.05 ± 1.3 43.5 ± 1.7 45.2 ± 0.88 pFedME 91.95 ± 2.1 93.39 ± 1.2 95.62 ± 0.5 48.46 ± 1.5 64.57 ± 2.1 75.11 ± 1.2 32.4 ± 2.2 36.3 ± 2.0 41.8 ± 1.7 non-Bayesian KD 89.1 ± 0.4 92.5 ± 0.2 93.2 ± 0.3 33.9 ± 1.3 53.2 ± 1.5 69.8 ± 1.0 26.1 ± 2.0 35.2 ± 1.2 42.7 ± 0.8 (Bayesian with Homogeneous Architectures) pFedGP 86.15 ± 1.3 90.59 ± 1.7 94.92 ± 0.3 45.62 ± 2.2 56.24 ± 1.8 72.89 ± 0.7 47.06 ± 1.3 53.1 ± 1.2 54.54 ± 0.2 pFedBayes 94.0 ± 0.2 94.6 ± 0.1 95.5 ± 0.3 58.7 ± 1.1 64.6 ± 0.8 78.3 ± 0.5 39.51 ± 1.8 41.43 ± 0.4 47.67 ± 1.1 FOLA 91.74 ± 1.0 92.87 ± 0.8 95.12 ± 0.6 43.29 ± 0.9 45.94 ± 0.7 67.98 ± 0.5 33.42 ± 1.3 48.8 ± 2.1 43.2 ± 1.6 Ours (Homo) 94.9 ± 1.0 95.72 ± 0.8 96.21 ± 0.3 70.6 ± 1.1 72.3 ± 0.6 79.7 ± 0.3 49.65 ± 1.4 55.4 ± 0.8 57.3 ± 0.8 Ours (Hetero) 93.1 ± 1.1 94.4 ± 0.2 95.9 ± 0.2 68.17 ± 2.0 71.73 ± 1.3 78.7 ± 0.7 47.5 ± 1.4 49.10 ± 1.1 51.1 ± 0.7 Ours (Hetero-DP) 89.82 ± 2.3 90.21 ± 1.6 91.43 ± 1.4 54.9 ± 1.91 61.83 ± 1.4 74.3 ± 1.6 43.7 ± 2.3 44.5 ± 1.7 47.0 ± 1.5 (DP-Baseline) DP-FedAvg 80.1 ± 1.7 85.2 ± 1.8 86.2 ± 1.7 35.17 ± 0.8 50.22 ± 1.1 74.6 ± 1.2 26.5 ± 0.3 30.7 ± 1.4 32.4 ± 0.6 Training parameters and Evaluation We run all the algorithms for 200 global communication rounds and report the accuracy on the test dataset at the end of the 200th round. The number of local epochs is set to 20 and the size of AD is kept as 2000. Each client is allowed to train its personal model for a fixed number of epochs, which is kept to 50 in experiments, before entering the collaboration phase. The hyper-parameters of the training procedure are tuned on a set-aside validation set. At the beginning of each global communication round, for optimizing the prior parameters at each client according to Equation 4, we use an Adam optimizer with learning rate=0.0001 and run the prior optimization procedure for 100 steps. Then with the optimized prior we train the local BNN using Bayes-by-Backprop, with Adam optimizer, learning rate = 0.001 and batch size = 128. The noise effect γ is selected after fine-tuning and kept to be 0.7. For these experiments, the aggregation weight wj for each client j used to compute ¯Φ(X) is set to 1/N, and the AD is obtained by using a separated subset of the dataset in consideration. All the models are trained on a 4 GPU machine with GeForce RTX 3090 GPUs and 24GB per GPU memory. For evaluation, we report the classification accuracy obtained by running the trained models on test datasets from the MNIST, CIFAR10 and CIFAR100 datasets. Baselines We compare our method against the standard non-Bayesian FL algorithms and Bayesian-FL methods that build personalized models for clients. We also show results of differentially private FedAvg algorithm under similar privacy guarantee to provide perspective on the privacy. Apart from local training where all clients train independent models locally without collaboration, the non-Bayesian FL baselines include - i) FedAvg, ii) FedProx, iii) pFedME (which uses personalized models on each client using Monreau envelopes in loss). We also compare our method to other 8 baselines that use an auxiliary dataset for collaboration in non-Bayesian FL, namely i) FedAUX, which uses federated distillation for achieving collaboration in FL, we do not use FedMD [36] as a baseline since it requires a labelled auxiliary dataset. Further, we create a baseline corresponding to the non-Bayesian version of our method that works with knowledge distillation and call it nonBayesian KD. The Bayesian FL baselines include - i) pFedGP, a Gaussian process based approach that trains common deep kernels across clients and personal tree-based GPs for classification, ii) pFedBayes, which uses a variational inference-based approach for personalized FL by training personal models which are close to the aggregated global models, iii) FOLA, bayesian method using Gaussian product for model aggregation. And lastly, the DP baseline includes - i) DP-FedAvg, the FedAvg algorithm with gradient clipping and noise addition to the gradient at each client. The size of the AD is changed to 1000, number of local epochs to 40 and global communication rounds to 100 for DP-based experiments. For all the experiments, the hyper-parameters were obtained by tuning on a held-out validation dataset. We used our own implementation of the pFedBayes algorithm since the source code was not publicly available but we could not compare against FedPop due to the lack of some implementation details and publicly unavailable code. (a) (b) (c) Figure 1: Performance comparison of our method with baselines under different types and varying degree of heterogeneity for CIFAR-10 dataset with 20 clients. Figure (a) is for heterogeneity in compute capacity across clients under non-IID data setting, figure (b) for compute heterogeneity under IID setting, and figure (c) for heterogeneity in data resources. When a fraction of clients in the setting have low computing resources, the baselines being homogeneous can only train smaller models on all the clients as shown by constant performance. The results show that our method is more tolerant to both model heterogeneity and data heterogeneity across clients. 5.2 Results The performance of our method and the baselines under the non-IID data setting are reported in Table 2. Under the non-IID setting, we report the results corresponding to different dataset sizes on each client. To recall, in the small, medium, and full settings, each client has access to 50, 100, and all training data points per class respectively. We observe that our method with homogeneous architectures across clients outperforms all other baselines. Moreover, when we consider the performance of our method under a heterogeneous setting by considering 30% of the total clients to be small capacity, it is evident that our method is better than the higher capacity homogeneous baselines for more complex tasks like in CIFAR-10 and CIFAR-100. On average, our method achieves about 6% performance improvement over the baselines in the small and medium data settings. Figure 1 compares the performance of our method with the highest-performing baselines under model, data and statistical types of heterogeneity. Since our method can work with heterogeneous clients, we see that just by the proposed collaboration and having higher capacity clients in the FL ecosystem, the lower capacity clients are able to gain about 10% increase in their performance. Also, the performance degradation of our method with a change in the number of clients with limited data resources is more graceful as compared to the baselines. In an additional experiment intended to compare the performance of the baseline methods with additional data, we trained the priors for baseline methods’ encoders using the unlabeled data, AD, before starting their own prescribed FL procedure. We observed that the performance of the baseline methods does not change on doing this because the FL procedure that they incorporate forgets all the prior existing local knowledge at the client side. A similar result was also reported in [55]. The superior performance of our method could be attributed to the innovative and effective collaboration achieved by first distilling peer knowledge in the form of the aggregated output on the 9 AD, and then ensuring that this knowledge is successfully transferred to each client by specifying priors in the functional-space of the client model. Furthermore, the parameter in Equation 3 allows the clients the flexibility to choose the amount of global knowledge that needs to be incorporated, providing flexibility on the degree of personalization. 6 Discussion This paper introduced a novel method for personalized Bayesian learning in heterogeneous FL settings and demonstrated that it is able to outperform existing approaches under different types of heterogeneous situations, while also providing a privacy guarantee and calibrated responses. The experiments show that the method is particularly useful for clients with lower data and lower compute resources as they can benefit the most by the presence of other, more powerful clients in the ecosystem. While our method assumes the availability of a small, unlabelled auxiliary dataset at the server, it is typically a very mild requirement as such data can often be obtained from several open sources on the web. In many cross-silo and cross-device applications, the server often possesses its dataset alongside private data from clients. For example, hospitals with access to patient records may combine this data with private patient data collected from individual devices such as wearables or sensors for FL, source code generation applications might leverage open-source code along with private code repositories from developers, etc. [6, 7] also mention use-cases where such data is available in real-world. Recent advances in generative AI have made creation of synthetic data for training a much easier task. The privacy analysis on the method provides an intuitive and a rigorous guarantee with various tunable knobs that can be adjusted to achieve the desired privacy-utility trade-off. And while the application explored in the proposed work consists of image related tasks, both the proposed framework and the privacy analysis are generic and independent of specific training algorithms, therefore resulting in its wide applicability in various applications across data modalities. Also, while Bayesian methods are inherently more computationally expensive as they have to maintain distributions rather than point estimates, this extra work is invaluable in many applications where uncertainty quantification is important, for example to help engineers account for uncertainties in material properties, loading conditions, and manufacturing processes, leading to safer and more reliable designs [53]. The recent use of transformer based Bayesian methods [76, 42, 62] in varied applications indicate that the proposed framework can be also applied to settings where much larger neural networks are required. 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Global-local bayesian transformer for semantic correspondence, 2023. 15 Supplement for “A Bayesian Approach for Personalized Federated Learning in Heterogeneous Settings” In this supplementary material, we first go over the preliminaries of Bayesian learning methods, followed by the pseudo-code of the algorithm used for training our framework. Then, we provide definitions and results used in the privacy analysis of the method along with the proof of our privacy budget theorem. We show model calibration metrics and present results demonstrating our method is well-calibrated. We also discuss the details about the alignment dataset, AD, its affect on the performance, include additional experimental results and discuss the communication and computation cost of the procedure. A Bayesian Learning Consider a learning setting where we are trying to train a neural network on a dataset X. The aim of this setting is thus to obtain the set of weights, denoted by W, for the corresponding neural network that best fits the data. We could also view a neural network as a model that outputs IP(y\|x, W) which is the distribution of the label y for a given data point x under the weights W, for classification this would be the output of the softmax function. Now, the weights of the network can be learnt by Maximum Likelihood Estimation (MLE) for a given set of datapoints X = (xi, yi)n i=1 by solving the following optimization problem. WMLE = arg max W X i logP(yi\|xi, W) This optimization could be solved by gradient descent based methods and obtains a point estimate of the weight vector, denoted by WMLE. The Bayesian learning methods, on the other hand, obtain a posterior distribution on the weights given the training data, IP(W\|X), which as opposed to the point estimates denotes the joint distribution of all the weight parameters of the network over the set of values they are likely to take under the observed data and the prior information encoded in the prior distribution. The predictions for any new data point, x, are then obtained by taking expectation of the prediction under the posterior distribution, y = Ew∼IP(W\|X)[IP(y\|x, w)]. Exact inference of the posterior distribution, however, is intractable for neural networks. Variational inference is a traditional approximation method used to obtain an approximation of the posterior weight distribution, and it has also been shown to work for neural networks [29]. Specifically, variational inference tries to learn a simpler parameterized distribution q(w\|θ) from a family of distributions Q by optimizing the parameters θ such that the new distribution q(w\|θ∗) obtained for the optimal value of θ is close to the true posterior distribution IP(W\|X). Precisely, the optimization problem looks like θ∗= arg min θ:q(W\|θ)∈Q KL[q(W\|θ)\|\|IP(W\|X)] (6) = arg min θ:q(W\|θ)∈Q Z q(W\|θ)log q(W\|θ) IP(W)IP(X\|W) (7) = arg min θ:q(W\|θ)∈Q KL[q(W\|θ)\|\|p(W; ψ)] −Eq(W\|θ)[logIP(X\|W)] (8) where p(W; ψ) signifies the prior distribution over weights W parameterized by ψ. The prior distribution is typically used to encode any previously available information about the weights of the network. The above given objective is the same objective as in Equation 5 that is used for local training in our method. B Algorithm The pseudo-code of the algorithm used in the FedBNN method is included in the Algorithm 1. The Algorithm 1 works in the setting when there is a server connected to N clients with each client i having local dataset X i of size ni drawn from the local data distribution Di, and the server has an auxilliary unlabelled dataset called AD. The output of the algorithm is the set of personalized models Φi parameterized by Wi for each client i. All Wi’s, instead of being point estimates, are determined by a posterior distribution IP(Wi\|.) which is learnt from the data via variational inference. As mentioned in the Section 3.2, the learning procedure first optimizes the prior parameters by minimizing Equation 4 and then learns the posterior parameters keeping the prior fixed by minimizing Equation 5. C Privacy Analysis Some known results on differential privacy that are used to determine the privacy loss of our algorithm are given in this section and then the proof of the Theorem 4.1 is presented. 16 Algorithm 1 FedBNN Algorithm Input: number of clients N, number of global communication rounds E, number of local epochs e, weight vector [w1, w2, . . . wN], noise parameter γ Output: Personalized BNNs {Φi\|i ∈[1, N]}, parameterized by Wi ∼IP(Wi\|X) Server Side X = AD for t = 1 to E do Select a subset of clients Nt for each selected client i ∈Nt do Φi(X) = LocalTraining(t, ¯Φ(X)(t−1), X) end for ¯Φ(X)(t) = PNt j=1 wjΦj(X) end for Return Φ1(E), Φ2(E) . . . ΦN(E) LocalTraining(t, ¯Φ(X)(t−1), X) Run inference on X to obtain Φi(X) Generate Φcorrected i (X) = γ ¯Φ(X)(t−1) + (1 −γ)Φi(X) for each prior epoch do Minimize CrossEntropy(Φcorrected i (X), Φi(X)) to obtain prior parameters ψ of the BNN Φi end for for each local epoch do Minimize KL[q(Wi\|θ)\|\|p(Wi; ψ∗)] −Eq(Wi\|θ)[logIP(X i\|Wi)] over {θ : q(Wi\|θ) ∈Q} to obtain θ∗ end for IP(Wi\|X) ≈q(Wi\|θ∗) Obtain m Monte-carlo samples W(j) i : j ∈[1, m] from IP(Wi\|X) Compute Φi(X) = 1 m Pm j=1 Φi(X; W(j) i ) Return Φi(X) ……. Client 1 Client N prior prior posterior posterior local data local data ഥΦ 𝐴𝐷= σ 𝑤𝑖Φ𝑖(𝐴𝐷) Figure 2: A schematic overview of our method: Local BNN on each client obtain a posterior distribution over local parameters using the prior distribution and local data. These local models generate outputs on the AD using their respective posterior distributions and share these outputs with the server. The server aggregates these outputs and distributes the aggregated output on the AD back to all clients, guiding the prior distribution on each client. These updated prior distributions then further guide the learning of the posterior distributions. 17 Definition C.1 ((ϵ, δ)- Differential Privacy). A randomized algorithm M is (ϵ, δ)-DP if for any two neighboring datasets D and D′ that differ in at most one data point, the output of the algorithm M on D and D′ is bounded as IP[M(D) ∈S] ≤eϵIP[M(D′) ∈S] + δ, ∀S ⊆Range(M). A generalization of differential privacy is known as concentrated differential privacy(CDP). And an alternative form of concentrated differential privacy called zero-concentrated differential privacy(zCDP) was proposed to enable tighter privacy analysis [12]. We will also use the zCDP notion of privacy for our analysis. The relationship between standard DP and zCDP is shown below. Proposition C.2 ((ϵ, δ)-DP and ρ-zCDP). For a randomized algorithm M to satisfy (ϵ, δ)-DP, it is sufficient for it to satisfy ϵ2 4log 1 δ -zCDP. And a randomized algorithm M that satisfies ρ-zCDP, also satisfies (ϵ′, δ)-DP where ϵ′ = ρ + q 4ρlog 1 δ . As opposed to the notion of DP, the zCDP definition provides tighter bounds for the total privacy loss under compositions, allowing better choice of the noise parameters. The privacy loss under the serial composition and parallel composition incurred under the definition of zCDP was proved by [71] and is recalled below. Proposition C.3 (Sequential Composition). Consider two randomized mechanisms, M1 and M2, if M1 is ρ1-zCDP and M2 is ρ2-zCDP, then their sequesntial composition given by (M1(), M2()) is (ρ1 + ρ2)-zCDP. Proposition C.4 (Parallel Composition). Let a mechanism M consists of a sequence of k adaptive mechanisms, (M1, M2, . . . Mk) working on a randomized partition of the D = (D1, D2, . . . Dk), such that each mechanism Mi is ρi-zCDP and Mt : Qt−1 j=1 Oj × Dt →Ot, then M(D) = (M1(D1), M2(D1), . . . Mk(Dk)) is maxi ρi-zCDP. After computing the total privacy loss by an algorithm using the tools described above, we can determine the variance of the noise parameter σ for a set privacy budget. The relationship of the noise variance to privacy has been shown in prior works by [19, 71] and is given below. Definition C.5 (L2 Sensitivity). For any two neighboring datasets, D and D′ that differ in at most one data point, L2 sensitivity of a mechanism M is given by maximum change in the L2 norm of the output of M on these two neighboring datasets ∆2(M) = sup D,D′ \|\|M(D) −M(D′)\|\|2. Proposition C.6 (Gaussian Mechanism). Consider a mechanism M with L2 sensitivity ∆, if on a query q, the output of M is given as M(x) = q(x) + N(0, σ2), then M is ∆2 2σ2 -zCDP. Equipped with the above definitions and results, we now re-state the bound on the privacy loss of our algorithm and provide a proof below. Theorem C.7 (Privacy Budget). The proposed algorithm is (ϵ, δ)-differentially private, if the total privacy budget per global communication round per query is set to ρ = ϵ2 4EKlog 1 δ for E number of global communication rounds and K number of queries to the algorithm per round. Proof. After using Gaussian mechanism on each client and adding noise to each coordinate of Φi(AD), the local mechanism at each client becomes ρ-zCDP for ρ = ∆2 2σ2 . Since each client outputs the logit representation for each input, i.e., the normalized output of the clients, ∆2 ≤2. The sensitivity, denoted as ∆, is defined in Definition C.4 in the paper which defines L2-sensitivity as the maximum change in the L2 norm of the algorithm’s output between two neighboring datasets differing in at most one data point. Let D and D′ be two neighboring datasets that differ in one data point present at the ith row (without loss of generality), and let Φ(D(i, :)) be the nc (number of classes) dimensional output probabilities from the model Φ for the ith row datapoint in D and Φ(D′(i, :)) be the output probabilities for the ith row datapoint in D′. The L2 sensitivity of Φ is ∆(Φ) = \|\|Φ(D) −Φ(D′)\|\|2 Since all other data-points between D and D′ are identical, the L2 sensitivity of Φ becomes ∆(Φ) = \|\|Φ(D(i, :)) −Φ(D′(i, :))\|\|2 Now, Φ(D(i, :)) and Φ(D′(i, :)) are both probability distributions, therefore it can be seen that the squared L2 norm of their difference is bounded by 2, i.e., ∆(Φ)2 ≤2 (the maximum occurs when Φ(D(i, :))k = 1 and Φ(D(i, :))l = 1 for two separate indices k ̸= l). 18 Now, suppose in each global communication round we make K queries to each client, then by sequential composition C.3, we get EKρ, for E number of global communication rounds. By parallel composition C.4, the total privacy loss for all N clients is the maximum of the loss on each client and therefore remains EKρ. Relating it to the (ϵ, δ)-DP from C.2, we get ρ = ϵ2 4EKlog 1 δ for any δ > 0. D Uncertainty Quantification and Calibration Model calibration is a way to determine how well the model’s predicted probability estimates the model’s true likelihood for that prediction. Well-calibrated models are much more important when the model decision is used in critical applications like health, legal etc. because in those cases managing risks and taking calculated actions require a confidence guarantee as well. Visual tools such as reliability diagrams are often used to determine if a model is calibrated or not. In a reliability diagram, model’s accuracy on the samples is plotted against the confidence. A perfectly calibrated model results in an identity relationship. Other numerical metrics that could be used to measure model calibration include Expected Calibration Error (ECE) and Maximum Calibration Error (MCE). ECE measures the expected difference between model confidence and model accuracy whereas MCE measures the maximum deviation between the accuracy and the confidence. The definitions and empirical formulas used for calculating ECE and MCE are as given below. ECE = E ˆ P [IP( ˆY = Y \| ˆP = p) −p], MCE = max p∈[0,1] \|IP( ˆY = Y \| ˆP = p) −p\|. Empirically, ECE = M X i=1 \|Bi\| n \|accuracy(Bi) −confidence(Bi)\|, MCE = max i∈[1,M] \|accuracy(Bi) −confidence(Bi)\|, where Bi is a bin with set of indices whose prediction confidence according to the model falls into the range i−1 M , i M . Figure 3 shows the reliability diagram along with the ECE and MCE scores for our method measured on MNIST and CIFAR-10 dataset in the non-IID data setting. (a) Dataset: CIFAR-10, ECE: 0.070, MCE: 0.134 (b) Dataset: MNIST, ECE: 0.032, MCE: 0.156 Figure 3: Reliability diagrams and scores showing model calibration. Figure (a) is for the results corresponding to the CIFAR-10 dataset and Figure (b) for MNIST dataset. For the next analysis, we consider an approach similar to analysis on uncertainty quantification done in exemplar works in this area [35, 48]. One of the key requirement of reliable estimates and uncertainty quantification is ensuring high confidence in correct predictions and low confidence in incorrect predictions. To assess whether the proposed method meets this criterion, we train our method using standard MNIST train dataset, and test it on the MNIST test dataset as well as on an out-of-distribution dataset composed of NotMNIST10 (featuring images of alphabets instead of digits). Then, we compute the entropy of the predictive distribution (distribution over output class probabilities) for each dataset and visualize this entropy in the provided Figure ??. In the first row corresponding to the in-distribution dataset, both our Bayesian model and the non-Bayesian model exhibit 19 low entropy, as expected. However, for the out-of-distribution test dataset, while the non-Bayesian method demonstrates low entropy, our method yields high entropy. This observation implies that the non-Bayesian method tends to be overly confident in its predictions on unknown classes, which could pose significant risks in practical scenarios, especially in critical applications. Figure 4: Distribution of the entropy of class-probability distributions across different clients demonstrating the confidence of methods in predicting on in-distribution vs out-of-distribution data. E Alignment Dataset (AD) Figure 5: Ablation study comparing the affect of AD size on the performance. The included results are for CIFAR-10 dataset in the small data setting with non-IID partitions and heterogeneous clients. In FedBNN, the alignment dataset (AD) is used to achieve collaboration across clients. Since the only assumption on AD is for it to be of the same domain as the target application, there is no practical constraint on obtaining the AD in real-world settings. In many cases it could be obtained from web, for example images from common datasets in Huggingface, texts from Wikipedia, Reddit etc. Furthermore, the server having its own dataset in addition to the private data on the clients is common in various cross-silo and cross-device applications. For instance, hospitals with access to patient records may combine this data with private patient data collected from individual devices such as wearables or sensors for federated learning, source code generation applications might leverage open-source code along with private code repositories from developers to enhance generative models, self-driving car companies may collect their own data and utilize it alongside private data collected from customers’ vehicles, and many more. The use of AD is not different from how several other methods use an additional dataset for augmentation, and several other methods have used a labelled or an unlabelled auxiliary 20 Table 3: Effect of varying distribution of AD on the clients’ performance for the non-IID seeting with CIFAR-10 dataset and 20 clients where each client has data for the 5 different classes. Architecture Setting Local Training CIFAR10(10) CIFAR10(8) CIFAR10(5) CIFAR10(2) SVHN Homogeneous Architectures 64.3± 0.36 72.7 ± 0.15 69.7 ± 0.28 68.8 ± 0.97 67.2 ± 1.5 70.1 ± 0.18 Heterogeneous Architectures 61.2± 0.17 71.6 ± 0.93 68.4 ± 0.80 68.8 ± 1.4 68.1 ± 1.9 69.3 ± 0.8 dataset to improve the performance and privacy of the FL algorithms [52, 36, 55]. The effect of size of AD on the performance of models is demonstrated in Figure 5 for CIFAR-10 dataset in the small data and non-IID setting. In that figure, we observe that when the size of AD is small the performance of the model is low but as the size of AD increases the performance increases up to a point and becomes constant afterwards. The number of data points in AD that are required to achieve good improvement in the model performance is small and practical. We also vary the distribution of the AD being used and test the final performance of the models and report it in Table 3. We run these experiments on 20 clients for CIFAR-10 dataset where each client had access to only 5 of the 10 classes and each client belonged to the medium data setting. For the first experiment, we use a held-out dataset from the CIFAR-10 data as AD but vary the composition of the dataset by changing the distribution of the classes present in the AD, for example, CIFAR10(10) is composed of all 10 classes present in the CIFAR-10 dataset but CIFAR10(2) is composed of only 2 out of the 10 classes present in the AD and likewise. We also test the performance of our method when a significantly different dataset SVHN consisting of the colored house number images is used. Table 3 suggests that the performance of the method even with different datasets as AD always improves and that the gain between local training and the proposed procedure is better highlighted in the heterogeneous architecture settings, since there local client capacities and model architectures differ significantly and clients are able to utilize the peer knowledge to learn better models locally. We observed that even for different and dissimilar data distributions in AD, it is possible to obtain a value for the parameter γ such that the final performance of the local client model with collaboration is better than the model independently trained locally on the client. The best results for CIFAR-10 classification are seen when AD is composed of a held-out set from all 10 classes of CIFAR-10 denoted as CIFAR10(10) which is as expected. Then, as the composition of AD is changed from 10 classes to random 8, 5 and 2 classes of CIFAR-10 (denoted as CIFAR10(8), CIFAR10(5) and CIFAR10(2) respectively) the performance keeps on decreasing. We see that the performance of the same task with SVHN as AD is only strictly better than CIFAR10(2) as AD. We believe that the SVHN dataset as AD works better than the CIFAR10(2) because it provides more variability in the data distribution. A similar observation was also recorded in FedAUX [55] which also uses additional unlabelled data for knowledge distillation, which noted that the out-of-domain unlabelled data for distillation can perform even better. Moreover, the parameter γ controls the amount of global knowledge to be incorporated on each client and with appropriately set γ, the AD also helps achieve regularization for the local client model such that the local models do not overfit to the relatively smaller local datasets and generalize better. F Communication and Computation Efficiency Communication Cost In FedBNN, each global communication round requires that the server sends the alignment dataset to all the clients and the clients upload the outputs of their respective models on the common dataset AD. Since AD is a publicly available dataset, AD could be transmitted to the clients by specifying the source and the indices, and does not really needs to be communicated across the channel. The client output on AD, on the other hand, depends on the number of instances in AD, let’s call it K, therefore, the total communication cost in each round of our method is O(K). As shown in Figure 5, having K = 2000 gives a good performance. The communication cost between the clients and the server, thus, is also invariant of the number of model parameters which tends to run in millions. This allows our method to be much more communication efficient as compared to the conventional FL algorithms and other Bayesian FL methods that transmit model parameters or parameter distributions in each communication round, making it practically more useful. Computation Cost Similarly, the computation cost of a FL procedure involves the costs incurred in local training at the individual clients and the cost of aggregation at the server, both of which are discussed below. • Server-side computation cost The server side computation cost arises from the need to aggregate knowledge obtained from individual clients. In the state-of-the-art bayesian FL algorithms, the server aggregates posterior distributions for each weight parameter in the neural network obtained from various clients. The number of such weight parameters typically run in millions. In our method we do not aggregate the parameter distributions but achieve collaboration by aggregating the client outputs on the AD (with size 2000), thus the server side computation cost in our method is many orders 21 Table 4: Performance comparison as a function of the privacy guarantee. Privacy (ϵ) per round Test Accuracy ≈1 75.5 % ≈0.1 71.3 % ≈0.01 68.6 % ≈0.001 62.2 % ≈0.0001 59.6 % Table 5: Test accuracy comparison with more number of clients (500) in the setting. Method Test Accuracy pFedGP 53.2 ± 0.4 pFedBayes 52.9 ± 0.8 Ours(Homo) 56.1 ± 0.3 Ours(Hetero) 54.7 ± 1.0 of magnitude lower than the conventional methods and does not depend on the number of model parameters. This makes our method much more efficient and scalable than existing federated bayesian solutions. • Client-side computation cost The client-side computation cost is mostly determined by the cost of training a Bayesian Neural Network at the client side, which in turn depends on the type of inference procedure used for obtaining the posterior distribution over weights for each parameter of the neural network. In the proposed work, the method used for inference is Bayes by Backprop which uses the gradient computations similar to backpropagation(BP) algorithm to obtain the posterior distributions where the posterior distributions are characterized by the mean and standard deviation. A re-parameterization trick is used to compute the mean and std of the distributions from the backpropagated gradients. Thus the cost for obtaining the posterior distributions is similar to the cost of backpropagation. Moreover, since the method only uses gradient updates, the optimizations used for SGD like asynchronous SGD etc. could be readily used for obtaining the posteriors. An unrelated but similar algorithm in [28] does probabilistic backpropagation to train BNNs and shows that the average run time of probabilistic BP is not higher than that of BP. To summarize, the communication cost and the server-side computation cost of the proposed method is orders of magnitude lower than that of the other Bayesian baseline methods. On the other hand, the client-side computation cost is determined by the inference procedure used to obtain the posterior distributions and for which Bayes by Backprop provides an efficient mechanism. Several works in the recent past have discussed the use of related Bayesian inference based methods for training uncertainty-aware transformers [76, 62, 42] proving that Bayesian methods are not limited to use in simpler models. And therefore, our framework can also be extended to apply in settings where much larger neural networks are required. G Additional Experiments Privacy vs Performance Since the amount of noise required to be added to the client’s outputs via the Gaussian Mechanism is directly proportional to the guaranteed privacy, we test the affect of the privacy guarantee on the performance of the proposed framework by comparing the performance of the method with varying ϵ and δ = 10−4. The results are reported in Table 4. We observe that, as expected, when we reduce the amount of privacy loss in each iteration by adding more noise to the clients’ outputs going to the server, the performance of the method drops. However the drop in performance in all the cases is not drastic as the clients can tune the level of personalization or global knowledge required by appropriately setting the parameter γ in Equation 3. More clients To test the performance of the proposed method when a large number of clients are involved in the setup, we did additional experiments with 500 clients and non-IID setting with 5 classes per client in the medium data setting on the CIFAR-10 dataset where in each communication round only 10% of the clients are selected for participation and γ = 0.7. The obtained results at the end of 200th communication round are reported in Table 5. We observe that the homogeneous version of our method is better than the baselines by a significant margin and the heterogeneous version is slightly better than the baselines. 22 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We support all the claims made in the abstract and introduction by thorough experimental evaluation in the paper. Please refer Section 5 for details. 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4,462	Towards Open-Vocabulary Semantic Segmentation Without Semantic Labels Heeseong Shin1 Chaehyun Kim1 Sunghwan Hong2 Seokju Cho1 Anurag Arnab†,3 Paul Hongsuck Seo†,2 Seungryong Kim†,1 1KAIST 2Korea University 3Google Research {hsshin98, kchyun, seokju.cho, seungryong.kim}@kaist.ac.kr1 {sung_hwan, phseo}@korea.ac.kr2 aarnab@google.com3 Abstract Large-scale vision-language models like CLIP have demonstrated impressive openvocabulary capabilities for image-level tasks, excelling in recognizing what objects are present. However, they struggle with pixel-level recognition tasks like semantic segmentation, which additionally require understanding where the objects are located. In this work, we propose a novel method, PixelCLIP, to adapt the CLIP image encoder for pixel-level understanding by guiding the model on where, which is achieved using unlabeled images and masks generated from vision foundation models such as SAM and DINO. To address the challenges of leveraging masks without semantic labels, we devise an online clustering algorithm using learnable class names to acquire general semantic concepts. PixelCLIP shows significant performance improvements over CLIP and competitive results compared to captionsupervised methods in open-vocabulary semantic segmentation. Project page is available at https://cvlab-kaist.github.io/PixelCLIP (b) image-level semantic labels “photo of a {corgi} running in a {grass} field” image (a) pixel-level semantic labels model tree dog grass image model image model (c) without semantic labels (ours) mask generation VFM (e.g. SAM, DINO) global semantic clustering [CLASS1] [CLASS2] [CLASS3] learnable classes Figure 1: Illustration of different approaches for open-vocabulary semantic segmentation. In contrast to existing methods utilizing (a) pixel-level semantic labels [1, 2, 3, 4, 5, 6] or (b) image-level semantic labels [7, 8, 9, 10, 11, 12], we leverage unlabeled masks as supervision, which can be freely generated from vision foundation models such as SAM [13] and DINO [14]. 1 Introduction Semantic segmentation is a fundamental task in computer vision where the goal is to identify class labels for each pixel within the given image. However, segmentation datasets often require extensive human effort to obtain densely-annotated semantic labels, limiting their scalability. In this regard, recent advances in large-scale pre-trained vision-language models, e.g. CLIP [15] and ALIGN [16], †Corresponding authors 38th Conference on Neural Information Processing Systems (NeurIPS 2024). clustered SAM masks DINO masks clustered DINO masks SAM masks Figure 2: Visualization of masks from vision foundation models. We visualize the masks generated by SAM [13] and by clustering image features from DINO [14]. Although such models can freely generate fine-grained masks, the resulting masks can be too small or incomplete to have semantic meaning. To address this over-segmentation issue, we employ online clustering [18] of the masks into semantically meaningful groups defined globally for given images. have facilitated open-vocabulary semantic segmentation [1, 3, 2, 17, 4, 6], which aims to generalize semantic segmentation into unbounded range of classes. Despite showing remarkable generalization capabilities, they still require pixel-level semantic labels for leveraging the image-level pre-trained vision-language models for semantic segmentation. Recently, several studies [11, 12, 7, 8] have pioneered open-vocabulary semantic segmentation without densely-annotated semantic labels. These studies often utilize image-level semantic labels, such as image captions, to enhance the pre-trained vision-language models like CLIP for semantic segmentation. However, image captions typically provide information about what is in the image, but without where it is. Since CLIP is already effective in recognizing what the objects are, this causes models to only implicitly learn object locations, leading to sub-optimal performance or requiring millions of image-caption pairs to compensate for this weak supervision [8, 7]. Instead, we focus on informing CLIP about where objects are located to address the missing information. In this study, we propose a novel approach to achieve open-vocabulary semantic segmentation without leveraging semantic labels, but through guiding the pre-trained vision-language models, such as CLIP, on where to look. We leverage recent vision foundation models (VFMs), such as DINO [14] and SAM [13], to partition images into fine-grained regions to indicate where to look. Consequently, we explore methods to effectively leverage these masks for fine-tuning the image encoder of CLIP. In contrast to existing works that leverage semantic labels [6, 19, 7], we do not have any captions or class names that can be fed to the text encoder of CLIP. To leverage its knowledge, we devise a method that employs prompt learning [20, 21] on the text encoder of CLIP to construct learnable classes. Setting the learnable classes as a centroid, we propose applying the online clustering algorithm [18, 22] along the given masks to gather them into semantically meaningful groups, as shown in Fig. 2. We keep these learnable classes global across the entire images, which guides the learnable classes to contain the general semantic concepts. Despite the absence of semantic labels, our method is able to jointly leverage the image encoder and text encoder of CLIP during training, successfully achieving dense open-vocabulary recognition. Our framework, called PixelCLIP, achieves significant improvements to CLIP, on average of +16.2 mIoU in open-vocabulary semantic segmentation. Moreover, despite not using any semantic labels, PixelCLIP shows competitive performance in comparison to image-level supervised methods using captions [7, 9, 10], demonstrating the effectiveness of unlabeled masks for. We further show the effectiveness of PixelCLIP for classifying masks from various open-vocabulary segmentation models, which can be simply done by replacing the CLIP within existing methods. We also provide extensive ablation studies to validate our choices, with a detailed analysis of our method. We summarize our contribution as follows: • We propose a novel formulation of learning from images without semantic label for openvocabulary semantic segmentation by leveraging masks generated from DINO and SAM to fine-tune vision-language models. • We propose to globally cluster semantically similar masks by employing an online clustering algorithm, while learning class prompts for representing semantic clusters. • We demonstrate significant gains in open-vocabulary semantic segmentation, even surpassing methods leveraging image-level semantic labels, and provide thorough ablation studies with analysis to validate our framework. 2 2 Related Work 2.1 Open-vocabulary semantic segmentation Open-vocabulary semantic segmentation [2, 23] aims to label each pixel within an image into an unbounded range of classes. In this regard, recent works [1, 17, 2, 6, 24] aim to generalize to classes unseen during training through leveraging pre-trained vision-language models, such as CLIP [15]. Despite their remarkable performance, they leverage per-pixel semantic labels during their training, which requires expensive cost to annotate. Instead, we focus on the weakly-supervised setup, where the goal is to zero-shot transfer to segmentation task without densely-annotated class labels [11, 12, 25, 5, 26, 7, 27], utilizing image-level labels as supervision or even no labels at all. In this regard, recent studies [11, 12, 25, 5] leverage image caption as supervision. GroupViT [11] and ViL-Seg [12] are pioneering works for identifying groups or clusters emerging from captions. Along with the advance of vision-language models, SegCLIP [9] and TCL [7] leverage pre-trained CLIP and learn additional decoder modules to learn dense vision-language alignment. PACL [8] learns additional embedding layers to enhance the patch-level alignment in vision-language models and SAM-CLIP [10] attempts to merge SAM [13] and CLIP [15] into a unified model by additionally leveraging unlabeled mask data from SAM. Apart from these approaches, we avoid employing any semantic labels [26, 28], but leverage vision foundation models to obtain masks as a source for supervision to fine-tune the CLIP image encoder for achieving open-vocabulary semantic segmentation. 2.2 Fine-tuning vision-language models for dense prediction Recent large-scale pre-trained vision-language models have shown its effectiveness for jointly understanding images and language [29, 15, 16]. Notably, CLIP [15], trained with web-scale image-caption pairs, has been widely popularized for transferring its open-vocabulary recognition capabilities to various downstream tasks [26, 30, 31, 32]. However, despite its success in image-level tasks like image classification, CLIP tends to struggle in dense prediction tasks [17, 26, 6], such as object detection and semantic segmentation. This originates from CLIP being trained from image-level supervision being captions, hence exhibits bias towards the global image rather than fine-grained regions within the image [17]. While non-learnable approaches, such as MaskCLIP [26] show improvements by slightly modifying the architecture, CLIP still shows limited capabilities in dense predictions in comparison to its global understanding. To address this, OWL-ViT [33] directly fine-tunes pre-trained vision and text encoders to downstream open-vocabulary detection task, and CAT-Seg [6] introduces a cost aggregation scheme for finetuning the encoders of CLIP for semantic segmentation. Alternatively, ZegCLIP [19] and Xu et al. [3] implement prompt tuning [21, 20] for tuning the image and text encoders of CLIP. Instead of fine-tuning the full model, they learn prompt tokens that serve as a global prefix for the encoders of CLIP. While such methods show remarkable results from fine-tuning the encoders of CLIP for dense downstream tasks, they require densely annotated detection and segmentation data for training. 2.3 Vision foundation models With the advent of large-scale learning enabled by scalable vision backbone architectures [34, 35] and vast amounts of data, diverse vision foundation models are emerging in the field of computer vision. In this regard, self-supervised methods [36, 37, 38, 39] have demonstrated the effectiveness of its rich visual representations for various downstream tasks. Especially, DINO [14] exerted strengths in fine-grained semantic recognition [40, 41], making it highly effective for object detection and image segmentation. Moreover, DINO features have also been demonstrated for yielding fine-grained masks within the image through applying the k-means clustering with its features [42, 43, 44]. On the other hand, the segment anything model (SAM) [13] has demonstrated its capability for generating fine-grained, high-quality segmentation masks for any object in an image. Through its selfannotation pipeline, SAM has collected an unprecedented amount of mask annotation for achieving its capabilities. While we can freely leverage SAM to obtain detailed masks in any given image, we mainly utilize the pre-computed masks within the collected dataset, SA-1B. Both DINO and SAM, however, yield unlabeled masks without semantic labels as both models are also trained without semantic labels, presenting a challenge for leveraging their masks for achieving dense vision-language recognition. 3 image encoder momentum update text encoder image feature image feature mask decoder A photo of [C1] … A photo of [Ci] … A photo of [Ck] … … … learnable class prompts similarity map class features online cluster assignment masks image assigned class feature binary mask loss image encoder mask feature … … C M Sinkhorn-Knopp mask prediction image encoder text encoder image feature open-vocabulary prediction image text features A photo of “animal” in the scene A photo of “tree” in the scene A photo of “grass” in the scene C training with image-mask pairs open-vocabulary inference ℇ! "ℇ! 𝑓! 𝑓" 𝑓# ℇ$ 𝒟 ℇ! ℇ$ ̅𝑓! 𝑀!# &𝑀 𝑀 𝐼 frozen parameters M masked pooling C cosine similarity learnable parameters Figure 3: Illustration of our overall framework. We provide illustration of PixelCLIP, utilizing unlabeled images and masks for fine-tuning the image encoder of CLIP, enabling open-vocabulary semantic segmentation. We note that the momentum image encoder and the mask decoder are only leveraged during training, and inference is only done with image and text encoders of CLIP. 3 Methodology In this section, we first establish our problem formulation of learning dense vision-language alignment from images paired with masks, generated from vision foundation models. Next, we discuss the challenges of leveraging masks as supervision for fine-tuning the image encoder of CLIP and finally, present our methodology of semantic clustering of masks to address the challenges. 3.1 Preliminaries Given an input image I ∈RH×W ×3, open-vocabulary semantic segmentation [6, 7] aims to label each pixel within an image with classes given in free-form text. As a training signal, semantic labels offer a set of S textual descriptions for a semantic class T = {Ti}S i=1 related to I. This can be directly utilized with the CLIP text encoder EL(·) to obtain text features fT = EL(T) ∈RS×d, where d is the hidden dimension. Dense image features fI = EV (I) ∈Rh×w×d, where h × w is the output feature resolution, are then extracted. We finally obtain dense image-text similarity map MIT ∈Rh×w×S: M _{I T} (x , y, n )= \ fr ac {f_I(x , y) \ cdot f_T(n)}{\\|f_I(x, y)\\|\\|f_T(n)\\|}. \label {mask} (1) This can be interpreted as soft binary masks predicted from image and text features of CLIP, and be supervised with binary mask loss Lmask in a pixel-level manner to fine-tune CLIP [6]. 3.2 Integrating masks into CLIP features In this work, we do not have any access to T, but are only given unlabeled masks M ∈RH×W ×N, where N denotes the number of masks for the given image I. Hence, we devise methods to predict masks by incorporating M into CLIP features. We aim to fine-tune the CLIP image encoder EV (·) through leveraging unlabeled masks M as supervision. Since M is generated from vision foundation models, e.g. DINO or SAM, this presents us with the challenge of not having any semantic labels. In order to integrate masks into CLIP, a straightforward approach would be employing the masks M with the CLIP image feature map fI to obtain per-mask CLIP features. While there could be various methods to extract regional CLIP features [26, 45, 5], we apply mask pooling over fI to obtain mask 4 pooled features fM = MaskPool(fI, M) ∈RN×d. Consequently, we can leverage fM to obtain image-mask similarity map denoted MIM ∈Rh×w×N: M_{I M} (x , y, n )= \ frac {f_I(x , y) \cdot f_M(n)}{\\|f_I(x, y)\\|\\|f_M(n)\\|}. \label {mask} (2) This allows us to supervise the model with a binary mask loss Lmask for fine-tuning CLIP with given image I and unlabeled masks M. In practice, since MIM has the same resolution as the feature map from the CLIP image encoder fI, we employ a light-weight decoder D to mitigate the resolution gap between MIM and M, as shown in Fig. 3. This can be written as D : Rh×w →Rh′×w′, where h′ × w′ is resolution for the upsampled mask. Therefore, the output of the model can be updated as ˜ M = D(M). 3.3 Semantic clustering of masks Upon using mask pooled CLIP image features fM to predict MIM, however, we find the masks generated from DINO and SAM to often over-segment the image, resulting in too small or incomplete masks as seen in Fig. 2. This would require CLIP to forcefully discriminate regions that are semantically similar, impeding the training process. In this regard, we propose to group semantically similar masks into clusters and predict based on the clusters rather than individual masks. Moreover, we aim to define this cluster globally, which is shared across the entire training process rather than for each image or iteration. This would be analogous to constructing pixel-level semantic labels, where a fixed set of classes defined over the dataset is equivalent to each cluster. However, the difference is that there is no pre-defined set of classes that we can define the clusters with. While we could heuristically pre-define such classes, we describe our learnable method for globally clustering masks into semantically meaningful groups. Online clustering via learnable class prompts. To globally cluster masks into semantic categories, we propose representing these clusters using CLIP text features as centroids for clustering mask features. Given that the CLIP text encoder is trained with a broad understanding of natural language semantics, we expect these clusters to capture meaningful semantics by leveraging its comprehensive pre-trained knowledge. In this regard, we take a learnable approach, where each cluster is defined by class-specific learnable prompts fed into the CLIP text encoder. Unlike existing prompt learning methods, which typically focus on learning a task-specific prefix [20, 21, 3], we aim to learn prompt tokens that represent each class. For instance, in the sentence “A photo of an object”, traditional prompting methods would learn the tokens for the “A photo of a” prefix, whereas our method focuses on learning the token for the “object.” Specifically, given the number of clusters k, we can define prompt tokens as C ∈Rk×l×de, where l is the token length of the prompt and de is the dimension of the token embeddings. From this, we can utilize the CLIP text encoder EL(·) to obtain a set of class features fC = EL(P ∗, C) ∈Rk×d in the form of CLIP text features, where P ∗is a fixed template for the CLIP text encoder, such as “A photo of a {} in the scene." While we could assign each mask fM with fC in a winner-takes-all manner, we desire the classes to encode general semantics across all images. Therefore, we assume that we can equally divide m masks within a minibatch [18, 14], into k clusters given a sufficient amount of masks. Consequently, we aim to find an assignment Q ∈Rk×m + based on the image-text similarity between the mask pooled features fM and the class text features, which can be defined as: \ max _{Q \ i n \m a thcal {Q}} \ math r m {Tr } (Q^\ top { F _{M}^\top }{f_C}) + \varepsilon H(Q), \quad \text {s.t.} \quad Q \in \mathbb {R}_{+}^{k \times m}, \quad Q^\top \mathbbm {1}_k = \frac {1}{m}\mathbbm {1}_m, \quad Q \mathbbm {1}_m = \frac {1}{k} \mathbbm {1}_k, \label {eq:clsuter} (3) where FM is the set of all m mask features fM within the minibatch, and 1k denotes the k-dimensional vector of ones. H is the entropy function, H(Q) = −P ij QijlogQij with ε as a hyperparameter. The solution Q from Eq. 3 is an assignment matrix defining which of the k clusters each m mask should belong to, hence ε determines the smoothness of this mapping Q by scaling the entropy regularization from H. The equipartition constraint, Q⊤1k = 1 m1m, Q1m = 1 k1k encourages the class features fC to be selected at least m/k times on average, allowing to learn general concepts represented by the masks within the dataset. In practice, with the soft assignment relaxation [46], Q 5 can be solved as follows: Q = \te xt {d i ag} ( u ) \exp \left ( \frac {{F_{M}^\top }{f_C}}{\varepsilon } \right ) \text {diag}(v), (4) where u ∈Rk, v ∈Rm denote renormalization vectors, which can be efficiently computed by Sinkhorn-Knopp algorithm [46]. Finally, we can re-write the prediction of our model to be a cosine-similarity map between fI and fC: M_{I C} (x , y, i )= \ frac {f_I(x , y) \cdot f_C(i)}{\\|f_I(x, y)\\|\\|f_C(i)\\|}, (5) thereby predicting masks for fC(i) being the i-th class feature from fC, which we have obtained from clustering mask pooled features fM. Accordingly, ground truth masks M are also clustered according to Q by converting it into hard assignment with the argmax operator [47, 22]. This can be written as ¯ M ∈Rk×H×W where ¯ Mi is the union of masks assigned into the cluster represented by i-th learned class fC(i). Momentum encoder for integrating mask features. Since we jointly optimize the CLIP image encoder EV (·) as well as the learnable class feature fC, we may experience instability during our training process, or forgetting of the pre-trained knowledge [48]. To stabilize the training, we keep a momentum encoder [39, 38] for obtaining fM as seen in Fig. 3. Therefore, we update fM as fM = MaskPool( ¯EV (I), M), where ¯EV is the momentum encoder of the CLIP image encoder, updated with momentum γ. This can be denoted as θ′ ¯V ←γθ′ ¯V + (1 −γ)θV , where θ ¯V ′, θV are model parameters of ¯EV and EV , respectively. 4 Experiments 4.1 Implementation details For training, we employ per-pixel binary cross-entropy loss as Lmask to jointly train all of the components [6]. For all our experiments, we use a single text prompt “A photo of {} in the scene” for P ∗, including for our learnable class prompts while training and for inference, we apply prompt ensemble strategy [30] with 7 additional prompts originally curated from CLIP [15]. We train our model on SA-1B [13] dataset, where we randomly sample 5% of the images. We train for 10000 iterations with a batch size of 48 for all experiments. For experiments using masks from DINO, we obtain masks with k-means clustering where we set k = 16. For experiments using masks from SAM, we use the unlabeled mask annotation in the SA-1B dataset. Without specification, we report results on ConvNeXt-B [49] backbone with mask annotation from SAM, which takes approximately 6 hours to train with 4 NVIDIA A6000 GPUs. We provide more details in the supplementary materials. 4.2 Experimental setting Following Cha et al. [7], we evaluate our model on zero-shot transfer to semantic segmentation on the validation sets of COCO-Stuff [50], ADE-20K [51], PASCAL-Context [52], PASCAL VOC [53], and CityScapes [54]. For CLIP [15], we apply MaskCLIP [26] for ViT backbone for extracting image features, and remove the global pooling layer for OpenCLIP [55] with ConvNeXt [49] backbone. We note that we do not apply any post-processing to the predictions and for the compared methods. For the evaluation metric, we employ the mean Intersection over Union (mIoU). 4.3 Results Open-vocabulary semantic segmentation. We provide results for quantitative comparisons in Tab. 1. We first compare with CLIP, and demonstrate remarkable gains in all benchmarks, bringing in an average of +16.2 mIoU improvement. Since we do not have comparable baselines without leveraging semantic labels, we further provide a comparison with image-level supervised methods [7, 9, 10]. Surprisingly, PixelCLIP surpasses TCL [7] and SegCLIP [9] in all benchmarks while using only a fraction of the images without semantic labels. Furthermore, we show competitive performance compared to SAM-CLIP, which uses not only 40 million image-level semantic labels, but also leverages the SA-1B dataset on a similar scale to our framework. 6 Table 1: Quantitative comparison on open-vocabulary semantic segmentation. We compare in open-vocabulary semantic segmentation with vision-language models, as well as image-level supervised methods. : Images were seen during training. †: Masks from SA-1B [13] were used. Method Training Dataset Backbone Additional VFM Evaluation Dataset Labels COCO-St. ADE-150 Context CityScapes VOC GroupViT [11] CC12M [56], YFCC15M [57] ViT-S/16 15.3 9.2 23.4 11.1 79.7 CLIPpy [25] HQITP-134M [25] ViT-B/16 13.5 52.2 OVSegmentor [58] CC4M [58] ViT-B/16 DINO 20.4 53.8 CLIP [15] WIT-400M [15] ViT-B/16 16.5 13.2 25.6 14.9 73.9 OpenCLIP [55] LAION-2B [59] ConvNeXt-B 12.8 13.1 16.5 16.2 34.8 Training with additional image-level semantic labels SegCLIP [9] COCO [60], CC12M [56] ViT-B/16 Captions CLIP 26.5 24.7 52.6 TCL [7] CC3M, CC12M [56] ViT-B/16 Captions CLIP 19.6 14.9 30.3 23.1 77.5 SAM-CLIP [10] Merged-41M [10] ViT-B/16 Captions CLIP, SAM 17.1 29.2 60.6 Training without additional semantic labels ZeroSeg [28] ImageNet-1K [61] ViT-B/16 CLIP 20.2 20.4 40.8 ViT-B/16 CLIP, DINO 22.2 17.4 34.3 22.9 83.8 ViT-B/16 CLIP, SAM† 23.6 18.7 37.9 27.2 85.9 ConvNeXt-B CLIP, DINO 20.2 19.4 32.7 30.0 62.9 PixelCLIP (Ours) 5% SA-1B [13] (0.5M) ConvNeXt-B CLIP, SAM† 21.4 20.3 35.4 34.8 67.2 Table 2: Quantitative comparison on zero-shot mask classification. We compare the results for mask classification using ground truth masks and generated masks from ZegFormer [1] and FC-CLIP [24]. To evaluate zero-shot mask classification from CLIP, we report the results from the zero-shot branch for both methods. VLM Method Backbone Evaluation Dataset COCO-St. ADE-150 Context CityScapes VOC OpenCLIP [55] Zegformer [1] ConvNeXt-B 15.3 19.1 24.7 26.5 51.8 PixelCLIP (Ours) Zegformer [1] ConvNeXt-B 23.9 (+8.6) 21.5 (+2.4) 38.5 (+13.8) 34.2 (+7.7) 71.5 (+19.7) OpenCLIP [55] FC-CLIP [24] ConvNeXt-L 37.3 27.4 42.8 35.8 91.4 PixelCLIP (Ours) FC-CLIP [24] ConvNeXt-L 46.8 (+9.5) 30.1 (+2.7) 52.2 (+9.4) 48.1 (+12.3) 90.7 (-0.7) OpenCLIP [55] Ground Truth ConvNeXt-B 23.8 30.2 31.4 32.8 68.3 PixelCLIP (Ours) Ground Truth ConvNeXt-B 34.2 (+10.4) 34.6 (+4.4) 51.2 (+18.4) 41.4 (+8.6) 85.4 (+17.1) Zero-shot mask classification. We provide results for evaluating mask classification in Tab. 2. We consider ZegFormer [1] and FC-CLIP [24] as baselines since they first predict masks, then employ CLIP as a zero-shot mask classifier within their framework, and also provide results with ground-truth masks to simulate having oracle mask predictions. For all methods, we apply masked pooling to CLIP image feature map to classify masks. For ZegFormer [1] and FC-CLIP [24], reported results are only from the zero-shot prediction branch to solely ablate our gains. We highlight that PixelCLIP can be readily applied to existing frameworks that leverage CLIP as a zero-shot mask classifier, and bring instantaneous improvements by simply replacing the model and weights of CLIP. Qualitative results. We provide qualitative results for open-vocabulary semantic segmentation in Fig. 4 compared with results from CLIP, highlighting the dense open-vocabulary recognition capabilities of our framework. We further provide qualitative results in the supplementary materials. 4.4 Ablation studies In Tab. 3, we show ablation studies on open-vocabulary semantic segmentation to validate our design choices. We report results without prompt ensembling for ablations, and also report results from OpenCLIP [55] as a baseline. Component analysis. In Tab. 3 (a), we provide results for ablating our key components. Notably, we observe that without global semantic clustering of masks, the framework collapses and loses the pre-trained knowledge of CLIP. This validates the challenge presented by leveraging unlabeled masks and demonstrates the crucial role of our proposed clustering approach. Moreover, we observe constant improvements over all datasets with our learnable class prompt, proving our approach of leveraging the text encoder of CLIP to define the clusters in the form of prompt learning. We also observe constant gains with the momentum encoder for extracting mask pooled features fM. Number of clusters. In Tab. 3 (b), we compare the results of the variants of the proposed method by varying the number of clusters k. We find that scaling k does not necessarily guarantee performance boosting, but it generally improves until k is set to 64 and tends to degrade as k grows. Considering that with an extremely large number for k, we can assign each of the masks to individual clusters(e.g. 1 billion for SA-1B.) This scenario would virtually be identical to not having semantic clustering as 7 (a) Ours (b) CLIP Figure 4: Comparison between PixelCLIP and CLIP. We provide qualitative comparison on ADE20K [51] dataset with PixelCLIP and CLIP. We demonstrate the dense visual recognition capabilities achieved from fine-tuning CLIP, whereas CLIP shows results with significant noise. Table 3: Ablation studies. We show results on open-vocabulary semantic segmentation for validating our design choices. We also report results from OpenCLIP [55] as baseline in the results. Component Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline 12.8 13.1 16.5 16.2 34.8 Ours 21.1 20.2 34.2 33.2 66.0 w/o Semantic Clustering 0.8 2.1 4.2 4.4 6.0 w/o CLIP Text Encoder 17.9 18.5 29.9 28.9 53.5 w/o Class Prompt 18.2 18.8 30.1 28.1 54.4 w/o Momentum 19.4 18.5 28.8 27.2 58.2 (a) Component analysis. We validate the core components of our framework by ablating each components. Notably, global clustering of masks shows its importance for facilitating the framework. k Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline 12.8 13.1 16.5 16.2 34.8 32 19.8 19.4 33.0 31.3 60.5 64 21.1 20.2 34.2 33.2 66.0 128 21.0 20.3 33.5 30.1 64.1 256 21.3 20.4 33.6 30.0 64.1 512 21.2 20.2 32.7 29.8 62.7 (b) Number of clusters. For varying k, we find that scaling k larger than 64 does not show much improvements, while k = 32 also show competitive results. l Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline 12.8 13.1 16.5 16.2 34.8 1 20.2 19.7 32.7 30.8 64.5 4 21.1 20.2 34.2 33.2 66.0 10 20.4 19.6 33.2 30.2 63.3 20 19.9 19.7 32.6 33.8 62.8 (c) Length of learnable prompt token. For varying l, we find that l = 4 shows best overall performance. Text. Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline 12.8 13.1 16.5 16.2 34.8 Ours 21.1 20.2 34.2 33.2 66.0 COCO 19.5 17.7 30.0 24.8 63.3 (d) Effects of utilizing learnable classes. We compare our method of learnable class prompts to having fixed set of classes from COCO-Stuff [50]. seen in Table. 3 (a), and progressively growing k would slowly converge to this scenario. We further provide analysis in 4.5, studying the different aspects from varying k. Length to represent learnable class prompts. Tab. 3 (c) compares the effects of varying the length of the learnable class prompt, l. We find that l = 1 shows lower scores in comparison to other lengths. We can interpret this as only describing a class with a single word, whereas having multiple words would better describe the depicted class. However, for l = 4 and larger, we find that increasing l does not result in a gain of performance, hence, we adopt l = 4 as default. Effects of learnable prompt token. Finally, we compare PixelCLIP to having a pre-defined set of classes instead of using learnable prompt tokens. Specifically, we use 171 classes from COCOStuff [50], and do not apply online clustering for assignment when utilizing classes from COCO-Stuff, as it already yields text features with semantic meanings. We find apparent improvements over all the datasets as shown in Tab. 3 (d). We speculate that since the classes defined in COCO-Stuff are heuristically chosen, it is hard to ideally encompass various semantics and concepts that may appear in images, hence restricting the perception of the model to the finite set of classes. 8 (a) 𝑘= 64 (b) 𝑘= 128 (c) 𝑘= 64 (d) 𝑘= 128 learned init COCO-St. Figure 5: Visualization of learned class prompts. We visualize the text features from our learned class prompts, as well as text features from classnames of COCO-Stuff with t-SNE visualization in (a-b). We also visualize images inferenced with the learned class prompts in (c-d). 4.5 Analysis Learnable class prompt. We further analyze the learned class prompt in Fig. 5 (a-b) with t-SNE visualization on the text features encoded from the learned class prompts, as well as text features obtained from class names of COCO-Stuff. Since we initialize the class prompt tokens as random tokens, we observe that they are in a skewed distribution in the initial state. However, the learned prompts show that they are well-dispersed among the text features from COCO-Stuff, indicating that the class prompts have well-learned diverse semantic concepts within the text features. We observe well-distributed features both for k = 64 and k = 128. Since the learned prompts should act as implicit class names, we visualize the results from inference with learned class prompts in Fig. 5 (c-d). Although both k = 64 and k = 128 show similar performance when evaluated, we observe that the prompts have learned more fine-grained semantics for k = 128. We generally observe human parts to be well distinguished; this could come from the SA-1B dataset, as there are numerous images with fine-grained masks representing human parts as annotations. (a) ! = 64 (b) ! = 128 Figure 6: Visualization of interpreting learned text prompt. We provide visualization on results for predicting with learned class prompts, then mapping the results to classes in the dataset with the highest similarity to the prompt. Interpreting learned classes. Considering the learned class prompts represent semantic concepts, we further study the learned embeddings by mapping each class embeddings to class names in COCO-Stuff with the highest cosine-similarity score. Fig. 6 shows results when we first inference the image features with learned class prompts, then map the results with the closest COCO classes. We can observe that with k = 128, as the prompt learns more diverse semantics, we observe more accurately mapped classes. However, we still see predictions with large disparity to the actual ground truth. We leave a more in-depth analysis of the learned classes for future investigation. 5 Conclusion In this paper, we introduced PixelCLIP, a framework for leveraging unlabeled images and masks for fine-tuning the pre-trained vision-language models for open-vocabulary semantic segmentation. To address the unique challenges posed by incorporating unlabeled masks generated by vision foundation models into our framework, we propose global semantic clustering of the masks, with learnable class prompts to represent each cluster. We demonstrated PixelCLIP to show remarkable improvements to CLIP and its applicability to existing methods, providing instantaneous improvements, as well as surpassing methods that leverage image-level semantic labels such as image captions. 9 Acknowledgement. This research was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (RS-2019II190075, RS-2024-00509279, RS-2020-II201819, RS-2024-00398115, Research on the reliability and coherence of outcomes produced by Generative AI) and the Culture, Sports, and Tourism R&D Program through the Korea Creative Content Agency grant funded by the Ministry of Culture, Sports and Tourism (RS-2024-00348469, RS-2023-00266509), and National Research Foundation of Korea (RS-2024-00346597). References [1] Jian Ding, Nan Xue, Gui-Song Xia, and Dengxin Dai. Decoupling zero-shot semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11583–11592, 2022. [2] Golnaz Ghiasi, Xiuye Gu, Yin Cui, and Tsung-Yi Lin. Scaling open-vocabulary image segmentation with image-level labels. 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AdamW [65] optimizer is used with a learning rate of 2 · 10−4 for the decoder, 2 · 10−5 for the prompt tokens and 2 · 10−6 for CLIP, with weight decay set to 10−4. Prompt tokens are initialized as random word tokens with l = 4, and k = 64 as default. We use GPU implementation [66] of k-means clustering for our experiments with DINO masks. For EV , we apply CutOut [67] and color augmentations [68] during training. For the prompt ensemble strategy during inference, we use the prompts curated originally from CLIP [15] in their repository, which results in total of 8 text prompt as follows: “itap of a {}.”, “a bad photo of the {}.”, “a origami {}.”, “a photo of the large {}.”, “a {} in a video game.”, “art of the {}.”, “a photo of the small {}.”, “a photo of a {} in the scene”. B Additional Experiments Table 4: Quantitative results with various backbones. We show results on open-vocabulary semantic segmentation of various CLIP backbones with the addition of ViT-B from SigLIP [69] and ConvNeXt-L. Method Backbone Evaluation Dataset COCO-St. ADE-150 Context CityScapes VOC SigLIP [69] ViT-B/16 [62] 12.4 11.8 18.3 19.2 46.8 PixelCLIP (Ours) ViT-B/16 [62] 20.0 (+7.6) 19.2 (+7.4) 33.1 (+14.8) 31.6 (+12.4) 72.3 (+25.5) CLIP [26] ViT-B/16 [62] 16.5 13.2 25.6 14.9 73.9 PixelCLIP (Ours) ViT-B/16 [62] 21.4 (+4.9) 16.7 (+3.5) 34.9 (+9.3) 23.8 (+8.9) 83.1 (+9.2) OpenCLIP [55] ConvNeXt-B [49] 12.8 13.1 16.5 16.2 34.8 PixelCLIP (Ours) ConvNeXt-B [49] 21.1 (+8.3) 20.2 (+7.1) 34.2 (+17.7) 33.2 (+17.0) 66.0 (+31.2) OpenCLIP [55] ConvNeXt-L [49] 16.9 15.2 22.9 17.1 57.2 PixelCLIP (Ours) ConvNeXt-L [49] 24.8 (+7.9) 22.6 (+7.4) 39.4 (+16.5) 34.3 (+17.2) 78.9 (+21.7) B.1 Results on Different Backbones In Tab. 4, we show results for PixelCLIP when applied to different backbones. We note that since the ViT backbone has a larger output feature resolution scale compared to ConvNeXt models, we set the input image resolution to match the output feature resolution, and report results without prompt ensembling. In general, we observe noticeable gains across all backbones, with CLIP ViTB/16 outperforming ConvNeXt-B on several datasets. Through testing with various pre-trained CLIP models, we demonstrate that our method can effectively fine-tune CLIP for dense prediction regardless of the backbone architecture. 15 Table 5: Additional experiments on prompt ensembling. We show results on open-vocabulary semantic segmentation with prompt ensembling being used during only training, only inference, or both. The default setting of prompt ensembling only being used during inference is highlighted in gray. Prompt Ensembling Evaluation Dataset Training Inference COCO-St. ADE-150 Context CityScapes VOC 21.4 16.7 34.9 23.8 83.1 ✓ 23.6 (+2.2) 18.7 (+2.0) 37.9 (+3.0) 27.2 (+3.4) 85.9 (+2.8) ✓ 21.6 (+0.2) 17.1 (+0.4) 35.1 (+0.2) 24.9 (+1.1) 82.9 (-0.2) ✓ ✓ 23.7 (+2.3) 19.2 (+2.5) 37.9 (+3.0) 28.1 (+4.3) 85.5 (+2.4) B.2 Analysis on Prompt Ensembling In Tab. 5, we show results with prompt ensembling being applied during only training, only inference, and both. We report results with ViT-B/16 using SA-1B masks as supervision. Although prompt ensembling does bring slight gains when enabled during training, the computation for optimizing learnable class prompts scales along with the number of prompts used, increasing the training time and the memory consumption. On the other hand, applying prompt ensembling during inference only adds negligible cost as they can be computed once and be cached, but shows much significant gains compared to when applied training. Therefore, we adopt prompt ensemlbing only during inference, but noticing that the performance can be maximized with better prompts during training. In this regard we can better results with better prompt design or a learnable prefix to accompany the learnable class prompts, which we leave for future investigation. B.3 Additional Ablation Studies Table 6: Additional ablation studies. We show results on open-vocabulary semantic segmentation with a larger number of clusters and different training datasets. γ Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline [55] 12.8 13.1 16.5 16.2 34.8 0.99 19.9 19.5 32.5 29.5 62.6 0.999 21.1 20.2 34.2 33.2 66.0 0.9999 20.4 19.7 32.0 29.9 63.0 (a) Varying momentum γ. We show additional results for varying γ for the momentum update. Dataset Evaluation Dataset COCO ADE-150 Context CityScapes VOC Baseline [55] 12.8 13.1 16.5 16.2 34.8 COCO-St. [50] 24.1 21.9 36.8 30.2 71.0 SA-1B [13] 21.1 20.2 34.2 33.2 66.0 (b) Different training dataset. We show results for leveraging ground-truth masks from COCO-Stuff while removing its class labels. B.3.1 Ablation on the momentum update rate γ In Tab. 6 (a), we show results for varying γ for the momentum update. While having the momentum encoder generally shows improvements, we find γ = 0.999 to show the best results for updating the momentum encoder. B.3.2 Ablation on training dataset In Tab. 6 (b), we show results for training with mask annotation from COCO-Stuff [50]. For COCO-Stuff, we remove the ground truth class labels and utilize them as unlabeled masks, and other hyperparameters are set identically with k = 64. Although the masks from COCO-Stuff show better results across all datasets, we highlight that the SA-1B [13] dataset mostly consists of automatically generated masks from SAM, whereas COCO-Stuff has human annotated masks from expert annotators. 16 (a) 𝑘= 64 (b) 𝑘= 128 (c) 𝑘= 256 Figure 7: Visualization on COCO-Stuff with learned class prompts. We provide results with learned classes with different k up to 256. (a) Ours (b) CLIP Figure 8: Visualization on ADE-20k We provide qualitative comparison on ADE-20K [51]. C Additional Qualitative Results We provide qualitative results of visualization on ADE-20K [51], PASCAL-Context [52] in Fig. 8 and Fig. 9. D Additional Visualization In Fig. 7, we show visualization on COCO-Stuff by classifying the image features with our learned class prompts for varying k. From the first and the second row, we can observe that with larger numbers of k, different parts of human are segmented into fine-grained regions whereas k = 64 has 17 (a) Ours (b) CLIP Figure 9: Visualization on PASCAL-Context. We provide qualitative comparison on PASCALContext [52]. more coarse regions. Especially for k = 256, in the second row, we observe the glasses, hair, and hands all classified into different classes with our learned prompt. On the other hand, we also observe cases where a small number of k struggles to differentiate visual concepts in the last row, where the animals are partially grouped with k = 64 and show better groups for k = 128 and k = 256. This could indicate that with only a small number of clusters, several fine-grained visual concepts that may not be seen often in the dataset to be grouped as a whole, whereas independent clusters could be assigned with a larger number of k, allowing fine-grained recognition of semantics. E Limitations Although we aim to fine-tune the image encoder of CLIP for adapting to dense predictions, we initialize the mask encoder within our framework with pre-trained weights of CLIP, which yields poor results for classifying masks when applying mask pooling to its features. Consequently, the noisy mask features in the earlier stage of training may result in sub-optimal performance. While there could be alternative methods to extract per-mask CLIP image features, we consider mask pooling to be sufficient to show meaningful improvements to CLIP and consider such exploration for future directions. F Broader Impact Our framework facilitates open-vocabulary semantic segmentation through leveraging visionlanguage models, hence the recognition capabilities of our method rely on the pre-trained knowledge of the vision-language models. Considering that large-scale pre-trained vision-language models [15, 16, 55] leverage web-crawled data within its training, the models may exhibit wrongful behaviors from bias or corrupted data from the internet which calls for future research to address. 18 NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. 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4,463	Stability and Generalizability in SDE Diffusion Models with Measure-Preserving Dynamics Weitong Zhang1 Chengqi Zang2 Liu Li1 Sarah Cechnicka1 Cheng Ouyang1,3 Bernhard Kainz1,4 1 Imperial College London, UK, 2 University of Tokyo, JP, 3 University of Oxford, UK 4Friedrich-Alexander University Erlangen-Nürnberg, GER weitong.zhang20@imperial.ac.uk Abstract Inverse problems describe the process of estimating the causal factors from a set of measurements or data. Mapping of often incomplete or degraded data to parameters is ill-posed, thus data-driven iterative solutions are required, for example when reconstructing clean images from poor signals. Diffusion models have shown promise as potent generative tools for solving inverse problems due to their superior reconstruction quality and their compatibility with iterative solvers. However, most existing approaches are limited to linear inverse problems represented as Stochastic Differential Equations (SDEs). This simplification falls short of addressing the challenging nature of real-world problems, leading to amplified cumulative errors and biases. We provide an explanation for this gap through the lens of measure-preserving dynamics of Random Dynamical Systems (RDS) with which we analyse Temporal Distribution Discrepancy and thus introduce a theoretical framework based on RDS for SDE diffusion models. We uncover several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduce a novel score-based diffusion framework, the Dynamics-aware SDE Diffusion Generative Model (D3GM). The Measure-preserving property can return the degraded measurement to the original state despite complex degradation with the RDS concept of stability. Our extensive experimental results corroborate the effectiveness of D3GM across multiple benchmarks including a prominent application for inverse problems, magnetic resonance imaging. 1 Introduction Diffusion probabilistic models [53, 54, 52] have demonstrated impressive performance across various image generation tasks, primarily by modeling a diffusion process and then learning an associated reverse process. Among the many commonly used approaches [63], diffusion models that incorporate the concept of score functions [28, 55] can capture the intrinsic random fluctuations of the forward diffusion process, positioning them as a good choice for in-depth analysis. Score-based generative models (SGMs) entail gradually diffusing images towards a noise distribution, and then generating samples by chaining the score functions at decreasing noise levels with score-based sampling approaches. One such example of an SGM with a score-based sampling technique, known as score matching [54], has gained popularity for density estimation [16]. It employs methods such as 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Langevin dynamics [21, 43, 30] and SDEs [29, 37, 31, 71] to simulate the underlying probability distribution of training samples. However, vanilla unconditional SGMs can be extended to inverse problems by leveraging an implicit prior distribution, based on the available counterpart measurement, subjected to corruption and/or noise. To this end transitionary SGMs enable an iterative recovery of the data from this noisy counterpart, instead of relying on Gaussian white noise as a starting point [66, 37, 39, 12, 20, 34]. Intuitively, leveraging priors and a generative capacity into transitionary SGMs offers the possibility of exploring high quality reconstruction and restoration and gaining better performance. However, current transitionary SGMs that incorporate priors have largely overlooked the unreliable quality of the prior and its measurement. Empirically, transitionary SGMs cannot always be trusted in terms of stability and efficiency, especially in a regime of non-uniformly distributed noise or corrupted signal quality [9, 66, 37]. Hence, the exploitation of transitional learning within SGMs does not come without costs as their advantages vanish in limited data quality settings. Theoretical understanding is notably lacking in this field with the following fundamental open problem: Can we realize reliable transitionary diffusion processes in practice for inverse problems with a theoretical guarantee? While recent works have started to lay down a theoretical foundation for these models, a detailed understanding is still lacking. Current best practice advocates for smaller initialisation values (e.g., noise schedule [66, 20], instead of large values [15, 61] to ensure that the forward dynamics brings the diffusion sufficiently close to a known prior and simple noise distribution. However, a proper choice of the values conditioned on the prior within a theoretical framework should be preferred for a better approximation of the score-matching objective and higher computational efficiency. To fully facilitate the power of reversion and generation of transitionary SGMs and to mitigate the influence of low-quality measurements for solving inverse problems, this paper provides a measurepreserving dynamics of random dynamical system (RDS) perspective as a promising way to obtain reliable reversion and generation. Notably, our ‘measure’ is not only the observations (e.g., degraded images), but also represents the invariant probability measure (distribution) of the RDS. This allows to consider the concept of an RDS stability and to frame challenging degradation learning within a measure-preserving dynamical system. Thus, we can start from a transitionary SGM interpretation of diffusion models and connect RDS to the SDE in transitionary SGMs. The pitfalls (e.g., Instability) are discussed in Sect. 3 and further implications can be found in the Appendix. Transitionary SGMs have not been fully explored before, and we provide a theoretical interpretation of a stationary process as a possible solution. Our D3GM framework is abstracted from transitionary SGMs. The key to our framework is a stationary process following measure-preserving dynamics to ensure the stability and generalizability of the diffusion, as well as reducing the influence of accumulated error, distribution bias and degradation discrepancy. Our contributions can be summarised as follows: 1. Temporal Distribution Discrepancy: We conduct a rigorous theoretical examination of the instability issue of transitionary SGMs, measured as Temporal Distribution Discrepancy (i.e., lower bound of modeling error). This analysis sheds light on critical aspects related to stability and generalizability1, effectively addressing an unexplored fundamental gap in the understanding of solving challenging inverse problems with SDEs. 2. D3GM Framework: We propose a solution, D3GM, and an explanation from measure-preserving dynamics of Random Dynamical Systems (RDS). ‘Measure’ includes both measurements (degraded image) and invariant measures (distribution) of RDS, which allows complex degradation learning and enhances restoration and reconstruction accuracy. 3. Thorough Evaluation: Our contributions are substantiated by extensive validation. We demonstrate the practical benefits of our D3GM framework across various benchmarks, including challenging tasks such the reconstruction of Magnetic Resonance Imaging (MRI) data. We address the instability of diffusion models for inverse problems under domain shift and concept drift (unknown and heterogeneous degradation). This leads to what we believe is a completely novel view on the theoretical foundation of how the degradation process is modelled. The result is an approach that is more in line with the original intention of the theory of diffusion. We 1Generalizability refers to the extent to which out-of-distribution and domain shift impacts the fidelity of the restoration process. Stability in SDE diffusion models is demonstrated by its resilience to degradation beyond its domain and its consistent ability to restore high-quality images. 2 chose inverse problems as a relevant application area to demonstrate our ideas but also included a variety of challenging problem settings to explore the generalizability of D3GM. To the best of our knowledge, no other method can handle a diverse range of challenging tasks like real-world dehazing, compressed MRI reconstruction, blind MRI super-resolution, etc. with a unified underlying theoretical framework. In Tab. 1 we illustrate the key differences of D3GM compared with SGMs and transitionary SGMs. 2 Preliminaries SGMs. We will follow the typical construction of the diffusion process x(t), t ∈[0, T] with x(t) ∈Rd. Concretely, we want x(0) ∼p0(x), where p0 = pdata , and x(T) ∼pT , where pT is a tractable distribution that can be sampled. In this work, we consider the score-based diffusion form of the SDE [55]. Consider the following Itô diffusion process defined by an SDE: dx = f(x, t)dt + g(t)dW , (1) where f : Rd 7→Rd is the drift coefficient of x(t), g : R 7→R is the diffusion coefficient coupled with the standard d-dimensional Wiener process w ∈Rd. By carefully choosing ¯f, g, one can achieve a spherical Gaussian distribution as t →T. For the forward SDE in Eq. 1, there exists a corresponding reverse-time SDE [3, 55]: dx = [f(x, t) −g(t)2 ∇x log pt(x) \| {z } score function ]dt + g(t)dW , (2) where dt is the infinitesimal negative time step, and w is the Brownian motion running backwards. The score function ∇x log pt(x) is in general intractable and thus SDE-based diffusion models approximate it by training a time-dependent neural network under a score function [57, 28]. Transitionary SGMs. [17, 37, 66, 34, 20, 12] leverage a transitionary iterative denoising paradigm for the inverse problems. In inverse problems, such as super-resolution, we have an (nonlinear, partial, and noisy) observation y of the underlying high-quality signal x. The mapping x 7→y is many-to-one, posing an ill-posed problem. In this case, a strong prior on x is needed for finding a realistic solution. Formally, the general form of the forward (measurement) model is: y = A (x) + n, y, n ∈Rn, x ∈Rd, (3) where A(·): Rd 7→Rn2, oftentimes n ≪d is the forward measurement operator and n is the measurement noise, assuming n ∼N 0, σ2I . While sharing a similar aim of bridging y and x in transitionary SGMs, different mathematical frameworks have been used: [17] employs Inversion by Direct Iteration; [37, 34, 20] model it as a Mean-reverting SDE. Transitionary SGM has become an increasingly important line of SDE research due to the applicability on images with theoretical guarantees. However, they often perfrom poorly in real-world scenarios. To provide a theoretical investigation of this gap, we interpret Transitionary SGM as Ornstein-Uhlenbeck (OU) process. This perspective allows us to understand the random fluctuations in image degradation as stochastic processes, providing a foundation to integrate random dynamical systems (RDS) with the diffusion process as a natural extension of the SDE framework involving the OU process. The Measure-preserving property is introduced from the perspective of RDS: The distribution can still return to the original state despite severe degradation. Our approach constructs a bridge from measure-preserving dynamical system to transitionary SGM through measure-preserving dynamics, and highlights the Temporal Distribution Discrepancy in Sect. 3. Subsequently, we address this issue of instability: by incorporating a measure-preserving strategy into the solution of inverse problems, which is detailed in Sect. 4. This covers counterpart modeling, bridging a transition from uncertain diffusion modeling to deterministic solutions, yielding significant improvements in both performance and efficiency as demonstrated in Sect. 5. More details can be found in Sect. 6 and Appendix. 3 Instability Analysis: Transitionary SGMs with Corrupted SDE Diffusion Ornstein-Uhlenbeck (OU) process. An OU process is a common case in transitionary SGMs, where xt is defined using an SDE: dxt = −θtxtdt + σtdWt. Wt is standard Brownian motion. A drift term µ can be introduced: 2MRI signals are defined on Cn and Cd. We demonstrate in Sect. 5 that our approach is applicable to MRI. 3 Table 1: Differences between state-of-the-art SDE diffusion-based approaches. Model p(X0) p(X1) Theorem Foundation Properties TDD (Prop.2) Attractor Operator Dir. Inverse problem solver SGM [53] pA N(0, I) SDE diffusion No prior Unsolvable No subset attracts one-sided IR-SDE [37] pA pB(·\|X0, µ) Mean-reverting SDE Instability Limited Gaussian one-sided Lin/Non-Lin I2SB [34] δa pB(·\|X0) Schrodinger Bridge Strict Prior Limited No subset attracts one-sided Lin/ Non-Lin D3GM (ours) pA pB(·\|X0, µ, τ) Measure-Preserving RDS Stability Robust N(µ, τ 2σ2I) two-sided Lin/ Non-Lin/Blind dxt = θt(µ −xt)dt + σtdWt, (4) where µ denotes state mean, reflecting the expected state of the measurement (e.g., corrupted image [37], noisy speech [59]) over time. θt, σt are time-dependent parameters. The drift term corrects deviations from the constant µ, effectively pulling the process towards µ (t →∞) with Stability (in Appx. G) as opposed to pure noise in Eq. 1. Measure-preserving Dynamics in SDE Diffusion. The solution of the above SDE can be represented by a continuous-time random dynamical system φ defined on a complete separable metric space (X, d). (See precise definition of RDS in Appx. C). More generally, we can extend the RDS to a two-sided solution operator with a flow map. The base flow driven by Brownian motion can be written as W (t, ϑs(ω)) = W(t + s, ω) −W(s, ω). Proposition 1 After extending the solution of the OU process to RDS, the measure-preserving RDS φ should meet the property φ(t, s; ω)x = φ (t −s, 0; ϑsω) x. However, OU processes with timevarying coefficients usually do not satisfy this property. In this situation, the system breaks the forward-reverse processes, making it difficult to maintain stability. Intuition 1. A two-sided measure-preserving random dynamical (MP-RDS) system formulation enables us to use the Poincare recurrence theorem [45], (see precise statement in Appx.D), intuitively, with a two-sided MP-RDS φt, the Poincare recurrence theorem ensures that the system φt starts from terminal condition xT , run backward in time, will hit a region (x0 −ϵ, x0 + ϵ) for small ϵ in finite time, where x0 is the high-quality image. Example 1. Following Intuition 1. and Prop. 1, suppose that the OU’s θt follows a cosine schedule, such that θt = cos(t) for 0 ≤t ≤T, then for some 0 ≤s ≤t ≤T, φ(t, s; ω)x ̸= φ (t −s, 0; ϑsω) x because the change of θt w.r.t. time is not uniform. The OU-process instability exists due to Temporal Distribution Discrepancy (Prop. 2). At a high level, Proposition 1 can be extended to show that there exists a compact attracting set at any −∞< t < ∞, and this convention has allowed us to characterize the attractor K(ω) = N(µ, σ2/2θ). The closed-form distribution for y can be complex and may not be tractable depending on the particular scenarios of the actual image degradation process µ. The modification of σ and θ is used to regularize the perturbation and attempt to close the distribution. However, these injections might bypass the stationary process. More details can be found in Appx. D. Instability-Temporal Distribution Discrepancy. Given the process OU (xt, µ; t, θ) with Eq. 4, where xT ̸= x∞for finite T, indicates that the perturbed state cannot move towards the degraded LQ image and fails in matching the theoretical distribution. This inherent discrepancy further causes bias in the estimation of µt, which gradually accumulates into error in the reverse process. Temporal Distribution Discrepancy is illustrated by Proposition 2 (proof can be found in Appx. E): Proposition 2 Given Eq. 3 and Eq. 4, and assume that the score function is bounded by C in L2 norm, then the discrepancy between the reference and the retrieved data is, with probability (1 −δ) at least: ∥x0 −OU(x0, µ; T, θ)∥2 2 ≥\| (x0 −µ)2 −σ2 T /2θT e−2¯θT + σ2 T /2θT −σ2 max Cσ2 max + d + 2 p −d · log δ −2 log δ \|, (5) 4 Intuition 2. Intuitively, Proposition 2 provides a theoretical measurement on how the difference between finite iteration distribution and the asymptotic distribution of the OU-process in L2 could further enlarge the discrepancy between the retrieved image and the actual HQ image. Discrepancies are typically explained by complex degradation vs. monotonic modeling. For a noisy inverse problem, the retrieved data with any finite T depends on σt, µt, λ, T, ¯K (Lipschitz constant). This proposition also correlates to [39], where the lower bound of the distance between the high quality image and retrieved image in L2 norm in our model is further enlarged by this discrepancy, which correlates to the term (x0 −µ)2 −(σ2 T /2θT ) e−2¯θT + σ2 T /2θT . Another way to further minimize this bound is through the term e−2¯θT with [0, T] normalized to [0, 1]. What we refer to as θ-schedule corresponds to the exact functional form of θt, several schedules can be set here, e.g., constant, linear, cosine, and log. At a high level, the discrepancy between the reference and the retrieved data stems from the divergence between the forwarded final state and the low quality image. Eq. 5 can be factored into three constituent parts: the data residual, the stationary disturbance, and the random noise. While conditional diffusion generation entails a trade-off between variability and faithfulness [66], the persistent discrepancy within the residual has a significant impact on the generalizability of solving the transitionary tasks. This also establishes a connection with SDEdit [39] and CCDF [12]. When fitting inverse problems involving paired data into diffusion models, while accounting for deviations and degradation, inserting them into Eq. 4 directly may not be the most effective strategy. During sampling and inference with the degraded input y, the discrepancy identified in Prop. 2 intensifies. The complex degradations in y exacerbate the divergence from the expected µ distribution, significantly impacting the accuracy of the restored data ˆx0. More details are in Appx. F. 4 Towards Stability: Measure-preserving Dynamics in SDE Diffusion In Sect. 3, we extrapolate and theorize the Temporal Distribution Discrepancy on µ and x in the diffusion model for the inverse problem. Our key idea is to combine the stationary process to alleviate the Temporal Distribution Discrepancy problem following the measure-preserving dynamics from RDS. Recall that our ‘measure’ is not only the measurements (i.e., degraded image), but also represents the invariant measure (distribution) of the RDS. We begin by describing the forward and reverse processes of the D3GM, which serves as a stable bridge between the quality data and the counterpart measurement. We adapt score-based training methods to estimate this SDE. Following this, we describe the essential constructions for preserving the stationary process in the diffusion model and solving for Temporal Distribution Discrepancy on an orthogonal basis compared to current transitionary diffusion models. Measure-preserving Dynamics with the Stationary Process. Following Prop. 1, in a SDE Diffusion from 0 to T, the corresponding ‘attractors’ (states) can be viewed as N µ + (x0 −µ)e−¯θt, σ2 t (1 −e−2¯θt)/2θt . We can guide SDE Diffusion towards a stable and robust solution based on the properties of Measure-preserving in RDS. It can be extended to impose that for every t, σ2 t 2θt = λ2, where λ is the variance of the designated stationary measure forward process. This convention allows us to reduce the regularization on two variables σt, θt to just one variable to satisfy the property of the measure-preserving dynamics in the asymptotic sense, i.e., limt→∞φ(t, s; ω)x = limt→∞φ (t −s, 0; ϑsω) x. This convention allows us to characterize the attractor of the system as K(ω) = N(µ, λ2). The definition and constraint of the attractor are significant; without imposing this constraint, the measure-preserving property cannot be maintained, and the system would degrade into a Coefficient Decoupled SDE (Coe. Dec. SDE), we also analyse this in Fig. 2 and Tab. 11 in Appx. H. 5 High Quality Image Output Perturbed Image Forward Process Reverse Process Stationary Process Measure-preserving Dynamics Distribution Retrieval Low Quality Image D𝟑𝑮𝑴 GT LQ Rain Removal Real Dense Haze Removal MRI Reconstruction MRI Super-resolution 16x 8x Figure 1: Dynamics-aware SDE Diffusion Generative Model (D3GM). When extending transitionary SDEs to random dynamical systems (RDS), their measure-preserving property should be kept to maintain stability. This corresponds to driving the SDE towards the drift term µ (LQ). There is a Temporal Distribution Discrepancy which results from the gap between the forward estimation xT and the low quality image in the SDE. With the distribution aligned between xT and µ, the SDE can be made more robust to inverse problems. Reconstruction results for low quality (LQ) images after application of our D3GM method, on different tasks, compared to the ground truth (GT) on two domains - The frequency domain: MRI Reconstruction (undersampling factor 8x, 16x, frequency masks are colored red); MRI Super-resolution (up-scaling factor of X4, cross-domain evaluation). The image domain: Real Dense Haze Removal; Rain Removal (light, heavy). Example 2. When σ θ →∞, the attractor becomes excessively large, reducing the significance of µ. SDEs exhibiting this behavior are defined as Coefficient Decoupled SDEs. In practice, µ demonstrates non-infinite properties as an input image, while the corresponding sigma and theta are indeed unconstrained. In such decoupled forms, the coefficients of the attractor size increases σ µ, diminishing the significance of µ. Based on Prop. 2, since the temporal distribution discrepancy always exists as long as the running time T is finite, and we want the final state xT to be as close as possible to the distribution of x∞. Therefore, we introduce τ such that given T, the distribution of xT follows N(µ(1 −e¯θT ) + x0e¯θT , τ 2λ2(1 −e2¯θT )), and x∞follows N(µ, τ 2λ2), with τ > 1, we increase the possibility of a sample ˜xT from xT to become closer to the distribution of x∞, and thus serves as a plausible initial state for the reverse process. We can control how much to close the distributions, either by increasing the stiffness τ at the cost of potentially destabilizing the reverse process, or by decreasing τ to further smooth the density functions of both distributions at the cost of more reverse iterations. By connecting the inverse problem with the analysis above, we clarify the discrepancy in the stationary modeling process from measure-preserving dynamics and thereby improve the generalization of diffusion processes and the accuracy of the reverse process. This is particularly important for accommodating the diversity of degradation states and to ensure accurate sampling. Forward Process. We describe the forward process as: dxt = θt(µ−xt)dt+τσtdWt, parameterized by τ to calibrate the SDE modeling, µ is the state mean. The parameters θt and σt, both being timedependent and strictly positive, correspond to the rate of mean reversion and the stochastic volatility, respectively. The selection of θt and σt offers flexibility in Tab. 2, c.p., Sect. 3. Considering the trade-off between complexity and effectiveness, Cos has been chosen for both θt and σt. This aims at capturing complex temporal dynamics in a computationally tractable manner, thereby optimizing the balance between the performance and calculation convenience. 6 Table 2: µt(xt, t), vt(xt, t) solutions with various θt, σt. µt(xt, t) vt(xt, t) Lin eθ x2 2 x0 + 1 −eθ x2 2 y τ 2λ2 1 −eθx2 y Log e θ k log(1+ekt)x0 + 1 −e θ k log(1+ekt) y τ 2λ2 1 −e 2θ k log(1+ekt) y Cos e −θ t−sin(θt) θ x0 + 1 −e −θ t−sin(θt) θ y τ 2λ2 1 −e −2θ t−sin(θt) θ y Quad eθ x3 3 x0 + (1 −eθ x3 3 )y τ 2λ2(1 −e2θ x3 3 )y In the forward process, the mean µt approaches the low-quality image with E (xt) = µ, while the variance tends toward the stationary variance var (xt) = τ 2σ2 2θ . Essentially, the forward SDE transitions the high-quality image to a low-quality counterpart infused with Gaussian noise. The discretized SDE for the forward process is xti = xti−1 + θti−1(µ −xti−1)∆t + τσti−1∆Wi. We employ a transition strategy utilizing a varied stationary variance. Additionally, we execute an unconditional update, which operates without the need for matching in the reverse process. These not only allow image corruption but also provides effective adaptability for improvements. Reverse Process. The reverse process aims at reconstructing the original image by gradually denoising a low quality image. It utilizes the score of the marginal distribution, denoted as ∇x log ˆpt(x), and is governed by: dxt = θt(µ −xt) −τ 2σ2 t ∇x log ˆpt(x) dt + τσtdc Wt. (6) The reverse-time D3GM process of Eq. 6 can be found in Appx. D. This closely mirrors the forward process and incorporates an additional drift term proportional to the score of the marginal distribution. The ground truth score for this process, necessary for training our generative model, is: ∇x log ˆpt(x \| x0) = −xt −µt(x) vt , (7) where µt(x) represents the random attractor of the process at time t, and vt is the variance. Our training objective is defined as the minimization of the expected discrepancy between the predicted and true scores over the data distribution: θ∗= arg min θ Et,(x0,y),z,xt h w ∥Sθ(xt, y, t) −∇xt log p0t(xt \| x0, y)∥2 2 i , (8) where w = −1/τ 2 is a time-dependent weighting function, and Sθ denotes the score network parameterized by θ which approximates the score of the marginal distribution. The optimization is conducted over the network parameters θ, under the expectation with respect to the time variable t, the initial image x0, noisy image xt, and data y. 5 Experiments Experimental Settings: We evaluate D3GM on various challenging restoration and reconstruction problems. We initially analyze our method by examining its performance with closely related diffusion formulation variants. Subsequently, we benchmark D3GM against the state-of-the-art techniques in these domains. For comprehensive evaluation across all experiments, we report the PSNR [25] and SSIM [58] for pixel- and structural-level alignment, LPIPS [72] for measuring perceptual variance. An in-depth description of our implementation is provided in Appx. G. 5.1 Stability: Illustrations of the Measure-preserving Dynamics within Diffusion Models SGM, and Transitionary SGMs vs. D3GM: We perform qualitative and quantitative analyses using variants of closely related formulations for Prop. 1 and 2 and evaluate across (A) SGMs and (B) transitionary SGMs. (A) uses a common score-based SDE, (B) uses a Coefficient Decoupled SDE (e.g., variance exploding SDE with the drift term µ) according to Prop. 1 and OU SDE, alongside our D3GM. 7 Table 3: Quantitative results for Rain100H and Rain100L.(best in bold and second best underlined) Rain100H Rain100L Method PSNR↑ SSIM↑ LPIPS↓ PSNR↑ SSIM↑ LPIPS↓ JORDER [64] 26.25 0.835 0.197 36.61 0.974 0.028 IRCNN [37] 29.12 0.882 0.153 33.17 0.958 0.068 PreNet [49] 29.46 0.899 0.128 37.48 0.979 0.020 MPRNet [68] 30.41 0.891 0.158 36.40 0.965 0.077 MAXIM [56] 30.81 0.903 0.133 38.06 0.977 0.048 VPB (CD) [73] 30.89 0.885 0.051 38.12 0.968 0.023 IR-SDE (OU) [37] 31.65 0.904 0.047 38.30 0.981 0.014 D3GM 32.41 0.912 0.040 38.40 0.982 0.013 Table 4: Quantitative results for OHAZE and Dense-Haze. O-HAZE Dense-Haze Methods PSNR↑ SSIM↑ PSNR↑ SSIM↑ DCP [23] 16.78 0.653 12.72 0.442 DehazeNet [6] 17.57 0.770 13.84 0.430 GFN [50] 18.16 0.671 GDN [36] 18.92 0.672 14.96 0.536 MSBDN [18] 24.36 0.749 15.13 0.555 FFA-Net [46] 22.12 0.770 15.70 0.549 AECR-Net [60] 15.80 0.466 SGID-PFF [4] 20.96 0.741 12.49 0.517 Restormer [67] 23.58 0.768 15.78 0.548 Dehamer [22] 25.11 0.777 16.62 0.560 MB-TF [47] 25.31 0.782 16.44 0.566 D3GM 26.23 0.786 15.85 0.551 MSBDN LQ FFA-Net SGID-PFF Dehamer MB-TF 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) GT Dense-Haze O-HAZE (a) O-HAZE and Dense-Haze. 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) GT IR-SDE MPRNet PReNet JORDER LQ (b) Rain100H and Rain100L. Figure 3: Qualitative results for (a) deraining and (b) dehazing. OU SDE Coef. Dec. SDE SGM 𝑫𝟑𝑮𝑴 t = 0.00 t = 0.25 t = 0.50 t = 0.75 t = 1.00 Figure 2: Sampling trajectories of SGM, transitionary SGMs: Coef. Dec., OU SDE, and D3GM. Following Tab. 1, VPB [73] can be regarded as a Coefficient Decoupled SDE, and IR-SDE [37] as an OU SDE. Our results in Fig. 2 illustrate that D3GM converges stably towards the expected distribution, unlike other methods which exhibit instability or deviation. This highlights the reliance of other techniques, e.g., scorebased SDEs, on retrospective measurement consistency corrections. Simulated Deraining: We evaluated D3GM together with the state-of-the-art deraining strategies: (1) OU SDE method IR-SDE [37], Coefficient Decoupled (CD) VPB [73] and other CNNs [64, 49, 68, 56]. We use two of the most renowned synthetic raining datasets: Rain100H [65] and Rain100L [65]. Rain100H contains 1800 pairs of images with and without rain, along with 100 test pairs. As for Rain100L, it consists of 200 pairs for training and 100 pairs for testing. We present results based on the PSNR, SSIM, and LPIPS metrics. Quantitative results from the two raining datasets are presented in Tab. 3. Based on both distortion and perceptual metrics, D3GM is capable of generating the most realistic and high fidelity results as shown in Fig. 3b. 5.2 Generalizability: D3GM for real-world data Case Study 1: Dehazing. We utilize the real-world datasets O-HAZE [2], Dense-Haze [1], which contain 45 and 55 paired images, respectively. We use the last 5 images of each dataset as the testing set and the rest as the training set following the common split of other methods. Results are shown in Tab. 4 and Fig. 3a. Our work improves results on O-HAZE both quantitatively and qualitatively. Smaller improvements are observed on Dense-Haze. This can be attributed to the severe signal corruption of the Dense-Haze data. A combination with tailored task-specific, Transformer-based 8 methods [47, 22] might lead to further performance gains for such data. Such extensions are beyond the focus of this paper. Qualitatively D3GM achieves excellent visual results (Fig. 1 and Fig. 3a). Case Study 2: MRI Reconstruction. MRI data is represented in the complex-valued frequency domain, which is distinctly different from the natural image domain. We utilized the fastMRI dataset [69], containing single-channel, complex-valued MRI samples. Implementation details can be found in Appx. G. For a robust comparison, we benchmarked against a diverse set of deep learningbased state-of-the-art reconstruction methods. Although our method does not have a task-specific design, we still get comparable performance (more details and results are provided in Appx. H). Fig. 1 illustrates our reconstruction results from masked k-space data for 8x and 16x acceleration, i.e., under-sampling for faster data acquisition. Table 5: Quantitative results for fastMRI dataset with acceleration rates x8 and x16. Undersampling factor 8x Undersampling factor 16x Method PSNR↑ SSIM↑ LPIPS↓ PSNR↑ SSIM↑ LPIPS↓ ZeroFilling 22.74 0.678 0.504 20.04 0.624 0.580 D5C5 [51] 25.99 0.719 0.291 23.35 0.667 0.412 DAGAN [62] 25.19 0.709 0.262 23.87 0.673 0.317 SwinMR [27] 26.98 0.730 0.254 24.85 0.673 0.327 DiffuseRecon [44] 27.40 0.738 0.286 25.75 0.688 0.362 CDiffMR [26] 27.26 0.744 0.236 25.77 0.707 0.293 D3GM 27.92 0.740 0.175 25.26 0.701 0.153 Table 6: Quantitative results for IXI MRI SR on unseen datasets. X4 HH Guys IOP Methods PSNR↑ SSIM↑ PSNR↑ SSIM↑ IXI T2w EDSR [33] 23.03 0.700 25.10 0.727 SFM [19] 23.28 0.711 25.18 0.731 PDM [70] 22.89 0.709 27.93 0.851 ACT [48] 22.80 0.707 26.38 0.826 CST [14] 23.70 0.714 28.55 0.837 D3GM 25.13 0.799 28.60 0.863 Case Study 3: MRI Super-resolution (SR). IXI3 dataset is the largest benchmark considered in our MRI SR evaluation. Clinical MRI T2-weighted (T2w) scans are collected from three hospitals with different imaging protocols: HH, Guys, and IOP. For investigating our cross-domain generalization and robustness, a challenging task for both MRI SR and natural image restoration, we trained on HH data with k-space truncation, and tested on Guys and IOP with kernel degradation with an up-scaling factor of X4. More details can be found in Appx. G. The methods are tested under unseen data conditions, including different acquisition parameters, MRI scanners (different vendors and field-strengths) and unseen degradations. With D3GM we are able to demonstrate varying degrees of improvement, as well as generalizability to the discrepancy within the training domain and across the test domain as shown in Tab. 6. Qualitative results are shown in Fig. 11 and further results across domains in Appx. H. 6 Discussion and limitations Other works, like VPB (CD) [73], I2SB [34] are based on diffusion bridges assuming that clean and degraded images are already close. Thus, the tractability of the reverse process heavily relies on the validity of the assumed Dirac delta distribution. IR-SDE [37] employs the mean-reverting SDE theorem based on running the reverse SDE with instability. Since unstable errors accumulate in each step, this model will eventually become unable to learn the transformation, e.g., degradation. DPS [9] and CDDB [10] assume that the degradation process is known, or linear operations are directly used to simulate the degradation process, which limits the generalizability of the method. In contrast, D3GM is built on the theorem of Measure-preserving RDS, which bridges clean and degraded image distributions while taking both degradation and measurements into account. Moreover, D3GM can be extended to a two-sided solution operator (tractability) with a flow map according to Prop. 1. Limitations. Even though our results are better than others when the degradation process is very severly corrupted (e.g., real dehazing), the overall quality of the restored image is still limited, which is consistent with Prop. 2. This might be alleviated via guiding the sampling process with priors and enhanced µ, such as posterior sampling or degradation maps on the data manifold, but such approaches are still limited as shown in Tab. 7 and Appx I. Computational Complexity vs. Performance. Tab. 7 highlights that prior work is often tailor-made for a specific subset of tasks and thus also generalisation-limited for challenging environments in practice. D3GM’s focus on a generic robust solutions from an RDS perspective can mitigate this, while maintaining en-par performance with task-specific approaches in Tab. 8. 3http://brain-development.org/ixi-dataset/ 9 Table 7: Comparison to tailor-made approaches. Real Dehazing Diffu. for Inverse Problems. Transitionary SGMs. Methods DPS [9] CDDB [10] I2SB [34] IR-SDE [37] D3GM PSNR↑ 18.63 21.55 21.51 24.52 26.23 SSIM↑ 0.448 0.591 0.583 0.691 0.786 Table 8: D3GM vs. recent deraining works. Deraining rain200H Model Complexity Methods PSNR↑ SSIM↑ Param. FLOPs DRSformer [7] 32.17 0.933 33.7M 242.9G D3GM 32.21 0.925 36.5M 104.7G 7 Conclusion The proposed D3GM framework enhances the stability and generalizability of SDE-based diffusion methods for challenging inverse problems. Our approach, grounded in measure-preserving dynamics of random dynamical systems, ensures broad applicability and relevance. We demonstrate D3GM’s effectiveness across various benchmarks, including challenging tasks like MRI reconstruction. Acknowledgements: This work was supported by the JADS programme and UK Research and Innovation [UKRI Centre for Doctoral Training in AI for Healthcare grant number EP/S023283/1]. 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Denoising diffusion bridge models. arXiv preprint arXiv:2309.16948, 2023. 13 Appendix A Broader Impacts Our proposed method offers significant advancements in the restoration of degraded images with minimal risk of hallucinations due to stability guarantees and with applications including MRI reconstruction and super-resolution. In the clinical domain, the adoption of our method for MRI reconstruction must adhere to stringent regulatory and approval processes. The results generated by our model should serve as an auxiliary tool to assist healthcare professionals in their diagnostic and treatment decisions, rather than as a standalone diagnostic tool. B Intuition of Measure-Preserving Dynamics in SDE: Consider the analogy of a stretched rubber band, which naturally seeks to return to its original but does so with a lot of oscillations when released. This elastic behavior parallels the dynamics of the OU process, where deviations from a mean state are counteracted by a restorative force, guiding the system back towards equilibrium (i.e., final state), with random perturbations. Our process models the noise as a stochastic component that fluctuates around a stationary process and improve the OU process with RDS. Measure-preserving dynamics ensure that while the image undergoes transformations during the denoising process, the overall statistical properties remain consistent (i.e., invariant image features), which cannot be satisfied by vanilla OU processes or previous approaches (Tab. 1). C Mathematical Preliminaries Consider a probability space (Ω, F, P) accompanied by a standard Brownian motion Wt. A stochastic process xt over the interval 0 ≤t ≤T can be formulated by the following stochastic differential equation (SDE): dxt = b(t, xt)dt + σ(t, xt)dWt (9) Definition 1 Filtration A collection of sigma-fields, F := {Ft, 0 ≤t ≤T}, is termed a filtration if: 1. Ft ⊂F is a sub-σ-field for every t ∈[0, T]; 2. If 0 ≤t1 < t2 ≤T, then Ft1 ⊂Ft2. Here, Ft represents the information set at time t. Definition 2 Strong Solution A process x, which is F-progressively measurable, is considered a strong solution to the SDE given by Eq. 1 if: Z T 0 (\|b(t, xt)\|2 + \|σ(t, xt)\|2)dt < ∞ almost surely. This is captured by: xt = x0 + Z t 0 b(s, xs)ds + Z t 0 σ(s, xs)dWs, ∀t ∈[0, T] (10) Definition 3 Lipschitz Continuity For an N-dimensional stochastic process xt over t ∈[0, ∞), adapted to the filtration F, function w(xt) exhibits Lipschitz continuity in x if: 1. w(xt) is F-measurable with the requisite dimensions. 14 2. There exists a non-negative constant K such that, for every xt and xs, \|w(xt) −w(xs)\| ≤K\|xt −xs\|. In numerous diffusion models, Lipschitz continuity is inherent, ensuring the existence and uniqueness of the solution to the stochastic process. However, for clarity in network design, the emphasis on Lipschitz continuity ensures the foundation of neural networks remains consistent. Definition 4 Random Dynamical System A random dynamical system(RDS) consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space, we first discuss one fundamental element of our RDS, the base flow. Let (Ω, F, P) be a probability space, the noise space. Define the base flow ϑ : R × Ω→Ωas follows: for each "time" s ∈R, let ϑs : Ω→Ωbe a measure-preserving measurable function: P(E) = P ϑ−1 s (E) for all E ∈F and s ∈R Suppose also that 1. ϑ0 = idΩ: Ω→Ω, the identity function on Ω; 2. for all s, t ∈R, ϑs ◦ϑt = ϑs+t. That is, ϑs, s ∈R, forms a group of measure-preserving transformation of the noise (Ω, F, P) Now we are ready to define the random dynamical system(RDS). let (X, d) be a complete separable metric space, the phase space. Let φ : R × Ω× X →X be a (B(R)⊗F ⊗B(X), B(X)) measurable function such that 1. for all ω ∈Ω, φ(0, ω) = idX : X →X, the identity function on X; 2. for (almost) all ω ∈Ω, (t, x) 7→φ(t, ω, x) is continuous; 3. φ satisfies the (crude) cocycle property: for almost all ω ∈Ω, φ (t, ϑs(ω)) ◦φ(s, ω) = φ(t + s, ω) In the case of random dynamical systems driven by a Wiener process W : R × Ω→X, the base flow ϑs : Ω→Ωwould be given by W (t, ϑs(ω)) = W(t + s, ω) −W(s, ω). Theorem 1 Existence and Uniqueness If the initial condition x0 ∈L2 is a random variable that’s independent of W and both µ(0, x0) and σ(0, x0) ∈H2, then, provided there exists a constant K > 0 that satisfies: \|b(t, x) −b(t, y)\| + \|σ(t, x) −σ(t, y)\| ≤K\|x −y\|, ∀t ∈[0, T], x, y ∈Rn (11) (an attribute also recognized as Lipschitz continuity), a unique strong solution to Eq. 1 exists in H2 for every T > 0. Additionally: E sup t≤T \|xt\|2 ≤C(1 + E\|x0\|2)eCT (12) holds true, where the constant C depends on both T and K. D Preliminaries and Proof for Proposition 1 D.1 Proof for reverse-time D3GM process: There is a one-to-one and onto correspondence between the stochastic differential equation and the Kolmogorov equation for p (xt, t \| xs, s) , t ⩾s, which describes the evolution of the underlying probability distribution. Consequently, there should be a one-to-one and onto correspondence between a reverse-time equation for ˜xt and a Kolmogorov equation for p(xt, t\|xs, s), s ⩾t dxt = θt(µ −xt)dt + τσdWt We have the corresponding Kolmogorov backward equation given by −∂p (xs, s \| xt, t) ∂t =θt(µ −xt) · ∂p (xs, s \| xt, t) ∂xt + 1 2τ 2σ2 t · ∂2p (xs, s \| xt, t) ∂x2 t , (13) 15 The unconditioned Kolmogorov forward equation is given by −∂p (xs, s \| xt, t) ∂t =∂(θt(µ −xt) · p (xs, s \| xt, t)) ∂xt −1 2 · ∂2 τ 2σ2 t · p (xs, s \| xt, t) ∂x2 t , (14) See [3] for more details on Kolmogorov equations. Bayes rule gives p (xt, t, xs, s) = p (xs, s \| xt, t) p (xt, t) We plug this result into 13, which gives us the Kolmogorov equation −∂ ∂tp (xt, t, xs, s) = ∂ ∂xt ¯f (xt, t) p (xt, t, xs, s) + 1 2 ∂2 p (xt, t, xs, s) · τ 2σ2 t ∂x2 t (15) and the expression for ¯f is given by ¯f (xt, t) = θt(µ −xt) − 1 p (xt, t) ∂ ∂xt p (xt, t) τ 2σ2 t = θt(µ −xt) −τ 2σ2 t log ∂ ∂xt [p(xt, t)] (16) Therefore, we have that the reverse process corresponds to the Kolmogorov equation 16 is given by dxt = θt(µ −xt) −τ 2σ2 t log ∇xpt(xt) + τ 2σ2 t d ¯Wt Definition 5 Different from deterministic dynamical systems, random dynamical systems usually consider a pullback attractor rather than a forward attractor due to the non-autonomousness introduced by the random noise. The pullback attractor (or random global attractor) A(ω) for the RDS φ we defined in 1 is a P-almost surely unique random set such that: 1. A(ω) is a random compact set: A(ω) ⊆X is almost surely compact and ω 7→d(x, A(ω)) is a (F, B(X))-measurable function for every x ∈X 2. A(ω) is invariant: for all φ(t, ω)(A(ω)) = A (ϑtω) almost surely; 3. A(ω) is attractive: for any deterministic bounded set B ⊆ X, limt→+∞d (φ (t, ϑ−tω) (B), A(ω)) = 0 almost surely. B(X) denotes the Borel σ-algebra generated by the space X where the RDS is defined. Definition 6 Poincare Recurrence Theorem Let (X, Σ, µ) be a finite measure space and let f : X →X be a measure-preserving transformation. then we have that for any E ∈Σ, the set of those points x of E for which there exists N ∈N such that f n(x) /∈E for all n > N has zero measure. In other words, almost every point of E returns to E. In fact, almost every point returns infinitely often; i.e. µ ({x ∈E : there exists N such that f n(x) /∈E for all n > N}) = 0. D.2 Analysis of Proposition 1 Proposition 1 After extending the solution of the OU process to RDS, the measure-preserving flow map of the solution should meet the property φ(t, s; ω)x = φ (t −s, 0; ϑsω) x. However, OU processes with time-varying coefficients are usually not satisfied for this property(can be referred to as time-homogeneity) and thus the stability of the system breaks. The forward process in SDE notation dxt = θt(µ −xt)dt + σtdWt 16 and the solution to the above SDE is given by xt = µ + (xs −µ)e−¯θs:t + Z t s σze−¯θz:tdWz where ¯θs:t = R t s θzdz. This solution above can be represented by a continuous random dynamical system (RDS) φ defined on a complete separable metric space (X, d), where the noise is chosen from a probability space (Ω, F, P). More details can be found in [13]. More generally, we can extend the RDS to two-sided, infinite time, define a flow map or (solution operator) φ : R×Ω×Rd →Rd by φ (t, s, \|ω, x0) := x (t, s\|ω, x0) with ω ∈Ω, −∞< s ⩽t < ∞. The base flow driven by Brownian motion can be explicitly written as W (t, ϑs(ω)) = W(t+s, ω)− W(s, ω). Now suppose that: 1. The flow map ϑt, t ∈R is a measure-preserving transformations of (Ω, F, P), with the property that for all s < t and x ∈X, φ(t, s; ω)x = φ (t −s, 0; ϑsω) x, P-a.s. (17) 2. (i) φ(t, r; ω)φ(r, s; ω)x = φ(t, s; ω)x for all s ⩽r ⩽t and x ∈X; (ii) φ(t, s; ω) is continuous in X, for all s ⩽t. (iii) for all s < t and x ∈X, the mapping ω 7→φ(t, s; ω)x is measurable from (Ω, F) to (X, B(X)); and (iv) for all t, x ∈X, and P-a.e. ω, the mapping s 7→φ(t, s; ω)x is right continuous at any point. Where B(X) denotes the σ-algebra generated by X. Under assumptions (i), (ii), (iii), (iv) and suppose that for P-a.e. ω there exists a compact attracting set K(ω) at time 0, i.e., such that for all bounded sets B ⊂X, d(φ(0, s; ω)B, K(ω)) →0 as s →−∞ We can see that the attractor of this system is defined in the pullback sense, such that time is rewind backward before iterating forward. Moreover, the reverse process with any starting time t to s is defined as the RDS going backward in time φ(s, t\|ϑtω, xt) start from the time t realization and run backwards to s. the above proposition can be extended to show that there exists a compact attracting set at any −∞< t < ∞. and this convention has allowed us to characterize the attractor K(ω) = N(µ, λ2), when the time becomes finite, for example from 0 to T, the random attractors can be abstractly viewed as N µ + (xs −µ)e−¯θ0:t, λ(1 −e−¯θ0:t) . Moreover, an important assumption in the 17 is usually not satisfied for OU processes with timevarying coefficients, therefore, we impose that for every t, σ2 t 2θt = λ2, where λ is a constant, and will be the asymptotic variance of the forward process. This convention has allowed us to reduce the regularization on two variables σt, θt to just one variable to satisfy 17; and this convention has allowed as to characterize the attractor K(ω) = N(µ, λ2), when the time becomes finite, for example from 0 to T, the random attractors can be abstractly viewed as the Gaussian measure N µ + (xs −µ)e−¯θ0:t, λ(1 −e−¯θ0:t) . E Proof for Proposition 2 Proposition 2 Given Eq. 3 and Eq. 4, and assume that the score function is bounded by C in L2 norm, then the discrepancy between the reference and the retrieved data is, with probability at least (1 −δ): 17 ∥x0 −OU(x0, µ; T, θ)∥2 2 ≥\| (x0 −µ)2 −σ2 T /2θT e−2¯θT + σ2 T /2θT −σ2 max Cσ2 max + d + 2 p −d · log δ −2 log δ \|, (18) where x0 ˆx0 are the quality reference and sampling data. For a noisy inverse problem scenario, the retrieved data with any finite T always indicates difference depends on σt, µt, λ, T, ¯K, where ¯K is the Lipschitz constant for the reverse process. we have that the absolute value between the theoretical expectation and actual expectation after T period is given by ∥µ −E(ˆxT )∥= ∥(x0 −µ)e−¯θT ∥> 0 Similarly, the difference between theoretical variance and T-period variance also has a strictly positive difference. where ¯θt = R t 0 θsds. Therefore, with finite T, the final state of the forward process can only reach a ˆxT rather than the theoretical stationary distribution, which we denote by x∞, we denote the retrieved image after T-periods from theoretical stationary distribution by ˆx0, x0 by the ground truth HQ image, ˆxT the true distribution after T iteration, ∥xQ −fOUn(xQ, µ; t0, )∥= ∥ˆxT −x∞−(ˆx0) −x∞∥2 2 ≥∥∥ˆxT −x∞∥2 2 −∥ˆx0 −x∞∥2 2∥2 2 (19) Inside the norm, the first term is bounded below, since both ˆxT and x∞both follow a normal distribution and are independent of each other, the difference between those two random variables, we denote by zT , that follows a normal distribution N µ + (x0 −µ)e−¯θt −µ, λ2(1 −e−2¯θt) + λ2 =N (x0 −µ)e−¯θt, λ2(2 −e−2¯θt) (20) Therefore, we could rewrite the equality as: ∥xQ −fOU(xQ, µ; t0)∥ ≥ ∥zT ∥2 2 − Z 0 T −dσ2 t dt ∇x log pt(x) dt + dwt 2 2 2 2 = (x0 −µ)2e−2¯θt + λ2(2 −e−2¯θt) −Cσ4 max − Z 0 T r dσ2 t dt dwt 2 2 2 2 (21) Since we require λ = σ2 t 2θt , we can find a σmax such that σt < σmax for all t. The last term only concerns the random noise, according to [32], we have that the last term is equivalent to the squared L2 norm of a random variable from a Wiener process at time t = 0, with marginal distribution being ϵ ∼N 0, σ2 T I . The squared L2 norm of ϵ divided by σ2 T is a χ2-distribution with d-degrees of freedom, we have the following one-sided tail bound, according to [32] Pr ∥ϵ∥2 2/σ2 (t0) ≥d + 2 p d · −log δ −2 log δ ≤exp(log δ) = δ Therefore, with probability 1 −δ, the HQ image and retrieved image has a lower bound of,(d is the number of dimension for x0), \| (x0 −µ)2 −λ2 e−2¯θT + 2λ2 −σ2 max Cσ2 max + d + 2 p −d · log δ −2 log δ \| A SDE modeling that has suffered complex degradation is considered ’corrupted’: 18 Example 1. Based on the Prop. The conditional SDE diffusion is composed of three general stages, forward, backward, and sampling. Both time-reversal processes have been declared unstable under the degradation process µ according to the Prop. 2. The following advantages would be offered by our measure-preserving dynamical system when formulated as SDEs: Intuition 1. A two-sided measure-preserving random dynamical(MP-RDS) system formulation enables us to use the Poincare recurrence theorem, intuitively, with a two-sided MP-RDS φt, the Poincare recurrence theorem ensures that the system φt starts from terminal condition xT , run backwards in time, will hit a region (x0 −ϵ, x0 + ϵ) for small ϵ in finite time, where x0 is the high-quality image. F Temporal Distribution Discrepancy during Sampling Theorem 2 Suppose that both the drift bt(x) and diffusion σt(x) term of a stochastic process xt is Lipschitz continuous with some constant K, moreover, x ∈L2 (F, R) is a solution to the SDE dxt = b(t, x)dt + σ(t, x)dWt and initial condition b0, σ0 then we have that E h \|xT −x0\|2i ≤CI2 0, (22) such that I2 0 := E   Z T 0 \|b0\| dt !2 + Z T 0 \|σ0\|2 dt   and C depends only on T, K, which is the running time and Lipschitz constant Firstly, we have the following relationship: xT ≤\|x0\| + Z T 0 \|b(t, x)\| dt + sup 0≤t≤T Z t 0 σ(s, x) dWs , \|xT −x0\| ≤ Z T 0 \|b(t, x)\| dt + sup 0≤t≤T Z t 0 σ(s, x) dWs . (23) Squaring both sides, taking expectations, and applying the Burkholder-Davis-Gundy inequality, we get: E h \|xT −x0\|2i ≤CE   Z T 0 \|b(t, xt)\| dt !2 + sup 0≤t≤T Z t 0 σ(s, xs) dBs 2  ≤CE   Z T 0 [\|b0\| + \|xt\|] dt !2 + Z T 0 \|σ(t, xt)\|2 dt   ≤CE   Z T 0 \|b0\| dt !2 + Z T 0 h \|σ0\|2 + \|xt\|2i dt  . (24) Remark: It should be noted that the constant C, which depends on T and K, varies from line to line. Next, we show that for any ε > 0, there exists a constant Cε > 0 such that: 19 sup 0≤t≤T E h \|xt\|2i ≤εE h \|x∗ T \|2i + CεI2 0. (25) Applying Itô’s formula, we get: d \|xt\|2 = h 2xtb(t, xt) + \|σ(t, xt)\|2i dt + 2xtσ(t, xt) dBt. (26) Considering the martingale property of the third term, integrating, and taking expectation on both sides, we have: E h \|xt\|2i = E \|x0\|2 + Z t 0 h 2xsb(s, xs) + \|σ(s, xs)\|2i ds ≤E \|x0\|2 + Z t 0 h C \|xs\|2 + 2 \|xs\| \|b0\| + C \|σ0\|2i ds ≤C Z t 0 E h \|xs\|2i ds + 2E " xT Z T 0 \|b0\| ds # + CI2 0. (27) Using Gronwall’s inequality, we can prove (25) with the result above. Substituting (25) into (24) completes the proof, showing that the distance between xT and x0 in the L2-sense is bounded by C(K, T). With theorem 1 in hand, since the OU process is Lipschitz continuous, then the reverse process for the OU process is also Lipschitz continuous. Now, suppose that the Lipschitz constant for the reverse process is given by ¯K. then we have that in L2-norm, the distance between any final state xT ∈N(µ, λ) and the initial state(HQ image) is bounded by a constant that only depends on time T and Lipschitz constant ¯K, and the initial condition for the drift and diffusion term, where we denote as C( ¯K, T, µ, λ), which will be written as C( ¯K, T) for short. Now, since the time T is finite, where theoretically only when T →∞would xT converge to the theoretical stationary distribution, thus, if we denote the sample from T time steps by ˆxT , and xT by the sample from the theoretical stationary distribution, then we have E[xT −ˆxT )] > ϵ(T, K, µ, λ), note that the K here is the Lipschitz constant for the forward process, and this distance strictly decrease in T. Suppose that in inference, when the ground truth x0 is unknown, the distance between ground truth x0 and sample from the theoretical distribution xT is bounded from below by \|\|x0 −x∞\|\| > C( ¯K, T)I2 0 + ϵ (28) Then, we can see that the gap between xT and ˆxT increases such bound, which is \|\|x0 −ˆxT \|\| ≥\|\|x0 −x∞−(ˆxT −x∞)\|\| > C( ¯K, T)I2 0 + ϵ(T, K, µ, λ) where T is the time step for the forward process, and ϵ is some strictly positive constant. Now suppose that ˆxt is the solution to the reverse process, which runs for T periods in total, then we have that for t ∈[0, T], denote x0, ˆx0 as the original HQ image and the final state of the reverse process, respectively. xQuality −OU xQuality, µ; t0, θ 2 2 = ∥x0 −ˆx0∥= ∥x0 −ˆxT −(ˆx0 −ˆxT )∥≥ ∥∥x0 −ˆxT ∥−∥(ˆx0 −xT )∥∥> ϵ + C( ¯K, T)I2 0 −C( ¯K, T)I2 0 = ϵ (29) Therefore, the bias created in inference depends on the LQ image µ, stationary variance, Lipschitz constant, and the time steps T. 20 G Stable in Probability Definition 7 Stable in Probability Given a probability space (Ω, F, P) and a standard Brownian motion Wt, a general form of SDE for a stochastic process xt, 0 ≤t ≤T is given by dxt = b(t, x)dt + σ(t, x)dWt (30) Such that the Lipschitz condition is satisfied, for both b(t, x), σ(t, x), A solution x(t, ω) ≡0 is said to be stable in probability for t ≥0 if for any s ≥0 and ε > 0 lim x0→0 P sup t>s \|xs,x(t)\| > ε = 0. (31) It says that the sample path of the process issuing from a point x at time s will always remain within any prescribed neighbourhood of the origin with probability tending to one as x →0. In practice, this property ensures that the perturbation from the initial state caused by a stable process is bounded for all t with probability one. For example, the OU process 4 admits a unique unconditional stationary solution provided by theorem 19, however, in this example, without specifying the value that determines the stationary variance, i.e., σ, θ. If for large σ and a sample ˆx from the stationary distribution of the OU process, we have that P(ˆx > µ ± σ 2θ) > 0 (32) This means that a sample from the stationary distribution of the forward process could deviate largely from µ, thus making the result no different from the traditional VE(variance exploding) diffusion models that are defined as dxt = σtdWt because the variance could be set arbitrarily large if no restriction is specified. Therefore, how should such a problem be approached most easily? The Lyapunov theorem for stability has provided an easy way, without explicitly solving the SDEs, to ensure stability just from the coefficients. Then we provide the main theorem that ensures the stability of SDE, first, we give the definition of positive definite in the Lyapunov sense Definition 8 Let K denote the family of all continuous nondecreasing functions µ : R+ →R+such that µ(0) = 0 and µ(r) > 0 if r > 0. For h > 0, let Sh = {x ∈Rn : \|x\| < h}. A continuous function V (x, t) defined on Sh × [t0, ∞) is said to be positive-definite (in the sense of Lyapunov) if V (0, t) ≡0 and, for some µ ∈K, V (x, t) ≥µ(\|x\|) for all (x, t) ∈Sh × [t0, ∞) Then we will use the convention of the Lyapunov quadratic function, such that Definition 9 Lyapunov quadratic function V is given V (xt) = xT t Qxt, where Q is a symmetric positive-definite matrix. Theorem 3 The function LV LV (xt) =xT t Qb (t, xt) + b (t, xt)T Qxt+ σ (t, xt)T Qσ (t, xt) , (33) is negative-definite in some neighbourhood of xt = 0 for t ≥t0, with respect to system 7. Then the trivial solution of equation 7 is stochastically asymptotically stable. 21 Since this theorem is important and the proof will be intuitive in explaining why such a condition could ensure stability, the proof will be put here. Proof: First, we compute dV (x), which is the instantaneous growth of the Lyapunov quadratic function, gives dV (xt) =V (xt + dxt) −V (xt) = xT t + dxT t Q (xt + dxt) −xT t Qxt =xT t Qb (t, xt) dt + xT t Qσ (t, xt) dBt+ b (t, xt)T dtQxt + σ (t, xt)T dBtQxt+ σ (t, xt)T Qσ (t, xt) dt Then, take expectation, we can get E {dV (xt)} = xT t Qb (t, xt) dt + b (t, xt)T Qxtdt+ σ (t, xt)T Qσ (t, xt) dt = LV (xt) dt Then, if we assume that LV (x) −LV (xt) ≥kV (xt) such that k is a constant, then d dtE {V (xt)} ≤−kE {V (xt)} , E {V (xt)} ≤exp(−kt). (34) As can be seen from the proof, the operator LV as the function of the SDE xt is the expectation of the dV (xt), and the negative semi-definiteness can be regarded as requiring dV (xt) to be a contraction. This can be understood from 34 such that d dtE(V (xt))/E(V (xt)) < −k for k > 0 H Implementation details Model Implementation: Our exploration into mitigating Temporal Distribution Discrepancy in diffusion models employs two neural network architectures, each catering to different dataset complexities. We utilize the adopted UNet, a staple in DDPM [24] and DDIM[53] frameworks, chosen for its widespread use and strong benchmarking capabilities. By solving the discrepancy through this established structure, we achieve state-of-the-art results on synthetic data, showcasing the potential of improving transanary SDE diffusion models in terms of Temporal Distribution Discrepancy and stationary process. Additionally, the use of this prevalent architecture allows for comprehensive analysis and discussion. Real-world data with its inherent complexity, such as combined degradations, large resolutions, and extensive interdependencies, requires an architecture beyond the conventional UNet. Our improved model incorporates Squeeze-and-Excitation [38] and NAF [8], explicitly designed to capture intricate feature interrelations. While these models do not seek to innovate the architectural paradigm, it provides a solid baseline that provides better feature extraction ability than UNet performance in demanding scenarios. Additional details: The U-Net we adopted is similar to DDPM as described in [37, 11, 9], where the improved model incorporates Squeeze-and-Excitation [38] to replace the Attention module within NAF [8]. EDSR [33] is employed as the base model for TTA-based comparison methods in MRI super-resolution. In different downstream tasks, we follow the common setting of the latest compared methods: Deraining [37], Real Dehazing [47], MRI Reconstruction [26], and MRI super-resolution [14] and [33]. Below are more specific details about MRI reconstruction and MRI 22 Table 9: Datasets parameters and split setting. IOP details of the scan parameters are not available. Datasets Data setting Collection Sequence parameters Source domain Subjects Slices Hospitals Scanner Repetition time Echo train length Matrix size Receiver coil HH (IXI Brain) 184 60 Hammersmith Philips 3T 5725 16 192 x 187 Single Target domain Guys (IXI Brain) 30 60 Guy’s Hospital Philips 1.5T 8178 16 Unnormalized Single IOP (IXI Brain) 30 60 Institute of Psychiatry GE 1.5T Unknown Single Method PSNR↑ SSIM↑ LPIPS↓ WD UNet 27.31 0.8322 0.227 SN UNet 28.63 0.8687 0.144 UNet 31.03 0.9001 0.058 Table 10: Evaluation of the Lipschitz continuity. superresolution. For most of the experiments, the training patch-size is set to 128x128 with a batch size of 16. We utilize the Adam optimizer with β1 = 0.9, β2 = 0.999, a learning rate of 10−4 with a decay strategy. Our models are trained on three RTX 6000 GPUs for about four days, each with 40GB of memory. Random seed is 42. All mathematical variants follow the cosine schedule as per [42]. The variance λ is set to 10 for the OU and stationary processes. In the coef. decoupled SDE, σ is kept decoupled and does not vary with θ. We observed that the adaptability of τ to tasks is contingent upon the task’s corruption for model stability and generalizability. τ is a hyperparameter as one of the ways to introduce Measure-preserving Dynamics to shape more stable SDE diffusion. It is informed by the deviation between the degraded image and the expected high-quality image. For tasks with moderate intra-domain deviations, τ is set to 2, while for tasks with substantial cross-domain and degradation discrepancies, a larger τ is used. This metric is based on RDS settings of different degradations, rather than learned. A MLP can be used to learn τ as adaptive metric, although it is not comparable with obvious improvements. SGM, and Transitionary SGMs vs. D3GM: We perform qualitative 2 and quantitative 11 analyses using variants of closely related formulations for Prop. 1 and 2 and evaluate across (A) SGMs and (B) transitionary SGMs. (A) uses a common score-based SDE, (B) uses a Coefficient Decoupled SDE (e.g., variance exploding SDE with the drift term µ) according to Prop. 1 and OU SDE, alongside our D3GM. MRI Reconstruction: We applied 584 proton-density weighted knee MRI scans without fat suppression. These were subsequently partitioned into a training set (420 scans), a validation set (64 scans), and a testing set (100 scans). For each scan, we extracted 20 coronal 2D singlechannel complex-valued slices, predominantly from the central region with the uniform size of 320 x 320. Our test set differs from the 200 reported in [26], possibly as a result of the change in the official versions of FastMRI. We subjected all experiments to Cartesian under-sampling masks with undersampling factor 8x and 16x. In the undersampled MRI scans, the acquisition process entails sampling a fractional subset of the Fourier-space (k-space), typically governed by a mask along the dimension of undersampling rate. Undersampling inherently induces aliasing artifacts in the resultant images. Considering the domain deviation, we did not supplement the extra test set into the final 100 samples. Otherwise, we stayed consistent with the details of the paper. For a robust comparison, we benchmarked against a diverse set of deep learning-based state-of-the-art reconstruction methods, including CNN-based approaches such as D5C5 [51] and DAGAN [62], the Transformer-based SwinMR [27], diffusion model-inspired DiffuseRecon [44], and the CDiffMR [26]. Quantitative Table 11: Exemplary deraining results of (A) SGM, (B) transitionary SGMs: Coefficent Decoupled SDE, OU SDE, and D3GM. Method PSNR↑ SSIM↑ LPIPS↓ A SGM 27.27 0.840 0.144 Coef. Dec. SDE 26.18 0.826 0.205 B OU SDE 30.58 0.900 0.051 D3GM (ours) 32.41 0.912 0.040 23 results in Tab. 5 exhibit a differing trend from the image domain. The task-specific diffusion models achieve better results and are more capable of capturing the complex degradation that occurs in the frequency domain. Generally, MRI uses a mask in the phase-encoding direction (the shortest anatomical direction [41]) to model the complex degradation caused by undersampling. In the knee data, based upon the uncertainty of clinical diagnosis (longitudinal artifacts can significantly confuse the diagnosis of meniscal injury), we also masked the frequency direction, which will cause resolution reduction, deterioration of image features, and longitudinal artifacts, more qualitative results can be found in the next section. MRI Super-resolution: IXI4 contains clinical T1- and T2-weighted scans from three hospitals with different imaging protocols: HH, Guys, and IOP. We selected 184 HH T2 subjects as the sourcedomain data [train/val/test ratio: 7:1:2], and 30 subjects each from Guys and IOP as two target-domain datasets without degradation, acquisition parameters, datasets, and patient-wise crossovers. The central 60 slices were selected. We consider two benchmark ideas: (1) Blind SR degradation methods in frequency SFM [19] and spatial PDM [70] domain. (2) Source-free SR adaptation ACT [14] using external priors and testtime adaptation proposed initially for MRI CST [14] with data consistency in the source-domain. We are concerned that existing work on robustness and generalization focuses on medical image segmentation and natural images, and rarely on super-resolution and reconstruction of medical images. Employing the available evaluation criteria in this inevitable problem in practice makes it difficult to reflect the performance of D3GM. Therefore, domain-aware datasets isolation standard for public datasets is designed for the SRR adaptation to cross-domain data rather than same-domain in a source-free manner. The publicly available data were split into several subsets based on hospitals, scanner, acquisition parameters, modality, and anatomy as illustrated in Table 9. Explicit reference standards implicitly correspond to various degradation patterns, thus enabling the isolation of natural degradation patterns in source training domain and target-testing domain. In addition to this, different artificial degradation patterns for the subsets in the training and testing domains are employed: K-space truncation downsampling was applied to obtain LR data in the source domain and a kernel degradation [5] was applied in target domain. We reproduce two types of benchmark ideas on the top of EDSR [33] backbone to achieve the multipurpose goal: (1) Repurposed blind SR (BSR) for cross-domain data: We utilized BSR degradation methods in frequency SFM [19] and spatial PDM [70] domain. (2) Test-time adaptation (TTA): We compare to a source-free TTA method ACT [14] using external priors and a second TTA method proposed initially for MR reconstruction CST [14] with cycle consistency at source-domain. The settings and adaptation strategies of the comparison methods were used directly. Lipschitz Continuity for Stationary Process: We hypothesize that ensuring Lipschitz continuity of the neural network is pivotal for the convergence and stability of the diffusion process, particularly in the context of achieving a stationary process. Theoretically, Lipschitz continuity offers two key benefits: (1) it mitigates the impact of small perturbations in input data or model parameters, thus safeguarding against excessive output variability which could cause numerical instabilities or result in an ill-posed problem, and (2) it guarantees the existence of a unique solution to the diffusion process, underpinning the reliability and convergence of the numerical methods employed to solve these equations. To instantiate these theoretical benefits within our architecture, we integrated spectral normalization (SN) [40] and weight decay (WD) [35] into a U-Net structured score network. In the case of SN, it is achieved by rescaling each layer’s weight matrix W (l) by its spectral norm, σmax(W (l)), to obtain a normalized weight ˜W (l) = W (l) σmax(W (l)). By doing so, we intend to control the overall Lipschitz constant of the network for robust score matching within our diffusion model framework. A quantitive result can be seen in Tab. 10. 4http://brain-development.org/ixi-dataset/ 24 𝒙𝟎 𝒙" 𝒚 𝒚’ 𝒙$𝟎 𝓐 𝓐 𝒕−𝟏 →𝒕 𝒕 → 𝒕−𝟏 Basin of attraction Stability Figure 4: Reverse Initialization with Basin of attraction. I Additional Insights Basin of Attraction in Reverse Sampling: Within our diffusion framework, we establish a forward process with variety to accommodate a wide range of potential corruptions, ensuring the desired final distribution close to the expected distribution. We consider that our diffusion models inherently encompass the forward operator A within their structure. The transition from high to low-quality images is implicitly encoded in the diffusion pathway, hence the additional forward operator guidance might not contribute supplementary information, which we initially hypothesized would enhance the reverse process. Thus, the keypoint transfers from the A to the initial y′ (Detailed analysis based on Prop. 2 is provided in Appx. F). Also we found this problem fall into the Basin of Attraction (BA) in dynamical system as shown in Fig. 4. BA can be interpreted as the quality of the attractor of the degraded image in the reverse process here. A BA-guided initialization might be a more effective approach for posterior sampling in transitional SDE diffusion models. By initializing the reverse process closer to the expected solution, we may bypass the initial hurdle of distribution mismatch. J More Qualitative Results 25 GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) Figure 5: Deraining results with light rain images of our method. 26 GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) Figure 6: Deraining results with heavy rain images of our method. 27 GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) Figure 7: Dehazing results with real hazy images of our method. 28 8 X 16 X 8 X 16 X LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑃ℎ𝑎𝑠𝑒−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 Figure 8: MRI reconstruction results with undersampling rate x8 and x16, on Frequency-encoding and Phase-encoding directions. 29 HH 4X GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) Figure 9: MRI super-resolution results with in-domain images of our method. 30 GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) Guys 4X IOP 4X Figure 10: MRI super-resolution results with cross-domain (different imaging devices and degradation methods) images of our method. 31 8 X 16 X LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑃ℎ𝑎𝑠𝑒−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) GT LQ 𝑫𝟑𝑮𝑴 (𝐨𝐮𝐫𝐬) 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑃ℎ𝑎𝑠𝑒−𝑒𝑛𝑐𝑜𝑑𝑖𝑛𝑔 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 Figure 11: MRI reconstruction results with undersampling rate x8 and x16 of our method, on Phaseencoding and Frequency-encoding directions. 32 NeurIPS Paper Checklist 1. 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4,464	Du-IN: Discrete units-guided mask modeling for decoding speech from Intracranial Neural signals Hui Zheng,2,4, Hai-Teng Wang,1, Wei-Bang Jiang3, Zhong-Tao Chen1, Li He1, Pei-Yang Lin1, Peng-Hu Wei5, Guo-Guang Zhao5, Yun-Zhe Liu†,1,4 1Beijing Normal University, 2Peking University, 3Shanghai Jiao Tong University, 4Chinese Institute for Brain Research, 5Capital Medical University, Xuanwu Hospital, Beijing *Equal contribution,†yunzhe.liu@bnu.edu.cn Abstract Invasive brain-computer interfaces with Electrocorticography (ECoG) have shown promise for high-performance speech decoding in medical applications, but less damaging methods like intracranial stereo-electroencephalography (sEEG) remain underexplored. With rapid advances in representation learning, leveraging abundant recordings to enhance speech decoding is increasingly attractive. However, popular methods often pre-train temporal models based on brain-level tokens, overlooking that brain activities in different regions are highly desynchronized during tasks. Alternatively, they pre-train spatial-temporal models based on channel-level tokens but fail to evaluate them on challenging tasks like speech decoding, which requires intricate processing in specific language-related areas. To address this issue, we collected a well-annotated Chinese word-reading sEEG dataset targeting languagerelated brain networks from 12 subjects. Using this benchmark, we developed the Du-IN1 model, which extracts contextual embeddings based on region-level tokens through discrete codex-guided mask modeling. Our model achieves stateof-the-art performance on the 61-word classification task, surpassing all baselines. Model comparisons and ablation studies reveal that our design choices, including (i) temporal modeling based on region-level tokens by utilizing 1D depthwise convolution to fuse channels in the ventral sensorimotor cortex (vSMC) and superior temporal gyrus (STG) and (ii) self-supervision through discrete codex-guided mask modeling, significantly contribute to this performance. Overall, our approach – inspired by neuroscience findings and capitalizing on region-level representations from specific brain regions – is suitable for invasive brain modeling and represents a promising neuro-inspired AI approach in brain-computer interfaces. Code and dataset are available at https://github.com/liulab-repository/Du-IN. 90 橙汁 0.05 电脑 0.1 苹果 Du-IN 苹果 0.8 音乐 0.01 61-word classification sEEG recordings subj-01 subj-02 subj-03 subj-04 subj-05 subj-08 subj-09 subj-10 subj-11 subj-07 subj-06 subj-12 Figure 1: Overall illustration of sEEG decoding setup and comparison with SOTA baselines. 1Du-IN refers to the phonetic transcription of "讀音" (i.e., pronunciation) in Chinese. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1 Introduction Brain signals refer to the biometric information collected from the brain. Their patterns provide valuable insights toward understanding the physiological functions of the brain and the mechanism of related diseases, leading to various applications, including speech decoding [13, 19, 37], sleep cognition research [35, 55], neurological disorders detection [28, 54], and so on. Due to the high signal-noise ratio, invasive recording methods (e.g., stereoElectroEncephaloGraphy (sEEG), ElectroCorticoGraphy (ECoG)) usually reveal these underlying mechanisms better than non-invasive recording methods. Many previous works [29, 19] have shown that decoding speech from EEG signals is difficult, and the performance is limited. Compared with ECoG, sEEG imposes less trauma on patients and provides more stereotactic information from specific brain regions. Although some studies [37, 36] have recently shown promise for building high-performance speech decoders based on ECoG, there are few attempts made to explore the potential of sEEG-based speech decoding. Modeling intracranial neural signals, especially sEEG, has gained significant attention, but several issues remain unresolved. Current research on modeling neural signals is divided into two lines based on the basic modeling units (e.g., channel-level tokens or group-level tokens2). Some studies [54, 28] utilize shared embedding blocks to embed single channels into channel-level tokens, neglecting the specificity of brain computation [8]; then they adopt spatial-temporal integration to model spatial relationships among them, attempting to regain the precise state of the brain. However, these methods mainly focus on channel-level classification tasks, e.g., seizure detection, yet fail to validate them on more challenging group-level classification tasks, e.g., speech decoding. Other studies [19, 21] fuse all channels (across the brain) to build brain-level tokens, overlooking the brain’s desynchronization nature [7]; then they adopt temporal modeling to capture the rapid process of brain dynamics. Besides, labeling data at scale in medical experiments is often impractical or costly, emphasizing the need to maximize label efficiency. Hence, developing an efficient pre-training framework that draws on prior neuroscience findings is highly appealing, as it can make the most of abundant unlabeled data. The primary challenge in modeling intracranial neural signals lies in extracting meaningful tokens, requiring careful consideration of two key factors. (1) Temporal scale. Since intracranial neural signals have high temporal resolution and signal-noise ratio, these tokens must capture rapid dynamic changes in brain activity. (2) Spatial scale. Considering the brain’s desynchronization nature, these tokens should correctly capture the information of each brain region for further integration and, if needed, decouple different parts of brain dynamics within each brain region. To better assess how well different models capture the intricate processing within each brain region, we can evaluate these methods on tasks mainly involving a few brain regions. Since speech mainly involves specific brain regions related to vocal production, as demonstrated in Section 2.1, we utilize speech decoding tasks to evaluate which model can effectively extract information from specific brain regions. Since there are too few open source sEEG language datasets [1, 49], we collected a well-annotated Chinese word-reading sEEG dataset (vocal production), including 12 subjects, which makes up for the problem of missing sEEG recordings in language tasks. Inspired by neuroscientific findings, we systematically demonstrate the locality and specificity of brain computation and propose the Du-IN model to solve the abovementioned issues. Compared to other existing methods for modeling brain signals, Du-IN achieves SOTA performance on the 61-word classification task, demonstrating the effectiveness of our model in extracting meaningful tokens that can capture both the rapid changes and the precise state of specific brain regions. It marks a promising neuro-inspired AI approach [42, 41] in BCI. To sum up, the main contributions of our work comprise: 1. A well-annotated Chinese word-reading sEEG dataset, addressing the lack of sEEG language dataset. The dataset will be publicly available. 2. Demonstration of brain-specific computation – achieving the best decoding performance only requires about one electrode in specific brain regions (i.e., vSMC, STG). 3. A novel framework for sEEG speech decoding – Du-IN, which learns region-level contextual embeddings through discrete codex-guided mask modeling. 4. SOTA performance on the sEEG speech decoding task – Du-IN achieves 62.70% top-1 accuracy on the 61-word classification task, surpassing all other baselines. 2The term "group-level" includes "brain-level" and "region-level," and is distinct from "channel-level." 2 2 Related Works 2.1 Neural Basis of Language Function Neuroscientific research [5, 17, 43] in the past has extensively explored brain regions supporting language functionality. In neuroscience, the investigation into language functionality related to speech has been categorized into two main streams: one dedicated to semantic processing and the other to vocal production. Previous studies [4, 43] have shown that brain regions associated with semantic processing primarily include left inferior frontal gyrus (IFG), left anterior temporal lobe (ATL), and bilateral middle temporal gyrus (MTG). As for vocal production, which is also the focus of our work, it is predominantly governed by motor information related to language articulation, primarily involving ventral sensorimotor cortex (vSMC), bilateral superior temporal gyrus (STG), and bilateral dorsal laryngeal motor cortex (dLMC) [5, 17, 9]. Our analysis results based on our collected word-reading sEEG dataset also confirm this point, as illustrated in Figure 4. 2.2 Language Decoding in BCI The keys to decoding natural language from brain signals are (1) high-quality recordings, and (2) well-designed models with good representations. Compared to non-invasive recordings (e.g., EEG), invasive recordings manifest advantages in providing detailed information about specific brain regions with a high signal-noise ratio. Since speech mainly involves some specific brain regions, obtaining detailed recordings of these brain regions will significantly enhance the decoding performance. Existing works [13, 37, 21] have shown the great potential of building a high-performance decoder based on invasive recordings. The other key is well-designed models with good representations. Existing work for brain-to-language representations can be classified into two categories: self-supervision or alignment with representation models pre-trained on other modalities (e.g., text, audio). BrainBERT [49] learns general embeddings through self-supervised mask modeling. DeWave [19] introduces discrete codex encoding and aligns neural representations with text embeddings from BART [32], thus enhancing the extraction of semantic processing-related information from EEG recordings. Metzger et al. [36] align neural representations with acoustic embeddings to improve the extraction of vocal production-related information from ECoG recordings. 2.3 Self-supervised Learning in BCI In recent years, self-supervised pre-training has made significant progress in natural language processing [16, 39, 6] and computer vision [3, 25, 11]. However, its potential in BCI is far from being explored. BrainBERT (for sEEG) [49] embeds single channels into channel-level tokens and utilizes mask modeling to learn general representations. Brant (for sEEG) [54, 53], PopT (for sEEG) [10] and some works (for EEG) [28, 23] further adopt spatial-temporal integration to model spatial relationships among them. Some works (for EEG) [31, 20, 50, 22] take the other way – fusing all channels (across the whole brain) to build brain-level tokens, and it uses self-supervised learning to learn contextual representations. Considering the difference among brain regions, MMM (for EEG) [52] further splits channels into different groups to build region-level tokens. All existing pre-training methods for sEEG primarily pre-train spatial-temporal models based on channel-level tokens yet only evaluate them on channel-level classification tasks, e.g., seizure detection. However, unlike EEG pre-training methods, their effectiveness over more challenging group-level classification tasks, e.g., speech decoding. Besides, there is no standard channel configuration for sEEG recordings, unlike EEG recordings, which makes modeling spatial relationships in sEEG more challenging. 3 Method The overall architecture of Du-IN is illustrated in Figure 2, where the raw sEEG signals are fused across channels to build region-level tokens and further encoded for downstream tasks. 3 3.1 Task Definition Due to the lack of open-source sEEG datasets related to language tasks, we follow the experimental design outlined by Moses et al. [37] to collect a well-annotated Chinese word-reading sEEG dataset (vocal production). During the experiment, each subject speaks aloud 61 pre-determined Chinese words 50 times; see Appendix A for more details. We formulate the multi-channel sEEG signals as X ∈RC×T , where C is the number of sEEG channels and T is the total timestamps. The associated word label is denoted as y ∈Y, where Y represents the set of 61 pre-determined words. In summary, this dataset comprises paired sEEG-word data (⟨X, y⟩), and the model aims to decode the corresponding word y from a sequence of raw sEEG signals X. 3.2 Model Architecture Linear Conv Batch Norm X3 Spatial Encoder Temporal Embedding K Add & Norm FeedForward Add & Norm Attention LN V Q LN Transformer Encoder Output Embedding time channel patch Input sEEG Signals TI'4 TI'5 TI'6 F'9 ei eti epi Figure 2: The overall architecture of Du-IN Encoder. Du-IN Encoder is used as an encoder in all Du-IN models (i.e., Du-IN VQ-VAE, Du-IN MAE, Du-IN CLS (classification)), see Appendix C for more details. We introduce the Du-IN Encoder, a general architecture for sEEG speech decoding tasks that can deal with any input sEEG signals with arbitrary time length, as shown in Figure 2. The key operation for archiving this is segmenting the sEEG signals into patches, inspired by patch embeddings in images [18]. For each sample X, we use a W-length window without overlap to segment it into patches, obtaining X = {xi ∈RC×W \|i = 1, ..., N}, where N = ⌊T W ⌋is the number of patches. Spatial Encoder. As each sEEG patch has multiple channels, it is vital to fuse different channels to extract meaningful features before patch-wise interaction by self-attention. We employ a spatial encoder, which consists of a linear projection and several convolution blocks, to encode each sEEG patch into a patch embedding. The linear projection transforms the raw sEEG signals into the hidden neural space, and its weights are utilized for subsequent analysis. The convolution block is composed of a 1D depthwise convolution layer and a batch normalization layer [27]. We denote the output patch embeddings from the spatial encoder as Ep = {ep i ∈Rd\|i = 1, ..., N}, (1) where d is the dimension of the embeddings. Temporal Embedding. In order to enable the model to be aware of the temporal information of patch embeddings, we utilize the parameter-free position embeddings introduced in [48], i.e., Et = {et 1, ..., et tmax}. Note that tmax is the hyperparameter determining the maximum number of time patches and tmax ≥N. Given one arbitrary patch embedding ei in Equation 1 from the spatial encoder, we add the corresponding temporal embedding to it: Einit = {ep i + et i\|i = 1, ..., N}, (2) which forms the input embeddings Einit for the Transformer Encoder. Transformer Encoder. Finally, the sequence of embeddings will be directly fed into the Transformer encoder [48] to get the final encoded E = {ei ∈Rd\|i = 1, ..., N}. To make the training of the Transformer more stable and efficient, we incorporate some modifications [14] inspired by LaBraM [28]. We add layer normalization to the queries and keys before the dot-product attention mechanism, which avoids over-large values in attention logits: Attention(Q, K, V ) = softmax(LN(Q)LN(K)T √dhead )V, (3) where dhead is the dimension of attention head and LN denotes layer normalization [2]. For downstream classification tasks, we flatten the output embeddings followed by a classification head. 4 3.3 Du-IN VQ-VAE Training Prior to pre-training Du-IN through mask modeling, we need to tokenize the sEEG patches into discrete tokens. We introduce vector-quantized neural signal regression, which is trained by reconstructing the original sEEG signals, as shown in Figure 3. The key components are the Du-IN Encoder, which encodes the raw sEEG samples into embeddings, and the Du-IN Regressor, which reconstructs the original sEEG signals. The idea is basically inspired by VQ-VAE [47], which encodes images into discrete latent embeddings. Input sEEG Signals TI'4 TI'5 TI'6 F'9 Du-IN Encoder Neural Codex Lookup Replace Du-IN Regressor Reconstructed sEEG Signals TI'4 TI'5 TI'6 F'9 Du-IN VQ-VAE Training Mask TI'4 TI'5 TI'6 F'9 Symmetric Mask TI'4 TI'5 TI'6 F'9 Du-IN MAE Training Du-IN Encoder 729 69 42 660 6 534 Token Prediction Head 69 660 6 729 42 534 M M M M M M (a) (b) l2-norm l2-norm ei cj zi zq(ei) Figure 3: Overview of Du-IN VQ-VAE training and Du-IN MAE training. (a). We train the Du-IN Encoder in the Du-IN VQ-VAE to discretize sEEG signals into discrete neural tokens by reconstructing the original sEEG signals. (b). During the training of Du-IN MAE, part of sEEG patches are masked while the objective is to predict masked tokens from visible patches. Du-IN Encoder. We define a neural codex C = {cj\|j = 1, ..., Ncodex} ∈RNcodex×dcodex, where Ncodex is the number of the discrete neural embeddings and dcodex is the dimension of each embedding. Given a sEEG sample X, the Du-IN Encoder, illustrated in Figure 2, first encodes it to embeddings E = {ei ∈Rd\|i = 1, ..., N}. After that, we utilize a linear projection zc to get the mapped embeddings zc(E) = {zc(ei) ∈Rdcodex\|i = 1, ..., N} in the codex space. Then, the codex looks up the nearest neighbor of each embedding zc(ei) in the neural codex C. This procedure can be formulated as zq(E) = {zq(ei)\|i = 1, ..., N}, zq(ei) = czi, zi = arg min j \|\|ℓ2(zc(ei)) −ℓ2(cj)\|\|2, (4) where ℓ2 represents ℓ2 normalization and zq(ei) is the quantized vector after the quantizer. This is equivalent to finding the closest neural embedding by cosine similarity and such ℓ2 normalization improves the codex utilization [38]. Du-IN Regressor. The Du-IN Regressor consists of a Transformer decoder and a stack of transposed convolution layers. Given a sequence of the vector-quantized embeddings Z = {zi\|i = 1, ..., N}, the Du-IN Regressor convert these discrete embeddings back into raw sEEG signals ˜ X = {˜xi\|i = 1, ..., N}. The mean squared error (MSE) loss is utilized to guide the regression. The total loss for training the Du-IN VQ-VAE is defined as: Lvqvae = N X i=1 h \|\|˜xi −xi\|\|2 2 + \|\|sg[zc(ei)] −zq(ei)\|\|2 2 + β\|\|zc(ei) −sg[zq(ei)]\|\|2 2 i , (5) 5 where sg represents the stop-gradient operation, which is an identity at the forward pass and has zero gradients. To stabilize the codex update, we use the exponential moving average strategy [47]. 3.4 Pre-training Du-IN Masked sEEG Modeling. To enforce Du-IN learning contextual representations, we propose masked sEEG modeling. The whole procedure is presented in Figure 3. As illustrated in Figure 2, given a sEEG sample X, the spatial encoder first transforms it to patch embeddings Ep = {ep i \|i = 1, ..., N}. Given these patch embeddings Ep, around 50% of patch embeddings are patch-wisely chosen and masked. The masked position is termed as M. Then, a shared learnable embedding e[M] ∈Rd is used to replace the original patch embeddings: Em = {em i \|i = 1, ..., N}, em i = mi ⊙e[M] + (1 −mi) ⊙ep i , (6) where δ(·) is the indicator function and mi = δ(i ∈M). After that, the masked embeddings Em will be added by temporal embeddings, and then fed into the Transformer encoder. The output embeddings E will be used to predict the indices of the corresponding codes from the codex in the Du-IN VQ-VAE through a linear classifier: p(zi\|ei) = softmax(Linear(ei)), (7) The training loss of mask modeling is defined as: LM = − X i∈M mi ⊙log p(zi\|ei). (8) Symmetric Masking. Inspired by LaBraM [28], we further introduce a symmetric masking strategy to improve training efficiency. We calculate the inverse of the generated mask M, obtaining ˆ M. Similarly, we use the new mask ˆ M to perform the mask modeling, obtaining the mask modeling loss Lsym M . The total loss for pre-training the Du-IN model (i.e., Du-IN MAE model) is defined as: Lmae = LM + Lsym M . (9) 4 Experiments 4.1 Dataset Due to the lack of open-source sEEG datasets related to language tasks, we follow the experimental design outlined by Moses et al. [37] to collect a well-annotated Chinese word-reading sEEG dataset (vocal production), including 12 subjects. The subjects undergo a surgical procedure to implant 7 to 13 invasive sEEG electrodes, each with 72 to 158 channels, in their brain. For each subject, the dataset contains 15 hours of 2000Hz recordings, 3 hours of which are task recordings. Pre-training dataset. For each subject, the pre-training dataset contains all sEEG recordings (with about 54 million timestamps) of that subject. To stabilize computing resource usage, the time length of sEEG sample X is set to 4 seconds. Downstream dataset. For each subject, 3 hours of the sEEG recordings are task recordings. The sEEG signals are segmented into about 3000 3-second samples, each of which is paired with the corresponding word label (from 61 pre-determined words). 4.2 Implementation Details Preprocess. We first filter the sEEG signals between 0.5Hz and 200Hz to remove low-frequency noise. Then, a notch filter of 50Hz is applied to avoid power-line interference. After that, all sEEG signals are resampled to 1000Hz and bi-polar re-referenced [33]. Finally, we perform z-score normalization on each channel to guarantee normalized data scales across all channels. 6 Model Configurations. The length of the sEEG patch is 100ms, resulting in 40 patches per sample in the pre-training dataset and 30 patches per sample in the downstream dataset. The "Spatial Encoder" contains one linear projection and three 1D convolution layers, transforming the original sEEG patches into patch embeddings with d = 160. The following "Transformer Encoder" contains an 8-layer Transformer encoder with model dimension d = 160, inner dimension (FFN) dff = 320, and 8 attention heads. See Appendix C for more details. Pre-training. During the pre-training, we use either all sEEG recordings (15 hours) or the sEEG recordings without task recordings (12 hours) to train the Du-IN VQ-VAE and Du-IN MAE models. To enhance the robustness of the learned codex and representations, we further use data augmentation described in Appendix D. For each subject, the model is pre-trained on a Linux system with 2 CPUs (Intel Xeon Gold 6230 40-Core Processor) and 1 GPU (NVIDIA Tesla V100 32GB) for ∼1.2 days. Fine-tuning. During the downstream evaluation, we split the task recordings into training, validation, and testing splits with a size roughly proportional to 80%, 10%, and 10%. All experiments are conducted on the same machine with the same set of random seeds. The train/validation/test splits are the same across different models. We also use data augmentation, as described in Appendix D, to make the most of the gathered dataset. We employ cross-entropy loss (multi-class classification) as the training loss. Our experiments are conducted on one V100 GPU by Python 3.11.7 and PyTorch 2.1.2 + CUDA 12.3. The best models are trained based on the training set, selected from the validation set according to accuracy, and finally evaluated on the test set. For model comparison, we report the average and standard error values (of all subjects) on six different random seeds to obtain comparable results. For the results of the subject-wise evaluation, we report the average and standard deviation values (of each subject) in Appendix K. 4.3 Channel Contribution and Selection As demonstrated in Section 2.1, previous neuroscience studies reveal that vocal production predominantly engages specific brain regions. Given the sparse distribution of implanted sEEG electrodes (each containing 8-16 channels), it’s vital to exclude redundant electrodes unrelated to vocal production, thus improving decoding performance. We retain electrodes implanted in relevant brain regions and evaluate the performance based on the remaining electrodes. Table 1 demonstrates that excluding approximately 85% electrodes even leads to a dramatic increase in decoding performance. Table 1: The performance of Du-IN with or without electrode selection. Methods # of Channels (Averaged) Accuracy (%) ± Ste (%) Du-IN (w/o electrode selection) 109.75 30.12±5.64 Du-IN (w/ electrode selection) 12.25 55.92±4.96 To further understand the detailed contribution of each channel, we analyze the weights of linear projection in the spatial encoder. In detail, we calculate the contribution scores of channels per subject and organize them accordingly, as described in Appendix H. Figure 4 demonstrates that (1) the brain regions effective for speech decoding align with findings from previous neuroscience research, and (2) our model achieves optimal decoding performance with approximately 10 channels, 80% of which originate from the same electrode. To streamline, we utilize these top 10 channels (selected according to train-set) for both pre-training and downstream evaluation. 4.4 Comparasion with Other Models Table 2 presents the results of our Du-IN model and the advanced baselines that are designed for either brain signals or general time series. See Appendix B and Appendix C.3 for detailed descriptions of models. The results demonstrate that our Du-IN model outperforms all baselines. It’s worth noting that the models (i.e., the foundation models designed for brain signals) that adopt spatial-temporal integration to model spatial relationships among channel-level tokens perform worse than the models that adopt temporal modeling based on region-level tokens, challenging the generalizability of current strategies to model spatial relationships among channels with Transformer. 7 right left (a) 0.75 1 0.5 0.25 0 (b) Figure 4: The channel contribution analysis. (a). The channel contribution map. (b). The effect of the number of channels (sorted according to channel contribution scores) on decoding performance. Table 2: The performance of different methods (with the best in bold and the second underlined). Methods Token Level Config Model Size Accuracy (%) ± Ste (%) PT1 MS2 TS-TCC[20] Region ! % 0.32M 24.85±4.42 CNN-BiGRU[37] Region % 0.54M 32.04±5.45 EEG-Conformer[44] Region % 2.34M 45.82±4.66 Neuro-BERT[50] Region ! % 2.14M 49.51±4.43 DeWave[19] Region % 5.70M 32.43±4.48 BrainBERT[49] Channel ! % 43.58M 6.72±1.59 BrainBERT[49] Channel ! ! 43.58M 7.50±1.76 Brant[54] Channel ! % 69.35M 11.16±3.56 Brant[54] Channel ! ! 69.35M 12.42±4.10 LaBraM[28] Channel ! % 6.85M 11.53±2.63 LaBraM-PopT[28, 10] Channel ! ! 6.85M 11.78±2.70 Du-IN Region % 4.38M 56.29±5.20 Du-IN (vqvae+vq) Region ! % 4.38M 44.17±4.04 Du-IN (vqvae) Region ! % 4.38M 58.24±4.83 Du-IN (mae) Region ! % 4.38M 62.70±4.69 Du-IN (poms) Region ! ! 5.18M 59.18±4.63 1 PT: Whether the model is pre-trained before evaluation. 2 MS: Whether the model is pre-trained across multiple subjects. As BrainBERT [49] doesn’t consider the spatial relationships among channels, we mainly focus on understanding why Brant [54], LaBraM [28] and LaBraM-Popt [28, 10] fail to effectively capture the discriminative features on the speech decoding task. These models typically build channel-level tokens by segmenting non-overlapping patches with large receptive fields (e.g., 1 second) from single channels. However, this approach makes it challenging to capture the rapid process of brain dynamics. Moreover, while these models further utilize Transformer to capture the spatial relationships among these tokens, they do not encourage region-level embeddings, either through their architecture [52] or their pre-training objective [10]. Therefore, the effectiveness of building brain foundation models based on these spatial-temporal backbones is still under exploration, especially for cognitive tasks (e.g., speech decoding), which are of great value in the field of neuroscience. Besides, unlike LaBraM [28], Brant doesn’t introduce spatial embeddings to identify the spatial location of each channel. Since the electrodes are sparsely distributed in the brain and the raw sEEG signals on the same electrode are highly correlated, it’s fairly easy to identify their spatial relationships through their values. As demonstrated in iTransformer [34], this modeling approach is well suited for detecting time-delay events, e.g., seizure detection. For speech decoding tasks, 8 sEEG often requires bi-polar re-reference (or Laplacian re-reference) to remove the high correlations among channels, thus avoiding model overfitting [49]. Once the correlations among channels have been removed, Brant will lose the ability to model spatial relationships among channels. For other baselines that use temporal modeling based on region-level tokens, we provide a detailed explanation of their performance differences as follows. TS-TCC [20] tokenizes raw sEEG signals into region-level tokens with a stack of 1D depthwise convolution blocks, but it lacks a temporal Transformer for further integration over time. CNN-BiGRU [37] introduces a stack of GRU layers on top of these tokens to perform temporal integration. EEG-Conformer [44] introduces a temporal Transformer to better integrate global temporal information, which makes it outperform CNNBiGRU. However, EEG-Conformer tokenizes raw sEEG signals with the temporal-spatial convolution, applying the same convolutional kernel across different channels, which overlooks the specificity of brain computation [8]. This also raises a challenge for the effectiveness of current sEEG foundation models, which rely on shared convolution blocks across individual channels. Neuro-BERT [50] further introduces mask modeling to learn contextual embeddings, which makes it outperform EEGConformer. DeWave [19] utilizes the Conformer model [24] for tokenization, which involves more parameters but is less effective than 1D depthwise convolution. 4.5 Ablation Study Self-Supervision Initialization. As illustrated in Figure 3, the Du-IN model entails a two-stage pre-training process, wherein both the Du-IN VQ-VAE model and the Du-IN MAE model are trained. Previous studies utilize different strategies [19, 12, 28] to leverage these pre-trained models to enhance the performance of downstream tasks. Here, we evaluate these different strategies for comparison; see Appendix C.3 for detailed definitions. Table 2 shows that initializing weights from the Du-IN MAE model captures contextual embeddings effectively, resulting in the highest decoding performance. Pre-training with/without Downstream Datasets. During the pre-training stage, we hope that the Du-IN VQ-VAE model can extract general tokens of that brain region, thus guiding the Du-IN MAE model to learn general representations that are not specific to any particular task. Although no label data is used during the pre-training stage, to eliminate the influence of the pre-training data on downstream tasks, we compare the results with or without incorporating the downstream task dataset into the pre-training stage. Table 3 shows a slight performance drop when excluding downstream datasets. However, the decoding performance is still higher than the baseline performance without pre-training, which means that the degradation is mainly due to the decrease of the pre-training dataset. We hope that, with more pure recordings, our model can achieve better decoding performance. Table 3: Ablation study on whether pre-training with the downstream dataset (DD) or not. Methods Pre-training Dataset Size Accuracy (%) ± Ste (%) Du-IN (mae w/o DD) 12 hours per subject 60.02±4.34 Du-IN (mae w/ DD) 15 hours per subject 62.70±4.69 Discrete Codex. During the Du-IN VQ-VAE training stage, the Du-IN VQ-VAE model encodes sEEG patches into discrete codes and then reconstructs the original signal from these codes. We evaluate performance against varying codex sizes (512 to 8192) to ascertain if codex size affects the quality of the learned codex. As illustrated in Figure 5, while extremely small codex size lacks representation diversity, extremely large codex size often leads to codex collapse. We suspect that our existing training data might not be adequate for larger codex sizes. Furthermore, our experiments suggest that the model performs optimally when the codex dimension, denoted as dcodex = 64, is slightly less than the model dimension, d = 160, yielding a more effective regularization effect. Perception Time Window. We also conduct the ablation study on the model structure for the spatial encoder described in Section 3.2. As the spatial encoder transforms the sEEG signals within a given patch to a patch embedding, it compresses the sEEG signals for perception. As described in Section 4.2, the model utilizes a receptive field of 100ms. We conduct an ablation study of different receptive fields and report it in Figure 5. The model performance notably drops with a receptive field smaller than 60ms and gradually declines as the receptive field exceeds 160ms. The model reaches a 9 small peak around 100ms to 140ms. We think this phenomenon is rational since sEEG is known for its ability to capture the rapid dynamics of specific brain regions precisely. (b) (a) (c) Figure 5: Ablation study on different codex sizes, codex dimensions, and receptive fields. 5 Limitations Despite Du-IN’s enhancements in speech decoding via discrete codex-guided mask modeling, it is still restricted to close-set speech decoding tasks (i.e., the word set only includes 61 pre-determined words). However, a parallel to our work, Feng et al. [21], which follows previous works [26, 45], build an acoustic-inspired model that can decode arbitrary Chinese words by predicting syllable components (initials, finals, tones). Although their method requires a large amount of labeled data, their experimental design mirrors ours closely. The difference lies in the requirement for the subject to repeat syllable components, instead of entire words. Therefore, with slight modifications, our model can support open-set speech decoding tasks. Additionally, the experiments in this paper are restricted to the vocal production part of language decoding, i.e., speech decoding. A more interesting but difficult task is to decode language from the semantic level, in which large language models have been wildly used to improve the model performance [46, 19]. However, due to the locality of sEEG recordings, it is still under exploration whether sEEG recordings can fully capture semantic-related information across brain regions. 6 Conclusion This paper proposes Du-IN, a framework for speech decoding, which learns contextual embeddings through discrete codex-guided mask modeling on specific brain regions. To evaluate our model, we collect a well-annotated Chinese word-reading sEEG dataset to address the lack of sEEG language dataset. Inspired by neuroscientific findings, we analyze the effective brain regions for speech decoding and achieve the best decoding performance with about one electrode in specific brain regions, which dovetails with the past neuroscientific research on language. Comprehensive experiments demonstrate that our model outperforms both supervised and sEEG-based self-supervised baselines, effectively capturing the intricate processing within specific brain regions. It marks a promising neuro-inspired AI approach in BCI. In the end, we hope our work can have implications for future developments in sEEG-based self-supervised models with more consideration over how to build the basic representation units so that the model can maximally benefit from the pre-training stage. 7 Broader Impacts Our method advances the feasibility of invasive BCI technology by being the first to demonstrate speech decoding using a single sEEG electrode, which holds significant potential for clinical applications. For patients who have lost their ability to communicate or perform daily tasks due to neurological conditions like locked-in syndrome or amyotrophic lateral sclerosis (ALS), our approach offers a less invasive alternative to technologies like ECoG or microelectrode arrays, thereby reducing the risk of brain damage. 10 Acknowledgements This study was supported by the National Science and Technology Innovation 2030 Major Program (2022ZD0205500), the National Natural Science Foundation of China (32271093), the Beijing Natural Science Foundation (Z230010, L222033), and the Fundamental Research Funds for the Central Universities. We would like to extend our sincere appreciation to Dr. Zhi-Feng Yue for his coordination and support in securing the computing resources essential for this study. Besides, we also sincerely appreciate the LaBram [28] team for their valuable discussion on the visual design of Figure 2 and Figure 3. Ethics Statement Experiments that contribute to this work were approved by IRB. All subjects consent to participate. All electrode locations are exclusively dictated by clinical considerations. Our informed consent signing process is as follows: 1. If the experimental participants are adults and have full civil capacity, we will ask them to sign a written informed consent after the participants have fully informed consent; 2. If the experimental participants are minors or do not have full civil capacity, we will ask the participant’s legal guardian to sign a written informed consent after the participants and their legal guardians have fully informed consent. Our informed consent form includes the following points: 1. Contact information of research institutions and researchers; 2. Research direction and purpose; 3. Risks involved in the research; 4. Personal information, data and usage methods to be used in the research; 5. Privacy protection statement (all personal identification information (PII) will not be disclosed); 6. Data storage statement (retained after deleting all personal identification information (PII)); 7. Voluntary statement of participants; 8. Statement that participants can withdraw unconditionally at any time. Our data storage and protection procedures include the following processes: 1. Our data collection, transfer, and analysis tasks are only completed by researchers who have signed relevant confidentiality agreements; 2. The collected raw data will be copied twice as soon as possible, one copy to a storage computer that is not connected to the Internet and encrypted, and the other copy to a mobile hard disk and encrypted and stored offline; 3. The use of the data is only authorized to the research leader and the main researchers (less than 5 people), among which the main researchers can only access data that does not contain personal identification information (PII); 4. 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Due to the lack of open-source sEEG datasets related to language tasks, we follow the experimental design outlined by Moses et al. [37] to collect a well-annotated Chinese word-reading sEEG dataset, including 12 subjects (9 male, 3 female; aged 15-53, µ 27.8, σ 10.4) with pharmacologically intractable epilepsy. In the word-reading task, the subject speaks aloud individual words from a 61-word set while we simultaneously record his brain activity (measured by sEEG) and voice. The word set is chosen based on the following criteria: • The versatility of the words in generating a range of sentences. • The simplicity of using the words to express fundamental caregiving requirements. • The diversity of word pronunciations to cover as many Chinese pronunciation combinations as possible. A list of the words contained in this 61-word set is provided in Table 4. All data are collected as a series of "blocks" (25 blocks in total), with each block lasting about 10 minutes and consisting of multiple trials. During each block of this task, all words (from the 61-word set) are presented individually twice, leading to a total of 122 trials. Each trial in a block of this task starts with one word shown on the screen in white text. After 0.5 seconds, the text will turn green and remain on the screen for 2 seconds. This color transition from white to green represents the go cue for each trial, and the subject is instructed to speak the word aloud as soon as the text turns green. Afterward, the text will be replaced with a blank screen with a centered cross. After 0.5 seconds, the task continues to the next trial. The word presentation order is randomized within each task block. Besides, we also collected non-task recordings of subjects in their daily life. Apart from sleep periods, there are roughly 12 hours of non-task recordings during wakefulness. In summary, for each subject, we collect about 15 hours of sEEG recordings, of which 3 hours are task recordings. 16 Table 4: The Chinese words and their corresponding English translations in the 61-word set. Words Translations Words Translations Words Translations 嘴巴 mouth 菠萝 pineapple 帮助 help 把 get 朋友 friend 脸盆 washbasin 平静 calm 漂亮 pretty 衣服 clothes 豆腐 tofu 米饭 rice 放在 put on 面条 noodle 毛巾 towel 关门 close the door 电脑 computer 凳子 stool 小刀 knife 头疼 headache 软糖 gummies 醋 vinegar 青菜 vegetables 厕所 toilet 葱花 chopped green onion 手机 cell phone 篮球 basketball 钢琴 piano 心情 mood 丝瓜 loofah 蒜泥 garlic paste 怎样 how 香肠 sausage 需要 need 你 you 拿 hold 橙汁 orange juice 找 look for 猪肉 pork 吃 eat 穿 wear 是 be 家人 family 热水 hot water 护士 nurse 换药 change dressing 喝 drink 口渴 thirsty 看 look 碗 bowl 鱼块 steak 感觉 feel 给 give 玩 play 问题 problem 外卖 takeouts 有 have 音乐 music 预约 reserve 汤圆 sweet dumpling 愿意 willing 我 I 17 B Details of Baselines In experiments, we compare our model to the existing supervised or self-supervised methods on brain signals. The details of these baseline models are given here: • TS-TCC[20]: A self-supervised model that consists only of a CNN module to capture local features. This model learns robust temporal and discriminative representations from time series by designing a tough cross-view prediction task and a contextual contrasting module. Since sEEG is a unique type of time series, this model is suitable to serve as a baseline for comparison. • CNN-BiGRU[37]: A supervised model that consists of both CNN module and Bi-GRU module, to capture contextual features from EEG signals. This model is mainly designed for ECoG-based vocal production tasks, similar to ours. Since ECoG and sEEG are both intracranial neural signals of the brain, this model is suitable to serve as a baseline for comparison. • EEG-Conformer[44]: A supervised model that consists of both CNN module and Transformer module, to encapsulate local and global features in a unified EEG classification framework. EEG-Conformer is mainly designed for EEG-based motor imagination tasks. Since the data modes of EEG and sEEG are similar, and the signals primarily pertain to vocal production, this model is suitable to serve as a baseline for comparison. • Neuro-BERT[50]: A self-supervised model that consists of both CNN module and Transformer module, to encapsulate local and global features. This model learns robust contextual representations from EEG by introducing mask modeling. Since the data modes of EEG and sEEG are similar, this model is suitable to serve as a baseline for comparison. • DeWave[19]: A supervised model that consists of both Conformer module [24] and Transformer module, to encapsulate local and global features for language decoding. We adopt its encoder, which consists of a 6-layer Conformer and a 6-layer Transformer. Then, we add a classification head, which is also used in our model, for downstream word classification. Since DeWave is also designed for language decoding, this model is suitable to serve as a baseline for comparison. • BrainBERT[49]: A self-supervised model for sEEG recordings that bridges modern representation learning approaches to neuroscience. BrainBERT builds universal representation based on the superlet spectrograms of one single sEEG channel without modeling the spatial relationships among channels. Since the downstream tasks for BrainBERT are also related to language decoding (e.g., sentence-onset detection, speech vs. non-speech detection, etc.), this model is suitable to serve as a baseline for comparison. • Brant[54]: A self-supervised model for sEEG recordings that can capture both long-term temporal dependency and spatial correlation from neural signals. Brant is mainly designed for medicine, serving as a sEEG foundation model. Although Brant mainly evaluates its performance on the low-level modeling tasks [51] (e.g., neural signal forecasting, imputation, etc.), Brant achieves SOTA performance on some high-level modeling tasks (e.g., seizure detection). As a foundation model in sEEG pre-training field, this model is suitable to serve as a baseline for comparison. • LaBraM[28]: A self-supervised model for EEG recordings that learns generic representations with tremendous EEG data. LaBraM serves as an EEG foundation model, achieving SOTA performance on various downstream EEG tasks. Since the spatial embeddings are pre-defined according to the EEG caps, LaBraM can only be trained within one subject under the sEEG setting. Since the data modes of EEG and sEEG are similar, this model is suitable to serve as a baseline for comparison. • LaBraM+PopT[28, 10]: A self-supervised model based on LaBraM, simply replacing the learnable spatial embeddings with hard-coded spatial embeddings from PopT [10] to enable multi-subject pre-training under the sEEG setting. The detailed implementations of these baseline models are given here: • For the TS-TCC method [20], the hyper-parameters are optimized for better performance, as they also have different hyper-parameter settings for different datasets in their original 18 implementation. The data samples are resampled to 400Hz. The sizes of convolution kernels are changed to {25, 8, 8} (other attempts include {8, 8, 8}, {15, 8, 8}, {20, 8, 8}, and {30, 8, 8}); the sizes of pooling kernels are changed to {10, 2, 2} (other attempts include {2, 2, 2}, {5, 2, 2}, and {20, 2, 2}); the numbers of pooling strides are changed to {10, 2, 2} (other attempts include {2, 2, 2}, {5, 2, 2}, and {20, 2, 2}). All other hyper-parameters are the same as the original implementation. • For the CNN-BiGRU method [37], the hyper-parameters are the same as the original implementation. The data samples are resampled to the specified sampling rate. • For the EEG-Conformer method [44], the hyper-parameters are the same as the original implementation. The data samples are resampled to the specified sampling rate. • For the Neuro-BERT method [50], the hyper-parameters are optimized for better performance, as they also have different hyper-parameter settings for different datasets in their original implementation. The data samples are sampled to 400Hz. The sizes of convolution kernels are changed to {40,} (other attempts include {20,} and {80,}); the numbers of convolution strides are changed to {40,} (other attempts include {20,} and {80,}). • For the DeWave method [19], the hyper-parameters are the same as the original implementation. The data samples are resampled to the specified sampling rate. • For the BrainBERT method [49], the hyper-parameters are optimized for better performance. We change the "nperseg" and "noverlap" arguments of "scipy.signal.stft" function from {400, 350} to {1600, 1400} (other attempts include {200, 175}, {800, 700} and {3200, 2800}). • For the Brant method [54], the hyper-parameters are optimized based on the Brant-Tiny model for better performance. We change the length of the patch segment from 6 seconds to 1 second. Besides, we change the linear embedding layer to the convolution embedding layer, which is also used in LaBraM [28]. The numbers of convolution filters are {96, 96, 96} (other attempts include {192, 192, 192}); the sizes of convolution kernels are {9, 9, 3} (other attempts include {19, 9, 3} and {9, 9, 3}); the numbers of convolution strides are {5, 5, 1} (other attempts include {10, 5, 1}) and {5, 5, 2}). • For the LaBraM method [28], the hyper-parameters are the same as the original implementation of the LaBraM-Base model. The data samples are resampled to the specified sampling rate. • For the LaBraM-PopT method [28, 10], the hyper-parameters are the same as the original implementation of the LaBraM-Base model. The data samples are resampled to the specified sampling rate. When evaluating the decoding performance of these baseline models, we follow the same experiment setup of the Du-IN CLS model: • For one subject, we split the downstream dataset into training, validation, and testing splits with a size roughly proportional to 80%, 10%, and 10%. • The data samples are 3 seconds with the specified sampling rate corresponding to each model. • The samples in the train-set are augmented following the pipeline defined in Appendix D. For the self-supervised methods, the pre-training setup follows the original setup of each model: • For the TS-TCC model, we use all sEGG recordings for each subject to pre-train it. The data samples are 4 seconds. • For the Neuro-BERT model, we use all sEGG recordings for each subject to pre-train it. The data samples are 4 seconds. • For the BrainBERT model, we use around 180 hours of sEEG recordings from either each subject or 12 subjects for pre-training. This pre-training dataset is larger than the one (approximately 45 hours) used in the original paper. The data samples are 4 seconds. • For the Brant model, we also use all sEEG recordings from either each subject or 12 subjects to pre-train it. While the total pre-training dataset is smaller than the one (around 2700 hours) used in the original paper, the number of subjects (i.e., the number of sEEG location configurations) is greater than in the original paper. The data samples are 4 seconds. 19 • For the LaBraM model, we use all sEGG recordings for each subject to pre-train it. The data samples are 4 seconds. • For the LaBraM-PopT model, we use all sEEG recordings from 12 subjects to pre-train it. The data samples are 4 seconds. 20 C Model Details C.1 Du-IN VQ-VAE The architecture of the Du-IN VQ-VAE model contains three parts: (1) Du-IN Encoder, (2) Vector Quantizer, and (3) Du-IN Regressor. The overall architecture of "Du-IN Encoder" is shown in Figure 2. The "Vector Quantizer" is implemented similarly in LaBraM[28]. The "Du-IN Regressor" contains: • Transformer Decoder: A stack of Transformer layers. • Time Regression Head: A stack of 1D Transposed Convolution layers and one linear projection layer. The hyperparameters for Du-IN VQ-VAE training are shown in Table 5. Table 5: The hyperparameters for Du-IN VQ-VAE training. Module Sub-Module Name Value Du-IN Encoder Spatial Encoder Linear Projection 10 →16 # of Input Channels {16,128,128} # of Output Channels {128,128,16} Kernel Size {19,3,3} Stride {10,1,1} Padding {9,1,1} Transformer Encoder # of Transformer Layers 8 Hidden Size 160 MLP Size 320 MLP Dropout Ratio {0.2,0.} # of Attention Heads 8 Attention Head Size 64 Attention Dropout Ratio 0.2 Vector Quantizer Codex Size 2048 × 64 Embedding-to-Codex Projection 160 →160(Tanh) →64 Codex-to-Embedding Projection 64 →160 Du-IN Regressor Transformer Decoder # of Transformer Layers 4 Hidden Size 160 MLP Size 320 MLP Dropout Ratio {0.2,0.} # of Attention Heads 8 Attention Head Size 64 Attention Dropout Ratio 0.2 Time Regression Head # of Input Channels {160,128,128,128,128} # of Output Channels {128,128,128,128,16} Kernel Size {3,3,10,9,19} Stride {1,1,10,1,10} Padding Output Padding Linear Projection 16 →10 Optimizer Batch Size 64 Maximum Learning Rate 3e-4 Minimum Learning Rate 5e-5 Learning Rate Scheduler Cosine Optimizer Type AdamW Adam β (0.9, 0.99) Weight Decay 0.01 Total Epochs 400 Warm-up Epochs 40 21 C.2 Du-IN MAE The architecture of the Du-IN MAE model contains two parts: (1) Du-IN Encoder, and (2) Token Prediction Head. The overall architecture of the "Du-IN Encoder" is shown in Figure 2. The hyperparameters of "Du-IN Encoder" are the same as those in Du-IN VQ-VAE. It’s worth noting that when training Du-IN MAE, the weights of the "Du-IN Encoder" are randomly initialized, instead of loaded from the pre-trained Du-IN VQ-VAE model. The hyperparameters for Du-IN MAE training are shown in Table 6. Table 6: The hyperparameters for Du-IN MAE training. Module Sub-Module Name Value Token Prediction Head Linear Projection 160 →2048 Optimizer Batch Size 64 Maximum Learning Rate 3e-4 Minimum Learning Rate 5e-5 Learning Rate Scheduler Cosine Optimizer Type AdamW Adam β (0.9, 0.99) Weight Decay 0.05 Total Epochs 400 Warm-up Epochs 40 C.3 Du-IN CLS The architecture of the Du-IN CLS model contains two parts: (1) Du-IN Encoder, and (2) Label Prediction Head. The overall architecture of the "Du-IN Encoder" is shown in Figure 2. The hyperparameters of "Du-IN Encoder" are the same as those in Du-IN VQ-VAE. It’s worth noting that the "Du-IN Encoder" weights in Du-IN CLS can be loaded from either the pre-trained Du-IN MAE or the pre-trained Du-IN VQ-VAE. In the ablation experiments shown in Table 2, our models have different suffixes: • Du-IN: The original Du-IN CLS model. All weights of this model are randomly initialized. • Du-IN (vqvae+vq): The weights of the "Du-IN Encoder" in the Du-IN CLS model are loaded from the pre-trained Du-IN VQ-VAE. When fine-tuning it on the downstream task, the "Vector Quantizer" in the pre-trained Du-IN VQ-VAE is inserted between "Du-IN Encoder" and "Label Prediction Head". This is the same operation in DeWave[19]. • Du-IN (vqvae): The weights of the "Du-IN Encoder" in the Du-IN CLS model are loaded from the pre-trained Du-IN VQ-VAE. This is the same operation in EEGFormer [12]. • Du-IN (mae): The weights of the "Du-IN Encoder" in the Du-IN CLS model are loaded from the pre-trained Du-IN MAE. This is the same operation in LaBraM [28]. • Du-IN (poms): The weights of the "Du-IN Encoder" in the Du-IN CLS model are loaded from the Du-IN MAE, which is pre-trained on multiple subjects. The modification of the Du-IN VQ-VAE and the Du-IN MAE to support multi-subject pre-training includes (1) initializing different spatial encoders for different subjects and (2) sharing the same transformer encoder and neural codex. The "Label Prediction Head" is an MLP with one hidden layer, flattens the output embedding sequence from upstream, and maps this feature embedding to the final prediction through MLP. The hyperparameters for Du-IN CLS training are shown in Table 7. 22 Table 7: The hyperparameters for Du-IN CLS training. Module Sub-Module Name Value Label Prediction Head Flatten Linear Projection 30 × 160 →128(ReLU) →61 Optimizer Batch Size 32 Maximum Learning Rate 2e-4 Minimum Learning Rate 5e-6 Learning Rate Scheduler Cosine Optimizer Type AdamW Adam β (0.9, 0.99) Weight Decay 0.05 Total Epochs 200 Warm-up Epochs 20 23 D Data Augmentation To enhance the robustness of learned representations during both the pre-training and fine-tuning stages, we apply data augmentation in both datasets. Pre-training Dataset. In our implementation, we segment sEEG recordings into 8-second samples with a 4-second overlap. When fetching a sample, we randomly select a starting point between 0 and 4 seconds, then extract a 4-second sample beginning from that point. Downstream Dataset. Since a trial lasts for 3 seconds, employing the jittering mentioned above leads to the blending of information from other trials. In our implementation, we segment sEEG recordings into 3-second samples. When fetching a sample, we randomly choose a shift step between 0 and 0.3 seconds, then shift the sample either to the left or right, padding it with zeros. E Du-IN Pre-training Analysis The pre-training of Du-IN can be interpreted as the training of a variational autoencoder [30, 3]. Let x denote the original sEEG signal, ˜x the corrupted sEEG by mask, and z the neural tokens. Considering the evidence lower bound (ELBO) of the log-likelihood p(x\|˜x), i.e., recovering the original sEEG signal from its corrupted version: X (xi,˜xi)∈D log p(xi\|˜xi) ≥ X (xi,˜xi)∈D Ezi∼qϕ(z\|xi)[log pψ(xi\|zi)] \| {z } Neural Token Reconstruction −DKL[qϕ(z\|xi), pθ(z\|˜xi)] , (10) where (1) qϕ(z\|x) denotes the Du-IN Encoder in the Du-IN VQ-VAE that obtains neural tokens; (2) pψ(x\|z) decodes the original sEEG signal given input neural tokens; (3) pθ(z\|˜x) recovers the neural tokens based on the masked sEEG signal, which is our Du-IN pre-training task. The whole framework is optimized through a two-stage procedure as [47, 40]. For the first stage, we train the Du-IN Encoder in the Du-IN VQ-VAE as a discrete variational autoencoder by minimizing the reconstruction loss −Ezi∼qϕ(z\|xi)log pψ(˜xi\|zi) with a uniform prior. For the second stage, we set qϕ as well as pψ fixed and learn the prior pθ by minimizing the loss DKL. For simplicity, qϕ(z\|xi) is defined as a one-point distribution with the most likely neural tokens ˆzi = arg max z qϕ(z\|xi). Consequently, we can rewrite Equation 10 as X (xi,˜xi)∈D log p(xi\|˜xi) ≥ X (xi,˜xi)∈D Ezi∼qϕ(z\|xi)[log pψ(˜xi\|zi)] \| {z } Neural Token Reconstruction + log pθ(ˆzi\|˜xi) \| {z } Masked sEEG Modeling , (11) where the first term is the objective for vector-quantized neural signal regression in the first stage (i.e., the Du-IN VQ-VAE model), and the second term is the objective for Du-IN pre-training in the second stage (i.e., the Du-IN MAE model). F Visualization of Vector-Quantized sEEG Regression We further visualize how the sEEG signals are reconstructed. As depicted in Figure 7, although some details are missing, the overall trend of the signals is reconstructed well. Meanwhile, there is a stable decrease in the reconstruction loss during training, which indicates the discrete codex does learn high-level information from sEEG signals. G Visualization of Mask sEEG Modeling Figure 8 demonstrates the convergence curves of the total pre-training loss and masked sEEG modeling accuracy of the Du-IN MAE model. We observe that there is a stable decrease in the mask modeling loss, and the mask modeling accuracy achieves about 20%, which is similar to [28]. 24 Original sEEG signals Reconstructed sEEG signals (a) (b) Figure 7: The visualization of Vector-Quantized sEEG Regression. (a). The reconstruction loss curve during the training process of the Du-IN VQ-VAE model. (b). The visualization of reconstructed sEEG signals. (a) (b) Figure 8: The loss curve and accuracy curve during the training process of the Du-IN MAE model. H Channel Contribution Analysis For each subject, after training the Du-IN model (with random initialization) on the downstream dataset, we utilize the weights W ∈RC×D of linear projection in the spatial encoder to calculate the contribution scores S of channels: S = {si\|i = 1, ..., C}, si = 1 D D X j=1 \|Wij\|, (12) where C is the number of channels, D is the dimension of projected embedding and \| · \| gets the absolute value. Then, we normalize S using its maximum value to ensure it falls within the [0,1] range. Finally, given the variability in model performance across subjects, we further adjust the channel contribution scores based on the decoding performance of that subject, i.e., S = {si · p\|i = 1, ..., C}, where p represents the decoding performance of that subject. After calculating the channel contribution scores of all subjects, we project them to the standard brain template according to the MNI (Montreal Neurological Institute) locations of channels, using Nilearn 0.9.2. Since the electrodes are sparsely distributed within the brain, we use Scipy 1.8.1 to interpolate and smooth the channel contribution matrix and use NiLearn to plot the channel contribution map demonstrated in Figure 4 (a). With the sorted channels within each subject, we evaluate the effect of the number of channels on the decoding performance. For each subject, we evaluate the Du-IN model with {5, 10, 15, 20, 30, 60} channels (sorted by channel contribution scores), and the averaged performance (across subjects) is demonstrated in Figure 4 (b). 25 I Effectiveness of Region-Specific Channel Selection DeWave [19] successfully reconstructs 128-channel EEG signals with the same setting of vectorquantizer. However, this is not the case under the sEEG setting, which is shown in Table 8. It’s worth noting that sEEG signals are fundamentally different from EEG signals due to (1) the high information density and (2) the high specificity of different regions. Due to the desynchronization nature [7] of the brain during awake tasks, only specific brain regions are related to tasks. Therefore, only after region-specific channel selection, the Du-IN VQ-VAE model can successfully reconstruct the original signals, thus identifying the fine-grained state of brain regions. Table 8: Ablations to validate the effectiveness of region-specific channel selection. Settings Setting 1 Setting 2 Setting 3 MSE 0.2969±0.0376 0.5211±0.0492 0.9673±0.0148 1 Setting 1: Select top-10 channels relevant to speech decoding for neural signal reconstruction. 2 Setting 2: Randomly select 10 channels for neural signal reconstruction. 3 Setting 3: Use all channels (109.75 channels on average) for neural signal reconstruction. J Additional Group-Wise Evaluation The cross-entropy loss of different methods from each subject is provided in Table 9, with the best in bold and the second underlined. For model comparison, we report the average and standard deviation values (within each subject) on six different random seeds to obtain comparable results. "Std" means standard deviation. Table 9: The cross-entropy loss of different methods (with the best in bold and the second underlined). Methods Token Level Config Model Size Cross-Entropy ± Ste PT1 MS2 TS-TCC[20] Region ! % 0.32M 3.8871±0.3072 CNN-BiGRU[37] Region % 0.54M 4.0294±0.7621 EEG-Conformer[44] Region % 2.34M 3.8165±0.3456 Neuro-BERT[50] Region ! % 2.14M 3.6416±0.4360 DeWave[19] Region % 5.70M 4.1891±0.5722 BrainBERT[49] Channel ! % 43.58M 4.6254±0.1984 BrainBERT[49] Channel ! ! 43.58M 4.6190±0.2132 Brant[54] Channel ! % 69.35M 4.7962±0.7082 Brant[54] Channel ! ! 69.35M 5.0294±1.0621 LaBraM[28] Channel ! % 6.85M 4.8591±0.2723 LaBraM-PopT[28, 10] Channel ! ! 6.85M 4.6564±0.1893 Du-IN Region % 4.38M 3.5083±0.3003 Du-IN (vqvae+vq) Region ! % 4.38M 3.7244±0.3104 Du-IN (vqvae) Region ! % 4.38M 3.4309±0.2781 Du-IN (mae) Region ! % 4.38M 3.3707±0.2882 Du-IN (poms) Region ! ! 5.18M 3.4429±0.2754 1 PT: Whether the model is pre-trained before evaluation. 2 MS: Whether the model is pre-trained across multiple subjects. 26 K Subject-Wise Evaluation The detailed performance of different methods from each subject is provided in Table 10, Table 11, and Table 12, with the best in bold and the second underlined. For model comparison, we report the average and standard deviation values (within each subject) on six different random seeds to obtain comparable results. "Std" means standard deviation. Table 10: The performance of different methods from subjects (01-04). Methods Config Accuracy (%) ± Std (%) PT1 MS2 subj-01 subj-02 subj-03 subj-04 TS-TCC[20] ! % 26.90±1.72 61.57±1.21 6.65±1.09 20.66±0.79 CNN-BiGRU[37] % 46.46±4.03 68.06±1.56 4.35±0.44 17.68±3.88 EEG-Conformer[44] % 58.41±1.03 69.82±1.22 19.50±1.71 49.65±2.38 Neuro-BERT[50] ! % 60.44±2.23 72.97±1.47 28.38±4.23 52.76±3.41 DeWave[19] % 43.31±5.80 57.12±3.01 3.70±5.35 33.52±3.98 BrainBERT[49] ! % 5.30±0.90 23.49±1.29 2.74±0.40 5.18±0.38 BrainBERT[49] ! ! 6.76±0.64 25.64±1.23 2.97±0.66 5.09±0.44 Brant[54] ! % 8.64±1.59 47.97±1.30 3.06±0.36 2.74±0.68 Brant[54] ! ! 7.47±2.83 54.26±1.63 3.34±0.38 4.15±1.35 LaBraM[28] ! % 12.55±1.17 39.20±1.53 3.54±0.47 13.30±1.19 LaBraM-PopT[28, 10] ! ! 14.14±1.28 39.50±1.35 3.28±0.55 12.77±1.72 Du-IN % 71.25±1.44 77.99±0.87 23.04±4.76 59.91±4.58 Du-IN (vqvae+vq) ! % 50.15±3.80 62.79±4.67 20.72±2.15 48.24±2.65 Du-IN (vqvae) ! % 72.36±1.55 79.16±1.12 29.21±2.38 63.83±1.83 Du-IN (mae) ! % 78.60±0.79 83.61±0.38 38.80±2.52 70.98±0.81 Du-IN (poms) ! ! 73.23±0.67 79.12±1.07 28.89±1.96 64.24±1.29 1 PT: Whether the model is pre-trained before evaluation. 2 MS: Whether the model is pre-trained across multiple subjects. 27 Table 11: The performance of different methods from subjects (05-08). Methods Config Accuracy (%) ± Std (%) PT1 MS2 subj-05 subj-06 subj-07 subj-08 TS-TCC[20] ! % 34.53±1.34 9.73±0.97 24.83±1.15 20.08±1.60 CNN-BiGRU[37] % 51.26±4.93 31.52±1.48 47.75±1.12 24.64±4.44 EEG-Conformer[44] % 65.44±1.31 31.06±2.58 47.89±1.86 42.12±2.08 Neuro-BERT[50] ! % 71.61±1.97 36.63±2.15 50.23±2.63 43.24±1.36 DeWave[19] % 45.20±3.08 26.88±1.92 38.24±2.15 29.15±3.28 BrainBERT[49] ! % 9.59±0.90 2.67±0.31 4.79±0.64 5.10±0.59 BrainBERT[49] ! ! 11.28±1.17 3.00±0.47 5.31±0.55 5.22±0.76 Brant[54] ! % 24.10±2.14 5.09±1.27 7.74±1.46 8.66±1.07 Brant[54] ! ! 28.83±1.93 5.28±1.48 9.70±1.85 8.93±1.39 LaBraM[28] ! % 15.52±1.48 6.63±0.33 12.65±0.50 7.41±0.86 LaBraM-PopT[28, 10] ! ! 17.73±1.11 5.81±1.61 12.90±1.26 6.41±1.85 Du-IN % 77.60±1.20 41.91±1.80 59.63±2.20 52.35±2.18 Du-IN (vqvae+vq) ! % 63.46±2.28 34.84±1.98 45.20±2.44 40.14±1.77 Du-IN (vqvae) ! % 78.56±1.24 43.29±1.67 62.29±1.49 54.10±1.34 Du-IN (mae) ! % 81.56±1.11 46.90±1.02 65.45±1.74 59.09±0.98 Du-IN (poms) ! ! 76.91±1.38 47.01±1.76 61.68±0.95 55.17±1.37 1 PT: Whether the model is pre-trained before evaluation. 2 MS: Whether the model is pre-trained across multiple subjects. Table 12: The performance of different methods from subjects (09-12). Methods Config Accuracy (%) ± Std (%) PT1 MS2 subj-09 subj-10 subj-11 subj-12 TS-TCC[20] ! % 37.75±1.22 5.71±0.38 35.72±0.95 14.12±0.68 CNN-BiGRU[37] % 44.03±5.88 7.11±0.71 28.44±3.42 13.17±3.41 EEG-Conformer[44] % 56.51±1.98 22.22±1.07 57.10±2.03 29.87±1.44 Neuro-BERT[50] ! % 54.12±4.11 24.66±1.28 62.99±0.93 36.07±2.36 DeWave[19] % 41.98±4.60 6.22±0.94 44.60±3.35 19.22±2.95 BrainBERT[49] ! % 6.82±1.42 2.55±0.49 8.73±0.92 3.71±1.02 BrainBERT[49] ! ! 7.20±1.37 2.49±0.43 10.60±1.22 4.41±0.88 Brant[54] ! % 6.82±1.44 2.84±0.26 8.76±1.55 7.53±1.29 Brant[54] ! ! 6.46±1.66 3.00±0.31 9.82±1.71 7.82±1.66 LaBraM[28] ! % 8.97±0.52 3.50±0.30 7.92±0.61 7.19±0.54 LaBraM-PopT[28, 10] ! ! 9.35±1.09 3.91±0.31 7.84±0.92 7.74±1.24 Du-IN % 66.39±0.47 27.07±2.24 73.56±1.09 44.76±3.74 Du-IN (vqvae+vq) ! % 60.06±1.61 22.05±1.76 50.31±4.69 32.06±3.28 Du-IN (vqvae) ! % 67.18±1.22 31.06±1.59 72.41±1.98 45.38±2.26 Du-IN (mae) ! % 69.18±1.96 34.23±1.17 75.52±1.27 48.54±0.56 Du-IN (poms) ! ! 70.71±1.48 36.90±1.34 72.80±1.38 43.53±3.20 1 PT: Whether the model is pre-trained before evaluation. 2 MS: Whether the model is pre-trained across multiple subjects. 28 L Effectiveness of Vector-Quantized Neural Signal Prediction To verify the effectiveness of vector-quantized neural signal prediction, we elaborate on two types of experimental settings as illustrated in Table 13. The comparison between Du-IN and Setting 1 demonstrates that the codex is effective for masked sEEG modeling. The comparison between Du-IN and Setting 2 demonstrates that introducing the codex can prevent the model from focusing too much on reconstructing details, thus enabling the Du-IN MAE to learn better contextual embeddings. Table 13: Ablations to validate the effectiveness of vector-quantized neural signal prediction. Model Du-IN (mae) Setting 1 Setting 2 Acc. (%) ± Ste (%) 62.70±4.69 60.92±4.38 58.72±5.02 1 Setting 1: We directly predict output embeddings of the Du-IN Encoder in the Du-IN VQ-VAE by maximizing cosine similarity instead of predicting the discrete neural tokens from the codex. 2 Setting 2: We discard the Du-IN Encoder in the Du-IN VQ-VAE and directly reconstruct raw EEG patches by minimizing MSE loss. M Ablation on Mask Ratio In this experiment, we conduct different mask ratio settings to explore its impact. It is noted that we introduce the symmetric masking strategy, so we only need to validate half of the mask ratios. As the mask ratio is set to r, the symmetric masking will mask 1 −r proportion of sEEG patches. The ablation results are provided in Table 14. It can be induced that the best mask ratio is 0.5 (0.5) for our dataset. Table 14: Ablations to explore the impact of mask ratios. Mask Ratio 0.5 (0.5) 0.4 (0.6) 0.3 (0.7) 0.2 (0.8) 0.1 (0.9) Acc. (%) ± Ste (%) 62.70±4.69 60.58±4.33 59.58±4.98 58.92±4.07 58.55±3.94 N Ablation on Pre-training Epochs The impact of the number of pre-training epochs (of the Du-IN VQ-VAE model) is demonstrated in Table 15. We use the checkpoints according to the specified epochs to pre-train the Du-IN MAE model for 400 epochs. Once the reconstruction loss of the Du-IN VQ-VAE model converges, the Du-IN VQ-VAE model can extract the state of the brain region well, thus leading to better performance. The impact of the number of pre-training epochs (of the Du-IN MAE model) is demonstrated in Table 16. We use the checkpoints according to the specified epochs for downstream classification. Once the mask modeling loss of the Du-IN MAE model converges, the Du-IN MAE model learns robust contextual embeddings, thus leading to better performance. Table 15: Ablations to explore the impact of the pre-training epochs (of the Du-IN VQ-VAE model). # of Epochs 5 10 50 100 400 Acc. (%) ± Ste (%) 50.02±4.91 52.29±5.09 61.09±4.28 62.59±4.32 62.70±4.69 Table 16: Ablations to explore the impact of the pre-training epochs (of the Du-IN MAE model). # of Epochs 5 10 50 100 400 Acc. (%) ± Ste (%) 57.87±4.58 58.12±4.49 61.89±4.62 62.47±4.77 62.70±4.69 29 O Subject-Wise Electrode Locations We provide detailed information on the locations of the implanted sEEG electrodes for each subject. Red channels are the top 10 channels (selected through channel contribution analysis) for both pre-training and downstream evaluation, as described in Section 4.3. As the majority of subjects have sEEG electrodes implanted on only one side of their brains to locate the source of epilepsy, we provide side views of either the left or right brain areas here. subj-04 L R subj-01 L R L subj-02 R subj-03 L R Figure 9: Electrode locations from subjects (01-04). 30 subj-05 L R L subj-06 R subj-07 L R subj-08 L R Figure 10: Electrode locations from subjects (05-08). 31 subj-09 L R subj-10 L R L subj-11 R L subj-12 R Figure 11: Electrode locations from subjects (09-12). 32 P Subject-Wise Selected Channels The MNI coordinates, and brain region labels (according to Harvard-Oxford cortical and subcortical structural atlases [15]) for selected channels are listed below. The channels for each subject are arranged in descending order based on their contribution scores. Table 17: The MNI coordinates and brain region labels of selected channels from subjects (01-04). Subjects MNI coordinate Brain Region Du-IN Accuracy (%) x y z subj-01 -57 -16 19 Central Opercular Cortex L 71.25 -61 -16 20 Postcentral Gyrus L -54 -16 17 Central Opercular Cortex L -67 -15 23 Postcentral Gyrus L -25 -30 49 White L -64 -15 21 Postcentral Gyrus L -51 -17 16 Central Opercular Cortex L -22 -31 49 White L -48 -17 15 Central Opercular Cortex L -18 -32 49 White L subj-02 33 -27 7 White R 77.99 37 -28 7 Heschls Gyrus R 30 -27 6 White R 48 -29 9 Planum temporale R 51 -29 10 Planum temporale R 41 -28 8 White R 46 -3 -9 Insula R 55 -29 11 Planum temporale R 49 -3 -7 Planum temporale R 43 -3 -10 Insula R subj-03 -38 -30 6 White L 23.04 -58 -31 4 Superior Temporal Gyrus L -31 -30 7 White L -35 -30 7 White L 49 -11 1 Heschls Gyrus R 48 -36 26 Parietal Operculum Cortex R 42 -11 -1 Insula R -55 -31 5 Superior Temporal Gyrus L -41 -30 6 Planum temporale L 45 -11 0 Heschls Gyrus R subj-04 -44 -10 32 Precentral Gyrus L 59.91 -45 -11 35 Precentral Gyrus L -53 -6 -1 Planum temporale L -44 -9 28 Precentral Gyrus L -46 -12 39 Precentral Gyrus L -52 -5 2 Planum temporale L -43 -8 24 White L -38 -3 6 Insula L -53 -7 -5 Superior Temporal Gyrus L -16 43 25 White L 33 Table 18: The MNI coordinates and brain region labels of selected channels from subjects (05-08). Subjects MNI coordinate Brain Region Du-IN Accuracy (%) x y z subj-05 -48 -16 33 Postcentral Gyrus L 77.60 -46 15 29 Postcentral Gyrus L -43 -14 22 White L -51 -17 39 Postcentral Gyrus L -44 -15 26 White L -53 -17 43 Postcentral Gyrus L -50 -16 36 Postcentral Gyrus L -41 -14 19 Insula L -55 -18 46 Postcentral Gyrus L -49 -37 9 White L subj-06 56 -1 10 Central Opercular Cortex R 41.91 58 -4 4 Planum temporale R 52 4 21 Precentral Gyrus R 53 3 17 Precentral Gyrus R 57 -2 7 Central Opercular Cortex R 54 1 14 Precentral Gyrus R 51 6 24 Precentral Gyrus R 64 -25 -4 Middle Temporal Gyrus R 21 -1 11 White R 49 7 28 Precentral Gyrus R subj-07 -38 -18 2 Insula L 59.63 -44 -23 1 Heschls Gyrus L -41 -21 1 Heschls Gyrus L -35 -16 2 Insula L -50 -28 0 Superior Temporal Gyrus L -47 -26 1 Planum temporale L -52 -30 0 Superior Temporal Gyrus L -26 -16 40 White L -42 0 -8 Insula L -39 -22 41 Postcentral Gyrus L subj-08 -40 -20 4 Heschls Gyrus L 52.35 -43 -21 4 Heschls Gyrus L -37 -20 4 Insula L -50 -22 4 Heschls Gyrus L -47 -21 4 Heschls Gyrus L -55 3 24 Precentral Gyrus L -64 -24 3 Superior Temporal Gyrus L -61 -23 3 Superior Temporal Gyrus L -54 -22 3 Planum temporale L -52 2 22 Planum temporale L 34 Table 19: The MNI coordinates and brain region labels of selected channels from subjects (09-12). Subjects MNI coordinate Brain Region Du-IN Accuracy (%) x y z subj-09 -58 -13 21 Postcentral Gyrus L 66.39 -55 -12 19 Central Opercular Cortex L -53 -11 17 Central Opercular Cortex L -50 -10 15 Central Opercular Cortex L -61 -15 23 Postcentral Gyrus L -63 -16 25 Postcentral Gyrus L -45 -8 10 Central Opercular Cortex L -47 -9 12 Central Opercular Cortex L -42 -7 8 Insula L 52 -2 26 Precentral Gyrus R subj-10 -34 -47 41 Supramarginal Gyrus L 27.07 -42 -55 41 Angular Gyrus L -39 -53 41 Angular Gyrus L -37 -50 41 Supramarginal Gyrus L -25 -37 42 White L -44 -58 41 Angular Gyrus L -27 -39 42 White L -18 -41 48 Postcentral Gyrus L -34 -34 -23 Temporal Fusiform Cortex L -13 -40 43 Precuneous Cortex L subj-11 39 -22 3 Heschls Gyrus R 73.56 36 -23 3 Insula R 47 -22 2 White R 32 -23 4 White R 43 -22 2 Planum temporale R 53 -9 16 Central Opercular Cortex R 57 -8 17 Postcentral Gyrus R 50 -10 16 Central Opercular Cortex R 61 -21 0 Superior Temporal Gyrus R 39 -14 13 Insula R subj-12 45 -20 11 Heschls Gyrus R 44.76 38 -20 12 Insula R 42 -20 11 Heschls Gyrus R 49 -20 10 Heschls Gyrus R 53 -19 10 Heschls Gyrus R 60 -19 9 Planum temporale R 56 -19 9 Planum temporale R 60 -58 3 Middle Temporal Gyrus R 56 -57 3 Middle Temporal Gyrus R 35 -21 12 Insula R 35 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: Three of the four contributions mentioned at the end of the "Introduction" section are explicitly included. The contribution related to neuroscience-inspired analysis is simplified as "inspired by neuroscience findings" at the end of the "Abstraction" section. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We provide a separate "Limitations" section; see Section 5 for more details. 3. Theory Assumptions and Proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [NA] Justification: The paper does not include theoretical results. 4. 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4,465	SLED: Self Logits Evolution Decoding for Improving Factuality in Large Language Models Jianyi Zhang1, Da-Cheng Juan2, Cyrus Rashtchian2, Chun-Sung Ferng2, Heinrich Jiang2, Yiran Chen1 1 Duke University, 2 Google Research Project Website Abstract Large language models (LLMs) have demonstrated remarkable capabilities, but their outputs can sometimes be unreliable or factually incorrect. To address this, we introduce Self Logits Evolution Decoding (SLED), a novel decoding framework that enhances the truthfulness of LLMs without relying on external knowledge bases or requiring further fine-tuning. From an optimization perspective, our SLED framework leverages the latent knowledge embedded within the LLM by contrasting the output logits from the final layer with those from early layers. It then utilizes an approximate gradient approach to enable latent knowledge to guide the self-refinement of outputs, thereby effectively improving factual accuracy. Extensive experiments have been conducted on established benchmarks across a diverse range of model families (LLaMA 2, LLaMA 3, Gemma) and scales (from 2B to 70B), including more advanced architectural configurations such as the mixture of experts (MoE). Our evaluation spans a wide variety of tasks, including multi-choice, open-generation, and adaptations to chain-of-thought reasoning tasks. The results demonstrate that SLED consistently improves factual accuracy by up to 20% compared to existing decoding methods while maintaining natural language fluency and negligible latency overhead. Furthermore, it can be flexibly combined with other decoding methods to further enhance their performance. 1 Introduction Large Language Models (LLMs) have achieved remarkable breakthroughs in recent years, demonstrating exceptional performance across various domains [1, 2, 35, 36, 44, 47, 48]. However, a significant challenge associated with LLMs is their tendency to hallucinate or distort the truth, resulting in outputs that are not factual [15, 17, 65]. This issue of hallucination undermines the reliability and trustworthiness of LLMs in practical applications. A popular strategy for improving the LLM factuality involves refining the decoding process [43, 53]. Real-world Factuality Distribution LLM Output Distribution LLM Latent Knowledge Explicit Training Implicit Learning Factuality Decoding What the LLM knows What the LLM tells Inference Figure 1: Factuality decoding overview. Decoding focuses on how the model selects the next token during the generation process, which can significantly influence the factual accuracy of the output. The decoding methods can be cost-effective since (a) they do not rely on external knowledge and (b) no additional training is required. Furthermore, decoding methods can be synergistically combined with other techniques aimed at improving the LLM factuality, such as retrieving information from external knowledge bases [24, 25], various fine-tuning strategies for better alignment [46, 48], or ensemble learning methods [10]. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). The capital of British Columbia province is 8th layer 16th layer 24th layer N-th layer (final layer) Vancouver Victoria Richmond Surrey … … … Early-exit logits: Early-exit logits: softmax softmax Early-exit logits: Latent knowledge 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 Contrast logits to uncover the latent knowledge False True … … Utilize 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 to estimate the real-world distribution and steer the self-evolution of 𝒍𝒐𝒈𝒊𝒕𝒔𝑵 𝒍𝒐𝒈𝒊𝒕𝒔𝟐𝟒 𝒍𝒐𝒈𝒊𝒕𝒔𝟏𝟔 𝒍𝒐𝒈𝒊𝒕𝒔𝟖 Original logits: 𝒍𝒐𝒈𝒊𝒕𝒔𝑵 Self-evolved logits: * 𝒍𝒐𝒈𝒊𝒕𝒔𝑵= 𝒍𝒐𝒈𝒊𝒕𝒔𝑵−𝜶⋅𝜵𝑲𝑳𝓟𝒍𝒂𝒕𝒆𝒏𝒕, 𝓟𝒍𝒐𝒈𝒊𝒕𝒔𝑵 LLM … Vancouver Victoria Richmond Surrey Figure 2: Illustration of our Self Logits-Evolution Decoding (SLED) workflow. Recent studies [22, 26, 42, 50] suggest that LLMs sometimes have learned the factual content based on extensive pretraining or fine-tuning, although they fail to produce the correct answer when a user queries the model. This has inspired the development of several factuality decoding methods [7, 26, 27, 64] to reveal what the model implicitly "knows." Figure 1 summarizes the underlying mechanism of these factuality decoding methods. The LLMs’ output distribution is derived by applying the softmax function to the output logits from the final layer. During the training phase, this distribution is optimized based on the real-world factuality distribution represented by the training dataset. However, during the inference phase, "what the LLM tells" might still contain factual errors, which implies a discrepancy between the output distribution and the real-world factuality distribution. While the real-world distribution remains inaccessible during the inference phase, the model’s latent knowledge ("what the model knows") may have implicitly learned some factual content correctly during the training phase [22, 50]. Therefore, a key challenge for factuality decoding strategies lies in effectively harnessing the latent knowledge embedded within LLMs to refine the output distribution (logits) during inference. To address this challenge, we propose Self Logits Evolution Decoding (SLED), a novel factuality decoding approach that leverages the latent knowledge within LLMs by contrasting the final layer’s logits with early layers’ logits. During the decoding process, as LLMs progress from early to final layers, they progressively incorporate factual information stored in each layer into the output. SLED tracks this evolution process to unearth the latent knowledge within LLMs, and enables the “selfevolution” of the output distribution further to align it more closely with real-world facts. Furthermore, our approach recognizes that the latent knowledge within LLMs, while valuable, may not always be perfect. Therefore, instead of simply replacing the original outputs with this latent knowledge, SLED integrates it into the original logits through an operation similar to “single-step gradient descent” over the output logits during the inference time. This operation minimizes the Kullback-Leibler (KL) divergence between the latent knowledge distribution and the output distribution, effectively balancing the two and mitigating potential drawbacks such as overfitting or biased outputs. Figure 2 illustrates the SLED workflow, highlighting how SLED optimizes the output logits, leading to a more factual output distribution. We evaluate SLED on various LLMs (e.g., LLaMA 2 [48], LLaMA 3 [1], Gemma [31]) and benchmarks to demonstrate its state-of-the-art performance in layer-wise contrastive decoding methods. In summary, our main contributions are: • We propose SLED, a novel decoding method that aligns LLMs outputs with factual knowledge without requiring an external knowledge base or fine-tuning data. • We conduct extensive experiments across a range of LLMs, with varying configurations and scales. The results demonstrate that SLED consistently improves factual accuracy on various tasks and benchmarks, including multiple-choice, open-ended generation, and chain-of-thought reasoning tasks. • SLED can be flexibly integrated with other factuality decoding methods to enhance their effectiveness further. • We provide a new interpretable perspective for understanding layer-wise contrastive decoding methods, paving the way for further developments in factuality decoding. 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 2 4 6 8 10 Loss 7B 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 0 2 4 6 8 10 Loss 13B 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Layer Index 0 2 4 6 8 10 12 Loss 70B Figure 3: We analyze the next-token predictions of three LLaMA-2-base models using the logits from each layer individually. This analysis is performed on 200 true claims from the FACTOR dataset. The results verify that the logits distribution at the final layer is closer to the real-world distribution than all the early layers in terms of KL divergence. 2 Self Logits Evolution Decoding A large language model, equipped with N layers and a vocabulary V = [v1, v2, . . . , vd], typically generates text in the next-token prediction fashion. For each given prefix, the model computes the logits at the final (N -th) layer, logitsN ≜(ℓ(1,N), ℓ(2,N), . . . , ℓ(d,N)), which are obtained by applying a linear transformation to the hidden states of the final layer, projecting the high-dimensional hidden state vectors onto the space of the vocabulary size. Subsequently, the output distribution PlogitsN at the final (N -th) layer for the next token is derived by applying softmax function on the logits, PlogitsN ≜(p(1,N), . . . , p(d,N)) = softmax (logitsN /τ) , where τ is the temperature parameter. Therefore, for each p(i,N) (1 ≤i ≤d), we have p(i,N) = exp(ℓ(i,N)/τ)/S, where S = Xd j=1 exp(ℓ(j,N)/τ). Similarly, we can also derive the logits from early layers by applying the same linear transformation mentioned above to their hidden states. For any early layer n (n < N), we denote its logits as logitsn ≜(ℓ(1,n), . . . , ℓ(d,n)) and the corresponding distribution as Plogitsn ≜(p(1,n), . . . , p(d,n)). 2.1 Logits Evolution To improve factual accuracy, it is crucial that the correct token vi receives a higher value of logitsN to ensure a higher probability value p(i,N) in the output distribution PlogitsN . From a mathematical perspective, this means aligning the model’s output distribution PlogitsN closely with the real-world factuality distribution Preal. Specifically, we can formulate this goal as optimizing the following loss function L regarding the logits: L(logits) ≜KL(Preal, Plogits), where logits = (ℓ1, ..., ℓd), Plogits = softmax(logits/τ) (1) We describe the above optimization as Logits Evolution. Interestingly, the training of LLMs also aims at minimizing the divergence (typically the KL divergence, as the training loss function is often the cross-entropy loss) between the ground truth Preal and the output distribution PlogitsN . During the training phase, the logits evolution is driven externally by the real-world distribution Preal presented in the training dataset, and the corresponding solution is logits = logitsN . However, Preal is not accessible during the inference phase. To address this challenge, SLED utilizes the model’s latent knowledge to estimate Preal and enables "self-evolution" of the logits. We denote the estimation as Platent and the self logits evolution can be achieved by the following gradient-descent operation: ] logitsN = logitsN −α · ∇logitsN KL(Platent, PlogitsN ). (2) The parameter α, termed the Evolution Rate, governs the magnitude of adjustments applied to logitsN in the direction of the gradient ∇logitsN KL(Platent, PlogitsN ). In the following Section 2.2 and 2.3, we discuss how we derive the Platent as the estimation of the real-world distribution Preal. 3 2.2 Estimate Preal by Tracking the Logits Evolution Direction throughout Layers The core principle of our method involves leveraging the difference between each early layer’s logits and the final layer’s logit, logitsn −logitsN to approximate the gradient of KL(Preal, Plogits) at logits = logitsn. Then we estimate Preal based on this approximation. This is inspired by a new perspective of interpreting the training phase of LLMs as the evolution of logits described in Problem 1. As mentioned above, the solution derived by the training phase is the final layer’s logits logits = logitsN , since the final layer’s logitsN directly engage with the real-world distribution Preal through the loss function in training. This implies that we can generally consider the final logits logitsN to be a better solution than the logits from an early layer logitsn, with KL(Preal, PlogitsN ) < KL(Preal, Plogitsn). We present some examples in Figure 3 to demonstrate this. Based on this discussion, if we contrast the final layer’s logits with the early layer’s logits, we can consider the direction (orientation) of logitsn −logitsN can approximately align with the direction of the gradient ∇logitsKL(Preal, Plogits)\|logits=logitsn. To further verify this motivation, we calculate the cosine similarity between logitsn −logitsN and ∇logitsnKL(Preal, Plogitsn) for thousands of tokens across different models in Figure 7. We find that the majority of these values are positive, which means that the directions of these two vectors are close. Hence, for each early layer n, we propose to maximize the following function of cosine similarity and derive the P(n) latent to estimate the Preal. P(n) latent = arg max P CosSim(logitsn −logitsN , ∇logitsnKL(P, Plogitsn), 0 (3) 2.3 Achieving the Self Logits Evolution in Three Phases Based on the above analysis, we can introduce the procedures of SLED: First, we estimate P(n) latent for each early layer n using the gradient approximation in Section 2.2. Subsequently, we apply a weighted average on {P(n) latent} across all early layers n < N to derive Platent, which serves as the final estimation of the real-world distribution. Finally, we apply Platent in Equation 2 to facilitate the self-evolution of logitsN , thereby derive the updated logits, ] logitsN . logitsn −logitsN in direction ≈ ∇logitsnKL(Preal, Plogitsn) Phase 1 ====⇒ Estimate P(n) latent Phase 2 =====⇒ Ensemble Platent Phase 3 ===========⇒ Self-evolution in Eq 2 ] logitsN Phase 1: An exhaustive search for an exact solution to the complex optimization problem (Equation 3) is computationally impractical. We can reduce the solution space by the following. Suppose the real-world factuality distribution dictates that the next word to be generated is the i-th token vi from the vocabulary V. Thus Preal = Pei, where Pei represents a standard basis vector (one-hot vector) with the i-th component set to 1 and all other components set to 0. Then, we can simplify the aforementioned optimization problem by limiting the solution space to {Pei}d i=0 and decide which token i should be selected. The corresponding gradient when P = Pei has the following formulation. Proposition 1. The gradient of KL(Pei, Plogits) at logits = logitsn is: ∇logitsnKL(Pei, Plogitsn) = (Plogitsn −Pei)/τ = p(1,n), . . . , p(i,n) −1, . . . , p(d,n) /τ (4) We calculate the cosine similarity between the gradient ∇logitsnKL(Pei, Plogitsn) and the difference logitsn −logitsN for each token in the vocabulary V. Then we select the Pei∗of which the gradient is closest to logitsn −logitsN as the estimation P(n) latent. Mathematically, this involves selecting i∗ according to the following criterion i∗= arg max 1≤i≤d ¯m(n) i , where ¯m(n) i = max CosSim(logitsn −logitsN , Plogitsn −Pei), 0 , and adopting P(n) latent = Pei∗as the "hard estimation" of Preal. Drawing from the concept of hard and soft targets in label smoothing and knowledge distillation, we further extend it to the "soft estimation", P(n) latent = (m(n) 1 , . . . , m(n) i , . . . , m(n) d )/m(n), where m(n) i = ( ¯m(n) i )2 and m(n) = Xd i=1 m(n) i We square { ¯m(n) i } to moderately amplify their differences. Prior studies prove that soft targets usually offer stronger generalization capabilities, more information, and more robustness to noise than hard targets [13, 34, 45, 59, 62]. Hence, we adopt the soft estimation in lieu of the hard estimation. 4 Eliza's rate per hour for the first 40 hours she works each week is $10. She also receives an overtime pay of 1.2 times her regular hourly rate. If Eliza worked for 45 hours this week, how much are her earnings for this week? Eliza's regular hourly rate is $10. For 40 hours, her earnings are 40 x $10 = $400. For 5 hours of overtime, her earnings are 5 x $10 = $50. So her total earnings for the week are $400 + $50 = $450. The answer is $450. (Wrong) Eliza's regular hourly rate is $10. For 40 hours, her earnings are 40 x $10 = $400. For 5 hours of overtime, her earnings are 5 x $10 x 1.2 = $60. So her total earnings for the week are $400 + $60 = $460. The answer is $460. (Correct) 𝒕!𝟐 𝒕!𝟏 𝒕𝟎 𝒕𝟏 𝒕𝟐 𝒕𝟑 𝓟𝒍𝒐𝒈𝒊𝒕𝒔𝟑𝟐 1 0 = $ 5 0 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟑𝟏 1 0 = 1 2 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟐𝟔 1 0 x 1 . 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟐𝟏 1 0 x 1 . 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟏𝟔 1 0 x 1 . 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟏𝟏 1 0 x 1 . 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟔 1 0 x 1 . 2 𝓟𝒍𝒂𝒕𝒆𝒏𝒕 𝟏 1 0 x 1 . 2 For 5 hours of overtime, her earnings are 5 x 𝒕!𝟐 𝒕!𝟏 𝒕𝟎 𝒕𝟏 𝒕𝟐 𝒕𝟑 LLaMA-2-7B-Chat + SLED LLaMA-2-7B-Chat Question (The ground truth is $460) #Layer 𝒔𝒏 values across layers Figure 4: An example from GSM8K demonstrating SLED’s mechanism. SLED derives the estimations P(n) latent by contrasting final layer’s logits logitsN with early layers’ logits {logitsn}. We list the token with the highest probability value from the P(n) latent for different early layers. As shown, SLED downplays incorrect tokens by assigning lower weights s(n) to the corresponding P(n) latent. Conversely, if the estimation is correct, the weights are relatively larger. The parameter evaluation scale is set to 2. Phase 2: We ensemble P(n) latent across all layers by computing a weighted average of the set {P(n) latent} and adopt it as the final estimation of the Platent: Platent = XN n=0 s(n)P(n) latent, where s(n) = m(n)/( XN n=0 m(n)) This estimation suggests that the weight s(n) of certain layer n will be larger if the corresponding gradient approximation logitsn −logitsN is more closely aligned with the gradients {∇logitsnKL(Pei, Plogitsn)} for the tokens in the vocabulary. This in turn amplifies the influence of layer n on the final estimation, which is a desirable effect in our method. Figure 4 demonstrates that SLED can downplay incorrect tokens based on the gradient alignment. One can further validate that for each component mi in the final estimation Platent ≜(m1, m2, . . . , md), the following relationship holds: mi = PN n=0 m(n) i /(PN n=0 Pd j=1 m(n) j ). This property simplifies the description in Algorithm 1. Phase 3: Applying Platent in Equation 2 enables us to derive the gradient necessary for steering the self-evolution on the final layer’s logits logitsN . Proposition 2. The gradient of KL(Platent, Plogits) at logits = logitsN is: ∇logitsN KL(Platent, PlogitsN ) = (PlogitsN −Platent)/τ = p(1,N) −m1, . . . , p(d,N) −md /τ Then we can derive the self-evolved logits ] logitsN ] logitsN ≜(˜ℓ(1,N), . . . , ˜ℓ(i,N), . . . , ˜ℓ(d,N)), where ˜ℓ(i,N) = ℓ(i,N) −α(p(i,N) −mi)/τ. (5) 2.4 Computational Complexity and Design Decisions For each layer, computing CosSim(logitsn −logitsN , Plogitsn −Pei) for every token vi in the vocabulary V needs O(d2) operations. To reduce the computational complexity, we select only a subset VIk, where the token vi ∈VIk has the top-k highest logits in the final layer. In this scenario, we only initiate the self-evolution in Equation 2 of the logits corresponding to these top-k tokens. For the remaining tokens, which have lower probabilities, their logits are adjusted to a very lower numerical value, e.g., −1000. This strategy significantly reduces the computational complexity, while maintaining focus on the most relevant tokens. We name the parameter k, as Evolution Scale, since it determines the number of top-probability tokens active for self-evolution. Q 2.1: Why SLED contrast the final layer with all the early layers, instead of picking one premature layer to contrast based on JSD? DoLa selects a subset of early layers to form a candidate set. Then it calculates the Jensen-Shannon Divergence (JSD) between the final layer and each layer in this set. Their strategy is to choose the 5 Algorithm 1 Self Logits Evolution Decoding 1: Initialization: LLM with N layers, inputs, evolution rate α, evolution scale k > 0, η ≪0, temperature parameter τ, and the one-hoc vectors {Pei} defined in Section 2.3. 2: Feed the inputs into the LLM to obtain the logits logitsn = (ℓ(1,n), . . . , ℓ(d,n)) and probabilities Plogitsn = (p(1,n), . . . , p(d,n)) = softmax(logitsn/τ) at each layer n, where n ≤N. 3: Identify the tokens with the top-k largest values in logitsN and denote their indices by Ik. 4: for each early layer n, (n < N) do 5: Compute differences for top-k logits logitsn −logitsN . 6: Calculate m(n) i = max CosSim(logitsn −logitsN , Plogitsn −Pei), 0 2 , i ∈Ik. 7: end for 8: Compute weighted average mi = PN n=1 m(n) i PN n=1 P j∈Ik m(n) j across different layers for each i ∈Ik. 9: for each i from 1 to d do 10: Set ˜ℓ(i,N) = ℓ(i,N) −α τ (p(i,N) −mi) if i ∈Ik else Set ˜ℓ(i,N) = η ≪0. 11: end for 12: Output: The self-evolved logits are ] logitsN = (˜ℓ(1,N), . . . , ˜ℓ(i,N), . . . , ˜ℓ(d,N)). layer with the highest JSD as the premature layer, and the chosen layer will be contrasted with the final layer to update probabilities. Obviously, if this strategy is reasonable, a larger candidate set should lead to a better choice of the premature layer and, consequently, enhanced overall performance. However, a paradoxical finding from their experimental results, which our tests also confirm in the discussion in Section 3.5, is that a larger candidate set for DoLa leads to decreased performance. Specifically, when the candidate set for DoLa ranged from 0 to 32 layers for LLaMA-2-7B-Base, the performance was inferior compared to a smaller set of 0 to 16 layers. This fundamental flaw indicates that selecting a good candidate set remains a challenge when applying DoLa. In contrast, our method does not face this concern as it applies an ensemble approach to all early layers. It is also important to note that our method works well even when only contrasting the final layer with part of the early layers, as demonstrated in Section 3.5 and B, proving the robustness of our approach. Q 2.2: Why not use Platent directly as the model’s output distribution? It is crucial to understand that Platent is merely an estimation of the real-world distribution based on the model’s latent knowledge instead of the exact Preal. Consequently, relying solely on Platent, similar to DoLa, might lead to inaccuracies, as the latent knowledge can be imperfect. The original logits logitsN are still important as they are refined directly by real-world data during training. The evolution rate α in Equation 2, serves to balance this trade-off, enabling a reciprocal enhancement between Platent and the original logitsN . More ablation studies are provided in Section 3.5 and B. Q 2.2: Considering that SLED adopts logitsn −logitsN as the estimation of the gradient, why not directly apply it in Equation 2? It is important to note that while logitsn −logitsN is unconstrained, the gradients estimated in Equation 2 (e.g., p(1,N) −m1, . . . , p(d,N) −md) are constrained within [−1, 1]. Thus, direct substitution could lead to a mismatch in magnitudes and might also introduce unexpected noise. Proper normalization and subsequent aggregation of estimations from different layers are precisely what our method addresses in Section 2.2 and 2.3. Further analysis is provided in Section B. 3 Experiments As a novel layer-wise contrastive decoding approach, we first benchmark SLED against the stateof-the-art approach DoLa [7] across a diverse range of model families (LLaMA 2, LLaMA 3, Gemma) and model scales (from 2B to 70B), including the more advanced mixture of experts (MoE) architecture, as detailed in Section 3.2 and 3.3. The results showcase notable factuality improvements across a variety of tasks, including multi-choice, open-generation, and adaptations to chain-of-thought reasoning tasks. Then, in Section 3.4, we integrate our method with other established factuality decoding techniques, illustrating that SLED can further enhance their performance. In Section 3.5, we further conduct in-depth studies on mitigating the repetition issue, layer selection, various parameter settings, and latency overhead to gain more comprehensive insights into SLED’s performance. We 6 Table 1: Comparison on LLaMA 2 model family. The best results are in bold for each dataset/metric. SLED outperforms DoLa and vanilla greedy decoding. Model&Method TruthfulQA (MC) FACTOR TruthfulQA (Open-Ended) CoT MC1 MC2 MC3 %Truth %Info %T*I %Reject StrQA GSM8K LLaMA-2-7B-Base 33.17 59.42 31.78 58.15 32.80 90.09 23.99 8.45 60.96 14.03 +DoLa 32.56 63.03 30.57 62.49 35.74 95.23 32.31 2.57 60.61 14.71 +SLED (ours) 34.15 62.57 31.89 67.27 55.81 94.61 52.87 0.12 61.31 15.01 LLaMA-2-7B-Chat 35.62 57.46 32.07 56.78 59.24 78.95 38.68 17.50 63.67 21.08 +DoLa 33.41 61.93 30.35 56.65 58.02 87.03 45.78 13.10 64.32 21.00 +SLED (ours) 37.08 63.86 32.90 64.70 67.07 88.13 55.69 11.02 64.67 21.15 LLaMA-2-13B-Base 33.69 62.75 31.74 63.69 31.21 91.55 23.26 7.96 66.07 28.66 +DoLa 29.25 62.13 30.29 57.08 37.58 92.41 30.11 7.47 65.55 18.88 +SLED (ours) 34.15 63.62 31.89 70.91 38.31 94.85 33.29 5.02 66.81 29.34 LLaMA-2-13B-Chat 36.47 63.05 32.77 62.06 60.34 86.54 47.12 13.59 69.87 36.47 +DoLa 34.52 63.24 31.48 58.08 60.22 90.33 51.16 9.67 67.90 34.57 +SLED (ours) 37.09 63.75 32.60 67.50 63.65 95.23 58.87 5.26 69.96 36.54 LLaMA-2-70B-Base 33.66 61.10 32.33 72.78 55.45 62.55 18.48 36.74 75.20 56.33 +DoLa 26.93 60.33 29.42 61.92 60.95 70.62 32.07 17.72 73.45 43.37 +SLED (ours) 35.13 64.92 33.52 77.49 59.24 82.99 43.70 13.10 75.20 57.09 LLaMA-2-70B-Chat 35.98 64.18 32.99 69.07 49.57 81.27 31.33 29.13 77.25 54.59 +DoLa 31.58 54.40 32.31 58.28 61.44 77.97 39.90 21.28 74.41 49.05 +SLED (ours) 38.31 66.71 34.66 73.98 62.55 84.70 47.74 14.98 77.38 54.81 also extend our analysis with additional ablation studies and results across more benchmarks in Section B and D in the Appendix, and provide several examples of generated text as the qualitative study in Section C. 3.1 Experimental Setup Benchmarks We compare our method with baselines on several multiple-choice and open-ended generation tasks. For multiple-choice question tasks, we use the TruthfulQA [29] and FACTOR (Wiki) [33] datasets to assess the LLMs’ factuality in short-answer/long-paragraph scenario, respectively. For open-ended generation tasks, we adopt TruthfulQA [29] and tasks involving chain-of-thought reasoning [52]: StrategyQA [12] and GSM8K [8]. Models & Baselines We evaluate the performance of SLED on six LLaMA-2 models [48] ({7B,13B,70B}-Base, {7B,13B,70B}-Chat), four LLaMA-3 family models [1] ({8B,70B}-Base, {8B,70B}-IT), two Gemma models (2B,7B), two MoE models (Mixtral-8×7B, Mixtral-8×7BIT) [18]. We adopt the following baselines: 1) standard decoding (greedy decoding or sampling depending on the tasks), 2) DoLa [7], 3) Inference Time Intervention (ITI) [26], 4) Activation Decoding (AD) [4], 5) Contrastive Decoding (CD) [27], and 6) Induce-then-Contrast Decoding (ICD) [64]. Metrics We adopt the factual accuracy evaluation implemented in [7] for multiple-choice tasks and chain-of-thought reasoning tasks. For the open-ended generation task on TruthfulQA, we follow the evaluation procedure in [7, 29], using “finetuned-GPT3-judge”s to measure the truthfulness, informativeness, and rejection rate of generated outputs respectively. 3.2 Evaluation on a Broad Range of LLM Benchmarks Multiple-Choices Tasks The objective of these tasks is to employ decoding methods that enable LLMs to assign higher probabilities to correct completions/answers over incorrect alternatives. We demonstrate the effectiveness of SLED for both Short-Answer Factuality on the TruthfulQA and Long-Paragraph Factuality on the FACTOR dataset. For both DoLa and our SLED, we contrast the results from the final layer against all preceding layers. We randomly sample approximately 5% of the data for validation regarding parameter selection. The results, as shown in Table 1, indicate that SLED achieves superior outcomes in almost all metrics across six LLaMA-2 models. Notably, SLED 7 Table 2: Using SLED with other LLM families also improves the factuality. Model FACTOR TruthfulQA Model FACTOR TruthfulQA MC1 MC2 MC3 MC1 MC2 MC3 LLaMA-3-8B 64.33 33.78 63.00 32.59 Mixtral-8×7B 71.41 35.13 49.98 34.17 +DoLa 68.04 33.29 63.35 32.16 +DoLa 58.28 32.44 35.91 33.68 +SLED (ours) 68.67 35.13 64.09 32.50 +SLED (ours) 74.92 35.86 57.26 32.96 LLaMA-3-8B-IT 59.49 38.92 68.16 36.50 Mixtral-8×7B-IT 70.51 37.94 62.51 35.25 +DoLa 61.06 35.86 65.30 33.78 +DoLa 56.15 32.19 39.17 33.76 +SLED (ours) 67.17 42.23 69.03 37.97 +SLED (ours) 75.55 41.73 68.52 37.70 LLaMA-3-70B 78.72 35.62 65.66 34.18 Gemma-2B 50.87 23.38 37.16 17.42 +DoLa 77.56 33.29 64.83 32.81 +DoLa 32.93 26.07 48.97 26.55 +SLED (ours) 80.83 37.58 66.19 34.11 +SLED (ours) 57.05 25.21 50.20 26.94 LLaMA-3-70B-IT 73.95 44.80 70.29 41.02 Gemma-7B 60.42 31.58 47.63 22.75 +DoLa 71.51 38.43 68.70 35.21 +DoLa 36.07 25.21 43.14 26.13 +SLED (ours) 76.85 48.35 74.03 43.16 +SLED (ours) 65.56 32.31 49.88 25.22 achieves better performance under the MC1/MC3 metrics on TruthfulQA, which are more sensitive to fluctuations and pose a greater challenge. For long sentences in FACTOR, our method shows improvements over baselines by 5-13%. These results not only underscore the benefits of our method for factuality but also demonstrate its robustness across different lengths of text. Open-Ended Generation Tasks In open-ended settings, we prompt the model to generate answers for the same questions from TruthfulQA, following the settings outlined in [29, 7, 27]. In Table 1, we compare the performance of six LLaMA-2 models using standard greedy decoding, (greedy) DoLa, and (greedy) SLED. All the generated answers are then evaluated by a fine-tuned GPT-3 model for both truthfulness and informativeness scores. Considering that a 100% truthful score can be easily achieved by simply responding with ’I have no comment,’ which would result in a 0% informative score and thus is not very useful, we have introduced additional metrics—%Truth × Info and the rejection ratio %Reject —to demonstrate that SLED is a mutual-gains approach to achieve better both truthful and informative scores. We have improved the overall %Truth x Info scores by 3-20% across different models and reduced the rejection ratio by up to 95%. These enhancements demonstrate that our method effectively avoids the ’rejection pitfall,’ making it more helpful. Adaptation to Chain-of-thought Reasoning Tasks Although the StrategyQA and GSM8K tasks are also open-ended and require factual accuracy, the primary focus here is to evaluate how different decoding methods adapt to the Chain-of-Thought (COT) approach for handling complex reasoning tasks. We maintain a repetition penalty of 1, as we will discuss the repetition flaws associated with DoLa in Section 3.5. StrategyQA demands multi-hop reasoning, and as shown in Table 1, our method boosts accuracy across six models, whereas DoLa generally worsens it without a repetition penalty. GSM8K, a benchmark for math word problems that require arithmetic reasoning, also shows consistent accuracy improvement with SLED in 7B, 13B and 70B models. 3.3 Evaluation Across Diverse LLM Configurations As discussed above and shown in Table 1, our method, SLED, demonstrates strong generalization capabilities across the LLaMA-2 model family, proving robust from 7B to 70B model sizes. In Table 2, we further showcase SLED’s impressive performance on the more recent LLaMA-3 family models, both at 8B and 70B sizes, in terms of long paragraph factuality and short answer factuality. Interestingly, SLED is also applicable to different pre-trained models, such as Gemma at both 2B and 7B sizes, and can even be adapted to the increasingly popular Mixture of Experts (MoE) architectures. These results confirm the exceptional adaptability of our method across various LLM configurations. 3.4 Evaluation on Integrating SLED with Other LLM Factuality Decoding Methods SLED exclusively focuses on contrasting differences between layers without altering other parts of the model. Thus, it remains compatible with other techniques that incorporate additional strategies or utilize auxiliary models. This compatibility allows SLED to be seamlessly integrated into existing 8 Table 3: Comparison of decoding strategies on TruthfulQA datasets. SLED can also be seamlessly combined with other decoding strategies to improve performance further. Model LLaMA-2-7B-base LLaMA-2-7B-chat Method AD AD +DoLa AD +SLED AD AD +DoLa AD +SLED ITI ITI +SLED ICD ICD +SLED MC1 32.80 25.58 33.29 35.37 33.41 36.23 36.60 43.33 46.32 46.87 MC2 59.59 39.06 62.55 58.14 50.31 63.15 65.62 65.75 69.08 72.09 MC3 31.05 17.89 31.80 31.84 23.15 32.23 34.89 37.66 41.25 43.64 Model LLaMA-2-13B-base LLaMA-2-13B-chat Method AD AD +DoLa AD +SLED CD CD +SLED AD AD +DoLa AD +SLED CD CD +SLED MC1 33.90 24.72 33.90 30.11 33.78 36.84 34.72 36.35 28.15 36.47 MC2 62.93 37.74 63.69 50.31 63.22 63.75 50.42 64.83 54.87 64.93 MC3 31.61 17.66 31.38 28.18 32.21 32.69 23.83 32.85 29.75 33.39 Table 4: Accuracy of LLaMA 2 13B Base on StrategyQA with Varying Repetition Penalties Metric Method 1 1.02 1.04 1.06 1.08 1.1 1.2 2 Accuracy(%) DoLa 65.55 65.98 66.37 65.98 65.59 66.37 67.16 66.64 SLED (Ours) 66.81 69.39 68.51 68.47 67.07 65.72 60.87 54.75 Repetition-4(%) DoLa 7.63 7.19 6.45 5.98 5.50 5.10 3.73 2.05 SLED (Ours) 3.73 2.45 1.89 1.36 1.05 0.69 0.20 0.10 Repetition-Sen(%) DoLa 2.16 2.04 1.66 1.37 1.12 0.89 0.23 0.03 SLED (Ours) 0.88 0.39 0.10 0.02 0.03 0 0 0 methods, enhancing factuality further without the need for modifications to SLED. We integrate SLED with the following approaches: ITI, AD, CD and ICD. Table 3 shows that SLED leads to accuracy improvements from 1% to 12% across four LLaMA-2 models. 3.5 Ablation Studies and Analysis Mitigating Repetition Issues Table 4 demonstrates that our method, SLED, effectively addresses a significant issue in DoLa: repetitive content in open-ended generation tasks. Our approach outperforms DoLa without the need for excessive repetition penalty. While a slight increase in the repetition penalty further enhances the performance of our method, excessive penalties, such as 1.1, tend to degrade it. This suggests that SLED does not inherently require heavy adjustments for repetition issues. In contrast, DoLa’s performance improves with higher penalties (e.g., 1.1, 1.2, 2), indicating a more critical need for addressing repetitive content. We also employ two intuitive metrics, Repetition-4 and Repetition-Sen, to gauge the severity of repetition issues, following prior research [55]. Regardless of the repetition penalty imposed, our method consistently exhibits lower repetition rates. Table 7 includes some examples of generated text to illustrate this further. Layer Selection As discussed in Section 2.4, how to choose a good candidate set is still a paradoxically difficult task when applying DoLa. Our method does not exhibit this issue. Instead of selecting a single premature layer from the candidate set like DoLa, SLED contrasts the final layer with all layers in the candidate set and then ensembles all the results. Figure 5 shows that setting a larger candidate set, such as all the 32 layers for LLaMA-2-7B-Base, yields better performance than focusing solely on either the first [0, 16) or second half [16, 32). This implies that our layer-wise contrast approach captures more useful information in a more scientific manner. Furthermore, our tests confirm the robustness of our method even when the candidate set is minimal, such as a single layer, consistently demonstrating strong performance. Our settings mirror those of DoLa. Parameter Analysis We next investigate the impact of parameters — evolution rate α and evolution scale k — on the performance of SLED using a subset of the FACTOR dataset. We test evolution rates from {0.01, 0.1, 1, 2, 5, 10} and evolution scale values from {5, 10, 20, 50}. Without extreme 9 Figure 5: Evaluating using different premature layers for SLED and DoLa on a 10% subset of the GSM8K dataset. Contrasting all layers for SLED is better than using only the first half [0, 16) or the second half [16, 32). Hence, there are no improvements for SLED from strategic layer subset selection. 0 5 10 15 20 25 30 Premature Layer 8 10 12 14 16 18 20 Accuracy Greedy Ours [16,32) Ours [0,16) Ours [0,32) Ours (single) DoLa [0,32) DoLa [16,32) DoLa [0,16) 7B Greedy Ours (single) Ours [0,16) Ours [16,32) Ours [0,32) DoLa [0,16) DoLa [16,32) DoLa [0,32) 0 5 10 15 20 25 30 35 40 Premature Layer 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 Accuracy Greedy Ours [20,40) Ours [0,20) Ours [0,40) Ours (single) DoLa [20,40) DoLa [0,20) DoLa [0,40) 13B Greedy Ours (single) Ours [0,20) Ours [20,40) Ours [0,40) DoLa [0,20) DoLa [20,40) DoLa [0,40) Figure 6: WE explore the impact of evolution scale and rate based on the factual accuracy of a subset of the FACTOR dataset. (G: Greedy, D: DoLa) 61.5 5 0.01 61.5 0.1 61.5 1 61.5 2 59.5 5 53.0 10 65.5 10 64.5 67.5 68.0 60.5 50.0 66.5 20 66.5 66.0 67.0 64.5 51.5 65.5 50 64.5 64.5 64.5 57.5 41.5 52(G) 58(D) Evolution Rate Evolution Scale (a) LLaMA 2 7B Base 67.0 5 0.01 66.5 0.1 67.5 1 69.0 2 63.0 5 51.0 10 70.0 10 70.5 70.5 72.0 69.0 54.5 72.5 20 72.0 71.5 72.5 68.0 48.5 68.5 50 69.5 69.5 71.0 65.0 45.5 54.5(D) 64(G) Evolution Rate (b) LLaMA 2 7B Chat 54.0 5 0.01 54.0 0.1 56.0 1 57.5 2 56.0 5 49.0 10 65.0 10 64.5 66.0 67.5 64.5 56.5 62.0 20 62.5 65.0 63.5 65.0 48.0 58.0 50 56.0 59.0 60.5 58.5 45.0 55(D) 57(G) Evolution Rate (c) LLaMA 2 13B Base 54.0 5 0.01 54.0 0.1 56.0 1 57.5 2 56.0 5 49.0 10 65.0 10 64.5 66.0 67.5 64.5 56.5 62.0 20 62.5 65.0 63.5 65.0 48.0 58.0 50 56.0 59.0 60.5 58.5 45.0 56.5(D) 61(G) Evolution Rate (d) LLaMA 2 13B Chat evolution rates (e.g., 10), our method performs well, confirming its robustness. As analyzed in our methodology and Eq. 2, the evolution rate balances the logit distribution (PN ) with the latent knowledge distribution (Platent). A lower evolution rate works better for larger models (13B) and chat models as their logits already better represent real-world distributions. Latency Our method, SLED, does not incur significant latency overhead. The latencies presented in Table 5 demonstrate that our method, SLED, just increases the decoding time of DoLa by factors ranging from 0.1% to 10%. Notably, even with an atypical setting such as evolution scale = 100, which is seldom used, the increase remains around 10%. The latency for DoLa and SLED is much higher compared to the vanilla greedy decoding because we set all early layers as candidate layers set for both DoLa and SLED for a fair comparison. Table 5: Latency (ms/token) comparison across different configurations. (ES: evolution scale) Model Greedy DoLa SLED (ES=5) SLED (ES=20) SLED (ES=50) SLED (ES=100) LLaMA-2-7B 23.64 29.93 30.41 31.15 32.70 34.63 LLaMA-2-13B 30.41 39.57 39.61 41.14 43.30 45.09 LLaMA-2-70B 82.63 136.42 138.33 140.24 143.12 148.85 4 Conclusion We introduced Self Logits Evolution Decoding (SLED), which is a new method to improve accuracy and factuality without requiring external knowledge (e.g., RAG) or fine-tuning (e.g., SFT). The key idea is to optimize the output logits based on the LLMs’ latent knowledge to improve factuality during inference. On several datasets, SLED achieved the SOTA results, improving over the vanilla decoding and DoLa. We also show that SLED does not increase the inference time significantly and that it can be combined with other factuality decoding methods. 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Many techniques use additional knowledge graphs or fine-tuning data to increase factuality by updating the model parameters for this goal. One method is Retrieval-Augmented Generation (RAG) to use external knowledge to improve generation [3, 5, 9, 25]. Another option is to use post-generation retrieval and editing for improving attribution [11]. Other directions that use additional training or preference data are supervised fine-tuning (SFT) [38, 46], RLHF [37], DPO [40] or self-rewarding [58]. Complementary to these approaches, we wish to improve the LLM output distribution directly without needing any additional data. Decoding and Factuality Decoding For each prefix, the LLM generates a probability distribution for the next token on a fixed vocabulary list, and a decoding method determines how the next token is derived based on the estimated distribution. Decoding methods were initially developed to enhance the fluency and coherence of text generation, such as Beam Search (BS), which maintains the k most probable sequences at each time step. Common decoding methods also include Diverse Beam Search (DBS) [49], Contrastive Decoding [27], Top-p Sampling [14] and so on. Recently, the potential of decoding has extended beyond merely improving text readability, with some factuality decoding methods being proposed. These methods modify the generation process to focus on truthful statements rather than unsupported claims during the inference phase, aiming to reduce hallucinations. Notable recent works include Inference-Time Intervention (ITI) [26], InducedContrastive Decoding [64], Decoding by Contrasting Layers (DoLa) [7] and so on. ITI adjusts model activations during inference by following learned directions across a limited number of attention heads to improve truthfulness. Some researchers have extended previous Contrastive Decoding [27] methods to improve factual accuracy, such as Frustratingly Easy Model Decoding [56] and InducedContrastive Decoding [64], leveraging differences between expert and amateur models. Most closely related to our work is DoLa, which also employs contrasting logits from different layers. However, significant distinctions exist: Firstly, our method diverges in how to utilize those differences between logits to extract latent knowledge. Secondly, whereas DoLa directly substitutes the original output distribution with the latent knowledge distribution, our approach recognizes potential inaccuracies in this estimated distribution and adopts gradient descent within an optimization framework to integrate the model’s latent knowledge with the original output. Limitations. As we continue to refine our approach, several aspects of our method can be further developed and enhanced. Our method, SLED, achieves better factuality at the cost of operating slightly slower. Ideally, we could improve the output logits without incurring any computational cost compared to performing inference on the base LLM model. Another aspect is that currently, our experimental results support the superiority of SLED on multiple datasets. Parameter optimization using Bayesian methods [54, 30, 60, 61, 66], evolutionary algorithms [16, 32, 41] or reinforcement learning [67, 6, 19, 39] might also lead to more robust performance. It would also be ideal to back up our results with more theoretical analysis of SLED. B Additional Analysis and Ablation Studies Justification on the Gradients Approximation of SLED in Section 2.2 To further support our method’s mechanism, which utilizes logitsn −logitsN to approximate the gradient of KL(Preal, Plogits) at logits = logitsn, we manually calculate the Cosine_similarity(logitsn − logitsN , ∇logitsKL(Preal, Plogits)\|logits=logitsn) among thousands of tokens and layers. We plot the density function for different models. We find that the majority of these values are positive, demonstrating that the directions of these two vectors are very close. Hence, our gradient approximation strategy in Section 2.2 is reasonable. Further Ablation Studies for Section 2.4 We design the following two ablation studies to support our claims in Section 2.4. The first study, referred to as ’Ablation 1’, directly employs Platent as 16 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Cosine Similarity 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Density Llama-2-7B-chat Llama-2-13B-chat Llama-2-70B-chat Llama-3-8B-Instruct Llama-3-70B-Instruct Mixtral-8x7B-Instruct Figure 7: We collect 10k pairs of (logitsn −logitsN , ∇logitsnKL(Preal, Plogitsn)) on different tokens in FACTOR and different early layers. We calculate their cosine similarity values and draw the density function for each LLM. Most of the pairs have positive Cosine similarity values, which verifies that the approximation strategy of SLED is reasonable. the output distribution as discussed in Q 2.2. The second study, denoted as ’Ablation 2’, involves directly scaling the differences, {logitsn −logitsN }, to constrain their magnitudes within [−1, 1]. Then, we simply average these scaled differences across different layers and apply them to Equation 2 as mentioned in Q 2.3. The results presented in Table 6 demonstrate that the design of our SLED is reasonable. FACTOR TruthfulQA MC1 MC2 MC3 LLaMA-2-7B-Chat + Ablation 1 63.59 25.21 51.09 26.25 + Ablation 2 62.73 33.66 39.83 31.47 + SLED 65.16 37.08 63.86 32.90 LLaMA-2-13B-Chat + Ablation 1 66.70 27.05 52.72 28.46 + Ablation 2 66.29 37.33 45.00 31.98 + SLED 67.06 37.09 63.75 32.60 Table 6: Performance comparison of ablation studies and SLED on FACTOR and TruthfulQA. C Qualitative Studies We present some examples from the StrategyQA dataset in Table 7 to illustrate that our method addresses the repetition issue of DoLa. D Further Results from Open-ended Generation Task Benchmarks We have conducted additional experiments on more realistic open-ended generations datasets, HotPotQA [57], Natural Question (NQ) [23], TriviaQA [21]. We adopt the Exact Match(EM) and the F1 score. Different from the setting in the Section 3, we adopt [0, 2, 4, 6, 8, 10, 12, 14] as candidate layers for LLaMA 2 7B Chat model and [0, 2, 4, 6, 8, 10, 12, 14, 18] as candidate layers for LLaMA 2 13B Chat model for both DoLa and SLED. Our method still has robust performance across different datasets and metrics. 17 Q: Can you buy a fair trade laptop? DoLa A laptop is a portable computer. A portable computer is a handheld computer. A handheld computer is a palmtop computer. A palmtop computer is a PDA. A PDA is a personal digital assistant. A personal digital assistant is a PDA. A PDA is a handheld computer. A handheld computer is a palmtop computer. A palmtop computer is a PDA. A PDA is a personal digital assistant. A personal digital assistant is a PDA. Thus, a PDA is a PDA. A PDA is a PDA. A PDA is a PDA. A PDA is a PDA. A PDA is a PDA. A PDA is a PDA ... SLED Fair Trade is a certification for coffee. Laptops are not coffee. Thus, you cannot buy a fair trade laptop. So the answer is no. Q: Can you see the moon in Wembley Arena? DoLa Wembley Arena is in London, England. The moon is a celestial body, and the moon is in the sky. The moon is in the sky, and Wembley Arena is in London, England. London, England is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena is on Earth, and the moon is in the sky. The moon is in the sky, and Wembley Arena is on Earth. Wembley Arena SLED Wembley Arena is in London. London is in the Northern Hemisphere, and the moon is in the Southern Hemisphere. Thus, you cannot see the moon in Wembley Arena. So the answer is no. Q: Does Thiago Moises May 13 2020 submission move hypothetically hurt Achilles? DoLa Thiago Moises is a MMA fighter. He is 1.73 m tall, and Achilles is about 0.00000000000000000000000000000000000000000000000000000000000000000000... SLED Thiago Moises’s submission was a rear naked choke, which is a blood choke. A blood choke can cause a fighter to pass out. Thus, Thiago Moises’s submission could hypothetically hurt Achilles. So the answer is yes. Table 7: We present three examples of the generated text from LLaMA-2-13B-Base on StrategyQA dataset. SLED method can mitigate the repetition issue. E Implementation Details We conducted all experiments utilizing NVIDIA A100 GPUs and implemented our method based on the following repositories: DoLa1, AD2, and ICD3. For decoding responses from the LLMs on TruthfulQA, StrategyQA, and GSM8K, we employed greedy decoding. The models were operated with 16-bit floating-point precision and a batch size of 1. For the LLaMA 2 models sized 7B, 13B, and 70B, we utilized 1, 1, and 3 GPUs respectively. Cross-GPU inference, involving model weight sharding, was facilitated by the Hugging Face Accelerate package4. 1https://github.com/voidism/DoLa 2https://github.com/hkust-nlp/Activation_Decoding/tree/main 3https://github.com/HillZhang1999/ICD?tab=readme-ov-file 4https://github.com/huggingface/accelerate 18 Table 8: Performance comparison on HotPotQA, Natural Question (NQ) and TriviaQA. Model HotpotQA NQ TriviaQA EM F1 EM F1 EM F1 LLaMA 2 7B Chat 19.6 20.1 21.8 20.4 44.4 44.3 + DoLa 20.4 21.3 23.5 21.5 45.2 45.3 + SLED (ours) 20.9 21.5 24.4 22.2 47.6 46.3 Llama 2 13B Chat 23.8 21.7 33.1 28.9 63.0 60.9 + DoLa 24.5 23.2 33.1 28.9 63.2 61.5 + SLED (ours) 25.0 24.5 34.6 31.6 63.3 62.2 Regarding the details in Section 3.4, we evaluate the 7B-chat model for ITI, as the checkpoint is publicly available. Combining ITI with SLED results in better performance compared to using ITI alone. AD employs an entropy-based metric to measure the ‘sharpness’ of in-context hidden states and incorporates it into the decoding process. Combining AD with SLED surpasses both the original AD and its combination with DoLa across four model types. For CD, we have conducted experiments in two distinct configurations: (i) the LLaMA 2 13B base model is contrasted with that of the Llama 2 7B base model, and (ii) the LLaMA 2 13B chat model and the LLaMA 2 7B chat model are compared. Applying SLED to the larger models (13B) boosts performance beyond vanilla CD. ICD contrasts a trustworthy 7B model with a fine-tuned, untrustworthy 7B model, and again, applying SLED on the trustworthy 7B model improves factual accuracy further. F Additional Results of DoLa Table 9 presents some additional results of DoLa across various benchmarks. 5 Specifically, DoLa in Table 9 selects a subset of early layers as candidates for calculating the Jensen-Shannon Divergence (JSD) instead of using all layers. For example, for the LLaMA 2 7B Chat model, layers [0, 2, 4, 6, 8, 10, 12, 14] are designated as candidate layers. Notably, a specific trick implemented in DoLa is omitting the post-softmax step on logits for the TruthfulQA multiple-choice task to enhance accuracy. This trick is not applied to the vanilla greedy decoding in Table 9. In contrast, for the results presented in our Tables 1, 2, and 3, this technique is also been applied to vanilla greedy decoding to ensure a fair comparison. Table 9: The Performance of DoLa Across Various Benchmarks Model TruthfulQA GSM8K StrQA MC1 MC2 MC3 LLaMA-2-7B-Base 28.40 43.39 20.52 14.03 60.96 + DoLa 31.21 62.12 29.73 14.63 60.74 LLaMA-2-13B-Base 29.01 44.27 20.71 28.66 66.07 + DoLa 29.38 63.95 33.63 28.81 66.59 LLaMA-2-70B-Base 37.70 53.60 27.36 56.33 75.20 + DoLa 27.05 60.26 31.64 56.94 74.93 LLaMA-2-7B-Chat 33.66 51.29 24.91 21.08 63.67 + DoLa 33.29 60.86 29.77 20.55 64.37 LLaMA-2-13B-Chat 35.37 53.31 26.71 36.47 69.87 + DoLa 31.95 62.44 31.23 35.79 69.48 LLaMA-2-70B-Chat 37.33 56.33 27.94 54.59 77.25 + DoLa 31.33 54.48 34.43 54.44 76.86 5These results are provided by Yung-Sung Chuang. 19 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: This paper introduces Self Logits Evolution Decoding and shows that it improves the factual accuracy on benchmark datasets. This is the main contribution and is reflected in the abstract and introduction. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: The limitations are discussed in Section A. 3. Theory Assumptions and Proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [NA] Justification: This paper introduces a new decoding algorithm for large language models, and validates its performance empirically on benchmark datasets. Guidelines: 4. 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4,466	Solving Sparse & High-Dimensional-Output Regression via Compression Renyuan Li Department of Industrial Systems Engineering & Management National University of Singapore renyuan.li@u.nus.edu Zhehui Chen Google zhehuichen@google.com Guanyi Wang Department of Industrial Systems Engineering & Management National University of Singapore guanyi.w@nus.edu.sg Abstract Multi-Output Regression (MOR) has been widely used in scientific data analysis for decision-making. Unlike traditional regression models, MOR aims to simultaneously predict multiple real-valued outputs given an input. However, the increasing dimensionality of the outputs poses significant challenges regarding interpretability and computational scalability for modern MOR applications. As a first step to address these challenges, this paper proposes a Sparse & High-dimensional-Output REgression (SHORE) model by incorporating additional sparsity requirements to resolve the output interpretability, and then designs a computationally efficient twostage optimization framework capable of solving SHORE with provable accuracy via compression on outputs. Theoretically, we show that the proposed framework is computationally scalable while maintaining the same order of training loss and prediction loss before-and-after compression under arbitrary or relatively weak sample set conditions. Empirically, numerical results further validate the theoretical findings, showcasing the efficiency and accuracy of the proposed framework. 1 Introduction Multi-Output Regression (MOR) problem [8, 44] is a preponderant tool for factor prediction and decision-making in modern data analysis. Compared with traditional regression models that focus on a scalar output for each sample, MOR aims to predict multiple outputs y ∈RK simultaneously based on a given input x ∈Rd, i.e., y := arg min u∈Y dist(u, bg(x)) with bg := arg min g∈G 1 n n X i=1 ℓ(yi, g(xi)) , where we use {(xi, yi)}n i=1 to denote its given sample set with xi ∈Rd i-th input feature vector and yi ∈RK corresponding output vector, define ℓ: RK × RK →R as the loss function, dist : RK × RK →R as some prediction/distance metric, Y as the structure/constraint set for multiple outputs, and G as the candidate set for predicting model g : Rd →RK. Hence, MOR and its variants have been used for numerous regression tasks with structure requirements on multi-dimensional outputs arising from real applications, such as simultaneous estimation of biophysical parameters from remote sensing images [40], channel estimation through the prediction of several received signals [35], the grounding (e.g., factuality check [16]) in the Large Language Model (LLM, [34, 11]) era, to name but a few. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). In this paper, we are interested in the interpretability issue of high-dimensional outputs obtained from modern MOR tasks. One typical example is raised from algorithmic trading. In particular, in algorithmic trading, MOR helps to construct the portfolio [31] from a large number of financial instruments (e.g., different stocks, futures, options, equities, etc [19]) based on given historical market and alternative data [22]. To be concise, a high-dimensional output in this example could be viewed as a “decision”, where every component denotes the investment for the corresponding financial instruments. Thus, other than accuracy, quantitative researchers prefer outputs with only a few instruments to enhance interpretability for the underlying decision-making reasons, which naturally introduces a sparse output condition. Similar scenarios apply to other applications, including offline reinforcement learning in robotics[33], discovering genetic variations based on genetic markers[25]. As a result, the dramatic growth in output dimensions gives rise to two significant challenges: 1. High-dimensional-output impedes human interpretation for decision-making; 2. Approaches with better computational scalability are desired for training & predicting MOR. Upon these challenges, a conceptual question that motivates this research is: How to design a framework that predicts output with enhanced interpretability, better computational scalability, and provable accuracy under a modern high-dimensional-output setting? Generally speaking, this paper provides an affirmative answer as a first step to the above question. Before presenting the main contributions, let us first introduce the model that will be studied in this paper. Unlike the classical MOR model, we further assume that given outputs are of high-dimensional (i.e., d ≪K), and to address the interpretability issue, these outputs have at most s non-zero components, i.e., ∥yi∥0 ≤s, for all i ∈[n] with some pre-determined sparsity-level s(≪K). Based on such given samples, this paper proposes the (uncompressed) Sparse & High-dimensional-Output REgression (SHORE) model that aims to predict an interpretable high-dimensional output y (i.e., s-sparse in this paper) via any input feature vector x. In particular, to be concise and still capture the essential relationship, the proposed (uncompressed) SHORE model predicts y from x under a linear model, i.e., y = arg min∥y∥0≤s dist(y, bZx) for some distance metric (see Section 3.1, prediction stage) and the linear regression bZ is obtained by solving the following linear regression problem: bZ := arg min Z∈RK×d bLn(Z) := 1 n∥Y −ZX∥2 F , (1) where X := (x1 \| · · · \| xn) ∈Rd×n is the input matrix and Y := (y1 \| · · · \| yn) ∈RK×n is the corresponding column-sparse output matrix. 1.1 Contributions and Paper Organization This paper makes the following three main contributions: 1. We propose a two-stage computationally efficient framework for solving SHORE model. Specifically, the first training stage offers a computationally scalable reformulation on solving SHORE through compression in the output space. The second prediction stage then predicts high-dimensional outputs from a given input by solving a specific sparsity-constrained minimization problem via an efficient iterative algorithm. 2. We show that for arbitrarily given samples, the training loss in the first stage with compression is bounded by a 1 + δ multiplicative ratio of the training loss for the original one (1) with some positive constant δ. Additionally, the proposed iterative algorithm in the second stage exhibits global geometric convergence within a neighborhood of the ground-truth output, with a radius proportional to the given sample’s optimal training loss. Furthermore, if all samples are drawn from a light-tailed distribution, the generalization error bound and sample complexity remain in the same order for SHORE with output compression. This finding indicates that the proposed framework achieves improved computational efficiency while maintaining the same order of generalization error bounds statistically. 3. We conduct rich numerical experiments that validate the theoretical findings and demonstrate the efficiency and accuracy of the proposed framework on both synthetic and real-world datasets. In summary, this paper studies the SHORE model through computational and statistical lenses and provides a computationally scalable framework with provable accuracy. 2 The paper is organized as follows: Section 2 reviews related literature; Section 3 presents our proposed framework and provides theoretical results on sample complexity and generalization error bounds; Section 4 compares the proposed method with existing baselines in a suite of numerical experiments on both synthetic and real instances. Concluding remarks are given in Section 5. Notation. Given a positive integer n, we denote [n] := {1, . . . , n}. We use lowercase letters a as scalars and bold lowercase letters a as vectors, where ai is its i-th component with i ∈[d], and bold upper case letters A as matrices. Without specific description, for a m-by-n matrix A, we denote Ai,j as its (i, j)-th component, A⊤ i,: as its i-th row, A:,j as its j-th column. For a symmetric square matrix A, we denote λmax(A), λmin(A) and λi(A) as its maximum, minimum and i-th largest eigenvalue, respectively. We denote ∥a∥1, ∥a∥2, ∥a∥∞, ∥A∥F , ∥A∥op as the ℓ1, ℓ2, ℓ∞-norm of a vector a, the Frobenius norm and the operator norm of a matrix A, respectively. We denote I(·) as the indicator function, ∥a∥0 := Pd i=1 I(ai ̸= 0) as the ℓ0-norm (i.e., the total number of nonzero components), supp(a) := {i ∈[d] \| ai ̸= 0} as the support set. We denote VK s := {y ∈RK \| ∥y∥0 ≤s} as a set of s-sparse vectors, B2(c; ρ) := {y ∈RK \| ∥y −c∥2 ≤ρ} as a closed ℓ2-ball with center c and radius ρ, N(µ, σ2) as a Gaussian distribution with mean µ and covariance σ2. For two sequences of non-negative reals {fn}n≥1 and {gn}n≥1, we use fn ≲gn to indicate that there is a universal constant C > 0 such that fn ≤Cgn for all n ≥1. We use standard order notation fn = O(gn) to indicate that fn ≲gn and fn = eOτ(gn) to indicate that fn ≲gn lnc(1/τ) for some universal constants τ and c. Throughout, we use ϵ, δ, τ, c, c1, c2, . . . and C, C1, C2, . . . to denote universal positive constants, and their values may change from line to line without specific comments. 2 Literature Review Multi-output regression (MOR) and its variants have been studied extensively over the past decades. In this section, we focus on existing works related to our computational and statistical results. Computational part. Existing computational methods for solving MOR can be, in general, classified into two categories [8], known as problem transformation methods and algorithm adaptation methods. Problem transformation methods (e.g., Binary Relevance (BR), multi-target regressor stacking (MTRS) method [37], regression chains method [37]) aim to transform MOR into multiple single-output regression problems. Thus, any state-of-the-art single-output regression algorithm can be applied, such as ridge regression [15], regression trees [9], and etc. However, these transformation methods ignore the underlying structures/relations between outputs, which leads to higher computational complexities. In contrast, algorithm adaptation methods focus more on the underlying structures/relations between outputs. For instance, [36] investigates input component selection and shrinkage in multioutput linear regression; [1] later couples linear regressions and quantile mapping and thus captures joint relationships among variables. However, the output dimension considered in these works is relatively small compared with modern applications, and their assumptions concerning low-dimensional structure of outputs are hard to verify. To overcome these shortages, we consider high-dimensional-output regression with only an additional sparsity requirement on outputs. Statistical part. There are numerous works concerning statistical properties of traditional or multioutput regressions. [18] gives sharp results on "out-of-sample" (random design) prediction error for the ordinary least squares estimator of traditional linear regression. [45] proposes an empirical risk minimization framework for large-scale multi-label learning with missing outputs and provides excess risk generalization error bounds with additional bounded constraints. [28] investigates the generalization performance of structured prediction learning and provides generalization error bounds on three different scenarios, i.e., Lipschitz continuity, smoothness, and space capacity condition. [27] designs an efficient feature selection procedure for multiclass sparse linear classifiers (a special case for SHORE with sparsity-level s = 1), and proves that the proposed classifiers guarantee the minimax generalization error bounds in theory. A recent paper [42] studies transfer learning via multi-task representation learning, a special case in MOR, which proves statistically optimistic rates on the excess risk with regularity assumptions on the loss function and task diversity. In contrast with these works, our contributions concentrate on how generalization error bounds change before and after the compression under relatively weak conditions on the loss function and underlying distributions. Specific results in MLC. MLC is an important and special case for MOR with {0, 1}-valued output per dimension, i.e., y ∈{0, 1}K, and thus, in this paragraph, we use labels to replace 3 outputs. Here, we focus on dimensionality reduction techniques on outputs, in particular, the compressed sensing and low-rank conditions on the output matrix Y . The idea of compressed sensing rises from signal processing, which maps the original high-dimensional output space into a smaller one while ensuring the restricted isometry property (RIP). To the best of our knowledge, the compressed sensing technique is first used in [17] to handle a sparse expected output E[y\|x]. Later, [39, 12] propose Principle Label Space Transformation (PLST) and conditional PLST through singular value decomposition and canonical component analysis respectively. More recently, many new compression approaches have been proposed, such as robust bloom filter [13], log time log space extreme classification [23], merged averaged classifiers via hashing [32], etc. Additionally, computational efficiency and statistical generalization bounds can be further improved when the output matrix Y ensures a low-rank condition. Under such a condition, [45] provides a general empirical risk minimization framework for solving MLC with missing labels. Compared with the above works, this paper studies MOR under a sparse & high-dimensional-output setting without additional correlation assumptions or low-rank assumptions for output space, and then provides a complete story through a computational and statistical lens. 3 Main Results 3.1 Two-Stage Framework This subsection presents a general framework for solving SHORE and then the computational complexity for the proposed framework with/without compression. Given a set of training samples {(xi, yi)}n i=1 as described in Section 1, the framework can be separated into two stages: (compressed) training stage & (compressed) prediction stage. Training stage. In the first training stage, the framework finds a compressed regressor by solving a linear regression problem with compressed outputs. In particular, the framework compresses the original large output space (K-dim) to a smaller "latent" output space (m-dim) by left-multiplying a so-called "compressed" matrix Φ ∈Rm×K to outputs. Thus, the compressed version of training stage in SHORE can be represented as follows, c W := arg min W ∈Rm×d bLΦ n (W ) := 1 n∥ΦY −W X∥2 F . (2) We would like to point out that the idea of compressing the output space into some smaller intrinsic dimension has been used in many existing works, e.g., [17, 39, 12] mentioned in Section 2. Prediction stage. In the second prediction stage, given any input x ∈Rd, the framework predicts a sparse output by by solving the following prediction problem based on the learned regressor c W in the training stage, by(c W ) := arg min y ∥Φy −c W x∥2 2 s.t. y ∈VK s ∩F, (3) where VK s is the set of s-sparse vectors in RK, and F is some feasible set to describe additional requirements of y. For example, by letting F be RK, RK + , {0, 1}K, the intersection VK s ∩F denotes the set of s-sparse output, non-negative s-sparse output, {0, 1}-valued s-sparse output, respectively. We use by(c W ) (shorthanded in by) to specify that the predicted output is based on the regressor c W . To solve the proposed prediction problem (3), we utilize the following projected gradient descent method (Algorithm 1), which could be viewed as a variant/generalization of existing iterative thresholding methods [6, 21] for nonconvex constrained minimization. In particular, step 4 incorporates additional constraints from F other than sparsity into consideration, which leads to non-trivial modifications in designing efficient projection oracles and convergence analysis. Later, we show that the proposed Algorithm 1 ensures a near-optimal convergence (Theorem 2 and Theorem 4) while greatly reduces the computational complexity (Remark 2) of the prediction stage for solving compressed SHORE. Before diving into theoretical analysis, we first highlight the differences between the proposed prediction stage (3), general sparsity-constrained optimization (SCO), and sparse regression in the following remark. Remark 1. Proposed prediction stage v.s. General SCO: To be clear, the SCO here denotes the following minimization problem min∥α∥0≤k ∥Aα −β∥2 2. Thus, the prediction stage is a special case 4 of general SCO problem. In particular, the predicted stage takes a random projection matrix Φ with restricted isometry property (RIP) to be its A and uses c W x with c W obtained from the compressed training-stage to be its β. As a result (Theorem 2 and Theorem 4), the proposed Algorithm 1 for prediction stage ensures a globally linear convergence to a ball with center by (optimal solution of the prediction-stage) and radius O(∥Φby −c W x∥2), which might not hold for general SCO problems. Proposed prediction stage v.s. Sparse regression: Although the proposed prediction stage and sparse high-dimensional regression share a similar optimization formulation min∥β∥0≤k ∥Y −X⊤β∥2 2, the proposed prediction stage (3) is distinct from the sparse regression in the following parts: (1) Underlying Model: Most existing works about sparse high-dimensional regression assume that samples are i.i.d. generated from the linear relationship Y = X⊤β∗+ ϵ with underlying sparse ground truth β∗. In the proposed prediction stage, we do not assume additional underlying models on samples if there is no further specific assumption. The problem we studied in the predicted stage takes the random projection matrix Φ with restricted isometry property (RIP) as its X⊤(whereas X⊤in sparse regression does not ensure RIP), and uses c W x with c W obtained from the compressed training-stage as its Y . (2) Problem Task: The sparse regression aims to recover the sparse ground truth β∗given a sample set {(xi, yi)}n i=1 with n i.i.d. samples. In contrast, the task of the proposed prediction stage is to predict a sparse high-dimensional output by given a random projection matrix Φ and a single input x. As a quick summary, some typical and widely used iterative algorithms [38, 3, 4, 29] for sparse regression cannot be directly applied to the proposed prediction stage. Then, we provide the computational complexity with and without the compression for the proposed two-stage framework to complete this subsection. Remark 2. Training stage: Conditioned on XX⊤is invertible, the compressed regressor c W has a closed form solution c W = ΦY X⊤(XX⊤)−1 with overall computational complexity O(Kmn + mnd + nd2 + d3 + md2) ≈O(Kmn). Compared with the computational complexity of finding bZ from the uncompressed SHORE (1) O(Knd + nd2 + d3 + Kd2) ≈O(K(n + d)d), solving c W enjoys a smaller computational complexity on the training stage if m ≪d. In later analysis (see Section 3.2), m = O(δ−2 · s log( K τ )) with some predetermined constants δ, τ and sparsity-level s ≪d, thus in many applications with large output space, the condition m ≪d holds. Prediction stage: The computational complexity of each step-3 in Algorithm 1 is O(Km + K + Km + K) ≈O(Km). The projection in step-4 is polynomially solvable with computational complexity O(K min{s, log K}) (see proof in Appendix A.5.1). Thus, the overall computational complexity of Algorithm 1 is O(K(m + min{s, log K})T). Compared with the complexity O(K(d + min{s, log K})) of predicting by from the uncompressed SHORE (1), the compressed version enjoys a smaller complexity on the prediction stage if (m + min{s, log K})T ≪d + min{s, log K}. (4) In later analysis (see Theorem 2), since m = O(δ−2 · s log( K τ )) with predetermined constants δ, τ, sparsity-level s ≪d, and T = O(log[ ∥by−v(0)∥2 ∥Φby−c W x∥2 ]) from inequality (5) , we have condition 4 holds. Whole computational complexity: Based on the analysis of computational complexity above, we conclude that when the parameters (K, d, m, T) satisfies K > K1/3 > d ≫O(δ−2 log(K/τ) · T) = mT, the compressed SHORE enjoys a better computational complexity with respect to the original one (1). 5 Algorithm 1 Projected Gradient Descent (for Second Stage) Input: Regressor c W , input sample x, stepsize η, total iterations T 1: Initialize point v(0) ∈VK s ∩F. 2: for t = 0, 1, . . . , T −1: do 3: Update ev(t+1) = v(t) −η · Φ⊤(Φv(t) −c W x). 4: Project v(t+1) = Π(ev(t+1)) := arg minv∈VK s ∩F ∥v −ev(t+1)∥2 2. 5: end for Output: v(T ). 3.2 Worst-Case Analysis for Arbitrary Samples We begin this subsection by introducing the generalization method of the compressed matrix Φ. Assumption 1. Given an m-by-K compressed matrix Φ, all components Φi,j for 1 ≤i ≤m and 1 ≤j ≤K, are i.i.d. generated from a Gaussian distribution N(0, 1/m). Before presenting the main theoretical results, let us first introduce the definition of restricted isometry property (RIP, [10]), which is ensured by the generalization method (Assumption 1). Definition 1. (V, δ)-RIP: A m-by-K matrix Φ is said to be (V, δ)-RIP over a given set of vectors V ⊆RK, if, for every v ∈V, (1 −δ)∥v∥2 2 ≤∥Φv∥2 2 ≤(1 + δ)∥v∥2 2. In the rest of the paper, we use (s, δ)-RIP to denote (VK s , δ)-RIP. Recall VK s = {v ∈RK \| ∥v∥0 ≤s} is the set of s-sparse vectors. Remark 3. From Johnson-Lindenstrauss Lemma [43], for any δ ∈(0, 1), any τ ∈(0, 1), and any finite vector set \|V\| < ∞, if the number of rows m ≥O δ−2 · log( \|V\| τ ) , then the compressed matrix Φ generated by Assumption 1 satisfies (V, δ)-RIP with probability at least 1 −τ. Now, we are poised to present the first result on training loss defined in (2). Theorem 1. For any δ ∈(0, 1) and τ ∈(0, 1), suppose compressed matrix Φ follows Assumption 1 with m ≥O( 1 δ2 · log( K τ )). We have the following inequality for training loss ∥ΦY −c W X∥2 F ≤(1 + δ) · ∥Y −bZX∥2 F , holds with probability at least 1 −τ, where bZ, c W are optimal solutions for the uncompressed (1) and compressed SHORE (2), respectively. The proof of Theorem 1 is presented in Appendix A.1. In short, Theorem 1 shows that the optimal training loss for the compressed version is upper bounded within a (1 + δ) multiplicative ratio with respect to the optimal training loss for the uncompressed version. Intuitively, Theorem 1 implies that SHORE remains similar performances for both compressed and compressed versions, while the compressed version saves roughly O(Kn(d −m) + Kd2) computational complexity in the training stage from Remark 2. Moreover, the lower bound condition on m ≥O( 1 δ2 · log( K τ )) ensures that the generated compressed matrix Φ is (1, δ)-RIP with probability at least 1 −τ. For people of independent interest, Theorem 1 only needs (1, δ)-RIP (independent with the sparsity level) due to the unitary invariant property of Φ from Assumption 1 (details in Appendix A.1). Additionally, due to the inverse proportionality between m and δ2, for fixed K and τ, the result can be written as ∥ΦY −c W X∥2 F ≤ 1 + O(1/√m) · ∥Y −bZX∥2 F , which is verified in our experiments 4. We then present the convergence result of the proposed Algorithm 1 for solving prediction problem (3). Theorem 2. For any δ ∈(0, 1) and τ ∈(0, 1), suppose the compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )). With a fixed stepsize η ∈( 1 2−2δ, 1), the following inequality ∥by −v(t)∥2 ≤ct 1 · ∥by −v(0)∥2 + c2 1 −c1 · ∥Φby −c W x∥2 holds for all t ∈[T] simultaneously with probability at least 1 −τ, where c1 := 2 −2η + 2ηδ < 1 is some positive constant strictly smaller than 1, and c2 := 2η √ 1 + δ is some constant. 6 The proof of Theorem 2 is given in Appendix A.2. Here, the lower bound condition on the number of rows m ensures that the generated compressed matrix Φ is (3s, δ)-RIP with probability at least 1 −τ by considering a δ/2-net cover of set V = VK 3s ∩B2(0; 1) from Johnson-Lindenstrauss Lemma [43]. Moreover, since the number of rows m required in Theorem 2 is greater than the one required in Theorem 1, term ∥Φby −c W x∥2 can be further upper bounded using the uncompressed version (1 + δ)∥by −bZx∥2 with probability at least 1 −τ. Then we obtain a direct corollary of Theorem 2: suppose ∥by −v(0)∥2 > ∥Φby −c W x∥2, if t ≥t∗:= O log ∥by −v(0)∥2/∥Φby −c W x∥2 . log (1/c1) , (5) the proposed Algorithm 1 guarantees a globally linear convergence to a ball B(by; O(∥Φby −c W x∥2)). In contrast with OMP used in [17] for multi-label predictions, Theorem 2 holds for arbitrary sample set without the so-called bounded coherence guarantee on Φ. Moreover, as reported in Section 4, the proposed prediction method (Algorithm 1) has better computational efficiency than OMP. 3.3 Generalization Error Bounds for IID Samples This subsection studies a specific scenario when every sample (xi, yi) is i.i.d. drawn from some underlying subGaussian distribution D over sample space Rd × VK s . Specifically, we use ED x y = µx µy =: µ and VarD x y = Σxx Σxy Σyx Σyy =: Σ ⪰0(d+K)×(d+K) to denote its mean and variance, respectively. Let ξx := x −µx, ξy := y −µy, ξ := (ξ⊤ x , ξ⊤ y )⊤ be centered subGaussian random variables of x, y, (x⊤, y⊤)⊤, respectively. Let a, b ∈{x, y}, we use Mab := ED[ab⊤] = Σab + µaµ⊤ b , c Mab := 1 n Pn i=1 ai(bi)⊤to denote the population second (cross-)moments and empirical second (cross-)moments, respectively. Then, the population training loss is defined as min Z∈RK×d L(Z) := E(x,y)∼D ∥y −Zx∥2 2 with its optimal solution Z∗:= MyxM −1 xx . Similarly, given a Φ, the compressed training loss is given by LΦ(W ) := E(x,y)∼D ∥Φy −W x∥2 2 with optimal solution W∗:= ΦMyxM −1 xx . We then define the following assumption: Assumption 2. Let D be σ2-subGaussian for some positive constant σ2 > 0, i.e., the inequality ED[exp(λu⊤ξ)] ≤exp λ2σ2/2 holds for any λ > 0 and unitary vector u ∈Rd+K. Moreover, the covariance matrix Σxx is positive definite (i.e., its minimum eigenvalue λmin(Σxx) > 0). Remark 4. Assumption 2 ensures the light tail property of distribution D. Note that in some real applications, e.g., factuality check [31], algorithmic trading [16], one can normalize input and output vector to ensure bounded ℓ2-norm. Under such a situation, Assumption 2 is naturally satisfied. Our first result in this subsection gives the generalization error bounds. Theorem 3. For any δ ∈(0, 1) and τ ∈(0, 1 3), suppose compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )), and Assumption 2 holds, for any constant ϵ > 0, the following results hold: (Matrix Error). The inequality for matrix error M 1/2 xx c M −1 xx M 1/2 xx op ≤4 holds with probability at least 1 −2τ as the number of samples n ≥n1 with n1 := max 64C2σ4 9λ2 min(Mxx) (d + log(2/τ)) , 322∥µx∥2 2σ2 λ2 min(Mxx) 2 √ d + p log(1/τ) 2 , where C is some fixed positive constant used in matrix concentration inequality of operator norm. (Uncompressed). The generalization error bound for uncompressed SHORE satisfies L( bZ) ≤ L(Z∗) + 4ϵ with probability at least 1 −3τ, as the number of samples n ≥max{n1, n2} with n2 := max ( 4(∥Z∗∥2 F + K) · d + 2 p d log(K/τ) + 2 log(K/τ) ϵ , 4∥µy −Z∗µx∥2 2 · d ϵ ) . 7 (Compressed). The generalization error bound for the compressed SHORE satisfies LΦ(c W ) ≤ LΦ(W∗) + 4ϵ with probability at least 1 −3τ, as the number of sample n ≥max{n1, en2} with en2 := max ( 4(∥W∗∥2 F + ∥Φ∥2 F ) · d + 2 p d log(m/τ) + 2 log(m/τ) ϵ , 4∥Φµy −W∗µx∥2 2 · d ϵ ) . The proof of Theorem 3 is presented in Appendix A.3. The proof sketch mainly contains three steps: In Step-1, we represent the difference L( bZ) −L(Z∗) or LΦ(c W ) −LΦ(W∗) as a product between matrix error (in Theorem 3) and rescaled approximation error (see Appendix A.3 for definition), i.e., L( bZ) −L(Z∗) or LΦ(c W ) −LΦ(W∗) ≤(matrix error) × (rescaled approximation error); Step-2 controls the upper bounds for matrix error and rescaled approximation error separately, using concentration for subGuassian variables; for Step-3, we combine the upper bounds obtained in Step-2 and complete the proof. Based on the result of Theorem 3, ignoring the logarithm term for τ, the proposed generalization error bounds can be bounded by L( bZ) ≤L(Z∗) + eOτ max{∥Z∗∥2 F , ∥µy −Z∗µx∥2 2, K} · d n , LΦ(c W ) ≤LΦ(W∗) + eOτ max{∥W∗∥2 F , ∥Φµy −W∗µx∥2 2, ∥Φ∥2 F } · d n . Remark 5. To make a direct comparison between the generalization error bounds of the uncompressed and the compressed version, we further control the norms ∥W∗∥2 F , ∥Φµy −W∗µx∥2 2, ∥Φ∥2 F based on additional conditions on the compressed matrix Φ. Recall the generalization method of the compressed matrix Φ as mentioned in Assumption 1, we have the following event E1 := Φ ∈Rm×K ∥W∗∥2 F = ∥ΦZ∗∥2 F ≤(1 + δ)∥Z∗∥2 F ∥Φµy −W∗µx∥2 F = ∥Φ(µy −Z∗µx)∥2 F ≤(1 + δ)∥µy −Z∗µx∥2 F holds with probability at least 1 −τ due to the RIP property for a fixed matrix. Moreover, since every component Φi,j is i.i.d. drawn from a Gaussian distribution N(0, 1/m), using the concentration tail bound for chi-squared variables (See Lemma 1 in [26]), we have the following event E2 := ( Φ ∈Rm×K ∥Φ∥2 F ≤K + 2 r K log(1/τ) m + 2 log(1/τ) m ) holds with probability at least 1 −τ. Conditioned on these two events E1 and E2, the generalization error bound of the compressed version achieves the same order (ignoring the logarithm term of τ) as the generalization error bound of the uncompressed version. That is to say, LΦ(c W ) ≤(1 + δ) · L(Z∗) + eOτ max{∥Z∗∥2 F , ∥µy −Z∗µx∥2 2, K} · d n holds with probability at least 1 −5τ. Comparing with existing results on generalization error bounds mentioned in Section 2, we would like to emphasize that Theorem 4 guarantees that the generalization error bounds maintain the order before and after compression. This result establishes on i.i.d. subGaussian samples for the SHORE model without additional regularity conditions on loss function and feasible set as required in [45]. Additionally, we obtained a O(Kd/n) generalization error bound for squared Frobenius norm loss function L or LΦ, which is smaller than O(K2d/n) as presented in [Theorem 4, [45]]. We then give results on prediction error bounds. Theorem 4. For any δ ∈(0, 1) and any τ ∈(0, 1/3), suppose the compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )), and Assumption 2 holds. Given any learned regressor c W from training problem (2), let (x, y) be a new sample drawn from the underlying distribution D, we have the following inequality holds with probability at least 1 −τ: ED[∥by −y∥2 2] ≤ 4 1 −δ · ED[∥Φy −c W x∥2 2], where by is the optimal solution from prediction problem (3) with input vector x. 8 The proof of Theorem 4 is presented in Appendix A.4. Theorem 4 gives an upper bound of ℓ2norm distance between by and y. Since ∥v(T ) −y∥2 ≤∥v(T ) −by∥2 + ∥by −y∥2, combined with Theorem 2, we have ED[∥v(T ) −y∥2 2] ≤O(ED[∥Φy −c W x∥2]) when T ≥t∗defined in (5) (see Appendix A.5.5), where the final inequality holds due to the optimality of by. Hence, we achieve an upper bound of ℓ2-norm distance between v(T ) and y as presented in Theorem 4, see Remark 6. Remark 6. . For any δ ∈(0, 1) and τ ∈(0, 1/3), suppose compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )), and Assumption 2 holds, for any constant ϵ > 0, the following inequality holds with probability at least 1 −3τ: ED[∥v(T ) −y∥2 2] ≤O(ED[∥Φy −c W x∥2]) ≤O(LΦ(W∗) + 4ϵ). 4 Numerical Experiments In this section, we conduct numerical experiments on two types of instances (i.e., synthetic data sets and real data sets) to validate the theoretical results and illustrate both efficiency and accuracy of the proposed prediction method compared with typical existing prediction baselines, i.e., Orthogonal Matching Pursuit (OMP, [46]), Fast Iterative Shrinkage-Thresholding Algorithm (FISTA, [2]) and Elastic Net (EN, [47]). Due to the space limit, we put the implemented prediction method (Algorithm 2) in Appendix A.6.1, aforementioned existing prediction baselines in Appendix A.6.2, experiment setting details and results for real data in Appendix A.7. Performance measures. Given a sample (x, y) with input x and corresponding true output y, we use v to denote the predicted output obtained from any prediction method, and measure the numerical performances based on the following three metrics: 1. For a ground truth y with sparsity-level s, the metric precision over selected supports, i.e., Precision@s := 1 s\|supp(v) ∩supp(y)\| measures the percentage of correctly identified supports in the predicted output; 2. The metric output difference, i.e., Output −diff := ∥v −y∥2 2 measures the ℓ2-norm distance between the predicted output and the ground-truth; 3. For any given MOR c W and compressed matrix Φ, the metric prediction loss, i.e., Prediction −Loss := ∥Φv −c W x∥2 2 computes the prediction loss with respect to c W x. Synthetic data generation procedure. The synthetic data set is generated as follows: Every input xi for i ∈[n] is i.i.d. drawn from a Gaussian distribution N(µx, Σxx), where its mean vector µx and covariance matrix Σxx are selected based on the procedures given in Appendix A.6.3. For any given sparsity-level s, underlying true regressor Z∗∈RK×d, and Signal-to-Noise Ratio (SNR), the groundtruth yi (corresponding with its given input xi) is generated by yi = ΠVK s ∩F Z∗xi + ϵi , where ϵi ∈RK is a i.i.d. random noise drawn from the Gaussian distribution N(0K, SNR−2∥Z∗xi∥∞·IK). Parameter setting. For synthetic data, we set input dimension d = 104, output dimension K = 2 × 104, and sparsity-level s = 3. We generate in total n = 3 × 104, i.i.d. samples as described above, i.e., Ssyn := {(xi, yi)}3×104 i=1 with SNR−1 ∈{1, 0.32, 0.032} to ensure the signal-to-noise decibels (dB, [14]) takes values on dB := 10 log(SNR2) ∈{0, 10, 30}. We select the number of rows for compressed matrix Φ by m ∈{100, 300, 500, 700, 1000, 2000}. For computing the empirical regressor c W ∈Rm×d, we first split the whole sample set Ssyn into two non-overlap subsets, i.e., a training set Stra with 80% and a testing set Stest with rest 20%. The regressor c W is therefore obtained by solving compressed SHORE (2) based on the training set Stra with a randomly generated compressed matrix Φ. For evaluating the proposed prediction method, Algorithm 2, we pick a fixed stepsize η = 0.9, F = RK + , and set the maximum iteration number as T = 60, and run prediction methods over the set Stest. Hardware & Software. All experiments are conducted in Dell workstation Precision 7920 with a 3GHz 48Cores Intel Xeon CPU and 128GB 2934MHz DDR4 Memory. The proposed method and other methods are solved using PyTorch version 2.3.0 and scikit-learn version 1.4.2 in Python 3.12.3. Numerical Results & Discussions. The results are demonstrated in Figure 1, which does not include the results from the Elastic Net and OMP due to relatively much longer running time. 9 Figure 1: Numerical results on synthetic data. In short, each dot in the figure represents the average value of 10 independent trials (i.e., experiments) of compressed matrices Φ(1), . . . , Φ(10) on a given tuple of parameters (K, d, n, SNR, m). The shaded parts represent the empirical standard deviations over 10 trials. In the first row, we plot the ratio of training loss after and before compression, i.e., ∥ΦY −c W X∥2 F /∥Y −bZX∥2 F versus the number of rows m. It is obvious that the ratio converges to one as m increases, which validates the result presented in Theorem 1. In the second row, we plot percision@3 versus the number of rows. As we can observe, the proposed algorithm outperforms CD and FISTA. Based on Figure 1, we observe that the proposed algorithm enjoys a better computational cost and accuracy on most metrics. The running time for the proposed algorithm and baselines are reported in Table 2 (see in Appendix A.7), which further demonstrates the efficiency of the proposed algorithm. The implemented code could be found on Github https://github.com/from-ryan/Solving_ SHORE_via_compression. 5 Conclusion and Future Directions In conclusion, we propose a two-stage framework to solve Sparse & High-dimensional-Output REgression (SHORE) problem, the computational and statistical results indicate that the proposed framework is computationally scalable, maintaining the same order of both the training loss and prediction loss before and after compression under relatively weak sample set conditions, especially in the sparse and high-dimensional-output setting where the input dimension is polynomially smaller compared to the output dimension. In numerical experiments, SHORE provides improved optimization performance over existing MOR methods, for both synthetic data and real data. We close with some potential questions for future investigation. The first is to extend our theoretical results to nonlinear/nonconvex SHORE frameworks [24]. The second direction is to improve existing variable reduction methods for better scalability while maintaining small sacrificing on prediction accuracy, e.g., new design and analysis on randomized projection matrices. The third direction is to explore general scenarios when high dimensional outputs enjoys additional geometric structures [30] from real applications in machine learning or operations management other than s-sparsity and its variants as discussed in the paper. 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We have the following inequality for training loss ∥ΦY −c W X∥2 F ≤(1 + δ) · ∥Y −bZX∥2 F holds with probability at least 1 −τ, where bZ, c W are optimal solutions for the uncompressed (1) and compressed SHORE (2), respectively. Proof. Given a set of n samples {(xi, yi)}n i=1, and a matrix Φ generated from Assumption 1, we have ∥ΦY −c W X∥2 F ≤∥ΦY −Φ bZX∥2 F = ∥Φ(Y −bZX)∥2 F , where the inequality holds due to the optimality of c W . Let Y ′ = Y −bZX. Consider the singular value decomposition: Y ′ = UY ′ΣY ′V ⊤ Y ′, and then we have ∥Φ(Y −bZX)∥2 F = ∥ΦY ′∥2 F = ∥ΦUY ′ΣY ′V ⊤ Y ′∥2 F = ∥ΦUY ′ΣY ′∥2 F . Set ˜Φ := ΦUY ′. Since the generalization method for Φ ensures its (1, δ)-RIP with probability 1 −τ (from the Johnson-Lindenstrauss Lemma), and UY ′ is a real unitary matrix, then Lemma 1 for unitary invariant shows that ˜Φ is also (1, δ)-RIP with probability 1 −τ. Now, using ˜Φ is (1, δ)-RIP and all columns in ΣY ′ has at most one non-zero component, we have ∥ΦUY ′ΣY ′∥2 F = ∥˜ΦΣY ′∥2 F ≤(1 + δ)∥ΣY ′∥2 F = (1 + δ)∥Y ′∥2 F . Combining the above inequalities together implies ∥ΦY −c W X∥2 F ≤(1 + δ)∥Y ′∥2 F = (1 + δ) · ∥Y −bZX∥2 F , which completes the proof. A.2 Proof of Theorem 2 Recall the following theorem in the main text. Theorem 2. For any δ ∈(0, 1) and τ ∈(0, 1), suppose the compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )). With a fixed stepsize η ∈( 1 2−2δ, 1), the following inequality ∥by −v(t)∥2 ≤ct 1 · ∥by −v(0)∥2 + c2 1 −c1 · ∥Φby −c W x∥2 holds for all t ∈[T] simultaneously with probability at least 1 −τ, where c1 := 2 −2η + 2ηδ < 1 is some positive constant strictly smaller than 1, and c2 := 2η √ 1 + δ is some constant. Proof. Suppose Assumption 1 holds, then the randomized compressed matrix Φ ensures (3s, δ)-RIP with probability at least 1 −τ (see Remark 3). Thus, to complete the proof, it is sufficient to show the above inequality holds for all t ∈[T] under such a compressed matrix Φ with (3s, δ)-RIP. We conclude that the proof is, in general, separated into three steps. Step-1. We establish an upper bound on the ℓ2-norm distance between the current point and the optimal solution. Due to the optimality of the projection (step-4 in Algorithm 1), we have the following inequality ∥ev(t+1) −v(t+1)∥2 2 ≤∥ev(t+1) −v∥2 2 holds for all v ∈VK s ∩F, which further implies that ∥ev(t+1) −v + v −v(t+1)∥2 2 ≤∥ev(t+1) −v∥2 2 ⇔∥v −v(t+1)∥2 2 ≤2⟨v −ev(t+1), v −v(t+1)⟩ holds for all v ∈VK s ∩F. 14 Step-2. We show one-iteration improvement based on the above inequality. Since by ∈VK s ∩F, we can replace v by by, which still ensures the above inequality. Set ∆(t) := by −v(t) for all t ∈[T]. Based on the updating rule (step-3 in Algorithm 1), ev(t+1) = v(t) −η · Φ⊤(Φv(t) −c W x). Thus, the above inequality can be written as ∥∆(t+1)∥2 2 ≤2⟨by −v(t) + η · Φ⊤(Φv(t) −c W x), ∆(t+1)⟩ = 2⟨∆(t), ∆(t+1)⟩+ 2η⟨Φv(t) −c W x, Φ∆(t+1)⟩ = 2⟨∆(t), ∆(t+1)⟩−2η⟨Φ∆(t), Φ∆(t+1)⟩+ 2η⟨Φby −c W x, Φ∆(t+1)⟩. (6) Using Lemma 2 with ∆(t) ∥∆(t)∥2 , ∆(t+1) ∥∆(t+1)∥2 , ∆(t) ∥∆(t)∥2 + ∆(t+1) ∥∆(t+1)∥2 , ∆(t) ∥∆(t)∥2 − ∆(t+1) ∥∆(t+1)∥2 all 3s-sparse vectors, and Φ a (3s, δ)-RIP matrix, we have −2η Φ ∆(t) ∥∆(t)∥2 , Φ ∆(t+1) ∥∆(t+1)∥2 ≤2δη −2η ∆(t) ∥∆(t)∥2 , ∆(t+1) ∥∆(t+1)∥2 , which implies −2η⟨Φ∆(t), Φ∆(t+1)⟩≤2δη∥∆(t)∥2∥∆(t+1)∥2 −2η⟨∆(t), ∆(t+1)⟩. Inserting the above result into inequality (6) gives ∥∆(t+1)∥2 2 ≤(2 −2η)⟨∆(t), ∆(t+1)⟩+ 2δη∥∆(t)∥2∥∆(t+1)∥2 + 2η⟨Φby −c W x, Φ∆(t+1)⟩ (i) ≤(2 −2η + 2ηδ)∥∆(t)∥2∥∆(t+1)∥2 + 2η∥Φby −c W x∥2∥Φ∆(t+1)∥2 ≤(2 −2η + 2ηδ)∥∆(t)∥2∥∆(t+1)∥2 + 2η √ 1 + δ∥Φby −c W x∥2∥∆(t+1)∥2, where the above inequality (i) requests η < 1 to ensure the inequality (2 −2η)⟨∆(t), ∆(t+1)⟩≤ (2 −2η)∥∆(t)∥2∥∆(t+1)∥2 holds. Therefore, dividing ∥∆(t+1)∥2 on both side implies ∥∆(t+1)∥2 ≤(2 −2η + 2ηδ)∥∆(t)∥2 + 2η √ 1 + δ∥Φby −c W x∥2, (7) which gives the one-step improvement. Step-3. Combine everything together. To ensure contractions in every iteration, we pick stepsize η such that 2 −2η + 2ηδ ∈(0, 1) with η < 1, which gives η ∈ 1 2(1−δ), 1 . Using the above inequality (7) for one-step improvement, we have ∥by −v(t)∥2 ≤(2 −2η + 2ηδ)t · ∥by −v(0)∥2 + 2η √ 1 + δ 2η −2ηδ −1∥Φby −c W x∥2, which completes the proof. A.3 Proof of Theorem 3 Recall the following theorem in the main text. Theorem 3. For any δ ∈(0, 1) and τ ∈(0, 1 3), suppose compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )), and Assumption 2 holds, for any constant ϵ > 0, the following results hold: (Matrix Error). The inequality for matrix error M 1/2 xx c M −1 xx M 1/2 xx op ≤4 holds with probability at least 1 −2τ as the number of samples n ≥n1 with n1 := max 64C2σ4 9λ2 min(Mxx) (d + log(2/τ)) , 322∥µx∥2 2σ2 λ2 min(Mxx) 2 √ d + p log(1/τ) 2 , 15 where C is some fixed positive constant used in matrix concentration inequality of operator norm. (Uncompressed). The generalization error bound for uncompressed SHORE satisfies L( bZ) ≤ L(Z∗) + 4ϵ with probability at least 1 −3τ, as the number of samples n ≥max{n1, n2} with n2 := max ( 4(∥Z∗∥2 F + K) · d + 2 p d log(K/τ) + 2 log(K/τ) ϵ , 4∥µy −Z∗µx∥2 2 · d ϵ ) . (Compressed). The generalization error bound for the compressed SHORE satisfies LΦ(c W ) ≤ LΦ(W∗) + 4ϵ with probability at least 1 −3τ, as the number of sample n ≥max{n1, en2} with en2 := max ( 4(∥W∗∥2 F + ∥Φ∥2 F ) · d + 2 p d log(m/τ) + 2 log(m/τ) ϵ , 4∥Φµy −W∗µx∥2 2 · d ϵ ) . Proof. Let us start with the uncompressed version. Step-1. Note that the optimal solutions for population loss and empirical loss are Z∗= MyxM −1 xx and bZ = Y X⊤ n XX⊤ n −1 =: c Myx c M −1 xx , respectively. Thus, the generalization error bound is L( bZ) −L(Z∗) = ∥bZ −Z∗∥2 Mxx = ∥( bZ −Z∗)M 1/2 xx ∥2 F . Note that ( bZ −Z∗)M 1/2 xx = c Myx c M −1 xx −MyxM −1 xx M 1/2 xx = c Myx −MyxM −1 xx c Mxx c M −1 xx M 1/2 xx = bE[yx⊤−Z∗xx⊤] c M −1 xx M 1/2 xx = bE[(y −Z∗x)x⊤c M −1/2 xx ] c M −1/2 xx M 1/2 xx , where we use bE[·] to denote the empirical distribution. Then, the above generalization error bound can be upper-bounded as follows ( bZ −Z∗)M 1/2 xx F = bE[(y −Z∗x)x⊤c M −1/2 xx ] c M −1/2 xx M 1/2 xx F ≤ bE[(y −Z∗x)x⊤c M −1/2 xx ] F \| {z } rescaled approximation error c M −1/2 xx M 1/2 xx op \| {z } matrix error . Step-2. Next, we provide upper bounds on these two terms bE[(y −Z∗x)x⊤c M −1/2 xx ] F and c M −1/2 xx M 1/2 xx op in the right-hand-side separately. For matrix error term c M −1/2 xx M 1/2 xx op, we have c M −1/2 xx M 1/2 xx 2 op = M 1/2 xx c M −1 xx M 1/2 xx op . Due to Assumption 2, the centralized feature vector ξx := x −µx ensures the following inequality ED exp λv⊤ξx ≤exp λ2∥v∥2 2σ2 2 for all v ∈Rd. Consider the empirical second moment of x, c Mxx = n X i=1 xi(xi)⊤ n = n X i=1 ξi x(ξi x)⊤ n + µx n X i=1 ξi x n !⊤ + n X i=1 ξi x n ! µ⊤ x + µxµ⊤ x . 16 Since ξ1 x, . . . , ξn x are i.i.d. σ2-subGaussian random vector with zero mean and covariance matrix Σxx, then based on Lemma 3, there exists a positive constant C such that for any τ ∈(0, 1), PD   n X i=1 ξi x(ξi x)⊤ n −Σxx op ≤Cσ2 max (r d + log(2/τ) n , d + log(2/τ) n ) ≥1 −τ, and based on Lemma 4, for any τ ∈(0, 1), PD n X i=1 ξi x n 2 ≤4σ √ d + 2σ p log(1/τ) √n ! ≥1 −τ. Let ∆xx := Pn i=1 ξi x(ξi x)⊤ n −Σxx and ¯ξ := Pn i=1 ξi x n , then c Mxx can be represented by c Mxx = Σxx + µxµ⊤ x \| {z } =:Mxx +∆xx + µx ¯ξ⊤+ ¯ξµ⊤ x , and thus we haveM −1/2 xx c MxxM −1/2 xx = Id + M −1/2 xx ∆xx + µx ¯ξ⊤+ ¯ξµ⊤ x M −1/2 xx . Then the minimum eigenvalue of M −1/2 xx c MxxM −1/2 xx can be lower bounded as follows λmin M −1/2 xx c MxxM −1/2 xx ≥1 − M −1/2 xx ∆xx + µx ¯ξ⊤+ ¯ξµ⊤ x M −1/2 xx op ≥1 − M −1/2 xx ∆xxM −1/2 xx op − M −1/2 xx µx ¯ξ⊤+ ¯ξµ⊤ x M −1/2 xx op ≥1 − Cσ2 λmin(Mxx) r d + log(2/τ) n − 2∥µx∥2 λmin(Mxx) 4σ √ d + 2σ p log(1/τ) √n , where the final inequality holds with probability at least 1−2τ by inserting the above non-asymptotic bounds. Then, we have M 1/2 xx c M −1 xx M 1/2 xx op = λ−1 min M −1/2 xx c MxxM −1/2 xx ≤ " 1 − Cσ2 λmin(Mxx) r d + log(2/τ) n − 2∥µx∥2 λmin(Mxx) 4σ √ d + 2σ p log(1/τ) √n #−1 holds with probability at least 1 −2τ. It is easy to observe that as n ≥n1 := max 64C2σ4 9λ2 min(Mxx) (d + log(2/τ)) , 322∥µx∥2 2σ2 λ2 min(Mxx) 2 √ d + p log(1/τ) 2 , we have M 1/2 xx c M −1 xx M 1/2 xx op ≤4 holds with probability 1 −2τ. For rescaled approximation error term bE[(y −Z∗x)x⊤c M −1/2 xx ] F , we first compute variance proxy for the subGaussian vector y −Z∗x. Note that the j-th component of the subGaussian vector y −Z∗x can be written as [y −Z∗x]j = (−[Z∗]⊤ j,: \| e⊤ j ) x y , where [Z∗]⊤ j,: is the j-th row of Z∗, e⊤ j is a K-dimensional vector with j-th component equals to one and rest components equal to zero. Thus, it is easy to observe that the ℓ2-norm square of (−[Z∗]⊤ j,: \| e⊤ j ) is ∥[Z∗]j,:∥2 2 + 1, and therefore, based on the Assumption 2, we have ED exp λ[y −Z∗x]j −λ[µy −Z∗µx]j ≤exp λ2 · (∥[Z∗]j,:∥2 2 + 1)σ2/2 , 17 i.e., a subGaussian with variance proxy σ2 j := (∥[Z∗]j,:∥2 2 + 1)σ2. Thus the rescaled approximation error can be upper-bounded by bE[(y −Z∗x)x⊤c M −1/2 xx ] 2 F = K X j=1 bE[[y −Z∗x]jx⊤c M −1/2 xx ] 2 2 ≤2 K X j=1 bE[([y −Z∗x]j −[µy −Z∗µx]j)x⊤c M −1/2 xx ] 2 2 \| {z } =:T 1 j +2 K X j=1 bE[[µy −Z∗µx]jx⊤c M −1/2 xx ] 2 2 \| {z } =:T 2 j . We control term T 1 j for all j ∈[K] separately using Lemma 5 as follows: For all τ ∈(0, 1), we have PD T 1 j ≤σ2 j (d + 2 p d log(K/τ) + 2 log(K/τ)) n ! ≥1 −τ/K. For the term T 2 j , we have T 2 j = 1 n n X i=1 [µy −Z∗µx]j(xi)⊤c M −1/2 xx 2 2 = [µy −Z∗µx]j √n c M −1/2 xx x1 √n \| · · · \| c M −1/2 xx xn √n 1n 2 2 = [µy −Z∗µx]2 j n c M −1/2 xx x1 √n \| · · · \| xn √n 1n 2 2 . Now let x1 √n \| · · · \| xn √n = UxDxV ⊤ x be the singular value decomposition of the matrix x1 √n \| · · · \| xn √n , the above ℓ2-norm can be further written as [µy −Z∗µx]2 j n c M −1/2 xx x1 √n \| · · · \| xn √n 1n 2 2 (i) = [µy −Z∗µx]2 j n 1⊤ n Vx Id 0d×(n−d) 0(n−d)×d 0(n−d)×(n−d) V ⊤ x 1n (ii) = [µy −Z∗µx]2 j n · d, where the equality (i) holds due to the definition of empirical matrix c Mxx = 1 n Pn i=1 xi(xi)⊤, the equality (ii) holds due to the unitary property of matrix Vx. Combining the above two parts implies bE[(y −Z∗x)x⊤c M −1/2 xx ] 2 F = K X j=1 bE[[y −Z∗x]jx⊤c M −1/2 xx ] 2 2 ≤2 K X j=1 T 1 j + 2 K X j=1 T 2 j (iii) ≤2 K X j=1 σ2 j (d + 2 p d log(K/τ) + 2 log(K/τ)) n + 2 K X j=1 [µy −Z∗µx]2 j n · d = 2(∥Z∗∥2 F + K) · d + 2 p d log(K/τ) + 2 log(K/τ) n + 2∥µy −Z∗µx∥2 2 · d n 18 with inequality (iii) holds with probability at least 1 −τ. Still, it is easy to observe that for any positive constant ϵ, as n ≥n2 := max ( 4(∥Z∗∥2 F + K) · d + 2 p d log(K/τ) + 2 log(K/τ) ϵ , 4∥µy −Z∗µx∥2 2 · d ϵ ) , we have bE[(y −Z∗x)x⊤c M −1/2 xx ] 2 F ≤ϵ holds with probability at least 1 −τ. Step-3. Combining two upper bounds together, if n ≥max{n1, n2}, the following inequality for generalization error bound L( bZ) −L(Z∗) ≤ bE[(y −Z∗x)x⊤c M −1/2 xx ] 2 F · M 1/2 xx c M −1 xx M 1/2 xx op ≤4ϵ holds with probability at least 1 −3τ. We then study the compressed version. Step-1’. Similarly, its optimal solutions for population loss and empirical loss are W∗= ΦMyxM −1 xx and c W = ΦY X⊤ n XX⊤ n −1 =: Φ c Myx c M −1 xx , respectively. Thus, the generalization error bound is LΦ(c W ) −LΦ(W∗) = ∥c W −W∗∥2 Mxx = ∥(c W −W∗)M 1/2 xx ∥2 F . Still, we have (c W −W∗)M 1/2 xx = bE[(Φy −W∗x)x⊤c M −1/2 xx ] c M −1/2 xx M 1/2 xx , and therefore, the generalization error bound can be upper-bounded by (c W −W∗)M 1/2 xx 2 F ≤ bE[(Φy −W∗x)x⊤c M −1/2 xx ] 2 F c M −1/2 xx M 1/2 xx 2 op . Step-2’. Next, we provide upper bounds on these two terms bE[(Φy −W∗x)x⊤c M −1/2 xx ] F and c M −1/2 xx M 1/2 xx op in the right-hand-side separately. Note that for the matrix error term c M −1/2 xx M 1/2 xx op, we could use the same upper bounded as mentioned in the proof of uncompressed version. Now, to give the upper bound on the rescaled approximation error term bE[(Φy −W∗x)x⊤c M −1/2 xx ] F , we first compute the variance proxy for the subGaussian vector Φy −W∗x, which is eσ2 j := (∥[W∗]j,:∥2 2 + ∥Φj,:∥2 2)σ2 for all j ∈[m]. Thus, the rescaled approximation error for the compressed version can be upper bounded by bE[(Φy −W∗x)x⊤c M −1/2 xx ] 2 F = m X j=1 bE[[Φy −W∗x]jx⊤c M −1/2 xx ] 2 2 ≤2 m X j=1 bE[([Φy −W∗x]j −[Φµy −W∗µx]j)x⊤c M −1/2 xx ] 2 2 \| {z } =: e T 1 j + 2 m X j=1 bE[[Φµy −W∗µx]jx⊤c M −1/2 xx ] 2 2 \| {z } =: e T 2 j . Still using Lemma 5, for all τ ∈(0, 1), we have PD eT 1 j ≤eσ2 j (d + 2 p d log(m/τ) + 2 log(m/τ)) n ! ≥1 −τ/m. 19 For the term eT 2 j , following the same proof procedures of the uncompressed version implies eT 2 j = [Φµy −W∗µx]2 j n c M −1/2 xx x1 √n \| · · · \| xn √n 1n 2 2 = [Φµy −W∗µx]2 j n · d Therefore, the rescaled approximation error for the compressed version is upper-bounded by bE[(Φy −W∗x)x⊤c M −1/2 xx ] 2 F ≤2(∥W∗∥2 F + ∥Φ∥2 F ) · d + 2 p d log(m/τ) + 2 log(m/τ) n + 2∥Φµy −W∗µx∥2 2 · d n with probability at least 1 −τ. Similarly, it is easy to get that for any positive constant ϵ, as n ≥en2 := max ( 4(∥W∗∥2 F + ∥Φ∥2 F ) · d + 2 p d log(m/τ) + 2 log(m/τ) ϵ , 4∥Φµy −W∗µx∥2 2 · d ϵ ) , we have bE[(Φy −W∗x)x⊤c M −1/2 xx ] 2 F ≤ϵ holds with probability at least 1 −τ. Step-3’. Combining two upper bounds together, if n ≥max{n1, en2}, the following inequality for generalization error bound LΦ(c W ) −LΦ(W∗) ≤ bE[(Φy −W∗x)x⊤c M −1/2 xx ] 2 F · M 1/2 xx c M −1 xx M 1/2 xx op ≤4ϵ holds with probability at least 1 −3τ. A.4 Proof of Theorem 4 Recall the following theorem in the main text. Theorem 4 For any δ ∈(0, 1) and any τ ∈(0, 1/3), suppose the compressed matrix Φ follows Assumption 1 with m ≥O( s δ2 log( K τ )), and Assumption 2 holds. Given any learned regressor c W from training problem (2), let (x, y) be a new sample drawn from the underlying distribution D, we have the following inequality holds with probability at least 1 −τ: ED[∥by −y∥2 2] ≤ 4 1 −δ · ED[∥Φy −c W x∥2 2], where by is the optimal solution from prediction problem (3) with input vector x. Proof. Due to the optimality of by, we have Φby −c W x 2 2 ≤ Φy −c W x 2 2 ⇔ Φby −Φy + Φy −c W x 2 2 ≤ Φy −c W x 2 2 ⇔∥Φby −Φy∥2 2 ≤2⟨Φby −Φy, Φy −c W x⟩ ⇒∥Φby −Φy∥2 2 ≤2 ∥Φby −Φy∥2 Φy −c W x 2 ⇔∥Φby −Φy∥2 ≤2 Φy −c W x 2 ⇔∥Φby −Φy∥2 2 ≤4 Φy −c W x 2 2 ⇒(1 −δ)∥by −y∥2 2 ≤4 Φy −c W x 2 2 where the final ⇒holds due to the (3s, δ)-RIP property of the compressed matrix Φ with probability at least 1 −τ. Taking expectations on both sides implies (1 −δ)ED[∥by −y∥2 2] ≤4ED[∥Φy −c W x∥2 2], which completes the story. 20 A.5 Technical Lemma A.5.1 Proof of Claim Proposed in Remark 2 Proof. Let us discuss the computational complexity for F to be RK, RK + , {0, 1}K separately. Given a fixed ˜v, • If F = RK, the projection method arg minv∈VK s ∩F ∥v −˜v∥2 2 can be reformulate using the following mixed-integer programming (MIP), (v∗, z∗) := arg minv,z PK p=1 zp(vp −˜vp)2 s.t. PK p=1 zp ≤s , with v∗the output of the projection method. Sorting the absolute values {\|˜vp\|}K p=1 in decreasing order such that \|˜v(1)\| ≥· · · ≥\|˜v(K)\|, the output v∗of the proposed projection is [v∗]j = ˜vj if j ∈{(1), . . . , (s)} 0 o.w. , with computational complexity O(K min{s, log K}). • If F = RK + , the projection method arg minv∈VK s ∩F ∥v −˜v∥2 2 can be reformulate using the following mixed-integer programming (MIP), (v∗, z∗) := arg minv,z PK p=1 zp(vp −˜vp)2 s.t. PK p=1 zp ≤s vp ≥0 ∀p ∈[K] . Sorting {˜vp}K p=1 in decreasing order such that ˜v(1) ≥· · · ≥˜v(K), the output v∗of the proposed projection is [v∗]j = ˜vj · I(˜vj > 0) if j ∈{(1), . . . , (s)} 0 o.w. , with computation complexity O(K min{s, log K}). • If F = {0, 1}K, the projection method minv∈VK s ∩F ∥v −˜v∥2 2 presented in step-4 of Algorithm 1 can be represented as minz PK p=1(1 −zp)˜v2 p + zp(˜vp −1)2 s.t. PK p=1 zp ≤s zp ∈{0, 1} ∀p ∈[K] = minz PK p=1 ˜v2 p −zp(2˜vp −1) s.t. PK p=1 zp ≤s zp ∈{0, 1} ∀p ∈[K] . Sort {2˜vp −1}K p=1 in decreasing order such that 2˜v(1) −1 ≥2˜v(2) −1 ≥· · · ≥2˜v(K) −1, then, the optimal z∗can be set by z∗ p = I(2vp −1 > 0) if p ∈{(1), . . . , (s)} 0 o.w. . For computational complexity, computing the sequence {2vp −1}K p=1 needs O(K), picking the top-s elements of the above sequence requires O(K min{s, log K}), setting the optimal solution z∗ needs O(s), and thus the total computational complexity is O(K) + O(K min{s, log K}) + O(s) = O(K min{s, log K}). 21 A.5.2 Lemma for the Proof of Theorem 1 Lemma 1. (Unitary invariant). Let Φ ∈Rm×d be a randomized compressed matrix as described in Assumption 1, and U ∈Rd×d be a real unitary matrix. Then we have ˜Φ = ΦU is (1, δ)-RIP with probability at least 1 −τ. Proof. Note that (i, j)-th component of ˜Φ can be represented as ˜Φi,j = Φi,:U:,j = Pd ℓ=1 Φi,ℓUℓ,j. Since every component Φi,j in Φ is i.i.d. drawn from N(0, 1/m), we have ˜Φi,j = d X ℓ=1 Φi,ℓUℓ,j ∼N 0, d X ℓ=1 1 mU 2 ℓ,j ! = N(0, 1/m). Now, we need to show that any two distinct components ˜Φi1,j1 and ˜Φi2,j2 in ˜Φ are independent. It is easy to observe that ˜Φi1,j1 and ˜Φi2,j2 are independent when i1 ̸= i2 since Φi1,: and Φi2,: are independent. If i1 = i2 = i, then the following random vector satisfies ˜Φi,j1 ˜Φi,j1 = Φi,:U:,j1 Φi,:U:,j2 ∼N 0 0 , 1/m 0 0 1/m . That is to say, ˜Φi1,j1 and ˜Φi2,j2 are jointly Gaussian distributed and uncorrelated, which shows that ˜Φi1,j1 and ˜Φi2,j2 are independent. Combining the above together, we have ˜Φ is a randomized matrix with component i.i.d. from N(0, 1/m). Based on the existing result ([7], Theorem 1.5), when m ≥C1 · δ−2[ln(eK) + ln(2/τ)] for any δ > 0 and τ ∈(0, 1), we have ˜Φ ensures (1, δ)-RIP with probability at least 1 −τ. A.5.3 Lemma for the Proof of Theorem 2 Lemma 2. For any integer parameter s(≤d) and positive parameter δ ∈(0, 1), let Φ ∈Rm×d be a (s, δ)-RIP matrix. For u1, u2, if u1, u2, u1 + u2, u1 −u2 are all s-sparse, then the following inequality holds −2δ(∥u1∥2 2 + ∥u2∥2 2) + 4⟨u1, u2⟩≤4⟨Φu1, Φu2⟩≤2δ(∥u1∥2 2 + ∥u2∥2 2) + 4⟨u1, u2⟩. Proof. Since u1, u2, u1 + u2, u1 −u2 are s-sparse, we have (1 −δ)(∥u1 + u2∥2 2) ≤⟨Φ(u1 + u2), Φ(u1 + u2)⟩≤(1 + δ)(∥u1 + u2∥2 2) (8) (1 −δ)(∥u1 −u2∥2 2) ≤⟨Φ(u1 −u2), Φ(u1 −u2)⟩≤(1 + δ)(∥u1 −u2∥2 2) (9) Subtracting (9) from (8) gives −2δ(∥u1∥2 2 + ∥u2∥2 2) + 4⟨u1, u2⟩≤4⟨Φu1, Φu2⟩≤2δ(∥u1∥2 2 + ∥u2∥2 2) + 4⟨u1, u2⟩, which completes the proof. A.5.4 Lemma for the Proof of Theorem 3 Lemma 3. Let ξ1, . . . , ξn be n i.i.d. σ2-subGaussian random vectors with a zero mean and a covariance matrix Σ. Then, there exists a positive constant C such that for all τ ∈(0, 1), P   n X i=1 ξi(ξi)⊤ n −Σ op ≤Cσ2 max (r d + log(2/τ) n , d + log(2/τ) n ) ≥1 −τ. Proof. Lemma 3 is a direct corollary from [Theorem 6.5, [41]]. It is easy to observe that the proposed Lemma 3 holds by setting the parameter δ listed in [Theorem 6.5, [41]] as min{1, c p ln(2/τ)/n} with c some positive constant. Lemma 4. Let ξ1, . . . , ξn be n i.i.d. σ2-subGaussian random vectors with a zero mean and a covariance matrix Σ. Then for any τ ∈(0, 1), we have P n X i=1 ξi n 2 ≤4σ √ d + 2σ p log(1/τ) √n ! ≥1 −τ. 22 Proof. We show this lemma by discretizing the unit ℓ2-norm ball B2(0; 1). Let N1/2 be a 1 2-minimum cover of B2(0; 1) with its cardinality \|N1/2\| ≤5d. Since for any vector ξ ∈Rd, we always have ∥ξ∥2 = max ∥v∥2≤1⟨v, ξ⟩≤ max v′∈N1/2⟨v′, ξ⟩+ max ∥v′′∥2≤1/2⟨v′′, ξ⟩= max v′∈N1/2⟨v′, ξ⟩+ 1 2 max ∥v′′∥2≤1⟨v′′, ξ⟩, then ∥ξ∥2 ≤2 maxv′∈N1/2⟨v′, ξ⟩. Therefore, for any σ2-subGaussian random vector, P(∥ξ∥2 ≥t) ≥P max v′∈N1/2⟨v′, ξ⟩≥t/2 ≤\|N1/2\| · exp −t2 8σ2 ≤5d · exp −t2 8σ2 , which implies that ∥ξ∥2 ≤4σ √ d + 2σ p log(1/τ) with probability at least 1 −τ. Now, since ¯ξ = Pn i=1 ξi n is a σ2/n-subGaussian random vector, inserting this variance proxy into the above inequality gives the desired result. Lemma 5. Let η(x) be a zero-mean, σ2 η-subGaussian random variable. Let x1, . . . , xn be n i.i.d. σ2-subGaussian random vectors (may not zero-mean) as described in Assumption 2. Conditioned on c Mxx = 1 n Pn i=1 xi(xi)⊤≻0d×d, for any τ ∈(0, 1), we have P bE[η(x)x⊤c M −1/2 xx ] 2 2 ≤σ2 η(d + 2 p d log(1/τ) + 2 log(1/τ)) n ! ≥1 −τ. Proof. The proof of Lemma 5 can be found in [Lemma 5, [18]]. A.5.5 Discussion after Theorem 4 Since ∥v(T ) −y∥2 ≤∥v(T ) −by∥2 + ∥by −y∥2, combined with Theorem 2, we have ED[∥v(T ) −y∥2 2] ≤2ED[∥v(T ) −by∥2 2] + 2ED[∥by −y∥2 2] ≤O(ED[∥Φby −c W x∥2]) + 8 1 −δ · ED[∥Φy −c W x∥2 2] ≤O(ED[∥Φy −c W x∥2]). A.6 Additional Numerical Experiments on Synthetic Data A.6.1 Implemented Prediction Method The implemented prediction method is presented as follows, see Algorithm 2. Comparing with Algorithm 1 proposed in the main content, it adds an additional stopping criteria ∥v(t) −v(t−2)∥2 0.01 + ∥v(t)∥2 < 10−6 to ensure an earlier stop than Algorithm 1. Note, in later numerical experiments, we use the terminology ‘early stopping’ to denote that the iteration generated by the prediction algorithm satisfies the above additional stopping criteria within the maximum iteration number, i.e. T = 60 (as listed in Section 4). A.6.2 Discussions on Baselines Baselines. We compare our proposed prediction method with the following baselines. Orthogonal Matching Pursuit. Orthogonal Matching Pursuit(OMP) is a greedy prediction algorithm. It iteratively chooses the most relevant output and then performs least-squares and updates the residuals. The built-in function ‘OrthogonalMatchingPursuit’ from Python package ‘Sklearn.Linear_model’ is used in the experiment. Correlation Decoding. Correlation decoding is a standard decoding algorithm. It computes the multiplication of the transpose of compression matrix Φ and the learned regressor c W . For any test point x, the algorithm predicts the top s labels in Φc W x orded by magnitude. 23 Algorithm 2 Implemented Projected Gradient Descent (for Second Stage) Input: Regressor c W , input sample x, stepsize η, total iterations T 1: Initialize point v(0) ∈VK s ∩F and t = 0. 2: while t < T and ∥v(t) −v(t−2)∥2/(0.01 + ∥v(t)∥2) > 10−3: do 3: Update ev(t+1) = v(t) −η · Φ⊤(Φv(t) −c W x). 4: Project v(t+1) = Π(ev(t+1)) := arg minv∈VK s ∩F ∥v −ev(t+1)∥2 2. 5: Update t := t + 1. 6: end while Output: v(T ). Elastic Net. The elastic net is a regression method that combines both the ℓ1-norm penalty and the ℓ2-norm penalty to guarantee the sparsity and the stability of the prediction. The parameters for the ℓ1-norm penalty and ℓ1-norm penalty in Elastic Net are set to be 0.1 through the experiments. The built-in function ‘ElasticNet’ from Python package ‘Sklearn.Linear_model’ is used in the experiment. Fast Iterative Shrinkage-Thresholding Algorithm. The fast iterative shrinkage-thresholding algorithm is an advanced optimization method designed to efficiently solve certain classes of unconstrained convex optimization problems. It utilizes momentum-like strategies to speed up convergence rates, which is particularly effective for minimizing functions that may be non-smooth. A.6.3 Procedures on Generating Mean & Covariance In this paper, we give exact procedures on selecting µx and Σxx as mentioned in Section 4. • The mean vector µx is selected as follows. We first generate a d-dimensional vector µ from the standard d-dimensional Gaussian distribution N(0d, Id). Then we set [µx]j = \|µ\|j for all j ∈[d] by taking absolute values over all components. • The covariance matrix Σxx is selected as follows. We first generate a d-by-d matrix A, where every component of A is i.i.d. generated from the standard Gaussian distribution N(0, 1). Then the covariance matrix Σxx is set to be Σxx := 1 dA⊤A + 1 2Id. A.7 Additional Numerical Experiments on Real Data In this subsection, we do experiments on real data and compare the performance of the proposed prediction method (see Algorithm 2) with four baselines, i.e., Orthogonal Matching Pursuit (OMP, [46]), Correlation Decoding (CD,[20]), Elastic Net (EN, [47]), and Fast Iterative Shrinkage-Thresholding Algorithm (FISTA,[2]) see Appendix A.6.2 for detailed explanations. Real data. We select two benchmark datasets in multi-label classification, Wiki10-31K and EURLex-4K[5] due to their sparsity property. Table 1 shows the details for the datasets. Input Dim Output Dim Training set Test set Dataset d K n d K n d K EURLex-4K 5,000 3,993 17,413 236.69 5.30 1,935 240.96 5.32 Wiki10-31K 101,938 30,938 14,146 673.45 18.64 6,616 659.65 19.03 Table 1: Statistics and details for training and test sets, where d, K denote their averaged non-zero components for input and output, respectively. 24 Figure 2: The figure reports the prediction running time (measured in seconds) on synthetic data with early stopping by the proposed algorithm under different compressed output dimensions. As we can observe, the running time first decreases dramatically, then increases almost linearly with respect to m . Such a phenomenon has occurred since the max number of iterations is 60 in the implemented prediction method with early stopping, which is relatively large; As m increases but is still less than 500, the actual number of iterations drops dramatically due to early stopping criteria; After passes 500, the actual number of iterations stays around 10, and then the running time grows linearly as dimension increases. Parameter setting. In prediction stage, we choose s ∈{1, 3} for EURLex-4K and s ∈{3, 5} for Wiki10-31K. We choose the number of rows m ∈{100, 200, 300, 400, 500, 700, 1000, 2000} on both EURLex-4K and Wiki10-31. Ten independent trials of compressed matrices Φ(1), . . . , Φ(10) are implemented for each tuple of parameters (s, m) on both datasets. Empirical running time. Here, we report the running time of the proposed algorithm and baselines on both synthetic and real datasets, see Table 2. Dataset Prop.Algo. OMP CD Elastic Net FISTA Synthetic Data ≈1 second 200-400 seconds <1 second 700-900 seconds <3 seconds EURLex-4K <1 second 20-80 seconds <1 second ≈1 second Wiki10-31K <5 seconds 500-700 seconds <5 seconds 5-10 seconds Table 2: Time Complexity Comparison for each prediction Numerical Results & Discussions. Figure 2 further illustrates that the computational complexity increases linearly with respect to the growth of compressed output dimension m on synthetic data, when m is greater than 500 to ensure the convergence of the prediction algorithm (see Remark 2). For real data, Figure 3 and Figure 4 present the results of their accuracy performances. In particular, the accuracy grows relatively stable with respect to m when the compression matrix satisfies the RIP-property with high probability. Besides, based on the results presented in Figure 3, Figure 4, and Table 2, we observe that the proposed algorithm slightly outperforms the baselines on precision as s increases while enjoys a significant better computational efficiency, especially on large instances, which demonstrate the stability of the proposed algorithm. 25 Figure 3: This figure reports the numerical results on real data – EURLex-4K. Each dot in the figure represents 10 independent trials (i.e., experiments) of compressed matrices Φ(1), . . . , Φ(10) on a given tuple of parameters (s, m). The curves in each panel correspond to the averaged values for the proposed Algorithm and baselines over 10 trials; the shaded parts represent the empirical standard deviations over 10 trials. In the first row, we plot the output distance versus the number of rows. In the second row, we plot the precision versus the number of rows, and we cannot observe significant differences between these prediction methods. 26 Figure 4: This figure reports the numerical results on real data – Wiki10-31K. Similar to the plot reporting on EURLex-4K above, each dot in the figure represents 10 independent trials (i.e., experiments) of compressed matrices Φ(1), . . . , Φ(10) on a given tuple of parameters (s, m). The curves in each panel correspond to the averaged values for the proposed algorithm and baselines over 10 trials; the shaded parts represent the empirical standard deviations over 10 trials. Similarly, in the first row, the precision of the proposed algorithm outperforms the FISTA especially when s is small. In the second & third rows for output difference and prediction loss, there are only slight improvement on the proposed algorithm than CD of output difference. 27 NeurIPS Paper Checklist 1. 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4,467	Robustly overfitting latents for flexible neural image compression Yura Perugachi-Diaz Vrije Universiteit Amsterdam y.m.perugachidiaz@vu.nl Arwin Gansekoele Centrum Wiskunde & Informatica awg@cwi.nl Sandjai Bhulai Vrije Universiteit Amsterdam s.bhulai@vu.nl Abstract Neural image compression has made a great deal of progress. State-of-the-art models are based on variational autoencoders and are outperforming classical models. Neural compression models learn to encode an image into a quantized latent representation that can be efficiently sent to the decoder, which decodes the quantized latent into a reconstructed image. While these models have proven successful in practice, they lead to sub-optimal results due to imperfect optimization and limitations in the encoder and decoder capacity. Recent work shows how to use stochastic Gumbel annealing (SGA) to refine the latents of pre-trained neural image compression models. We extend this idea by introducing SGA+, which contains three different methods that build upon SGA. We show how our method improves the overall compression performance in terms of the R-D trade-off, compared to its predecessors. Additionally, we show how refinement of the latents with our bestperforming method improves the compression performance on both the Tecnick and CLIC dataset. Our method is deployed for a pre-trained hyperprior and for a more flexible model. Further, we give a detailed analysis of our proposed methods and show that they are less sensitive to hyperparameter choices. Finally, we show how each method can be extended to three- instead of two-class rounding. 1 Introduction Image compression allows efficient sending of an image between systems by reducing their size. There are two types of compression: lossless and lossy. Lossless image compression sends images perfectly without losing any quality and can thus be restored in their original format, such as the PNG format. Lossy compression, such as BPG Bellard (2014), JPEG Wallace (1992) or JPEG2000 Skodras et al. (2001), loses some quality of the compressed image. Lossy compression aims to preserve as much of the quality of the reconstructed image as possible, compared to its original format, while allowing a significantly larger reduction in required storage. Traditional methods Wallace (1992); Skodras et al. (2001), especially lossless methods, can lead to limited compression ratios or degradation in quality. With the rise of deep learning, neural image compression is becoming a popular method Theis et al. (2017); Toderici et al. (2017). In contrast with traditional methods, neural image compression methods have been shown to achieve higher compression ratios and less degradation in image quality Ballé et al. (2018); Minnen et al. (2018); Lee et al. (2019). Additionally, neural compression techniques have shown improvements compared to traditional codecs for other data domains, such as video. Agustsson et al. (2020); Habibian et al. (2019); Lu et al. (2019). In practice, neural lossy compression methods have proven to be successful and achieve state-of-theart performance Ballé et al. (2018); Cheng et al. (2020); Minnen et al. (2018); Lee et al. (2019). These models are frequently based on variational autoencoders (VAEs) with an encoder-decoder structure 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Kingma and Welling (2013). The models are trained to minimize the expected rate-distortion (R-D) cost: R + λD. Intuitively, one learns a mapping that encodes an image into a compressible latent representation. The latent representation is sent to a decoder and is decoded into a reconstructed image. The aim is to train the compression model in such way that it finds a latent representation that represents the best trade-off between the length of the bitstream for an image and the quality of the reconstructed image. Even though these models have proven to be successful in practice, they do have limited capacity when it comes to optimization and generalization. For example, the encoder’s capacity is limited which makes the latent representation sub-optimal Cremer et al. (2018). Recent work Campos et al. (2019); Guo et al. (2020); Yang et al. (2020) proposes procedures to refine the encoder or latents, which lead to better compression performance. Furthermore, in neural video compression, other work focuses on adapting the encoder Aytekin et al. (2018); Lu et al. (2020) or finetuning a full compression model after training to improve the video compression performance van Rozendaal et al. (2021). The advantage of refining latents Campos et al. (2019); Yang et al. (2020) is that improved compression results per image are achieved while the model does not need to be modified. Instead, the latent representations for each individual image undergo a refining procedure. This results in a latent representation that obtains an improved bitstream and image quality over its original state from the pre-trained model. As mentioned in Yang et al. (2020), the refining procedure for stochastic rounding with Stochastic Gradient Gumbel Annealing (SGA) considerably improves performance. In this paper, we introduce SGA+, an extension of SGA that further improves compression performance and is less sensitive to hyperparameter choices. The main contributions are: (i) showing how changing the probability space with more natural methods instead of SGA boosts the compression performance, (ii) proposing the sigmoid scaled logit (SSL), which can smoothly interpolate between the approximate atanh, linear, cosine and round, (iii) demonstrating a generalization to rounding to three classes, that contains the two classes as a special case, and (iv) showing that SGA+ not only outperforms SGA on a similar pre-trained mean-scale hyperprior model as in Yang et al. (2020), but also achieves an even better performance for the pre-trained models of Cheng et al. (2020). Further, we show how SSL outperforms baselines in an R-D plot on the Kodak dataset, in terms of peak signal-to-noise ratio (PSNR) versus the bits per pixel (BPP) and in terms of true loss curves. Additionally, we show how our method generalizes to the Tecnick and CLIC dataset, followed by qualitative results. We analyze the stability of all functions and show the effect of interpolation between different methods with SSL. Lastly, we analyze a refining procedure at compression time that allows moving along the R-D curve when refining the latents with another λ than a pre-trained model is trained on Gao et al. (2022); Xu et al. (2023). The code can be retrieved from: https://github.com/yperugachidiaz/flexible_neural_image_compression. 2 Preliminaries and related work In lossy compression, the aim is to find a mapping of image x where the distortion of the reconstructed image ˆx is as little as possible compared to the original one while using as little storage as possible. Therefore, training a lossy neural image compression model presents a trade-off between minimizing the length of the bitstream for an image and minimizing the distortion of the reconstructed image Ballé et al. (2017); Lee et al. (2019); Minnen et al. (2018); Theis et al. (2017). Neural image compression models from Ballé et al. (2017); Cheng et al. (2020); Minnen et al. (2018); Theis et al. (2017), also known as hyperpriors, accomplish this kind of mapping with latent variables. An image x is encoded onto a latent representation y = ga(x), where ga(·) is the encoder. Next, y is quantized Q(y) = ˆy into a discrete variable that is sent losslessly to the decoder. The reconstructed image is given by: ˆx = gs(ˆy), where gs(·) represents the decoder. The rate-distortion objective that needs to be minimized for this specific problem is given by: L = R + λD = Ex∼px [−log2 pˆy(ˆy)] \| {z } rate +λ Ex∼px [d(x, ˆx)] \| {z } distortion , (1) where λ is a Lagrange multiplier determining the rate-distortion trade-off, R is the expected bitstream length to encode ˆy and D is the metric to measure the distortion of the reconstructed image ˆx 2 compared to the original one x. Specifically for the rate, px is the (unknown) image distribution and pˆy represents the entropy model that is learned over the data distribution px. A frequently used distortion measure for d(x, ˆx), is the mean squared error (MSE) or PSNR. In practice, the latent variable y often consists of multiple levels in neural compression. Namely, a smaller one named z, which is modeled with a relatively simple distribution p(z), and a larger variable, which is modeled by a distribution for which the parameters are predicted with a neural network using z, the distribution p(y\|z). We typically combine these two variables into a single symbol y for brevity. Furthermore, a frequent method of quantizing Q(·) used to train hyperpriors consists of adding uniform noise to the latent variable. 2.1 Latent optimization Neural image compression models have been trained over a huge set of images to find an optimal encoding. Yet, due to difficulties in optimization or due to constraints on the model capacity, model performance is sub-optimal. To overcome these issues, another type of optimizing compression performance is proposed in Campos et al. (2019); Yang et al. (2020) where they show how to find better compression results by utilizing pre-trained networks and keeping the encoder and decoder fixed but only adapting the latents. In these methods, a latent variable y is iteratively adapted using differentiable operations at test time. The aim is to find a more optimal discrete latent representation ˆy. Therefore, the following minimization problem needs to be solved for an image x: arg minˆy [−log2 pˆy(ˆy) + λd(x, ˆx)] . (2) This is a powerful method that can fit to a test image x directly without the need to further train an entire compression model. 2.2 Stochastic Gumbel Annealing Campos et al. (2019) proposes to optimize the latents by iteratively adding uniform noise and updating its latents. While this method proves to be effective, there is still a difference between the true ratedistortion loss ( ˆL) for the method and its discrete representation ˆy. This difference is also known as the discretization gap. Therefore, Yang et al. (2020) propose the SGA method to optimize latents and show how it obtains a smaller discretization gap. SGA is a soft-to-hard quantization method that quantizes a continuous variable v into the discrete representation for which gradients can be computed. A variable v is quantized as follows. First, a vector vr = (⌊v⌋, ⌈v⌉) is created that stacks the floor and ceil of the variable, also indicating the rounding direction. Next, the variable v is centered between (0, 1) where for the flooring: vL = v −⌊v⌋and ceiling: vR = ⌈v⌉−v. With a temperature rate τ ∈(0, 1), that is decreasing over time, this variable determines the soft-to-hardness where 1 indicates training with a fully continuous variable v and 0 indicates training while fully rounding variable v. To obtain unnormalized log probabilities (logits), the inverse hyperbolic tangent (atanh) function is used as follows: logits = (−atanh(vL)/τ, −atanh(vR)/τ). (3) To obtain probabilities a softmax is used over the logits, which gives the probability p(y) which is the chance of v being floored: p(y = ⌊v⌋), or ceiled: p(y = ⌈v⌉). This is approximated by the Gumbel-softmax distribution. Then, samples are drawn: y ∼Gumbel-Softmax(logits, τ) Jang et al. (2016) and are multiplied and summed with the vector vr to obtain the quantized representation: ˆv = P i (vr,i ∗yi). As SGA aids the discretization gap, this method may not have optimal performance and may not be as robust to changes in its temperature rate τ. Besides SGA, Yang et al. (2020) propose deterministic annealing Agustsson et al. (2017), which follows almost the same procedure as SGA, but instead of sampling stochastically from the Gumbel Softmax, this method uses a deterministic approach by computing probabilities with the Softmax from the logits. In practice, this method has been shown to suffer from unstable optimization behavior. 2.3 Other methods While methods such as SGA aim to optimize the latent variables for neural image compression at inference time, other approaches have been explored in recent research. Guo et al. (2021) proposed a soft-then-hard strategy alongside a learned scaling factor for the uniform noise to achieve better 3 0.0 0.2 0.4 0.6 0.8 1.0 v 0.0 0.2 0.4 0.6 0.8 1.0 probs Probability space SSL (a=2.3) Cosine SSL (a=1.33) Linear Atanh (a) Probability space for several SGA+ methods, along with atanh. Solid lines denote the probability of flooring ⌊v⌋and dotted lines the probability of ceiling ⌈v⌉. 1.5 1.0 0.5 0.0 0.5 1.0 1.5 v 0.0 0.2 0.4 0.6 0.8 1.0 probs Probability space p(y = v ) p(y = v 1) p(y = v + 1) r = 0.9, n = 1 r = 1, n = 3 (b) Three-class rounding for the extended version of linear. Solid lines denote two-class rounding, dashed lines denote three-class rounding and dotted lines indicate the smoothness. Figure 1: Probability space for (a) Two-class rounding (b) Three-class rounding compression and a smoother latent. These methods are used to fine-tune network parameters but not the latents directly. Zhu et al. (2022) proposed using Swin-transformer-based coding instead of ConvNet-based coding. They showed that these transforms can achieve better compression with fewer parameters and shorter decoding times. van Rozendaal et al. (2021) proposed to also fine-tune the decoder alongside the latent for video compression. While accommodating the additional cost of saving the model update, they demonstrated a gain of ∼1dB. Zhang et al. (2021) and Dupont et al. (2021) proposed using implicit neural representations for video and image compression, respectively. He et al. (2022) proposed an improved context model (SCCTX) and a change to the main transform (ELIC) that achieve strong compression results together. El-Nouby et al. (2023) revisited vector quantization for neural image compression and demonstrated it performs on par with hyperprior-based methods. Li et al. (2020) proposed a method to incorporate trellis-coded quantization in neural codecs. While these approaches change the training process, our work differs in that we only consider the inference process. Balcilar et al. (2023) proposes latent shift, a method that can further optimize latents using the correlation between the gradient of the reconstruction error and the gradient of the entropy. 3 Methods As literature has shown, refining the latents of pre-trained compression models with SGA leads to improved compression performance Yang et al. (2020). In this section, we extend SGA by introducing SGA+ containing three other methods for the computation of the unnormalized log probabilities (logits) to overcome issues from its predecessor. We show how these methods behave in probability space. Furthermore, we show how the methods can be extended to three-class rounding. 3.1 Two-class rounding Recall from SGA that a variable v is quantized to indicate the rounding direction to two classes and is centered between (0,1). Computation of the unnormalized log probabilities is obtained with atanh from Equation (3). Recall, that in general the probabilities are given by a softmax over the logits with a function of choice. As an example, for SGA the logits are computed with atanh. The corresponding probabilities for rounding down is then equal to: eatanh(vL) eatanh(vL)+eatanh(vR) . Then looking at the probability space from this function, see Figure 1a, the atanh function can lead to sub-optimal performance when used to determine rounding probabilities. The problem is that gradients tend to infinity when the function approaches the limits of 0 and 1, see Appendix A for the proof that gradients at 0 tend to ∞. This is not ideal, as these limits are usually achieved when the discretization gap is minimal. In addition, the gradients may become larger towards the end of optimization. Further analyzing the probability space, we find that there are a lot of possibilities in choosing probabilities for rounding to two classes. However, there are some constraints: the probabilities need to be monotonic functions, and the probabilities for rounding down (flooring) and up (ceiling) need to sum up to one. Therefore, we introduce SGA+ and propose three methods that satisfy the above constraints and can be used to 4 overcome the sub-optimality that the atanh function suffers from. We opted for these three as they each have their own interesting characteristics. However, there are many other functions that are also valid and would behave similarly to these three. We will denote the probability that v is rounded down by: p(y = ⌊v⌋), (4) where y represents the random variable whose outcome can be either rounded down or up. The probability that v is rounded up is conversely: p(y = ⌈v⌉) = 1 −p(y = ⌊v⌋). Linear probabilities To prevent gradient saturation or vanishing gradients completely, the most natural case would be to model a probability that linearly increases or decreases and has a gradient of one everywhere. Therefore, we define the linear: p(y = ⌊v⌋) = 1 −(v −⌊v⌋). (5) It is easy to see that: p(y = ⌈v⌉) = v −⌊v⌋. In Figure 1a, the linear probability is shown. Cosine probabilities As can be seen in Figure 1a, the atanh tends to have gradients that go to infinity for v close to the corners. Subsequently, a method that has low gradients in that area is by modeling the cosine probability as follows: p(y = ⌊v⌋) = cos2 (v −⌊v⌋)π 2 . (6) This method aids the compression performance compared to the atanh since there is less probability of overshooting the rounding value. Sigmoid scaled logit There are a lot of possibilities in choosing probabilities for two-class rounding. We introduced two probabilities that overcome sub-optimality issues from atanh: the linear probability from Equation (5), which has equal gradients everywhere, and cosine from Equation (6), that has little gradients at the corners. Besides these two functions, the optimal probability might follow a different function from the ones already mentioned. Therefore, we introduce the sigmoid scaled logit (SSL), which can interpolate between different probabilities with its hyperparameter a and is defined as follows: p(y = ⌊v⌋) = σ(−aσ−1(v −⌊v⌋)), (7) where a is the factor determining the shape of the function. SSL is exactly the linear for a = 1. For a = 1.6 and a = 0.65 SSL roughly resembles the cosine and atanh. For a →∞the function tends to shape to (reversed) rounding. Note that the main reason behind the linear version is the fact that it is the only function with constant gradients which is also the most robust choice, the cosine version is approximately mirrored across the diagonal of the −atanh(x) which shows that it is more stable compared to the −atanh(x), and the reason behind the SSL is that it is a function that can interpolate between all possible functions and can be tuned to find the best possible performance when necessary. 3.2 Three-class rounding As described in the previous section, the values for v can either be floored or ceiled. However, there are cases where it may help to round to an integer further away. Therefore, we introduce three-class rounding and show three extensions that build on top of the linear probability Equation (5), cosine probability Equation (6), and SSL from Equation (7). The probability that v is rounded is denoted by: p(y = ⌊v⌉) ∝f3c(w\|r, n), where w = v −⌊v⌉is centered around zero. Further, we define the probability that v is rounded +1 and rounded −1 is respectively given by: p(y = ⌊v⌉−1) ∝f3c(w −1\|r, n) and p(y = ⌊v⌉+ 1) ∝f3c(w + 1\|r, n). Recall, that vL, vR ∈[0, 1], whereas w ∈[−0.5, 0.5]. Defining w like this is more helpful for the 3-class since it has a center class. The general notation for the three-class functions is given by: f3c(w\|r, n) = f(clip(w · r))n, (8) where clip(·) clips the value at 0 and 1, r is the factor determining the height and steepness of the function and power n controls the peakedness of the function. Note that n can be fused with temperature τ together, to scale the function. This only accounts for the computation of the logits and not to modify the Gumbel temperature, therefore, τ still needs a separate definition. 5 0 500 1000 1500 2000 # Optimization iterations 0.60 0.64 0.68 0.72 True R-D Loss ( ) STE Uniform noise Atanh SSL (a) True R-D loss 0 500 1000 1500 2000 # Optimization iterations 0.10 0.05 0.00 0.05 0.10 Difference R-D Loss ( ) (b) Difference loss 0 500 1000 1500 2000 # Optimization iterations 31.2 31.6 32.0 32.4 PSNR (c) PSNR 0 500 1000 1500 2000 # Optimization iterations 0.285 0.290 0.295 0.300 BPP (d) BPP Figure 2: Performance plots of (a) True R-D Loss (b) Difference in loss (c) PSNR (d) BPP. Extended linear Recall that the linear probability can now be extended to three-class rounding as follows: flinear(w) = \|w\|. (9) A special case is f3c,linear(w\|r = 1, n = 1), where the function is equivalent to the linear of the two-class rounding from Equation (5). For r < 1 this function rounds to three classes, and for n ̸= 1 this function is not linear anymore. In Figure 1b, three-class rounding for the extension of Equation (5) can be found. As can be seen, solid lines denote the special case of two-class rounding with r = 1 and n = 1, dashed lines denote three-class rounding with r = 0.9 and n = 1 and dotted lines denote the two-class rounding with r = 1 and n = 3, which shows a less peaked function. For an example of two- versus three-class rounding, consider the case where we have variable v = −0.95. For two-class rounding there is only the chance of rounding to −1 with p(y = ⌊v⌉) (red solid line), a chance to round to 0 with p(y = ⌊v⌉+ 1) (green solid line) and zero chance to round to −2 with p(y = ⌊v⌉−1) (yellow solid line). For three-class rounding, with r = 0.9 and n = 1, when v = −0.95 we find a high chance to round to −1 with p(y = ⌊v⌉) (red dashed line) and a small chance to round to 0 with p(y = ⌊v⌉+ 1) (green dashed line) and a tiny chance to round to −2 with p(y = ⌊v⌉−1) (yellow dashed line). Extended cosine Similarly, we can transform the cosine probability from Equation (6) to three-class rounding: fcosine(w) = cos \|w\|π 2 . (10) When f3c,cosine(w\|r = 1, n = 2), this function exactly resembles the cosine for two-class rounding, and for r < 1 this function rounds to three classes. Extended SSL Additionally, SSL from Equation (7) can be transformed to three-class rounding as follows: fSSL(w) = σ −aσ−1 (\|w\|) , (11) where a is the factor determining the shape of the function. When f3c,SSL(w\|r = 1, n = 1), this function exactly resembles the two-class rounding case, and for r < 1, the function rounds to three classes. Recall that this function is capable of exactly resembling the linear function and approximates the cosine from two-class rounding for a = 1 and a = 1.6, respectively. 4 Experiments In this section, we show the performance of our best-performing method in an R-D plot and compare it to the baselines. Further, we evaluate and compare the methods with the true R-D loss performance ( ˆL), the difference between the method loss and true loss (L −ˆL), and corresponding PSNR and BPP plot that respectively expresses the image quality, and cost over t training steps. Finally, we show how our best-performing method performs on the Tecnick and CLIC dataset and show qualitative results. Following Yang et al. (2020), we run all experiments with temperature schedule τ(t) = min(exp{−ct}, τmax), where c is the temperature rate determining how fast temperature τ is decreasing over time, t is the number of train steps for the refinement of the latents and τmax ∈(0, 1) determines how soft the latents start the refining procedure. Additionally, we refine the latents for t = 2000 train iterations, unless specified otherwise. See Section 4.2 for the hyperparameter settings. 6 Implementation details We use two pre-trained hyperprior models to test SGA+, for both models we use the package from CompressAI Bégaint et al. (2020). The first model is similar to the one trained in Yang et al. (2020). Note, that the refining procedure needs the same storage and memory as theirs. In Appendix C, implementation details and results of this model can be found. All experiments in this section are run with a more recent hyperprior-based model which is based on the architecture of Cheng et al. (2020). Additionally, this model reduces the R-D loss for atanh on average by 7.6% and for SSL by 8.6%, compared to the other model. The model weights can be retrieved from CompressAI Bégaint et al. (2020). The models were trained with λ = {0.0016, 0.0032, 0.0075, 0.015, 0.03, 0.045}. The channel size is set to N = 128 for the models with λ = {0.0016, 0.0032, 0.0075}, refinement of the latents on Kodak with these models take approximately 21 minutes. For the remaining λ’s channel size is set to N = 192 and the refining procedure takes approximately 35 minutes. We perform our experiments on a single NVIDIA A100 GPU. Baseline methods We compare our methods against the methods that already exist in the literature. The Straight-Through Estimator (STE) is a method to round up or down to the nearest integer with rounding bound set to a half. This rounding is noted as ⌊·⌉. The derivative of STE for the backward pass is equal to 1 Bengio et al. (2013); Van Den Oord et al. (2017); Yin et al. (2019). The Uniform Noise quantization method adds uniform noise from u ∼U(−1 2, 1 2) to latent variable y. Thus: ˆy = y + u. In this manner ˆy becomes differentiable Ballé et al. (2017). As discussed in Section 2.2, we compare against Stochastic Gumbel Annealing, which is a soft-to-hard quantization method that quantizes a continuous variable v into a discrete representation for which gradients can be computed. 4.1 Overall performance Figure 3a shows the rate-distortion curve, using image quality metric PSNR versus BPP, of the base model and for refinement of the latents on the Kodak dataset with method: STE, uniform noise, atanh and SSL. We clearly see how SSL outperforms all other methods. In Appendix B.1a, the results after t = 500 iterations are shown, where the performance is more pronounced. To show how each of the methods behave, we take a closer look at the performance plots shown in Figure 2. The refinement results are obtained from the pre-trained model, trained with λ = 0.0075. The true R-D loss in Figure 2a shows that STE performs worse and has trouble converging, which is reflected in the R-D curve. Uniform noise quickly converges compared to atanh and SSL. We find that SSL outperforms all other methods, including atanh in terms of the lowest true R-D loss at all steps. Looking at the difference between the method loss and true loss Figure 2b, we find that both SSL and atanh converge to 0. Yet, the initial loss difference is smaller and smoother for SSL, compared to atanh. Additionally, uniform noise shows a big difference between the method and true loss, indicating that adding uniform noise overestimates its method loss compared to the true loss. Tecnick and CLIC To test how our method performs on other datasets, we use the Tecnick Asuni and Giachetti (2014) and CLIC dataset. We run baselines atanh and the base model and compare against SSL with a = 2.3. Figure 3b shows the corresponding R-D curves on the Tecnick dataset. Note that Appendix B.3 contains the results of the CLIC dataset which shows similar behavior as on Tecnick. We find that both refining methods improve the compression performance in terms of the R-D trade-off. Additionally, our proposed method outperforms atanh and shows how it improves performance significantly at all points. The R-D plot after t = 500 iterations for both datasets can be found in Appendix B.1. Note, that these results show even more pronounced performance difference. BD-Rate and BD-PSNR To evaluate the overall improvements of our method, we computed the BD-Rate and BD-PSNR Bjontegaard for Kodak, Tecnick and CLIC in Table 1. We observe that across the board SSL achieves an improvement in both BD-PSNR and BD-Rate. After 500 steps, SSL achieves almost double the BD-Rate reduction as atanh. The difference between the two becomes smaller at 2000 steps. This underlines the faster convergence behavior of the SSL method. 4.2 Qualitative results In Figure 4, we demonstrate the visual effect of our approach. We compare the original image with the compressed image from base model with λ = 0.0016, refinement method atanh, and SSL. For 7 Table 1: Pairwise Comparison between atanh and SSL of BD-PSNR and BD-Rate. BD-PSNR (dB) BD-Rate (%) 500 steps 2000 steps 500 steps 2000 steps Kodak Tecnick CLIC Kodak Tecnick CLIC Kodak Tecnick CLIC Kodak Tecnick CLIC Base vs SSL 0.50 0.57 0.56 0.82 0.95 0.89 -10.30 -11.60 -13.18 -16.23 -18.77 -20.11 Base vs Atanh 0.26 0.28 0.31 0.69 0.79 0.78 -5.52 -5.91 -7.37 -13.82 -15.93 -17.70 Atanh vs SSL 0.24 0.28 0.26 0.14 0.16 0.11 -5.04 -5.97 -6.20 -2.86 -3.34 -2.76 the image compressed by SSL, we observe that there are less artifacts visible in the overall image. For instance, looking at the window we see more texture compared to the base model and atanh method. Hyperparameter settings Refinement of the latents with pre-trained models, similar to the one trained in Yang et al. (2020), use the same optimal learning rate of 0.005 for each method. Refinement of the latents with the models of Cheng et al. (2020) use a 10 times lower learning rate of 0.0005. Following Yang et al. (2020), we use the settings for atanh with temperature rate τmax = 0.5, and for STE we use the smaller learning rate of 0.0001, yet STE still has trouble converging. Note that, we tuned STE just as the other baselines. However, the STE method is the only method that has a lot of trouble converging. Even with smaller learning rates, the method performed poorly. The instability of training is not only observed by us, but is also observed in Yang et al. (2020); Yin et al. (2019). For SGA+, we use optimal convergence settings, which are a fixed learning rate of 0.0005, and τmax = 1. Experimentally, we find approximately best performance for SLL with a = 2.3. 5 Analysis and Societal Impact In this section we perform additional experiments to get a better understanding of SGA+. An in-depth analysis shows the stability of each proposed method, followed by an experiment that expresses changes in the true R-D loss performance when one interpolates between functions. Further, we evaluate three-class rounding for each of our methods. Finally, we show how SGA+ for semi-multirate behavior improves the performance of its predecessor and we discuss the societal impact. Note, the results of the additional experiments are obtained from the model trained with λ = 0.0075. 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 PSNR Kodak (2000 Steps) Base model STE Uniform noise Atanh SSL (a) Kodak 0.1 0.2 0.3 0.4 0.5 0.6 Bits per pixel (BPP) 30 32 34 36 38 40 PSNR Tecnick (2000 Steps) Base model Atanh SSL (b) Tecnick 0.0 0.2 0.4 0.6 0.8 1.0 Bits per Pixel (bpp) 26 28 30 32 34 36 38 PSNR (dB) Semi-Multi-Rate Performance (500 Steps) Base model Atanh : 0.0016 SSL : 0.0016 Atanh : 0.0032 SSL : 0.0032 Atanh : 0.0075 SSL : 0.0075 Atanh : 0.015 SSL : 0.015 Atanh : 0.03 SSL : 0.03 Atanh : 0.045 SSL : 0.045 (c) Semi-multi-rate Figure 3: R-D performance for SSL on (a) Kodak with the baselines, (b) Tecnick with the base model and atanh and (c) Kodak for semi-multi-rate behavior with atanh. Best viewed electronically. (a) Original Image (b) Base Model (0.19 BPP vs 25.75 PSNR) (c) atanh (0.19 BPP vs 25.91 PSNR) (d) SSL (0.19 BPP vs 25.98 PSNR) Figure 4: Qualitative comparison of a Kodak image from pre-trained model trained with λ = 0.0016. Best viewed electronically. 8 Table 2: True R-D loss for different τmax settings of: atanh(v), linear, cosine and SSL with a = 2.3. The lowest R-D loss per column is marked with: ↓. Note that the function containing atanh is unnormalized. Function \τmax 0.2 0.4 0.6 0.8 1.0 exp atanh(v) 0.6301 0.6273 0.6267 0.6260 0.6259 1 −v (linear) 0.6291 ↓ 0.6229 ↓ 0.6225 0.6222 0.6220 cos2( vπ 2 ) 0.6307 0.6233 0.6194 ↓ 0.6186 0.6187 σ(−aσ−1(v)) 0.6341 0.6233 0.6196 0.6181 ↓ 0.6175 ↓ exp atanh(v) 0.0010 0.0044 0.0073 0.0079 0.0084 1 −v (linear) 0 0 0.0031 0.0041 0.0045 Temperature sensitivity Table 2 represents the stability of atanh and the SGA+ methods, expressed in true R-D loss, for different τmax settings for the temperature schedule. As can be seen, the most optimal setting is around τmax = 1 for each of the methods. In the table we find that the linear function is least sensitive to changes in τmax. To further examine the stability of the linear function compared to atanh, we subtract the best τmax, column-wise, from the linear and atanh of that column. Also taking into account the sensitivity results of the one of Yang et al. (2020) in Appendix C.2 we find in general, that overall performance varies little compared to the best τmax settings of the other methods and has reasonable performance. While SSL has the largest drop in performance when reducing τmax, it achieves the highest performance overall for higher values of τmax. If there is no budget to tune the hyperparameter of SGA+, the linear version is the most robust choice. Further, we evaluated the necessity of tuning both the latents and hyper-latents. When only optimizing the latents with the linear approach for τmax = 1, we found a loss of 0.6234. This is a difference of 0.0012, which implies that optimizing the hyper-latent aids the final loss. Table 3: True R-D loss of two- versus three-class rounding for SGA+ with the extended version of the linear, cosine, and SSL method at iteration 2000 and in brackets after 500 iterations. Function \Rounding Two Three f3c,linear(w\|r = 0.98, n = 2.5) 0.6220 (0.6594) 0.6175(0.6435) f3c,cosine(w\|r = 0.98, n = 3) 0.6187 (0.6516) 0.6175 (0.6449) f3c,sigmoidlogit(w\|r = 0.93, n = 2.5) 0.6175 (0.6445) 0.6203 (0.6360) Interpolation Table 4 represents the interpolation between different functions, expressed in true R-D loss. In Appendix B.2 the corresponding loss plots can be found. Values for a < 1 indicate methods that tend to have larger gradients for v close to the corners, while high values of a represent a method that tends to a (reversed) step function. The smallest a = 0.01 diverges and results in a large loss value compared to the rest. For a = 2.3 we find a lowest loss of 0.6175 and for a = 5 we find fastest convergence compared to the rest. Comparing these model results with the model results with the one of Yang et al. (2020), see Appendix C.2, we find that the Cheng et al. (2020) model obtains more stable curves. Table 4: True R-D loss results for the interpolation between different functions by changing a of the SSL. a R-D Loss 0.01 0.7323 0.3 0.6352 0.65 (approx atanh) 0.6260 0.8 0.6241 1 (linear) 0.6220 1.33 0.6199 1.6 (approx cosine) 0.6186 2.3 0.6175 ↓ 5 0.6209 Three-class rounding In Table 3, the true R-D loss for two versus three-class rounding can be found at iteration t = 2000 and in brackets t = 500 iterations. For each method, we performed a grid search over the hyperparameters r and n. As can be seen in the table, most impact is made with the extended version of the linear of SGA+, in terms of the difference between the two versus threeclass rounding at iteration t = 2000 with loss difference 0.0045 and t = 500 with 0.0159 difference. In Appendix B.3 a loss plot of the two- versus three- class rounding for the extended linear method can be found. Concluding, the three-class converges faster. For the extended cosine version, there is a smaller difference and for SSL we find that the three-class extension only boosts performance when run for t = 500. In Appendix C.2, the three-class experiments for the pre-trained model similar to Yang et al. (2020) can be found. We find similar behavior as for the model of Cheng et al. (2020), whereas with the linear version most impact is made, followed by the cosine and lastly SSL. The phenomenon that three-class rounding only improves performance for t = 500 iterations for SSL may be due to the fact that SSL is already close to optimal. Additionally, we ran an extra experiment to asses what percentage of the latents are assigned to the 3-classes. This is run with the best settings for the linear version f3c,linear(w\|r = 0.98, n = 2.5). At the first iteration, the probability is distributed as follows: p(y = ⌊v⌉) = 0.9329, for p(y = ⌊v⌉−1) = 0.0312, and p(y = ⌊v⌉+ 1) = 0.0359. This indicates that the class probabilities are approximately 3.12 for class −1 and 3.6 for class +1. This is a lot when taking into account that many samples are taken for a large dimensional latent. In 9 conclusion, three-class rounding may be attractive under a constraint optimization budget, possibly because it is easier to jump between classes. Additionally, for the extended linear three-class rounding also results in faster convergence. Semi-multi-rate behavior An interesting observation is that one does not need to use the same λ during refinement of the latents, as used during training. This is also mentioned in Gao et al. (2022) for image and Xu et al. (2023) for video compression. As a consequence of this approach, we can optimize to a neighborhood of the R-D curve without the need to train a new model from scratch. We experimented and analyze the results with methods: atanh and SSL. Figure 3c shows the performance after t = 500 iterations for the model of Cheng et al. (2020). We find that SSL is moving further along the R-D curve compared to atanh. Note the refinement does not span the entire curve and that the performance comes closer together for running the methods longer, see Appendix B.4. For future work it would be interesting how SGA+ compares to the methods mentioned in Gao et al. (2022); Xu et al. (2023), since SSL outperforms atanh. Societal impact The improvement of neural compression techniques is important in our data-driven society, as it allows quicker development of better codecs. Better codecs reduce storage and computing needs, thus lowering costs. However, training these codes requires significant computational resources, which harms the environment through power consumption and the need for raw materials. 6 Conclusion and Limitations In this paper we proposed SGA+, a more effective extension for refinement of the latents, which aids the compression performance for pre-trained neural image compression models. We showed how SGA+ has improved properties over SGA and we introduced SSL that can approximately interpolate between all of the proposed methods. Further, we showed how our best-performing method SSL outperforms the baselines in terms of the R-D trade-off and how it also outperforms the baselines on the Tecnick and CLIC dataset. Exploration of SGA+ showed how it is more stable under varying conditions. Additionally, we gave a general notation and demonstrated how the extension to three-class rounding improves the convergence of the SGA+ methods. Lastly, we showed how SGA+ improves the semi-multi-rate behavior over SGA. In conclusion, especially when a limited computational budget is available, SGA+ offers the option to improve the compression performance without the need to re-train an entire network and can be used as a drop-in replacement for SGA. Besides being effective, SGA+ also comes with some limitations. Firstly, we run each method for 2000 iterations per image. In practice this is extremely long and time consuming. We find that running the methods for 500 iterations already has more impact on the performance and we would recommend doing this, especially when a limited computational budget is available. Future work may focus on reducing the number of iterations and maintaining improved performance. Note, higher values for a flatten out quickly, but they achieve much better gains with low-step budgets. Further, the best results are obtained while tuning the hyperparameter of SSL and for each of our tested models this lead to different settings. Note, that the experiments showed that the linear version of SGA+ is least sensitive to hyperparameter changes and we would recommend using this version when there is no room for tuning. Additionally, although three-class rounding improves the compression performance in general, it comes with the cost of fine-tuning extra hyperparameters. Finally, it applies for each method that as the temperature rate has reached a stable setting, the performance will be less pronounced, the longer you train, but in return requires extra computation time at inference. 10 Acknowledgments This work was carried out on the Dutch national e-infrastructure with the support of SURF Cooperative. 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Recall that the probability is given by a 2-class softmax is defined by: K(v) = ef(v) ef(v) + eg(v) , where f(v) = −atanh(v) and g(v) = −atanh(1 −v). We will study the softmax for the first class 0, since the softmax is symmetric this also holds for the second class. The problem is that the gradients of the function K(v) will tend to ∞for both v →1 but also for v →0. Here we show that the gradients also tend ∞for v →0, via the normalization with the term g(v). First, take the derivative to v: dK(v) dv = dK(v) df(v) · df(v) dv + dK(v) dg(v) · dg(v) dv , where dK(v) df(v) = K(v)(1 −K(v)) and dK(v) dg(v) = −K(v) eg(v) ef(v)+eg(v) . Recall that datanh(v) dv = 1 1−v2 therefore df(v) dv = − 1 1−v2 and dg(v) dv = 1 1−(1−v)2 . Plugging this in and computing dK(v) dg(v) gives us: dK(v) dv = K(v)(1 −K(v)) − 1 1 −v2 −K(v) · eg(v) ef(v) + eg(v) · 1 1 −(1 −v)2 . Taking the limit to 0, (recall that limv→0 K(v) = 1, limv→0 ef(v) = 1 and limv→0 eg(v) = 0) allows the following simplifications: lim v→0 0 · − 1 1 −02 −1 · eg(v) 1 + 0 · 1 1 −(1 −v)2 For simplicity we substitute q = 1 −v (when q →1, then v →0) which will result in the following: lim v→0 −e−atanh(1−v) · 1 1 −(1 −v)2 = lim q→1 −e−atanh(q) · 1 1 −q2 , Recall, −atanh(q) = −1 2 ln 1+q 1−q, so e−atanh(q) = 1/ q 1+q 1−q thus: −lim q→1 r 1 −q 1 + 1 · 1 1 −q2 = −lim q→1 s 1 2 (1 −q) (1 −q2)2 Since 1 2 is a constant and limx→∞ √x = ∞the final step is to simplify and solve: −lim q→1 s (1 −q) (1 −q2)2 = −lim q→1 s −1 (q −1)(q + 1)2 = −∞. This concludes the proof that the gradients tend to −∞for v →0. 14 B Additional results In this appendix, additional experimental results for refinement of the latents with the pre-trained models of Cheng et al. (2020) can be found. Difference across runs Although we do not report the standard deviation for every run, the difference across runs is very small. For the model of Cheng et al. (2020) with λ = 0.0032, running five refinement procedures for 2000 iterations, results in a mean of 0.3969 and a standard deviation of 2.41 · 10−5. B.1 Additional overall performance Figure B.1 show the R-D curve for the Kodak dataset. We observe that atanh is similar to uniform noise at t = 500 iterations, while SSL manages to achieve better R-D gains. After t = 2000 iterations SLL achieves the best R-D trade-off, but the gain is a bit smaller compared to atanh at t = 500 iterations. Tecnick and CLIC Figure B.2 and Figure B.3 show the R-D results for respectively, Tecnick and CLIC at t = {500, 2000} iterations. We observe that SSL achieves best performance at t = 2000, compared to t = 500 iterations. However, running the method for 2000 iterations is in practice very long. Looking at the results at t = 500 iterations we see that there is already great improvement of performance for SSL, compared to the base model for both Tecnick and CLIC dataset. 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 PSNR Kodak (500 Steps) Base model STE Uniform noise Atanh SSL (a) Kodak (t = 500) 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 PSNR Kodak (2000 Steps) Base model STE Uniform noise Atanh SSL (b) Kodak (t = 2000) Figure B.1: Comparison of atanh and SSL on the Kodak dataset for t = {500, 2000} iterations. B.2 Interpolation In Figure B.4a the true loss, difference in loss, BPP and PSNR curves can be found. As can be seen for a = 0.01, the function diverges and a = 0.3 does not seem to reach stable behavior, both resulting in large loss values. For a ≥1, the difference in losses start close to zero (see Figure C.7b). SSL with a = 5 results in the fastest convergence and quickly finds a stable point but ends at a higher loss than most methods. B.3 Two- versus three- class rounding Besides an improved RD performance for the three-class rounding, we also found that three-class rounding leads to faster convergence. In Figure B.5 a true loss plot over the iterations for the linear method, can be found. As one can see, the three-class converges faster and especially on 500 iterations, boosts performance which makes it attractive under a constraint budget. 15 0.1 0.2 0.3 0.4 0.5 0.6 Bits per pixel (BPP) 30 32 34 36 38 40 PSNR Tecnick (500 Steps) Base model Atanh SSL (a) Tecnick (t = 500) 0.1 0.2 0.3 0.4 0.5 0.6 Bits per pixel (BPP) 30 32 34 36 38 40 PSNR Tecnick (2000 Steps) Base model Atanh SSL (b) Tecnick (t = 2000) Figure B.2: Comparison of atanh and SSL on the Tecnick dataset for t = {500, 2000} iterations. 0.1 0.2 0.3 0.4 0.5 0.6 Bits per pixel (BPP) 30 32 34 36 38 40 PSNR CLIC (500 Steps) Base model Atanh SSL (a) CLIC (t = 500) 0.1 0.2 0.3 0.4 0.5 0.6 Bits per pixel (BPP) 30 32 34 36 38 40 PSNR CLIC (2000 Steps) Base model Atanh SSL (b) CLIC (t = 2000) Figure B.3: Comparison of atanh and SSL on the CLIC dataset for t = {500, 2000} iterations. B.4 Semi-multi-rate behavior In Figure B.6, we have plotted the R-D curve of the base model (lime green line) and its corresponding R-D curves, obtained when refining the latents with the proposed λ’s. For each model trained using λ ∈{0.0016, 0.0032, 0.0075, 0.015, 0.03, 0.045}, we run atanh and SSL with a = 2.3 for t = 2000 iterations for all λ ∈{0.0004, 0.0008, 0.0016, 0.0032, 0.0075, 0.015, 0.03, 0.045, 0.06, 0.09}. We depicted the base curve alongside the curves for each base model. We observe that the improved performance by SSL is especially noticeable at t = 500 iterations in Figure B.6a. 16 500 1000 1500 2000 # Optimization iterations 0.60 0.64 0.68 0.72 True R-D Loss ( ) a=0.01 a=0.3 a=0.65 a=0.8 a=1 a=1.33 a=1.6 a=5 a=2.3 (a) True R-D Loss 0 500 1000 1500 2000 # Optimization iterations 0.1 0.0 0.1 0.2 Difference R-D Loss ( ) (b) Loss difference: L −ˆL 0 500 1000 1500 2000 # Optimization iterations 31.2 31.6 32.0 32.4 PSNR (c) PSNR 0 500 1000 1500 2000 # Optimization iterations 0.285 0.290 0.295 0.300 0.305 BPP (d) BPP Figure B.4: Interpolation performance plots of different a settings for SLL (a) True R-D Loss (b) Difference in loss (c) PSNR (d) BPP. 0 500 1000 1500 2000 # Optimization iterations 0.60 0.64 0.68 0.72 True R-D Loss ( ) Linear (2-class) Linear (3-class) Figure B.5: True R-D loss curves for two- versus three-class rounding of the linear method. 0.0 0.2 0.4 0.6 0.8 1.0 Bits per Pixel (bpp) 26 28 30 32 34 36 38 PSNR (dB) Semi-Multi-Rate Performance (500 Steps) Base model Atanh : 0.0016 SSL : 0.0016 Atanh : 0.0032 SSL : 0.0032 Atanh : 0.0075 SSL : 0.0075 Atanh : 0.015 SSL : 0.015 Atanh : 0.03 SSL : 0.03 Atanh : 0.045 SSL : 0.045 (a) t = 500 iterations 0.0 0.2 0.4 0.6 0.8 1.0 Bits per Pixel (bpp) 26 28 30 32 34 36 38 PSNR (dB) Semi-Multi-Rate Performance (2000 Steps) Base model Atanh : 0.0016 SSL : 0.0016 Atanh : 0.0032 SSL : 0.0032 Atanh : 0.0075 SSL : 0.0075 Atanh : 0.015 SSL : 0.015 Atanh : 0.03 SSL : 0.03 Atanh : 0.045 SSL : 0.045 (b) t = 200 iterations Figure B.6: R-D performance on Kodak of the Cheng et al. (2020) model when varying the target λ. Best viewed electronically. 17 C Mean-Scale Hyperprior To make a clear comparison we trained a similar mean-scale hyperprior as in Yang et al. (2020). Therefore, we use the architecture of Minnen et al. (2018), except for the autoregressive part as a context model. Instead, we use the regular convolutional architecture of Ballé et al. (2018). The model package for the mean-scale hyperprior is from CompressAI Bégaint et al. (2020). The details and results of this model can be found in this section. Similar as for Cheng et al. (2020) we run all experiments with temperature schedule τ(t) = min(exp{−ct}, τmax). Additionally, we refine the latents for t = 2000 train iterations, unless specified otherwise. Table C.1: True R-D loss results for the interpolation between different functions by changing a of the SSL. a R-D Loss 0.01 1.15 0.3 0.7528 0.65 (approx atanh) 0.7410 0.8 0.7396 1 (linear) 0.7386 1.33 0.7380 ↓ 1.6 (approx cosine) 0.7382 2.25 0.7388 5 0.7415 Implementations details The pre-trained mean-scale hyperpriors are trained from scratch on the full-size CLIC 2020 Mobile dataset Toderici et al. (2020), mixed with the ImageNet 2012 dataset Russakovsky et al. (2015) with randomly cropped image patches taken of size 256 × 256. For ImageNet, only images with a size larger than 256 for height and width are used to prevent bilinear up-sampling that negatively affects the model performance. During training, each model is evaluated on the Kodak dataset Kodak. The models were trained with λ = {0.001, 0.0025, 0.005, 0.01, 0.02, 0.04, 0.08}, with a batch size of 32 and Adam optimizer with a learning rate set to 1e−4. The models are trained for 2M steps, except for model λ = 0.001, which is trained for 1M steps and model λ = 0.08, which is trained for 3M steps. Training runs took half a week for the 1M step model, around a week for the 2M step models, and around 1.5 weeks for the larger 3M step model. We ran all models and methods on a single NVIDIA A100 GPU. Further, the models for λ = {0.04, 0.08} are trained with 256 hidden channels and the model for λ = 0.001 is trained with 128 hidden channels. The remaining models are trained with hidden channels set to 192. C.1 R-D Performance We evaluate our best-performing method SSL on the Kodak and Tecnick datasets, by computing the R-D performance, average over each of the datasets. The R-D curves use image quality metric PSNR versus BPP on the Kodak and Tecnick dataset. Recall that as base model we use the pre-trained mean-scale hyperprior, trained with λ = {0.001, 0.0025, 0.005, 0.01, 0.02, 0.04, 0.08}. For SSL we choose a = 4 3 as we found that this setting achieves the best R-D loss overall at 2000 iterations. This is a lower setting compared to the model by Cheng et al. (2020). The hyperparameters are similar to what we reported for Cheng et al. (2020) but with two main differences. We found that we could increase the learning rate by a factor 10 to 0.005 for atanh and SGA+. We also found that a lower a ∈[1.3, 1.4] was optimal. Kodak Figure C.2 shows the R-D curve for refining the latents, evaluated on Kodak. We compare SLL against baselines: STE, uniform noise, atanh and the base model at iteration t = 500 (see Figure C.2a) and after full convergence at t = 2000 (see Figure C.2b). As can be seen in Figure C.2a, STE performs slightly better than the base model, while after t = 2000 iterations the method performs worse, this is also reflected in the corresponding true loss curve for λ = 0.01 (see Figure C.1a), which diverges. Remarkably, for the smallest λ = 0.001, STE performs better than at t = 500. Adding uniform noise results in better performance when running the method longer. Comparing the R-D curves, Figure B.1 to Figure C.2, we find that most impact is made at t = 500 iterations. However, for the model similar to Yang et al. (2020) the performance lay closer to the performance of atanh, than for the model of Cheng et al. (2020). Tecnick Figure C.3 shows the R-D curve when refining latents on the Tecnick dataset, after t = 500 and t = 2000 iterations. As can be seen in the plot, we find that the longer the methods run, the closer the performance lies to each other. The improvement by SSL compared to atanh is greater for Tecnick than for Kodak. 18 0 500 1000 1500 2000 # Optimization iterations 0.725 0.750 0.775 0.800 0.825 0.850 True R-D Loss ( ) STE Uniform noise Atanh SSL (a) True R-D loss 0 500 1000 1500 2000 # Optimization iterations 0.10 0.05 0.00 0.05 0.10 Difference R-D Loss ( ) (b) Difference loss 0 500 1000 1500 2000 # Optimization iterations 32.0 32.5 33.0 33.5 34.0 PSNR (c) PSNR 0 500 1000 1500 2000 # Optimization iterations 0.38 0.40 0.42 0.44 BPP (d) BPP Figure C.1: Performance plots of (a) True R-D Loss (b) Difference in loss (c) PSNR (d) BPP. 0.00 0.25 0.50 0.75 1.00 1.25 Bits per pixel (BPP) 26 28 30 32 34 36 38 40 PSNR Kodak (500 Steps) Base model STE Uniform noise Atanh SSL (a) R-D curve at iteration t = 500 0.00 0.25 0.50 0.75 1.00 1.25 Bits per pixel (BPP) 26 28 30 32 34 36 38 40 PSNR Kodak (2000 Steps) Base model STE Uniform noise Atanh SSL (b) R-D curve at iteration t = 2000 Figure C.2: R-D performance on Kodak of the base mean-scale hyperprior model, STE, Uniform noise, SGA atanh and SSL with a = 4 3. CLIC Figure C.4 shows the R-D curve when refining latents on the CLIC dataset, after t = 500 and t = 2000 iterations. Similar to the previous results, we find that the longer the methods run, the closer the performance lies to each other and that running the method shorter already gives better performance, compared to the base model. BD-Rate Gain In Table C.2, we computed the change in BD-PSNR and BD-rate for the mean-scale hyperprior model. We observe that SSL is slightly better than atanh, although the difference between them is smaller than reported on the Cheng et al. (2020) model. The gap between 500 steps and 2000 steps is also smaller compared to the results in 1. C.2 Analysis In this appendix, we analyze additional experiments for the model, similar to those in Yang et al. (2020). The results for the analysis are obtained from a pre-trained model trained with λ = 0.01. 19 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 40 PSNR Tecnick (500 Steps) Base model Atanh SSL (a) R-D curve at iteration t = 500 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 40 PSNR Tecnick (2000 Steps) Base model Atanh Sigmoid logit (b) R-D curve at iteration t = 2000 Figure C.3: R-D performance on Tecnick of the base mean-scale hyperprior model, SGA atanh and SSL with a = 4 3. Best viewed electronically. 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 40 PSNR CLIC (500 Steps) Base model Atanh SSL (a) R-D curve at iteration t = 500 0.0 0.2 0.4 0.6 0.8 1.0 Bits per pixel (BPP) 28 30 32 34 36 38 40 PSNR CLIC (2000 Steps) Base model Atanh SSL (b) R-D curve at iteration t = 2000 Figure C.4: R-D performance on CLIC of the base mean-scale hyperprior model, SGA atanh and SSL with a = 4 3. Best viewed electronically. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Bits per Pixel (bpp) 26 28 30 32 34 36 38 40 PSNR (dB) Semi-Multi-Rate Performance (500 Steps) Base model Atanh : 0.001 SSL : 0.001 Atanh : 0.0025 SSL : 0.0025 Atanh : 0.005 SSL : 0.005 Atanh : 0.01 SSL : 0.01 Atanh : 0.02 SSL : 0.02 Atanh : 0.04 SSL : 0.04 Atanh : 0.08 SSL : 0.08 (a) R-D semi-multi-rate curve at iteration t = 500 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Bits per Pixel (bpp) 26 28 30 32 34 36 38 40 PSNR (dB) Semi-Multi-Rate Performance (2000 Steps) Base model Atanh : 0.001 SSL : 0.001 Atanh : 0.0025 SSL : 0.0025 Atanh : 0.005 SSL : 0.005 Atanh : 0.01 SSL : 0.01 Atanh : 0.02 SSL : 0.02 Atanh : 0.04 SSL : 0.04 Atanh : 0.08 SSL : 0.08 (b) R-D semi-multi-rate curve at iteration t = 2000 Figure C.5: R-D performance on Kodak of the base mean-scale hyperprior model, SGA atanh and SSL with a = 4 3. Each point is optimized with a different target λ ∈ {0.001, 0.0025, 0.005, 0.01, 0.02, 0.04, 0.08}. 20 Table C.2: Pairwise Comparison of BD-PSNR and BD-Rate for the Kodak, Tecnick, and CLIC dataset on the Mean-Scale Hyperprior Model. BD-PSNR (dB) BD-Rate (%) 500 steps 2000 steps 500 steps 2000 steps Kodak Tecnick CLIC Kodak Tecnick CLIC Kodak Tecnick CLIC Kodak Tecnick CLIC Base vs SSL 0.68 1.21 1.03 0.91 1.50 1.33 -13.52 -22.77 -21.87 -17.69 -28.17 -27.86 Base vs Atanh 0.59 1.06 0.89 0.87 1.46 1.30 -11.90 -20.20 -19.14 -17.03 -27.49 -27.28 Atanh vs SSL 0.09 0.15 0.14 0.04 0.04 0.03 -1.80 -3.22 -3.14 -0.82 -1.03 -0.74 0 500 1000 1500 2000 # Optimization iterations 0.725 0.750 0.775 0.800 0.825 0.850 True R-D Loss ( ) Loss ssl (lr=0.02) ssl (lr=0.01) ssl (lr=0.005) atanh (lr=0.02) atanh (lr=0.01) atanh (lr=0.005) Figure C.6: True R-D loss curves for different learning rates settings for method SSL and atanh. Table C.3: True R-D loss for different learning settings of: atanh and SSL with a = 4 3. At t = 2000 iterations and in brackets t = 500 iterations Lr \Method SSL atanh 0.02 0.7386 (0.7506) 0.7491 (0.7627) 0.01 0.7375 (0.7498) 0.7411 (0.7540) 0.005 (base) 0.7380 (0.7521) 0.7408 (0.7570) Learning rates We run SSL and atanh with higher learning rate settings of 0.02 and 0.01 and compare it to the results obtained with learning rate 0.005. Figure C.6 shows the corresponding loss curves and Table C.3 shows the corresponding loss values at t = {500, 2000} iterations. We find that for a learning rate of 0.01 the gap between atanh versus SSL at 500 iterations is only around 14.3% smaller and it remains pronounced, while with a learning rate of 0.01 the gap at 2000 iterations is around 28.6% better. This concludes that SSL benefits more than atanh at 2000 iterations, with a higher learning rate. More interestingly, for a learning rate of 0.02, atanh diverges whereas SSL still reaches comparable performance. This highlights the sensitivity of atanh. Table C.4: True R-D loss for different τmax settings of: atanh(v), linear, cosine and SSL with a = 4 3. Lowest R-D loss per column is marked with: ↓. Note that the function containing atanh is unnormalized. Function \τmax 0.2 0.4 0.6 0.8 1.0 exp atanh(v) 0.7445 ↓ 0.7408 0.7412 0.7416 0.7418 1 −v (linear) 0.7458 0.7406 ↓ 0.7390 ↓ 0.7386 0.7386 cos2( vπ 2 ) 0.7496 0.7414 0.7393 0.7387 0.7384 σ(−aσ−1(v)) 0.7578 0.7409 0.7391 0.7383 ↓ 0.7380 ↓ exp atanh(v) 0 0.0002 0.0022 0.0033 0.0038 1 −v (linear) 0.0013 0 0 0.0003 0.0006 Temperature sensitivity Table C.4 represents the stability of atanh and the SGA+ methods, expressed in true R-D loss, for different τmax settings for the temperature schedule. As can be seen, the most optimal setting is with τmax = 1 for each of the SGA+ methods. atanh obtains equal loss for τmax ∈[0.4, 0.5]. In general, we find that the linear method of SGA+ is least sensitive to changes in τmax and has equal loss between τmax ∈ [0.7, 1]. To further examine the stability of the linear function compared to atanh, we subtract the best τmax, column-wise, from the linear and atanh of that column. We now see that the linear function is not only least sensitive to changes in τmax, but overall varies little compared to the best τmax settings of the other methods. While the SSL has the largest drop in performance when reducing τmax, it achieves the highest performance overall for higher values of τmax. 21 Interpolation Table C.1 presents the true R-D loss results for the interpolation with different a settings for SSL for the mean-scale hyperprior model. In Figure C.7, the corresponding overall performance of the methods can be found. As can be seen in Figure C.7a, for a = {0.01, 0.30}, the functions diverge, resulting in large loss values. For a = 0.65, we find that the loss curve is slightly unstable at the beginning of training, which can be seen in the bending of the curve, indicating non-optimal settings. This may be due to the fact that we run all methods with the same τmax = 1 for a fair comparison. Additionally, note that SSL with a = 0.65 obtains a true R-D loss of 0.7410 compared to 0.7418 for atanh with the same settings. This is due to the fact that SSL, especially in the tails of the probability, is slightly more straight-curved compared to the atanh when looking at its probability space. Remarkably, for a ≥1, the difference in losses start close to zero (see Figure C.7b). SSL with a = 5 results in the fastest convergence and quickly finds a stable point but ends at a higher loss than most methods. Table C.5: True R-D loss of two versus three-class rounding for SGA+ with the extended version of the linear, cosine, and SSL method at iteration 500 and in brackets after 2000 iterations. Function \Rounding Two Three f3c,linear(w\|r = 0.98, n = 1.5) 0.7552 (0.7386) 0.7517(0.7380) f3c,cosine(w\|r = 0.98, n = 2) 0.7512 (0.7384) 0.7513 (0.7379) f3c,sigmoidlogit(w\|r = 0.93, n = 1.5) 0.7524 (0.7380) 0.7504 (0.7380) Three-class rounding Table C.5 shows the true R-D loss for two- versus three-class rounding, at iteration t = 500 and in brackets t = 2000 iterations. For each method, we performed a grid search over the hyperparameters r and n. Additionally, for the extended SSL, we also performed a grid search over a and found the best setting to be a = 1.4. As can be seen in the table, most impact is made with the extended version of the linear of SGA+, in terms of the difference between the two versus three-class rounding at iteration t = 500 with loss difference 0.0035 and t = 2000 with 0.0006 difference. There is a small difference at t = 500 for the extended cosine version. In general, we find that running models longer results in convergence to similar values. SSL converges to equal values for two- and three-class rounding. Semi-multi-rate behavior Similar as in Appendix B.4, we experimented with different values for λ to obtain a semi-multi-rate curve. For every pre-trained model, we ran SSL and atanh using λ ∈{0.001, 0.0025, 0.005, 0.01, 0.02, 0.04, 0.08}. In Figure C.5, we have plotted the R-D curve of the base model (lime green line) and its corresponding R-D curves, obtained when refining the latents with the proposed λ’s for atanh and SSL. As can be seen, running the methods for t = 500 iterations, SSL obtains best performance. While the longer you train, the closer together the performance will be. 22 0 500 1000 1500 2000 # Optimization iterations 0.725 0.750 0.775 0.800 0.825 0.850 True R-D Loss ( ) a=0.01 a=0.30 a=0.65 a=0.8 a=1 a=1.6 a=2.25 a=5 a=1.3 (a) True R-D Loss 0 500 1000 1500 2000 # Optimization iterations 0.1 0.0 0.1 0.2 Difference R-D Loss ( ) (b) Loss difference: L −ˆL 0 500 1000 1500 2000 # Optimization iterations 32.2 32.9 33.6 PSNR (c) PSNR 0 500 1000 1500 2000 # Optimization iterations 0.40 0.41 0.42 0.43 0.44 BPP (d) BPP Figure C.7: Interpolation performance plots of different a settings for SLL (a) True R-D Loss (b) Difference in loss (c) PSNR (d) BPP under the mean-scale hyperprior model. 23 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: See the Methods and Experiments sections. 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4,468	Parameter Symmetry and Noise Equilibrium of Stochastic Gradient Descent Liu Ziyin Massachusetts Institute of Technology, NTT Research ziyinl@mit.edu Mingze Wang Peking University mingzewang@stu.pku.edu.cn Hongchao Li The University of Tokyo lhc@cat.phys.s.u-tokyo.ac.jp Lei Wu Peking University leiwu@math.pku.edu.cn Abstract Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries – a broad subclass of continuous symmetries present in the model architecture or loss function – interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a “Noether flow”) of the parameters θ along the degenerate direction to a unique initializationindependent fixed point θ∗. These points are referred to as the noise equilibria because, at these points, noise contributions from different directions are balanced and aligned. Then, we show that the balance and alignment of gradient noise can serve as a novel alternative mechanism for explaining important phenomena such as progressive sharpening/flattening and representation formation within neural networks and have practical implications for understanding techniques like representation normalization and warmup. 1 Introduction Stochastic gradient descent (SGD) and its variants have become the cornerstone algorithms used in deep learning. In the continuous-time limit, the algorithm can be written as [19, 13, 21, 32, 9]: dθt = −∇L(θt)dt + √ 2σ2Σ(θt)dWt, (1) where Σ(θ) is the covariance matrix of gradient noise (Section 3) with the prefactor σ2 = η/(2S) modeling the impact of a finite learning rate η and batch size S; Wt denotes the Brownian motion. When σ = 0, Eq. (1) corresponds to gradient descent (GD)1. However, SGD and GD can exhibit significantly different behaviors, often converging to solutions with significantly different levels of performance [31, 39, 44, 22, 49]. Notably, even when σ2 ≪1, where we expect a close resemblance between SGD and GD over finite time [19], their long-time behaviors still differ substantially [26]. These observations indicate that gradient noise can bias the dynamics significantly, and revealing its underlying mechanism is thus crucial for understanding the disparities between SGD and GD. Contribution. In this paper, we study the how of SGD noise biases training through the lens of symmetry. Our key contributions are summarized as follows. We show that 1. when symmetry exists in the loss function, the dynamics of SGD can be precisely characterized and is different from GD along the degenerate direction; 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1“Gradient descent” and “gradient flow” are used interchangeably as we work in the continuous-time limit. 2. the treatment of common symmetries, including the rescaling and scaling symmetry in deep learning, can be unified in a single theoretical framework that we call the exponential symmetry; 3. for any θ, every exponential symmetry implies the existence of a unique and attractive fixed point along the degenerate direction for SGD; 4. symmetry and balancing of noise can serve as novel mechanisms for important deep learning phenomena such as progressive sharpening/flattening and latent representation formation. Figure 1: An example of a 2d loss function with scale invariance: ℓ(θ) = ℓ(λθ) for a scalar λ and θ ∈R2. Because of the symmetry, the gradient ∇ℓ must be tangential to the circles whose center is the origin. This implies that the norm ∥θ∥does not change during gradient flow training. However, when the training is stochastic or discretetime, SGD must move outward. If the model starts at θt, it must move to a larger circle. As an illustrative example, this loss function has a unique and attractive fixed point: ∥θ∥= ∞. SGD will diverge after training under scale invariance. Also, see Remark 4.4 for a discussion of the difference between discrete-time and continuous-time dynamics. See Figure 1 for an illustration of how symmetry leads to a systematic flow of SGD. This work is organized as follows. We discuss the most relevant works in Section 2. The main theoretical results are presented in Section 4. We apply our theory to understand specific problems and present numerical results in Section 5. The last section concludes this work. All the proofs are presented in the Appendix. 2 Related Works The dynamics of SGD in the degenerate directions of the loss landscape is a poorly understood problem. There are two closely related prior works. Ref. [48] studies the dynamics of SGD when there is a simple rescaling symmetry and applies it to derive the stationary distribution of SGD for linear networks. Our result is more general because rescaling symmetry is the simplest case of exponential symmetries2. Another related work is Ref. [20], which studies a different special case of exponential symmetry, the scale invariance, and in the presence of weight decay. Their analysis assumes the existence of the fixed point of the dynamics, which we proved to exist. Also related is the study of conservation laws under gradient flow [34, 18, 24, 43, 40], which we will discuss more closely in Section 4. However, these works do not take the stochasticity of training into account. Comparing with these results that assume no stochasticity, our result could suggest that SGD converges to initialization-independent solutions, whereas the GD finds solutions are strongly initialization-dependent. In addition, Section D extends our main result to discrete-time SGD. 3 Preliminaries Setup and Notations. Let ℓ∶Ω× Z ↦R denote the per-sample loss, with Ωand Z denoting the parameter and sample space, respectively. Here, z ∈Z includes both the input and label and accordingly. We use Ez = E to denote the expectation over a given training set. Therefore, L(θ) = Ez[ℓ(θ,z)] is the empirical risk function. The covariance of gradient noise is given by Σ(θ) = Ez[∇ℓ(θ,z)∇ℓ(θ,z)⊺] −∇L(θ)∇L(θ)⊺. Additionally, we use Σv(θ) ∶= Ez[∇vℓ(θ,z)∇vℓ(θ,z)⊺] −∇vL(θ)∇vL(θ)⊺to denote the covariance of gradient noise impacting on the subset of parameters v. Denote by (θt)t≥0 the trajectory of SGD or GD. For any h ∶Ω↦R, we write ht = h(θt) and ˙h(θt) = d dth(θt) for brevity. When the context is clear, we also use ℓ(θ) to denote ℓ(θ,z). Symmetry. The per-sample loss ℓ(⋅,⋅) is said to possess the Q-symmetry if ℓ(θ,z) = ℓ(Qρ(θ),z),∀ρ ∈R, (2) where (Qρ)ρ∈R is a set of continuous transformation parameterized by ρ ∈R. Without loss of generality, we assume Q0 = id. The most common symmetries exist within the model f, namely fθ is invariant under certain transformations of θ. However, our formalism is slightly more general in the sense that it is also possible for the model to be variant while the per-sample loss remains unchanged, which appears in self-supervised learning [50], for example. 2These are known as “continuous symmetries.” Prior works also studied SGD training under discrete symmetries [45, 3], which are different from continuous symmetries. 2 4 Continuous Symmetry and Noise Equilibria Taking the derivative with respect to ρ at ρ = 0 in Eq. (2), we have 0 = ∇θℓ(θ,z) ⋅J(θ), (3) where J(θ) = dQρ(θ) dρ ∣ρ=0. Denote by C be the antiderivative of J, that is, ∇C(θ) = J(θ). Then, taking the expectation over z in (3) gives the following conservation law for GD solutions (θt)t≥0: ˙C(θt) = 0. (4) Essentially, this is a consequence of Noether’s theorem [25], and C will be called a “Noether charge” in analogy to theoretical physics. The conservation law (4) implies that the GD trajectory is constrained on the manifold {θ ∶C(θ) = C(θ0)}. We refer to Ref. [18] for a study of this type of conservation law under the Bregman Lagrangian [16]. 4.1 Noether Flow in Degenerate Directions In this paper, we are interested in how C(θt) changes, if it changes at all, under SGD. By Ito’s lemma, we have the following Noether flow (namely, the flow of the Noether charge): ˙C(θt) = σ2Tr[Σ(θt)∇2C(θt)], (5) where ∇2C denotes the Hessian matrix of C. The derivation is deferred to Appendix B. By definition, Σ(θt) is always positive semidefinite (PSD). Thus, we immediately have: if ∇2C is PSD throughout training, C(θt) is a monotonically increasing function of time. Conversely, if ∇2 θC is negative semidefinite (NPD), C(θt) is a monotonically decreasing function of time. The existence of symmetry implies that (with suitable conditions of smoothness) any solution θ resides within a connected, loss-invariant manifold, defined as Mθ ∶= {Qρ(θ) ∶ρ ∈R}. We term directions within this manifold as “degenerate directions” since movement along them does not change the loss value. Notably, the biased flow (5) suggests that SGD noise can drive SGD to explore within this manifold along these degenerate directions since the value of C(θ) for θ ∈Mθ can vary. 4.2 Exponential symmetries Now, let us focus on a family of symmetries that is common in deep learning. Since the corresponding conserved quantities are quadratic functions of the model parameters, we will refer to this class of symmetries as exponential symmetries. Definition 4.1. (Qρ)ρ is said to be a exponential symmetry if J(θ) ∶= d dρQρ(θ)∣ρ=0 = Aθ for a symmetric matrix A. This implies when ρ ≪1, Qρ = id + ρA + o(ρ). In the sequel, we also use the words “A-symmetry” and “Q-symmetry” interchangeably since all properties of Qρ we need can be derived from A. This definition applies to the following symmetries that are common in deep learning: • Rescaling symmetry: Qρ(a,b) = (a(ρ+1),b/(ρ+1)), which appears in linear and ReLU networks [7, 48]. In this symmetry, A = diag(Ia,−Ib), where I is the identity matrix with dimensions matching that of a and b. • Scaling symmetry: Qρθ = (ρ + 1)θ, which exists whenever part of the model normalized using techniques like batch normalization [14], layer normalization [2], or weight normalization [29]. In this case, A = I. • Double rotation symmetry: This symmetry appears when parts of the model involve a matrix factorization problem, where for an arbitrary invertible matrix B ℓ= ℓ(UW) = ℓ(UBB−1W). Writing the exponential symmetry for this case is a little cumbersome. We need first to view U and W as a single vector, and the exponential transformation is given by a block-wise diagonal matrix diag(B,...,B,B−1,...,B−1). See Section 5.1 for more detail. It is possible for only a subset of parameters to have a given symmetry. Mathematically, this corresponds to the case when A is low-rank. It is also common for ℓto have multiple exponential symmetries at once, often for different (but not necessarily disjoint) subsets of parameters. For example, a ReLU network has a different rescaling symmetry for every hidden neuron. 3 It is obvious that under this Q symmetry, the Noether charge has a simple quadratic form: C(θ) = θ⊺Aθ. (6) Moreover, the interplay between this symmetry and weight decay can be explicitly characterized in our framework. To this end, we need the following definition. Definition 4.2. For any γ ∈R, we say ℓγ(θ,x) ∶= ℓ(θ,x) + γ∥θ∥2 has the Q symmetry as long as ℓ(θ,x) has the Q symmetry. For the SGD dynamics that minimizes Lγ(θ) = Ex[ℓγ(θ,x)], it follows from (5) that ˙C(θt) = −4γC(θt) + σ2Tr[Σ(θt)A] =∶G(θt). (7) Thus, a positive γ always causes ∣C(θt)∣to decay, and the influence of symmetry is determined by the spectrum of A. Denote by A = ∑j µjnjn⊺ j the eigendecomposition of A. Then, Tr[Σ(θt)A] = ∑ i∶µi>0 µin⊺ i Σ(θt)ni + ∑ j∶µj<0 µjn⊺ jΣ(θt)nj. This gives a clear interpretation of the interplay between SGD noise and the exponential symmetry: the noise along the positive directions of A causes C(θt) to grow, while the noise along the negative directions causes C(θt) to decay. In other words, the noise-induced dynamics of C(θt) is determined by the competition between the noise along the positive- and negative-eigenvalue directions of A. Time Scales. The above analysis implies that the dynamics of SGD can be decomposed into two parts: the dynamics that directly reduce loss, and the dynamics along the degenerate direction of the loss, which is governed by Eq (5). These two dynamics have essentially independent time scales. The first part is independent of the σ2 in expectation, whereas the time scale of the dynamics in the degenerate directions depends linearly on σ2. The first time scale term is due to the dynamics of empirical risk minimization. The second time scale tequi is the time scale for Eq. (5) to reach equilibrium, which is irrelevant to direct risk minimization. When the parameters are properly tuned, term is of order 1, whereas tequi is proportional to σ2 = η/(2S). Therefore, when σ2 is large, the parameters will stay close to the equilibrium point early in the training, and one can expect that ˙C(θt) is approximately zero after tequi. In line with Ref. [20], this can be called the fast-equilibrium phase of learning. Likewise, when σ2 ≪1, the approach to equilibrium will be slower than the actual time scale of risk minimization, and the dynamics in the degenerate direction only take off when the model has reached a local minimum. This can be called the slow-equilibrium phase of learning. 4.3 Noise Equilibrium and Fixed Point Theorem It is important and practically relevant to study the stationary points of dynamics in Eq. (7). Formally, the stationary point is reached when −γC(θ)+ηTr[Σ(θ)A] = 0. Because we make essentially no assumption about ℓ(θ) and Σ(θ), one might feel that it is impossible to guarantee the existence of a fixed point. Remarkably, we prove below that a fixed point exists and is unique for every connected degenerate manifold. To start, consider the exponential maps generated by A: eλAθ ∶= lim ρ→0(I + ρA + o(ρ))λ/ρθ, which applies the symmetry transformation to θ for λ/ρ times. Then, it follows that if we apply Qρ transformation to θ infinitely many times and for a perturbatively small ρ, ℓ(θ) = ℓ(eλAθ). (8) Thus, the exponential symmetry implies the symmetry with respect to an exponential map, a fundamental element of Lie groups [11]. Note that exponential-map symmetry is also an exponential symmetry by definition. For the exponential map, the degenerate direction is clear: for any λ, θ connects to eλAθ without any loss function barrier. Therefore, the degenerate direction for any exponential symmetry is unbounded. Now, we prove the following fixed point theorem, which shows that for every exponential symmetry and every θ, there is one and only one corresponding fixed point in the degenerate direction. 4 Theorem 4.3. Let the per-sample loss satisfy the A-exponential symmetry and θλ ∶= exp[λA]θ. Then, for any θ and any γ ≥0,3 (1) G(θλ) (Eq. (7)) and −C(θλ) are monotonically decreasing functions of λ; (2) there exists a λ∗∈R ∪{±∞} such that G(θλ∗) = 0; (3) in addition, if G(θλ) ≠0, λ∗is unique and G(θλ) is strictly monotonic; (4) in addition to (3), if Σ(θ) is differentiable, λ∗(θ) is a differentiable function of θ. Remark 4.4. It is now worthwhile to differentiate gradient flow (GF), GD, SGD, and stochastic gradient flow (SGF). Technically, one can prove that the same result holds for discrete-time GD and SGD in expectation, and GF is the only of the four algorithms that do not obey this theorem (See Section D), and so one could argue that the discrete step size is the essential cause of noise balance. Mathematically, the SGF can be seen as a model of the leading order effect of having a finite step size and thus also share this effect (remember that the Ito Lemma contains a second-order term in dθ). That being said, there is a practical caveat: in practice, we find it much easier for models to reach these fixed points with SGD than with GD, and so it is fair to say that this effect is the most dominant when gradient noise is present. Part (1), together with Part (2), implies that the unique stationary point is essentially attractive. This is because ˙C decreases with λ while C increases with it. Let C∗= C(θλ∗). Thus, C(θ) −C∗ always have the opposite sign of λ∗, while d dtC(θ) will have the same sign. Conceptually, this means that C will always move to reduce its distance to C∗. Assuming that C∗is a constant in time (or close to a constant, which is often the case at the end of training), Part (1) implies that d dt(C(θ) −C∗) ∝−sgn(C(θ) −C∗), signaling a convergence to C(θ) = C∗. In other words, SGD will move to restore the balance if it is perturbed away from λ∗= 0. If the matrix ΣA is wellbehaved, one can indeed establish the convergence to the fixed point in the relative distance even if C∗is mildly divergent due to diffusion. Because this part is strongly technical and our focus is on the fixed points, we leave the formal statement and its discussion to Appendix B.3. Theorem 4.5. (Informal) Let C∗follow a drifted Brownian motion and ΣA satisfy two well-behaved conditions. Then, either C −C∗→0 in L2 or (C −C∗)2/(C∗)2 →0 in probability. Parts (2) and (3) show that a unique fixed point exists. We note that it is more common than not for the conditions of uniqueness to hold because there is generally no reason for Tr[Σ(θ)A] or Tr[θθ⊺A] to vanish simultaneously, except in some very restrictive subspaces. One major (perhaps the only) reason for the first trace to vanish is when the model is located at an interpolation minimum. However, interpolation minima are irrelevant for modern large-scale problems such as large language models because the amount of available text for training far exceeds the size of the largest models. Even when the interpolation minimum exists, the unique fixed point should still exist when the training is not complete. See Figure 1. Part (4) means that the fixed points of the dynamics is well-behaved. If the parameter θ has a small fluctuation around a given location, C will also have a small fluctuation around the fixed point solution. This justifies approximating C by a constant value when θ changes slowly and with small fluctuation. Fixed point as a Noise Equilibrium. Let θ∗be a fixed point of (7). It must satisfy 4γC(θ∗) = σ2Tr[Σ(θ∗)A]. (9) Hence, a large weight decay leads to a small ∣C(θ∗)∣, whereas a large gradient noise leads to a large ∣C(θ∗)∣. When there is no weight decay, we get a different equilibrium condition: Tr[Σ(θ∗)A] = 0, which can be finite only when A contains both positive and negative eigenvalues. This equilibrium condition is equivalent to ∑i∶µi>0 µin⊺ i Σ(θ∗)ni = −∑j∶µj<0 µjn⊺ jΣ(θ∗)nj. Namely, the overall gradient fluctuation in the two different subspaces specified by the symmetry A must balance. We will see that the main implication of this result is that the gradient noise between different layers of a deep neural network should be balanced at the end of training. Conceptually, Theorem 4.3 suggests the existence of a special type of fixed point for SGD, which the following definition formalizes. Definition 4.6. θ is a noise equilibrium for a nonconstant function C(θ) if ˙C(θ) = 0 under SGD. 5 Applications Now, we analyze the noise equilibria of a few important problems. These examples are prototypes of what appears frequently in deep learning practice and substantiate our arguments with numerical 3A similar result can be proved for the discrete-time SGD. See Section D. 5 Figure 2: Comparison between GD and SGD for matrix factorizations. Left: Example of a learning trajectory. The convergence speed is almost exponential-like in experiments. Mid: evolution of 10 individual elements of ∆ij ∶= (U ⊺ΓUU −WΓW W ⊺)ij. As the theory shows, they all move close to zero and fluctuate with a small variance. Right: Converged solutions of SGD agree with the prediction of Theorem 5.2, but are an order of magnitude away from the solution found by GD, even if they start from the same init. examples. In addition, an experiment with the scale invariance in normalized tanh networks is presented in Appendix A.1. 5.1 Generalized Matrix Factorization Exponential symmetry is also observed when the model involves a (generalized) matrix factorization. This occurs in standard matrix completion problems [33] or within the self-attention of transformers through the key and query matrices [35]. For a (generalized) matrix factorization problem, we have the following symmetry in the objective: ℓ(U,W,θ′) = ℓ(UA,A−1W,θ′) (10) for any invertible matrix A and symmetry-irrelevant parameters θ′. We consider matrices A that are close to identity: A = I + ρB + O(ρ2), and A−1 = I −ρB + O(ρ2). Therefore, for an arbitrary symmetric B, we have a conserved quantity for GD: CB(θ) = Tr[UBU ⊺] −Tr[W ⊺BW]. This conservation law can also be written in the matrix form, which is a well-known result for GD [8, 24]: (WtW ⊺ t −U ⊺ t Ut) = (W0W ⊺ 0 −U ⊺ 0 U0). (11) For SGD, applying (5) gives the following proposition. Proposition 5.1. Suppose the symmetry (10) holds. Let U = (˜u1,⋯, ˜ud2)⊺∈Rd2×d, W = ( ˜w1,⋯, ˜wd0) ∈Rd×d0, where ˜ui, ˜wj ∈Rd. Let CB(θ) = Tr[UBU ⊺]−Tr[W ⊺BW] for any symmetric matrix B ∈Rd×d. Then, for SGD, we have ˙CB(θt) = σ2 ⎛ ⎝ d2 ∑ i=1 Tr[Σ˜ui(θt)B] − d0 ∑ j=1 Tr[Σ ˜ wj(θt)B]⎞ ⎠. This dynamics is analytically solvable when U ∈R1×d and W ∈Rd×1. In this case, taking B = Ek,l+ El,k where Ei,j denotes the matrix with entries of 1 at (i,j) and zeros elsewhere. For this choice of B, we obtain that CB(θ) = WkWl −UkUl, and applying the results we have derived, it is easy to show that for some random variable r: ˙CB(θt) = −Var[r(θt)]CB(θt), which signals an exponential decay. For common problems, Var[r(θt)] > 0 [48]. Since the choice of B is arbitrary, we have that WkWl →UkUl for all k and l. The message is rather striking: SGD automatically converges to a solution where all neurons output the same sign (sgn(Ui) = sgn(Uj)) at an exponential rate. 5.2 Balance and Stability of Matrix Factorization As a concrete example, let us consider a two-layer linear network (this can also be seen as a variant of standard matrix factorizations): ℓγ = ∥UWx −y∥2 + γ(∥U∥2 F + ∥W∥2 F ). (12) where x ∈Rdx is the input data, and y = y′ + ϵ ∈Rdy is a noisy version of the label. The ground truth mapping is linear and realizable: y′ = U ∗W ∗x. The second moments of the input and noise are denoted as Σx = E[xx⊺] and Σϵ = E[ϵϵ⊺], respectively. Note that this problem is essentially identical to a matrix factorization problem, which is not only a theoretical model of neural networks but also an important algorithm frequently in use for recommender systems [41]. The following theorem gives the fixed point of Noether flow. 6 Figure 3: A two-layer linear network after training. Here, the problem setting is the same as Figure 8. The theoretical prediction is computed from Theorem 5.2. Left: balance of the norm is only achieved when ϕx = 1, namely, when the data has an isotropic covariance. We also test SGD with a small weight decay (10−4), which is sufficiently small that the solution we obtained for SGD without SGD still holds approximately. In contrast, training with GD + WD always converges to a norm-balanced solution. Right: the sharpness of the converged model trained with SGD. We see that for some data distributions, SGD converges to a sharper solution, whereas it converges to flatter solutions for other data distributions. This flattening and sharpening effect are both due to the noise-balance effect of SGD. Here, we find that the systematic error between experiment and theory is due to the use of a finite learning rate and decreases as we decrease η. Theorem 5.2. Let r = UWx −y be the prediction residual. For all symmetric B, ˙CB = 0 if WΓW W ⊺= U ⊺ΓUU, (13) where ΓW = E[∥r∥2xx⊺] + 2γI, ΓU = E[∥x∥2rr⊺] + 2γI. See Figure 8 for the convergence of SGD to this solution under different learning rate, batch size and width. The equilibrium condition takes a more suggestive form when the model is at the global minimum, where U ∗W ∗x−y = ϵ. Assuming that ϵ and x are independent and that there is no weight decay, we have: W ¯ΣxW ⊺= U ⊺¯ΣϵU (14) Here, the bar over the matrices indicates that they have been normalized by their traces: ¯Σ = Σ/Tr[Σ]. The matrices ΓW and ΓU simplifies because at the global minimum, ri = ϵi and so E[∥x∥2rr⊺] = Tr[Σx]Σϵ and E[∥r∥2xx⊺] = Tr[Σϵ]Σx. This condition should be compared with the alignment condition for GD in Eq. (11), where the alignment is entirely determined by the initialization and perfect alignment is achieved only if the initialization is perfectly aligned. This condition simplifies further if both ¯Σx and ¯Σϵ are isotropic, where the equation simplifies to WW ⊺/dx = U ⊺U/dy. Namely, the two layers will be perfectly aligned, and the overall balance depends only on input and output dimensions. Figure 2-Left shows an experiment that shows that the two-layer linear net is perfectly aligned after training. Here, every point corresponds to the converged solution of an independent run with the same initialization and training procedures but different values of Σϵ. In agreement with the theory, the two layers are aligned according to Theorem 5.2 under SGD, but not under GD. In fact, GD finds solutions that are more than an order of magnitude away from SGD. Noise Driven Progressive Sharpening and Flattening. This result implies a previously unknown mechanism of progressive sharpening and flattening, where, during training, the stability of the algorithm steadily improves (during flattening) or deteriorates (during sharpening) [39, 15, 6]. To see this, we first derive a metric of sharpness for this model. Proposition 5.3. For the per-sample loss (12), let S(θ) ∶= Tr[∇2L(θ)]. Then, S(θ) = dy∥WΣ1/2 x ∥2 F + ∥U∥2 F Tr[Σx]. The trace of the Hessian is a good metric of the local stability of the GD and SGD algorithm because the trace upper bounds the largest Hessian eigenvalue. Let us analyze the simplest case of an autoencoding task, where the model is at the global minimum. Here, Σx ∝Idx, Σϵ ∝Idy. For a random Gaussian initialization with variance σ2 W and σ2 U, the trace at initialization is, in expectation, Sinit = dydTr[Σx](σ2 W + σ2 U). At the end of the training, the model is close to the global minimum and satisfies Proposition 5.3. Here, the rank of U and W matters and is upper bounded by min(d,dx), and at the global minimum, U and W are full-rank (equal to min(d,dx)), and all the singular values are 1. Thus, {Sinit = dxd(σ2 U + σ2 W )Tr[Σx], Send = 2min(d,dx)Tr[Σx]. (15) 7 The change in the sharpness during training thus depends crucially on the initialization scheme. For Xavier init, σ2 U = (dy + d)−1 and σ2 W = (d + dx)−1, and so Sinit ≈Send (but Sinit is slightly smaller). Thus, for the Xavier init., the sharpness of loss experiences a small sharpening during training. For Kaiming init., σ2 U = 1 and σ2 W = d−1 x . Therefore, it always holds that Sinit ≥Send, and so the stability improves as the training proceeds. The only case when the Kaiming init. does not experience progressive flattening is when d = dx = dy, which agrees with the common observation that training is easier if the widths of the model are balanced [12]. See Figure 4 for an experiment. In previous works, the progressive sharpening happens when the model is trained with GD [6]; our theory suggests an alternative mechanism for it. Figure 4: Dynamics of the stability condition S during the training of a rank-1 matrix factorization problem. The solid lines show the training of SGD with Kaiming init. When the learning rate (η = 0.008) is too large, SGD diverges (orange line). However, when one starts training at a small learning rate (0.001) and increases η to 0.008 after 5000 iterations, the training remains stable. This is because SGD training improves the stability condition during training, which is in agreement with the theory. In contrast, the stability condition of GD and that of SGD with a Xavier init increases only slightly. Also, note that both Xavier and Kaiming init. under SGD converges to the same stability condition because the equilibrium is unique. A practical technique that the theory explains is using warmup to stabilize training in the early stage. This technique was first proposed in Ref. [10] for training CNNs, where it was observed that the training is divergent if we start the training at a fixed large learning rate ηmax. However, this divergent behavior disappears if we perform a warmup training, where the learning rate is increased gradually from a minimal value to ηmax. Later, the same technique is found to be crucially useful for training large language models [27]. Our theory shows that the gradient noise can drive Kaiming init. to a stabler status where a larger learning can be applied. Flat or Sharp? Prior works have often argued that SGD prefers flatter solutions to sharper ones (e.g., see Ref. [42]). The exact solution we found, however, implies a subtle picture: for some datasets, SGD prefers sharper solutions, while for others, SGD prefers flatter solutions. Therefore, there is no causal relationship between SGD training and the sharpness of the found solution. See Figure 3 for the dependence of the flatness on the data distribution. A related question is whether SGD noise creates a similar effect as weight decay training. The answer is also negative: weight decay always prefers smaller norms and, thus, norm-balanced solutions, which are not necessarily noise-aligned solutions. Figure 3 shows that SGD can also lead to unbalanced solutions, unlike weight decay. 5.3 Noise-Aligned Solution of Deep Linear Networks Here, we apply our result to derive the exact solution of an arbitrarily deep and wide deep linear network, which has been under extensive study due to its connection in loss landscape to deep neural networks [4, 5, 17, 23, 47, 38]. Deep linear networks have also been a major model for understanding the implicit bias of GD [1]. The per-sample loss for a deep linear network can be written as: ℓ(θ) = ∥WD...W1x −y∥2, (16) where Wi is an arbitrary dimensional matrix for all i. The global minimum is realizable: y = V x+ϵ, for i.i.d. noises ϵ. Because there is a double rotation symmetry between every two neighboring matrices, the Noether charge can be defined with respect to every such pair of matrices. Let Bi be a symmetric matrix; we define the charges to be CBi = W T i BiWi. The noise equilibrium solution is given by the following theorem. Theorem 5.4. Let WD...W1 = V . Let V ′ = √ΣϵV √Σx such that V ′ = LS′R is its SVD and d = rank(V ′). Then, for all i and all Bi, a noise equilibrium for CBi at the global minimum is √ ΣϵWD = LΣDU ⊺ D−1, Wi = UiΣiU ⊺ i−1, W1 √ Σx = U1Σ1R, (17) for i = 2,⋯,D −1. Ui are arbitrary matrices satisfying U T i Ui = Id×d, and Σi are diagonal matrices such that Σ1 = ΣD = ( d TrS′ ) (D−2)/2D √ S′, Σi = (TrS′ d ) 1/D Id×d. (18) 8 Figure 5: Norms of weights of multilayer deep linear network during training on MNIST without weight decay. We see that the intermediate layers converge to the same norm during training, whereas the input and output layers are different because they are determined by the input and output noise. This effect is robust against different initializations. This agrees with our analysis for deep linear nets (Theorem 5.4). Left: initializing all layers with the same norm. Right: initializing all layers at randomly different norms. This solution has quite a few striking features. Surprisingly, the norms of all intermediate layers are balanced: Tr[Σ2 1] = Tr[Σ2 i ] = (TrS′)2/Dd1−2/D. (19) All intermediate layers are thus rescaled orthogonal matrices aligned with the neighboring matrices and the only two matrices that process information are the first and the last layer. See Figure 5 for an illustration of this effect. This explains an experimental result first observed in Ref. [30], where the authors showed that the neural networks find similar solutions when the model is initialized with the standard init., where there is no alignment at the start, and with the aligned init. Thus, the balance and alignment between different layers in the neural networks can be attributed to the rescaling symmetry between each pair of matrices. 5.4 Approximate Symmetry and Bias of SGD Lastly, let us consider what happens if the loss function only has an approximate symmetry. As a minimal model, let us consider the following loss function: ℓ= ℓ1(θ,x) + ζℓ2(θ). Here, ℓ1 has the A-symmetry, whereas ℓ2(θ) has no symmetry nor randomness and so ℓ2 does not affect Σ at all. ζ determines the relative strength between the two terms. In totality, ℓno longer has the A-symmetry. As before, let CA = θ⊺Aθ. Then, ˙CA(θ) = −ζ(∇ℓ2)⊺Aθ∗+ σ2Tr[Σ(θ)A], whose fixed point is ζ(∇ℓ2)⊺Aθ∗= σ2Tr[Σ(θ)A]. (20) This equilibrium condition thus depends strongly on how large ζ is. When ζ is small, we see that SGD still favors the fixed point given by Theorem 4.3, but with a first-order correction in ζ. Conversely, if ζ is large and σ2 is small, we can expand around a local minimum of the loss function θ∗, and so the fixed point becomes ζ(θ −θ∗)⊺H(θ∗)Aθ∗= σ2Tr[Σ(θ∗)A] + O(σ2∥θ −θ∗∥+ ∥θ −θ∗∥2), (21) where H is the Hessian of ℓ2. Certainly, this implies that SGD will stay around a point that deviates from the local minimum by an O(σ2) amount. This stationary point potentially has many solutions. For example, one class of solution is when θ −θ∗is an eigenvector of H with eigenvalue h∗> 0 and eigenvector n, we can denote s = (θ −θ∗)⊺n and obtain a direct solution of s: s = σ2Tr[Σ(θ∗)A] ζh∗n⊺Aθ∗ . (22) This deviation disappears in the limit σ2 →0. Therefore, this implicit regularization effect is only a consequence of SGD training and is not present under GD. With this condition, one can obtain a clear expression of the deviation of the quantity C from its local minimum value C∗∶= (θ∗)⊺Aθ∗. We have that C(θ) = C∗+ 2(θ −θ∗)⊺Aθ∗+ O(∥θ −θ∗∥2) = C∗+ 2 σ2 ζh∗Tr[ΣA]. (23) Thus, our results in the previous section still apply. The quantity C will be systematically larger than the local minimum values of C if the approximate symmetry matrix A is PD. It is systematically smaller if A is ND. When A contains both positive and negative eigenvalues, the deviation of C 9 Figure 6: The latent representations of a two-layer tanh net trained under SGD (left) are similar across different layers, in agreement with the theory. However, the learned representations are dissimilar under GD (right). Here, we plot the matrices W ¯ΣxW (first and third plots) and U ¯ΣϵU (second and fourth plots). Note that the quantity W ¯ΣxW is equal to the covariance of the preactivation representation of the first layer. This means that SGD and GD learn qualitatively different features after training. Also, see Appendix A.4 for other activations. This mechanism also complements the recent result in Ref. [46], which proposes a physics-inspired theory showing that gradient noise is a key factor in determining the latent representation of neural networks. depends on the local gradient fluctuation balancing condition. When the smallest eigenvalue of H is close to zero (which is true for common neural networks), the dominant factor that biases C occurs in this space. Therefore, it is not bad to approximate the deviation as C(θ) ≈C∗+2σ2Tr[ΣA]/hmin, where hmin is the smallest eigenvalue of the Hessian at the local minimum. In reality, ζ is neither too large nor too small, and one expects that the solution favored by SGD is an effective interpolation between the true local minimum and the fixed point favored by the symmetries. A set of experiments is shown in Figure 6, where we compare the latent representation of a twolayer tanh net with the prediction of 5.2. This is a natural example because fully connected networks are believed to be approximated by deep linear networks because they have the same connectivity patterns. We thus compare the prediction of Theorem 5.2 with the experimental results of nonlinear networks. Here, the task is a simple autoencoding task, where x ∈R40 and y = x + ϵ. x is sampled from an isotropic Gaussian, and ϵ is an independent non-isotropic (but diagonal) Gaussian noise such that Var[ϵ1] = 5 and Var[ϵi] = 1 for i ≠1. We train with SGD or GD for 104 iterations. The experimental results show that if trained with SGD, the learned representation agrees with the prediction of Theorem 5.2 well, whereas under GD, the model learned a completely different representation. This suggests that our result may be greatly useful for understanding the structures of latent representations of trained neural networks because the quantity W ¯ΣxW has a clean interpretation as the normalized covariance matrix of pre-activation hidden representation. Also, this result is not a special feature of tanh networks. Appendix A.4 also shows that the same phenomenon can be observed for swish [28], ReLU, and leaky-ReLU nets. 6 Conclusion In this work, we have studied how continuous symmetries affect the learning dynamics and fixed points of SGD. The result implies that SGD converges to initialization-independent solutions at the end of training, in sharp contrast to GD, which converges to strongly initialization-dependent solutions. We constructed the theoretical framework of exponential symmetries to study the special tendency of SGD to stay close to a special fixed point along the constant directions of the loss landscape. We proved that every exponential symmetry leads to a mapping of every parameter to a unique and essentially attractive fixed point. This point also has a clean interpretation: it is the point where the gradient noises of SGD in different subspaces balance and align. Because of this property, we termed these fixed points the “noise equilibria.” The advantage of our result is that it only relies on the existence of symmetries and is independent of the particular definitions of model architecture or data distribution. A limitation of our work is that we only focus on the problems that exponential symmetries can describe. It would be important to extend the result to other types of symmetries in the future. Another interesting future direction is to study these noise equilibria of more advanced models, which may deepen both our understanding of deep learning and neuroscience. Acknowledgement Lei Wu is supported by the National Key R&D Program of China (No. 2022YFA1008200) and National Natural Science Foundation of China (No. 2288101). Hongchao Li is supported by Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-leading Innovative Graduate Study (WINGS) Program, the University of Tokyo. 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This shows that the evolution of each layer can be vastly different, but the total norm of the parameters with the scaling symmetry is always monotonically increasing. Also, note that for net B, each layer also has the rescaling symmetry, and so the norm of each layer for net-B is also increasing. In contrast, net-A does not have layer-wise symmetry, and the individual norms can be either increasing or decreasing. A Additional Experiments A.1 Scale Invariance The scale invariance appears when common normalization techniques such as batch normalization [14] and layer normalization [2] are used. Let ℓ(θ,x) denote a per-sample loss such that for any ρ ∈R+: ℓ(θ,x) = ℓ(ρθ,x), where θ ∈Rd. For this symmetry, A = I. Thus, by Eq. (7), we have during SGD training that d dt∥θt∥2 = −γ∥θt∥2 + σ2Tr[Σ(θt)]. (24) Thus, without weight decay, the parameter norm increases monotonically and even diverges, particularly for under-parameterized models where the gradient noise is typically non-degenerate. Here, we numerically compare two networks trained on GD and SGD: Net-A: f(x) = ∑j uj ∥w∥tanh(w⊺ j x/∥u∥F ); and Net-B: f(x) = ∑j uij ∥u∥tanh(w⊺ j x/∥w∥F ). Here, w and u are matrices and wj denotes the j-th row of w and uj denotes the j-th column of u. The two networks are different functions of u and w. However, both networks have the global scale invariance: if we scale both U and W by an arbitrary positive scalar ρ, the network output and loss function remain unchanged for any sample x. We train these two networks on simple linear Gaussian data with GD or SGD. Figure 7 shows the result. Clearly, for SGD, both networks have a monotonically increasing norm, whereas the norm remains unchanged when the training proceeds with GD. What’s more, Net-B has two additional layer-wise scale invariances where one can scale only u (or only w) by ρ while keeping the loss function unchanged. This means that both layers will have a monotonically increasing norm, which is not the case for Net-A. Recent works have studied the dynamics of SGD under the scale-invariant models when weight decay is present [36, 20]. Our result shows that the model parameters will diverge without weight decay, leading to potential numerical problems. Combining the two results, the importance of having weight decay becomes clear: it prevents the divergence of models. A.2 Experiment Detail for Alignment Dynamics of Matrix Factorization Here, we give the details for the experiment in Figure 2. We train a two-layer linear net with d0 = d2 = 30 and d = 40. The input data is x ∼N(0,1), and y = x+ϵ, where ϵ is i.i.d. Gaussian with unit variance. At the end of SGD training, every element of the matrix U ⊺ΓUU is close to that of WΓW W ⊺, and they are therefore very well aligned. Such a phenomenon does not happen for GD. A.3 Convergence of Matrix Factorization to the Noise Equilibrium See Figure 8. 14 Figure 8: The convergence of matrix factorization to the noise equilibria is robust against different hyperparameter settings. The task is an autoencoding task where y = x ∈R100. The distribution of x is controlled by a parameter ϕx: x1∶50 ∼N(0, ϕx), x51∶100 ∼N(0, 2 −ϕx). This directly controls the overall covariance of x. The output noise covariance is set to be identity. Unless it is the independent variable, η, S and d are set to be 0.1, 100 and 2000, respectively. Left: using different learning rates. Mid: different data dimension: dx = dy = d. Right: different batch size S. A.4 Alignment in Nonlinear Networks Here, we complement the experiment in the main text with other types of activations. The experimental setting is exactly the same except that we switch the activation to swish, ReLU, and LeakyReLU. This shows that the prediction of Proposition 13 may have a surprisingly wide applicability. See Figure 9. 15 Figure 9: Activation patterns of nonlinear networks trained with SGD (Upper to lower: ReLU, leaky-ReLU, swish). Left: WΓW W ⊺. Right: U ⊺ΓUU. The similarity between the two matrices is striking. 16 B Proofs B.1 Ito’s Lemma and Derivation of Eq. (5) Let a vector Xt follow the following stochastic process: dXt = µt dt + Gt dWt (25) for a matrix Gt. Then, the dynamics of any function of Xt can be written as (Ito’s Lemma) df(Xt) = (∇⊺ Xfµt + 1 2Tr[G⊺ t ∇2f(Xt)Gt])dt + ∇f(Xt)⊺Gt dWt. (26) Applying this result to quantity C(θ) under the SGD dynamics, we obtain that dC = (∇⊺C∇L + σ2 2 Tr[Σ(θ)∇2C])dt + ∇⊺C √ σ2Σ(θ)dWt, (27) where we have used µt = ∇L, Gt = √ σ2Σ(θ). By Eq. (3), we have that ∇⊺C∇L = E[∇⊺C∇ℓ] = 0, (28) and ∇⊺CΣ = E[∇⊺C∇ℓ∇⊺ℓ] −E[∇⊺C∇ℓ]E[∇⊺ℓ] = 0. (29) Because Σ(θ) and √ Σ(θ) share eigenvectors, we have that ∇⊺C √ σ2Σ(θ) = 0. (30) Therefore, we have derived: dC = σ2 2 Tr[Σ(θ)∇2C]dt. (31) B.2 Proof of Theorem 4.3 We first prove a lemma that links the gradient covariance at θ to the gradient covariance at θλ. Lemma B.1. Tr[Σ(θλ)A] = Tr[e−2λAΣ(θ)A]. (32) Proof. By the definition of the exponential symmetry, we have that for an arbitrary λ, ℓ(θ) = ℓ(eλAθ). (33) Taking the derivative of both sides, we obtain that ∇θℓ(θ) = eλA∇θλℓ(θλ), (34) The standard result of Lie groups shows that eλA is full-rank and symmetric, and its inverse is e−λA. Therefore, we have e−λA∇θℓ(θ) = ∇θλℓ(θλ). (35) Now, we apply this relation to the trace of interest. By definition, Σ(θλ) = E[∇θλℓ(θλ)∇⊺ θλℓ(θλ)] (36) = e−λAΣ(θ)e−λA. (37) Because eλA is a function of A, it commutes with A. Therefore, Tr[Σ(θλ)A] = Tr[e−λAΣ(θ)e−λAA] (38) = Tr[e−2λAΣ(θ)A]. (39) Now, we are ready to prove the main theorem. 17 Proof. First of all, it is easy to see that C(θλ) is a monotonically increasing function of λ. By definition, C(θλ) = θT eλAAeλAθ (40) = θT (Z+ + Z−)θ, (41) where we have decomposed the matrix eλAAeλA = Z+ + Z−into two symmetric matrices such that Z+ only contains nonnegative eigenvalues, and Z−only contains nonpositive eigevalues. Because eλA commute with A, they share the eigenvectors. Using elementary Lie algebra shows that the eigenvalues of Z+ are a+eλa+ and that of Z−are a−eλa−, where a+ ≥0 and a0 ≤0. This implies that θT Z+θ and θT Z−θ are monotonically increasing functions of λ. Now, by Lemma B.1, we have Tr[Σ(θλ)A] = Tr[e−2λAΣ(θ)A]. (42) Similarly, the regularization term is γθ⊺ λAθλ = γTr[θθ⊺Ae2λA]. (43) Now, by assumption, if G(θλ) ≠0, we have either Tr[Σ(θ)A] ≠0 or Tr[θθ⊺A] ≠0. If Tr[Σ(θ)A] = θ⊺Aθ = 0, we have already proved item (2) of the theorem. Therefore, let us consider the case when either (or both) Tr[Σ(θ)A] ≠0 or θ⊺Aθ ≠0 Without loss of generality, we assume γ ≥0, and the case of γ < 0 follows an analogous proof. In such a case, we can write the trace in terms of the eigenvectors ni of A: −γθ⊺ λAθλ + ηTr[Σ(θλ)A] = η ∑ µi>0 e−2λ∣µi∣∣µi∣σ2 i + γ ∑ µi<0 e−2λ∣µi∣∣µi∣˜θ2 i ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I1(λ) −⎛ ⎝η ∑ µi<0 e2λ∣µi∣∣µi∣σ2 i + γ ∑ µi>0 e2λ∣µi∣∣µi∣˜θ2 i ⎞ ⎠ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I2(λ) =∶I(λ), where µi is the i-th eigenvalue of A, ˜θi = (n⊺ i θi)2, σ2 i = n⊺ i Σ(θ)ni ≥0 is the norm of the projection of Σ in this direction. By definition, I1 is either a zero function or strictly monotonically increasing function with I1(−∞) = +∞,I1(+∞) = 0 Likewise, I2 is either a zero function or a strictly monotonically increasing function with I2(−∞) = 0,I2(+∞) = +∞. By the assumption Tr(Σ(θ)A) ≠0 or Tr(θθ⊺A) ≠0, we have that at least one of I1 and I2 must be a strictly monotonic function. • If I1 or I2 is zero, we can take λ to be either +∞or −∞to satisfy the condition. • If both I1 and I2 are nonzero, then I = I1−I2 is a strictly monotonically decreasing function with I(−∞) = +∞and I(+∞) = −∞. Therefore, there must exist only a unique λ∗∈R such that I(λ∗) = 0. For the proof of (4), we denote the multi-variable function J(θ;λ) ∶= G(θλ). Given that Σ(θ) is differentiable, ∂J ∂θ exists. It is easy to see that ∂J ∂λ is continuous. Moreover, for any θ and λ = λ∗(θ), −∂J 2∂λ = η ∑ µi>0 e−2λ∣µi∣∣µi∣2σ2 i + γ ∑ µi<0 e−2λ∣µi∣∣µi∣2˜θ2 i + η ∑ µi<0 e2λ∣µi∣∣µi∣2σ2 i + γ ∑ µi>0 e2λ∣µi∣∣µi∣2˜θ2 i ≠0. Consequently, according to the Implicit Function Theorem, the function λ∗(θ) is differentiable. Additionally, ∂λ ∂θ = − ∂J ∂θ ∂J ∂λ . 18 B.3 Convergence First of all, notice an important property, which follows from Theorem 4.3: λ∗= 0 if and only if C −C∗= 0. Lemma B.2. For all θ(t), ˙C(θ) λ∗(θ) ≥{2σ2Tr[Σ(θ∗)A2 +] if λ∗> 0; 2σ2Tr[Σ(θ∗)A2 −] if λ∗< 0. (44) Proof. As in the main text, let θ∗denote θλ∗, C = C(θ) and C∗= C(θ∗). Thus, dC dt = σ2Tr[Σ(θ)A] (45) = σ2Tr[Σ(θ∗)e2λ∗AA], (46) where the second equality follows from Lemma B.1. One can decompose A as a sum of two symmetric matrices A = QΣ+Q⊺ ´¹¹¹¹¹¹¸¹¹¹¹¹¹¶ ∶=A+ +QΣ−Q⊺ ´¹¹¹¹¹¹¸¹¹¹¹¹¹¶ ∶=A− , (47) where Q is an orthogonal matrix, Σ+ (Σ−) is diagonal and contains only non-negative (non-positive) entries. Note that by the definition of λ∗, we have Tr[Σ(θ∗)A] = 0 and, thus, Tr[Σ(θ∗)A+] = −Tr[Σ(θ∗)A−]. (48) Thus, Tr[Σ(θ)A] = Tr[Σ(θ∗)e2λ∗AA] (49) = Tr[Σ(θ∗)(e2λ∗A −I)A] (50) = Tr[Σ(θ∗)(e2λ∗A+ −I)A+] + Tr[Σ(θ∗)(e2λ∗A−−I)A−]. (51) Using the inequality I + A ≤eA (namely, that eA −I −A is PSD), we obtain a lower bound Tr[Σ(θ)A] ≥2λ∗Tr[Σ(θ∗)A2 +] −Tr[Σ(θ∗)(I −e2λ∗A−)A−] (52) If λ∗> 0, Tr[Σ(θ∗)(e2λ∗A−−I)A−] < 0, Tr[Σ(θ)A] ≥2λ∗Tr[Σ(θ∗)A2 +]. (53) Likewise, there is an upper bound, which simplifies to the following form if λ∗< 0: Tr[Σ(θ)A] ≤2λ∗Tr[Σ(θ∗)A2 −]. (54) This finishes the proof. Lemma B.3. For any θ, −C −C∗ λ∗ ≤{2(θ∗)⊺A2 +θ∗ if λ∗> 0; 2(θ∗)⊺A2 −θ∗ if λ∗< 0. (55) Proof. The proof is conceptually similar to the previous one. By definition, we have C −C∗= (θ∗)⊺e−2λ∗AAθ∗−(θ∗)⊺Aθ∗ (56) = (θ∗)⊺A(e−2λ∗A −I)θ∗ (57) = (θ∗)⊺A+(e−2λ∗A+ −I)θ∗+ (θ∗)⊺A−(e−2λ∗A−−I)θ∗. (58) By the inequality I + A ≤eA, we have an upper bound C −C∗≥−2λ∗(θ∗)⊺A2 +θ∗+ (θ∗)⊺A−(e−2λ∗A−−I)θ∗. (59) If λ∗> 0, (θ∗)⊺A−(e−2λ∗A−−I)θ∗≥0, C −C∗≥−2λ∗(θ∗)⊺A2 +θ∗. (60) Likewise, if λ∗< 0, one can prove a lower bound: C −C∗≤−2λ∗(θ∗)⊺A2 −θ∗. (61) 19 Combining the above two lemmas, one can prove the following corollary. Corollary B.4. ˙C C −C∗≤ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ −σ2 Tr[Σ(θ∗)A2 +] (θ∗)⊺A2 +θ∗, if C −C∗< 0; −σ2 Tr[Σ(θ∗)A2 −] (θ∗)⊺A2 −θ∗, if C −C∗> 0. (62) Now, one can prove that as long as C∗is not moving too fast, C converges to C∗in mean square. Lemma B.5. Let the dynamics of C∗be a drifted Brownian motion: dC∗= µdt + sdW, where W is a Brownian motion with variance s2. If there exists c0 > 0 such that Tr[Σ(θ∗)A2 +] (θ∗)⊺A2 +θ∗ ≥c0 and Tr[Σ(θ∗)A2 −] (θ∗)⊺A2 −θ∗ > c0, E(C −C∗)2 ≤2µ2 + s2 2σ4c2 0 = O(1). (63) Proof. By assumption, ˙C C −C∗≤−σ2c0. (64) Let us first focus on the case when C −C∗> 0. By the definition of C∗and Ito’s lemma, d(C −C∗) ≤−σ2c0(C −C∗)dt −µdt −sdW. (65) Let Z = eσ2c0t(C −C∗), we obtain that dZ = eσ2c0td(C −C∗) + σ2c0eσ2c0t(C −C∗)dt (66) ≤−µeσ2c0tdt −seσ2c0tdW. (67) Its solution is given by Z ≤−µeσ2c0t σ2c0 −s∫eσ2c0tdW. (68) Alternatively, if C −C∗< 0, we let Z = eσ2c0t(C∗−C), and obtain Z ≤µeσ2c0t σ2c0 + s∫eσ2c0tdW. (69) Thus, E[Z2] ≤µ2e2σ2c0t σ4c2 0 + s2 ∫e2σ2c0tdt (70) = µ2e2σ2c0t σ4c2 0 + s2e2σ2c0t 2σ2c0 . (71) where we have used Ito’s isometry in the first line. By construction, E[(C −C∗)2] ≤2µ2 + s2 2σ4c2 0 . (72) The proof is complete. Let →p denote convergence in probability. One can now prove the following theorem, the convergence of the relative distance to zero in probability. Theorem B.6. Let the assumptions be the same as Lemma. (B.5). Then, if s = µ = 0, E[(C − C∗)2] →0. Otherwise, (C −C∗)2 (C∗)2 →p 0. (73) 20 Proof. By Lemma B.5 and Markov’s inequality: Pr(∣C(t) −C∗(t)∣> t1/4) →0. (74) Now, consider the distribution of (C∗)2. Because C∗is a Gaussian variable with mean µt and variance s2t, we have that Pr(∣C∗∣> √ t) →1. (75) Now, Pr(∣C(t) −C∗(t)∣/∣C∗∣> t−1/4) ≥Pr(∣C(t) −C∗(t)∣> t1/4&∣C∗∣< √ t) (76) ≥max(0,Pr(∣C(t) −C∗(t)∣> t1/4) + Pr(∣C∗∣< √ t) −1) (77) →0, (78) where we have used the Frechet inequality in the second line. This finishes the proof. B.4 Proof of Proposition 5.1 Proof. Recall that U = (˜u1,⋯, ˜ud2)⊺∈Rd2×d, W = ( ˜w1,⋯, ˜wd0) ∈Rd×d0, where ˜ui, ˜wi ∈Rd. θ = vec(U,W) = (˜u⊺ 1,⋯, ˜u⊺ d2, ˜w⊺ 1,⋯, ˜w⊺ d0)⊺∈R(d2+d0)d. ˙C = ηTr(Σ(θ)∇2 θθC), For ∇2 θθC, it holds that ∇2 ˜ui,˜ujC = {B, i = j; 0, otherwise. , ∇2 ˜ wi, ˜ wjC = {−B, i = j; 0, otherwise. ∇2 ˜ui, ˜ wjC = 0. Therefore, Tr[Σ(θ)∇2 θθC] = n ∑ i=1 Tr[∇θℓi∇θℓ⊺ i ∇2 θθC] = n ∑ i=1 ( d2 ∑ k=1 Tr[∇˜ukℓi∇˜ukℓ⊺ i B] − d0 ∑ l=1 Tr[∇˜ wlℓi∇˜ wlℓ⊺ i B]) = d2 ∑ k=1 Tr[Σ(˜uk)B] − d0 ∑ l=1 Tr[Σ( ˜wl)B]. The proof is complete. B.5 Proofs of Proposition 5.3 Proof. The loss function is ℓ= ∥UWx −y∥2 + γ(∥U∥2 F + ∥W∥2 F ). Let us adopt the following notation: U = (˜u1,⋯, ˜udy)⊺∈Rdy×d, W = ( ˜w1,⋯, ˜wdx) ∈Rd×dx, where ˜ui, ˜wi ∈Rd. θ = vec(U,W) = (˜u⊺ 1,⋯, ˜u⊺ dy, ˜w⊺ 1,⋯, ˜w⊺ dx)⊺∈R(dx+dy)d. Due to ∇˜uiℓ= Wx(˜u⊺ i Wx −yi) + 2γ˜ui, ∀i ∈[dy]; ∇˜ wjℓ= dy ∑ i=1 ˜uixj (˜u⊺ i Wx −yi) + 2γ ˜wj, ∀j ∈[dx]; the diagonal blocks of the Hessian ∇2 θθℓhave the following form: ∇2 ˜ui,˜uiℓ= Wxx⊺W ⊺+ 2γI, ∀i ∈[dy]; ∇2 ˜ wj, ˜ wjℓ= x2 j dy ∑ i=1 ˜ui˜u⊺ i + 2γI, ∀j ∈[dx]. 21 The trace of the Hessian is a good metric of the local stability of the GD and SGD algorithm because the trace upper bounds the largest Hessian eigenvalue. For this loss function, the trace of the Hessian of the empirical risk is S(U,W) ∶= Tr[∇2 θθℓ−2γI] = dy ∑ i=1 Tr[Wxx⊺W ⊺] + dx ∑ j=1 Tr[x2 j dy ∑ i=1 ˜ui˜u⊺ i ] =dyTr[WΣxW ⊺] + ∥U∥2 F Tr[Σx] = dy∥WΣ1/2 x ∥2 F + ∥U∥2 F Tr[Σx], where Σx = xx⊺. C Proof of Theorem 5.2 Proof. First, we split U and W like U = (u1,⋯,ud) ∈Rdy×d and W = (w⊺ 1,⋯,w⊺ d)⊺∈Rd×dx. The quantity under consideration is CB = Tr[UBU ⊺] −Tr[W ⊺BW] for an arbitrary symmetric matrix B. What will be relevant to us is the type of B that is indexed by two indices k and l such that ⎧⎪⎪⎨⎪⎪⎩ B(k,l) ij = B(k,l) ji = 1 if i = k and j = l or i = l and j = k; B(k,l) ij = 0 otherwise. (79) Specifically, for k,l ∈[d], we select B(k,l) i,j = δi,kδj,l + δi,lδj,k in CB. With this choice, for an arbitrary pair of k and l, CB(k,l) = u⊺ kul −w⊺ kwl. and W ⊺B(k,l)W = wkw⊺ l + wlw⊺ k, (80) UB(k,l)U ⊺= uku⊺ l + ulu⊺ k. (81) Therefore, E dy ∑ i=1 Tr[Σ(˜ui)B(k)] = E dy ∑ i=1 (˜u⊺ i Wx −yi)2Tr[Wxx⊺W ⊺B(k)] (82) = E[∥r∥2Tr[Wxx⊺W ⊺B(k,l)]] (83) = Tr[E[∥r∥2xx⊺]W ⊺B(k)W] (84) = Tr[Σ′ W (wkw⊺ l + wlw⊺ k)] (85) = 2w⊺ kΣ′ wwl (86) where we have defined ri = ˜u⊺ i Wx −yi and Σ′ W = E[∥r∥2xx⊺]. Likewise, we have that E dx ∑ j=1 Tr[Σ( ˜wj)B(k)] = E dx ∑ j=1 x2 jTr ⎡⎢⎢⎢⎢⎣ ⎛ ⎝ dy ∑ i=1 ˜ui(˜u⊺ i Wx −yi)⎞ ⎠ ⎛ ⎝ dy ∑ i=1 (˜u⊺ i Wx −yi)˜u⊺ i ⎞ ⎠B(k) ⎤⎥⎥⎥⎥⎦ = Tr[E[∥x∥2U ⊺rr⊺UB(k,l)]] = Tr[Σ′ UUB(k,l)U ⊺] = 2u⊺ kΣ′ uul. where we have defined Σ′ u = E[∥x∥2rr⊺]. Therefore, we have found that for arbitrary pair of k and l ˙CB(k,l) = −2γ(u⊺ kul −w⊺ kwl) + 2η(w⊺ kΣ′ wwl −u⊺ kΣ′ uul). (87) The fixed point of this dynamics is: w⊺ kΣwwl = u⊺ kΣuul. (88) 22 where Σw = ηΣ′ w + γI and ΣU = ηΣ′ u + γI. Because this holds for arbitrary k and l, the equation can be written in a matrix form: WΣwW ⊺= U ⊺ΣuU. (89) Let V = UW. To show that a solution exists for an arbitrary V . Let W ′ = W√Σw and U ′ = √ΣuU, which implies that U ′W ′ = √ ΣuV √ Σw ∶= V ′, (90) and W ′(W ′)⊺= (U ′)⊺U ′. (91) Namely, U ′ and (W ′)⊺must have the same right singular vectors and singular values. This gives us the following solution. Let V ′ = LSR be the singular value decomposition of V ′, where L and R are orthogonal matrices an S is a positive diagonal matrix. Then, for an arbitrary orthogonal matrix F, the following choice of U ′ and W ′ satisfies the two conditions: {U ′ = L √ SF; W ′ = F ⊺√ SR. (92) This finishes the proof. D Discrete-Time GD and SGD In fact, our results hold in a similar form for discrete-time GD and SGD. Let us focus on the exponential symmetries with the symmetric matrix A. The following equation holds with probability 1: 0 = ∇θℓ(θ,z) ⋅Aθ. (93) For discrete-time SGD, it is notationally simpler and without loss of generality to regard ℓ(θ) as the minibatch-averaged loss, which is the notation we adopt here. This is because if a symmetry holds for every per-sample loss, then it must also hold for every empirical average of these per-sample losses. The dynamics of SGD gives ∆θt = −η∇θℓ(θt,z). (94) This means that ∆θt ⋅J(θ) = 0. (95) Therefore, we have that ∆Ct = 2∆θ⊺ t Aθt + ∆θ⊺ t A∆θt (96) = ∆θ⊺ t A∆θt. (97) Therefore, ∆Ct = η2Tr[˜Σd(θ)A], (98) where ˜Σd(θ) = ∇θℓ(θ)∇⊺ θℓ(θ) is by definition PSD. Already, note the similarity between Eq. (98) and its continuous-time version. The qualitative discussions carry over: if A is PSD, Ct increases monotonically. Now, while the first-order terms in η also vanish in the r.h.s, the problem is that the r.h.s. becomes stochastic because Σd(θt) is different for every time step. However, one can still analyze the expected flow and show that the expected flow (over the sampling of minibatches) is zero at a unique point in a way similar to the continuous-time limit of the problem. Therefore, we define Gd(θt) = Ez[∆Ct], (99) Σd(θt) = Ez[˜Σd]. (100) We can now prove the following theorem. Note that this theorem applies for any batch size, and so it applies to both SGD and GD. Theorem D.1. (Discrete-time fixed point theorem of SGD.) Let the per-sample loss satisfy the A exponential symmetry and θλ ∶= exp[λA]θ. Then, for any θ and any γ ∈R, 23 (1) Gd(θλ) is a monotonically decreasing function of λ; (2) there exists a λ∗∈R ∪{±∞} such that Gd(θλ) = 0; (3) in addition, if Tr[Σd(θ)A] ≠0 or Tr[θθ⊺A] ≠0, λ∗is unique and Gd(θλ) is strictly monotonic; (4) in addition to (3), if Σd(θ) is differentiable, λ∗(θ) is a differentiable function of θ. Proof. Similarly, let us establish the relationship between ∇ℓ(θ) and ∇ℓ(exp(λA)). By the definition of the exponential symmetry, we have that for an arbitrary λ, ℓ(θ) = ℓ(eλAθ). (101) Taking the derivative of both sides, we obtain that ∇θℓ(θ) = eλA∇θλℓ(θλ), (102) The standard result of Lie groups shows that eλA is full-rank and symmetric, and its inverse is e−λA. Therefore, we have e−λA∇θℓ(θ) = ∇θλℓ(θλ). (103) Now, we apply this relation to the trace of interest. By definition, Σd(θλ) = E[∇θλℓ(θλ)∇⊺ θλℓ(θλ)] (104) = e−λAΣd(θ)e−λA. (105) Because eλA is a function of A, it commutes with A. Therefore, Tr[Σd(θλ)A] = Tr[e−λAΣd(θ)e−λAA] (106) = Tr[e−2λAΣd(θ)A]. (107) Similarly, the regularization term is γθ⊺ λAθλ = γTr[θθ⊺Ae2λA] (108) Now, if Tr[Σd(θ)A] = θ⊺Aθ = 0, we have already proved item (2) of the theorem. Therefore, let us consider the case when either (or both) Tr[Σd(θ)A] ≠0 or θ⊺Aθ ≠0 Without loss of generality, we assume γ ≥0, and the case of γ < 0 follows an analogous proof. In such a case, we can write the trace in terms of the eigenvectors ni of A: −γθ⊺ λAθλ + ηTr[Σd(θλ)A] = η ∑ µi>0 e−2λ∣µi∣∣µi∣σ2 i + γ ∑ µi<0 e−2λ∣µi∣∣µi∣˜θ2 i ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I1(λ) −⎛ ⎝η ∑ µi<0 e2λ∣µi∣∣µi∣σ2 i + γ ∑ µi>0 e2λ∣µi∣∣µi∣˜θ2 i ⎞ ⎠ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ I2(λ) =∶I(λ), where µi is the i-th eigenvalue of A, ˜θi = (n⊺ i θi)2, σ2 i = n⊺ i Σd(θ)ni ≥0 is the norm of the projection of Σd in this direction. By definition, I1 is either a zero function or strictly monotonically increasing function with I1(−∞) = +∞,I1(+∞) = 0 Likewise, I2 is either a zero function or a strictly monotonically increasing function with I2(−∞) = 0,I2(+∞) = +∞. By the assumption Tr(Σd(θ)A) ≠0 or Tr(θθ⊺A) ≠0, we have that at least one of I1 and I2 must be a strictly monotonic function. • If I1 or I2 is zero, we can take λ to be either +∞or −∞to satisfy the condition. • If both I1 and I2 are nonzero, then I = I1−I2 is a strictly monotonically decreasing function with I(−∞) = +∞and I(+∞) = −∞. Therefore, there must exist only a unique λ∗∈R such that I(λ∗) = 0. The proof of (4) follows from the Implicit Function Theorem, as in the continuous-time case. The final question is this: what does it mean for θ to reach a point where Gd(θ) = 0? An educated guess can be made: the fluctuation in C does not vanish, but the flow takes C towards this vanishing flow point – something like a Brownian motion trapped in a local potential well [37]. However, it is difficult to say more without specific knowledge of the systems. 24 E Proof of Theorem 5.4 We first prove the following theorem, which applies to an arbitrary parameter that are not necessarily local minima of of the loss. Theorem E.1. Let r = WD⋯W1x −y, ξi+1 ∶= WD⋯Wi+2 and hi ∶= Wi−1⋯W1. For all layer i, the equilibrium is achieved at W ⊺ i+1ξ⊺ i+1Ci 0ξi+1Wi+1 = WihiCi 1h⊺ i W ⊺ i , (109) where Ci 0 = E[∥hix∥2rr⊺],Ci 1 ∶= E[∥ξ⊺ i+1r∥2xx⊺]. Or equivalently, ξ⊺ i Ci 0ξi = hi+1Ci 1h⊺ i+1. (110) Proof. By Proposition 5.1, d dtCi B = σ2 (ETr[ ∂ℓ ∂Wi+1 B ( ∂ℓ ∂Wi+1 ) ⊺ ] −ETr[( ∂ℓ ∂Wi ) ⊺ B ∂ℓ ∂Wi ]). (111) The derivatives are ∂ℓ ∂Wi+1 = ξ⊺ i+1r(Wihix)⊺, (112) ∂ℓ ∂Wi = ξ⊺ i+1W ⊺ i+1r(hix)⊺. (113) Therefore, the two terms on R.H.S of Eq. (111) are given by ETr[ ∂ℓ ∂Wi+1 B ( ∂ℓ ∂Wi+1 ) ⊺ ] = ETr[ξ⊺ i+1r(Wihix)⊺B(Wihix)r⊺ξi+1], = E∥ξ⊺ i+1r∥2Tr[hixx⊺h⊺ i W ⊺ i BWi] (114) ETr[( ∂ℓ ∂Wi ) ⊺ B ∂ℓ ∂Wi ] = Tr[W ⊺ i+1ξ⊺ i+1r(hix)⊺B(hix)r⊺ξi+1Wi+1] = E[∥hix∥2Tr[W ⊺ i+1ξ⊺ i+1rr⊺ξi+1Wi+1B]]. (115) Because the matrix B is arbitrary, we can let Bi,j = δi,kδj,l + δi,lδj,k. Then, the two terms become ETr[ ∂ℓ ∂Wi+1 B ( ∂ℓ ∂Wi+1 ) ⊺ ] = 2E[∥ξ⊺ i+1r∥2 ˜w⊺ i,khixx⊺h⊺ i ˜wi,l], (116) ETr[( ∂ℓ ∂Wi ) ⊺ B ∂ℓ ∂Wi ] = 2E[∥hix∥2 ˜w⊺ i+1,kξ⊺ i+1rr⊺ξi+1 ˜wi+1,l]. (117) Here, we define the vectors Wi = ( ˜w⊺ i,1,⋯, ˜w⊺ i,d)⊺and Wi+1 = ( ˜wi+1,1,⋯, ˜wi+1,d). Because Eq. (116) and (117) hold for arbitrary k,l, we have ETr[ ∂ℓ ∂Wi+1 B ( ∂ℓ ∂Wi+1 ) ⊺ ] = 2WihiE[∥ξ⊺ i+1r∥2xx⊺]h⊺ i W ⊺ i , (118) ETr[( ∂ℓ ∂Wi ) ⊺ B ∂ℓ ∂Wi ] = 2W ⊺ i+1ξ⊺ i+1E[∥hix∥2rr⊺]ξi+1Wi+1. (119) For Eq. (111) to be 0, we must have WihiE[∥ξ⊺ i+1r∥2xx⊺]h⊺ i W ⊺ i = W ⊺ i+1ξ⊺ i+1E[∥hix∥2rr⊺]ξi+1Wi+1, (120) which is Eq. (109). The proof is complete. We are now ready to prove Theorem 5.4. 25 Proof. It suffices to specialize Theorem E.1 to the global minimum. At the global minimum, we can define r = W ∗ D⋯W ∗ 1 x −y = ϵ. (121) Then, Eq. (109) can be written as W ⊺ i+1 W ⊺ i+2⋯W ⊺ DΣϵWD⋯Wi+2 Tr[W ⊺ i+2⋯W ⊺ DΣϵWD⋯Wi+2]Wi+1 = Wi Wi−1⋯W1ΣxW ⊺ 1 ⋯W ⊺ i−1 Tr[Wi−1⋯W1ΣxW ⊺ 1 ⋯W ⊺ i−1]W ⊺ i . (122) To solve Eq. (122), we substitute WD and W1 with W ′ 1 = W1 √Σx and W ′ D = √ΣϵWD, which transform Eq. (122) into W ⊺ i+1 W ⊺ i+2⋯W ′⊺ D W ′ D⋯Wi+2 Tr[W ⊺ i+2⋯W ′⊺ D W ′ D⋯Wi+2]Wi+1 = Wi Wi−1⋯W ′ 1W ′⊺ 1 ⋯W ⊺ i−1 Tr[Wi−1⋯W ′ 1W ′⊺ 1 ⋯W ⊺ i−1]W ⊺ i . (123) The global minimum condition can be written as W ′ DWD−1⋯W2W ′ 1 = √ ΣϵV √ Σx ∶= V ′. (124) Then, we can decompose the matrices W ′ 1,⋯,W ′ D as W ′ D = LΣDU ⊺ D−1, Wi = UiΣiU ⊺ i−1(i ≠1,D), W ′ 1 = U1Σ1R, (125) where ΣD,⋯,Σ1 ∈Rd×d, L ∈Rdy×d, Ui ∈Rdi×d, R ∈Rd×dx with d ∶= rank(V ′) and arbitrary di. The matrices Ui satisfy U ⊺ i Ui = Id×d. By substituting the decomposition into Eq. (123), we have Σi+1⋯ΣDΣD⋯Σi+1 Tr[Σi+2⋯ΣDΣD⋯Σi+2] = Σi⋯Σ1Σ1⋯Σi Tr[Σi−1⋯Σ1Σ1⋯Σi−1]. (126) Since these diagonal matrices commute with each other, we can see Σi = cId×d. Then we move on to fix the parameter c. By taking i = 1 and i = D −1 in Eq. (126), we obtain Σ2 2⋯Σ2 D Tr[Σ2 3 ...Σ2 D] = c2 Σ2 D Tr[Σ2 D] = Σ2 1 d , (127) Σ2 D d = Σ2 1⋯Σ2 D−1 Tr[Σ2 1 ...Σ2 D−1] = c2 Σ2 1 Tr[Σ2 1], (128) where d represents the dimension of the learning space. By taking trace to both sides of Eqs. (127) and (128), we can see Tr[Σ2 1] = Tr[Σ2 D] and hence, Σ1 = ΣD. The parameter c is given by c = √ Tr[Σ2 1] d . (129) With the SVD decomposition V ′ = LS′R, we have Σ2 1cD−2 = S′. (130) Therefore, the solutions for c and Σ1 are c = (TrS′ d ) 1/D , Σ1 = √ S′ c(D−2)/2 = ( d TrS′ ) (D−2)/2D √ S′. (131) The scaling of the diagonal matrices are shown as Tr[Σ2 1] = d1−2/D(TrS′)2/D, Tr[Σ2 i ] = (TrS′)2/Dd1−2/D = Tr[Σ2 1]. (132) The proof is complete. 26 NeurIPS Paper Checklist 1. 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4,469	PSL: Rethinking and Improving Softmax Loss from Pairwise Perspective for Recommendation Weiqin Yang † ‡ Zhejiang University tinysnow@zju.edu.cn Jiawei Chen ∗† ‡ § Zhejiang University sleepyhunt@zju.edu.cn Xin Xin Shandong University xinxin@sdu.edu.cn Sheng Zhou Zhejiang University zhousheng_zju@zju.edu.cn Binbin Hu Ant Group bin.hbb@antfin.com Yan Feng † ‡ Zhejiang University fengyan@zju.edu.cn Chun Chen † ‡ Zhejiang University chenc@zju.edu.cn Can Wang † § Zhejiang University wcan@zju.edu.cn Abstract Softmax Loss (SL) is widely applied in recommender systems (RS) and has demonstrated effectiveness. This work analyzes SL from a pairwise perspective, revealing two significant limitations: 1) the relationship between SL and conventional ranking metrics like DCG is not sufficiently tight; 2) SL is highly sensitive to false negative instances. Our analysis indicates that these limitations are primarily due to the use of the exponential function. To address these issues, this work extends SL to a new family of loss functions, termed Pairwise Softmax Loss (PSL), which replaces the exponential function in SL with other appropriate activation functions. While the revision is minimal, we highlight three merits of PSL: 1) it serves as a tighter surrogate for DCG with suitable activation functions; 2) it better balances data contributions; and 3) it acts as a specific BPR loss enhanced by Distributionally Robust Optimization (DRO). We further validate the effectiveness and robustness of PSL through empirical experiments. The code is available at https://github.com/Tiny-Snow/IR-Benchmark. 1 Introduction Nowadays, recommender systems (RS) have permeated various personalized services [1–4]. What sets recommendation apart from other machine learning tasks is its distinctive emphasis on ranking [5]. Specifically, RS aims to retrieve positive items in higher ranking positions (i.e., giving larger prediction scores) over others and adopts specific ranking metrics (e.g., DCG [6] and MRR [7]) to evaluate its performance. The emphasis on ranking inspires a surge of research on loss functions in RS. Initial studies treated recommendation primarily as a classification problem, utilizing pointwise loss functions (e.g., BCE [8], MSE [9]) to optimize models. Recognizing the inherent ranking nature of RS, pairwise loss Corresponding author. †State Key Laboratory of Blockchain and Data Security, Zhejiang University. ‡College of Computer Science and Technology, Zhejiang University. §Hangzhou High-Tech Zone (Binjiang) Institute of Blockchain and Data Security. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). functions (e.g., BPR [10]) were introduced to learn a partial ordering among items. More recently, Softmax Loss (SL) [11] has integrated contrastive learning paradigms [12, 13], augmenting positive items as compared with negative ones, achieving state-of-the-art (SOTA) performance. While SL has proven effective, it still suffers from two limitations: 1) SL can be used to approximate ranking metrics, e.g., DCG and MRR [11, 14], but their relationships are not sufficiently tight. Specifically, SL uses the exponential function exp(·) as the surrogate activation to approximate the Heaviside step function in DCG, resulting in a notable gap, especially when the surrogate activation takes larger values. 2) SL is sensitive to noise (e.g., false negatives [15]). Gradient analysis reveals that SL assigns higher weights to negative instances with large prediction scores, while the weights are rather skewed and governed by the exponential function. This characteristic renders the model highly sensitive to false negative noise. Specifically, false negative instances are common in RS, as a user’s lack of interaction with an item might stem from unawareness rather than disinterest [16–18]. These instances would receive disproportionate emphasis, potentially dominating the training direction, leading to performance degradation and training instability. To address these challenges, we propose a new family of loss functions, termed Pairwise Softmax Loss (PSL). PSL first reformulates SL in a pairwise manner, where the loss is applied to the score gap between positive-negative pairs. Such pairwise perspective is more fundamental to recommendation as the ranking metrics are also pairwise dependent. Recognizing that the primary weakness of SL lies in its use of the exponential function, PSL replaces this with other surrogate activations. While this extension is straightforward, it brings significant theoretical merits: • Tighter surrogate for ranking metrics. We establish theoretical connections between PSL and conventional ranking metrics, e.g., DCG. By choosing appropriate surrogate activations, such as ReLU or Tanh, we demonstrate that PSL achieves a tighter DCG surrogate loss than SL. • Control over the weight distribution. PSL provides flexibility in choosing surrogate activations that control the weight distribution of training instances. By substituting the exponential function with an appropriate surrogate activation, e.g., ReLU or Tanh, PSL can mitigate the excessive impact of false negatives, thus enhancing robustness to noise. • Theoretical connections with BPR loss. Our analyses reveal that optimizing PSL is equivalent to performing Distributionally Robust Optimization (DRO) [19] over the conventional pairwise loss BPR [10]. DRO is a theoretically sound framework where the optimization is not only on a fixed empirical distribution but also across a set of distributions with adversarial perturbations. This DRO characteristic endows PSL with stronger generalization and robustness against out-ofdistribution (OOD), especially given that such distribution shifts are common in RS, e.g., shifts in user preference and item popularity [16, 20, 21]. Our analyses underscore the theoretical effectiveness and robustness of PSL. To empirically validate these advantages, we implement PSL with typical surrogate activations (Tanh, Atan, ReLU) and conduct extensive experiments on four real-world datasets across three experimental settings: 1) IID setting [22] where training and test distributions are identically distributed [23]; 2) OOD setting [24] with distribution shifts in item popularity; 3) Noise setting [15] with a certain ratio of false negatives. Experimental results demonstrate the superiority of PSL over existing losses in terms of recommendation accuracy, OOD robustness, and noise resistance. 2 Preliminaries Task formulation. We will conduct our discussion in the scope of collaborative filtering (CF) [25], a widely-used recommendation scenario. Given the user set U and item set I, CF dataset D ⊂U × I is a collection of observed interactions, where each instance (u, i) ∈D means that user u has interacted with item i (e.g., clicks, reviews, etc). For each user u, we denote Pu = {i ∈I : (u, i) ∈D} as the set of positive items of u, while I \ Pu represents the negative items. The goal of recommendation is to learn a recommendation model, or essentially a scoring function f(u, i) : U × I →R that quantifies the preference of user u on item i accurately. Modern RS often adopts an embedding-based paradigm [26]. Specifically, the model maps user u and item i into d-dim embeddings u, v ∈Rd, and predicts their preference score f(u, i) based on embedding similarity. The cosine similarity is commonly utilized in RS and has demonstrated particular effectiveness [27]. Here we set f(u, i) = u·v ∥u∥∥v∥· 1 2, where the scaling factor 1 2 is introduced for faciliating analyses 2 and can be absorbed into the temperature hyperparameter (τ). The scores f(u, i) are subsequently utilized to rank items for generating recommendations. Ranking metrics. The Discounted Cumulative Gain (DCG) [6] is a prominent ranking metric for evaluating the recommendation quality. Formally, for each user u, DCG is calculated as follows: DCG(u) = X i∈Pu 1 log2(1 + πu(i)) (2.1) where πu(i) is the ranking position of item i in the ranking list sorted by the scores f(u, i). DCG quantifies the cumulative gain of positive items, discounted by their ranking positions. Similarly, the Mean Reciprocal Rank (MRR) [7, 28] is another popular ranking metric using the reciprocal of the ranking position as the gain, i.e., MRR(u) = P i∈Pu 1/πu(i). Additionally, other metrics such as Recall [29], Precision [29], and AUC [30] are also utilized in RS [29]. Compared to these metrics, DCG and MRR focus more on the top-ranked recommendations, thus attracting increasing attention in RS [11, 31]. In this work, we aim to explore the surrogate loss for DCG and MRR. Recommendation losses. To train recommendation models effectively, a series of recommendation losses has been developed. Recent work on loss functions can mainly be classified into three types: • Pointwise loss (e.g., BCE [8], MSE [9], etc.) formulates recommendation as a specific classification or regression task, and the loss is applied to each positive and negative instance separately. Specifically, for each user u, the pointwise loss is defined as Lpointwise(u) = − X i∈Pu log(φ+(f(u, i))) − X j∈I\Pu log(φ−(f(u, j))) (2.2) where φ+(·) and φ−(·) are the activation functions adapted for different loss choices. • Pairwise loss (e.g., BPR [10], etc.) optimizes partial ordering among items, which is applied to the score gap between negative-positive pairs. BPR [10] is a representative pairwise loss, which is defined as LBPR(u) = X i∈Pu X j∈I\Pu log σ(f(u, j) −f(u, i)) (2.3) where σ denotes the activation function that approximates the Heaviside step function. The basic intuition behind BPR loss is to let the positive instances have higher scores than negative instances. In practice, there are various choices of the activation function. For instance, Rendle et al. [10] originally uses the sigmoid function, and the resultant BPR loss can approximate AUC metric. • Softmax Loss (i.e., SL [11]) normalizes the predicted scores into a multinomial distribution [32] and optimizes the probability of positive instances over negative ones [33], which is defined as LSL(u) = − X i∈Pu log exp(f(u, i)/τ) P j∈I exp(f(u, j)/τ) ! (2.4) where τ is the temperature hyperparameter. SL can also be understood as a specific contrastive loss, which draws positive instances (u, i) closer and pushes negative instances (u, j) away [13]. 3 Analyses on Softmax Loss from Pairwise Perspective In this section, we aim to first represent the Softmax Loss (SL) in a pairwise form, followed by an analysis of its relationship with the DCG metric, where two limitations of SL are exposed. Pairwise form of SL. To facilitate the analysis of SL and to build its relationship with the DCG metric, we rewrite SL (cf. Equation (2.4)) in the following pairwise form: LSL(u) = X i∈Pu log  X j∈I exp(duij/τ)  , where duij = f(u, j) −f(u, i) (3.1) Equation (3.1) indicates that SL is penalized based on the score gap between negative-positive pairs, i.e., duij = f(u, j) −f(u, i). This concise expression is fundamental for ranking, as it optimizes the relative order of instances rather than their absolute values. 3 Connections between SL and DCG. We now analyze the connections between SL and the DCG metric (cf. Equations (2.1) and (3.1)), which could enhance our understanding of the advantages and disadvantages of SL. Our analysis follows previous work [11, 14], which begins by relaxing the negative logarithm of DCG with −log DCG(u) + log \|Pu\| ≤−log 1 \|Pu\| X i∈Pu 1 πu(i) ! ≤ 1 \|Pu\| X i∈Pu log πu(i) (3.2) where the first inequality holds due to log2(1 + πu(i)) ≤πu(i), and the second inequality holds due to Jensen’s inequality [34]. Note that the ranking position πu(i) of item i can be expressed as πu(i) = X j∈I I(f(u, j) ≥f(u, i)) = X j∈I δ(duij) (3.3) where δ(·) denotes the Heaviside step function, with δ(x) = 1 for x ≥0 and δ(x) = 0 for x < 0. Since δ(duij) ≤exp(duij/τ) holds for all τ > 0, we deduce that SL is a smooth upper bound of Equation (3.2), and thus serves as a reasonable surrogate loss for DCG and MRR metrics1. However, our analysis also reveals two limitations of SL: • Limitation 1: SL is not tight enough as a DCG surrogate loss. There remains a significant gap between the Heaviside step function δ(·) and the exponential function exp(·), especially when duij reaches a relatively large value, where exp(·) becomes substantially larger than δ(·). This gap is further exacerbated by the temperature τ. Practically, we find that the optimal τ is usually chosen to be less than 0.2 (cf. Appendix B.5.2). Given the explosive nature of exp(·), the gap becomes extremely large, potentially leading to suboptimal performance of SL in optimizing DCG. • Limitation 2: SL is highly sensitive to noise (e.g., false negative instances). False negative instances [15] are common in the typical RS. This is often due to the exposure bias [16], where a user’s lack of interaction with an item might stem from unawareness rather than disinterest. Unfortunately, SL is highly sensitive to these false negative instances. On one hand, these instances (u, j), which may exhibit patterns similar to true positive ones, are difficult for the model to differentiate and often receive larger predicted scores, thus bringing potentially larger duij for positive items i. As analyzed in Limitation 1, these instances can significantly enlarge the gap between SL and DCG due to the exponential function, causing the optimization to deviate from the DCG metric. Gradient analysis of SL. Another perspective to support the view of Limitation 2 comes from the gradient analysis. Specifically, the gradient of SL w.r.t. duij is ∂LSL(u) ∂duij = exp(duij/τ)/τ \|I\|Ej′∼I[exp(duij′/τ)] ∝exp(duij/τ)/τ (3.4) As can be seen, SL implicitly assigns a weight to the gradient of each negative-positive pair, where the weight is proportional to exp(duij/τ). This suggests that instances with larger duij will receive larger weights. While this property may be desirable for hard mining [11], which can accelerate convergence, it also means that false negative instances, which typically have larger duij, will obtain disproportionately large weights, as shown in the weight distribution of SL in Figure 1b. Therefore, the optimization of SL can be easily dominated by false negative instances, leading to performance drops and training instability. Discussions on DRO robustness and noise sensitivity. Recent work [15] claims that SL exhibits robustness to noisy data through Distributionally Robust Optimization (DRO) [19]. However, we argue that this is not the case. DRO indeed can enhance model robustness to distribution shifts, but it also increases the risk of noise sensitivity, as demonstrated by many studies on DRO [35, 36]. Intuitively, DRO emphasizes hard instances with larger losses, making noisy data contribute more rather than less to the optimization. This is also demonstrated from the experiments with false negative instances (cf. Figure 8 in [15]), where the improvements of SL over other baselines in Noise setting do not increase significantly but sometimes decay. 1Note that the middle term in Equation (3.2), i.e., −log 1 \|Pu\| P i∈Pu 1/πu(i) , is exactly −log MRR(u). Therefore, SL serves as an upper bound of the negative logarithm of DCG and MRR, and minimizing SL leads to the improvement of these ranking metrics. 4 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 duij 0.0 0.5 1.0 1.5 2.0 2.5 (duij) (duij) exp(duij) tanh atan relu (a) Surrogate activations. 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 duij 0 25 50 75 100 125 150 175 200 (u)/ duij SL PSL-tanh PSL-atan PSL-relu (b) Weight distributions. Figure 1: (a) Illustration of different surrogate activations. (b) The weight distribution of SL as compared with PSL using three different surrogate activations. Here we set τ = 0.2, which typically achieves optimal results in practice. 4 Methodology 4.1 Pairwise Softmax Loss Recognizing the limitations of SL, particularly its reliance on the unsatisfactory exponential function, we propose to extend SL with a more general family of losses, termed Pairwise Softmax Loss (PSL). In PSL, the exponential function exp(·) is replaced by other surrogate activations σ(·) approximating the Heaviside step function δ(·). For each user u, the PSL is defined as LPSL(u) = X i∈Pu log  X j∈I σ(duij)1/τ   (4.1) One might wonder why we apply the temperature outside the activation function (i.e., extending exp(duij)1/τ to σ(duij)1/τ )2 rather than within it (i.e., extending exp(duij/τ) to σ(duij/τ)). This subtlety will be elucidated later as we demonstrate that the form in Equation (4.1) offers superior properties over the alternative. Our PSL provides a flexible framework for selecting better activation functions, allowing the loss to exhibit improved properties compared to SL. We advocate for three activations, including PSL-tanh: σtanh = tanh(duij)+1, PSL-atan: σatan = arctan(duij)+1, and PSL-relu: σrelu = ReLU(duij+1). In the following, we will discuss the advantages of PSL and provide evidence for the selection of these surrogate activations. Advantage 1: PSL is a better surrogate for ranking metrics. To highlight the advantages of replacing exp(·) with alternative surrogate activations, we present the following lemma: Lemma 4.1. If the condition δ(duij) ≤σ(duij) ≤exp(duij) (4.2) is satisfied for any duij ∈[−1, 1], then PSL serves as a tighter DCG surrogate loss compared to SL. The proof is presented in Appendix A.1. This lemma reveals that PSL could be a tighter surrogate loss for DCG compared to SL. Additionally, it provides guidance on the selection of a proper surrogate activation — we may choose the activation that lies between exp(·) and δ(·). As demonstrated in Figure 1a, our chosen surrogate activations σtanh, σatan, and σrelu adhere to this principle. Advantage 2: PSL controls the weight distribution. The gradient of PSL w.r.t. duij is ∂LPSL(u) ∂duij = σ′(duij) · σ(duij)1/τ−1/τ \|I\|Ej′∼I[σ(duij′)1/τ] ∝σ′(duij) · σ(duij)1/τ−1/τ (4.3) 2Note that the equation exp(duij/τ) = exp(duij)1/τ holds. 5 This implies that the shape of the weight distribution is determined by the choice of surrogate activation. By selecting appropriate activations, PSL can better balance the contributions of instances during training. For example, the three activations advocated before can explicitly mitigate the explosive issue on larger duij (cf. Figure 1b), bringing better robustness to false negative instances. One might argue that adjusting τ in SL could improve noise resistance. However, such adjustments do not alter the fundamental shape of the weight distribution, which remains exponential. Furthermore, as we discuss subsequently, τ plays a crucial role in controlling robustness against distribution shifts. Thus, indiscriminate adjustments to τ may compromise out-of-distribution (OOD) robustness. Advantage 3: PSL is a DRO-empowered BPR loss. We establish a connection between PSL and BPR [10] based on Distributionally Robust Optimization (DRO) [19, 37]. Specifically, optimizing PSL is equivalent to applying a KL divergence DRO on negative item distribution over BPR loss (cf. Equation (2.3)), as demonstrated in the following theorem3: Theorem 4.2. For each user u and its positive item i, let P = P(j\|u, i) be the uniform distribution over I. Given a robustness radius η > 0, consider the uncertainty set Q consisting of all perturbed distributions Q = Q(j\|u, i) satisfying: (i) Q is absolutely continuous w.r.t. P, i.e., Q ≪P; (ii) the KL divergence between Q and P is constrained by η, i.e., DKL(Q∥P) ≤η. Then, optimizing PSL is equivalent to performing DRO over BPR loss, i.e., min Ei∼Pu log Ej∼I h elog(σ(duij))/τi \| {z } LPSL(u) ⇔min Ei∼Pu sup Q∈Q Ej∼Q(j\|u,i) [log σ(duij)] \| {z } LBPR-DRO(u) (4.4) where τ = τ(η) is a temperature parameter controlled by η. The proof is presented in Appendix A.2. Theorem 4.2 demonstrates how PSL, based on the DRO framework, is inherently robust to distribution shifts. This robustness is particularly valuable in RS, where user preference and item popularity may shift significantly. Therefore, PSL can be regarded as a robust generalization of BPR loss, offering better performance in OOD scenarios. In addition, Theorem 4.2 also gives insights into the rationality of PSL that differs from serving as a DCG surrogate loss, but rather as a DRO-empowered BPR loss: • Rationality of surrogate activations: The activation function in BPR is originally chosen as an approximation to the Heaviside step function [10]. Since PSL is a generalization of BPR as stated in Theorem 4.2, it is reasonable to select the activations in PSL that aligns with the ones in BPR. Interestingly, this principle coincides with our analysis from the perspective of DCG surrogate loss. • Rationality of the position of temperature: Theorem 4.2 also rationalizes the extension form that places the temperature on the outside rather than inside. For the outside form (i.e., σ(duij)1/τ), Theorem 4.2 holds, and the temperature τ can be interpreted as a Lagrange multiplier in DRO optimization, which controls the extent of distribution perturbation. However, for the inside form (i.e., σ(duij/τ)), Theorem 4.2 no longer holds, and it would be challenging to establish the relationship between PSL and BPR. • Rationality of pairwise perspective: Recent work such as BSL [15] also reveals the DRO property of SL (cf. Lemma 1 in [15]). However, we wish to highlight the distinctions between Theorem 4.2 and Wu et al. [15]’s analyses: 1) Wu et al. [15] views SL from a pointwise perspective and associates it with a specific, less commonly used pointwise loss. In contrast, our analyses adopt a pairwise perspective and establish a relationship between PSL and the widely used BPR loss. 2) We construct a link between two families of losses with flexible activation selections, and Wu et al. [15]’s analyses can be regarded as a special case within our broader framework. The above analyses underscore the advantages of PSL and provide the principles to select surrogate activations. Remarkably, PSL is easily implemented and can be integrated into various recommendation scenarios. This can be achieved by merely replacing the exponential function exp(·) in SL with another activation σ(·) surrogating the Heaviside step function, requiring minimal code modifications. 3Note that elog(σ(duij))/τ = σ(duij)1/τ holds, thus the PSL in Equation (4.4) is identical to the one in Equation (4.1). 6 4.2 Discussions Comparisons of two extension forms. In previous discussions, we highlight the advantages of the form that positions the temperature outside (i.e., σ(duij)1/τ) over the inside (i.e., σ(duij/τ)). As discussed in the analyses of Theorem 4.2, the outside form can be regarded as a DRO-empowered BPR, while the inside form cannot, which ensures the robustness of PSL against distribution shifts. Here we provide an additional perspective on the advantages of the outside form. In fact, the outside form facilitates the selection of surrogate activations. For instance, to ensure that PSL serves as a tighter DCG surrogate loss compared to SL (i.e., ensure Lemma 4.1 holds), the outside form only need to consider the condition (4.2) on the range of duij ∈[−1, 1]. However, for the inside form, this condition should be satisfied on the entire domain of the activation σ(·), which complicates the selection of activation functions. Therefore, the outside form is more flexible and easier to implement. We further provide empirical evidence in Appendix C.3, demonstrating that the inside form will lose the advantages of achieving tighter DCG surrogate loss, leading to compromised performance. Connections with other losses. We further discuss the connections between PSL and other losses: • Connection with AdvInfoNCE [38]: According to Theorem 3.1 in Zhang et al. [38], AdvInfoNCE can indeed be considered as a special case of PSL with σ(·) = exp(exp(·)). We argue that this activation is not a good choice as it would enlarge the gap between the loss and DCG. In fact, we have −log DCG ≤LPSL ≤LSL ≤LAdvInfoNCE (cf. Appendix A.3 for proof). While AdvInfoNCE may achieve good performance in some specific OOD scenarios as tested in Zhang et al. [38], we argue that AdvInfoNCE is a looser DCG surrogate loss and would be highly sensitive to noise (cf. Table 1 and Figure 2 in Section 5.2 for empirical validation). • Connection with BPR [10]: Besides the DRO relation stated in Theorem 4.2, we also derive the bound relation between BPR and PSL with the same activation, i.e., −log DCG ≤LPSL ≤ log LBPR (cf. Appendix A.3 for proof). This relation clearly demonstrates the effectiveness of PSL over BPR — performing DRO over BPR results robustness to distribution shifts, while also achieving a tighter surrogate of DCG, which is interesting (cf. Tables 1 and 2 in Section 5.2 for empirical validation). An intuitive explanation is that DCG focuses more on the higher-ranked items. Given that DRO would give more weight to the hard negative instances with larger prediction scores and higher positions, it would naturally narrow the gap between BPR and DCG. 5 Experiments 5.1 Experimental Setup Testing scenarios. We adopt three representative testing scenarios to comprehensively evaluate model accuracy and robustness, including: 1) IID setting: the conventional testing scenario where training and test data are randomly split and identically distributed; 2) OOD setting: to assess the model’s robustness on the out-of-distribution (OOD) data, we adopt a debiasing testing paradigm where the item popularity distribution shifts. We closely refer to Zhang et al. [20], Wang et al. [24], and Wei et al. [39], sampling a test set where items are uniformly distributed while maintaining the long-tail nature of the training dataset; 3) Noise setting: to evaluate the model’s sensitivity to noise, following Wu et al. [15], we manually impute a certain proportion of false negative items in the training data. The details of the above testing scenarios are provided in Appendix B.1. Datasets. Four widely-used datasets including Amazon-Book, Amazon-Electronic, Amazon-Movie [40, 41], and Gowalla [42] are used in our experiments. Considering the item popularity is not heavily skewed in the Amazon-Book and Amazon-Movie datasets, we turn to other conventional datasets, Amazon-CD [40, 41] and Yelp2018 [43], as replacements for OOD testing. All datasets are split into 80% training set and 20% test set, with 10% of the training set further treated as the validation set. The details of the above datasets are summarized in Appendix B.1. Metrics. We closely refer to Wu et al. [15] and Zhang et al. [38], adopting Top-K metrics including NDCG@K [6] and Recall@K [29] for performance evaluation, where NDCG is the normalized DCG, i.e., dividing DCG by the ideal value. Here we simply set K = 20 as in recent work [15, 38] while observing similar results with other choices. For more details, please refer to Appendix B.2. 7 Table 1: Performance comparison in terms of Recall@20 and NDCG@20 under the IID setting. The best result is bolded, and the blue-colored zone indicates that PSL is better than SL. Imp.% denotes the NDCG@20 improvement of PSL over SL. The marker "" indicates that the improvement is statistically significant (p-value < 0.05). Model Loss Amazon-Book Amazon-Electronic Amazon-Movie Gowalla Recall NDCG Recall NDCG Recall NDCG Recall NDCG MF [26] BPR [10] 0.0665 0.0453 0.0816 0.0527 0.0916 0.0608 0.1355 0.1111 LLPAUC [44] 0.1150 0.0811 0.0821 0.0499 0.1271 0.0883 0.1610 0.1189 SL [11] 0.1559 0.1210 0.0821 0.0529 0.1286 0.0929 0.2064 0.1624 AdvInfoNCE [38] 0.1557 0.1172 0.0829 0.0527 0.1293 0.0934 0.2067 0.1627 BSL [15] 0.1563 0.1212 0.0834 0.0530 0.1288 0.0931 0.2071 0.1630 PSL-tanh 0.1567 0.1225 0.0832 0.0535 0.1297 0.0941 0.2088 0.1646 PSL-atan 0.1567 0.1226 0.0832 0.0535 0.1296 0.0941 0.2087 0.1646 PSL-relu 0.1569 0.1227 0.0838 0.0541 0.1299 0.0945 0.2089 0.1647 Imp.% +1.40%* +2.31%* +1.72%* +1.42%* LightGCN [22] BPR [10] 0.0984 0.0678 0.0813 0.0524 0.1006 0.0681 0.1745 0.1402 LLPAUC [44] 0.1147 0.0810 0.0831 0.0507 0.1272 0.0886 0.1616 0.1192 SL [11] 0.1567 0.1220 0.0823 0.0526 0.1304 0.0941 0.2068 0.1628 AdvInfoNCE [38] 0.1568 0.1177 0.0823 0.0528 0.1292 0.0936 0.2066 0.1625 BSL [15] 0.1568 0.1220 0.0823 0.0526 0.1306 0.0943 0.2069 0.1628 PSL-tanh 0.1575 0.1233 0.0825 0.0532 0.1300 0.0947 0.2091 0.1648 PSL-atan 0.1575 0.1233 0.0825 0.0532 0.1300 0.0948 0.2091 0.1648 PSL-relu 0.1575 0.1233 0.0830 0.0536 0.1300 0.0953 0.2086 0.1648 Imp.% +1.12%* +1.98%* +1.22%* +1.24%* XSimGCL [45] BPR [10] 0.1269 0.0905 0.0777 0.0508 0.1236 0.0857 0.1966 0.1570 LLPAUC [44] 0.1363 0.1008 0.0781 0.0481 0.1184 0.0828 0.1632 0.1200 SL [11] 0.1549 0.1207 0.0772 0.0490 0.1255 0.0905 0.2005 0.1570 AdvInfoNCE [38] 0.1568 0.1179 0.0776 0.0489 0.1252 0.0906 0.2010 0.1564 BSL [15] 0.1550 0.1207 0.0800 0.0507 0.1267 0.0918 0.2037 0.1597 PSL-tanh 0.1567 0.1225 0.0790 0.0501 0.1308 0.0926 0.2034 0.1591 PSL-atan 0.1565 0.1225 0.0792 0.0502 0.1253 0.0917 0.2035 0.1591 PSL-relu 0.1571 0.1228 0.0801 0.0507 0.1313 0.0935 0.2037 0.1593 Imp.% +1.72%* +3.39%* +3.42%* +1.48%* Compared methods. Five representative loss functions are compared in our experiments, including 1) the representative pairwise loss BPR (UAI’09 [10]); 2) the SOTA recommendation loss Softmax Loss (SL) (TOIS’24 [11]) and its two DRO-enhancements AdvInfoNCE (NIPS’23 [38]) and BSL (ICDE’24 [15]); 3) another SOTA loss LLPAUC (WWW’24 [44]) that optimizes the Lower-Left Partial AUC. Refer to Appendix B.3 for more details about these baselines. Backbones. We also adopt three representative backbone models to evaluate the effectiveness of loss, including MF [26], LightGCN [22], and XSimGCL [45], see Appendix B.4 for more details. Hyperparameter settings. A grid search is utilized to find the optimal hyperparameters. For all compared methods, we closely refer to the configurations provided in their respective publications to ensure their optimal performance. As we also carefully finetune SL, the improvements of existing methods over it are not as significant as those presented in their papers. The hyperparameter settings are provided in Appendix B.5, where the detailed optimal hyperparameters for each method on each dataset and backbone are reported. 5.2 Performance Comparisons Results under IID setting. Table 1 presents the performance of our PSL compared with baselines. • PSL outperforms SL and other baselines. Experimental results demonstrate that PSL, with three carefully selected surrogate activations, consistently outperforms SL across all datasets and backbones, with only a few exceptions. For instance, on the MF backbone, compared to the marginal improvements or sometimes even degradation of AdvInfoNCE (-3%~0.5%) and BSL (0.0%~0.5%), PSL shows a significant enhancement over SL (1%~3%). Moreover, our PSL surpasses all compared baselines in most cases, clearly demonstrating its effectiveness. • PSL achieves tighter connections with ranking metrics. We observe that the results align well with our theoretical analyses of PSL’s Advantage 1 in Section 4. By replacing the exponential function with other suitable surrogate activations, PSL establishes a tighter relationship with ranking 8 Table 2: Performance comparison in terms of Recall@20 and NDCG@20 under the OOD setting with popularity shift (on MF backbone). The best result is bolded, and the blue-colored zone indicates that PSL is better than SL. Imp.% denotes the NDCG@20 improvement of PSL over SL. The marker "" indicates that the improvement is statistically significant (p-value < 0.05). Loss Amazon-CD Amazon-Electronic Gowalla Yelp2018 Recall NDCG Recall NDCG Recall NDCG Recall NDCG BPR [10] 0.0518 0.0318 0.0132 0.0069 0.0382 0.0273 0.0118 0.0072 LLPAUC [44] 0.1103 0.0764 0.0225 0.0134 0.0729 0.0522 0.0324 0.0210 SL [11] 0.1184 0.0815 0.0230 0.0142 0.1006 0.0737 0.0349 0.0224 AdvInfoNCE [38] 0.1189 0.0818 0.0228 0.0139 0.0927 0.0676 0.0348 0.0223 BSL [15] 0.1184 0.0815 0.0231 0.0142 0.1006 0.0738 0.0351 0.0225 PSL-tanh 0.1202 0.0834 0.0239 0.0146 0.1013 0.0748 0.0357 0.0228 PSL-atan 0.1202 0.0835 0.0239 0.0146 0.1013 0.0748 0.0358 0.0228 PSL-relu 0.1203 0.0839 0.0241 0.0149 0.1014 0.0752 0.0358 0.0229 Imp.% +3.01% +5.02%* +2.02%* +2.05%* 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.070 0.075 0.080 0.085 0.090 NDCG@20 Amazon-Book NDCG@20 Imp.% PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (a) Amazon-Book 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.025 0.030 0.035 0.040 NDCG@20 Amazon-Electronic NDCG@20 Imp.% PSL-relu SL 0% 2% 4% 6% 8% 10% 12% 14% Imp.% (b) Amazon-Electronic 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.050 0.055 0.060 0.065 0.070 NDCG@20 Amazon-Movie NDCG@20 Imp.% PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (c) Amazon-Movie Figure 2: Performance comparison of SL and PSL in terms of NDCG@20 with different false negative noise ratio (on MF backbone). We also present the relative improvements (i.e., Imp.%) achieved by PSL over SL. The complete results of other baselines are provided in Appendix C.1. metrics, thus achieving better NDCG performance (cf. Lemma 4.1). This is also empirically evident from the larger improvements in NDCG compared to Recall. In contrast, as discussed in Section 4.2, other baselines like AdvInfoNCE and BSL either widen the gap or fail to connect with the ranking metrics, resulting in slight improvements or even performance drops. Results under OOD setting. Table 2 presents the results in OOD scenarios with popularity shift. Given the consistent behavior across the three backbones, here we only report the results on MF. • PSL is robust to distribution shifts. Experimental results indicate that PSL has a strong robustness against distribution shifts, which is consistent with PSL’s Advantage 3 in Section 4. As can be seen, PSL not only outperforms all baselines (2%~5%), but also achieves more pronounced improvements than in IID setting, like on Amazon-Electronic (2.31% →5.02%) and Gowalla (1.42% →2.02%). This demonstrates the superior robustness of PSL to distribution shifts, as shown in Theorem 4.2. • PSL is a DRO-enhancement of more reasonable loss. Although both PSL and SL can be considered as DRO-enhanced losses (cf. Theorem 4.2), the original loss of our three PSLs before DRO-enhancement is more reasonable than that of SL, which degenerates from BPR loss to a linear triplet loss [46]. Therefore, we observe significant improvements of PSL over SL. Results under Noise setting. Figure 2 and Appendix C.1 presents the results with a certain ratio of imputed false negative noise. Specifically, we regard 10% of the positive items in the training set as false negative noise and allow the negative sampling procedure to have a certain probability p of sampling those items. We test the model performance with varying noise ratios p ∈{0.05, 0.1, 0.2, 0.3, 0.5}. • PSL has strong noise resistance. Experimental results demonstrate that as the noise ratio p increases, both the performance of SL and PSL decline. The performance decline rate of PSL is significantly smaller than that of other baselines, resulting in higher performance enhancement(> 10% when p = 0.5). These results indicate that PSL possesses stronger noise resistance than SL, which stems from our rational activation design, as discussed in PSL’s Advantage 2 in Section 4. 9 However, for DRO-enhanced losses such as AdvInfoNCE, the performance declines similarly to or even more quickly than SL (cf. Appendix C.1), which coincides with our theoretical analyses. 6 Related Work Model-related recommendation research. Recent years have witnessed flourishing publications on collaborative filtering (CF) models. The earliest works are mainly extensions of Matrix Factorization [26], building more complex interactions between embeddings [47], such as MF [26], LRML [48], SVD [49, 50], SVD++ [51], NCF [8], etc. In recent years, given the effectiveness of Graph Neural Networks (GNNs) [52–58] in capturing high-order relations, which align well with CF assumptions, GNN-based models have emerged and achieved great success, such as LightGCN [22], NGCF [55], LCF [59], APDA [60], etc. Building upon LightGCN, some works attempt to introduce contrastive learning [12, 61] for graph data augmentation, such as SGL [62] and XSimGCL [45], achieving SOTA performance in recommendation. Loss-related recommendation research. Existing recommendation losses can be primarily categorized into pointwise loss [8, 9], pairwise loss [10], and Softmax Loss (SL) [11], as discussed in Section 2. Given the effectiveness of SL, recently some researchers have proposed to enhance SL from different perspectives. For instance, BSL [15] aims to enhance the positive distribution robustness by leveraging Distributionally Robust Optimization (DRO); AdvInfoNCE [38] employs adversarial learning to enhance SL’s robustness; Zhang et al. [20] suggests incorporating bias-aware margins in SL to tackle popularity bias. Beyond these three types of losses, other approaches have also been explored in recent years. For example, Zhao et al. [63] introduces auto-loss, which utilizes automated machine learning techniques to search the optimal loss; Shi et al. [44] proposes LLPAUC to approximate Recall@K metric. The main concerns with these losses are their lack of theoretical connections to ranking metrics like DCG, which may result in them not consistently outperforming the basic SL. Moreover, both auto-loss and LLPAUC require iterative learning, leading to additional computational time and increased instability. 7 Conclusion and Limitations In this work, we introduce a new family of loss functions, termed Pairwise Softmax Loss (PSL). PSL theoretically offers three advantages: 1) it serves as a better surrogate for ranking metrics with appropriate surrogate activations; 2) it allows flexible control over the distribution of the data contribution; 3) it can be interpreted as a specific BPR loss enhanced by Distributionally Robust Optimization (DRO). These properties demonstrate that PSL has greater effectiveness and robustness compared to Softmax Loss. Our extensive experiments across three testing scenarios validate the superiority of PSL over existing methods. One limitation of both PSL and SL is inefficiency, as they require sampling a relatively large number of negative instances per iteration. 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Incorporating secondorder functional knowledge for better option pricing. Advances in neural information processing systems, 13, 2000. 13 A Theoretical Proofs A.1 Proof of Lemma 4.1 Lemma A.1 (Lemma 4.1). If the condition δ(duij) ≤σ(duij) ≤exp(duij) (4.2) is satisfied for any duij ∈[−1, 1], then PSL serves as a tighter DCG surrogate loss compared to SL. Proof of Lemma 4.1. For any τ > 0, Equation (4.2) indicates that δ(duij) ≤σ(duij)1/τ ≤exp(duij)1/τ (A.1) which means σ(·)1/τ is tighter than exp(·)1/τ approximating δ(·). According to Equations (3.2) and (3.3) in Section 3, we conclude that PSL is a tighter surrogate loss for DCG compared to SL. A.2 Proof of Theorem 4.2 Theorem A.2 (Theorem 4.2). For each user u and its positive item i, let P = P(j\|u, i) be the uniform distribution over I. Given a robustness radius η > 0, consider the uncertainty set Q consisting of all perturbed distributions Q = Q(j\|u, i) satisfying: (i) Q is absolutely continuous w.r.t. P, i.e., Q ≪P; (ii) the KL divergence between Q and P is constrained by η, i.e., DKL(Q∥P) ≤η. Then, optimizing PSL is equivalent to performing DRO over BPR loss, i.e., min Ei∼Pu log Ej∼I h elog(σ(duij))/τi \| {z } LPSL(u) ⇔min Ei∼Pu sup Q∈Q Ej∼Q(j\|u,i) [log σ(duij)] \| {z } LBPR-DRO(u) (4.4) where τ = τ(η) is a temperature parameter controlled by η. To prove Theorem 4.2, it suffices to prove the following lemma: Lemma A.3 (DRO under KL divergence). Given the loss term ℓ(x; θ) of input x and parameters θ, for any robustness radius η > 0, DRO under KL divergence is equivalent to optimizing a loss in the form of log E[exp(·)], i.e., min θ sup Q∈Q Ex∼Q[ℓ(x; θ)] ⇔min θ,τ>0 {τ log Ex∼P [exp(ℓ(x; θ)/τ)] + τη} (A.2) where the uncertainty set Q consists of all perturbed distributions Q constrained by KL divergence w.r.t. the original distribution P, i.e., Q = {Q ≪P : DKL(Q∥P) ≤η}. Lemma A.3, which was first proposed by Hu and Hong [37] with a complex proof, gives a closed-form solution for DRO under KL divergence. Here we provide an elegant proof based on the following general result about the ϕ-divergence DRO, which was first proposed by Shapiro [19]. Theorem A.4 (DRO under ϕ-divergence, [19]). Consider the DRO problem in ϕ-divergence Dϕ(Q∥P) = Z ϕ dQ dP dP (A.3) where ϕ : R →R+ = R+ ∪{∞} is a convex function such that ϕ(1) = 0 and ϕ(t) = +∞ for any t < 0. Then the inner maximization problem in DRO, i.e., supQ∈Q Ex∼Q[ℓ(x; θ)] with the uncertainty set Q = {Q ≪P : Dϕ(Q∥P) ≤η}, is equivalent to the following optimization problem: inf τ>0,µ {Ex∼P [(τϕ)∗(ℓ(x; θ) −µ)] + τη + µ} (A.4) where f ∗(y) = supx {yx −f(x)} is the Fenchel conjugate [64] for any convex function f : R →R. Proof of Theorem A.4. Let the likelihood ratio L(x) = dQ(x)/dP(x), then the inner maximization problem in DRO can be reformulated as sup L⪰0 {Ex∼P [L(x)ℓ(x; θ)] \| Ex∼P [ϕ(L(x))] ≤η, Ex∼P [L(x)] = 1} (A.5) 14 The Lagrangian of Equation (A.5) is L(L, τ, µ) = Ex∼P [L(x)ℓ(x; θ) −τϕ(L(x)) −µL(x)] + τη + µ (A.6) where τ ≥0 and µ are the Lagrange multipliers. Problem (A.5) is a convex optimization problem. One can easily check the Slater’s condition [65] by choosing L(x) ≡1, thus the strong duality [65] holds, and problem (A.5) is equivalent to the dual problem (A.7) of the Lagrangian (A.6): inf τ≥0,µ sup L⪰0 L(L, τ, µ) (A.7) Consider the inner maximization problem supL⪰0 L(L, τ, µ) in Equation (A.7), τη + µ is a constant and can be ignored. By the theorem of interchange of minimization and integration [64], we can interchange sup and expectation in Equation (A.7).Then supL⪰0 L(L, τ, µ) can be reformulated as Ex∼P sup L⪰0 {L(x)(ℓ(x; θ) −µ) −τϕ(L(x))} (A.8) The above problem can be rewritten by the Fenchel conjugate as Ex∼P [(τϕ)∗(ℓ(x; θ) −µ)] (A.9) Thus, problem (A.7) is equivalent to inf τ≥0,µ {Ex∼P [(τϕ)∗(ℓ(x; θ) −µ)] + τη + µ} (A.10) Finally, note that the condition τ ≥0 in problem (A.10) can be relaxed to τ > 0 without affecting the optimal value, thus problem (A.10) is equivalent to problem (A.4), which completes the proof. Lemma A.3 can be directly derived from Theorem A.4 as follows: Proof of Lemma A.3. KL divergence is a special case of ϕ-divergence with ϕ(x) = x log x, and the Fenchel conjugate of τϕ is (τϕ)∗(y) = sup x {yx −τx log x} = τey/τ−1 (A.11) By Theorem A.4, the DRO problem under KL divergence is equivalent to inf τ>0,µ n Ex∼P h τe(ℓ(x;θ)−µ)/τ−1i + τη + µ o = inf τ>0,µ n Ex∼P h eℓ(x;θ)/τi τe−µ/τ−1 + τη + µ o (A.12) We fix τ and solve the optimal value of µ as µ∗= τ log Ex∼P h eℓ(x;θ)/τi −τ (A.13) Therefore, by substituting the optimal µ∗in Equation (A.13) back to Equation (A.12), the original DRO problem is equivalent to inf θ,τ>0 n τ log Ex∼P h eℓ(x;θ)/τi + τη o (A.14) This completes the proof. Theorem 4.2 is a direct consequence of Lemma A.3, when setting the loss term ℓ(x; θ) as log σ(duij) (i.e., the pairwise loss term in BPR loss), P as the uniform distribution over I, Q as the perturbed distribution constrained by KL divergence w.r.t. P, and τ = τ(η) as the optimal value of Lagrange multiplier τ in Equation (A.2). This completes the proof of Theorem 4.2. 15 A.3 Proof of the Bound Connections between PSL and Other Losses in Section 4.2 Proof of the Bound Connections in Section 4.2. We have proved in Lemma 4.1 that −log DCG(u) + log \|Pu\| ≤ 1 \|Pu\| X i∈Pu log  X j∈I σ(duij)1/τ   (A.15) with any surrogate activation σ satisfying δ(duij) ≤σ(duij). Furthermore, if two surrogate activations σ1, σ2 satisfy σ1(duij) ≤σ2(duij) for any duij ∈[−1, 1], then the corresponding DCG surrogate losses satisfy the same inequality. Therefore, we have −log DCG ≤LPSL ≤LSL ≤LAdvInfoNCE (A.16) where the constant term is omitted for simplicity. Finally, we prove that BPR serves as a surrogate loss for DCG. Apply Jensen’s inequality to the RHS of Equation (A.15), we have 1 \|Pu\| X i∈Pu log  X j∈I σ(duij)1/τ  ≤log  1 \|Pu\| X i∈Pu X j∈I σ(duij)1/τ   (A.17) The RHS of Equation (A.17) is just log LBPR(u) −log \|Pu\| with the same surrogate activation σ in BPR. Equation (A.17) indicates that for any surrogate activation σ, the general PSL (including SL, BSL, and AdvInfoNCE) is always better than BPR with the same σ, i.e., −log DCG ≤LPSL ≤log LBPR (A.18) where the constant term is omitted for simplicity. This completes the proof. 16 B Experimental Details B.1 Datasets The six benchmark datasets used in our experiments are summarized in Table B.1. In dataset preprocessing, following the standard practice in Wang et al. [55], we use 10-core setting [66], i.e., all users and items have at least 10 interactions. We also remove the low-quality interactions, such as those with ratings (if available) lower than 3. After preprocessing, we split the datasets into 80% training and 20% test sets. In IID and Noise settings, we further randomly split a 10% validation set from training set for hyperparameter tuning. Table B.1: Statistics of datasets. All datasets are cleaned by 10-core setting. If the dataset is used in both IID and OOD settings, the statistics below are provided for the IID setting. Dataset #Users #Items #Interactions Density Amazon-Electronic [40, 41] 13,455 8,360 234,521 0.00208 Amazon-CD [40, 41] 12,784 13,874 360,763 0.00203 Amazon-Movie [40, 41] 26,968 18,563 762,957 0.00152 Gowalla [42] 29,858 40,988 1,027,464 0.00084 Yelp2018 [43] 55,616 34,945 1,506,777 0.00078 Amazon-Book [40, 41] 135,109 115,172 4,042,382 0.00026 The details of datasets are as follows: • Amazon [40, 41]: The Amazon dataset is a large crawl of product reviews from Amazon4. The 2014 version of Amazon dataset contains 142.8 million reviews spanning May 1996 - July 2014. We process four widely-used categories: Electronic, CD, Movie, and Book, with interactions ranging from 200K to 4M. • Gowalla [42]: The Gowalla dataset is a check-in dataset collected from the location-based social network Gowalla5, including 1M users, 1M locations, and 6M check-ins. • Yelp2018 [43]: The Yelp6 dataset is a subset of Yelp’s businesses, reviews, and user data, which was originally used in the Yelp Dataset Challenge. The 2018 version of Yelp dataset contains 5M reviews. The detailed dataset constructions in IID, OOD and Noise settings are as follows: • IID setting [22]: In the IID setting, the test set is randomly split from the original dataset. Specifically, the positive items of each user are split into 80% training and 20% test sets. Moreover, the training set is further split into 90% training and 10% validation sets for hyperparameter tuning. In the IID setting, the training and test sets are both long-tail. • OOD setting [20, 24, 39]: In the OOD setting, a 20% test set is uniformly sampled (w.r.t. items) from the original dataset, while the 80% training set remains long-tail. The OOD setting is used to simulate real-world online recommender systems. In order to avoid leaking information about the test set distribution, we do not introduce the validation set. • Noise setting [15]: In the Noise setting, the validation and test sets are split in the same way as the IID setting. However, we randomly sample 10% of the training set as the false negatives. In Noise training, the negative items will be sampled from the false negatives with a probability of p as the negative noise, where p ∈{0.05, 0.1, 0.2, 0.3, 0.5} is a.k.a. the noise ratio. All experiments are conducted on one NVIDIA GeForce RTX 4090 GPU and one AMD EPYC 7763 64-Core Processor. B.2 Metrics This section provides a detailed explanation of the recommendation metrics used or mentioned in our experiments. 4https://www.amazon.com/ 5https://en.wikipedia.org/wiki/Gowalla 6https://www.yelp.com/ 17 As stated in Section 5.1, we use Top-K recommendation [5]. It should be noted that for each user, the positive items in the training set will be masked and not included in the Top-K recommendations when evaluating, and the ground-truth positive items Pu only consist of those in the test set. For convenience, we denote the set of hit items in the Top-K recommendations for user u as Hu = {i ∈ Pu : πu(i) ≤K}. The recommendation metrics are defined as follows: • Recall@K [29]: The proportion of hit items among Pu in the Top-K recommendations, i.e., Recall@K(u) = \|Hu\|/\|Pu\|, and the overall Recall@K = Eu∼U[Recall@K(u)]. • NDCG@K [6]: The Discounted Cumulative Gain in the Top-K recommendations (DCG@K) is defined as DCG@K(u) = P i∈Hu 1/ log2(1 + πu(i)). Since the range of DCG@K will vary with the number of positive items \|Pu\|, we should consider to normalize DCG@K to [0, 1]. The Normalized DCG in the Top-K recommendations (NDCG@K) = DCG@K(u)/IDCG@K(u), where IDCG@K is the ideal DCG@K, i.e., IDCG@K(u) = Pmin{K,\|Pu\|} i=1 1/ log2(1 + i). The overall NDCG@K = Eu∼U[NDCG@K(u)]. • MRR@K [7, 28]: The Mean Reciprocal Rank (MRR) is originally defined as the reciprocal of the rank of the first hit item. Here we follow the definition of Argyriou et al. [28]’s to meet the requirements of multi-hit scenarios, i.e., MRR@K(u) = Ei∼Hu[1/πu(i)], and the overall MRR@K = Eu∼U[MRR@K(u)]. B.3 Baselines We reproduced the following losses as baselines in our experiments: • BPR [10]: A pairwise loss based on the Bayesian Maximum Likelihood Estimation (MLE). The objective of BPR is to learn a partial order among items, i.e., positive items should be ranked higher than negative items. Furthermore, BPR is a surrogate loss for AUC metric [10, 30]. In our implementation, we follow He et al. [22]’s setting and use the inner product as the similarity function for user and item embeddings. • LLPAUC [44]: A surrogate loss for Recall and Precision. In fact, LLPAUC is a surrogate loss for the lower-left part of AUC. In practice, LLPAUC is a min-max loss. • Softmax Loss (SL) [11]: A SOTA recommendation loss derived from the listwise MLE, i.e., maximizing the probability of the positive items among all items. The effectiveness of SL has been thoroughly reviewed in Sections 2 and 3. In fact, SL is a special case of PSL with surrogate activation σ = exp(·). • AdvInfoNCE [38]: A DRO-based modification of SL. AdvInfoNCE tries to introduce adaptive negative hardness to pairwise score duij in SL (cf. Equation (3.1)). In Zhang et al. [38]’s original design, AdvInfoNCE can be seen as a failure case of PSL with surrogate activation σ = exp(exp(·)), as discussed in Section 4.2. In practice, AdvInfoNCE is a min-max loss. • BSL [15]: A DRO-based modification of SL. BSL applies additional DRO on the positive term in the pointwise form of SL. The hyperparameter settings of each method are detailed in Appendix B.5. B.4 Backbones We implemented three popular recommendation backbones in our experiments, including • MF [26]: MF is the most basic but still effective recommendation model, which factorizes the useritem interaction matrix into user and item embeddings. All the embedding-based recommendation models use MF as the first layer. Specifically, we set the embedding size d = 64 for all settings, following the setting in Wang et al. [55]. • LightGCN [22]: LightGCN is an effective GNN-based recommendation model. LightGCN performs graph convolution on the user-item interaction graph, so as to aggregate the high-order interactions. Specifically, LightGCN simplifies NGCF [55] and only retains the non-parameterized graph convolution operator. In our experiments, we set the number of layers as 2, which aligns with the original setting in He et al. [22]. • XSimGCL [45]: XSimGCL is a novel recommendation model based on contrastive learning [12, 67]. Based on 3-layers LightGCN, XSimGCL adds a random noise to the output embeddings of each layer, and introduces the contrastive learning between the final layer and the l∗-th layer, 18 i.e., adding an auxiliary InfoNCE loss [61] between these two layers. Following the original Yu et al. [45]’s setting, the modulus of random noise between each layer is set as 0.1, the contrastive layer l∗= 1 (where the embedding layer is 0-th layer), the temperature of InfoNCE is set as 0.1, and the weight of the auxiliary InfoNCE loss is set as 0.2 (except for the Amazon-Electronic dataset, where the weight is set as 0.05). B.5 Hyperparameters B.5.1 Hyperparameter Settings Optimizer. We use Adam [68] optimizer for training. The learning rate (lr) is searched in {10−1, 10−2, 10−3}, except for BPR, where the lr is searched in {10−1, 10−2, 10−3, 10−4}. The weight decay (wd) is searched in {0, 10−4, 10−5, 10−6}. The batch size is set as 1024, and the number of epochs is set as 200. Following the negative sampling strategy in Wu et al. [15], we uniformly sample 1000 negative items for each positive instance in training. Loss. The hyperparameters of each loss are detailed as follows: • BPR: No other hyperparameters. • LLPAUC: Following Shi et al. [44]’s setting, the hyperparameters α ∈{0.1, 0.3, 0.5, 0.7, 0.9} and β ∈{0.01, 0.1} are searched. • Softmax Loss (SL): The temperature τ ∈{0.005, 0.025, 0.05, 0.1, 0.25} is searched. • AdvInfoNCE: The temperature τ is searched in the same space as SL. The other hyperparameters are fixed as the original setting in Zhang et al. [38]. Specifically, the negative weight is set as 64, the adversarial learning will be performed every 5 epochs, with the adversarial learning rate as 5 × 10−5. • BSL: The temperatures τ1, τ2 for positive and negative terms are searched in the same space as SL, respectively. • PSL: The temperature τ is searched in the same space as SL. B.5.2 Optimal Hyperparameters The hyperparameters we search include the learning rate (lr), weight decay (wd), and other hyperparameters: {α, β} for LLPAUC, {τ} for SL, AdvInfoNCE, and PSL, {τ1, τ2} for BSL. IID optimal hyperparameters. Table B.2 shows the optimal hyperparameters of IID setting, including four datasets (Amazon-Book, Amazon-Electronic, Amazon-Movie, Gowalla) and three backbones (MF, LightGCN, XSimGCL). OOD optimal hyperparameters. Table B.3 shows the optimal hyperparameters of OOD setting on MF backbone, including four datasets (Amazon-CD, Amazon-Electronic, Gowalla, Yelp2018). Noise optimal hyperparameters. The Noise setting uses the optimal hyperparameters of IID setting, as listed in Table B.2. We compare the performance of each method under different noise ratios p ∈{0.05, 0.1, 0.2, 0.3, 0.5} on MF backbone and four IID datasets (Amazon-Book, AmazonElectronic, Amazon-Movie, Gowalla). 19 Table B.2: Optimal hyperparameters of IID setting. Model Loss Amazon-Book Amazon-Electronic lr wd others lr wd others MF BPR 10−4 0 10−3 10−5 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.5, 0.01} AdvInfoNCE 10−2 0 {0.05} 10−1 0 {0.1} SL 10−1 0 {0.025} 10−2 0 {0.1} BSL 10−1 0 {0.25, 0.025} 10−1 0 {0.25, 0.1} PSL-tanh 10−1 0 {0.025} 10−2 0 {0.1} PSL-atan 10−1 0 {0.025} 10−2 0 {0.1} PSL-relu 10−1 0 {0.025} 10−2 0 {0.1} LightGCN BPR 10−3 0 10−2 10−6 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.5, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−2 0 {0.1} SL 10−1 0 {0.025} 10−2 0 {0.1} BSL 10−1 0 {0.25, 0.025} 10−2 0 {0.1, 0.1} PSL-tanh 10−1 0 {0.025} 10−2 0 {0.1} PSL-atan 10−1 0 {0.025} 10−2 0 {0.1} PSL-relu 10−1 0 {0.025} 10−2 0 {0.1} XSimGCL BPR 10−4 10−5 10−2 0 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.3, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.1} SL 10−1 0 {0.025} 10−2 0 {0.1} BSL 10−1 0 {0.025, 0.025} 10−1 0 {0.05, 0.1} PSL-tanh 10−2 0 {0.025} 10−1 0 {0.1} PSL-atan 10−2 0 {0.025} 10−1 0 {0.1} PSL-relu 10−1 0 {0.025} 10−1 0 {0.1} Model Loss Amazon-Movie Gowalla lr wd others lr wd others MF BPR 10−3 10−6 10−3 10−6 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.7, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.05} SL 10−1 0 {0.05} 10−1 0 {0.05} BSL 10−2 0 {0.25, 0.05} 10−1 0 {0.1, 0.05} PSL-tanh 10−1 0 {0.05} 10−1 0 {0.05} PSL-atan 10−1 0 {0.05} 10−1 0 {0.05} PSL-relu 10−1 0 {0.05} 10−1 0 {0.05} LightGCN BPR 10−3 0 10−3 0 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.7, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.05} SL 10−1 0 {0.05} 10−1 0 {0.05} BSL 10−1 0 {0.025, 0.05} 10−1 0 {0.025, 0.05} PSL-tanh 10−1 0 {0.05} 10−1 0 {0.05} PSL-atan 10−1 0 {0.05} 10−1 0 {0.05} PSL-relu 10−1 0 {0.05} 10−1 0 {0.05} XSimGCL BPR 10−4 10−4 10−4 0 LLPAUC 10−1 0 {0.3, 0.01} 10−1 0 {0.7, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.05} SL 10−2 0 {0.05} 10−2 0 {0.05} BSL 10−1 0 {0.025, 0.05} 10−1 0 {0.025, 0.05} PSL-tanh 10−1 0 {0.1} 10−1 0 {0.05} PSL-atan 10−1 0 {0.05} 10−1 0 {0.05} PSL-relu 10−2 0 {0.1} 10−1 0 {0.05} 20 Table B.3: Optimal hyperparameters of OOD setting. Model Loss Amazon-CD Amazon-Electronic lr wd others lr wd others MF BPR 10−2 10−6 10−2 10−6 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.7, 0.1} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.05} SL 10−1 0 {0.05} 10−1 0 {0.05} BSL 10−1 0 {0.05, 0.05} 10−1 0 {0.1, 0.05} PSL-tanh 10−1 0 {0.05} 10−1 0 {0.05} PSL-atan 10−1 0 {0.05} 10−1 0 {0.05} PSL-relu 10−1 0 {0.05} 10−1 0 {0.05} Model Loss Gowalla Yelp2018 lr wd others lr wd others MF BPR 10−3 0 10−3 0 LLPAUC 10−1 0 {0.7, 0.01} 10−1 0 {0.7, 0.01} AdvInfoNCE 10−1 0 {0.05} 10−1 0 {0.05} SL 10−1 0 {0.025} 10−1 0 {0.05} BSL 10−1 0 {0.25, 0.025} 10−1 0 {0.1, 0.05} PSL-tanh 10−1 0 {0.025} 10−1 0 {0.025} PSL-atan 10−1 0 {0.025} 10−1 0 {0.025} PSL-relu 10−1 0 {0.025} 10−1 0 {0.025} 21 C Supplementary Experiments C.1 Noise Results The Recall@20 and NDCG@20 results under Noise setting on four datasets (Amazon-Book, AmazonElectronic, Amazon-Movie, Gowalla) are shown in Figures C.1 to C.4. 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.07 0.08 0.09 0.10 0.11 Recall@20 Amazon-Book Recall@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (a) Amazon-Book (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.05 0.06 0.07 0.08 0.09 NDCG@20 Amazon-Book NDCG@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (b) Amazon-Book (NDCG@20) Figure C.1: Noise results on Amazon-Book dataset. 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.040 0.045 0.050 0.055 0.060 0.065 Recall@20 Amazon-Electronic Recall@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% 12% Imp.% (a) Amazon-Electronic (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.025 0.030 0.035 0.040 NDCG@20 Amazon-Electronic NDCG@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% 12% 14% Imp.% (b) Amazon-Electronic (NDCG@20) Figure C.2: Noise results on Amazon-Electronic dataset. 22 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.070 0.075 0.080 0.085 0.090 0.095 Recall@20 Amazon-Movie Recall@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% Imp.% (a) Amazon-Movie (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.050 0.055 0.060 0.065 0.070 NDCG@20 Amazon-Movie NDCG@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (b) Amazon-Movie (NDCG@20) Figure C.3: Noise results on Amazon-Movie dataset. 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.10 0.11 0.12 0.13 0.14 0.15 Recall@20 Gowalla Recall@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% Imp.% (a) Gowalla (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.08 0.09 0.10 0.11 NDCG@20 Gowalla NDCG@20 Imp.% AdvInfoNCE BSL PSL-atan PSL-relu SL 0% 2% 4% 6% 8% 10% 12% Imp.% (b) Gowalla (NDCG@20) Figure C.4: Noise results on Gowalla dataset. 23 C.2 PSL-softplus Results BPR uses Softplus [69] as log σ, i.e., σ(duij) = exp(duij) + 1, which is looser than SL. That is, this surrogate activation is not a suitable choice for PSL. We call this PSL variant as PSL-softplus. In this section, we conduct experiments to evaluate the performance of PSL-softplus with surrogate activation σ(duij) = exp(duij) + 1. The IID, OOD, and Noise results of PSL-softplus are shown in Tables C.4 and C.5, Figures C.5 to C.8, respectively. Results demonstrate that PSL-softplus is inferior to SL and three PSLs in all settings. This confirms our claim – the choice of surrogate activation σ is crucial, and an unreasonable or intuitive design will decrease the accuracy. Table C.4: IID results of PSL-softplus. The results of SL, PSL-tanh, PSL-atan, and PSL-relu have been listed in Table 1. The blue-colored results are better than PSL-softplus. Model Loss Amazon-Book Amazon-Electronic Amazon-Movie Gowalla Recall NDCG Recall NDCG Recall NDCG Recall NDCG MF SL 0.1559 0.1210 0.0821 0.0529 0.1286 0.0929 0.2064 0.1624 PSL-tanh 0.1567 0.1225 0.0832 0.0535 0.1297 0.0941 0.2088 0.1646 PSL-atan 0.1567 0.1226 0.0832 0.0535 0.1296 0.0941 0.2087 0.1646 PSL-relu 0.1569 0.1227 0.0838 0.0541 0.1299 0.0945 0.2089 0.1647 PSL-softplus 0.1536 0.1149 0.0826 0.0522 0.1280 0.0919 0.2053 0.1613 LightGCN SL 0.1567 0.1220 0.0823 0.0526 0.1304 0.0941 0.2068 0.1628 PSL-tanh 0.1575 0.1233 0.0825 0.0532 0.1300 0.0947 0.2091 0.1648 PSL-atan 0.1575 0.1233 0.0825 0.0532 0.1300 0.0948 0.2091 0.1648 PSL-relu 0.1575 0.1233 0.0830 0.0536 0.1300 0.0953 0.2086 0.1648 PSL-softplus 0.1536 0.1152 0.0814 0.0514 0.1296 0.0932 0.2053 0.1613 XSimGCL SL 0.1549 0.1207 0.0772 0.0490 0.1255 0.0905 0.2005 0.1570 PSL-tanh 0.1567 0.1225 0.0790 0.0501 0.1308 0.0926 0.2034 0.1591 PSL-atan 0.1565 0.1225 0.0792 0.0502 0.1253 0.0917 0.2035 0.1591 PSL-relu 0.1571 0.1228 0.0801 0.0507 0.1313 0.0935 0.2037 0.1593 PSL-softplus 0.1545 0.1161 0.0770 0.0484 0.1242 0.0894 0.1996 0.1557 Table C.5: OOD results of PSL-softplus. The results of SL, PSL-tanh, PSL-atan, and PSL-relu have been listed in Table 2. The blue-colored results are better than PSL-softplus. Loss Amazon-CD Amazon-Electronic Gowalla Yelp2018 Recall NDCG Recall NDCG Recall NDCG Recall NDCG SL 0.1184 0.0815 0.0230 0.0142 0.1006 0.0737 0.0349 0.0224 PSL-tanh 0.1202 0.0834 0.0239 0.0146 0.1013 0.0748 0.0357 0.0228 PSL-atan 0.1202 0.0835 0.0239 0.0146 0.1013 0.0748 0.0358 0.0228 PSL-relu 0.1203 0.0839 0.0241 0.0149 0.1014 0.0752 0.0358 0.0229 PSL-softplus 0.1169 0.0799 0.0232 0.0139 0.0909 0.0665 0.0346 0.0222 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.06 0.07 0.08 0.09 0.10 0.11 Recall@20 Amazon-Book Recall@20 PSL-relu PSL-softplus SL (a) Amazon-Book (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.05 0.06 0.07 0.08 0.09 NDCG@20 Amazon-Book NDCG@20 PSL-relu PSL-softplus SL (b) Amazon-Book (NDCG@20) Figure C.5: Noise results of PSL-softplus on Amazon-Book dataset. 24 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.035 0.040 0.045 0.050 0.055 0.060 0.065 Recall@20 Amazon-Electronic Recall@20 PSL-relu PSL-softplus SL (a) Amazon-Electronic (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.025 0.030 0.035 0.040 NDCG@20 Amazon-Electronic NDCG@20 PSL-relu PSL-softplus SL (b) Amazon-Electronic (NDCG@20) Figure C.6: Noise results of PSL-softplus on Amazon-Electronic dataset. 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.07 0.08 0.09 Recall@20 Amazon-Movie Recall@20 PSL-relu PSL-softplus SL (a) Amazon-Movie (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.045 0.050 0.055 0.060 0.065 0.070 NDCG@20 Amazon-Movie NDCG@20 PSL-relu PSL-softplus SL (b) Amazon-Movie (NDCG@20) Figure C.7: Noise results of PSL-softplus on Amazon-Movie dataset. 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.10 0.11 0.12 0.13 0.14 0.15 Recall@20 Gowalla Recall@20 PSL-relu PSL-softplus SL (a) Gowalla (Recall@20) 0.05 0.1 0.2 0.3 0.5 Noise Ratio p 0.07 0.08 0.09 0.10 0.11 NDCG@20 Gowalla NDCG@20 PSL-relu PSL-softplus SL (b) Gowalla (NDCG@20) Figure C.8: Noise results of PSL-softplus on Gowalla dataset. 25 C.3 Comparisons of Two Extension Forms In this section, we compare the two different extension forms from SL to PSL, i.e., outside form σ(duij)1/τ and inside form σ(duij/τ). As discussed in Section 4.2, the outside form scales in the value domain, while the inside form scales in the definition domain. Therefore, the inside form will lead to certain drawbacks: 1) the condition (4.2) must be satisfied over the entire duij ∈R to ensure a tighter DCG surrogate loss (cf. Lemma 4.1), which is hard to achieve; 2) the value of σ(duij/τ) and its gradient may be quickly exploded when τ →0, as the range of duij/τ is hard to control, which may cause numerical instability. To empirically compare the above two extension forms, we conduct experiments on MF backbone and four IID datasets. Specifically, since there exists serious numerical instability, we expand the range of τ to {0.005, 0.025, 0.05, 0.1, 0.25, 0.5, 1.0} for the inside form, where the outside form remains the same search space τ ∈{0.005, 0.025, 0.05, 0.1, 0.25}. The results are shown in Table C.6, demonstrating that the outside form is superior to the inside form in all cases. Table C.6: Extension forms comparisons on MF under IID setting. The blue-colored results are better than the counterpart. Form Loss Amazon-Book Amazon-Electronic Amazon-Movie Gowalla Recall NDCG Recall NDCG Recall NDCG Recall NDCG σ(duij)1/τ PSL-tanh 0.1567 0.1225 0.0832 0.0535 0.1297 0.0941 0.2088 0.1646 PSL-atan 0.1567 0.1226 0.0832 0.0535 0.1296 0.0941 0.2087 0.1646 PSL-relu 0.1569 0.1227 0.0838 0.0541 0.1299 0.0945 0.2089 0.1647 σ(duij/τ) PSL-tanh 0.1415 0.1041 0.0767 0.0494 0.0876 0.0590 0.1956 0.1507 PSL-atan 0.0307 0.0213 0.0453 0.0268 0.0363 0.0247 0.0982 0.0727 PSL-relu 0.1366 0.1053 0.0723 0.0452 0.1210 0.0855 0.1732 0.1304 26 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The main claims made in the abstract accurately conclude this paper’s contribution, including theoretical findings and experimental results. 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4,470	M3GPT: An Advanced Multimodal, Multitask Framework for Motion Comprehension and Generation Mingshuang Luo1,2,3, Ruibing Hou1∗, Zhuo Li4, Hong Chang1,3, Zimo Liu2, Yaowei Wang2,5, Shiguang Shan1,3 1Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS), Institute of Computing Technology, CAS, China 2Peng Cheng Laboratory, China, 3University of Chinese Academy of Sciences, China 4WeChat, Tencent Inc, 5Harbin Institute of Technology, Shenzhen mingshuang.luo@vipl.ict.ac.cn,{houruibing,changhong,sgshan}@ict.ac.cn albertzli@tencent.com,liuzm@pcl.ac.cn,wangyaowei@hit.edu.cn Abstract This paper presents M3GPT, an advanced Multimodal, Multitask framework for Motion comprehension and generation. M3GPT operates on three fundamental principles. The first focuses on creating a unified representation space for various motion-relevant modalities. We employ discrete vector quantization for multimodal conditional signals, such as text, music and motion/dance, enabling seamless integration into a large language model (LLM) with a single vocabulary. The second involves modeling motion generation directly in the raw motion space. This strategy circumvents the information loss associated with a discrete tokenizer, resulting in more detailed and comprehensive motion generation. Third, M3GPT learns to model the connections and synergies among various motion-relevant tasks. Text, the most familiar and well-understood modality for LLMs, is utilized as a bridge to establish connections between different motion tasks, facilitating mutual reinforcement. To our knowledge, M3GPT is the first model capable of comprehending and generating motions based on multiple signals. Extensive experiments highlight M3GPT’s superior performance across various motion-relevant tasks and its powerful zero-shot generalization capabilities for extremely challenging tasks. Project page: https://luomingshuang.github.io/M3GPT/. Can you generate a motion for the caption that a person performs a knee tuck to kick L? Can you generate a music for the dance? Can you fullfill the following motion? A person is performing a Hip Bounce Wrist Circle movement. Can you generate a dance for the music? Can you translate the motion into a caption? M3GPT Can you predict a motion for a given motion? Figure 1: M3GPT can handle core motion comprehension and generation tasks, including text-to-motion, motionto-text, music-to-dance, dance-to-music, motion prediction, and motion in-between. The motion sequences within the dashed-line areas are masked in the input. ∗Corresponding author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1 Introduction Motion comprehension and generation in multimodality are crucial for diverse applications, including AR/VR creation, video games, and virtual reality. Numerous studies [15, 51, 64, 52] focus on motion comprehension, including captioning 3D human motions and generating music from 3D human dances2. Recent advancements in AI [14, 44, 42, 48] have paved the way for motion generation, allowing for various control signals including textual descriptions, music pieces, and human poses. A significant shortcoming of most existing works is their focus on single-modality control signals, overlooking the potential for multimodal information integration. More importantly, the comprehension and generation of motions are predominantly studied in isolation. In reality, human motion cognition and communication indispensably require seamless transitions between any motion-relevant modalities. Therefore, it is vital to develop a unified framework for motion comprehension and generation that can efficiently utilize multiple signals simultaneously. Recent works [12, 62, 63, 58] have shown success in developing a unified multitask motion framework which integrates text-driven and audio-driven motion generation through a single architecture. Employing a large language model (LLM), [60] adeptly handles multimodal control signals, such as text and single-frame pose, to generate consecutive motions. Despite their promising performance in motion generation, these approaches often fall short in comprehending motion. MotionGPT [21], a recent innovation, constructs a unified motion-language model to generate plausible human motions and natural language descriptions through prompt instructions. However, MotionGPT focuses solely on a single non-motion modality, i.e., text. While aligning motion with one additional modality is relatively straightforward, integrating three or more modalities within a single framework and achieving bidirectional alignment among them to cover a broad range of modalities for motion comprehension and generation presents a formidable challenge. Two main challenges need to be solved for building a unified multimodal framework for motion comprehension and generation. The first is how to create a unified representation space across different motion-relevant modalities. MotionGPT [21] and SpeechGPT [54] separately treat motion and speech as specific language for seamlessly integrating with text. Inspired by these efforts [21, 54], we view both motion and music as distinct forms of language, facilitating better associations with text via LLMs. Specifically, akin to language, we compress raw motion and music into a sequence of discrete semantic tokens. By encoding motion, music, and language within a single vocabulary, we can build a unified representation space across these different modalities. The second is how to model the connections and synergies among various motion tasks. Different motion-relevant tasks are interconnected and can mutually enhance each other. Since text is the most familiar and well-understood modality for LLMs, we propose employing text as a bridge to establish connections between different motion tasks. Specifically, to better learn the complex music-to-dance task where both input and output modalities are unfamiliar to LLMs, we introduce two auxiliary tasks: musicto-text and text-to-dance, aimed at aligning music and dance modalities with the structured text embedding space. This strategy enables us to establish connections and synergies between musicto-dance and text-to-motion tasks, facilitating the alignment and collaboration of text, music, and motion/dance modalities across different tasks. In this work, we propose a uniform Multimodal, Multitask framework for Motion comprehension and generation, namely M3GPT, that leverages the strong language generation capability of LLMs for unifying various motion-relevant tasks, as depicted in Fig. 1. M3GPT comprises three tires. Firstly, M3GPT is equipped with multimodal tokenizers capable of compressing raw multimodal data, including motion, music, and text, into a sequence of discrete semantic tokens. These discrete representations allow the core LLM to unify motion comprehension and generation in an autoregressive manner, operating at the discrete semantic representation space. Secondly, different from [21, 60] that solely optimize LLM in discrete semantic space, we jointly train LLM and motion de-tokenizer, optimizing LLM in both discrete semantic space and raw continuous motion space. This operation enables the motion-space error signals from de-tokenizer to backpropagate to LLM, enhancing LLM’s ability to generate the details of motion. Thirdly, we construct paired text descriptions for music, and design two auxiliary music-to-text and text-to-dance tasks, which aid in aligning music and dance with the text embedding space. Also, we build up a shared tokenizer for motion and dance data 2In this paper, the term "motion" generally includes "dance." We distinguish them when referring to specific tasks or scenes, such as text-to-motion, and music-to-dance. 2 Methods T2M M2T A2D D2A M2M Random M Random T Random A TM2D[12] " % " % % " % % UDE[62] " % " % % " % % MotionGPT[60] " % % % " " % % MotionGPT[21] " " % % " " " % M3GPT (Ours) " " " " " " " " Table 1: Comparison of recent multimodal, multitask methods across various motion comprehension and generation tasks. T2M: text-to-motion; M2T: motion-to-text; A2D: music-to-dance; D2A: dance-to-music; M2M: motion-to-motion that includes motion prediction and motion in-between. Random M, Random T, and Random A represent the unconstrained generation of motion, text, and music3, respectively. to project them into a shared semantic space. These auxiliary tasks and shared tokenizer establish connections between music-to-dance and text-to-motion, enabling mutual reinforcement. We employ a multimodal pre-training + instruction-tuning pipeline to train M3GPT, enhancing inter-modal alignment and effectively aligning them with human intent. To our knowledge, M3GPT is the first approach to integrate six core tasks of motion comprehension and generation—text-tomotion, motion-to-text, music-to-dance, dance-to-music, motion prediction, and motion in-between— into a uniform framework. Extensive experiments demonstrate that M3GPT achieves competitive performance across multiple motion-relevant tasks. Additionally, through qualitative results, we demonstrate that M3GPT possesses powerful zero-shot generalization capabilities, e.g., long-term dance generation and music-text conditioned dance generation. 2 Related Work Motion comprehension and Generation. Many existing works focus on studying human appearance, pose, detection, attribute, part parsing and so on [61, 19, 45, 40, 23, 17]. This work focuses on studying human motion, including motion comprehension and motion generation. Motion comprehension involves two core tasks: motion-to-text and dance-to-music. Motion-to-text aims to describe human motion with natural language [37]. For example, recurrent networks have been used in [37] to accomplish this task. Dance-to-music involves creating a piece of music from a given dance [20, 27, 64]. For example, Zhun et al. [64] utilizes a generative adversarial network to generate music from dance videos. On the other hand, motion generation involves generating diverse human motions using multimodal inputs, such as text [44, 56, 15, 5, 57], music [27, 18, 52, 42] and incomplete motion [31, 1, 3]. Text-to-motion is one of the most important motion generation tasks. Recent works typically map text to motion using different architectures: diffusion model [57] and autoregressive transformer model [15]. Music-to-dance focuses on generating dance movements from music. For example, [42] predicts discrete token sequences conditioned on music, which are then used to regenerate the dance sequence. Motion Completion generates motion conditioning on partial motions, such as motion prediction [31, 1] and motion-in-between [34, 43]. Although these methods have shown promising results in various human motion tasks, most are limited to handling a single task. Until recently, some works [12, 21, 60, 62] attempt to integrate two or more tasks into a unified model, as shown in Tab. 1. However, these works either lack the ability of motion comprehension [62, 60] or fail to handle music modality [12, 21]. In this work, we propose a unified motion comprehension and generation framework that can handle multiple control signals simultaneously. Language Models and Multimodal. Large language models (LLMs) enabled by extensive datasets and model size, such as T5 [39], Flan-T5 [7], LLaMA [46], LLaMA-2 [47] and Vicuna [6], have demonstrated impressive comprehension and generation capabilities. Researchers have leveraged the capabilities of LLMs to handle multimodal tasks, expanding them to multimodal large language models (MLLMs). For example, AnyGPT [53] employs LLaMA-2 [47] to construct an any-to-any multimodal language model. NExT-GPT [50] employs Vicuna [6] with multimodal adaptors and diffusion decoders to perform tasks across arbitrary combinations of text, images, videos, and audio. Recently, the works [21, 60] attempt to use LLMs for motion-related tasks. [60] uses LLaMA [46] to build a general-purpose motion generator, which, however, lacks the ability to comprehend motion. [21] leverages T5 to construct a unified motion-language model, but cannot deal with music modality. 3Note: in this paper, the term "audio" specifically refers to "music". This designation is adopted to avoid confusion between the initial letter "M" shared by both "music" and "motion," which could lead to ambiguity when these modalities are represented by their initials. 3 Stage 1: Training of Multimodal Tokenizers Motion Tokenizer Motion De-tokenizer Stage 2: Pre-training LLM Stage 3: Instruction Tuning LLM Can you generate a motion for the caption that a person performs a knee tuck to kick L? Multimodal Vocabulary Music Tokenizer Music De-tokenizer Input Music Generated Music Text Codebook ... Input Motion Generated Motion Text Tokens ... <Token1> <Token2> Music Codebook ... Music Tokens ... <Token1> <Token2> <Token3> <Token4> <Token6> <Token6> ... ... Motion Codebook ... Motion Tokens ... ... ... <Token1> <Token2> <Token3> <Token4> <Token5> <Token6> LLM LLM Motion Tokenizer Music Tokenizer Task Instructions A person performs a knee tuck to kick L. Motion Tokenizer Motion Tokenizer Music Tokenizer A person is dancing Jazz. A person performs a knee tuck to kick L. A person is dancing Jazz. Music Detokenizer Motion Detokenizer Motion Detokenizer Motion Detokenizer Motion Detokenizer Training LLM among Multimodalities and Multitasks Same Data Frozen Fine-tune Figure 2: An overview of the M3GPT framework. M3GPT consists of multimodal tokenizers and a motionaware language model. The training process of M3GPT consists of three stages: multimodal tokenizers training, modality-alignment pre-training, and instruction tuning. 3 Method To enhance the comprehension and generation of motion-relevant modalities, we propose a unified multimodal framework named M3GPT. As illustrated in Fig. 2, M3GPT consists of multimodal tokenizers responsible for compressing raw motion and music data into discrete tokens (Sec. 3.1), and a motion-aware language model that learns to understand and generate motion tokens from LLMs by corresponding text and music (Sec. 3.2). To address motion-relevant tasks, we employ a three-stage training scheme encompassing multimodal tokenizers training, modality-alignment pre-training, and instruction tuning (Sec. 3.3). During the inference process, multimodal tokens are decoded back into their original representations by associated de-tokenizers (decoders of multimodal tokenizers), enabling various motion-relevant tasks to be executed via instructions (Sec. 3.4). 3.1 Multimodal tokenizers As shown in Fig. 2, Multimodal tokenizers aim to discretize continuous human motion and music into language-like tokens, allowing the three modalities to be unified within a single language model. 3D Human Motion Tokenizer. To represent motion in discrete semantic tokens, we build a 3D human motion tokenizer based on Vector Quantized Variational Autoencoders (VQ-VAE) following [12, 62, 21, 60]. The motion tokenizer consists of a motion encoder Em and a motion decoder Dm, along with a codebook Bm = b1, b2, . . . , bNm containing Nm discrete semantic vectors. Notably, to facilitate mutual enhancement between motion and dance data, we employ a shared tokenizer for both motions and dances, projecting them into a consistent and shared semantic space. Formally, given a 3D motion sequence m ∈RTm×dm, where Tm is the time length and dm is the dimensionality of each frame’s pose, the motion encoder Em that consists of several 1-D convolutional layers projects m to a latent embeddings z ∈RLm×d. Here, Lm is the time interval after downsampling and d is the latent dimension. Next, we transform z into a collection of codebook entries through discrete quantization. Specifically, the quantization process replaces each item of z with its nearest embedding in the codebook Bm, obtaining the quantized latent vectors e ∈RLm×d as follows: e = arg min bk∈Bm z −bk 2 . (1) The motion decoder Dm, which consists of several 1-D deconvolutional layers, projects the quantized embeddings back to raw motion space as ˆ m = Dm (e). Following [21, 60], the motion tokenizer can be trained by the reconstruction loss, embedding loss and commitment loss as follows: Lvq = ∥ˆ m −m∥1 + ∥sg [z] −e∥2 2 + β∥z −sg [e] ∥2 2 . (2) where sg [·] is the stop gradient, and β is the factor that adjusts the weight of the commitment loss. 4 After training the motion tokenizer, a motion sequence m can be represented as a sequence of discrete codebook-indices of quantized embedding vector, namely motion tokens qm ∈RLm, as follows: qm = arg min k∈{1,...,Nm} Em (m) −bk 2 . (3) Music Tokenizer. For the music data, we adopt the VQ-VAE in Jukebox [9] as the music tokenizer, which consists of a music encoder Ea, a music decoder Da and a music codebook Ba. Notably, the limited number of music samples in dance datasets makes it inadequate for training an effective music tokenizer. To leverage the strong representation ability of the tokenizer trained on the large-scale musical dataset, we use the pre-trained VQ-VAE from Jukebox [9], which has been trained on a dataset of 1.2 million songs. Specifically, we first segment each input music sample into 5-second music segments. Then, for each 5 seconds segment a ∈RTa×da, we use the pre-trained music tokenizer {Ea, Ba} to encode a into a sequence of discrete codebook-indices qa ∈RLa (namely music tokens) following Eq. 3. 3.2 Language Model Backbone Expanding Vocabulary. To incorporate multimodal discrete representations into a pre-trained LLM, we expand the original text vocabulary Vt in LLM with motion vocabulary Bm and music vocabulary Ba, forming a new unified vocabulary V = {Vt, Bm, Ba}. To accommodate the expanded vocabulary, we extend the corresponding embedding and prediction layer of LLM, where the newly incorporated parameters are initialized randomly. Unified Multimodal Language Model. Equipped with multimodal tokenizers, we can compress multimodal data into discrete token sequences. To be specific, employing the trained motion tokenizer and music tokenizer, the input motion m ∈RTm×dm and music a ∈RTa×da can be mapped into a sequence of discrete motion tokens qm ∈RLm and music tokens qa ∈RLa. Then equipped with a unified vocabulary V , we can formulate various motion-relevant tasks in a general format, where both input and output tokens come from the same vocabulary. These tokens can represent natural language, human motion, music, or any combination, depending on the specific task to be solved. This naturally enables the core LLM to unify motion comprehension and generation tasks in an autoregressive manner. Following [21], we employ T5 [39] as the language model backbone, which is pre-trained on 750 GB of text tokens. By leveraging this pre-trained large language model, we can harness its powerful modeling capabilities and generalizability to develop a more user-friendly, motion-related human-computer interaction model. 3.3 Training Strategy The training process is divided into three stages. The first stage is Multimodal Tokenizers Training, which focuses on learning the motion/music tokenizer to represent motion/music as discrete tokens. The second stage is Modality-Alignment Pre-training, which aims to align motion, music, and text modalities, and facilitate collaboration across different motion-relevant tasks. The third stage is Instruction Fine-Tuning, aimed at enhancing the model’s instruction-following capability. Stage1: Multimodal Tokenizers Training. We first train a motion tokenizer using the objective defined in Eq. 2. As for the music tokenizer, due to the limited music samples in existing dance datasets, we directly use the pre-trained VQ-VAE model from Jukebox [9]. This process allows any motion sequence and music to be represented as a sequence of tokens, enabling seamless integration with text within LLM. To ensure the stability of LLM training, the encoder of motion tokenizer and whole music tokenizer remain unchanged. Notably, we continue to optimize the decoder of motion tokenizer in subsequent training stages to further enhance the quality of generated motions. Stage2: Modality-Alignment Pre-training. To enable LLM to handle discrete modalities, we utilize paired motion corpus to train LLM in a next-token prediction task. This process aims to align the text, music, and motion modalities for unified reasoning in LLM. • Joint optimization of LLM and motion de-tokenizer. Human motion (especially dance) encompasses intricate details. Previous works [21, 60] keep the motion de-tokenizer fixed during training LLM, which hinders LLM’s ability to perceive the distribution and details of motions. Specifically, in the output space of LLM, different motion tokens are treated as independent classes; therefore, the cost of classifying a motion token as semantic-similar token and semantic-distant 5 token is the same. Apparently, relying solely on LLM’s autoregressive loss is insufficient for capturing the details of motion. To address this problem, we jointly optimize LLM and motion de-tokenizer in stage2 and stage3. This strategy enables the reconstruction error signals in raw motion space to backpropagate to LLM, enhancing LLM’s ability to generate the details of motion. With the goal of minimizing L1 loss between the predicted and real motion, we search for the motion’s token sequence that could minimize this L1 loss in original motion space. As the motion de-tokenizer continuously optimizes, the target motion’s token sequence, which supervises LLM training, dynamically changes. This dynamic adjustment reduces L1 loss progressively, achieving joint optimization. • Synergy learning of multitasks. Although aligning text with one additional modality is relatively straightforward, integrating multiple modalities (e.g., motion, text, and music) within a single framework poses a significant challenge. Additionally, as noted in [4], multitask joint training usually achieves inferior performance on each individual task compared to single-task training. This phenomenon is also observed in our text-to-motion task, as shown in Tab. 2. We argue that the large modality difference among different motion-relevant tasks (e.g., music-to-dance and text-to-motion) prevents the model from effectively establishing connections between these tasks. Thus it is difficult for the model to identify a common optimization direction that benefits all tasks. As ‘text’ serves as a highly semantic descriptor for other modalities and is the most familiar and well-modeled modality to LLM, we use ‘text’ as a bridge to align motion, text, and music data, thereby mitigating conflicts in aligning multiple modalities. Initially, we construct paired textual descriptions for music samples in the dance datasets. Specifically, we use the style annotations of the music to create paired texts, such as ‘a person is dancing Jazz’. Then, we construct two auxiliary tasks using the generated pairs of music and text, i.e., music-to-text and text-to-dance. Through these two auxiliary tasks, M3GPT implicitly learns to decompose the complex music-to-dance task into two simpler tasks music-to-text and text-to-dance. Additionally, with a shared tokenizer for motion and dance, text-to-dance and text-to-motion tasks occupy the same matching space, and thus can mutually reinforce each other. In this way, M3GPT builds the synergies between the two primary motion generation tasks, music-to-dance and text-to-motion, facilitating mutual reinforcement, as shown in Tab. 2. Combining the above analysis, we jointly train LLM and motion de-tokenizer using a mixture of motion comprehension and generation tasks, along with two auxiliary music-to-text and text-todance tasks. Besides the auxiliary tasks, we consider 2 basic motion comprehension tasks, i.e., motion-to-text and dance-to-music, and 4 basic motion generation tasks, i.e., text-to-motion, musicto-dance, motion prediction and motion in-between. Formally, for a specific task, we denote the source input consisting of a sequence of tokens as qs = qi s Ls i=1, the target output as qt = qi t Lt i=1, LLM predicts the probability distribution of potential next token at each step pθ qi t\|q<i t , qs in an autoregressive manner. For motion generation tasks, we add a reconstruction loss. Specifically, when the output tokens are motion tokens, we pass them to motion de-tokenizer to generate a motion sequence (denoted as ˆ m), where a reconstruction loss is then employed for guidance. Overall, during this training process, the objective is to maximize the log-likelihood of the data distribution and minimize the reconstruction error within raw motion space: L = Lt−1 X i=0 log pθ qi t\|q<i t , qs + λ ∥ˆ m −m∥1 , (4) where m denotes the ground-truth for ˆm generated by motion de-tokenizer, and λ is a hyperparameter to adjust the weight for reconstruction loss. Stage3: Instruction Fine-Tuning. To enhance the generalization and instruction-following capability of M3GPT, we construct a multimodal instruction dataset with resort to GPT-4, building upon existing motion datasets. Specifically, we define 11 core tasks, each comprising 200/50/50 training/validation/test instruction templates. For example, an instruction for text-to-motion task could be "Create a motion that complements the poetic elements in <Caption_Placeholder>", with <Caption_Placeholder> standing for any text sequence; an instruction for music-to-dance could be "Script a dance that adapts to the tempo shifts in <Audio_Placeholder>", with <Audio_Placeholder> standing for any music sequence. Further details are available in Appendix B.4. 6 3.4 Inference M3GPT During inference, we evaluate M3GPT’s performance across multiple motion-relevant tasks and datasets (Sec. 4.2 and Appendix (C, D). Also, we consider two challenging dance generation tasks to evaluate the zero-shot generalization ability of M3GPT: (1) Generating long-duration dances from long music. Long-duration dance generation involves creating uninterrupted, coherent dance sequences based on a single piece of music. Due to the limitation of computational cost and memory overload, we train M3GPT on the task of 5-second music-to-dance generation. Conversely, during inference, we can combine this training task music-todance to generate an initial 5-second dance segment, and an unseen zero-shot task music+dance-todance that recursively generates subsequent dance segments conditioned on both music and previously generated dance segments, to perform long-duration and coherent dance generation. (2) Generating dance controlled by both music and text. Integrating music and text as control signals in dance generation (music+text-to-dance) augments music-to-dance task with text modality. This process guides the generated dances to synchronize with particular actions described in input texts. Thanks to the tokenizer mechanism, M3GPT can seamlessly combine music and text in LLM’s input, enabling the integration of text instructions to produce a wide variety of dance sequences. 4 Experiments 4.1 Experimental setup Datasets and Preprocessing. We use a large-scale text-to-motion dataset: Motion-X [29], and two music-to-dance datasets: AIST++ [24] and FineDance [25]. Notably, the 3D pose annotations differ among these datasets, therefore, we standardize their processing for uniform usage. Specifically, we select 22 joints common to these datasets and preprocess the motion samples following [13], resulting in motion sequences with identical representation. Further details on datasets and preprocessing are provided in Appendix B.1. Evaluation Metrics. Different tasks employ distinct evaluation metrics. We use the most common evaluation metrics to assess the performance of M3GPT for each task. (1) Text-to-Motion. Following [21, 12], we use Frechet Inception Distance (FID), Diversity (Div), R-Precition that calculates the top-1 motion-to-text retrieval accuracy (R TOP1). (2) Motion-to-Text. Following [21], we use linguistic metrics like BLEU, CIDEr, along with R-Precision for evaluating motion-to-text alignment. (3) Music-to-Dance. Following [26, 52], we use FID, Diversity and Beat Align Score (BAS) on kinetic features [22] (denoted as "k") to evaluate the dance generation quality. Notably, as noted in [48], the geometric features [33] are unreliable as a measure of dance generation quality. So we only use the kinetic features for evaluation. (4) Dance-to-Music. Following [52], we use Beats Coverage Scores (BCS), Beats Hit Scores (BHS), and F1 score to evaluate music generation quality. (5) Motion Prediction and In-Between. Following [21], we use FID and Diversity to measure the consistency between the provided pose conditions and generated motion. More details and results on other evaluation metrics are provided in Appendix B.2 and C. Implementation Details. For motion tokenizer, we set the codebook size to 512. As for music tokenizer, we use the pre-trained VQ-VAE from Jukebox with a codebook size of 2048. In term of temporal downsampling rate, the motion encoder uses a rate of 4, while the music encoder uses a rate of 128. We utilize T5 base [39] as our language model backbone. For training the motion tokenizer, we use Adam as the optimizer with a batch size of 1000 and an initial learning rate of 10−4. To train the language model backbone, we employ the Adafactor_dev optimizer and use CosineAnnealingLR as the learning rate scheduler. The learning rate is set to 2 × 10−4 for pre-training stage, and 10−4 for instruction fine-tuning stage. For hyperparameter settings, λ in Eq. 4 is set to 0.2, and β in Eq. 2 is set to 0.02. All our experiments are conducted on 8 NVIDIA A40 GPUs. To evaluate the model’s performance across different platforms, we also test our trained M3GPT with T5-base on Ascend 910B NPUs. Further details on implementation and hyperparameter analysis are provided in Appendix E. 7 Table 2: Evaluation of synergy learning and joint optimization of LLM and motion de-tokenizer on Text-toMotion (Motion-X [29]) and Music-to-Dance (AIST++ [24]). T2M: Text-to-Motion. A2D: Music-to-Dance. T2D: Text-to-Dance. A2T: Music-to-Text. Trained single task refers to a model trained and tested on a single task. Pre-trained and Instruction-tuned indicate the model after pre-training (stage2) and instruction tuning (stage3), followed by direct testing on each task. The arrows (↑) indicate that higher values are better. The arrows (↓) indicate that smaller values are better. Bold indicates the best result. Methods Re-Optimizing motion de-tokenizer Text-to-Motion Music-to-Dance R TOP1 ↑ FID ↓ Div↑ FIDk ↓ Divk ↑ BAS ↑ Ground Truth 0.675 0.009 2.316 17.10 8.19 0.2374 Trained single task 0.645 0.081 2.124 83.33 5.18 0.1835 Trained single task ! 0.656 0.078 2.133 75.47 5.57 0.1884 T2M+A2D 0.564 0.094 2.080 51.26 6.73 0.2037 T2M+A2D ! 0.578 0.092 2.106 47.71 7.47 0.1958 T2M+A2D+T2D+A2T 0.617 0.093 2.110 42.70 7.54 0.2084 T2M+A2D+T2D+A2T ! 0.626 0.088 2.197 25.24 7.63 0.2217 M3GPT (Pretrained without T2D and A2T) 0.526 0.105 2.058 40.71 7.47 0.2030 M3GPT (Pretrained without T2D and A2T) ! 0.547 0.104 2.099 37.14 7.61 0.2005 M3GPT (Pre-trained) 0.598 0.089 2.218 32.71 7.43 0.2090 M3GPT (Pre-trained) ! 0.601 0.092 2.251 27.65 7.52 0.2105 M3GPT (Instruction-tuned) 0.606 0.091 2.251 28.46 7.49 0.2052 M3GPT (Instruction-tuned) ! 0.615 0.093 2.253 24.34 7.50 0.2056 4.2 Ablation Studies In this section, we conduct ablation studies to validate the effectiveness of our method. We use the same model architecture throughout the experiments. The ablation results are shown in Tab. 2. Effectiveness of joint optimization of LLM and motion de-tokenizer. Different from previous works [21, 60] that fix motion de-tokenizer during training LLM, we jointly optimize LLM and motion de-tokenizer in stage2 and stage3, as detailed in Sec. 3.3. As shown in Tab. 2, the joint optimization consistently brings performance gains across various metrics and most settings. Specifically, it largely enhances the fidelity of generated dances, reflected in a notable decrease in FIDk score. We also notice a minor increase (less than 0.003) in FID of text-to-motion task in M3GPT. The possible reason is that the motion patterns controlled by text are relatively simple, making LLM optimized solely in discrete semantic space adequate for text-to-motion. Conversely, dances involve greater complexity, necessitating the joint optimization of motion decoder to accurately capture intricate dance movements without compromising information. Effectiveness of synergy learning of multitasks. During the training of M3GPT, we introduce a synergy multitask learning strategy by constructing two auxiliary tasks: Text-to-Dance (T2D) and Music-to-Text (A2T), as detailed in Sec. 3.3. As shown in Tab. 2, the inclusion of T2D and A2T consistently brings performance gains across various metrics on both text-to-motion and music-to-dance tasks. Specifically, for music-to-dance, the FIDk score is decreased by nearly 10 points, indicating that the synergy learning helps generate more realistic dances. We argue that by incorporating these two auxilary tasks, M3GPT implicitly learns to decompose the complex music-to-dance into two simpler tasks, music-to-text and text-to-dance. This way, the text-to-motion task can assist in learning the music-to-dance task, thereby enhancing its performance. 4.3 Comparisons with State-of-the-arts In this section, we compare our M3GPT with state-of-the-arts on multiple core motion-relevant tasks. We respectively report the comparison results on text-to-motion dataset, Motion-X [29], and music-to-dance datasets, AIST++ [24] and FineDance [25]. More quantitative and qualitative results are provided in Appendix C and D. Also, in the supplementary material’s zip file, we provide the render videos of generated motions/dances and generated music files by our M3GPT. Main results on text-to-motion dataset. On the text-to-motion dataset, Motion-X, we evaluate M3GPT on 4 tasks, i.e., text-to-motion, motion-to-text, motion prediction, and motion in-between. The comparison results are shown in Tab. 3. As shown, M3GPT achieves competitive performance across all evaluated tasks, highlighting its capability to address diverse motion tasks in a single model. Also, for text-to-motion task, M3GPT (instruction-tuned only T2M), which combines multitask pretraining and instruction fine-tuning solely on T2M task, yields better performance than Trained single 8 Table 3: Comparison results on Motion-X [29] dataset. The evaluation metrics are computed using the encoder introduced in Appendix A. Empty columns of previous methods indicate that they can not handle the task. Instruction-tuned only T2M indicates the model that is initially pre-trained on multiple tasks, followed by instruction tuning solely on text-to-motion task. Methods Text-to-Motion Motion-to-Text Motion Prediction Motion In-between R TOP1↑ FID↓ Div↑ R TOP3↑ Bleu@4↑ CIDEr↑ FID↓ Div↑ FID↓ Div↑ Real 0.675±0.003 0.009±0.000 2.316±0.011 0.881 0.009 2.316 0.009 2.316 MLD [5] 0.612±0.003 0.122±0.008 2.267±0.018 T2M-GPT [55] 0.647±0.002 0.101±0.005 2.270±0.033 0.814 1.755 MotionDiffuse [57] 0.659±0.002 0.075±0.004 2.220±0.022 TM2T [15] 0.581±0.002 0.148±0.003 2.005±0.034 0.806 12.13 20.16 MDM [44] 0.472±0.008 0.078±0.000 2.133±0.012 1.028 1.746 0.831 1.768 MoMask [16] 0.668±0.003 0.074±0.004 2.241±0.016 0.626 1.884 MotionLCM [8] 0.658±0.005 0.078±0.003 2.206±0.026 MotionGPT[21] 0.659±0.003 0.078±0.001 2.166±0.026 0.840 11.21 31.36 0.701 1.818 0.648 1.875 Trained single task 0.656±0.002 0.078±0.002 2.133±0.012 0.767 10.14 22.92 0.774 1.778 0.692 1.810 M3GPT (Pre-trained) 0.601±0.002 0.092±0.002 2.251±0.012 0.834 11.00 24.12 0.707 1.874 0.604 1.879 M3GPT (Instruction-tuned) 0.615±0.003 0.093±0.002 2.253±0.026 0.845 11.50 42.93 0.682 1.838 0.612 1.900 M3GPT (Instruction-tuned only T2M) 0.661±0.003 0.076±0.002 2.273±0.026 Table 4: Comparison results on Motion-X [29] dataset based on Ascend 910B NPUs. Methods Text-to-Motion Motion-to-Text Motion Prediction Motion In-between R TOP1↑ FID↓ Div↑ R TOP3↑ Bleu@4↑ CIDEr↑ FID↓ Div↑ FID↓ Div↑ Trained single task 0.654±0.002 0.081±0.003 2.304±0.017 0.763 10.16 22.89 0.776 1.818 0.712 1.880 M3GPT (Pre-trained) 0.596±0.002 0.096±0.003 2.241±0.018 0.831 11.05 24.03 0.710 1.882 0.608 1.874 M3GPT (Instruction-tuned) 0.612±0.002 0.094±0.002 2.276±0.021 0.846 11.52 42.64 0.684 1.841 0.621 1.903 M3GPT (Instruction-tuned only T2M) 0.659±0.003 0.078±0.003 2.314±0.023 Table 5: Comparison results on AIST++ [24] and FineDance [25] datasets. Methods Music-to-Dance on AIST++ Music-to-Dance on FineDance Dance-to-Music on AIST++ FIDk ↓ Divk ↑ BAS ↑ FIDk ↓ Divk ↑ BAS ↑ BCS↑ BHS↑ F1↑ Real 17.10 10.60 0.2374 0.2120 FACT [24] 35.35 5.94 0.2209 113.38 3.36 0.1831 Bailando [42] 28.16 7.83 0.2332 82.81 7.74 0.2029 EDGE [48] 42.16 3.96 0.2334 94.34 8.13 0.2116 Lodge [26] 37.09 5.58 0.2423 45.56 6.75 0.2397 Foley [11] 96.4 41.0 57.5 CMT [10] 97.1 46.2 62.6 D2MGAN [64] 95.6 88.7 93.1 CDCD [65] 96.5 89.3 92.7 LORIS [52] 98.6 90.8 94.5 Trained single task 75.47 5.57 0.1884 128.37 6.48 0.2036 93.9 93.6 92.8 M3GPT (Pre-trained) 27.65 7.52 0.2105 92.35 7.67 0.2134 93.4 93.8 94.2 M3GPT (Instruction-tuned) 24.34 7.50 0.2056 86.47 7.75 0.2158 93.6 94.0 94.9 M3GPT (Instruction-tuned for single task) 23.01 7.85 0.2261 42.66 8.24 0.2231 94.3 94.0 95.0 Table 6: Comparison results on AIST++ [24] and FineDance [25] datasets based on Ascend 910B NPUs. Methods Music-to-Dance on AIST++ Music-to-Dance on FineDance Dance-to-Music on AIST++ FIDk ↓ Divk ↑ BAS ↑ FIDk ↓ Divk ↑ BAS ↑ BCS↑ BHS↑ F1↑ Trained single task 77.32 5.61 0.1860 134.66 6.52 0.2088 93.8 93.6 92.7 M3GPT (Pre-trained) 27.99 7.61 0.2102 91.39 7.71 0.2128 93.4 93.6 94.2 M3GPT (Instruction-tuned) 25.05 7.48 0.2072 88.25 7.76 0.2160 93.5 93.8 94.7 M3GPT (Instruction-tuned for single task) 23.68 7.83 0.2264 43.78 8.39 0.2225 94.3 94.0 94.9 task that only trains the model on T2M task. Tab. 4 presents the results of testing on the Ascend 910B NPUs, where M3GPT also achieves comparably good performance. These results demonstrate that multitask pre-training can enhance the performance of individual tasks across different computation platforms. Further results of M3GPT trained on NPUs will be presented later. Main results on music-to-dance datasets. On the music-to-dance datasets, AIST++ and FineDance, we evaluate M3GPT on 2 tasks, i.e., music-to-dance and dance-to-music. As shown in Tab. 5, in general, the performance of multitask pre-training and instruction fine-tuning in M3GPT outperforms single-task training, underscoring the effectiveness of multitask training for dance-relevant tasks. Also, M3GPT achieves competitive performance on most metrics. For music-to-dance, the best-performing 9 + “A person does a cartwheel.” <Dance> Generate <Music-to-Dance, 5s> + <Music+Dance-to-Dance> <Long Dance> Generate Frame148 Frame147 Frame146 Frame145 Frame144 Frame143 Frame149 Frame150 Frame151 Frame152 Frame153 Frame154 Frame155 Frame156 Frame157 Frame158 （a）（b） Figure 3: Qualitative results for long-term dance and music-text conditioned dance generation of M3GPT. method on the FineDance dataset is Lodge [26]. This approach features a specialized two-stage architecture for generating long-duration dance sequences, progressively refining movements from coarse to fine granularity using a diffusion model. On AIST++ dataset, M3GPT reports the best FIDk of 24.34 for music-to-dance task, the best BHS and F1 of 94.0 and 94.9 for dance-to-music task. The results in Tab. 6, tested on the Ascend 910B NPUs, also demonstrate that multitask training can enhance the performance of both music-to-dance and dance-to-music tasks. 4.4 Evaluation on Zero-Shot Tasks In this section, we explore M3GPT’s capabilities in handling zero-shot tasks, as mentioned in Sec. 3.4. Fig. 3 (a) shows the long-term dance generation. As seen, M3GPT can generate a coherent dance sequence by seamlessly integrating the music-to-dance and zero-shot music+dance-to-dance tasks. Fig. 3 (b) shows the generated 3D dance with both music and text instruction. We can see that M3GPT maintains plausible visual results in accordance with input text instructions (cartweel), underscoring its remarkable zero-shot generalization capability. 5 Conclusion In this paper, we present M3GPT, a unified framework for comprehending and generating motion aligned with both text and music modalities. By employing text as a bridge, we build connections and synergies between different motion-relevant tasks, facilitating mutual reinforcement. Additionally, the joint optimization of LLM and motion de-tokenizer further enriches the details of generated motion, enhancing overall motion generation quality. Our extensive evaluations across various motion-relevant tasks demonstrate the effectiveness of M3GPT in both motion comprehension and generation. Besides, M3GPT exhibits strong zero-shot generalization abilities, enabling it to handle previously unseen and challenging motion-relevant tasks. Limitations and Broader Impacts. Although our M3GPT has successfully processed signals from motion, text, and music modalities for motion comprehension and generation, it focuses on modeling human body movements, excluding hands and faces details. Future research can extend the scope of M3GPT to include hands and facial modeling. Acknowledgments. This work is sponsored by the National Natural Science Foundation of China (NSFC) under Grant 62306301, the National Postdoctoral Program for Innovative Talents under Grant BX20220310, and the CAAI-CANN Open Fund developed on the OpenI Community. It is also supported by the project of Peng Cheng Laboratory (PCL2023A08). References [1] Judith Butepage, Michael J Black, Danica Kragic, and Hedvig Kjellstrom. Deep representation learning for human motion prediction and classification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 6158–6166, 2017. 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As depicted in Fig. 4, the training of the text-motion alignment model is split into two phases: pre-training the motion auto-encoder and text-motion contrastive learning. During the pre-training phase, we use motion data from the text-to-motion, and dance data from the music-to-dance tasks. This stage employs a reconstruction loss to ensure the model achieves a robust initial state capable of extracting an expressive motion representation. In the text-motion contrastive learning phase, we utilize text-motion pair data from the text-to-motion task. To train a more robust text-motion alignment model, we employ text-motion training dataset data along with motion data reconstructed by Motion VQ-VAE for text-motion contrastive training. We incorporate an adapter MLP layer into both the motion and text encoders to align the dimensions of zm and zt at 512. This setup facilitates the alignment of text and motion in the representational space. The motion reconstruction loss Lrecon_motion for the pre-training stage and the contrastive learning loss LinfoNCE for the aligning stage are used to optimize this model, as follows, Lrecon_motion = ∥x −˜x∥2 (5) LinfoNCE = −1 N K X i=1 log exp(⟨z′ i, zt i⟩/τ) PK j=1 exp(⟨z′ i, zt j⟩/τ) ! (6) Em A person performs a knee tuck to kick L. Em Et Dm zm Input Motion Generated Motion Lrecon_motion MLP MLP LinfoNCE Frozen Train Stage 1: Pre-train Motion Auto-Encoder Stage 2: Text-Motion Contrastive Learning zm zt Figure 4: Pipeline of Text-Motion Alignment Model. The training of the text-motion alignment model includes two stages: pre-training motion auto-encoder and text-motion contrastive learning. B Details for Training and Evaluating B.1 Data Introductions and Preprocessing We leverage the largest available text-to-motion dataset, Motion-X [29], along with widely-used music-todance datasets, AIST++ [24] and FineDance [25], for our multitask training regimen. Motion-X is used for text-to-motion, motion-to-text, motion prediction, and motion in-between tasks, while AIST++ and FineDance datasets support both music-to-dance and dance-to-music tasks, and are also adapted for motion prediction and in-between tasks to enhance our training resources. Motion-X includes 15.6 million precise 3D whole-body SMPL-X [36] pose annotations across 81.1K motion sequences with sequence-level semantic text descriptions. AIST++ contains 1,409 dance motion sequences across 10 dance genres with SMPL [30] pose annotations, and FineDance provides 7.7 hours of dance, totaling 831,600 frames with SMPL-H [41] poses at 30 fps across 16 dance genres. Tab. 7 shows the training datasets and their corresponding sample numbers that we use to train our model. We standardize data across these datasets by selecting the 22 common joints and normalizing each motion sequence to face the same direction and to run at 30 fps. We use a processing technique consistent with prior research [14, 5, 21] that integrates joint velocities, positions, and rotations for consistent motion representation, facilitating effective utilization across tasks. For the music data, we use the Librosa toolkit [32] to load raw .wav data at a sampling rate of 22050 Hz, processed into features by the Jukebox encoder [9]. To optimize the use of these datasets, we strategically employ data from specific datasets for different tasks. During training for music and dance tasks, we randomly select 5-second segments from complete music tracks and corresponding dance 15 segments as training samples, setting the sample length to 6.25 seconds for motion prediction and in-between tasks with AIST++ and FineDance. When assessing music-to-dance on the FineDance dataset, we don’t evaluate all 5-second samples directly. Instead, we generate continuous long dance sequences using music-to-dance and motion prediction, then segment these into 30-second samples for evaluation to align with Lodge’s testing methodology. Table 7: The training datasets and sample numbers for different tasks. Tasks T2M M2T Motion Prediction/In-between A2D D2A A2T T2D Training dataset Motion-X Motion-X Motion-X/AIST++/FineDance AIST++/FineDance AIST++/FineDance AIST++/FineDance AIST++/FineDance Training samples number 64867 64867 64867/952/177 952/177 952/177 952/177 952/177 B.2 Comprehensive Evaluation Metrics Different tasks utilize specific evaluation metrics. We use the consistent evaluation metrics following prior research [14, 21, 12, 26, 64, 52]. • Text-to-Motion. We measure the discrepancy between generated and actual motion features using Frechet Inception Distance (FID), assess diversity (Div) among all generated motion sequences, and evaluate motion-text semantic correlation with R-precision. Multimodal Distance (MM Dist) quantifies the disparity between motions and texts. A specialized model developed for evaluating the text-to-motion task on the Motion-X dataset with 22 joints is detailed in Appendix A. • Motion-to-Text. We use linguistic metrics including BLEU [35], ROUGE [28], CIDEr [49], and BertScore [59], along with R-Precision and MM Dist to assess alignment between generated text and motion. • Music-to-Dance. We employ the evaluation framework recommended by FACT [24] and Bailando [42], utilizing FID, Diversity, and Beat Align Score (BAS) to gauge dance quality. In our paper, we use kinetic features to compute FID and Diversity. • Dance-to-Music. We use metrics from [52, 64] such as Beats Coverage Scores (BCS), Beats Hit Scores (BHS), F1 scores, Beats Coverage Std (CSD), and Beats Hit Std (HSD) to evaluate music generation quality. • Motion Prediction and Motion In-between. We use Average Displacement Error (ADE) and Final Displacement Error (FDE) to assess the quality of predicted motion. The text-motion alignment model aids in evaluating motion prediction performance. B.3 Distributed Training for Multitasks We employ a single-node multi-GPU distributed training strategy for M3GPT, distributing each task across separate GPUs to facilitate multitask training through shared model parameters. This method allows us to tailor the maximum token length of the LLM for each task, based on the longest sample sequence typical for that task. Specifically, we set the maximum LLM token length to 192 for tasks such as text-to-motion, motion-to-text, motion prediction, and motion in-between. For tasks involving the music modality, such as music-to-dance and dance-to-music, the maximum length is set at 980. This task-specific configuration enables us to optimize batch sizes effectively, thus maximizing GPU utilization. In our experimental setup, the batch size is set to 40 for the text-to-motion, text-to-dance, and motion-to-text tasks, and 4 for the music-to-dance, dance-to-music, and music-to-text tasks. For motion prediction and motion in-between tasks, the batch size is set to 45. We also establish the number of iterations for pre-training at 1,000,000, instruction fine-tuning at 200,000. This structured approach ensures that each task is optimally processed, enhancing the efficiency and effectiveness of our training regimen. B.4 Tasks for Pre-training and Instruction tuning As shown in Fig. 5, we define 11 core motion-related tasks for the instruction tuning of M3GPT, including text-to-motion, random-to-motion, motion-to-text, random-to-text, motion prediction, motion in-between, musicto-dance, dance-to-music, random-to-music, text-to-dance, and music-to-text. The tasks of text-to-dance and music-to-text were specifically constructed based on the music-to-dance datasets. Random-to-X represents the unconstrained generation of motion, text and music. In Tab. 8, we present a selection of command templates for each task, where <Motion_Placeholder>, <Caption_Placeholder>, and <Music_Placeholder> respectively represent the motion sequence (including dance sequence), textual description, and music segment from the training data. C Quantitative Results and Comparisons with SOTA Methods In this section, we compare the performance of our M3GPT with existing SOTA methods on a broader set of metrics for each task across three datasets: Motion-X [29], AIST++ [24], and FineDance [25]. Tab. 9 shows 16 M3GPT Text-to-Motion Motion-to-Text Music-to-Dance Dance-to-Music Motion-to-Motion Text-to-Dance Music-to-Text text-to-motion random-to-motion motion-to-text random-to-text dance-to-music random-to-music motion prediction motion in-between Figure 5: Tasks for M3GPT pre-training and instruction tuning. Random represents the unconstrained generation of motion/text/music in the corresponding task. Table 8: Examples of instruction templates for each task when instruction tuning M3GPT. Task Instruction template Output text-to-motion Design a motion that illustrates the emotions conveyed by <Caption_Placeholder>. <Motion_Placeholder> How could you express the resilience mentioned in <Caption_Placeholder> through motion? Can you develop a motion that captures the existential debate in <Caption_Placeholder>? random-to-motion Can you generate a motion randomly? <Motion_Placeholder> Please generate a random motion. Display a motion for me. motion-to-text What themes are explored through the motion in <Motion_Placeholder>? <Caption_Placeholder> Can you describe the motion <Motion_Placeholder> with texts? How would you interpret the actions depicted in <Motion_Placeholder>? random-to-text Can you generate a text description for motion randomly? <Caption_Placeholder> Give me a caption which describes a action. How can we describe a motion with texts? motion prediction What movements would suitably follow the thematic climax of <Motion_Placeholder>? <Motion_Placeholder> What steps might deepen the emotional expression seen in <Motion_Placeholder>? What movements could follow to resolve the suspense built in <Motion_Placeholder>? motion in-between What new character dynamics could the middle section of <Motion_Placeholder> explore? <Motion_Placeholder> Infer the type of dramatic climax that the masked section of <Motion_Placeholder> might contain. What potential themes of ascent or descent could be explored in the middle of <Motion_Placeholder>? music-to-dance Script a dance that adapts to the tempo shifts in <Music_Placeholder>. <Motion_Placeholder> Create a dance that would visually mimic the lyrical journey in <Music_Placeholder>. Compose a dance that explores the genre characteristics of <Music_Placeholder>. dance-to-music Can you design a music for this dance <Motion_Placeholder>? <Music_Placeholder> Please create a music based on this dance <Motion_Placeholder>. Arrange a symphony that captures the shifts in <Motion_Placeholder>. random-to-music Please generate a music for a dance randomly. <Music_Placeholder> Can you generate a music with dance style? Creat a music for any style dance. text-to-dance Please generate a dance based on the caption <Caption_Placeholder>. <Motion_Placeholder> Create a dance for the text <Caption_Placeholder>. Generate a dance that corresponds to the textual description <Caption_Placeholder>. music-to-text Describe the dance movements that correspond to the given music <Music_Placeholder>. <Caption_Placeholder> Describe the dance actions that match the provided music <Music_Placeholder>. Detail the dance steps associated with the specified music <Music_Placeholder>. the comparison of text-to-motion on Motion-X dataset. Tab. 10 shows the comparison of motion-to-text on Motion-X dataset. Tab. 11 shows the comparison of music-to-dance on AIST++ and FineDance datasets. Tab. 12 shows the comparison of dance-to-music on AIST++ and FineDance datasets. Tab. 13 and Tab. 14 shows the comparison of motion prediction and motion in-between on Motion-X dataset. Tab. 15 shows the comparison of music-to-text, text-to-dance and text-to-music on AIST++ dataset. Tab. 16 and Tab. 17 show the comparison of motion-related tasks among different size of T5. As shown in these tables, M3GPT can achieve competitive performance with SOTAs across all evaluated tasks. D Qualitative Results and Comparison with SOTA Methods Fig. 6 presents visualizations for a variety of tasks, including text-to-motion, motion-to-text, motion prediction, motion in-between, music-to-dance, long-term dance generation, and music-text conditioned dance generation. The visualization results show that our method can generate realistic results across various motion-relevant tasks. Fig. 7 presents the qualitative results between different methods for text-to-motion and music-to-dance. E Additional Experiments The performance of text-to-motion based on different λ. In Tab. 18, the performance of M3GPT on Motion-X dataset is analyzed across different values of λ for the text-to-motion task. The results indicate that as λ increases, the model’s recall precision (Top1, Top2, Top3) initially improves but subsequently declines. The optimal performance is achieved at λ = 0.2, as evidenced by the highest scores in RPrecision and modality metrics. 17 Further increases in λ to 0.3 and 0.4 lead to deteriorating performance, particularly in FID and RPrecision, suggesting that excessive λ values may result in over-regularization or reduced adaptability of the model. Table 9: Comparison of Text-to-Motion on Motion-X dataset [29]. The arrows (↑) indicate that higher values are better. The arrows (↓) indicate that smaller values are better. Bold and underline indicate the best and the second best result. Methods RPrecision ↑ FID ↓ MM Dist ↓ Div ↑ MModality ↑ Top1 Top2 Top3 Real 0.675±0.003 0.821±0.003 0.878±0.002 0.009±0.000 2.938±0.007 2.316±0.011 MDM [44] 0.472±0.008 0.616±0.005 0.704±0.003 0.161±0.006 5.404±0.031 2.234±0.015 2.241±0.043 MLD [5] 0.612±0.003 0.743±0.002 0.808±0.004 0.122±0.008 3.117±0.035 2.267±0.018 2.210±0.055 T2M-GPT [55] 0.647±0.002 0.785±0.004 0.845±0.003 0.101±0.005 3.046±0.028 2.270±0.033 2.226±0.036 MotionDiffuse [55] 0.659±0.002 0.802±0.004 0.865±0.002 0.075±0.004 2.944±0.004 2.220±0.022 2.102±0.036 Trained single task 0.656±0.002 0.795±0.001 0.843±0.001 0.078±0.000 2.942±0.001 2.133±0.012 2.046±0.052 M3GPT (Pre-trained) 0.601±0.002 0.751±0.003 0.803±0.002 0.092±0.002 2.945±0.001 2.251±0.012 2.188±0.074 M3GPT (Fine-tuned) 0.615±0.003 0.757±0.004 0.815±0.003 0.093±0.002 2.944±0.002 2.253±0.023 2.204±0.058 M3GPT(Fine-tuned only T2M) 0.661±0.003 0.804±0.004 0.861±0.003 0.076±0.002 2.940±0.002 2.273±0.026 2.131±0.032 Table 10: Comparison of Motion-to-Text on Motion-X [29]. Methods RPrecision ↑ MM Dist ↓ Bleu@1 ↑ Bleu@4 ↑ Rouge ↑ CIDEr ↑ BertScore ↑ Top1 Top2 Top3 Real 0.681 0.824 0.881 2.897 TM2T [15] 0.574 0.726 0.806 2.988 30.54 12.13 32.52 20.16 25.37 Trained single task 0.565 0.706 0.767 3.011 31.07 10.14 31.65 22.92 28.19 M3GPT (Pre-trained) 0.627 0.773 0.834 2.946 33.31 11.00 34.10 24.12 30.96 M3GPT (Fine-tuned) 0.631 0.783 0.845 2.950 34.27 11.50 34.55 42.93 31.49 Table 11: Comparison of Music-to-Dance on AIST++ [24] and FineDance [25]. Methods Music-to-Dance on AIST++ Music-to-Dance on FineDance FIDk ↓ Divk ↑ BAS↑ FIDk ↓ Divk ↑ BAS↑ Real 17.10 8.19 0.2374 9.73 0.2120 FACT [24] 35.35 5.94 0.2209 113.38 3.36 0.1831 Bailando [42] 28.16 7.83 0.2332 82.81 7.74 0.2029 EDGE [48] 42.16 3.96 0.2334 94.34 8.13 0.2116 Lodge [26] 37.09 5.58 0.2423 45.56 6.75 0.2397 Trained single task 75.47 5.57 0.1884 128.37 6.48 0.2036 M3GPT (Pre-trained) 27.65 7.52 0.2105 92.35 7.67 0.2134 M3GPT (Fine-tuned) 24.34 7.50 0.2056 86.47 7.75 0.2158 Table 12: Comparison of Dance-to-Music on AIST++ [24] and FineDance [25]. Methods Dance-to-Music on AIST++ Dance-to-Music on FineDance BCS ↑ CSD ↓ BHS ↑ HSD ↓ F1 ↑ BCS ↑ CSD ↓ BHS ↑ HSD ↓ F1 ↑ Foley [11] 96.4 6.9 41.0 15.0 57.5 CMT [10] 97.1 6.4 46.2 18.6 62.6 D2MGAN [64] 95.6 9.4 88.7 19.0 93.1 CDCD [65] 96.5 9.1 89.3 18.1 92.7 LORIS [52] 98.6 6.1 90.8 13.9 94.5 Trained single task 93.9 9.2 93.6 12.8 92.8 84.84 21.61 51.35 27.13 63.97 M3GPT (Pre-trained) 93.4 10.9 93.8 11.5 94.2 83.16 19.95 73.65 23.90 78.12 M3GPT (Fine-tuned) 93.6 10.1 94.0 10.6 94.9 84.10 18.36 74.66 23.45 78.23 18 Table 13: Comparison of Motion Prediction and Motion In-between on Motion-X [29]. Methods Motion Prediction Motion In-between FID ↓ Diversity ↑ ADE ↓ FDE ↓ FID ↓ Diversity ↑ ADE ↓ Ground Truth 0.009 2.316 0.009 2.316 MDM [44] 1.028 1.746 8.057 11.266 0.831 1.768 6.542 Trained single task 0.774 1.778 7.840 9.575 0.692 1.810 6.690 M3GPT-pretrain 0.707 1.874 6.954 8.684 0.604 1.897 5.692 M3GPT-finetune 0.682 1.838 6.898 8.091 0.612 1.900 5.649 Table 14: Comparison of MPJPE on Motion Prediction and Motion In-between on Motion-X [29]. Methods on Motion-X Motion Prediction (MPJPE ↓) Motion In-between (MPJPE ↓) T2M-GPT [55] 80.2 63.7 MoMask [16] 67.9 55.2 MotionGPT [21] 71.3 59.9 M3GPT (Instruction-tuned) 54.2 51.0 Table 15: Comparison of Music-to-Text (A2T), Text-to-Dance (T2D) and Text-to-Music (T2A) on AIST++. Methods on AIST++ Music-to-Text Text-to-Dance Text-to-Music Bleu@4 ↑ CIDEr ↑ R-TOP1 ↑ FID ↓ BCS ↑ BHS ↑ M3GPT (Single task training for A2T) 9.24 24.55 M3GPT (Single task training for T2D) 0.541 0.095 MusicLDM [2] 74.5 73.8 Mubert 73.3 73.0 M3GPT (Instruction-tuned) 11.95 28.88 0.588 0.077 74.5 74.7 Table 16: Comparison of Text-to-Motion and Motion-to-Text with different size of T5. Methods on Motion-X LLM Training time Text-to-Motion Motion-to-Text R-TOP1 ↑ FID ↓ Div ↑ R-TOP3 ↑ Bleu4 ↑ CIDEr ↑ M3GPT T5-small (60M) 5 days 0.598 0.096 2.202 0.822 10.43 38.22 M3GPT T5-base (220M) 7 days 0.615 0.093 2.253 0.845 11.50 42.93 M3GPT T5-large (770M) 8 days 0.619 0.090 2.256 0.848 11.64 43.05 Table 17: Comparison of Music-to-Dance and Dance-to-Music with different size of T5. Methods on AIST++ LLM Training time Music-to-Dance Dance-to-Music FIDk ↓ DIVk ↑ BAS ↑ BCS ↑ BHS ↑ M3GPT T5-small (60M) 5 days 28.05 5.96 0.2091 89.1 91.2 M3GPT T5-base (220M) 7 days 24.34 7.50 0.2056 93.6 94.0 M3GPT T5-large (770M) 8 days 23.26 7.54 0.2061 93.8 94.1 Table 18: Hyper-parameter analysis of λ. Comparison of Text-to-Motion on Motion-X [29] with different values of λ. For this ablation study, M3GPT is trained solely on the text-to-motion task to examine the impact of λ. This study is conducted during the pre-training stage. Methods RPrecision ↑ FID ↓ MM Dist ↓ Diversity → MModality ↑ Top1 Top2 Top3 Real 0.675±0.003 0.821±0.003 0.878±0.002 0.009±0.000 2.938±0.007 2.316±0.011 λ=0.0 0.645±0.002 0.778±0.003 0.826±0.002 0.081±0.002 2.944±0.002 2.124±0.011 2.025±0.021 λ=0.1 0.649±0.002 0.787±0.002 0.838±0.002 0.078±0.002 2.944±0.003 2.128±0.022 2.039±0.049 λ=0.2 0.656±0.002 0.795±0.001 0.843±0.001 0.078±0.000 2.942±0.001 2.133±0.012 2.046±0.052 λ=0.3 0.629±0.002 0.716±0.002 0.784±0.001 0.095±0.002 2.941±0.001 2.113±0.037 2.036±0.041 λ=0.4 0.573±0.001 0.725±0.002 0.793±0.002 0.102±0.002 2.942±0.002 2.090±0.007 2.030±0.086 19 A woman is doing the Box Swing To High Knees exercise. The guy is doing the Arm Leg Lift To Crunch L exercise. A man raises his left hand to touch his face Text-to-Motion Task A person performs a Wide To Narrow Squat. A man walks forward and moves something with his right hand. A person is performing a Hip Bounce Wrist Circle movement. Motion-to-Text Task Motion prediction based on given 20% frames Motion prediction based on given 20% frames Motion-to-Motion Tasks Motion in-between based on unmasked 50% frames Music-to-Dance Task + “A person does a cartwheel.” <Dance> Generate Music-Text Conditioned Dance Generation Task <Music-to-Dance, 5s> + <Motion Prediction, 6.25s> <Long Dance> Generate Frame148 Frame147 Frame146 Frame145 Frame144 Frame143 Frame149 Frame150 Frame151 Frame152 Frame153 Frame154 Frame155 Frame156 Frame157 Frame158 Frame148 Frame147 Frame146 Frame145 Frame144 Frame143 Frame149 Frame150 Frame151 Frame152 Frame153 Frame154 Frame155 Frame156 Frame157 Frame158 <Music-to-Dance, 5s> + <Motion Prediction, 6.25s> <Long Dance> Generate Long-Term Danc Generation Task + “A person turns a circle on the spot” Generate <Dance> Figure 6: The qualitative results for different motion comprehension and generation tasks. 20 “a person appears to have severe arm pain holding and slouching his left shoulder.” “a person walks forward and picks things up and puts them down with their hands.” “a person starts slowly walking forward and then jogs some before coming to a stop.” Real MDM MoMask M3GPT Bailando M3GPT (Break style，5s) (a) Qualitative comparison on text-to-motion task. 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4,471	Utilizing Human Behavior Modeling to Manipulate Explanations in AI-Assisted Decision Making: The Good, the Bad, and the Scary Zhuoyan Li Department of Computer Science Purdue University West Lafayette, IN, 47906 li4178@purdue.edu Ming Yin Department of Computer Science Purdue University West Lafayette, IN, 47906 mingyin@purdue.edu Abstract Recent advances in AI models have increased the integration of AI-based decision aids into the human decision making process. To fully unlock the potential of AIassisted decision making, researchers have computationally modeled how humans incorporate AI recommendations into their final decisions, and utilized these models to improve human-AI team performance. Meanwhile, due to the “black-box” nature of AI models, providing AI explanations to human decision makers to help them rely on AI recommendations more appropriately has become a common practice. In this paper, we explore whether we can quantitatively model how humans integrate both AI recommendations and explanations into their decision process, and whether this quantitative understanding of human behavior from the learned model can be utilized to manipulate AI explanations, thereby nudging individuals towards making targeted decisions. Our extensive human experiments across various tasks demonstrate that human behavior can be easily influenced by these manipulated explanations towards targeted outcomes, regardless of the intent being adversarial or benign. Furthermore, individuals often fail to detect any anomalies in these explanations, despite their decisions being affected by them. 1 Introduction Recent advances in AI models have significantly increased the integration of AI-based decision aids into human decision making process. The widespread adoption of such AI-based decision aids has opened up a new paradigm of human-AI collaboration–the AI model provides recommendations for a given decision making task, while human decision makers are responsible for making the final decisions. To fully unlock the potential of AI-based decision aids in enhancing human decision making, a few studies [1–5] have developed computational models to capture how humans factor AI recommendations into their decision-making process, and explored how these behavioral models can be utilized to improve human-AI team performance. For example, Vodrahalli, Gerstenberg, and Zou [6] developed a human behavior model to characterize the impact of AI predictions and confidence levels on human final decisions. This model was then utilized to adjust the model confidence displayed to people, with the objective of calibrating human trust in AI assistance. Meanwhile, the black-box nature of prevalent AI models has driven a greater integration of model explanations, generated through various explainable AI (XAI) methods [7–11], into AI-assisted decision making. These explanations seek to provide some insights into the the underlying decision rationales of AI models, assisting humans in evaluating the reliability of AI decisions and identifying the optimal strategies to rely on AI recommendations. However, many empirical studies [12–18], which evaluate the effectiveness of current XAI methods for improving people’s understanding 38th Conference on Neural Information Processing Systems (NeurIPS 2024). of the AI model [16, 19, 20] and supporting their calibrated trust in the model [13, 14, 20], have demonstrated that humans often struggle to use the explanations generated by these methods in the optimal way. Thus, despite designers’ expectations that XAI methods will positively shape human interaction with AI models, they often fall short of their intended goals, such as fostering appropriate levels of trust and reliance in AI-assisted decision-making. This could be because existing XAI methods do not account for human reactions, making them unadaptive to human cognitive processes. If this is the case, one may naturally wonder if it is feasible to quantitatively model how humans incorporate both AI recommendations and explanations into their decision making process. If so, can the quantitative understanding of human behavior obtained from the learned behavior models be utilized to directly manipulate AI explanations, thereby nudging human decision-makers towards making targeted decisions? To answer these questions, in this paper, we begin by training behavior models that characterize how AI recommendations and explanations are factored into human decisions based on the collected behavior datasets for various decision making tasks. Utilizing these learned models, we then adjust the AI explanations for different purposes through gradient-based optimizations. Our extensive human experiments across various tasks demonstrated that such human behavior modeling can bring forth both benefits and risks — when used for benign purposes, the human behavior models can inform the adjustment of AI explanations such that the manipulated explanations could significantly enhance the decision-making performance of human-AI teams in most tasks; however, the same behavior models can also be exploited by adversarial parties for adversarial purposes, such as increasing human decision-makers’ biases against a certain protected group and significantly decreasing the fairness level of humans’ final decisions across different groups. Finally, examining human perceptions of these manipulated explanations reveals the “scary” truth — while human decisions are easily swayed by these altered explanations, individuals generally fail to detect any anomalies in the explanations, underscoring a significant vulnerability in human-AI interaction. 2 Related Work 2.1 Computational Modeling of AI-assisted Decision Making There has been a surge of interests among researchers recently in computationally modeling human behavior in AI-assisted decision making [1–5, 21–24]. The goals of these studies for modeling human behavior are diverse, encompassing improving human-AI team performance through intelligent interventions or model recommendation adjustments [6, 25–29], deciding when to present AI powered code suggestion in programming [30], evaluating the utility of AI explanations to improve user understanding of AI model behavior [31–33], and deploying adversarial attacks on AI models to reduce human trust [34]. In this paper, we take a more holistic view and explore whether we can model how humans integrate both AI recommendations and explanations into their decision making process, and what the implications of such behavior modeling are. 2.2 Human-centered Evaluation of AI Explanations With the increasing use of AI technologies as decision aids for humans, a variety of explainable AI techniques have been developed to increase the interpretability of AI models [7–11, 33, 35–37]. To understand the effectiveness of these explanation methods, a growing body of empirical humancentered evaluations have been conducted to examine how AI explanations would affect the ways that humans perceive and interact with AI models [12–18, 31, 32, 38, 39]. These evaluations look into various aspects of impacts of AI explanations, such as the influence on people’s trust and reliance on the AI model [13, 14, 20], understanding of AI model [16, 19, 20], and the collaboration performance of the human-AI team [12, 40, 41]. Recently, some research has explored the modification of AI explanations to influence human behavior. For example, Lakkaraju and Bastani [42] demonstrated that handcrafting modifications in AI explanations—such as hiding sensitive features—can mislead human trust in AI models. Another study [43] found that aligning AI explanations with humans’ own decision rationales can increase agreement between human decisions and the AI model’s predictions. Different from previous work which required intensive handcrafting of AI explanations, in this paper, we explore whether it is possible to directly exploit the computational human behavior models to manipulate AI explanations, even without the access to AI models, with the goal of nudging human decisions towards targeted directions. 2 3 Methodology 3.1 Problem Formulation In this study, we explore the scenario of human-AI collaboration within the context of AI-assisted decision making, and we now formally describe it. Consider a decision making task represented by a n-dimension feature vector x ∈Rn, and y is the correct decision to make in this task. Specifically, in this study, we focus on decision making tasks with binary choices of decisions, i.e., y ∈{−1, 1}. The AI model’s recommendation on the decision task is represented as ym = M(x) , ym ∈{−1, 1}. Following the explainable AI methods like LIME[7] or SHAP[8], the AI model could also provide some “explanations” of its decision, e = E(M(x)), e ∈Rn, by showing the contributions of each feature to the decision. With all these information, the human decision maker (DM) needs to make the final decision yh ∈{−1, 1} by either accepting or rejecting AI model’s decision recommendation ym, which can be characterized by yh = H(x, ym, e). The goal of our study is to explore whether we can quantitatively model such decision making process—specifically, H(x, ym, e)—and whether this quantitative understanding of human behavior can be utilized to adjust AI explanations (i.e., change e to e′) without accessing to the original AI model M(·), thereby nudging human DMs to make the targeted decision ˆyh ∈{−1, 1}, denoted as ˆyh = H(x, ym, e′). 3.2 Modeling Human Behavior in AI-assisted Decision Making We first build computational models to characterize how humans integrate both AI recommendations and explanations into their decision process. Following previous works on modeling human behavior in different scenarios of AI-assisted decision making [23, 31], we adopted a two-layer neural network as the structure for modeling the human decision in this study: yh = Hwh(x, ym, e) = Hwh([x, ym, e, x ⊙e]) (1) The inputs to the behavior model include the task features x, the AI model’s prediction ym, the AI explanation e, and the interaction term between the task features and the AI explanation x ⊙e that reflects how humans may redirect their attention to the corresponding features highlighted by the AI explanation. Given the human behavior dataset D = {xi, ym i , ei, yh i }N i=1, we can employ the maximum log-likelihood estimation to learn the behavior model Hwh. 3.3 Manipulating AI Explanations through the Behavior Model We next proceed to explore how the quantitative understanding of human behavior from Hwh can be utilized to manipulate AI explanations. In particular, given the targeted decision ˆyh for the task instance x, we want to identify a new AI explanation e′ that maximizes the likelihood that human DMs make the targeted decision ˆyh according to the learned behavior model Hwh. In addition, to prevent the case where the manipulated explanations e′ has a very low level of fidelity [10], such as suggesting a recommendation that is inconsistent with the AI model’s prediction ym, we also impose a constraint that the new explanation e′ should still support the original AI recommendation ym. Since we assume no access to the original AI model, we define Lconsistency(e, ym) as a measurement of agreement consistency between the manipulated AI explanations and the AI recommendation: Lconsistency(e, ym) = 0 if sign (P i ei) = sign(ym), 1 otherwise. (2) Together, we use the following optimization problem to manipulate AI explanations: argmine′∈RnLbehavior(Hwh(x, ym, e′), ˆyh), subj. to Lconsistency(e′, ym) ≤0 (3) where Lbehavior is defined as the cross entropy function. Since exactly solving the above optimization problem is intractable, we used the gradient-based optimization to approximate it: e′ θt+1 = e′ θt −η∇e′ θ(Lbehavior(Hwh(x, ym, e′ θ), ˆyh) + λLconsistency(e′ θ, ym)) (4) where η is the step size, λ is the trade-off parameter, and e′ θ represents the parameterized explanations in the optimization process. We can iteratively optimize manipulated explanations e′ until Lbehavior is smaller than a threshold τ or reach the maximum number of rounds T. 3 4 Human Behavior Model Learning To develop the human behavior model for manipulating AI explanations, we first conduct a human subject experiment to collect human behavior data. 4.1 Decision Making Task and AI Assistance We consider four decision making tasks in this study: • Census Prediction (Tabular Data) [44]: This task was to determine a person’s annual income level. In each task, the human DM was presented with a profile with 7 features, including the person’s gender, age, education level, martial status, occupation, work type, and working hour per week. The subject was asked to decide whether this person’s annual income is higher or lower than $50k for each task. We trained a random forest model to make the income prediction, and the accuracy of the AI model was 76%. • Recidivism Prediction (Tabular Data) [45]: This task was to determine a person’s recidivism risk. In each task, the human DM was presented with a profile with 8 features, including their basic demographics (e.g., gender, age, race), criminal history (e.g., the count of prior non-juvenile crimes, juvenile misdemeanor crimes, juvenile felony crimes committed), and information related to their current charge (e.g., charge issue, charge degree). The subject was asked to decide whether this person would reoffend within two years. We trained a random forest model to make the prediction, and the accuracy of the AI model was 62%. • Bias Detection (Text Data) [46]: In this task, the human DM was presented with a text snippet and needed to decide whether it contained any bias. We fine-tuned a BERT [47] model to identify bias in the snippet, and the accuracy of the AI model is 79%. • Toxicity Detection (Text Data) [48]: In this task, the human DM was presented with a text snippet and needed to decide whether it contained any toxic content. We fine-tuned a BERT model to identify the toxic content, and the accuracy of the AI model is 86%. To understand how people respond to various AI explanations, we employed LIME and SHAP to explain the predictions made by the AI model. Additionally, we augment the LIME or SHAP explanations by either randomly masking out contributions from some features or amplifying contributions of some features (referred to as the “Augmented” explanations) to see how humans react to them. These explanations are provided with AI recommendations together to humans in decision making. 4.2 Experimental Procedure We posted our data collection study on the Prolific 1 to recruit human participants. Upon arrival, we randomly assigned each participant to one of the four decision making tasks and they needed to fill in an initial survey to report their demographic information and their knowledge of AI models and explanations. Participants started the study by completing a tutorial that described the decision making task that they needed to work on. To familiarize participants with the task, we initially asked them to complete five tasks independently without AI assistance. During these training tasks, we immediately provided the correct answer at the end of the task. After the completion of training tasks, participants moved on to the formal tasks. In the formal tasks, participants would receive one type of AI explanations among SHAP, LIME, or Augmented. Specifically, each participant was asked to complete a total of 15 tasks. In each task, participants were provided with the AI prediction and the explanations along with the task instance. They were then required to make their final decisions. Finally, participants were required to complete an exit survey to report their perceptions of the AI explanations they received during the study. They were asked to rate the alignment of AI explanations with their own rationale, as well as the usefulness, transparency, comprehensibility, satisfaction with the provided explanations, and their trust in the AI models, on a 5-point Likert scale. For the detailed survey questions, please refer to Appendix A.3. We offered a base payment of $1.2 and a potential bonus of $1 if the participant’s accuracy is above 85%. The study was open to US-based workers only, and each worker can complete the study once. 1https://www.prolific.com/ 4 Table 1: The number of subjects recruited in data collection for training behavior models, and the average accuracy of the human behavior model in 5-fold cross validation for each task. Census Recidivism Bias Toxicity Number of Participants 78 80 72 42 Model Accuracy 0.74 0.79 0.65 0.76 4.3 Training Results After collecting data on human behavior, we developed human behavior models for each type of task. For the human behavior models for two textual tasks—Toxicity Detection and Bias Detection—we employed the pretrained BERT encoder to extract features from the original sentences, which were then used as the task feature x in the human behavior model Hwh. We optimized these behavior models using Adam [49] with an initial learning rate of 1e −4 and a batchsize of each training iteration of 128. The number of training epochs is set as 10. Table 1 shows the number of participants recruited, as well as the average accuracy of the human behavior model evaluated through 5-fold cross validation for each task. We observed that the average accuracy of all human behavior models exceeds 0.65, which is considered to be reasonable. Consequently, we utilized these learned human behavior models to manipulate AI explanations in the following evaluations. 5 Evaluation I: Manipulating AI Explanations for Adversarial Purposes In our first evaluation, we adopted the role of an adversarial party to explore whether they could utilize the learned human behavior model to manipulate AI explanations. The manipulation goal was to nudge human DMs to be biased against certain protected groups in the decision making process. We are particularly interested in comparing the fairness level of human decision outcomes between human DMs who receive original explanations, such as SHAP or LIME, and those who receive manipulated explanations. Notably, all human DMs are provided with the same AI predictions for the same decision making task. Additionally, we also explore differences in human perceptions of original AI explanations versus manipulated AI explanations. Evaluation Metrics and Manipulating AI Explanations. Following previous work [50, 51], we used the false positive rate difference (i.e., FPRD) and the false negative rate difference (i.e., FNRD) to measure the fairness level of human decision outcomes—the closer these values are to zero, the more fair the decisions are. To manipulate AI explanations and nudge human DMs toward biasing against certain protected groups, we define the targeted human decision ˆyh for each task as follows: • Census Prediction: In this task, we considered a person’s sex as the protected attribute. The targeted human decision is defined as ˆyh = 1 (indicating a person’s annual income exceeds $50K) when xsex = male, and ˆyh = −1 (indicating a person’s annual income does not exceed $50K) when xsex = female. The fairness metrics can be computed as FPRD = FPRfemale −FPRmale, and FNRD = FNRfemale −FNRmale. • Recidivism Prediction: In this task, we considered the defendant’s race as the protected attribute. The targeted human decision is defined as ˆyh = 1 (indicating the defendant will reoffend) when xrace = black, and ˆyh = −1 (indicating the defendant will not reoffend) when xrace = white. The two fairness metrics can be computed as FPRD = FPRwhite − FPRblack, and FNRD = FNRwhite −FNRblack. • Bias Detection: In this task, we divided text snippets into groups based on their political leaning. The targeted human decision is defined as ˆyh = 1 (indicating the text is biased) when xleaning = democratic, and ˆyh = −1 (indicating the text is not biased) when xleaning = republican. The fairness metrics can be computed as FPRD = FPRrep −FPRdem, and FNRD = FNRrep −FNRdem. • Toxicity Detection: In this task, we divided text snippets into groups based on the victim of the text. The targeted human decision is defined as ˆyh = 1 (indicating the text is toxic) when xvictim = white, and ˆyh = −1 (indicating the text is non-toxic) when xvictim = black. The two fairness metrics can be computed as FPRD = FPRblack −FPRwhite, and FNRD = FNRblack −FNRwhite. 5 Table 2: The number of participants we recruited in the evaluation study, categorized according to the type of AI explanation they received and the task they were assigned to. Census Recidivism Bias Toxicity SHAP 86 89 60 88 LIME 65 71 59 85 Adversarially Manipulated 82 92 71 65 Benignly Manipulated 77 84 69 46 For the Bias Detection and Toxicity Detection tasks, the original datasets [46, 48] provide annotations for the political leanings of the sentences and the targeted victims of the text snippets, respectively. After determining the targeted decision ˆyh for each task instance, we then followed the gradient-based optimization procedure (i.e., Equation 4) to identify the manipulated explanation. We set the step size η as 0.01, the trade-off λ as 0.01, the optimization threshold τ as 0.1, and the maximum optimization number T as 100. For the initial AI explanation eθ0 at the start of the optimization process, we directly initialize e′ θ0 as e′ θ0 ∼U(−1, 1). We then repeated this optimization process for 5 times and took the average to use in the following human evaluations. For the examples of manipulated explanations, please refer to Appendix B.3. Data Collection. We followed the experimental procedure described in Section 4.2 to collect data on human responses to and perceptions of different AI explanations. We randomly assigned either SHAP or LIME explanations, or the manipulated explanations, to participants. Participants were required to complete 15 tasks with AI model predictions and the assigned explanations (SHAP or LIME or manipulated). We offered a base payment of $1.2 and a potential bonus of $1 if the participant’s accuracy is above 85%. Table 2 reports detailed statistics of the participants in each task. Below, we analyzed how the manipulated explanations affect fairness level of human decisions and how do humans perceive those explanations. 5.1 How do the adversarially manipulated explanations affect fairness level of human decisions? The fairness levels of participants’ decision outcomes under the manipulated explanation, SHAP explanation, and LIME explanation are presented in Figure 1. Visually, it appears that when human DMs are provided with manipulated explanations, both FPRD and FNRD scores of their decision outcomes tend to deviate more from zero compared to when DMs receive SHAP or LIME explanations. To examine whether these differences are statistically significant, we conducted regression analyses. Specifically, the focal independent variable was the type of explanation received by participants, while the dependent variables were the participants’ FPRD and FNRD scores. To minimize the impact of potential confounding variables, we included a set of covariates in our regression models, such as participants’ demographic background (e.g., age, race, gender, education level), their knowledge of AI explanations, their trust in AI models, and the FPRD or FNRD scores of the AI model decisions they received in the study. These covariates were selected based on prior HCI research [14, 20, 41] which empirically reveal how characteristics of human DMs may moderate the impacts of AI explanations on human decisions in AI-assisted decision making. Our regression results indicate that the adversarial party can significantly increase the level of unfairness in human decision outcomes with manipulated explanations through human behavior modeling. Specifically, when examining FPRD, we found that participants who received manipulated AI explanations made more unfair decisions compared to those who received SHAP or LIME explanations (p < 0.05) in the Census and Recidivism tasks. The difference was marginally significant (p < 0.1) in the Toxicity task. When examining FNRD, results show that participants who received manipulated explanations made decisions that were significantly more unfair than those who received SHAP or LIME explanations (p < 0.01) in the Bias task. 5.2 How do humans perceive the adversarially manipulated AI explanations? 6 (a) False Positive Rate Difference (b) False Negative Rate Difference Figure 1: Comparing average FPRD and FNRD of the human decision outcomes under the adversarially manipulated explanation, SHAP explanation, or LIME explanation. Error bars represent the 95% confidence intervals of the mean values. , , and ** denote significance levels of 0.1, 0.05, and 0.01, respectively. For both FPRD and FNRD, a value closer to zero indicates that the human decisions are more fair. (a) Transparency (b) Usefulness Figure 2: Comparing the average human perceived transparency and usefulness of the adversarially manipulated explanation, SHAP explanation, and LIME explanation. Error bars represent the 95% confidence intervals of the mean values. In Section 5.1, we found that the adversarial party can manipulate AI explanations to nudge human DMs toward making more unfair decisions compared to those who received the original AI explanations, aligning with the adversarial party’s intentions. To determine whether DMs could detect any abnormalities in the manipulated explanations, we examined how their perceptions of AI explanations varied among manipulated, SHAP, and LIME explanations. Figures 2a and 2b compare the average perceived transparency and usefulness of three types of AI explanations. Visually, there are no significant differences in how the explanations are perceived by people who received different explanations. We also applied regression models to predict human perceptions of these explanations by accounting for their demographic background (e.g., age, race, gender, education level), their knowledge of AI explanations and their trust in AI models. The regression results indicate that there are no significant differences in the perceived transparency and usefulness of manipulated explanations compared to SHAP or LIME explanations. Similar patterns were observed for perceptions of alignment, comprehensibility, satisfaction, and trust between the manipulated and unmanipulated explanations. While adversarially manipulated explanations significantly influence human decision making behavior, individuals generally do not detect abnormalities in the manipulated AI explanations across most tasks. For further details, please refer to Appendix B.2. 6 Evaluation II: Manipulating AI Explanations for Benign Purposes In the previous section, we found that the adversarial party could use the behavior model to manipulate AI explanations, thereby misleading humans into making unfavorable decisions against specific groups. Naturally, one might wonder could a third party also use behavior models to manipulate AI explanations for benign purposes, such as promoting more appropriate human reliance on AI models? For instance, can manipulated AI explanations lead humans to reject AI recommendations when the 7 (a) Accuracy (b) Overreliance (c) Underreliance Figure 3: Comparing the average accuracy, overreliance, and the underreliance of human decision outcomes under the benignly manipulated explanation, SHAP explanation, or LIME explanation. Error bars represent the 95% confidence intervals of the mean values. , , and ** denote significance levels of 0.1, 0.05, and 0.01 respectively. AI model decision is likely incorrect, and encourage acceptance when the decision is likely correct? We aim to explore the answers to this question in this section. Evaluation Metrics and Manipulating AI Explanations Following previous work [14, 41, 43], we used the accuracy, underreliance, and overreliance to measure human DMs’ appropriate reliance level on AI models. Underreliance refers to the fraction of tasks where the participant’s decision was different from the AI model’s decision when the AI model’s decision was correct. Overreliance refers to the fraction of tasks where the participant’s decision was the same as the AI model’s decision when the AI model’s decision was incorrect. To manipulate AI explanations for the promotion of more appropriate reliance on AI models, it is necessary to determine the reliability of AI model prediction on each task instance. Recent work [52, 53] has proposed methods to leverage the complementary strengths of humans and AI by combining human independent decisions and AI model, which is often shown to result in more accurate decisions than those made by either humans or AI models alone. Specifically, given the human independent decision yh independent, the AI model recommendation ym, and the task instance x, these methods learn models to combine yh independent and ym to produce a combined result: ycombine = CombineModel(yh independent, ym, x) (5) To see whether ycombine can yield better decisions compared to AI alone or human alone, we evaluated various combination models including the human-AI combination method [53] and several truth inference methods [54–57] used in crowdsourcing. Our results showed that the human-AI combination method [53] generally outperformed AI solo and independent human decision, as well as other combination methods. Thus, ycombine produced by the human-AI combination method [53] is defined as the targeted decision ˆyh for manipulating AI explanations. For detailed information on the evaluations of each combination method, please refer to the Appendix C.1. We again followed Equation 4 to manipulate the AI explanations. We set the step size η as 0.01, the trade-off λ as 0.01, the optimization threshold τ as 0.1, and the maximum optimization number T as 100, and the initial AI explanation e′ θ0 at the start of the optimization process is initialized as e′ θ0 ∼U(−1, 1). We repeated this optimization process for 5 times and took the average to use in the following experiments. For the examples of manipulated explanations, please refer to Appendix C.4. Data Collection. We recruited participants from Prolific once again to collect behavioral data under the benignly manipulated explanations, following the experimental procedure described in Section 4.2. We offered a base payment of $1.2 and a potential bonus of $1 if the participant’s accuracy is above 85%. Table 2 shows the detailed statistics of the participants we recruited for each task. Subsequently, we analyzed whether the benignly manipulated explanations can promote appropriate reliance of human DMs on AI models, as well as their perceptions of these AI explanations. 6.1 Can benignly manipulated explanations promote appropriate reliance of human DMs on AI models? Figures 3a, 3b, and 3c compare the average accuracy, overreliance, and underreliance of human decision outcomes under manipulated, SHAP, and LIME explanations, respectively. It is clear that providing human DMs with manipulated AI explanations leads to an increase in the accuracy of their decision outcomes for most of tasks. We subsequently conducted regression analyses to determine 8 (a) Transparency (b) Usefulness Figure 4: Comparing the average human perceived transparency and usefulness of the benignly manipulated explanations, SHAP explanations, and LIME explanations. Error bars represent the 95% confidence intervals of the mean values. whether these differences are statistically significant. The regression models incorporated a set of covariates, including participants’ demographic backgrounds (e.g., age, race, gender, education level), their knowledge of AI explanations, their trust in AI models, and the accuracy of the AI models. The regression results indicate that in the Census, Recidivism, and Bias tasks, substituting SHAP or LIME explanations with manipulated explanations significantly improves the accuracy of human-AI team. In contrast, for the Toxicity task, we observed no statistical difference, which could potentially be attributed to the high competence of humans in solving this task (e.g., when presented with SHAP or LIME explanations, the average decision accuracy of participants already exceeds 0.8, leaving limited room for further improvement). 6.2 How do humans perceive benignly manipulated AI explanations? In Section 5.2, we observed that it is challenging for humans to detect abnormalities in the adversarially manipulated explanations, even though they are unconsciously influenced by the manipulated explanations to make more unfair decisions. In this section, we revisit this question to investigate into whether humans’ perceptions of the manipulated explanations change, when they are manipulated for benign purposes. Figures 4a and 4b compare the average human perceived transparency and usefulness of the benignly manipulated explanations, SHAP explanations, and LIME explanations. Regression analyses reveal no statistically significant differences among the perceived transparency and usefulness of these three types of explanations. Similar trends were observed for other perceptual aspects of explanations, including perceived alignment, comprehensibility, satisfaction, and trust. For further results, please refer to Appendix C.3. 7 Conclusion and Limitations In this paper, we explore whether we can quantitatively model how humans incorporate both AI recommendations and explanations into their decision making process, and whether we can utilize the quantitative understanding of human behavior obtained from these learned models to manipulate AI explanations for both adversarial and benign purposes. Our extensive experiments across various tasks demonstrate that human behavior can be easily influenced by these manipulated explanations toward targeted outcomes, regardless of the intent being benign or adversarial. Despite the significant influence of these falsified explanations on human decisions, individuals typically fail to detect or recognize any abnormalities. Our study has several limitations. For example, it focuses on modeling and manipulating score-based explanations. Further research is needed to explore how to model how humans incorporate other types of explanations, such as example-based and rule-based explanations, and how these can be manipulated to influence human behavior as observed with score-based explanations in our study. Additionally, our study was limited to decision making tasks involving tabular and textual data, which are naturally suited to score-based explanations. Further explorations are needed to extend these findings to decision tasks with other data types (e.g., images). 9 Ethical Consideration This study was approved by the Institutional Review Board of the authors’ institution. Through our findings, we aim to draw the community’s attention to the ease with which third parties can manipulate AI explanations with the learned behavior models to influence human decision making. Users often lack the ability to accurately and appropriately interpret the AI explanations presented to them, yet their decision behavior is easily swayed by the manipulated AI explanations. Our findings highlight the critical importance of securing human-AI interaction data to prevent the misuse of human behavior models derived from it. Additionally, there is an urgent need to ensure that AI explanations provided to humans are more secure and inherently benign. Moreover, providing pre-education is essential to assist humans in establishing a proper understanding of AI explanations, which may potentially mitigate the risks of manipulation. In addition, our experiments are based on datasets that are publicly available; “correct” decisions for the tasks in these datasets are generally considered as recording the real-world ground truth. While these datasets are not intentionally biased toward any specific groups, we acknowledge that there might be implicit biases introduced to these datasets during the curation process, which are beyond our control. Importantly, we note that we made no alterations to the datasets that would introduce additional bias in our experiment. Acknowledgments We thank the support of the National Science Foundation under grant IIS-2229876 and IIS-2340209 on this work. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors alone. References [1] Gagan Bansal et al. “Is the most accurate ai the best teammate? optimizing ai for teamwork”. In: Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. 13. 2021, pp. 11405–11414. 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[57] Vikas C Raykar et al. “Learning from crowds.” In: Journal of machine learning research 11.4 (2010). 12 Table A.1: The average hourly payment received by participants in our study across four tasks. In the row “Number of Participants”, the number in parentheses indicates the number of invalid participants who did not pass the attention check questions. Recidivism Census Bias Toxicity Number of Participants 336 (16) 310 (25) 259 (20) 286 (17) Average Working Time (Minute) 6.67 6.25 6.98 6.34 Hourly Payment (Base) $10.8 $11.7 $10.2 $11.3 Hourly Payment (Base + Bonus) $11.9 $11.8 $11.4 $14.4 (a) Census prediction (b) Recidivism prediction Figure A.1: The task interfaces for the census prediction and recidivism prediction. A The Design of Human Study (Additional Details) A.1 Compensation Details To determine the appropriate payment level for each type of task, we first conducted a preliminary study to estimate the time workers might spend on the tasks. Our pilot study indicated that a base payment of $1.2 per task translates to an approximate hourly rate of $10. To provide greater transparency about the compensation received by participants in our formal study, Table A.1 summarizes the average hourly payment and the average time spent on each task. A.2 Task Interfaces Figure A.1a, A.1b, A.2a, and A.2b show the interfaces participants used in the Census Prediction, Recidivism Prediction, Bias Detection, and Toxicity Detection tasks, respectively. A.3 Exit Survey Questions In the study, after the main tasks, the participants need to fill in an exit survey to report their perceptions of presented AI explanations. The survey questions are detailed as: • Alignment: On a scale of 1-5, how well do you think the explanations align with your understanding of the problem? • Usefulness: On a scale of 1-5, how useful are the explanations in helping you make decisions? • Transparency: On a scale of 1-5, how well do you think the explanations reveal the AI model’s decision making process? • Comprehensibility: On a scale of 1-5, how easy is it for you to understand the explanations? • Satisfaction: On a scale of 1-5, how satisfied are you with the explanations provided by the AI model? • Trust: On a scale of 1-5, would you trust the AI model’s prediction or decision based on the explanations? 13 (a) Bias detection (b) Toxicity detection Figure A.2: The task interfaces for the bias detection and the toxicity detection. Table B.1: Agreement between the sum of feature importance in explanations and AI predictions, measured in terms of the Pearson correlation coefficient. Census Recidivism Bias Toxicity SHAP 0.88 0.97 0.88 0.91 LIME 0.41 0.40 0.76 0.78 Adversarially Manipulated 0.57 0.38 0.73 0.68 Benignly Manipulated 0.87 0.68 0.89 0.89 B Evaluation I: Manipulating AI Explanations for Adversarial Purposes (Additional Results) B.1 Visual Consistency of Explanations Table B.1 compares the agreement between the sum of feature importance in explanations and AI predictions, the Pearson correlation coefficients, measured in terms of Pearson correlation coefficient, for adversarially manipulated, LIME, and SHAP explanations, respectively. We observed that the visual consistency of the manipulated explanations is lower than that of SHAP but very close to that of LIME. B.2 Human Perceptions of Explanations Figures B.1a to B.1d compare the average human perceived alignment, comprehensibility, satisfaction with the provided explanations ,and the trust in the AI models under the under the adversarially manipulated explanation, SHAP explanation, or LIME explanation. In general, our findings indicate that there are no significant differences in people’s perceptions of the three explanations across four tasks, with the exceptions for the alignment and trust in the Toxicity task. Specifically, for the Toxicity task, participants perceived LIME explanations as aligning more closely with their own rationales than the adversarially manipulated explanations, with a marginally significant difference (p < 0.1). Furthermore, participants reported significantly greater trust in the AI models accompanied by LIME explanations compared to those with adversarially manipulated explanations (p < 0.01). B.3 Examples of Manipulated Explanations Figures B.2 to B.5 show the visual comparisons of adversarially manipulated, LIME, and SHAP explanations for the Census Prediction task, Recidivism Prediction task, Bias Detection task, and Toxicity Detection task, respectively. 14 (a) Alignment (b) comprehensibility (c) Satisfaction (d) Trust Figure B.1: Comparing the average human perceived alignment, comprehensibility, satisfaction with the provided explanations, and the trust in the AI models under the adversarially manipulated explanation, SHAP explanation, or LIME explanation. Error bars represent the 95% confidence intervals of the mean values. * and *** denote significance levels of 0.1 and 0.01 respectively. Figure B.2: The visual comparisons of adversarially manipulated, LIME, and SHAP explanations for the Census Prediction task. C Evaluation II: Manipulating AI Explanations for Benign Purposes (Additional Results) C.1 Combining Human Decisions and AI Predictions In the main paper, we aim to benignly manipulate AI explanations to encourage human DMs to rely more appropriately on AI models. Following previous research [52, 53], we combined independent human decisions with AI model predictions to determine the targeted decision for each task instance. We evaluated the human-AI combination method [53] and several truth inference methods used in crowdsourcing for truth discovery. We detailed the process of evaluation below. Table C.1: The average accuracy of the independent human behavior model through 5-fold validation for each task. Census Recidivism Bias Toxicity Accuracy 0.81 0.84 0.62 0.79 15 Figure B.3: The visual comparisons of adversarially manipulated, LIME, and SHAP explanations for the Recidivism Prediction task. Figure B.4: The visual comparisons of adversarially manipulated, LIME, and SHAP explanations for the Bias Detection task. Simulating Human Independent Decision. To understand how humans independently make decisions on each task instance, we first conducted a study again on the Prolific to collect independent human decision behavior data across four tasks. We recruited 40 participants for each task. Each recruited participant needed to complete 15 tasks. With the collection of human behavior data, we then fitted two-layer neural networks to simulate human independent decision behavior. For Toxicity Detection and Bias Detection textual tasks, we used the pretrained BERT encoder to initially extract features from the original sentences as the input to the independent behavior models. We optimized these independent behavior models using Adam [49] with an initial learning rate of 1e −4 and 16 Figure B.5: The visual comparisons of adversarially manipulated, LIME, and SHAP explanations for the Toxicity Detection task. (a) Alignment (b) comprehensibility (c) Satisfaction (d) Trust Figure C.1: Comparing the average human perceived alignment, comprehensibility, satisfaction with the provided explanations, and the trust in the AI models under the benignly manipulated explanation, SHAP explanation, or LIME explanation. Error bars represent the 95% confidence intervals of the mean values. a batchsize of each training iteration of 128. The number of training epochs is set as 10. The average accuracy of 5-fold validation for each model is reported in Table C.1, which we found to be satisfactory. We then utilized these fitted models to simulate independent human decisions yh independent in the human-AI combination process to determine the potentially better decisions. Comparing Combination Performance. We consider the human + AI combination method [53] and a few truth inference methods in crowdsourcing as baselines in the evaluation, including GLAD [55], CATD [56], LFC [57], EM [54], and MV [54]. These methods combine the human independent decisions yh independent predicted by the fitted independent human behavior models and AI model recommendations ym to produce combined decisions ycombine. The accuracy of each method on holdout task pools for each task to be used in the subsequent evaluation is reported in Table C.2. In general, we found that human + AI combination method outperforms other baselines. By integrating human decisions with AI predictions, this method shows superior performance to either AI solo or human solo across all four tasks. Consequently, we used the combined decisions ycombine from the human + AI combination method as the targeted decision ˆyh in subsequent experiments to manipulate explanations. C.2 Visual Consistency of Explanations Table B.1 compares the agreement between the sum of feature importance in explanations and AI predictions, the Pearson correlation coefficients, measured in terms of Pearson correlation coefficient, for benignly manipulated, LIME, and SHAP explanations, respectively. We observed that the visual consistency of the manipulated explanations is very close to that of SHAP and higher than that of LIME. C.3 Human Perceptions of Explanations Figures C.1a to C.1d compare the average human perceived alignment, comprehensibility, satisfaction with the provided explanations ,and the trust in the AI models under the under the adversarially 17 Table C.2: The accuracy of each method on the holdout task pools, used in following experiments to manipulate AI explanations. The best result in each row is highlighted in bold. Human Solo AI Solo Human + AI [53] GLAD [55] CATD [56] LFC [57] EM [54] MV [54] Census 0.61 0.73 0.76 0.62 0.74 0.61 0.60 0.69 Recidivism 0.54 0.58 0.61 0.59 0.61 0.58 0.59 0.62 Bias 0.55 0.80 0.81 0.66 0.65 0.67 0.69 0.66 Toxicity 0.76 0.86 0.86 0.78 0.79 0.81 0.77 0.82 Figure C.2: The visual comparisons of benignly manipulated, LIME, and SHAP explanations for the Census Prediction task. manipulated explanation, SHAP explanation, or LIME explanation. We found that there are no significant differences in people’s perceptions of the three explanations across four tasks on these aspects. C.4 Examples of Manipulated Explanations Figures C.2 to C.5 show the visual comparisons of benignly manipulated, LIME, and SHAP explanations for the Census Prediction task, Recidivism Prediction task, Bias Detection task, and Toxicity Detection task, respectively. 18 Figure C.3: The visual comparisons of benignly manipulated, LIME, and SHAP explanations for the Recidivism Prediction task. Figure C.4: The visual comparisons of benignly manipulated, LIME, and SHAP explanations for the Bias Detection task. 19 Figure C.5: The visual comparisons of benignly manipulated, LIME, and SHAP explanations for the Toxicity Detection task. 20 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] . Justification: In this paper, we aim to explore whether we can model how humans incorporate both AI predictions and explanations into their decision making process, and whether we can utilize the quantitative understanding of human behavior from these models to manipulate explanations, thereby nudging human decisions in AI-assisted decision making. Through extensive human subject experiments, we showed the good, bad, and the ugly side of this. The claims in the abstract and the introduction accurately reflect the paper’s contributions and scope. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] . Justification: We discuss the limitations of the work in the Conclusion section. 3. Theory Assumptions and Proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [NA] . Justification: This paper does not include theoretical results. 4. Experimental Result Reproducibility Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] . Justification: We report the settings of the hyper-parameters for training behavior models and manipulating AI explanations in the paper. 5. Open access to data and code Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [No] . Justification: We will ensure that the collected human behavior data is used responsibly by implementing strict access controls to minimize potential risks associated with unauthorized use or misuse. Individuals who wish to have access to the data or code must apply for permission, which will only be granted to those who meet the necessary authorization criteria. 6. Experimental Setting/Details Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] . Justification: We describe training and test details in the paper. 7. Experiment Statistical Significance Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] . 21 Justification: The experimental results are accompanied by the 95% confidence intervals, and the statistical significance tests. 8. Experiments Compute Resources Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] . Justification: We used one RTX 4060 for behavior model training and AI explanation manipulation. 9. Code Of Ethics Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? Answer: [Yes] . Justification: We checked the ethics guidelines, and the research conducted in the paper conform with the NeurIPS Code of Ethics. 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] . Justification: We discuss the societal impacts of the work in the Ethical Consideration section. 11. Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [No] . Justification: We will ensure that the collected human behavior data is used responsibly by implementing strict access controls to minimize potential risks associated with unauthorized use or misuse. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] . Justification: We cited all datasets used in the paper. 13. New Assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] . Justification: The paper does not release new assets. 14. Crowdsourcing and Research with Human Subjects Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [Yes] . Justification: We included the task interfaces, the experimental procedure, and the compensation structure in the paper. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects 22 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [Yes] . Justification: The study was approved by the Institutional Review Board of the authors’ institution. To minimize the risk of leaving our study participants with inappropriate impressions of manipulated AI explanations, upon the end of the study, we provided a debrief session directly through the Prolific chat system. Each participant who received manipulated AI explanations in the study were individually contacted through the Prolific platform. In the chat-based session, we clarified that the AI explanations were manipulated and did not accurately represent the underlying rationales of the AI models. We emphasized that these explanations were intentionally biased for the purpose of the study, based on the specific task participants were involved in. 23	2024	163
4,472	ParallelEdits: Efficient Multi-Aspect Text-Driven Image Editing with Attention Grouping Mingzhen Huang, Jialing Cai, Shan Jia, Vishnu Suresh Lokhande∗, Siwei Lyu∗ University at Buffalo, State University of New York, USA ducks city fireworks Source prompt: A striped brown cat sits on wooden planks against a wooden wall Source Image curled tail cheetah Direct Inversion Number of edited aspects Source prompt: a man sitting in a boat is silhouetted against the sunset with mountain in the background 1 2 3 4 1 2 3 4 beach necktie grey cat ball lawn curtain sunglasses wool rug bear bridge man sailboat evening glow castle bright sun InfEdit ParallelEdits (Ours) Figure 1: Multi-aspect text-driven image editing. Multiple edits in images pose a significant challenge in existing models (such as DirectInverison [1] and InfEdit [2]), as their performance downgrades with an increasing number of aspects. In contrast, our ParallelEdits can achieve precise multi-aspect image editing in 5 seconds. The symbol ⊗denotes a swap action, the symbol ⊕denotes an object addition action, and the symbol ⊖denotes an object deletion. Arrows (→) on the image highlight the aspects edited by our method. Abstract Text-driven image synthesis has made significant advancements with the development of diffusion models, transforming how visual content is generated from text prompts. Despite these advances, text-driven image editing, a key area in computer graphics, faces unique challenges. A major challenge is making simultaneous edits across multiple objects or attributes. Applying these methods sequentially for multi-aspect edits increases computational demands and efficiency losses. In this paper, we address these challenges with significant contributions. Our main contribution is the development of ParallelEdits, a method that seamlessly manages simultaneous edits across multiple attributes. In contrast to previous approaches, ParallelEdits not only preserves the quality of single attribute edits but also significantly improves the performance of multitasking edits. This is achieved through innovative attention distribution mechanism and multi-branch design that operates across several processing heads. Additionally, we introduce the PIE-Bench++ dataset, an expansion of the original PIE-Bench dataset, to better support evaluating image-editing tasks involving multiple objects and attributes simultaneously. This dataset is a benchmark for evaluating text-driven image editing methods in multifaceted scenarios. Codes are available at: https://mingzhenhuang.github.io/projects/ParallelEdits.html. Corresponding authors 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 1 Introduction Recently, text-driven image editing has experienced remarkable growth, driven by advances in diffusion-based image generative models. This technique involves modifying existing images based on textual prompts to alter objects, their attributes, and the relationships among various objects. The latest methods [3, 1, 4] can produce edited images that closely match the semantic content described in the prompts while keeping the rest of the image unchanged. Unlike early image editing approaches that required image matting to precisely extract foreground objects using alpha mattes [5], text-driven editing offers a less labor-intensive alternative. User-provided textual prompts guide the edits, with auxiliary inputs like masks facilitating localized modifications [6]. While these methods have showcased promising results, existing methods typically focus on editing a single aspect in the source image. An “aspect” refers to a specific attribute or entity within the textual prompt that describes the image and can be modified, such as object type, color, material, pose, or relationship. However, the ability to edit multiple aspects through text prompts is rarely explored. We introduce the concept of multi-aspect text-driven image editing to address this gap. Multi-aspect image editing is essential due to the rich and diverse content and structure of digital images, as well as the varied requirements of users. For example, it always occurs that users wish to modify multiple attributes or regions in an image, such as adding a necktie to a cat and changing the background wall to a beach (Fig. 1, Left), or removing a man and replacing a mountain with a castle in the right example. Unlike traditional editing methods (e.g., [1, 2]) that focus on a single aspect, multi-aspect editing allows users to manipulate various aspects simultaneously. Different from full text-to-image synthesis [7, 8], which involves creating content from scratch, multi-aspect editing works with the source image to ensure essential content preservation. It bridges the gap between single-aspect editing and full synthesis, catering to a wide range of editing scenarios. However, we observe that directly applying the single-aspect text-driven image editing methods in cases where multiple image aspects must be modified often does not yield satisfactory results. A straightforward solution to this problem is to apply the single aspect editing method sequentially – we can order the aspects to be modified and use a single-aspect editing method to change the aspects one by one. Although sequential applications of single-aspect text-driven image editing methods can modify multiple aspects of an image, they may introduce significantly higher computational overhead. More importantly, the order of the aspects modified may affect the quality – changes to later aspects may undo the early ones or accumulate the errors and artifacts, thus reducing the effectiveness of the final editing results, as the last two rows of Fig. 5 and Table 1 show. In this work, we introduce ParallelEdits as an efficient and effective solution to the problem of multi-aspect text-driven image editing. This method is based on a crucial insight that the editing step can occur in parallel with the image’s diffusion steps. Therefore, in ParallelEdits, we build image aspect editing into the diffusion steps to accelerate the editing process. ParrallelEdits is based on an architecture with a fixed number of additional branches dedicated to handling rigid, non-rigid, and style changes. This design ensures scalability independent of the number of prompt aspects altered. In addition, we employ an attention aggregator to accurately assess editing difficulty and route aspects to appropriate branches within the ParallelEdits framework, ensuring precise and efficient editing. To enable subsequent research and evaluation of multi-aspect text-driven image editing methods, we also build the PIE-Bench++ dataset, an extension of the PIE-Bench [1] that has 700 images with detailed text prompts and tailored to facilitate simultaneous edits across multiple image aspects. We propose evaluation metrics and benchmark different text-driven image editing methods on PIE-Bench++. The ParallelEdits outperforms the state-of-the-art image editing methods on PIE-Bench++. 2 Related Works Diffusion Models for Text-Driven Image Editing. Text-driven image editing aims to manipulate local regions of an image based on textual prompts. The editing has two main goals: ensuring the edits align with provided instructions and preserving essential content. Diffusion models [9] have gained popularity as a preferred image editing model for their capacity for generating highquality samples by incorporating diverse conditions, especially using text [10, 11, 2, 12–14, 1]. This involves transforming the images into the latent space and generating regions using diffusion models conditioned by the text prompt while ensuring accurate reconstruction of unmodified regions 2 during editing. To avoid the edited image deviating from original image, early text-driven image editing typically requires user-specified masks as additional condition [15–17] or training [18–20] to guided the editing process, which constrain their potential zero-shot application. To address this limitation, recent editing models, such as InfEdit [2], PnP [21], Direct Inversion [1] follow the work Prompt-to-Prompt (P2P) [3], which proposed to obtain an attention map from the cross attention process and either swap or refine the attention map from text prompt for image editing. This design automatically obtains the editing mask and only allows image editing using a text prompt. Another method, MasaCtrl [4], converts existing self-attention in diffusion models into mutual self-attention for non-rigid consistent image synthesis and editing, enabling to query correlated local contents and textures from source images for consistency. Multi-Aspect Image Editing. While current image editing models have shown promising results in their text-driven image editing benchmarks, we have observed that they work well on single-attribute editing while struggling to edit multiple aspects, especially when editing multiple objects (as shown in Fig. 1). We attribute this limitation to the following reasons. First, existing methods use the attention mask to direct where edits should be made. With multiple attributes, the editing area may expand significantly, incorporating extensive semantic information or scattered regions that are challenging to edit using a single mask. Second, employing a fixed mask from cross-attention maps struggles with edits involving changes in region size (such as pose adjustments), while using an adaptive mask faces challenges in maintaining edit fidelity. Therefore, integrating various attention masks for accurate multi-attribute editing presents a challenging technical problem. Early studies [22, 23] have employed GAN models such as StyleGAN2 [24] to edit multiple attributes in faces. The multiple-attribute editing is realized by training the GAN model with supervised multi-class training and a training dataset of image and attribute vector pairs. This solution heavily relies on the training sets and has limitations in generalizing to new editing types. Few recent works achieve multi-aspect editing with additional inputs: [25] leverages rich text to edit multiple objects and [26] pre-processes the image with grounding to localize multiple edited regions for multi-aspect editing. However, the editing performance highly relies on additional input beyond plain text, either from user input or other off-the-shelf models. A recent work [27] proposes an iterative multi-granular image editor, where a diffusion model can faithfully follow a series of image editing instructions from a user. However, this interactive editing pipeline will result in significant computational overhead. Image Editing with Multiple Branches. In the literature [4, 3], image editing processes have been conducted by implementing a dual-branch approach. This methodology involves segregating source and target branches throughout the editing process. Specifically, the source branch is reverted to z0, while the trajectory of the target branch is iteratively adjusted. By computing the distance from the source branch, the calibration of the target branch occurs at each time-step. Our observation underscores the disparity between the effectiveness of a dual branch in enhancing the editing process and its failure in multi-aspect editing. A singular target branch proves inadequate in calibrating fully from the source branch, leading to imperfect incorporation of all aspects into the image. Hence, our primary proposition advocates for multi-aspect editing by utilizing multiple target branches. Each target branch’s trajectory is meticulously calibrated, with simpler concepts addressed in the initial branches and more complex aspects deferred to subsequent ones. In the following section, we will delve deeper into this concept. 3 Diffusion-based Image Generation and Editing We are provided with an image sample x0 which transforms the latent space via an encoder/decoder pair E/D, such that z0 = E(x0). Here, z0 represents the latent representation of the image x0. With a slight abuse of notation, we approximate the reconstructed image ¯x0 as D(¯z0), where ¯z0 denotes the reconstructed version of z0. These operations are integral to the latent diffusion model [9]. The diffusion process constitutes two steps: the forward step incrementally adds zero-mean white Gaussian noise with time-varying variance to the latent vector z according to discrete-time t, zt = √αtz0 + √ 1 −αtϵ with ϵ ∼N(0, I), (1) α1:T represents a variance schedule for t drawn from the interval [1, T]. The variance schedule can be different, such as linear or cosine quadratic [28]. The backward step is an iterative process to remove Diffusion process is rigorously defined as a continuous-time stochastic differential equation, but in practice often implemented with discrete-time updates. 3 the noise from the data progressively. Using the same variance schedule α1:T as in the forward step, a noise schedule σ1:T and a parameterized noise prediction network ϵθ with coefficients cpred = √αt−1, cdir = p 1 −αt−1 −σ2 t , and cnoise = σt, the backward step corresponds to the following process: zt−1 = cpredfθ(zt, t) \| {z } predicting ¯z0 + cdirϵθ(zt, t) \| {z } adjust along zt + cnoiseϵt \| {z } random noise with ϵt ∼N(0, I) (2) The noise schedule σ1:T comprises hyperparameters requiring careful selection based on factors like image dimensions or desired performance [29][30]. In the framework of Denoising Diffusion Implicit Models (DDIM) [31], the function fθ is employed for the prediction and reconstruction of ¯z0, based on the input zt. Specifically, we have ¯z0 = fθ(zt, t) = 1 √αt zt − √1−αt √αt ϵθ(zt, t). Consistency Models for Inversion-free Image Editing. Consistency models [32, 33] have been introduced to expedite the generation process through a consistent distillation approach. These models exhibit a self-consistency property, ensuring that samples along the same trajectory map to the same initial point. Specifically, the function fθ is rendered self-consistent by satisfying fθ(zt, t) = z0 for a given sample zt at timestep t. As a result, the self-consistency property yields a closed-form solution for the noise predictor ϵθ. We denote this particular ϵθ as ϵcons, which is derived as ϵcons = zt−√αtz0 √1−αt . Since ϵcons is not parameterized and contains the ground-truth z0, Xu et al. [2] propose starting directly with random noise, i.e., zT ∼N(0, I), at the last time-step T, which is particularly advantageous for image-editing tasks as it eliminates the need for inversion from z0 to zT . Therefore, starting with zτ = zT ∼N(0, I), the sampling process proceeds as follows: 1 z = zτ −√1−ατ ϵcons τ √ατ . Where, ϵcons τ is given by zτ −√αtz0 √1−αt 2 Noise is added to zτ, i.e, zτ = √ατz + √1 −ατϵ where ϵ ∼N(0, I) After many iterations, the final output is z. Furthermore, [2] demonstrates that the dual-branch paradigm (involving a source and a target branch) used in image editing tasks can be executed in an inversion-free manner. We will delve into this, along with our method description, in Section 4.2.2. 4 Multi-Aspect Image Editing 4.1 Problem Definition The input to the multi-aspect image editing task includes a source image (Isrc), the source prompt, and a set of edits to be applied to the source image, indicating the changes from the source prompt to target prompt. A text prompt (whether source or target) comprises several independent tokens, of which only a subset is editable. We refer to these editable tokens as Aspects. Definition 4.1 (Aspect). We define an ith aspect Ai src in the source prompt (or the jth aspect Aj edt in the target prompt) as any entity that can be substituted, deleted, or inserted into the text prompt, resulting in a meaningful sentence structure. Several examples of tokens corresponding to aspects or not are given in Fig. 3. In other words, aspects correspond to single or multiple tokens representing object color, pose, material, content, background, image style, etc. An editing operation Ei→j between the editing pair (Ai src, Aj edt) as Ei→j ∈{⊗, ⊕, ⊖, ⊘}. Here, ⊗denotes a swap action, ⊕denotes an object addition action, ⊖ denotes object deletion, and ⊘indicates no change in the aspect. Such an editing operation can be inferred directly by appropriately mapping the source and target prompts, or it can be provided as metadata [3, 34]. The editing task is considered successful if the edited source image, Iedt, reflects the required edits while preserving the unaffected aspects of the original image. 4.2 Method Figure 2 outlines the overall pipeline of our method, which has three steps. In the first step (Sec. 4.2.1), we perform aspect grouping using attention maps generated by running a few iterations of the diffusion process. The aspects in the source image are put into up to N groups, each processed by a distinct branch. The second step (Sec. 4.2.2) demonstrates how each branch, which receives a specific group of aspects, performs inversion-free editing. In the last step (Sec. 4.2.3), we perform the necessary adjustments for enabling cross-branch interaction and elucidate the significance of such interaction. 4 K Q V Q K V Non-Rigid Editing sailboat evening glow ducks Aspect Grouping ducks evening glow Source prompt: a man sitting in a boat is silhouetted against the sunset with mountain in the background Target prompt: a man sitting in a sailboat is silhouetted against the evening glow with ducks on the water and bridge in the background Source Branch Global Editing Q K V K Q V Q K V K Q Q K V Q K V Edited image Q K V Source image Attention maps Branch 3 Self Attention Q K V Q K V Cross Attention bridge man man Branch 4 K Q Q K V V V Q K V Q K saiboat K Q Rigid Editing K V bridge Branch 1 Branch 2 Branch 5 Figure 2: Pipeline. Our method, ParallelEdits, takes a source image, source prompt, and target prompt as input and produces an edited image. The target prompt specifies the edits needed in the source image. Attention maps for all edited aspects are first collected. Aspect Grouping (see Section 4.2.1) categorizes each aspect into one of N groups (in the above figure, N = 5). Each group is then assigned a branch and the branch-level updates are detailed in Section 4.2.2. Each branch can be viewed either as a rigid editing branch, non-rigid editing branch, or global editing branch. Finally, adjustments to query/key/value at the self-attention and cross-attention layers are made, as illustrated in the figure and described in Section 4.2.3. 4.2.1 Aspect Grouping sailboat evening glow boat mountain man sunset Source prompt: Target prompt: in the background Branch 1 Branch 2 Branch 4 Branch 5 Global a man sitting in a boat is silhouetted against the sunset with mountain with ducks on the water and bridge in the background ducks < > Branch 3 < > bridge a man sitting in a sailboat is silhouetted against the evening glow Rigid Rigid Non-Rigid Non-Rigid Figure 3: Aspects and Aspect Grouping. In a text prompt, there are multiple independent tokens, with only some being editable, known as aspects and are underlined in the above example. These aspects can be added, deleted, or swapped between the source and target prompts. Pairs of source and target aspects are grouped into branches, and the methodology for aspect grouping is explained in Section 4.2.1. We would like to group aspects in a prompt into N distinct groups using the cross-attention maps of the diffusion UNet [35] to characterize the spatial layouts as in previous studies [36]. Given an editing operation Ei→j between the source aspect Ai src and the target aspect Aj edt, we obtain the corresponding attention maps from both the source and target prompts as ¯ Mi src and ¯ Mj edt, respectively. The attention map M is defined by the query feature ˆQ and key feature ˆK from the cross-attention as M = softmax ˆ Q ˆ KT √ d . The binarized attention map ¯ M is obtained by normalizing M and thresholding its values. Our aspect grouping proceeds in two steps, Step 1. Assign a type for every editing operation (Ei→j). We consider three possible types of edits, in line with previous works [4], namely a global edit, a local rigid edit or a local non-rigid edit. Rigid local edits, such as changing an object’s color or texture, do not alter the layout of objects. Conversely, non-rigid local edits modify the layout of objects, such as adding or deleting objects or changing object poses. Global edits affect background and style changes. The type assignment for the editing operation (Ei→j) is determined by the following rules: type(Ei→j) =      global edit ....................................................... γ( ¯ Mj edt) ≥βγ P{ ¯ Medt} non-rigid edit ϕ( ¯ Mi src, ¯ Mj edt) < λ rigid edit ϕ( ¯ Mi src, ¯ Mj edt) ≥λ ) local edit ... γ( ¯ Mj edt) < βγ P{ ¯ Medt} (3) Here, ϕ represent mIoU [37], while γ indicates the alpha mattes of attention maps. λ and β are tunable hyperparameters. For further details, please refer to the supplementary Sec. D. 5 Step 2. Categorize every editing operation (Ei→j) into N groups. For each editing operation (Ei→j) of a specific type, we assess whether ϕ( ¯ Mj edt, ¯ Mk edt) ≥λ to determine if there exists substantial overlap between any pair of attention maps of that type. If significant overlap is detected, the attention maps are grouped together. On the other hand, if attention maps are isolated like the "boat" and "mountain" in Fig. 3 are categorized into separate groups due to small overall. Therefore, we have a total of N groups. Each group has a dedicated branch, resulting in a total of N > 2 branches. 4.2.2 Inversion-Free Multi-Branch Editing We use a set of N branches indexed by n. These N branches are in addition to a source branch (also shown in Figure 2) that undergoes a DDCM sampling process [2]. The nth branch is calibrated to its (n −1)th branch, and the first branch is calibrated to the source branch. The N−way target branch calibration can occur simultaneously, saving significant compute time. For the DDCM sampling process of the nth branch, it has the form of Section 3, Step 1 : Updating nth branch z }\| { z(n)edt \| {z } edited latent = z(n)edt τ \| {z } noisy latent − √ 1 −ατ ϵ(n)edt τ −ϵ(n −1)edt τ \| {z } parameterized noise + ϵ(n)cons τ \| {z } consistency noise /√ατ (4) Let us break down Eq. 4 step by step. n = 1 representing the source branch, we have z(1)edt = zsrc and ϵ(1)edt τ = ϵsrc τ . Also, z(1)edt τ = zsrc τ , which at time step τ = τ1, is random noise drawn from N(0, I). Similarly, when n = N, z(N)edt represents the final calibrated/edited image containing all the required aspect edits after repeating for τ ∈{τ1, τ2, . . . τT } timesteps. The noise addition on any target branch remains the same as Step 2 , i.e., z(n)edt τ = √ατz(n)edt + √1 −ατϵ where ϵ ∼N(0, I). For 1 < n < N, we have ϵ(n)edt τ = ϵθ(z(n)edt τ , τ), where ϵθ represents a parameterized noise predictor network (details in the Appendix Sec. D). A key observation is that the difference in the parameterized noise at the nth branch and (n −1)th branch is utilized to calculate z(n)edt in (4). Finally, ϵ(n)cons τ is defined by ϵ(n)cons τ = (z(n)edt τ −√ατ ˆz(n −1)edt)/√1 −ατ. Unlike the dual-branch setup in [2], the reference initial input is the estimated latent from the previous branch at a previous diffusion denoising iteration as indicated by ˆz(n −1)edt. 4.2.3 Cross-Branch Interactions For rigid local branches, the cross-attention map Mi n from the previous branch is either switched or injected into the current branch, akin to the method used in P2P [3]. This approach facilitates local edits while preserving structural consistency. For non-rigid local branches, we observe that the query features in the shallow layers of UNet [35] can effectively query correlated local contents and textures from the prior branch’s latent features, ensuring consistency. Consequently, the key and value features from the prior branch are retained in the current branch to maintain consistent editing. We use a non-rigid editing branch to manage non-rigid local edits. In the current branch n, textures from the previous branch (n −1) are preserved by replacing the Kn−1 and Vn−1 features from the last branch with the Kn and Vn features in the current branch. Only the query features are preserved to maintain layout semantic correspondence. Additionally, the attention mask Mn−1 from the previous branch’s cross-attention layer is used to guide the editing process by adding it to Mn, thereby converting the object layout from Mn−1 to Mn. This step is crucial for object removal or shape modification edits, where the object mask is derived from the previous branch. For all global branches, there is no replacement of attention features or masks, and the attention mask is not used to guide the editing process, as the entire image is intended to be altered. 5 Experiments PIE-Bench++ Dataset. We introduce a new dataset, PIE-Bench++, derived from PIE-Bench [1] and dedicated to evaluate the performance of multi-aspect image editing. The PIE-Bench dataset contains 700 images and prompts with single-aspect editing including object-level manipulations (addition, 6 A Dalmatian dog is lying on the concrete in front of red flowers A Dalmatian dog with a collar is lying on the snow covered concrete in front of pink tulips bread on a wooden table with tomatoes and a napkin chocolate bread on a wooden table with strawberries, kiwi, and a napkin A grey horse walks through a field A chestnut horse with a harness walks through a field of flowers chocolate strawberry kiwi pink tulips snowcovered collar a painting of a woman in green blouse and jewelry a painting of a woman in green dress and jewelry blouse jewelry harness flowers chestnut a cartoon dog wearing sunglasses in a white background scarf eyeglasses a cartoon dog wearing eyeglasses and scarf in a white background bamboo bonsai plant and notebook on white table money tree plant and word 'Hello, World!' is written on the notebook on black table black table 'Hello, World!' money tree Figure 4: Qualitative results of ParallelEdits. We denote the edits in arrows with edit actions and aspects for each pair of images. The last image pair is a failure case of ParallelEdits. deletion, or alteration), attribute-level manipulations (changes in content, pose, color, and material), and image-level manipulations that modify background and overall style. Our PIE-Bench++ extends PIE-Bench by enabling multi-aspect edits: 57% of our dataset have two aspect edits per prompt, 19% have more than two edits, and the remaining 24% have a signle aspect edit. For additional details and examples of the PIE-Bench++ dataset, please refer to the supplementary material. Evaluation Metrics. We introduce two new metrics designed for evaluating multi-aspect text-driven image editing, alongside standard evaluation metrics. (a) Aspect Accuracy-LLaVA. Drawing inspiration from the remarkable capability of large vision language models in comprehending intricate semantics within images, we propose to innovatively leverage them as an “omniscient” agent equipped with extensive knowledge to understand various attributes of images. We use the LLaVA [38] model, trained on visual grounding tasks, to evaluate the accuracy of multi-aspect image editing. Given a text prompt with multiple aspects, such as “A [pink] [taxi] with [colorful] [flowers] on top”, we provide the following prompt with the edited image to the LLaVA model: “Does the image match the elements in [ ]: A [pink] [taxi] with [colorful] [flowers] on top? Return a list of numbers where 1 is matched and 0 is unmatched.” We then parse the returned list and compute its average to determine the aspect accuracy. We name this new evaluation metric as AspAcc-LLaVA. Examples and detailed explanations of this evaluation metric are available in the supplementary material. (b) Aspect Accuracy-CLIP. We also use the similarity of the CLIP [39] to evaluate if an attribute has been successfully edited. Given an edited image Iedt and the target prompt Pedt with k edited aspects Aedt, every time we remove an aspect Aj edt from Pedt and revert it back to Ai src as ˆPedt. We then extract the CLIP [39] similarity between the edited image Iedt and two prompts, i.e., s1 = CLIP(Iedt, Pedt) and s2 = CLIP(Iedt, ˆPedt). We expect s1 > s2 if the aspect Aj edt has been successfully edited. Thus, the aspect accuracy is ks k when a total of ks aspects have been successfully edited among k target edits. Note that in the case of an edited or added object that also involves changes in attributes (such as color or material), we consider it a successful edit only if both the object and its attributes have been successfully modified. We name this metric as AspAcc-CLIP. (c) Standard Metrics. Several standard metrics widely used for evaluating text-image similarity and image quality are considered, including PSNR, LPIPS [40], MSE, and SSIM [41]. We also use the CLIP [39] score to measure the image-text alignment performance. Additionally, the bi-directional CLIP (D-CLIP) score [42] is reported, which is formulated as follows: cos⟨CLIPimg (Iedt) −CLIPimg (Isrc), CLIPtext (Pedt) −CLIPtext(Psrc)⟩ 5.1 Quantitative Results We first conduct experiment on the PIE-Bench++ dataset to compare our method with the state-of-theart text-driven image editing methods combining their corresponding inversion method leads to best 7 Source Image Null-text Inversion DirectInversion InfEdit GPT-4V PnP* Source Image MasaCtrl Rich-text✭✭ GPT-4V InfEdit "A German Shepherd stands on the grass wearing a collar " "A German Shepherd stands on the grass with its mouth open and wears a leather collar among autumn leaves" Null-text Inversion "a man walking dog in a city with snow mountains in the background" Source Image Null-text Inversion DirectInversion InfEdit* Source Image MasaCtrl Rich-text✭✭ GPT-4V InfEdit "a man walking in a town with mountains in the background" "A bullfinch stands on a mossy branch against a blurred background" "A yellow bullfinch stands on flowers against a starry night background" GPT-4V "white dumplings on brown wooden bowl" "white cupcakes on black metal bowl" ParallelEdits (Ours) PnP* P2P P2P ParallelEdits (Ours) ParallelEdits (Ours) ParallelEdits (Ours) Figure 5: Qualitative results comparison. Current methods fail to edit multiple aspects effectively, even using sequential edits (noted as ). Methods marked with ⋆⋆taking additional inputs other than source image and plain text. StyleD MasaCtrl P2P DI NTI InfEdit PnP DI P2P* InfEdit* PnP* Ours CLIP (%) ↑ 24.02 23.37 24.00 24.40 24.03 24.44 24.90 22.80 25.13 25.17 25.39 25.70 D-CLIP (%) ↑ 8.43 7.68 11.43 13.23 12.08 11.02 11.83 2.74 8.30 11.77 11.85 20.70 Eff. (secs/sample) ↓ 382.98 12.70 33.72 29.70 145.29 2.22 32.51 100.98 121.32 11.82 122.81 4.98 AspAcc-CLIP (%) ↑ 32.37 34.05 26.14 31.95 42.19 42.38 44.91 28.23 38.96 42.38 48.20 51.05 AspAcc-LLaVA (%)↑53.79 55.79 55.04 54.42 59.80 60.55 61.36 46.24 55.21 61.90 63.80 65.19 Table 1: Comparison results in multi-aspect image editing on the PIE-Bench++ dataset. Computational efficiency is abbreviated as Eff., and * denotes the method using sequential editing. The best performance is highlighted in bold and the second best performance is underlined. performance, including DDIM+MasaCtrl [4], DDIM+Prompt-to-Prompt (P2P) [3], DDIM+Plug-andPlay (PnP) [21], StyleDiffusion (StyleD) [43]+P2P, Null-text Inversion (NTI) [34]+P2P, DirectInverison (DI)[1]+PnP, and InfEdit [2]. An intuitive way to improve off-the-shelf image editing methods is to apply the single-aspect editing method sequentially. We follow [27] to adapt existing image editing methods into sequential editing processes, where these methods are applied multiple times to achieve multi-aspect editing. Each time, only one aspect is edited. Table 1 presents the metrics in terms of text-image similarity (i.e., CILP and D-CLIP scores), computational efficiency, and aspect accuracy. Our ParallelEdits model outperforms all baselines in editing effectiveness, with a slightly longer runtime than the InfEdit model. Even though sequential editing better aligns the target prompt than their vanilla methods, it significantly increases computational overhead and may propagate editing errors over time. Moreover, although the sequential editing is conducted in the latent space, it would introduce more noise and artifacts to the edited image. Hence, their performance in all editing quality metrics was inferior to our method. 5.2 Qualitative Results Fig. 4 presents several examples of our method’s multi-aspect editing on the PIE-Bench++ dataset. The results demonstrate the effectiveness of our method in handling multiple and varied types of 8 PIE-Bench++ Ours DirectInversion P2P InfEdit Ours DirectInversion P2P InfEdit PIE-Bench++ PIE-Bench PIE-Bench Figure 6: Comparison across different numbers of editing aspects. We also include the comparison in PIE-Bench dataset. Our proposed method is robust to different numbers of editing aspects. Background Preservation Aspect Preservation% Methods PSNR↑ LPIPS×103↓ MSE×104 ↓ SSIM×102 ↑ CLIP↑ LLaVA ↑ P2P [3] 18.48 / 16.64 188.26 / 231.83 190.07 / 345.07 73.55 / 69.17 20.72 / 23.48 66.59 / 72.60 PnP [21] 22.73 / 21.54 103.16 / 120.87 75.97 / 102.47 80.73 / 78.85 20.79 / 25.59 75.65 / 78.77 InfEdit [2] 24.61 / 24.09 103.99 / 107.43 160.54 / 163.72 78.85 / 79.64 24.69 / 25.04 75.90 / 78.05 Ours 26.13 95.87 113.86 82.35 25.49 80.70 Table 2: Comparison results in terms of background and aspects preservation. The results from sequential editing is noted as green. ParallelEdits achieves state-of-the-art performance on multi-aspect editing while preserving the background and content consistency. edits across diverse image content. Fig. 5 further compares our method with several state-of-the-art models and one popular multi-modal large language model, GPT-4V [44], by providing the source image, source prompt, and target prompt to guide the image editing. The Rich-text [25] model differs from other models, which uses rich-text prompt to edit the image generated from the plain (source) text prompt. The results show that current image editing models even with sequential editing fail to edit multiple aspects, while multi-modal large language models fail to preserve the content of source image. Our method achieves visually convincing results by successfully editing different attributes with good content preservation. 5.3 Ablation Study and Analysis (a) Impact of Editing Aspect Number. We first examine the performance of our ParallelEdits and baseline methods on various editing aspect numbers by comparing CLIP and LLaVA-based aspect accuracies on the original PIE-Bench [1] and our PIE-Bench++ datasets. The bar charts in Fig. 6 show the outstanding performance of our method across all settings, including single-aspect editing on two datasets and multi-aspect editing. Takeaway: the proposed ParallelEdits demonstrates robustness across varying numbers of editing aspects. (b) Evaluation on Perservation. We follow [1] to evaluate the background preservation. We first use the PSNR, LPIPS [40], MSE and SSIM [41] to evaluate the background preservation. We measure that metric on a subset of images of our proposed PIE-Bench++ dataset where the background can be well defined in that image, e.g., no image style or background editing, and the background is visible after aspect editing. The results are shown in Table 2, where we compare our method with the top performance methods in Table 1. Moreover, we adopt the similar way as calculating the AspAcc-LLaVA to prompt LLaVA [38] for evaluating how the unchanged aspect preserves in the edited image. We also calculate the CLIP [39] score between the target image and the text prompt after removing all edited aspects. The results are reported in Table 2 noted as CLIP and LLaVA, respectively. Takeaway: preservation is even maintained in ParallelEdits. (c) Branches numbers and aspect grouping. To demonstrate the effectiveness of our multi-branch design and early aspect grouping, we design additional ablation studies for our method in threefold. (1) We only use one single non-rigid branch to conduct all edits; (2) we remove the aspect categorization process from the pipeline and use the same non-rigid branch for each edit; (3) we adopt one single branch for different type of edits without using any auxillary branches which results a total of three branches (also see Section B for more details). Takeaway: As shown in Table 3, the multi-branch design and aspect grouping play a significant role in enhancing the performance of our proposed ParallelEdits. 9 with aspect with aspect with auxillary Similarity % Aspect Accuracy % categorization grouping branch CLIP↑D-CLIP↑CLIP↑ LLaVA ↑ ParallelEdits × × × 24.32 10.45 40.97 57.67 × ✓ ✓ 25.14 11.97 46.66 58.37 ✓ × × 24.50 12.33 48.08 61.22 ✓ ✓ ✓ 25.70 20.70 51.05 65.19 Table 3: Ablation studies on branch numbers and aspect grouping. Change Add Delete Asepct Acc-CLIP Object Content Pose Color Material Background Style Object Object P2P [3] 33.13 20.00 25.83 34.17 31.67 30.63 19.38 22.29 11.88 MasaCtrl [4] 40.83 23.75 40.83 20.00 30.83 26.88 29.38 37.08 28.96 NTI [45] 48.13 41.25 23.75 51.25 24.17 51.25 22.50 40.42 32.08 DirectInversion [1] 40.63 26.25 23.33 40.00 25.42 32.50 25.00 30.00 20.83 InfEdit [2] 36.24 33.33 25.41 41.67 27.50 48.75 41.88 50.63 45.41 PnP [21] 44.38 27.29 27.91 49.17 32.91 52.50 55.63 44.38 42.08 ParallelEdits 51.46 44.16 39.58 60.00 47.50 60.00 50.00 56.04 52.08 Table 4: Comparison on each category in PIE-Bench++. Our ParallelEdits achieves the best performance on most of the categories from the dataset. (d) Performance comparison on each category. Recall that our dataset includes nine different categories for editing. We compare the performance of baseline models and our approach across the nine categories, as presented in Table 4. Takeaway: Our proposed ParallelEdits achieves state-of-theart performance across most categories. Limitations and Failure Cases. The proposed ParallelEdits has several limitations. First, it cannot handle the text editing in the image, as shown in the last image pair of Fig. 4. Second, ParallelEdits fails to edit dramatic background changes, as examples shown in the supplementary material. 6 Conclusion In this work, we propose a new research task, multi-aspect text-driven image editing, to modify multiple object types, attributes, and relationships. We introduce a dedicated method, ParallelEdits, to multi-aspect text-driven image editing as an effective and efficient solution to this problem. Due to the lack of evaluation benchmark, we introduce PIE-Bench++, an improved version of PIE-Bench [1] tailored for simultaneous multiple-aspect edits within images. ParallelEdits achieves better quality and performance than existing methods on proposed PIE-Bench++. Our work introduces ParallelEdits, a novel approach that adeptly handles multiple attribute edits simultaneously, preserving the quality of edits across single and multiple attributes through a unique attention grouping mechanism without adding computational complexity. There are several future works we would like to explore. First, different aspects of an image have a specific semantic order. Editing these aspects according to their intrinsic order will simplify the editing process. Secondly, the current ParallelEdits still has limitations, as shown in Fig. 4. It will be of interest to study approaches to improve these aspects. Ethics Statement. In anticipation of contributing to the academic community, we plan to make the dataset and associated code publicly available for research. Nonetheless, we acknowledge the potential for misuse, particularly by those aiming to generate misinformation using our methodology. We will release our code under an open-source license with explicit stipulations to mitigate this risk. These conditions will prohibit the distribution of harmful, offensive, or dehumanizing content or negatively representing individuals, their environments, cultures, religions, and so forth through the use of our model weights. Acknowledgement. This work was supported in part by the National Science Foundation (NSF) Projects under grants SaTC-2153112, No.1822190, and TIP-2137871. Prof. Lokhande thanks support provided by University at Buffalo Startup funds. We thank Sudhir Kumar Yarram for the insightful discussions on the project. 10 References [1] Xuan Ju, Ailing Zeng, Yuxuan Bian, Shaoteng Liu, and Qiang Xu. Direct inversion: Boosting diffusion-based editing with 3 lines of code. 2023. [2] Sihan Xu, Yidong Huang, Jiayi Pan, Ziqiao Ma, and Joyce Chai. Inversion-free image editing with natural language. arXiv preprint arXiv:2312.04965, 2023. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10748–10757, 2022. 13 Appendix A ParallelEdits: The Algorithm In this section we provide Algorithm 1: Early Aspect Grouping and Algorithm 2: ParallelEdits on a particular branch. These algorithms describe the overall idea behind ParallelEdits. They are also pictorially illustrated in Figures 2 and 3 of the main paper. Let us denote an arbitrary branch and the timestep in the diffusion process by n and t respectively. Firstly, in Algorithm 1, we demonstrate how Early Aspect Grouping is conducted over the attention maps. Recall that we refer to this as “early" aspect grouping because only a few steps (maximum of 5) are sufficient to perform the grouping. This phase of ParallelEdits takes as an input, the edit action set {Ei→j} and the corresponding cross-attention maps for every token Aj src, and outputs the grouped edit actions set ¯ Ac edt. Recall from Section 4 of the paper that Ei→j ∈{⊗, ⊕, ⊖, ⊘}, with ⊗denoting a swap action, ⊕denoting an add action, ⊖denoting aspect deletion, and ⊘indicating no change in the aspect. Once grouped edit actions set is computed, it is fed into Algorithm 1 to conduct multi-aspect editing and obtain the edited latent features. In Algorithm 2, we implement several operations on the attention masks, similar to the P2P method [3], and describe them as follows. Replace: Swapping token attention mask Mn−1 in the prompt from previous branch, overriding Mn; Refine: Injecting only the attention mask that corresponds to the unchanged part of the prompt from Mn−1 to Mn; Retain: Keeping the attention mask Mn unchanged. Algorithm 1 Early Aspect Grouping Input: Edit action set {Ei→j}, Cross attention maps {M} 1: rigid-edit ←{}, non-rigid-edit ←{}, global-edit ←{} 2: for Ai→j edt ∈{Ei→j} do 3: if γ( ¯ Mj edt) ≥βγ(P{ ¯ Medt}) then ▷This is a global edit 4: global-edit ←global-edit + {Ei→j} 5: else if ϕ( ¯ Mi src, ¯ Mj edt) < θ then ▷This is a rigid edit 6: for ¯ Ac edt ∈rigid-edit do 7: if mIoU( ¯ Ac edt, Ei→j ≥θ) then ▷¯ Ac edt is a set of grouped edit actions 8: ¯ Ac edt ←¯ Ac edt + Ei→j 9: else 10: rigid-edit ←rigid-edit + Ei→j 11: end if 12: end for 13: else if ϕ( ¯ Mi src, ¯ Mj edt) ≥θ then ▷This is a non-rigid edit 14: for ¯ Ac edt ∈non-rigid-edit do 15: if mIoU( ¯ Ac edt, Ei→j ≥θ) then 16: ¯ Ac edt ←¯ Ac edt + Ei→j 17: else 18: non-rigid-edit ←non-rigid-edit + Ei→j 19: end if 20: end for 21: end if 22: end for Output: Grouped edit actions set { ¯ Ac edt} B Some More Details on ParallelEdits In the literature [4, 3], image editing processes have been conducted through the implementation of a dual-branch approach. This method involves utilizing a source and target branches for editing. 14 Algorithm 2 ParallelEdits on a Particular Branch Input: Denoising UNet εθ, Grouped edit action ¯ Ac edt, ▷Output from early aspect grouping Latent feature in previous branch and previous timestep zt n−1, zt−1 n , Cross attention maps {M}, Self attention features Qn−1, Kn−1, Vn−1, Edit type list: rigid-edit, non-rigid-edit, global-edit 1: Mn ←εθ( ¯ Ac edt, zt−1 n , t −1) 2: if ¯ Ac edt ∈global-edit then ▷This is a global edit 3: retain(Mn) ▷Do not switch attention maps for global edits 4: else if ¯ Ac edt ∈non-rigid-edit then ▷This is a non-rigid edit 5: replace(Mn−1, Mn ) 6: else if ¯ Ac edt ∈rigid-edit then ▷This is a rigid edit 7: {Qn, Kn, Vn} ←{Qn, Kn−1, Vn−1} 8: refine(Mn−1, Mn ) 9: end if 10: ¯ Mn ←binarize(Pm≤n m=0 Mm) 11: zt n ←¯ Mn ⊙zt n + (1 −¯ Mn) ⊙zt n−1 Output: Latent feature zt n Specifically, the source branch is reverted to z0, while the trajectory of the target branch is iteratively adjusted. By computing the distance from the source branch and ϵcons with Latent Consistency Model [32], the target branch is calibrated at each time step. Our experiments, as seen in Section 5 of the main paper, show the ineffectiveness of a dual-branch procedure for multi-aspect editing tasks. Specifically, a single target branch is inadequate, leading to imperfection in the target image. Thereby we advocate multi-aspect editing through the use of multiple target branches. Each target branch handles a group of aspects, with simpler aspects such as non-rigid local edits directed to initial branches, and more complex aspects such as rigid local edits deferred to subsequent ones. Note that however, all the branches operate simultaneously. Auxiliary Rigid / Non-Rigid Branches. In the main paper, it was noted that there was one dedicated branch for each type of edit: non-rigid, rigid, and global edit. The Early Aspect Grouping algorithm 1 classifies aspects into these three categories. Our experiments revealed that sometimes, due to low overlap between attention maps, aspects may not always be grouped into dedicated rigid or non-rigid branches. In such cases, it becomes necessary to include an auxiliary branch to handle the ungrouped aspects. Therefore, ParallelEdits may involve a single rigid branch and additional auxiliary branches to manage ungrouped aspects, and similarly, a single non-rigid branch and supplementary auxiliary branches to address ungrouped aspects. An ablation study on auxiliary branches is provided in Table 3. C More Details on Evaluation Metrics In this section, we describe more details of our evaluation metrics. LLaVA aspect accuracy. We show how we leverage LLaVA [38] to evaluate the multi-aspect editing accuracy in Fig. 7 and Fig. 8, we also prompt LLaVA [38] for explanation to human readers. LLaVA [38] could provide detailed summary for the image and also explanations for the mismatched between edited aspects and image. Other evaluation metrics. Moreover, even though the Structure Distance [46] has been used in PIE-Bench [1] to evaluate the structure between source and target image while ignoring appearance information, it could not serve as a good evaluation metric for multi-aspect editing. This is due to the fact that the structure of multi-aspect edited target image may necessitate substantial modifications, particularly when it involves adding or removing multiple objects. 15 Does this image match the elements in []: "a [sailboat] is silhouetted against the [evening glow] with [snow] mountain in the backgrounds, [winter scene]. " Exactly follow the return format as a list where 1 is matched and 0 is unmatched, return list only. The list length is equal to 4. [1,1,1,1] Please provide some explanation. The image matches the elements in the description as follows: 1. A sailboat is present in the image, which is silhouetted against the evening glow. 2. The evening glow is visible in the sky, creating a beautiful backdrop for the sailboat. 3. The mountains in the background are covered with snow, which is consistent with the description. 4. The overall scene is a winter scene, as indicated by the snow-covered mountains and the calm, cold appearance of the water. Therefore, all elements in the description are matched by the image, resulting in a list of 4 ones. 🌋 LLaVA 🌋 LLaVA User Does this image match the elements in []: "a man [standing] in a boat is silhouetted against the sunset and [ducks] on the water with [Alps mountain] in the background" Exactly follow the return format as a list where 1 is matched and 0 is unmatched, return list only. The list length is equal to 4. [1,1,1,1] Please provide some explanation. The image matches all the elements in the given description: 1. A man is standing in a boat. 2. The man is silhouetted against the sunset. 3. There are ducks on the water. 4. The background features a mountain range, which could be described as Alps-like. Therefore, the elements in the description are all present in the image, and the list would be [1, 1, 1, 1]. 🌋 LLaVA 🌋 LLaVA Does this image match the elements in []: "a [boat] in a lake with a [house] and [ice mountain] in the background" Exactly follow the return format as a list where 1 is matched and 0 is unmatched, return list only. The list length is equal to 3. [1,1,1] Please provide some explanation. The image shows a boat on a lake with a house and a mountain in the background. The mountain appears to be covered with snow, which could be described as an "ice mountain." The elements in the brackets match the description provided: A boat is present in the image. A house is visible in the background. The mountain in the background is covered with snow, which could be described as an "ice mountain." Therefore, the elements in the brackets match the description, and the list would be [1, 1, 1]. 🌋 LLaVA 🌋 LLaVA Figure 7: Examples of prompting LLaVA for aspect accuracy measurement in cases of successful editing. LLaVA can effectively illustrate if and how the target image and edits are misaligned. 16 Does this image match the elements in []: "[money tree plant] and [word 'Hello, World!'] is written on the notebook on [black table]" Exactly follow the return format as a list where 1 is matched and 0 is unmatched, return list only. The list length is equal to 3. [1,0,0] Please provide some explanation. Certainly! Here's an explanation for the elements in the image and their match with the elements in the provided format: 1, "[money tree plant]" - The image shows a plant that is commonly known as a money tree. It is a type of bonsai tree with a twisted trunk and glossy green leaves. So, this element is matched. 2, "[word 'Hello, World!'] is written on the notebook" - The notebook in the image has the word "Hello, World!" written on it. However, the format specifies that the word should be written on the notebook, which is not the case here. The word is actually written on a whiteboard or a similar surface that is placed in front of the notebook. Therefore, this element is unmatched. 3, "[black table]" - The table in the image is not black; it is a light color, possibly gray or beige. So, this element is unmatched. Therefore, the list of matches would be [1, 0, 0]. 🌋 LLaVA 🌋 LLaVA Does this image match the elements in []: "a logo of [X] [with pink color] in a [light background]" Exactly follow the return format as a list where 1 is matched and 0 is unmatched, return list only. The list length is equal to 3. [1,1,0] Please provide some explanation. Certainly! The image you've provided features a logo that is composed of an X shape, which matches the first element in the description. The X is colored in a shade of pink, which matches the second element in the description. However, the background of the image is not light; it is dark, which does not match the third element in the description. Therefore, the list of matches would be: 1, Logo is an X - Matched 2, X is pink - Matched 3, Background is light - Not Matched This results in the list [1, 1, 0]. 🌋 LLaVA 🌋 LLaVA User Figure 8: Examples of prompting LLaVA for aspect accuracy measurement in cases of unsuccessful editing. D Implementation Details Our proposed ParallelEdits is based on the Latent Consistency Model [32], with the publicly available LCM † which is finetuned from Stable Diffusion v1.5. We then follow [2] to leverage their proposed inversion-free technique in ParallelEdits for image editing. During sampling, we perform LCM sampling [32] with 15 denoising steps, and the classifier-free guidance (CFG) is set to 4.0. ParallelEdits can control the editing strength by adjusting the CFG . There’s a trade-off between achieving satisfactory inversion and robust editing ability. A higher CFG tends to produce stronger editing effects but may lower inversion results and identity preservation. We also set the hyper-parameter θ as 0.9 and β as 0.8 in our experiments, where θ, β are used to determine the edit type of a given edit action. †https://huggingface.co/SimianLuo/LCMDreamshaperv7 17 Change Add Delete Object Content Pose Color Material Background Style Object Object #Edited Aspect 302 98 120 188 99 112 165 178 119 #Edited Token 316 155 227 205 116 175 424 507 381 Table 5: Summary of Editing Types and Categories in PIE-Bench++ dataset. There are 10 different categories in PIE-Bench++ and a total number of 700 images. In the inversion-free multi-branch editing approach, for 1 < n < N, the noise estimation is also conditioned on a text conditioning cn in branch n. This can be expressed as ϵ(n)edt τ = ϵθ(z(n)edt τ , τ, cn). Here, c1 corresponds to the source prompt, cN corresponds to the target prompt, and cn represents the prompt that includes all aspect edits up to branch n. E Additional Details of PIE-Bench++ E.1 PIE-Bench++ Details Unlike existing benchmarks that primarily focus on single-aspect edits, PIE-Bench++ is tailored to multiple aspect edits, reflecting the complexities inherent in real-world editing tasks. Our enhanced dataset, PIE-Bench++, builds upon the PIE-Bench [1] by incorporating 700 images across nine diverse categories, covering both natural and artificial scenes, with a significant focus on multi-aspect editing scenarios. Specifically, the Change Object category involves swapping objects in the scene with different yet reasonable alternatives. Add Object adds new elements to the scene. Delete Object focuses on removing objects, testing the model’s ability to erase elements seamlessly. Change Object Content alters the content of specific objects, such as changing the design on a shirt or the pattern on a wall. Change Object Pose includes changes in the shape of objects, humans, or animals. Change Object Color assesses the model’s ability to apply accurate color changes. Change Object Material evaluates the rendering of different textures and materials. Change Background involves editing scenarios where there is a distinct foreground object and a main background. This type of edit focuses on seamlessly integrating new background elements while preserving the integrity of the foreground object. Change Image Style involves the application of style transfer techniques to the entire image while ensuring the original content remains intact. For example, this could involve transforming a photograph to adopt a cartoon style. Each category is carefully curated to provide a comprehensive evaluation of the dataset’s multi-aspect editing capabilities, the summary of the dataset is shown in Table 5. E.2 Dataset Annotation The annotation process involves a primary annotator who labels the source prompt, describing the original image, and the target prompt, which outlines the desired modifications to generate the target image. The target prompt is carefully annotated to include all editing pairs expected to be reflected in the target image. Subsequently, a second annotator reviews the annotations for accuracy and consistency, ensuring the reliability of the dataset. The majority of target prompts in PIE-Bench++ feature at least two edited aspects. Nevertheless, within the categories that solely changing background and image styles, the number of edits is usually constrained to one or two aspects. This limitation is due to the intrinsic characteristics of these attributes, such as each image having only one background or style. Annotation format details. Each image in the dataset annotation is associated with key elements as shown in Fig. 9: a source prompt, a target prompt, an edit action, and a mapping of aspects. The edit action specifies the position index in the source prompt where changes are to be made, the type of edit to be applied, and the operation required to achieve the desired outcome. The aspect mapping connects objects undergoing editing to their respective modified attributes, enabling the identification of which objects are subject to editing. 18 "source_prompt": "a colorful bird standing on a branch", "edit_action": {"owl":{"position":2,"edit_type":1,"action":"bird"}, "brown":{"position":1,"edit_type":6,"action":"colorful"}, "flower":{"position":6,"edit_type":1,"action":"branch"}, "red":{"position":6,"edit_type":6,"action":"+"}}, "aspect_mapping": {"owl":["brown"],"flower":["red"]} Source Image "target_prompt": "a brown owl standing on a red flower", Text-based annotation "source_prompt": "a round cake with orange frosting on a wooden plate", "edit_action": {"square":{"position":1,"edit_type":4,"action":"bird"}, "strawberry frosting:{"position":4,"edit_type":6,"action":"orange frosting"}, "plastic":{"position":8,"edit_type":7,"action":"wooden"}}, "aspect_mapping": {"cake":["square","strawberry frosting"], "plate":["plastic"]} "target_prompt": "a square cake with strawberry frosting on a plastic plate", "source_prompt": "a slanted mountain bicycle on the road in front of a building", "edit_action": {"rusty":{"position":2,"edit_type":7,"action":"+"}, "motorcycle":{"position":3,"edit_type":1,"action":"bicycle"}, "fence":{"position":11,"edit_type":8,"action":"building"}, "on the road":{"position":4,"edit_type":3,"action":"-"}}, "aspect_mapping": {"motorcycle":["rusty"],"fence":["red"],"road":[]} "target_prompt": "a slanted rusty mountain motorcycle on the road in front of a fence", "source_prompt": "the galaxy over the durdle door", "edit_action": {"pink":{"position":1,"edit_type":6,"action":"+"}, "sunset":{"position":1,"edit_type":8,"action":"galaxy"}, "and rainbow":{"position":2,"edit_type":2,"action":"+"}}, "aspect_mapping": {"sunset":["pink"],"rainbow":[]} "target_prompt": "the pink sunset and rainbow over the durdle door", Figure 9: Annotation examples from PIE-Bench++. Each annotation containing a Source Prompt, Target Prompt, Edit Action, and Aspect Mapping. Edit action contains the specific instructions including the desired modification index in source prompt as position, edit type among 9 catergories and the action ∈{⊗, ⊕, ⊖}. The aspect mapping indicts the pair between object and attribute. 19 F Additional Qualitative Results We also provide more qualitative results in Fig. 10, showing the effectiveness of our proposed method in handling multi-aspect editing tasks. These examples showcase the model’s proficiency in executing intricate edits. For instance, as depicted in Fig. 10 (b), our method successfully removes a cup while accurately reconstructing the obscured parts of the lamp behind it. In Fig. 10 (a), the model demonstrates its ability to swap and add aspects, while preserving the composition of the scene. The results underscore the model’s adeptness in interpreting and executing sophisticated editing instructions, leading to visually consistent and contextually fitting edited images. Additional, we also provide the results for sequential editing methods with different editing order in Fig. 11 and Fig. 12. a bowl of strawberries and blueberries on a striped tablecloth a wooden bowl of strawberries, blueberries, and ice cream ball on a striped tablecloth with a fork fork wooden a wooden bowl of rice with a spoon in the kitchen a bowl of icecream with a spoon wooden rice kitchen illustration of a cup and a lamp on a table by a window illustration of a cup and a lamp on a table by a window with a moon outside cup moon luxury bedroom interior with marble wall and field outside luxury bedroom interior with stone wall and ocean outside a metal chair and a table sits in front of a wall with a floral pattern and a mirror a chair sits in front of a wall with a floral pattern stone wall ocean metal chair table mirror a white church sits on a hill in a field a pink house sits on a hill in a flower field with clouds on the sky clouds pink house flowers ice cream ball (a) (b) (c) (d) (e) (f) a lake with mountains in the background a boat in a lake with a house and ice mountains in the background "rustic" house, boat ice mountains a logo of bird shape in a black background a logo of X with pink color in a light background light logo X, pink color (g) (h) Figure 10: Qualitative results from ParallelEdits. ParallelEdits is able to swap, add and delete multiple aspects. The last image pair is a failure case of ParallelEdits. 20 cupcakes black bowl metal bowl Source Prompt: white dumplings on brown wooden bow Target Prompt: white cupcakes on black metal bowl white cat InfEdit Sequential InfEdit Sequential DDIM + PnP Sequential Source Prompt: A dog is laying down on a white background Target Prompt: A white cat wearing sunglasses is laying down on a white background” cat sunglasses DI + PnP Sequential Figure 11: Sequential editing using single-aspect text-driven image editing methods. The sequential editing might accumulate errors and undo previous edits. It also fails to edit significantly overlapped objects. ocean leather collar rose German Shepherd Source Prompt: A Golden Retriever holding a tulip sitting on the ground in front of fence Target Prompt: A German Shepherd wearing a leather collar holding a rose sitting on the ground in front of the ocean InfEdit Sequential rose rose German Shepherd German Shepherd ocean ocean leather collar leather collar ocean leather collar rose German Shepherd rose rose German Shepherd German Shepherd ocean ocean leather collar leather collar DDIM + PnP Sequential Figure 12: Sequential editing with different orders. Sequential editing with different orders can yield varying final results. Additionally, it may lead to error accumulation and potentially overwrite previous edits. 21 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We include the limitation and failure cases of the work in Sec. 5. 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4,473	Adversarial Moment-Matching Distillation of Large Language Models Chen Jia SI-TECH Information Technology jiachenwestlake@gmail.com Abstract Knowledge distillation (KD) has been shown to be highly effective in guiding a student model with a larger teacher model and achieving practical benefits in improving the computational and memory efficiency for large language models (LLMs). State-of-the-art KD methods for LLMs mostly rely on minimizing explicit distribution distance between teacher and student probability predictions. Instead of optimizing these mandatory behavior cloning objectives, we explore an imitation learning strategy for KD of LLMs. In particular, we minimize the imitation gap by matching the action-value moments of the teacher’s behavior from both on- and off-policy perspectives. To achieve this action-value moment-matching goal, we propose an adversarial training algorithm to jointly estimate the momentmatching distance and optimize the student policy to minimize it. Results from both task-agnostic instruction-following experiments and task-specific experiments demonstrate the effectiveness of our method and achieve new state-of-the-art performance. 1 Introduction Large language models (LLMs) like GPT-4 [1] and LLaMA [35] have revolutionized natural language processing, significantly enhancing the quality of text generation across various tasks. This success is largely due to the extensive scale of training data and the substantial increase in model parameters [19]. However, the high computational and memory requirements of these models present significant challenges for practical deployment. To address these issues, knowledge distillation (KD) [16] has emerged as a key technique. KD involves transferring knowledge from a large, complex teacher model to a smaller, more efficient student model, thereby maintaining high performance while reducing resource demands. Most distillation methods for auto-regressive text generation models, including LLMs, employ metrics of probability distribution distance, such as Kullback-Leibler (KL) divergence [20] and reverse KL divergence [14], aiming to align the token-level probability distributions between the teacher and student models. The distribution matching-based distillation methods can be viewed as behavior cloning on a decisionmaking problem from the perspective of imitation learning [24, 14, 2]. Based on this concept, early works based on the teacher-generated outputs [20] or a supervised dataset [30] can be viewed as an offpolicy approach. Recent works further incorporate an on-policy approach, training the student on its self-generated outputs [24], using KL-based divergence [14, 2, 21] and total variation (TV) distance [39]. Accordingly, such distribution matching-based methods face the sub-optimality problem. The objective functions aimed at aligning the probability distributions between the teacher and student models can be straightforward but cannot fully capture the goal of distilling language knowledge. First, intuitively, the correct output for an input can vary, and thus behavior cloning cannot capture the full knowledge of a teacher. Besides, there is no standardized definition for the quality of a generated output given an input, which makes it difficult to define the objective of knowledge distillation. This 38th Conference on Neural Information Processing Systems (NeurIPS 2024). �� ... �� ... �� ... �� ... �’ �� ∗ ℳ ~�� (�≤�, ��+1) ��(�≤�, �’) �� ... �� ... �≤� �� ... �� ... ��+1 �’ �� ∗ ℳ ~�∗ ��∗(�≤�, ��+1) ��∗(�≤�, �’) (a) On-policy distribution-matching distillation. (c) On-policy Q-value moment-match. distillation (ours). (b) Off-policy distribution-matching distillation. (d) Off-policy Q-value moment-match. distillation (ours). �� ∗ �∗ �∗ �∗ �’� �’�−1 ... �� ... ��−1 �’� ... �’�−1 �� ... ��−1 �≤�+1 ��+1 �≤� �≤�+1 �≤� �≤� �≤�+1 �≤�+1 ��+1 ��+1 Figure 1: The comparison between the distribution-matching-based distillation and the action-value moment-matching distillation is outlined. πθ and π∗denote the student policy and the teacher policy, respectively. For both on-policy (using student-generated outputs) and off-policy (using teachergenerated outputs) perspectives, our approach optimizes moment-matching of action-value functions (Q-functions) instead of minimizing the distribution distance measured by M = KL, RKL, TV, etc. imposes a significant limitation on the generalization performance of the student model through distillation. To address the aforementioned issues, we employ a reinforcement learning (RL) formulation for the auto-regressive text generation problem and utilize the definition of imitation gap to describe the high-level goal of knowledge distillation. Additionally, we address the imitation gap for KD by matching moments of the action-value function, which reflects the quality of token-level predictions for the entire output. In addressing the action-value function, we adopt the approach of Swamy et al. [33], considering a two-player minimax game between the language policy and the action-value functions, aiming to minimize an upper bound of the moment-matching objective. For this purpose, we introduce an adversarial training algorithm based on the policy gradient to jointly optimize the on-/off-policy objectives. Figure 1 illustrates the overall approach. Theoretically, we compare the moment-matching objective with other distribution-matching measurements such as step-wise TV distance and analyze the convergence rate of our algorithm to an ϵ-accurate stationary point for optimization. Empirically, we evaluate our approach on both the instruction-following dataset and three task-specific datasets for text summarization, machine translation, and commonsense reasoning. Results demonstrate that the proposed adversarial momentmatching approach effectively optimizes the moment-matching distance of the imitation gap and outperforms state-of-the-art KD methods and a range of distribution-matching-based methods. The code and implementation are released at https://github.com/jiachenwestlake/MMKD. 2 Related Work Distillation of large language models. There has been an increasing interest in knowledge distillation (KD) of auto-regressive LMs, especially concerning large language models (LLMs) [41, 42]. This process effectively transfers elicited knowledge from teacher LLMs to smaller student models, aiming to compress the large size of neural network parameters and make LLMs more efficient. Sequencelevel KD (SeqKD) [20] is a variation of supervised fine-tuning (SFT) in KD. It can be viewed as the simplest method for distillation of black-box LLMs by fine-tuning the student model with teachergenerated outputs. This method has been extensively used for LLMs and has achieved success [34, 6]. In contrast, distillation of white-box LLMs can make full use of internal information of the teacher model, such as logits [30, 39] and hidden states [23], for distribution alignment, making it more effective and efficient for KD. However, unlike previous work that explicitly clones the distribution of teacher LLMs into student models, this work learns an auxiliary Q-value function to guide KD. Distillation via distribution matching. Most promising results in the distillation of white-box LLMs are achieved by minimizing divergence between the probability distributions of the teacher model 2 and student models. Kullback-Leibler (KL) divergence, reverse Kullback-Leibler (RKL) divergence, and Jensen–Shannon (JS) divergence are three widely used KD objectives for auto-regressive LMs [39, 14, 2, 21, 41]. Wen et al. [39] have shown the equivalent formulations of sequence-level KL, RKL, JS divergences, and the step-wise terms. Additionally, they also present the strong performance of step-wise total variation (TV) distance for KD, which can upper bound the sequence-level term. As a result, most recent works focus on on-policy approaches for KD [2] and combine the realtime-generated outputs by students (on-policy) with the real-time-generated outputs by teachers (or from supervised datasets) (off-policy). Following this line, Gu et al. [14] further propose a policy gradient-based method to address the high variance issues of RKL-based methods while Ko et al. [21] propose a more efficient and effective method using a skew KL divergence loss and an adaptive off-policy approach. We also focus on a combination of on-policy and off-policy objectives for KD, but we introduce a more sophisticated moment-matching approach instead of directly using the well-studied distribution-matching metrics such as KL, RKL, JS divergences, and TV distance. Distillation via reinforcement learning. In a common formulation of RL in text generation [44, 26, 15], an auto-regressive model can be viewed as a language policy, making decisions on the next token (action) based on the currently generated sequence (state). From this perspective, KD corresponds to behavior cloning in imitation learning [20, 7, 14, 2]. For imitation learning in text generation, early works such as SeqGAN [44] and TextGAIL [40] utilize a generative adversarial framework to balance between the reward model, optimized by discriminating generated/real-word text, and the language policy, optimized by policy gradient-based methods using the reward model. Existing work on KD via imitation learning refers to ImitKD [24], which optimizes the student policy by learning from demonstrations of the teacher model. RL-based distillation can also be especially relevant for leveraging the feedback from the teacher to train student models [4, 9], in which teacher models are used to generate the feedback data for training a reward model. We build our method upon an RL-based imitation learning framework. However, unlike previous work [20, 14, 2], we propose an adversarial moment-matching approach to enhance behavior cloning. 3 Method 3.1 Notations and Definitions In this section, we consider the text generation task as a decision-making process and give a corresponding reinforcement learning (RL) formulation. Text generation. Given an input x, the auto-regressive generation task in our work aims to generate a sequence of tokens as the output (y1, . . . , yT ), where yt comes from a vocabulary V. For simplicity, we define y = (y0, y1, . . . , yT ) as the full input-output sequence, where y0 = x denotes the input. The generator is modeled by a conditional probability distribution pθ(y\|x) = ΠT −1 t=0 pθ(yt+1\|y≤t), where y≤t denotes the prefix (y0, y1, . . . , yt), t ∈{0, 1, . . . , T −1}. RL formulation. We model text generation as a finite-horizon, time-independent Markov decision process. At each time step t ∈{0, . . . , T −1}, the policy πθ takes an action (t): yt+1 ∈V based on the current state (t): y≤t ∈Y, transits to the next state (t + 1): y≤t+1 ∈Y and receives a reward (t): r(y≤t, yt+1) by a reward function r : Y × V →R. The policy corresponds to the generation model πθ(yt+1\|y≤t) = pθ(yt+1\|y≤t). We focus on a (conditional) trajectory {y1, y≤1, y2, . . . , y≤T −1, yT } =: τ ∼πθ\|x which refers to a sequence of state-action pairs generated by given an initial state y0 = x ∼px and then repeatedly sampling an action yt+1 ∼πθ(·\|y≤t) and obtain the next state y≤t+1 ∼T(·\|y≤t, yt+1)1 for T time steps. In such case, the probability of a (conditional) trajectory is formally represented as p(τ\|x, πθ) = ΠT −1 t=0 T(y≤t+1\|y≤t, yt+1)πθ(yt+1\|y≤t). We also define our value function and Q-value function as V πθ(y≤t) = Eτ(t)∼πθ\|y≤t hPT −1 t′=t γt′−tr(y≤t′, yt′+1) i and Qπθ(y≤t, yt+1) = Eτ(t)∼πθ\|y≤t,yt+1 hPT −1 t′=t γt′−tr(y≤t′, yt′+1) i , where γ ∈(0, 1) denotes the discounting factor. We define the RL objective in our generation task to maximize the performance J(πθ) = Ex∼pxEτ∼πθ\|x hPT −1 t=0 γtr(y≤t, yt+1) i . 1In text generation, the state-transition is commonly assumed to be deterministic [44, 26], i.e., T(y≤t+1\|y≤t, yt+1) = 1. 3 3.2 Knowledge Distillation as Moment-Matching Imitation Learning Based on the RL formulation of auto-regressive generation, we can view the goal of knowledge distillation at a high-level as to bridge the performance gap between the teacher policy and the student policy. Definition 1 (Imitation gap). We define the imitation gap between the teacher policy and student policy as: J(π∗) −J(πθ) = E x∼px τ∼π∗\|x "T −1 X t=0 γtr(y≤t, yt+1) # − E x∼px τ∼πθ\|x "T −1 X t=0 γtr(y≤t, yt+1) # , (1) From the perspective of imitation learning [33, 32], the objective of distillation from the teacher policy π∗to the student policy πθ can be represented as to minimize the imitation gap of Eq. (1) w.r.t. the parameters of student policy θ. A direct idea from Eq. (1) is to use moment matching over the reward to optimize the imitation gap [33]. However, we actually care about the long-term reward, at each time step, we should consider the accumulated reward in the future output rather than the immediate reward to the fitness of previous tokens (prefix). To this end, we can alternatively use the Q-value function (def. in §3.1) for each timestep to represent the overall reward from the current timestep to the last timestep. Similar to [33], we can apply the Performance Difference Lemma (PDL) [18, 3, 33] to expand the imitation gap in Eq. (1) into either off-policy or on-policy expressions. Proposition 1 (Off-policy bound of imitation gap [33]). Let FQ denote the set of Q-value functions induced by sampling actions from πθ, then we have: J(π∗) −J(πθ) ≤sup f∈FQ E x∼px τ∼π∗\|x "T −1 X t=0 γt f(y≤t, yt+1) − E y∼πθ(·\|y≤t) f(y≤t, y) !# \| {z } =:Loff(πθ,f) (2) In the following sections, we will use Loff(πθ, f) to represent the off-policy moment-matching objective of mitation learning for KD. The off-policy moment-matching objective in Proposition 1 only requires a collected dataset of teacher-generated trajectories to be evaluated and minimized. Proposition 2 (On-policy bound of imitation gap [33]). Let FQ∗denote the set of Q-value functions induced by sampling actions from π∗, then we have: J(π∗) −J(πθ) ≤ sup f∈FQ∗ E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) f(y≤t, y) −f(y≤t, yt+1) !# \| {z } =:Lon(πθ,f) (3) In the following sections, we will use Lon(πθ, f) to represent the on-policy moment-matching objective of an imitation learning for KD. Proof. See Appendix A.1 and Appendix A.2 for the complete derivations of Proposition 1 and Proposition 2, respectively. It is notable from Proposition 2 that the on-policy moment-matching objective requires interactions with the teacher to tell us what action they would take in any state visited by the student as well as on-policy samples from the student’s current policy τ ∼πθ\|x. In the remaining content of this section, we will explore the relationship between the momentmatching objectives and the existing distribution-matching objectives [39]. At the beginning, we draw a general formulation of the state-of-the-art methods for distillation of LLMs [39, 14, 2, 21] that rely on distribution-matching between the student’s and teacher’s predictions, through minimizing the step-wise probability distribution distance between the teacher policy and student policy. 4 Definition 2 (Generalized step-wise distribution distance). The off-policy and on-policy versions are defined as follows, doff M(πθ, π∗) := E x∼px τ∼π∗\|x "T −1 X t=0 γtM(π∗(·\|y≤t), πθ(·\|y≤t)) # ; (4) don M(πθ, π∗) := E x∼px τ∼πθ\|x "T −1 X t=0 γtM(π∗(·\|y≤t), πθ(·\|y≤t)) # , (5) where M(·, ·) denotes a distribution distance, consisting of total variation (TV) distance [39] and Kullback-Leibler (KL)-based divergence [14, 2]. Detailed definitions for these distances refer to Appendix A.3. For simplicity, we directly replace M with TV, KL, RKL, etc in the following sections. It is notable from Wen et al. [39] that the sequence-level KL, RKL and JS divergences can be equivalently represented as the step-wise terms, and the sequence-level TV distance can be upper bounded by the step-wise terms, which can be actually implemented by algorithms. To make a connection with the step-wise distribution distance (Definition 2), we use the following definition. Definition 3 (Distribution-matching formulation of moment-matching objectives). Based on Definition 2, we can re-formulate the off-policy and on-policy moment-matching (MM) objectives (Proposition 1 and Proposition 2, respectively) via step-wise distribution-matching, which can be defined as doff MM(πθ, π∗) and don MM(πθ, π∗) respectively, where the distance metric MM(·, ·) can be defined as follows, MMoff(on)(π∗(·\|y≤t), πθ(·\|y≤t))= E y∼π∗(·\|y≤t) h f off(on) ∗ (y≤t, y) i − E y∼πθ(·\|y≤t) h f off(on) ∗ (y≤t, y) i , Off-policy: f off ∗ = arg max f∈FQ Loff(πθ, f); On-policy: f on ∗ = arg max f∈FQ∗ Lon(πθ, f), (6) where Loff(πθ, f) and Lon(πθ, f) denote the off-policy and on-policy moment-matching objectives, which are defined in Proposition 1 and Proposition 2, respectively. Under Definition 3, we observe that the main difference between the moment-matching objectives and other step-wise distribution distance, e.g., TV distance and KL-based divergences in formulation comes from the optimal Q-value function f off(on) ∗ , aiming to maximize the discrepancy of its expectations based on π∗(·\|y≤t) v.s. πθ(·\|y≤t) for each step t ∈{0, 1, . . . , T −1}. To look deeper, we draw a connection between the moment-matching objectives and step-wise TV distance using the following corollary. Theorem 1 (Relationship between moment-matching objective and TV distance). Under a constrain of uniform boundness on the class of Q-value functions for off-/on-policy learning: FQ = FQ∗= {f : ∥f∥∞≤1}, the moment-matching objectives in Proposition 1 and Proposition 2 can be upper-bounded by the step-wise TV distance, Formally, we have J(π∗) −J(πθ) ≤ sup f:∥f∥∞≤1 Loff(πθ, f) ≤2doff TV(πθ, π∗); (7) J(π∗) −J(πθ) ≤ sup f:∥f∥∞≤1 Lon(πθ, f) ≤2don TV(πθ, π∗), (8) for the off-policy and on-policy perspectives, respectively. Proof. See Appendix A.4 for the complete derivation. We can observe from Theorem 1 that minimizing the step-wise TV distance can achieve suboptimal results compared to optimizing the moment-matching objectives Loff(πθ, f), Lon(πθ, f) for off-policy and on-policy imitation learning, which are defined in Proposition 1 and Proposition 2, respectively. Thus, optimizing the moment-matching objectives can potentially achieve better optimization results for imitation learning. 5 Algorithm 1: Adversarial training procedure Input: Dataset Dxy with inputs and ground-truth outputs Teacher policy π∗; Student policy πθ with initial parameters θ pretrained on Dxy; Off-policy Q-value function fϕ1 and on-policy Q-value function fϕ2 with initial parameters ϕ1 and ϕ2, respectively; Step sizes K (outer), N (inner); Learning rate η; Controlling factor α; Off-/on-policy combination factor β Output: The optimized student policy πθ∗ for k = 0, 1, 2, . . . , K −1 do for n = 0, 1, 2, . . . , N −1 do Sample an input x ∼Dx and generate an trajectory τ off ∼π∗\|x ϕ1 ←ϕ1 + αβη∇ϕ1 ˆLoff(τ off, θk, fϕ1) maximize Loff(πθk, fϕ1) in Eq. (9) Sample an input x ∼Dx and generate an trajectory τ on ∼πθ\|x ϕ2 ←ϕ2 + α(1 −β)η∇ϕ2 ˆLon(τ on, θk, fϕ2) maximize Lon(πθk, fϕ2) in Eq. (9) end Sample an input xk ∼Dx and generate trajectories τ off k ∼π∗\|xk and τ on k ∼πθ\|xk θk+1 ←θk −η −β ˆGoff(τ off k , θk) + (1 −β) ˆGon(τ on k , θk) minimize L(πθ, fϕ1, fϕ2) in Eq. (9) end 3.3 Adversarial Training Algorithm Optimization objective. As shown in previous work [14, 2, 21] incorporating both the off-policy and on-policy distillation benefits effectiveness and efficiency. We thus consider a training objective to jointly minimize the off-policy moment-matching objective in Proposition 1 and the on-policy moment-matching objective in Proposition 2. Both the off-/on-policy objectives can be optimized by viewing the learning procedure as solving a game. More specifically, we consider a two-player minimax game between the student policy and the Q-value functions. To this end, we initialize two small networks of a single-layer MLP to estimate the off-/on-policy Q-value functions, respectively. For example in a causal/seq-to-seq LM, the Q-value estimate module can be represented as fϕ1(2)(y≤t, y) = (hπθ t + voff(on) y )⊤woff(on) y for any action token y ∈V. This estimates the Q-value function by taking the current t ∈{0, 1, . . . , T −1} hidden step of a policy network hπθ t ∈RH (for next token prediction) to combine with the feature vector of the token voff(on) y ∈RH with a linear transformation by woff(on) y ∈RH for off(on)-policy learning. Here, H represents the hidden size and the additional parameter cost is O(H\|V\|) for Q-value estimation. Finally, combining off- and on-policy objectives with a factor β ∈(0, 1), the optimization problem can be represented as follows, min θ∈Θ max ϕ1,ϕ2∈Φ βLoff(πθ, fϕ1) + (1 −β)Lon(πθ, fϕ2) \| {z } =:L(πθ,fϕ1,fϕ2) , (9) where L(πθ, fϕ1, fϕ2) represents the overall training objective. To minimize the objective w.r.t the policy parameters θ, we use a policy gradient approach and derive the policy gradient in Appendix A.5, formally represented as follows, ∇L(πθ, fϕ1, fϕ2) = E x∼px −β E τ∼π∗\|x ˆGoff(τ, θ) + (1 −β) E τ ′∼πθ\|x ˆGon(τ ′, θ) s.t. ˆGoff(τ, θ) = T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t)fϕ1(y≤t, y) ; ˆGon(τ ′, θ) = T −1 X t=0 γt∇log πθ(y′ t+1\|y′ ≤t) ˆQfϕ2(y′ ≤t, y′ t+1), (10) where ˆQfϕ2 : Y × V →R denotes the empirical Q-value defined in Eq. (21). Besides, we use stochastic gradient ascent (SGA) to maximize the objective of L(πθ, fϕ1, fϕ2) w.r.t. parameters of the on-policy Q-value function ϕ1 and parameters of the off-policy Q-value function ϕ2. Training procedure. The goal is to achieve an equilibrium between minimizing the objective w.r.t. the parameters of student policy θ ∈Θ and maximizing the objective w.r.t. the parameters of on-policy and off-policy Q-value functions ϕ1, ϕ2 ∈Φ, formally defined as minθ maxϕ1,ϕ2 L(πθ, fϕ1, fϕ2) 6 (Eq. (9)). To this end, we use an adversarial training strategy in Algorithm 1, by starting from a student model fine-tuned on a dataset Dxy. In the training algorithm, we iteratively maximize the objective w.r.t. the parameters of Q-value functions fϕ1, fϕ2 and simultaneously minimize the objective w.r.t. the parameters of student policy πθ. In each iteration of policy updating, we first perform N steps of stochastic gradient ascent (SGA) w.r.t. the parameters of Q-value functions ϕ1, ϕ2. Then, the parameters of student policy θ are updated by stochastic gradient descent (SGD) with the estimated policy gradient with sampling policy gradients. 3.4 Convergence Analysis We further provide a convergence analysis for the algorithm proposed in §3.3. To deal with the challenges of non-convexity by certain reward structures, the algorithm is expected to obtain an ϵ-accurate stationary point of the policy parameters θ∗∈Θ, satisfying that E[∥∇L(θ∗)∥2] ≤ϵ. We focus on policy optimization and directly use the optimized off-/on-policy Q-value functions in each outer-loop iteration k ∈{0, 1, . . . , K −1}. We denote ϕ1(θk) = arg maxϕ1 Loff(πθk, fϕ1), ϕ2(θk) = arg maxϕ2 Lon(πθk, fϕ2) as the inner-loop optimized functions and use L(θk) := L(πθk, fϕ1(θk), fϕ2(θk)) (def. in Eq. (9)) for simplicity in this section. We start with the following standard assumption [45]. Assumption 1. Suppose that the optimized Q-value functions and the parameterized policy πθ satisfy the following conditions: (i) The uniformly boundness of off/on-policy Q-value functions optimized by Algorithm 1, i.e., ∥fϕ1∥∞, ∥fϕ2∥∞≤1. (ii) The B-Lipschitzness and the L-smoothness of the parameterized policy, i.e., for any stateaction pair (y≤t, yt+1) ∈Y × V at any time step t ∈{0, 1, . . . , T −1}, ∥∇log πθ(yt+1\|y≤t)∥≤B, for any θ ∈Θ, (11) ∥∇log πθ1(yt+1\|y≤t) −∇log πθ2(yt+1\|y≤t)∥≤L∥θ1 −θ2∥, for any θ1, θ2 ∈Θ (12) Theorem 2 (Convergence rate of Algorithm 1 to stationary points). Let {θk}1≤k≤K be the sequence of parameters of the policy πθk given by Algorithm 1. Let the learning rate η = 2 BL q 1−γT (1−γ)KLL . Under Assumption 1, we have min 0≤k≤K−1 E ∥∇L(θk)∥2 ≤O 1 √ K (13) Proof. See Appendix A.6 for the complete derivation. Theorem 2 illustrates that the output gradient norm square by Algorithm 1 can converge to a neighborhood around zero with the rate of 1/ √ K. Furthermore, leveraging a sufficient number of training iterations O(ϵ−2), Algorithm 1 can obtain an ϵ-accurate stationary point. This leads to the following corollary on the computational complexity of the training procedure. Corollary 1 (Computational complexity of Algorithm 1). We formalize the policy as a softmax function πθ with a linear transformation: softmax(θy≤t) for any y≤t ∈RH, where θ ∈R\|V\|×H and H denotes the hidden size. Then, to obtain an ϵ-accurate stationary point by Algorithm 1, the complexity of gradient computation is O(ϵ−2T\|V\|H(N + T + \|V\|)). Proof. See Appendix A.7 for the complete derivation. Corollary 1 shows that Algorithm 1 has a polynomial computational complexity w.r.t ϵ−2, N, \|V\|, H and T, to obtain an ϵ-accurate stationary point for optimizing the training objective in Eq. (9). 4 Experiments We consider task-agnostic instruction-following experiments and task-specific experiments, including text summarization, machine translation, and commonsense reasoning. We compare our approach 7 Table 1: Comparison with state-of-the-art KD methods on the instruction-following dataset using fine-tuned OpenLLaMA-7B as the teacher and fine-tuned OpenLLaMA-3B as the student. We format the best, the second best and worse than SFT results. The results based on GPT-2 are available in Appendix C.1. Method DollyEval SelfInst VicunaEval S-NI UnNI GPT-4 R-L GPT-4 R-L GPT-4 R-L R-L R-L OpenLLaMA2-7B (teacher) 58.8±1.2 32.5±0.4 56.7±0.8 21.6±0.2 46.2±0.6 22.6±0.5 36.3±0.5 38.5±0.2 SFT (student) 46.8±0.7 26.7±0.6 40.8±1.1 16.3±0.7 34.8±0.8 17.3±0.2 30.4±0.4 28.6±0.3 KD [16] 43.9±0.8 22.4±0.4 43.5±0.5 17.4±0.5 33.7±0.3 16.4±0.2 29.3±0.6 23.4±0.3 SeqKD [20] 50.2±0.6 26.2±0.4 46.8±0.3 15.8±0.5 38.8±1.2 18.0±0.6 29.7±0.3 27.8±0.1 ImitKD [24] 53.7±1.6 25.3±0.3 45.0±0.7 18.4±0.4 41.7±1.2 19.1±0.2 33.1±0.7 28.7±0.5 MiniLLM [14] 58.7±1.2 28.4±0.3 51.8±1.5 20.2±0.6 44.2±1.1 20.7±0.5 37.4±0.4 37.5±0.2 GKD [2] 57.6±1.0 27.5±0.3 52.4±1.2 20.9±0.3 45.5±0.8 19.3±0.5 36.8±0.6 34.8±0.3 DistiLLM [21] 59.2±1.2 29.5±0.2 53.4±1.0 20.8±0.7 46.3±0.9 20.4±0.3 37.2±0.1 38.2±0.1 Ours 59.8±0.8 30.7±0.4 54.2±1.2 21.7±0.5 47.8±0.7 21.4±0.4 38.7±0.4 39.1±0.3 with various KD baselines, including: SFT, which fine-tunes the student model on the supervised dataset Dxy; KD [16], which uses KL divergence on the supervised dataset Dxy; SeqKD [20], which applies SFT to the student model with teacher-generated outputs; ImitKD [24], which uses KL divergence on the student-generated outputs; MiniLLM [14], which uses RKL divergence with a policy gradient method; GKD [2], which uses JS divergence with an on-policy method; and DistiLLM [21], which uses an adaptive training method for off-policy optimization of a skew KL divergence. Additionally, we focus on step-wise distance optimization for KD and compare it with a range of well-known methods, including KL divergence, RKL divergence, JS divergence, and TV distance, as discussed by Wen et al. [39]. All the reported results are the average across three random seeds. 4.1 Task-Agnostic Distillation Experimental Setup. We follow the previous works [14, 21] for the implementation of the instructionfollowing experiment, aiming to evaluate the distilled model’s ability to handle diverse tasks presented in the form of instructions. We construct the training data from databricks-dolly-15k [8], where we randomly select 15K samples for training and equally split 500 samples for validation and testing. We evaluate the trained model on five instruction-following datasets: DollyEval, SelfInst [36], VicunaEval [6], S-NI [37], and UnNI [17]. Following the previous works [14, 21], we also add the OpenWebText [13] corpus, consisting of long-document plain text, for joint training with a language modeling task. This has been shown to effectively improve the performance of instruction tuning [14]. The evaluation metrics include ROUGE-L [25] and GPT-4 feedback with the same prompts as in [21]. More details on experimental setup refer to Appendix B. Main results. Table 1 illustrates the instruction-following performances. Compared with the SFT baseline, which indicates the student model without KD, KD and SeqKD hardly improve the performances. This indicates that using only supervised datasets or teacher-generated outputs does not benefit the KD of large language models. In contrast, utilizing the student-generated outputs with KL divergence [2], RKL divergence [14], and JS divergence [2] shows effectiveness for KD in the instruction-following task. State-of-the-art methods [14, 2, 21] tend to combine the studentgenerated outputs with the teacher-generated output or supervised dataset to further improve the results of KD. This shows that a mixture optimization of both on-policy and off-policy objectives can effectively improve the KD performance of large language models on the instruction-following task. In particular, we use an adversarial moment-matching method and optimize both on-policy and off-policy objectives for KD, thus achieving the best results on five test datasets with both GPT-4 feedback and ROUGE-L evaluations. 4.2 Task-Specific Distillation Experimental Setup. We evaluated the KD models on three tasks consisting of text summarization, machine translation, and reasoning. For the text summarization task, we follow Ko et al. [21] to conduct experiments on the SAMSum [12] dataset. For the machine translation tasks, we follow Ko et al. [21] to conduct experiments on the IWSLT’17 (en-de) [5] dataset. For the commonsense reasoning task, we conduct experiments on the StrategyQA dataset [11] with chain-of-thought augmentations 8 Table 2: Comparison with the state-of-the-art KD methods on text summarization, machine translation and commonsense reasoning datasets. We report the ROUGE-L, BLEU and accuracy for SAMSum, IWSLT’17 (en-de) and StrategyQA, respectively. We format the best, the second best and worse than SFT results. Method SAMSum IWSLT’17 (en-de) StrategyQA T5-Small T5-Base T5-Large T5-Small T5-Base T5-Large T5-Small T5-Base T5-Large T5-XL (teacher) 52.5±0.4 35.2±0.2 64.5±0.8 SFT (student) 40.6±0.2 47.3±0.3 49.8±0.2 21.5±0.1 30.1±0.0 33.7±0.1 52.4±0.5 57.5±0.8 60.7±0.8 KD [16] 39.2±0.4 46.5±0.3 47.4±0.3 21.7±0.1 29.8±0.2 31.7±0.1 49.7±0.3 55.3±0.1 59.2±0.5 SeqKD [20] 39.7±0.3 47.7±0.5 49.3±0.4 21.2±0.3 29.2±0.2 32.9±0.5 50.6±0.7 57.5±1.1 61.5±0.8 ImitKD [24] 41.8±0.3 48.6±0.7 51.2±0.5 22.2±0.3 28.7±0.6 34.1±0.2 53.8±0.8 59.7±0.5 61.7±0.6 GKD [2] 42.1±0.3 48.2±0.5 51.7±0.4 22.7±0.2 31.2±0.1 34.7±0.2 55.6±0.4 60.3±0.5 63.6±0.3 DistiLLM [21] 42.6±0.2 49.4±0.6 52.1±0.4 22.5±0.1 30.8±0.2 35.5±0.1 56.3±0.3 61.2±0.7 62.8±0.2 Ours 43.7±0.4 50.4±0.3 52.7±0.3 23.7±0.1 32.4±0.3 36.0±0.2 58.2±0.4 62.9±0.3 65.3±0.7 [38]. For all of the task-specific experiments, we use T5-XL [29] as the teacher model and T5-Large/Base/-Small as the student model. For the machine translation experiments, we employ a multilingual pretrained model, mT5 [43], to build the methods. For evaluation, we use ROUGE-L [25], BLEU [27], and accuracy as the performance metrics on SAMSum, IWSLT’17 (en-de), and StrategyQA, respectively. More details about the experimental setup refer to Appendix B. Main results. Table 2 displays the performances on three task-specific datasets. Since the original work of MiniLLM [14] does not consider these tasks, we thus do not make comparisons with MiniLLM. The performance trend is similar to the instruct-following results, revealing that KD of large language models for specific tasks also benefits from the combination of on-policy objectives with student-generated outputs and off-policy objectives with teacher-generated outputs or supervised datasets. Additionally, we observe that student models of different sizes all benefit from the KD methods to improve performance. Overall, our approach achieves the best results on all three task-specific datasets for student models of different sizes. This demonstrates the effectiveness of an adversarial moment-matching approach for KD of large language models on specific tasks. KL RKL JS TV Ours 20 22 24 26 28 30 32 ROUGE-L on-policy off-policy mixed (a) DollyEval. KL RKL JS TV Ours 43 44 45 46 47 48 49 50 51 52 ROUGE-L on-policy off-policy mixed (b) SAMSum. KL RKL JS TV Ours 26 27 28 29 30 31 32 33 34 BLEU on-policy off-policy mixed (c) IWSLT’17 (en-de). KL RKL JS TV Ours 54 56 58 60 62 64 Accuracy on-policy off-policy mixed (d) StrategyQA. Figure 2: Performance of difference step-wise distribution distances. 4.3 Analysis on Step-Wise Distance Optimization Comparison with distribution matching. We make comparisons with different step-wise distribution distances with a uniform formulation of Definition 2, considering the on-policy, offpolicy objectives as well as the joint form. Results on four tasks with a default combination factor β = 0.5 are shown in Figure 2. More instruct-following results are available in Appendix C.2 and results with different values of off-/on-policy combination factor are available in Appendix C.5. Compared with the KL divergence, RKL divergence, JS divergence and total variation distance, the proposed moment-matching distance achieves the best results under both the on-policy and off-policy training objectives, which shows that the proposed moment-matching approach is effective for KD of large language models. Besides, we observe that using a joint objective of both on-policy and off-policy can further significantly improve the performances. This shows that both on-policy and off-policy moment-matching objectives contribute to the minimization of the imitation gap and can thus benefit the KD of large language models. 9 0 2000 4000 6000 8000 10000 Step 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Loss (train) training loss on-policy distance don off-policy distance doff 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Distance (eval) (a) Training loss and don MM, doff MM against training step. DollyEval SelfInst VicunaEval S-NI UnNI Average KL RKL JS TV Ours 2.65 4.87 3.21 3.67 3.02 3.48 2.54 4.78 3.87 3.58 2.67 3.48 2.31 4.52 3.67 3.32 2.43 3.25 1.98 2.34 3.32 2.13 1.83 2.32 0.96 1.85 2.11 0.85 0.92 1.34 don on five test sets 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 (b) On-policy moment-matching distance don MM on the test sets. DollyEval SelfInst VicunaEval S-NI UnNI Average KL RKL JS TV Ours 2.05 2.79 2.21 2.07 2.02 2.23 1.84 3.28 2.37 2.18 1.67 2.27 1.78 2.62 2.15 1.68 1.43 1.93 1.53 1.94 2.42 1.13 1.23 1.65 0.75 1.45 1.11 0.72 0.62 0.93 doff on five test sets 1.0 1.5 2.0 2.5 3.0 (c) Off-policy moment-matching distance don MM on the test sets. Figure 3: Adversarial training procedure for optimizing the on-policy and off-policy momentmatching distances don MM, doff MM on the instruction-following dataset. Adversarial training procedure. We present the training loss and moment-matching distance against the adversarial training steps. As depicted in Figure 3 (a), the training loss initially increases within the first 0-1,000 steps, indicating that initially, the Q-value functions are stronger than the policy in maximizing the loss function L(πθ, fϕ1, fϕ2) in Eq. (9). Concurrently, the policy gradient method contributes to minimizing the training loss, which eventually converges to a much lower stable value. Additionally, both the on-policy and off-policy moment-matching distances don MM and doff MM decrease and eventually reach a low value with only minor fluctuations. For more results and details on experimental setups, please refer to Appendix C.3. Moment-matching distance optimization. We further illustrate the on-policy moment-matching distance don MM and the off-policy moment-matching distance doff MM (defined in Definition 3) optimized by different step-wise distances in Figure 3 (b) and (c), respectively. Interestingly, we observe that the total variation (TV) distance obtains the second-best results on average for both on-policy and off-policy distances. This finding suggests a similarity between the formulations of TV distance and moment-matching distances to some extent, as supported by the theoretical result of Theorem 1. Across all instruction-following test sets, our approach effectively optimizes both on-policy and off-policy moment-matching distances more than other step-wise distribution distances used in KD, including KL divergence, RKL divergence, JS divergence, and TV distance. This observation also underscores the effectiveness of our policy gradient methods. Extensive results on the task-specific datasets are available in Appendix C.4. 5 Conclusion In this work, we investigated a moment-matching approach for knowledge distillation of large language models. Specifically, we formulated knowledge distillation from a perspective of imitation learning and derived both on-policy and off-policy bounds for the imitation gap between the teacher model and student model via moment-matching distance. Additionally, we proposed an adversarial training algorithm to simultaneously estimate and minimize the joint objective of on-policy and off-policy moment-matching distances. In experiments, we evaluated the proposed algorithm on four instruction-following datasets and three task-specific datasets, comparing it with a range of state-of-the-art KD methods as well as four well-studied step-wise distribution distances for KD of auto-regressive models. Results demonstrate that our approach can effectively leverage the policy gradient method to optimize the moment-matching distance and achieve the best results across all datasets. Limitations and future work. The proposed adversarial training algorithm requires additional computational steps for the inner-loop gradient ascent, which may result in increased time complexity. Moreover, the proposed approach necessitates auxiliary networks to build the Q-value functions, which may incur additional memory costs. Besides, the experiments are conducted with limited LLM architectures, such as OpenLLaMA and T5. 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Similar to the proof of Performance Difference Lemma (PDL) [18, 3, 33], we have J(π∗) −J(πθ) = E x∼px τ∼π∗\|x "T −1 X t=0 γtr(y≤t, yt+1) # − E x∼px [V πθ(x)] = E x∼px τ∼π∗\|x "T −1 X t=0 γt r(y≤t, yt+1) + V πθ(y≤t) −V πθ(y≤t) # − E x∼px [V πθ(x)] = E x∼px τ∼π∗\|x "T −1 X t=0 γt r(y≤t, yt+1) + γV πθ(y≤t+1) −V πθ(y≤t) # = E x∼px τ∼π∗\|x "T −1 X t=0 γt r(y≤t, yt+1) + γEy≤t+1∼T (·\|y≤t,yt+1) V πθ(y≤t+1) −V πθ(y≤t) # (i) = E x∼px τ∼π∗\|x "T −1 X t=0 γt Qπθ(y≤t, yt+1) −V πθ(y≤t) # = E x∼px τ∼π∗\|x "T −1 X t=0 γt Qπθ(y≤t, yt+1) − E y∼πθ(·\|y≤t) Qπθ(y≤t, y) !# ≤sup f∈FQ E x∼px τ∼π∗\|x "T −1 X t=0 γt f(y≤t, yt+1) − E y∼πθ(·\|y≤t) f(y≤t, y) !# , where (i) follows from Bellman equation and noting that the transition probability T(·\|y≤t, yt+1) is deterministic in an auto-regressive text generation problem. This completes the proof. A.2 Proof of Proposition 2 Proof. Similar to the proof of Proposition 1, we have J(π∗) −J(πθ) = − E x∼px τ∼πθ\|x "T −1 X t=0 γtr(y≤t, yt+1) # + E x∼px [V π∗(x)] = E x∼px τ∼πθ\|x "T −1 X t=0 γt V π∗(y≤t) − r(y≤t, yt+1) + V π∗(y≤t) # + E x∼px [V π∗(x)] = E x∼px τ∼πθ\|x "T −1 X t=0 γt V π∗(y≤t) − r(y≤t, yt+1) + γV π∗(y≤t+1) # = E x∼px τ∼πθ\|x "T −1 X t=0 γt V π∗(y≤t) − r(y≤t, yt+1) + γEy≤t+1∼T (·\|y≤t,yt+1) V π∗(y≤t+1) # = E x∼px τ∼πθ\|x "T −1 X t=0 γt V π∗(y≤t) −Qπ∗(y≤t, yt+1) # = E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) Qπ∗(y≤t, y) −Qπ∗(y≤t+1, yt+1) !# ≤sup f∈FQ∗ E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) f(y≤t, y) −f(y≤t, yt+1) !# , 14 which completes the proof of Proposition 2. A.3 Existing Step-Wise Distribution Distance for Distillation Definition 4 (Step-wise distribution distances for distillation [39]). Following Wen et al. [39], we define four groups of well-studied probability distribution distances as follows, • Total variation (TV) distance. The token-level TV distance between the probabilities of teacher policy π∗and student policy πθ given the current state y≤t can be defined by the ℓ2-norm as follows, TV(πθ(·\|y≤t), π∗(·\|y≤t)) :=1 2 X y∈V π∗(y\|y≤t) −πθ(y\|y≤t) (14) • Kullback–Leibler (KL) divergence. The token-level KL divergence between the probabilities of teacher policy π∗and student policy πθ given the current state y≤t can be defined as follows, KL(πθ(·\|y≤t), π∗(·\|y≤t)) := X y∈V π∗(y\|y≤t) log π∗(y\|y≤t) πθ(y\|y≤t) (15) • Reverse Kullback–Leibler (RKL) divergence. The token-level RKL divergence between the probabilities of teacher policy π∗and student policy πθ given the current state boldsymboly≤t can be defined as follows, RKL(πθ(·\|y≤t), π∗(·\|y≤t)) := X y∈V πθ(y\|y≤t) log πθ(y\|y≤t) π∗(y\|y≤t) (16) • Jenson–Shannon (JS) divergence. The token-level JS divergence between the probabilities of teacher policy π∗and student policy πθ given the current state boldsymboly≤t can be defined based on the KL divergence and RKL divergence as follows, JS(πθ(·\|y≤t), π∗(·\|y≤t)) :=1 2KL(π∗, πθ + π∗ 2 ) + 1 2RKL(πθ, πθ + π∗ 2 ) (17) A.4 Proof of Theorem 1 Proof. We first derive an upper bound for the on-policy moment-matching objective of Eq. (3). Set FQ∗= {f : ∥f∥∞≤1}, and by the definition of L(πθ, f) in Eq. (3), we have sup f:∥f∥∞≤1 Lon(πθ, f) = sup f:∥f∥∞≤1 E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) f(y≤t, y) −f(y≤t, y) !# = sup f:∥f∥∞≤1 E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) f(y≤t, y) − E y∼πθ(·\|y≤t) f(y≤t, y) !# Then, we have sup f:∥f∥∞≤1 Lon(πθ, f) (i) ≤ E x∼px τ∼πθ\|x "T −1 X t=0 γt sup f:∥f∥∞≤1 E y∼π∗(·\|y≤t) f(y≤t, y) − E y∼πθ(·\|y≤t)) f(y≤t, y) !# (ii) = E x∼px τ∼πθ\|x   T −1 X t=0 γt X y∈V π∗(y\|y≤t) −πθ(y\|y≤t)  (iii) = 2don TV(πθ, π∗) (Def. 2 & 4), 15 where (i) follows from Jensen’s inequality, (ii) follows from [31] and (iii) follows from the definition of TV distance. Similarly, we can bound the off-policy version of Eq. (2) as follows, sup f:∥f∥∞≤1 E x∼px τ∼π∗\|x Loff(πθ, f) ≤ E x∼px τ∼π∗\|x   T −1 X t=0 γt X y∈V π∗(y\|y≤t) −πθ(y\|y≤t)   = 2doff TV(πθ, π∗) (Def. 4), which completes the proof of Theorem 1. A.5 Derivation of Policy Gradient in Eq. (10) Based on the definition of training objective in Eq. (9), we have ∇L(πθ, fϕ1, fϕ2) = β∇Loff(πθ, fϕ1) + (1 −β)∇Lon(πθ, fϕ2) (18) Based on the definition of Loff(πθ, fϕ1) in Eq. (2), we have ∇Loff(πθ, fϕ1) =∇ E x∼px τ∼π∗\|x "T −1 X t=0 γt f(y≤t, yt+1) − E y∼πθ(·\|y≤t) f(y≤t, y) !# = − E x∼px τ∼π∗\|x T −1 X t=0 γt∇ E y∼πθ(·\|y≤t) fϕ1(y≤t, y) = − E x∼px τ∼π∗\|x T −1 X t=0 γt X y∈V πθ(y\|y≤t)∇log πθ(y\|y≤t)fϕ1(y≤t, y) = − E x∼px τ∼π∗\|x T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t)fϕ1(y≤t, y) (19) Then, based on the definition of Lon(πθ, fϕ2) in Eq. (3), we have ∇Lon(πθ, fϕ2) =∇ E x∼px τ∼πθ\|x "T −1 X t=0 γt E y∼π∗(·\|y≤t) fϕ2(y≤t, y) −fϕ2(y≤t, yt+1) !# (i) = E x∼px τ∼πθ\|x "T −1 X t=0 γt∇log πθ(yt\|y≤t) T −1 X t′=t γt′−t E y∼π∗(·\|y≤t′) fϕ2(y≤t′, y) −fϕ2(y≤t′, yt′+1) !# , (20) where (i) follows from a standard derivation of gradient policy (c.f. [22]). For simplicity, set ˆQfϕ2(y≤t, yt+1) = T −1 X t′=t γt′−t E y∼π∗(·\|y≤t′) fϕ2(y≤t′, y) −fϕ2(y≤t′, yt′+1) ! (21) as the empirical Q-value given any draw of trajectory τ ∼πθ\|y0 = x, x ∼px in Eq. (20). Coming back to Eq. (18) and combining with Eq. (19) and Eq. (20), we have ∇L(πθ, fϕ1, fϕ2) = −β E x∼px τ∼π∗\|x "T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t)fϕ1(y≤t, y) # + (1 −β) E x∼px τ∼πθ\|x "T −1 X t=0 γt∇log πθ(yt+1\|y≤t) ˆQfϕ2(y≤t, yt+1) # # 16 Then, using the law of iterated expectations, we obtain the final formulation of policy gradient, ∇L(πθ, fϕ1, fϕ2) = E x∼px −β E τ∼π∗\|x T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t)fϕ1(y≤t, y) + (1 −β) E τ ′∼πθ\|x T −1 X t=0 γt∇log πθ(y′ t+1\|y′ ≤t) ˆQfϕ2(y′ ≤t, y′ t+1) , which completes the derivation of policy gradient in Eq. (10). A.6 Proof of Theorem 2 Lemma 1. Let ˆ∇L(θ) = −β ˆGoff(τ, θ) + (1 −β) ˆGon(τ ′, θ) denote the empirical policy gradient given any trajectories x ∼px, τ ∼π∗\|x, τ ′ ∼πθ\|x, where L(θ) := L(πθ, fϕ1, fϕ2) (def. in Eq. (9)) denote the objective w.r.t. the policy parameters θ given any off-/on-policy Q-value functions fϕ1 and fϕ2. Then, under Assumption 1, we have ∥ˆ∇L(θ)∥≤BL with BL = β(1 −γT )B 1 −γ + 2(1 −β)(1 −γT )2B (1 −γ)2 Proof. By triangle inequality, we have for any x ∼px, τ ∼π∗\|x, τ ′ ∼πθ\|x, ∥ˆ∇L(θ)∥≤β ˆGoff(τ, θ) + (1 −β) ˆGon(τ ′, θ) (22) By the formulation of off-policy gradient ∥ˆGoff(τ, θ)∥in Eq. (10) under the condition of optimized off-policy Q-value functions fϕ1 by Algorithm 1, we have ˆGoff(τ, θ) = T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t)fϕ1(y≤t, y) By Jensen’s inequality, we have ˆGoff(τ, θ) ≤ T −1 X t=0 γt E y∼πθ(·\|y≤t) ∇log πθ(y\|y≤t) fϕ1(y≤t, y) By Assumption 1, we have ˆGoff(τ, θ) ≤B T −1 X t=0 γt = B(1 −γT ) 1 −γ (23) Similarly, we can bound the on-policy gradient ∥ˆGon(τ ′, θ)∥by Jensen’s inequality as follows, ˆGon(τ ′, θ) ≤ T −1 X t=0 γt ∇log πθ(yt+1\|y≤t) ˆQfϕ2(y≤t, yt+1) Based on the definition of ˆQfϕ2(y≤t, yt+1) in Eq. (21) and by Jensen’s inequality, we have ˆQfϕ2(y≤t, yt+1) = T −1 X t′=t γt′−t E y∼π∗(·\|y≤t′) fϕ2(y≤t′, y) −fϕ2(y≤t′, yt′+1) ! ≤ T −1 X t′=t γt′−t E y∼π∗(·\|y≤t′) \|fϕ2(y≤t′, y)\| + \|fϕ2(y≤t′, yt′+1)\| ! Then, by Assumption 1 (i) that ∥fϕ2∥∞≤1, we have ˆQfϕ2(y≤t, yt+1) ≤2 T −1 X t′=t γt′−t ≤2 T −1 X t′=0 γt′ = 2(1 −γT ) 1 −γ 17 Thus, we have ˆGon(τ ′, θ) ≤2(1 −γT )2B (1 −γ)2 (24) Coming back to the bound of ∥ˆ∇L(θ)∥in Eq. (22), we combine it with Eq. (23) and Eq. (24). Then, we have ∥ˆ∇L(θ)∥≤β(1 −γT )B 1 −γ + 2(1 −β)(1 −γT )2B (1 −γ)2 \| {z } BL , which completes the proof of Lemma 1. Lemma 2. Under Assumption 1, the objective function L(θ) is LL-smooth such that for any θ, θ′ ∈Θ, L(θ) ≤L(θ′) + ⟨∇L(θ′), θ −θ′⟩+ 1 2LL∥θ −θ′∥2, with the constant LL = β (1 −γT )(B2 + L) 1 −γ + (1 −β)2(1 −γT )2 (1 −γ)2 γB2 1 −γ + L Proof. Under the definition of policy gradient in Eq. (10), for any θ1, θ2 ∈Θ, we have ∥∇L(θ1) −∇L(θ2)∥ = E x∼px −β E τ∼π∗\|x ˆGoff(τ, θ1) −ˆGoff(τ, θ2) + (1 −β) Eτ1∼πθ1\|x ˆGon(τ1, θ1) −Eτ2∼πθ2\|x ˆGon(τ2, θ2) Then, by Jensen’s inequality and triangle inequality, we have ∥∇L(θ1) −∇L(θ2)∥ ≤ E x∼px β E τ∼π∗\|x ˆGoff(τ, θ1) −ˆGoff(τ, θ2) \| {z } I1 + (1 −β) Eτ1∼πθ1\|x ˆGon(τ1, θ1) −Eτ2∼πθ2\|x ˆGon(τ2, θ2) \| {z } I2 (25) Based on the definition of off-policy gradient in Eq. (10) and using Jensen’s inequality, we have for any x ∼px, τ ∼π∗\|x, I1 = ˆGoff(τ, θ1) −ˆGoff(τ, θ2) ≤ T −1 X t=0 γt E y∼πθ1(·\|y≤t) ∇log πθ1(y\|y≤t)fϕ1(y≤t, y) − E y∼πθ2(·\|y≤t) ∇log πθ2(y\|y≤t)fϕ1(y≤t, y) (26) 18 Then, by triangle inequality, we have for any t ∈{0, 1, . . . , T −1}, E y∼πθ1(·\|y≤t) ∇log πθ1(y\|y≤t)fϕ1(y≤t, y) − E y∼πθ2(·\|y≤t) ∇log πθ2(y\|y≤t)fϕ1(y≤t, y) = X y∈V πθ1(y\|y≤t)∇log πθ1(y\|y≤t)fϕ1(y≤t, y) −πθ2(y\|y≤t)∇log πθ2(y\|y≤t)fϕ1(y≤t, y) ≤ X y∈V fϕ1(y≤t, y) πθ1(y\|y≤t) −πθ2(y\|y≤t) ∇log πθ1(y\|y≤t) + πθ2(y\|y≤t) ∇log πθ1(y\|y≤t) −∇log πθ2(y\|y≤t) (27) By Taylor expansion of πθ(y\|y≤t), we have that for any t ∈{0, 1, . . . , T −1}, πθ1(y\|y≤t) −πθ2(y\|y≤t) = (θ1 −θ2)⊤∇log π˜θ(y\|y≤t)π˜θ(y\|y≤t) ≤∥θ1 −θ2∥ ∇log π˜θ(y\|y≤t) π˜θ(y\|y≤t) ≤∥θ1 −θ2∥· B · π˜θ(y\|y≤t), where ˜θ is a vector lying between θ1 and θ2, i.e., there exists some λ ∈[0, 1] such that ˜θ = λθ1 + (1 −λ)θ2. Then, combining with Eq. (27), yields E y∼πθ1(·\|y≤t) ∇log πθ1(y\|y≤t)fϕ1(y≤t, y) − E y∼πθ2(·\|y≤t) ∇log πθ2(y\|y≤t)fϕ1(y≤t, y) ≤ X y∈V B2π˜θ(y\|y≤t)∥θ1 −θ2∥+ πθ2(y\|y≤t)L∥θ1 −θ2∥ =(B2 + L)∥θ1 −θ2∥ Then, combining with Eq. (26) yields I1 = ˆGoff(τ, θ1) −ˆGoff(τ, θ2) ≤(B2 + L)∥θ1 −θ2∥ T −1 X t=0 γt ≤(1 −γT )(B2 + L) 1 −γ ∥θ1 −θ2∥ (28) In addition, we can first bound I2 using Jensen’s inequality and triangle inequality, I2 = Eτ1∼πθ1\|x ˆGon(τ1, θ1) −Eτ2∼πθ2\|x ˆGon(τ2, θ2) ≤ T −1 X t=0 Z γt\| ˆQfϕ2(y≤t, yt+1)\| t−1 Y t′=0 πθ1(yt′+1\|y≤t′)∇log πθ1(yt′+1\|y≤t′) − t−1 Y t′=0 πθ2(yt′+1\|y≤t′)∇log πθ2(yt′+1\|y≤t′) dy≤1 · · · dy≤tdy1 · · · dyt By triangle inequality and the boundess of \| ˆQfϕ2(y≤t, yt+1)\| ≤2(1−γT ) 1−γ , we further have, I2 ≤2(1 −γT ) 1 −γ T −1 X t=0 Z γt t−1 Y t′=0 πθ1(yt′+1\|y≤t′) − t−1 Y t′=0 πθ2(yt′+1\|y≤t′) ∥∇log πθ1(yt′+1\|y≤t′)∥ + t−1 Y t′=0 πθ2(yt′+1\|y≤t′)∥∇log πθ1(yt′+1\|y≤t′) −∇log πθ2(yt′+1\|y≤t′)∥ dy≤1 · · · dy≤tdy1 · · · dyt (29) 19 By Taylor expansion of Qt−1 t′=0 πθ(yt′+1\|y≤t′), we have t−1 Y t′=0 πθ1(yt′+1\|y≤t′) − t−1 Y t′=0 πθ2(yt′+1\|y≤t′) = (θ1 −θ2)⊤ t−1 X t′=0 ∇π˜θ(yt′+1\|y≤t′) t−1 Y t′′=0,t′′̸=t′ π˜θ(yt′′+1\|y≤t′′) ≤∥θ1 −θ2∥ t−1 X t′=0 ∇log π˜θ(yt′+1\|y≤t′) t−1 Y t′′=0 π˜θ(yt′′+1\|y≤t′′) ≤∥θ1 −θ2∥· t · B · t−1 Y t′′=0 π˜θ(yt′′+1\|y≤t′′), where ˜θ denotes a vector lying between θ1 and θ2, i.e., there exists some λ such that ˜θ = λθ1 + (1 − λ)θ2. Coming back to the boundness of I2 in Eq. (29), we have I2 ≤2(1 −γT ) 1 −γ T −1 X t=0 Z γt B2t t−1 Y t′′=0 π˜θ(yt′′+1\|y≤t′′) + L t−1 Y t′=0 πθ2(yt′+1\|y≤t′) ! ∥θ1 −θ2∥dy≤1 · · · dy≤tdy1 · · · dyt =2(1 −γT ) 1 −γ T −1 X t=0 γt(B2t + L)∥θ1 −θ2∥≤2(1 −γT )2 (1 −γ)2 γB2 1 −γ + L ∥θ1 −θ2∥, (30) where the last inequality follows from the fact that T −1 X t=0 tγt = γ −TγT + (T −1)γT +1 (1 −γ)2 ≤γ −TγT +1 + (T −1)γT +1 (1 −γ)2 = γ(1 −γT ) (1 −γ)2 Then, combining Eq. (25) with the boundness of I1 in Eq. (28) and the boundness of I2 in Eq. (30), we obtain the final bound of ∥∇L(θ1) −∇L(θ2)∥ ≤ β (1 −γT )(B2 + L) 1 −γ + (1 −β)2(1 −γT )2 (1 −γ)2 γB2 1 −γ + L \| {z } LL ∥θ1 −θ2∥ (31) Next, we have for any θ, θ ∈Θ, L(θ) −L(θ′) −⟨∇L(θ′), θ −θ′⟩ ≤\|L(θ) −L(θ′) −⟨∇L(θ′), θ −θ′⟩\| ≤ Z (0,1) ⟨∇L(θ′ + t(θ −θ′)), θ −θ′⟩dt −⟨∇L(θ′), θ −θ′⟩ ≤ Z (0,1) ∥∇L(θ′ + t(θ −θ′)) −∇L(θ′)∥∥θ −θ′∥dt Then, by Eq. (31) and set θ1 = θ′ + t(θ −θ′) and θ2 = θ′, we have L(θ) −L(θ′) −⟨∇L(θ′), θ −θ′⟩ ≤ Z (0,1) LL∥θ −θ′∥2tdt = 1 2LL∥θ −θ′∥2, which completes the proof of Lemma 2. 20 We prove Theorem 2 as follows. Proof of Theorem 2. Let θt, θt+1, t ∈{0, 1, . . . , T −1} be adjacent parameters of policy πθt, πθt+1 given by Algorithm 1. Then, using Lemma 2 by setting θ = θk+1, θ′ = θk for any k ∈ {0, 1, . . . , K −1}, we have L(πθk+1, fϕ1(θk+1), fϕ2(θk+1)) ≤L(πθk, fϕ1(θk), fϕ2(θk)) + ⟨∇L(πθk, fϕ1(θk), fϕ2(θk)), θk+1 −θk⟩+ 1 2LL∥θk+1 −θk∥2 Following from the updating rule θk+1 = θk −η ˆ∇L(θk) and Lemma 1, we have ∥θk+1 −θk∥= η∥ˆ∇L(θk)∥≤ηBL Then, we have L(πθk+1, fϕ1(θk+1), fϕ2(θk+1)) ≤L(πθk, fϕ1(θk), fϕ2(θk)) −⟨∇L(πθk, fϕ1(θk), fϕ2(θk)), η ˆ∇L(θk)⟩+ 1 2η2B2 LLL We introduce a probability measure space (Ω, F, P) and then θk : Ω→Θ, k ∈{0, 1, . . . , K −1} can be viewed as a random variable on it. Let {σ(θk)}0≤k≤K−1 denote a sequence of increasing sigma-algebras such that σ(θ0) ⊂σ(θ1) ⊂· · · σ(θK−1) ⊂F, we define the conditional expectation E[ ˆ∇L(θk) \| σ(θk)] as E[ ˆ∇L(θk) \| σ(θk)] =Ex∼px h −βEτ∼π∗\|x ˆGoff(τ, θk) + (1 −β)Eτ ′∼πθk \|x ˆGon(τ ′, θk) i =∇L(πθk, fϕ1(θk), fϕ2(θk)), where the second equality follows from the unbiased estimation property in Eq. (10). Then, taking the conditional expectation, we have E[L(πθk+1, fϕ1(θk+1), fϕ2(θk+1))\|σ(θk)] ≤L(πθk, fϕ1(θk), fϕ2(θk)) −η∥∇L(πθk, fϕ1(θk), fϕ2(θk))∥2 + 1 2η2B2 LLL Taking total expectation, rearranging the terms and making average on k ∈{0, 1, . . . , K −1}, we have 1 K K−1 X k=0 E ∥∇L(πθk, fϕ1(θk), fϕ2(θk))∥2 ≤1 ηK K−1 X k=0 E[L(πθk, fϕ1(θk), fϕ2(θk))] −E[L(πθk+1, fϕ1(θk+1), fϕ2(θk+1))] + 1 2ηB2 LLL = 1 ηK L(πθ0, fϕ1(θ0), fϕ2(θ0)) −E[L(πθK, fϕ1(θK), fϕ2(θK))] + 1 2ηB2 LLL ≤2(1 −γT ) η(1 −γ)K + 1 2ηB2 LLL Let η = 2 BL q 1−γT (1−γ)KLL and denote L(θk) = L(πθk, fϕ1(θk), fϕ2(θk)) for simpilicity for any k ∈ {0, 1, . . . , K −1}, then we have min 0≤k≤K−1 E ∥∇L(θk)∥2 ≤1 K K−1 X k=0 E ∥∇L(θk)∥2 ≤2BL s (1 −γT )LL (1 −γ)K = O r 1 K ! (32) 21 A.7 Proof of Corollary 1 Proof. Let the convergence rate in Eq. (32) satisfy that 2BL q (1−γT )LL (1−γ)K ≤ϵ, then we have K ≥4(1 −γT )B2 LLL (1 −γ)ϵ2 , which indicates that when the iteration number of policy updating satisfies K := O(ϵ−2), it can reach an ϵ-accurate stationary point to optimize the objective in Eq. (9), such that min 0≤k≤K−1 E ∥∇L(θk)∥2 ≤ϵ For simplicity, we define the policy as a softmax function with a linear transformation of y≤t ∈RH with θ ∈R\|V\|×H. Formally, for any trajectory τ and any timestep t ∈{0, 1, . . . , T −1}, we have the probability of any y ∈V, πθ(y\|y≤t) = exp(θyy≤t) P y′∈V exp(θy′y≤t) (33) In the following, we will analyze the computational complexity in each policy updating iteration. First, we find that each inner-loop step of Q-value function updating has a gradient computation complexity of O(T\|V\|H) given the linear formulation of Q-value functions. Accordingly, N inner-loop steps in each policy updating iteration have a computational complexity of O(NT\|V\|H). Second, for policy gradient computation, since computational complexity of ∇log πθ(y\|y≤T ) is O(\|V\|H), the computation complexity of policy gradient computation is O(T\|V\|H(T + \|V\|)). Overall, the total gradient computational complexity is O(ϵ−2T\|V\|H(N + T + \|V\|)), which completes the proof of Corollary 1. B Experimental Setup We use NVIDIA A40 GPUs with 40GB RAM to conduct all the experiments. B.1 Instruction-Following Experiments Base models. We conduct experiments on both GPT-2 [28] and OpenLLaMA [10]. For the GPT-2 experiments, we use GPT-2 XL2 with 1.5B parameters to construct the teacher policy and GPT-23 with 117M parameters to construct the student policy. For the OpenLLaMA experiments, we use OpenLLaMA-7B4 with 6.7B parameters to construct the teacher policy and OpenLLaMA-3B5 with 2.7B parameters to construct the student model. Training details. We fine-tune the OpenLLaMA-7B teacher model and the OpenLLaMA-3B student models on the corresponding supervised dataset with 10,000 steps. The GPT-2 teacher and student models use the fine-tuned checkpoints by Gu et al. [14]. For the implementation of compared baselines, we use the code by Ko et al. [21] and re-run the results. The optimization protocol for KD training largely follows the previous work [14, 21]. In particular, we search for the learning rates among a finite set for each experiment to obtain the best result. The batch size for each experiments is seleted to make full use of the 40GB RAM of an A40 GPU. To handle the adversarial training, we choose the number of adversarial steps K = 5 and the adversarial control factor α = 0.1 based on the development experiments. We use a default off-/on-policy combination factor β = 0.5 for main experiments while exploring other values for analysis. The hyperparameters for training are listed in Table 3. 2https://huggingface.co/openai-community/gpt2-xl 3https://huggingface.co/openai-community/gpt2 4https://huggingface.co/openlm-research/open_llama_7b 5https://huggingface.co/openlm-research/open_llama_7b 22 Table 3: Hyperparameters for instruction-following experiments. Hyperparameter GPT-2 OpenLLaMA Max. Step Size (K) 10,000 10,000 Inner Step Size (N) 5 5 Batch Size (per GPU) {8, 16, 32} {4, 8} Dropout Rate 0.1 0.1 Controlling Factor (α) 0.1 0.1 Discounting Factor (γ) {0.90, 0.95, 0.99} {0.90, 0.95, 0.99} Combination Factor (β) {0, 0.5, 0.9, 0.99, 1.0} {0, 0.5, 0.9, 0.99, 1.0} Learning Rate (η) {5e−5, 1e−4, 5e−4} {5e−6, 1e−5, 5e−5} Warmup Steps 1,000 500 Weight Decay 1e−2 1e−2 Max Seq. Length 512 512 Sampling (top-p) 1.0 1.0 Sampling (temperature) 1.0 1.0 Evaluation Greedy Sampling Greedy Sampling #GPUs 2 4 B.2 Task-Specific Experiments Base models. For the text summarization and commonsense reasoning experiments, we use T5XL6 with 2.8B parameters to construct the teacher policy and construct the student policy with T5-Large7 (770M parameters), T5-Base8 (220M parameters) and T5-Small9 (60M parameters). For the machine translation experiments, we use mT5-XL [43] to construct the teacher policy and mT5-Large/-Base/-Small to construct the student policy. Training details. We initialize the corresponding teacher and student models using 10,000-step-finetuning checkpoints on the SAMSum dataset, 80,000-step-fine-tuning checkpoints on the IWSLT’17 (en-de) dataset and 3,000-step-fine-tuning checkpoints on the StrategyQA dataset. We largely follow Ko et al. [21] to set the hyperparameters for training. In particular, we search for the learning rate from a preset range to obtain the best result for each baseline and our method. The batch size is selected to make full use of the RAM of GPUs. We use a relatively larger maximum number of training steps for IWSLT’17 (en-de) experiments to satisfy sufficient convergences for the machine translation task. We use beam search for the evaluation of the IWSLT’17 (en-de) dataset. Table 4: Hyperparameters for three task-specific experiments. Hyperparameter SAMSum IWSLT’17 (en-de) StrategyQA Max. Step Size (K) 10,000 80,000 3,000 Inner Step Size (N) 5 2 5 Batch Size (per GPU) {16, 32, 64} {16, 32, 64} {16, 32, 64} Dropout Rate 0.0 0.3 0.1 Controlling Factor (α) 0.1 0.1 0.1 Discounting Factor (γ) {0.90, 0.95, 0.99} {0.90, 0.95, 0.99} {0.90, 0.95, 0.99} Combination Factor (β) {0, 0.5, 0.9, 0.99, 1.0} {0, 0.5, 0.9, 0.99, 1.0} {0, 0.5, 0.9, 0.99, 1.0} Learning Rate (η) {5e−5, 1e−4, 5e−4} {1e−4, 5e−4, 1e−3} {1e−4, 5e−4, 1e−3} Warmup Steps 1,000 4,000 300 Weight Decay 1e−2 1e−4 1e−2 Max. Seq. Length 1,024 512 1,024 Sampling (top-p) 1.0 1.0 1.0 Sampling (temperature) 1.0 1.0 1.0 Evaluation Greedy Sampling Beam Search Greedy Sampling #GPUs 2 4 1 6https://huggingface.co/google/t5-v1_1-xl 7https://huggingface.co/google/t5-v1_1-large 8https://huggingface.co/google/t5-v1_1-base 9https://huggingface.co/google/t5-v1_1-small 23 Table 5: Comparison with state-of-the-art KD methods on the instruction-following dataset using fine-tuned GPT-2 XL (1.5B) as the teacher model and fine-tuned GPT-2 (0.1B) as the student model. We format the best, the second best and worse than SFT results. Method DollyEval SelfInst VicunaEval S-NI UnNI GPT-4 R-L GPT-4 R-L GPT-4 R-L R-L R-L GPT-2 XL (teacher) 45.5±0.7 28.2±0.8 34.7±1.6 14.3±0.2 32.7±1.6 16.2±0.3 27.6±0.3 32.2±0.3 SFT (student) 29.8±1.2 23.4±0.2 20.2±0.7 10.3±0.5 17.8±0.9 14.6±0.4 16.1±0.3 18.2±0.6 KD [16] 29.5±0.8 23.8±0.3 18.0±1.0 12.3±0.2 17.2±0.7 15.2±0.4 20.8±0.5 22.5±0.3 SeqKD [20] 29.8±0.5 24.2±0.2 18.2±0.8 11.6±0.4 18.2±0.7 15.5±0.3 15.5±0.6 20.1±0.1 ImitKD [24] 26.4±0.6 22.7±0.5 18.2±0.5 11.5±0.4 18.6±0.4 14.5±0.3 18.2±0.3 21.8±0.4 MiniLLM [14] 30.2±1.2 24.3±0.3 20.5±0.3 13.2±0.3 20.5±0.7 18.5±0.3 22.7±0.3 23.5±0.2 GKD [2] 29.2±0.6 23.6±0.2 20.7±0.5 12.7±0.2 20.2±0.6 17.7±0.2 25.1±0.3 25.9±0.1 DistiLLM [21] 31.2±0.4 25.2±0.4 21.7±0.5 12.5±0.3 22.5±1.2 19.2±0.5 27.7±0.2 27.6±0.4 Ours 31.7±0.5 26.1±0.3 22.7±0.5 14.2±0.3 23.6±0.8 20.5±0.2 28.6±0.2 29.9±0.5 KL RKL JS TV Ours 20 22 24 26 28 30 32 ROUGE-L on-policy off-policy mixed (a) DollyEval. KL RKL JS TV Ours 16 17 18 19 20 21 22 23 ROUGE-L on-policy off-policy mixed (b) SelfInst. KL RKL JS TV Ours 15 16 17 18 19 20 21 22 23 ROUGE-L on-policy off-policy mixed (c) VicunaEval. KL RKL JS TV Ours 30 32 34 36 38 40 ROUGE-L on-policy off-policy mixed (d) S-NI. KL RKL JS TV Ours 26 28 30 32 34 36 38 40 ROUGE-L on-policy off-policy mixed (e) UnNI. Figure 4: Performance of difference step-wise distribution distances on five instruction-following datasets using OpenLLaMA-7B →OpenLLaMA-3B. C Additional Results C.1 Results Based on GPT-2 In addition to the experimental results based on OpenLLaMA for instruction-following tasks, we also conduct experiments based on GPT-2. Results are illustrated in Table 5. Compared with current state-of-the-art KD approaches, our method achieves the best results on five datasets with both GPT-4 feedback and ROUGE-L evaluations. C.2 Comparisons on Step-Wise Distribution Distance Figure 4 and Figure 5 illustrate performance comparison with well-studied step-wise distribution distance, including KL, RKL, JS divergences and TV distances. Results show that the optimization of proposed moment-matching objectives outperforms other step-wise distribution distances via either on-policy distillation or off-policy distillation. Besides, jointly using on-policy and off-policy moment-matching further improves the performances and achieves the best results on five instructionfollowing datasets with KD from the OpenLLaMA-7B to OpenLLaMA-3B model, and achieves the best results on three task-specific datasets with KD from the (m)T5-XL to (m)T5-Base model. 24 KL RKL JS TV Ours 43 44 45 46 47 48 49 50 51 52 ROUGE-L on-policy off-policy mixed (a) SAMSum. KL RKL JS TV Ours 26 27 28 29 30 31 32 33 34 BLEU on-policy off-policy mixed (b) IWSLT’17 (en-de). KL RKL JS TV Ours 54 56 58 60 62 64 Accuracy on-policy off-policy mixed (c) StrategyQA. Figure 5: Performance of difference step-wise distribution distances on three task-specific datasets using (m)T5-XL →(m)T5-Base. 0 2000 4000 6000 8000 10000 Step 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Loss (train) training loss on-policy distance don off-policy distance doff 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Distance (eval) (a) Instruct-following. 0 2000 4000 6000 8000 10000 Step 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Loss (train) training loss on-policy distance don off-policy distance doff 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Distance (eval) (b) SAMSum. 0 10000 20000 30000 40000 50000 Step 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 Loss (train) training loss on-policy distance don off-policy distance doff 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Distance (eval) (c) IWSLT’17 (en-de). 0 250 500 750 1000 1250 1500 1750 2000 Step 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Loss (train) training loss on-policy distance don off-policy distance doff 1.0 1.2 1.4 1.6 1.8 2.0 Distance (eval) (d) StrategyQA. Figure 6: Training loss and don MM, doff MM against training step on four datasets. C.3 Adversarial Training Procedure Figure 6 illustrates the training loss and on-/off-policy moment-matching distances against the training steps on the instruction-following dataset and three task-specific datasets. We can observe that the training losses on four datasets have a similar trend, increasing at the beginning and then converging to a relatively lower level. The trend of loss function aligns with the characteristics of adversarial training with gradient descent ascent. In contrast, both the on-policy moment-matching distance don MM and the off-policy moment-matching distance doff MM reduce as the number of training steps increases, which shows the effectiveness of our adversarial training approach for moment-matching. 25 DollyEval SelfInst VicunaEval S-NI UnNI Average KL RKL JS TV Ours 2.65 4.87 3.21 3.67 3.02 3.48 2.54 4.78 3.87 3.58 2.67 3.48 2.31 4.52 3.67 3.32 2.43 3.25 1.98 2.34 3.32 2.13 1.83 2.32 0.96 1.85 2.11 0.85 0.92 1.34 don on five test sets 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 (a) On-policy moment-matching distance. DollyEval SelfInst VicunaEval S-NI UnNI Average KL RKL JS TV Ours 2.05 2.79 2.21 2.07 2.02 2.23 1.84 3.28 2.37 2.18 1.67 2.27 1.78 2.62 2.15 1.68 1.43 1.93 1.53 1.94 2.42 1.13 1.23 1.65 0.75 1.45 1.11 0.72 0.62 0.93 doff on five test sets 1.0 1.5 2.0 2.5 3.0 (b) Off-policy moment-matching distance. Figure 7: Moment-matching via distribution-matching on the instruction-following dataset. KL RKL JS TV Ours Small Base Large 2.85 2.58 2.35 2.52 1.67 2.1 2.32 2.18 2.06 1.32 2.02 2.25 1.98 1.78 1.24 don on five test sets 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 (a) On-policy moment-matching distance on SAMSum. KL RKL JS TV Ours Small Base Large 2.55 2.38 2.25 2.32 1.5 2.22 2.23 2.08 1.85 1.02 1.97 2.17 1.78 1.93 1.23 don on IWSLT 17 (en-de) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (b) On-policy moment-matching distance on IWSLT’17 (en-de). KL RKL JS TV Ours Small Base Large 2.61 2.31 2.52 2.18 1.87 2.28 2.37 2.33 2.12 1.61 2.42 2.16 2.25 1.87 1.39 don on StrategyQA 1.4 1.6 1.8 2.0 2.2 2.4 2.6 (c) On-policy moment-matching distance on StrategyQA. KL RKL JS TV Ours Small Base Large 2.85 2.78 3.17 2.77 1.82 2.4 2.28 2.37 2.18 1.21 2.21 2.32 2.27 2.02 1.34 doff on five test sets 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 (d) Off-policy moment-matching distance on SAMSum. KL RKL JS TV Ours Small Base Large 2.08 2.51 2.14 1.95 1.12 1.65 2.32 1.87 1.23 0.72 1.87 2.1 1.72 1.32 0.68 doff on IWSLT 17 (en-de) 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 (e) Off-policy moment-matching distance on IWSLT’17 (en-de). KL RKL JS TV Ours Small Base Large 2.52 2.23 2.46 2.24 1.37 1.95 2.18 2.32 2.03 1.28 2.23 1.89 2.13 1.91 1.19 doff on StrategyQA 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (f) Off-policy moment-matching distance on StrategyQA. Figure 8: Moment-matching via distribution-matching on three task-specific datasets. C.4 Moment-Matching via Distribution Matching We investigate how the distribution-matching methods via KL, RKL, JS divergences or TV distance can optimize the moment-matching distance in Figure 7 and Figure 8. Results show that the proposed adversarial training algorithm is more effective in minimizing the moment-matching distance than the distribution-matching methods. C.5 Analysis on the Off-/On-Policy Combination Factor β We study the impact of on-policy and off-policy objectives with the combination factor β ∈ {0.00, 0.25, 0.50, 0.75, 1.00} in Eq. (9), which denotes a linear combination coefficient of the on-policy and off-policy objectives. We observe that if β = 0.00, only the on-policy objective contributes to policy learning. As it increases from 0 to 1, the influence of off-policy objective increases while that of the on-policy objective decreases. Finally, when β = 1.00, only the off-policy objective contributes to policy learning. We conduct experiments across four datasets. Specifically, we evaluate ROUGE-L for OpenLLaMA2-3B on the DollyEval dataset, ROUGE-L for T5-base on the SAMSum dataset, accuracy for T5-base on the IWSLT’17 dataset and accuracy for T5-base on 26 Table 6: Effects of the off-/on-policy combination factor β on four datasets. β 0.00 0.25 0.50 0.75 1.00 DollyEval 28.8±0.7 31.2±0.3 30.7±0.4 29.8±0.2 27.4±0.4 SAMSum 48.2±0.3 50.5±0.2 50.4±0.3 51.2±0.4 48.7±0.2 IWSLT’17 (en-de) 30.7±0.1 31.7±0.6 32.4±0.2 33.2±0.2 31.2±0.2 StrategyQA 59.7±0.4 61.4±0.2 62.9±0.4 62.7±0.4 60.8±0.3 the StrategyQA dataset. Results in Table 6 show that a combination of on-policy and off-policy objectives outperforms using either on-policy or off-policy objectives only across four datasets. NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] . 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4,474	Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms Dimitri Meunier∗ Gatsby Computational Neuroscience Unit University College London dimitri.meunier.21@ucl.ac.uk Zikai Shen∗ Department of Statistical Science University College London zikai.shen.22@ucl.ac.uk Mattes Mollenhauer Merantix Momentum mattes.mollenhauer@merantix-momentum.com Arthur Gretton Gatsby Computational Neuroscience Unit University College London arthur.gretton@gmail.com Zhu Li Department of Mathematics Imperial College London zli12@ic.ac.uk Abstract We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression and various implementations of gradient descent. Our contributions are twofold. First, we rigorously confirm the so-called saturation effect for ridge regression with vector-valued output by deriving a novel lower bound on learning rates; this bound is shown to be suboptimal when the smoothness of the regression function exceeds a certain level. Second, we present an upper bound on the finite sample risk for general vector-valued spectral algorithms, applicable to both well-specified and misspecified scenarios (where the true regression function lies outside of the hypothesis space), and show that this bound is minimax optimal in various regimes. All of our results explicitly allow the case of infinite-dimensional output variables, proving consistency of recent practical applications. 1 Introduction We investigate a fundamental topic in modern machine learning—the behavior and efficiency of learning algorithms for regression in high-dimensional and potentially infinite-dimensional output spaces Y. Given two random variables X and Y , we seek to empirically minimize the squared expected risk E(F) ∶= E[∥Y −F(X)∥2 Y] (1) over functions F in a reproducing kernel Hilbert space consisting of vector-valued functions from a topological space X to a Hilbert space Y. The study of this setting as an ill-posed statistical inverse problem is well established: see e.g. 46, 6, 53, 3, 5, 17. In this work, we study the setting when Y is high- or infinite-dimensional, since it has been less well covered by the literature, yet has many applications in multitask regression [7, 2] and infinite-dimensional learning problems, including ∗Equal Contribution. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). the conditional mean embedding [20, 21, 41], structured prediction [11, 12], causal inference [43], regression with instrumental and proximal variables [42, 35], the estimation of linear operators and dynamical systems [47, 37, 26, 38, 25], and functional regression [24]. Interestingly, the aforementioned infinite-dimensional applications typically use the classical ridge regression algorithm. Our goal is to motivate the use of alternative learning algorithms in these settings, while providing strong theoretical guarantees. Classically, the ill-posed problem (1) is solved via regularization strategies, which are often implemented in terms of so-called spectral filter functions in the context of inverse problems in Hilbert spaces [16]. When applied to the learning problem given by (1), these filter functions correspond to learning algorithms including ridge regression, a variety of different implementations of gradient descent, principal component regression, and other related methods (we refer the reader to 19 and 2 for overviews of the real-valued and vector-valued output variable case, respectively). Algorithms based on spectral filter functions when Y = R are studied extensively, see e.g. [5, 34]. To the best of our knowledge, the detailed behavior of this general class of algorithms has remained unknown when Y is a general Hilbert space, with the exception of a few results for special cases in the setting of ridge regression [6, 31]. Overview of our contributions. In this manuscript, we aim to theoretically understand vector-valued spectral learning algorithms. The contribution of our work is twofold: (i) we rigorously confirm the saturation effect of ridge regression for general Hilbert spaces Y (see paragraph below) in the context of lower bounds on rates for the learning problem (1) and (ii) we cover a gap in the existing literature by providing upper rates for general spectral algorithms in high- and infinite-dimensional spaces. Our results explicitly allow the misspecified learning case in which the true regression function is not contained in the hypothesis space. We base our analysis on the concept of vector-valued interpolation spaces introduced by [30, 31]. The interpolation space norms measure the smoothness of the true regression function, replacing typical source conditions found in the literature which only cover the well-specified case. To the best of our knowledge, these are the first bounds covering this general setting for vector-valued spectral algorithms. Saturation effect of ridge regression. The widely-used ridge regression algorithm is known to exhibit the so-called saturation effect: it fails to exploit additional smoothness in the target function beyond a certain threshold. This effect has been thoroughly investigated in the context of Tikhonov regularization in inverse problems [16, Chapter 5], but is generally reflected only in upper rates in the learning literature, see e.g. [34, 5]. Interestingly, existing lower bounds [6, 5, 31] are usually formulated in a more general setting and do not reflect this saturation effect, leaving a gap between upper and lower rates. We leverage the bias-variance decomposition paradigm to lower bound the learning risk of kernel ridge regression with vector-valued output, in order to close this gap. Learning rates of vector-valued spectral algorithms. Motivated by the fact that the saturation effect is technically unavoidable with vector-valued ridge regression, we proceed to study the generalization error of popular alternative learning algorithms. In particular, we provide upper rates in the vector-valued setting consistent with the known behavior of spectral algorithms in the real-valued learning setting, based on their so-called qualification property [5, 34]. In particular, we confirm that a saturation effect can be bypassed in high and infinite dimensions by algorithms such as principal component regression and gradient descent, allowing for a better sample complexity for high-smoothness problems. Furthermore, we study the misspecified setting and show that upper rates for spectral algorithms match the state-of-the-art upper rates for misspecified vector-valued ridge regression recently obtained by [31]. Those rates are optimal for a wide variety of settings. Moreover, we argue that applications of vector-valued spectral algorithms are easy to implement by making use of an extended representer theorem based on [2], allowing for the numerical evaluation based on empirical data—even in the infinite-dimensional case. Related Work. The saturation effect of regularization techniques in deterministic inverse problems is well-known. For example, [40, 36, 22] study the saturation effect for Tikhonov regularization and general spectral algorithms. In the kernel statistical learning framework, the general phenomenon of saturation is discussed by e.g. [3, 19]. Recent work by [29] investigates saturation effect in the learning context by providing a lower bound on the learning rate. To the best of our knowledge, however, all studies in the statistical learning context focus on the case when Y is real-valued. General upper bounds of kernel ridge regression with real-valued or finite-dimensional Y have been extensively studied in the literature (see e.g., [6, 50, 8, 17]), where minimax optimal learning 2 rates are derived. Recent work [30, 31] studies the infinite-dimensional output space setting with Tikhonov regularization and obtains analogous minimax optimal learning rates. [23] later study a setting where both the input and output space is the infinite dimensional Sobolev RKHS and establish the minimax optimal rate. For kernel learning with spectral algorithms, existing work (see e.g., [3, 5, 32, 34, 54, 28]) focuses on real-valued output space setting and obtains optimal upper learning rates depending on the qualification number of the spectral algorithms, where only [54, 28] consider the misspecified learning scenario where the target regression function does not lie in the hypothesis space. For vector-valued output spaces, [27] considers learning with vector-valued random features. However, general investigations of spectral algorithms for vector-valued output spaces are absent in the literature. Structure of this paper. This paper is structured as follows. In Section 2, we introduce mathematical preliminaries related to reproducing kernel Hilbert spaces, vector-valued regression as well as the concept of vector-valued interpolation spaces. Section 3 contains a brief review the so-called saturation effect and a corresponding novel lower bound for vector-valued kernel ridge regression. In Section 4, we investigate general spectral learning algorithms in the context of vector-valued interpolations spaces and provide our main result: upper learning rates for this setting. 2 Background and Preliminaries Throughout the paper, we consider a random variable X (the covariate) defined on a second countable locally compact Hausdorff space2 X endowed with its Borel σ-field FX , and the random variable Y (the output) defined on a potentially infinite dimensional separable real Hilbert space (Y,⟨⋅,⋅⟩Y) endowed with its Borel σ-field FY. We let (Ω,F,P) be the underlying probability space with expectation operator E. Let P be the push-forward of P under (X,Y ) and π and ν be the marginal distributions on X and Y, respectively; i.e., X ∼π and Y ∼ν. We use the Markov kernel p ∶ X × FY →R+ to express the distribution of Y conditioned on X as P[Y ∈A∣X = x] = ∫A p(x,dy), for all x ∈X and events A ∈FY, see e.g. [15]. We introduce some notation related to linear operators on Hilbert spaces and vector-valued integration; formal definitions can be found in Appendix A for completeness, or we refer the reader to [52, 14]. The spaces of Bochner square-integrable functions with respect to π and taking values in Y are written as L2(X,FX ,π;Y), abbreviated as L2(π;Y). We obtain the classical Lebesgue spaces as L2(π) ∶= L2(π;R). Throughout the paper, we write [F] or more explicitly [F]π for the π-equivalence class of (potentially pointwise defined) measurable functions from X to Y, which we naturally interpret as elements in L2(π;Y) whenever they are square-integrable. Let H be a separable real Hilbert space with inner product ⟨⋅,⋅⟩H. We write L(H,H′) as the Banach space of bounded linear operators from H to another Hilbert space H′, equipped with the operator norm ∥⋅∥H→H′. When H = H′, we simply write L(H) instead. We write S2(H,H′) as the Hilbert space of Hilbert-Schmidt operators from H to H′ and S1(H,H′) as the Banach space of trace class operators (see Appendix A for a complete definition). For two Hilbert spaces H,H′, we say that H is (continuously) embedded in H′ and denote it as H ↪H′ if H can be interpreted as a vector subspace of H′ and the inclusion operator i ∶H →H′ performing the change of norms with ix = x for x ∈H is continuous; and we say that H is isometrically isomorphic to H′ and denote it as H ≃H′ if there is a linear isomorphism between H and H′ which is an isometry. Tensor Product of Hilbert Spaces: Denote H ⊗H′ the tensor product of Hilbert spaces H, H′. The element x ⊗x′ ∈H ⊗H′ is treated as the linear rank-one operator x ⊗x′ ∶H′ →H defined by y′ →⟨y′,x′⟩H′x for y′ ∈H′. Based on this identification, the tensor product space H ⊗H′ is isometrically isomorphic to the space of Hilbert-Schmidt operators from H′ to H, i.e., H ⊗H′ ≃S2(H′,H). We will hereafter not make the distinction between these two spaces, and treat them as being identical. Remark 1 (1, Theorem 12.6.1). Consider the Bochner space L2(π;H) where H is a separable Hilbert space. One can show that L2(π;H) is isometrically identified with the tensor product space H ⊗L2(π), and we denote as Ψ the isometric isomorphism between the two spaces. See Appendix A for more details on tensor product spaces and the explicit definition of Ψ. 2Under additional technical assumptions, the results in this paper can also be formulated when X is a more general topological space. However, some properties of kernels defined on X such as the so-called c0-universality [10] simplify the exposition when X is a second countable locally compact Hausdorff space. 3 Scalar-valued Reproducing Kernel Hilbert Space (RKHS). We let k ∶X × X →R be a symmetric and positive definite kernel function and H be a vector space of functions from X to R, endowed with a Hilbert space structure via an inner product ⟨⋅,⋅⟩H. We say that k is a reproducing kernel of H if and only if for all x ∈X we have k(⋅,x) ∈H and for all x ∈X and f ∈H, we have f(x) = ⟨f,k(x,⋅)⟩H. A space H which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS; see e.g. 4). We denote the canonical feature map of H as ϕ(x) = k(⋅,x). We require some technical assumptions on the previously defined RKHS and kernel, which we assume to be satisfied throughout the text: 1. H is separable: this is satisfied if k is continuous, given that X is separable3; 2. k(⋅,x) is measurable for π-almost all x ∈X; 3. k(x,x) ≤κ2 for π-almost all x ∈X. The above assumptions are not restrictive in practice, as well-known kernels such as the Gaussian, Laplace, and Matérn kernels satisfy them on X ⊆Rd [48]. We now introduce some facts about the interplay between H and L2(π), which has been extensively studied by [44, 45], [13] and [51]. We first define the (not necessarily injective) embedding Iπ ∶H →L2(π), mapping a function f ∈H to its π-equivalence class [f]. The embedding is a well-defined compact operator as long as its Hilbert-Schmidt norm is finite. In fact, this requirement is satisfied since its Hilbert-Schmidt norm can be computed as [51, Lemma 2.2 & 2.3] ∥Iπ∥S2(H,L2(π)) = ∥k∥L2(π) ≤κ. The adjoint operator Sπ ∶= I∗ π ∶L2(π) →H is an integral operator with respect to the kernel k, i.e. for f ∈L2(π) and x ∈X we have [49, Theorem 4.27] (Sπf)(x) = ∫X k (x,x′)f (x′)dπ (x′). Next, we define the self-adjoint, positive semi-definite and trace class integral operators LX ∶= IπSπ ∶L2(π) →L2(π) and CX ∶= SπIπ ∶H →H. Vector-valued Reproducing Kernel Hilbert Space (vRKHS). Let K ∶X ×X →L(Y) be an operator valued positive-semidefinite (psd) kernel. Fix K, x ∈X, and h ∈Y, then (Kxh)(⋅) ∶= K(⋅,x)h defines a function from X to Y. The completion of Gpre ∶= span{Kxh ∣x ∈X,h ∈Y} with inner product on Gpre defined on the elementary elements as ⟨Kxh,Kx′h′⟩G ∶= ⟨h,K (x,x′)h′⟩Y, defines a vRKHS denoted as G. For a more complete overview of the vectorvalued reproducing kernel Hilbert space, we refer the reader to [9], [10] and [31, Section 2]. In the following, we will denote G as the vRKHS induced by the kernel K ∶X × X →L(Y) with K(x,x′) ∶= k(x,x′)IdY, x,x′ ∈X. (2) We emphasize that this family of kernels is the de-facto standard for high- and infinite-dimensional applications [20, 21, 41, 11, 12, 42, 35, 43, 37, 26, 38, 25, 24] due to the crucial representer theorem which gives a closed form solution for the ridge regression problem based on the data. We generalize this representer theorem to cover the general spectral algorithm case in Proposition 1. Remark 2 (General multiplicative kernel). Without loss of generality, we provide our results for the vRKHS G induced by the operator-valued kernel given by K(x,x′) = k(x,x′)IdY. However, with suitably adjusted constants in the assumptions, our results transfer directly to the more general vRKHS ̃G induced by the more general operator-valued kernel ̃ K(x,x′) ∶= k(x,x′)T where T ∶Y →Y is any positive-semidefinite self-adjoint operator. The precise characterization of the adjusted constants is given by [31, Section 4.1]. An important property of G is that it is isometrically isomorphic to the space of Hilbert-Schmidt operators between H and Y [31, Corollary 1]. Similarly to the scalar case we can map every element in G into its π−equivalence class in L2(π;Y) and we use the shorthand notation [F] = [F]π (see Definition 6 in Appendix A for more details). 3This follows from [49, Lemma 4.33]. Note that the Lemma requires separability of X, which is satisfied since we assume that X is a second countable locally compact Hausdorff space. 4 Theorem 1 (vRKHS isomorphism). For every function F ∈G there exists a unique operator C ∈S2(H,Y) such that F(⋅) = Cϕ(⋅) ∈Y with ∥C∥S2(H,Y) = ∥F∥G and vice versa. Hence G ≃S2(H,Y) and we denote the isometric isomorphism between S2(H,Y) and G as ¯Ψ. It follows that G can be written as G = {F ∶X →Y ∣F = Cϕ(⋅), C ∈S2(H,Y)}. 2.1 Vector-valued Regression We briefly recall the basic setup of regularized least-squares regression with Hilbert space-valued random variables. The squared expected risk for vector-valued regression is E(F) ∶= E[∥Y −F(X)∥2 Y] = ∫X×Y ∥y −F(x)∥2 Yp(x,dy)π(dx), (3) for measurable functions F ∶X →Y. The analytical minimizer of the risk over measurable functions is the regression function or the conditional mean function F⋆∈L2(π;Y) given by F∗(x) ∶= E[Y ∣X = x] = ∫Y y p(x,dy), x ∈X. Throughout the paper, we assume that E[∥Y ∥2 Y] < +∞, i.e., the random variable Y is squareintegrable. Note that this implies F⋆∈L2(π;Y). Our focus in this work is to approximate F∗with kernel-based regularized least-squares algorithms, where we pay special attention to the case when Y is of high or infinite dimension. We pick G as a hypothesis space of functions in which to estimate F∗. Note that by Theorem 1, minimizing the functional E on G is equivalent to minimizing the following functional on S2(H,Y), ¯E(C) ∶= E[∥Y −Cϕ(X)∥2 Y]. (4) It is shown in [38, Proposition 3.5 and Section 3.4] that the optimality condition can be written as CY X = C∗CX, C∗∈S2(H,Y), (5) where CY X ∶= E[Y ⊗ϕ(X)] is the cross-covariance operator. As discussed in full detail by [38], the problem (5) can be formulated as a potentially ill-posed inverse problem on the space of Hilbert– Schmidt operators. As such, a regularization is required; we introduce regularized solutions of this problem in Section 4 through the classical concept of spectral filter functions. 2.2 Vector-valued Interpolation Space and Source Condition We now introduce the background required in order to characterize the smoothness of the target function F∗, both in the well-specified setting (F∗∈G) and in the misspecified setting (F∗∉G). We review the results of [51] and [17] in constructing scalar-valued interpolation spaces, and [30] in defining vector-valued interpolation spaces. Real-valued Interpolation Space: By the spectral theorem for self-adjoint compact operators, there exists an at most countable index set I, a non-increasing sequence (µi)i∈I > 0, and a family (ei)i∈I ∈H, such that ([ei])i∈I 4 is an orthonormal basis (ONB) of ran Iπ ⊆L2(π) and (µ1/2 i ei)i∈I is an ONB of (kerIπ)⊥⊆H, and we have LX = ∑ i∈I µi⟨⋅,[ei]⟩L2(π)[ei], CX = ∑ i∈I µi⟨⋅,µ 1 2 i ei⟩Hµ 1 2 i ei (6) For α ≥0, the α-interpolation space [51] is defined by [H]α ∶= {∑ i∈I aiµα/2 i [ei] ∶(ai)i∈I ∈ℓ2(I)} ⊆L2(π), equipped with the inner product ⟨∑ i∈I ai(µα/2 i [ei]),∑ i∈I bi(µα/2 i [ei])⟩ [H]α = ∑ i∈I aibi, 4We recall that the bracket [⋅] denotes the embedding that maps f to its equivalence class Iπ(f) ∈L2(π). 5 for (ai)i∈I ,(bi)i∈I ∈ℓ2(I). The α-interpolation space defines a Hilbert space. Moreover, (µα/2 i [ei]) i∈I forms an ONB of [H]α and consequently [H]α is a separable Hilbert space. In the following, we use the abbreviation ∥⋅∥α ∶= ∥⋅∥[H]α. Vector-valued Interpolation Space: Introduced in [30], vector-valued interpolation spaces generalize the notion of scalar-valued interpolation spaces to vRKHS with a kernel of the form (2). Definition 1 (Vector-valued interpolation space). Let k be a real-valued kernel with associated RKHS H and let [H]α be the real-valued interpolation space associated to H with some α ≥0. The vector-valued interpolation space [G]α is defined as (refer to Remark 1 for the definition of Ψ) [G]α ∶= Ψ(S2([H]α,Y)) = {F ∣F = Ψ(C), C ∈S2([H]α,Y)}. The space [G]α is a Hilbert space equipped with the inner product ⟨F,G⟩α ∶= ⟨C,L⟩S2([H]α,Y) (F,G ∈[G]α), where C = Ψ−1(F), L = Ψ−1(G). For α = 0, we retrieve ∥F∥0 = ∥F∥L2(π;Y) = ∥C∥S2(L2(π),Y). Remark 3 (Interpolation space inclusions). Note that we have F∗∈L2(π;Y) since Y ∈L2(P;Y) by assumption. Furthermore, for 0 < β < α, [17, Eq. (7)] imply the inclusions [G]α ↪[G]β ↪[G]0 ⊆L2(π;Y). Under assumptions 1 to 3 and with X being a second-countable locally compact Hausdorff space, [G]0 = L2(π;Y) if and only if H is dense in the space of continuous functions vanishing at infinity, equipped with the uniform norm [31, Remark 4]. Remark 4 (Well-specified versus misspecified setting). We say that we are in the well-specified setting if F∗∈[G]1. In this case, there exists ¯F ∈G such that F∗= ¯F π−almost surely and ∥F∗∥1 = ∥¯F∥G, i.e. F∗admits a representer in G (see Remark 5 in Appendix A). When F∗∈[G]β for β < 1, F∗may not admit such a representation and we are in the misspecified setting, as [G]1 ⊆[G]β. Definition 1 and Remarks 3 and 4 motivate the use of following assumption on the smoothness of the target function: there exists β > 0 and a constant B ≥0 such that F∗∈[G]β and ∥F∗∥β ≤B. (SRC) We let C∗∶= Ψ−1(F∗) ∈S2([H]β,Y). (SRC) directly generalizes the notion of a so-called Höldertype source condition in the learning literature [6, 17, 32, 34] and allows to characterize the misspecified learning scenario. 2.3 Further Assumptions In addition to (SRC), we require standard assumptions to obtain the precise learning rates for kernel learning algorithms. We list them below. For constants D2 > 0 and p ∈(0,1] and for all i ∈I, µi ≤D2i−1/p. (EVD) For constants D1,D2 > 0 and p ∈(0,1) and for all i ∈I, D1i−i p ≤µi ≤D2i−1/p. (EVD+) (EVD) and (EVD+) are standard assumptions on the eigenvalue decay of the integral operator: they describe the interplay between the marginal distribution π and the RKHS H (see more details in 6, 17). (EVD+) is needed in order to establish lower bounds on the excess risk. Note that we have excluded the value p = 1 from (EVD+); indeed, p = 1 is incompatible with the assumption of a bounded kernel, a fact missed by previous works and of independent interest (see Appendix, Remark 7). For α ∈[p,1], the inclusion Iα,∞ π ∶[H]α ↪L∞(π) is continuous, and ∃A > 0 such that ∥Iα,∞ π ∥[H]α→L∞(π) ≤A. (EMB) Property (EMB) is referred to as the embedding property in [17]. It can be shown that it holds if and only if there exists a constant A ≥0 with ∑i∈I µα i e2 i (x) ≤A2 for π-almost all x ∈X [17, Theorem 9]. Since we assume k to be bounded, the embedding property always hold true when α = 1. 6 Furthermore, (EMB) implies a polynomial eigenvalue decay of order 1/α, which is why we take α ≥p. (EMB) is not needed when we deal with the well-specified setting, but is crucial to bound the excess risk in the misspecified setting. Finally, we assume that there are constants σ,R > 0 such that ∫Y ∥y −F∗(x)∥q Yp(x,dy) ≤1 2q!σ2Rq−2, (MOM) is satisfied for π-almost all x ∈X and all q ≥2. The (MOM) condition on the Markov kernel p(x,dy) is a Bernstein moment condition used to control the noise of the observations (see 6, 17 for more details). If Y is almost surely bounded, for example ∥Y ∥Y ≤Y∞almost surely, then (MOM) is satisfied with σ = R = 2Y∞. It is possible to prove that the Bernstein condition is equivalent to sub-exponentiality, see [38, Remark 4.9]. 3 Saturation Effect of Kernel Ridge Regression The most established way of learning F∗is by kernel ridge regression (KRR), which can be formulated as the following optimization problem: given a dataset D = {(xi,yi)}n i=1 independently and identically sampled from the joint distribution of X and Y , ˆFλ ∶= arg min F ∈G 1 n n ∑ i=1 ∥yi −F(xi)∥2 Y + λ∥F∥2 G, (7) where λ > 0 is the regularization parameter. The generalization error of vector-valued KRR is expressed as ˆFλ −F∗, and controlled in different norms: see [31] for an extensive study. We recall here a simplified special case of the key results obtained in this work. In the next Theorem, ≲,≳are inequality up to positive multiplicative constants that are independent of n. Theorem 2 (Upper and lower bounds for KRR in the well-specified regime). Let ˆFλ be the KRR estimator from (7). Furthermore, let the conditions (EVD+), (SRC) and (MOM) be satisfied for some 0 < p ≤1 and β ≥1. Then, with high probability we have ∥[ ˆFλn] −F∗∥ 2 L2(π;Y) ≲n− min{β,2} min{β,2}+p for a choice λn = Θ(n− 1 β+p ), and furthermore for all learning methods (i.e., measurable maps) of the form D →ˆFD, ∥[ ˆFD] −F∗∥2 L2(π;Y) ≳n− β β+p . Theorem 2 shows the minimax optimal learning rate for vector-valued KRR for β ∈[1,2]. However, when β > 2, the obtained upper bound saturates at n− 2 2+p , creating a gap with the lower bound. This phenomenon is referred to as the saturation effect of Tikhonov regularization, and has been well investigated in deterministic inverse problems [40]. In the case where Y is real-valued, [29] prove that the saturation effect cannot be avoided with Tikhonov regularization. Below, we give a similar but generalized bound on lower rates for the case that Y is a Hilbert space. For this result only, we assume that X is a compact subset of Rd. We give the proof in Appendix B. Theorem 3 (Saturation of KRR). Let X be a compact subset of Rd. Let λ = λ(n) be an arbitrary choice of regularization parameter satisfying λ(n) →0 as n →+∞and let ˆFλ be the KRR estimator from (7). We assume that the noise is non-zero and bounded below, i.e. there exists σ > 0, such that ∫Y ∥y −F∗(x)∥2 Yp(x,dy) ≥σ2, is satisfied for π-almost all x ∈X. We assume in addition and for this result only that k is Hölder continuous (see Definition 11 in the appendix), i.e., k ∈Cθ(X × X) for θ ∈(0,1]. Suppose that Assumptions (EVD+) and (SRC) hold with p ∈(0,1) and β ≥2. For τ ≥0, for sufficiently large n > 0, where the hidden index bound depends on τ, with probability greater than 1 −e−τ, there exists some constant cτ > 0 such that E[∥[ ˆFλ] −F∗∥ 2 L2(π;Y) ∣x1,...,xn] ≥cτn− 2 2+p . 7 The assumption that k is Hölder continuous is crucial in lower bounding the variance with a covering number argument. Kernels satisfying this assumption include Gaussian kernels, Laplace kernels and Matérn kernels. Theorem 3 clearly demonstrates that the learning rate from vector-valued KRR cannot reach the information theoretic lower rate given in Theorem 2. As discussed above, [29] propose a similar lower bound in the real-valued case, and we now highlight two fundamental differences with [29] in the proof. First, while both works adopt the same bias-variance decomposition, we need to lower bound the bias and the variance term with infinite-dimensional output in our setting. Second, we adopt a different and simpler approach in proving the lower bound, since there are a number of issues with the proof of [29], both in the treatment of the bias and of the variance. For a detailed comparison with the earlier work, and an explanation of the differences in our approach, please refer to Remark 6 in the Appendix. 4 Consistency and optimal rates for general spectral algorithms Regularized population solution: Our goal is to regularize (5) in such a way that we get a unique and well-defined solution that provides a good approximation to F∗. We first recall the concept of a filter function (i.e., a function on an interval which is applied on self-adjoint operators to each individual eigenvalue via the spectral calculus, see 16), that will allow to define a regularization strategy. One may think of the following definition as a class of functions approximating the inversion map x ↦1/x while still being defined for x = 0 in a reasonable way. We use the definition given by [34], but equivalent definitions can be found throughout the literature. Definition 2 (Filter function). Let Λ ⊆R+. A family of functions gλ ∶[0,∞) →[0,∞) indexed by λ ∈Λ is called a filter with qualification ρ ≥0 if it satisfies the following two conditions: 1. There exists a positive constant E such that, for all λ ∈Λ sup α∈[0,1] sup x∈[0,κ2] λ1−αxαgλ(x) ≤E (8) 2. There exists a positive constant ωρ < ∞such that sup α∈[0,ρ] sup λ∈Λ sup x∈[0,κ2] ∣rλ(x)∣xαλ−α ≤ωρ, with rλ(x) ∶= 1 −gλ(x)x. (9) Below, we give some standard examples which are discussed by e.g. [19, 5] in the context of kernel regression with scalar output variables, and in [2] for the vector-valued case. A variety of additional algorithms can be expressed in terms of a filter function. 1. Ridge regression. From the Tikhonov filter function gλ(x) = (x + λ)−1, we obtain the known ridge regression algorithm. In this case, we have E = ρ = ωρ = 1. 2. Gradient Descent. From the Landweber iteration filter function given by gk(x) ∶= τ k−1 ∑ i=0 (1 −τx)i for k ∶= 1/λ,k ∈N we obtain the gradient descent scheme with constant step size τ > 0, which corresponds to the population gradient iteration given by Fk+1 ∶= Fk −τ∇E(Fk) for k ∈N. In this case, we have E = 1 and arbitrary qualification with ωρ = 1 whenever 0 < ρ ≤1 and ωρ = ρρ otherwise. Gradient schemes with more complex update rules can be expressed in terms of filter functions as well [39, 32, 34]. 3. Kernel principal component regression. The truncation filter function gλ(x) = x−11[x ≥λ] yields kernel principal component regression, corresponding to a hard thresholding of eigenvalues at a truncation level λ. In this case we have E = ωρ = 1 for arbitrary qualification ρ. Population solution: Given a filter function gλ, we call gλ(CX)5 the regularized inverse of CX. We may think of the regularized inverse as approximating the pseudoinverse of CX (see e.g. [16]) when λ →0. We define the regularized population solution to (4) as Cλ ∶= CY Xgλ(CX) ∈S2(H,Y), Fλ(⋅) ∶= Cλϕ(⋅) ∈G. (10) 5gλ(CX) is defined with the rules of spectral calculus, see Definition 9 in the Appendix. 8 The solution arising from standard regularization strategies leads to well-known statistical methodologies. We refer to [16] for the background on filter functions in classical regularization theory. Empirical solution: Given the dataset D = {(xi,yi)}n i=1, the empirical analogue of (10) is ˆCλ ∶= ˆCY Xgλ( ˆCX), ˆFλ(⋅) ∶= ˆCλϕ(⋅) ∈G, (11) where ˆCY X, ˆCX are empirical covariance operators define as ˆCX ∶= 1 n n ∑ i=1 ϕ(xi) ⊗ϕ(xi) ˆCY X ∶= 1 n n ∑ i=1 yi ⊗ϕ(xi). Note that (11) is the regularized solution of the empirical inverse problem ˆCY X = ˆC ˆCX, ˆC ∈S2(H,Y), which arises as the optimality condition for minimizers on G of the empirical analogue of (3), given by En(F) ∶= 1 n ∑n i=1 ∥yi −F(xi)∥2 Y; see Proposition 2 in the Appendix for a proof. For the vector-valued kernel given in (2), it is well-known that ˆFλ can be computed in closed-form for the ridge regression estimator—even in infinite dimensions [47]. For general filter functions, an extended representer theorem is given by [2] in the context of finite-dimensional multitask learning: this approach works in infinite dimensions as well. We give the closed form solution based on [2] below (we include the proof in Appendix D.1). Proposition 1 (Representer theorem for general spectral filter). Let (K)ij = k(xi,xj), 1 ≤i,j ≤n denote the Gram matrix associated to the scalar-valued kernel k. We have ˆFλ(x) = n ∑ i=1 yiαi(x), α(x) = 1 ngλ (K n )kx ∈Rn, (kx)i = k(x,xi), 1 ≤i ≤n. (12) Example 1 (Conditional integration). Consider now a random variable Z taking values in a topological space Z on which we define a second RKHS H′ ⊆RZ with kernel ℓ∶Z × Z →R and canonical feature map ψ ∶Z →H′,z ↦ℓ(z,⋅). The conditional mean embedding [47, 20] is defined as F∗(x) ∶= E[ψ(Z) ∣X = x], x ∈X. We immediately see the link with vector-valued regression with Y = ψ(Z) and Y = H′. The conditional mean embedding allows us to compute the conditional expectation of any element of H′. Indeed, using the reproducing property, for f ∈H′, we have for all x ∈X, E[f(Z) ∣X = x] = ⟨f,E[ψ(Z) ∣X = x]⟩H′. Given a dataset {(xi,zi)}n i=1 6 and an estimate of the conditional mean embedding F∗with a spectral algorithm ˆFλ as in Eq. (11), and substituting the formula in Eq. (12), we obtain E[f(Z) ∣X = x] ≈ ⟨f, ˆFλ(x)⟩H′ = ∑n i=1⟨f,ψ(zi)⟩H′αi(x) = f ⊺ z α(x), where (fz)i = f(zi), 1 ≤i ≤n. Learning rates: We now give our main result, the learning rates for the difference between [ ˆFλ] and F∗in the interpolation norm, where Fλ and ˆFλ are given by (10) and (11) based on a general spectral filter satisfying Definition 2. The proof is deferred to Section C in the Appendix. Theorem 4 (Upper learning rates). Let ˆFλ be an estimator based on a general spectral filter with qualification ρ ≥0. Furthermore, let the conditions (EVD), (EMB), (MOM) be satisfied with 0 < p ≤α ≤1. With 0 ≤γ ≤1, if (SRC) is satisfied with γ < β ≤2ρ, we have 1. in the case β + p ≤α, let λn = Θ((n/logθ(n)) −1 α ) for some θ > 1, for all τ > log(6) and sufficiently large n ≥1, there is a constant J > 0 independent of n and τ such that ∥[ ˆFλn] −F∗∥ 2 γ ≤τ 2J ( n logθ n ) −β−γ α is satisfied with P n-probability not less than 1 −6e−τ. 6Note that this induces a dataset D = {(xi, ψ(zi))}n i=1 where we identify yi = ψ(zi). 9 2. in the case β + p > α, let λn = Θ(n− 1 β+p ), for all τ > log(6) and sufficiently large n ≥1, there is a constant J > 0 independent of n and τ such that ∥[ ˆFλn] −F∗∥ 2 γ ≤τ 2Jn−β−γ β+p is satisfied with P n-probability not less than 1 −6e−τ. Theorem 4 provides the upper rate for vector-valued spectral algorithms. In particular, in combination with the lower bound in Theorem 2, we see that vector-valued spectral algorithms with qualification ρ achieve an optimal learning rate when the smoothness β of the regression function is in the range (α−p,2ρ]. For algorithms with infinite ρ such as gradient descent and principal component regression, we confirm that they can exploit smoothness of the target function just as in the real-valued setting [3, 5, 30], while not suffering from saturation. For Tikhonov regularization, where ρ = 1, the rates recover the state-of-the-art results from [31]. Finally, we point out that obtaining minimax optimal learning rates for β < α −p still remains challenging even in the real-valued output scenario. Note however that for a large variety of RKHS, α is arbitrarily close to p and we obtain optimal rates for the whole range (0,2ρ]: we refer to [31, 54] for a detailed discussion. We provide a proof sketch for Theorem 4. The key technical challenge in extending the results of [31] to spectral filter functions lies in the analysis of the estimation error. The estimation error in γ−norm is bounded as ∥[ ˆCλ −Cλ]∥S2([H]γ,Y) ≤3λ−γ 2 ∥( ˆCλ −Cλ) ˆC 1 2 X,λ∥ S2(H,Y) (see Eq. (37) in Appendix C.3). We rely on the fact that IdH = ˆCXgλ( ˆCX) + rλ( ˆCX) (see Definition 2), to obtain the decomposition ˆCλ −Cλ = ( ˆCY X −Cλ ˆCX)gλ( ˆCX) −Cλrλ( ˆCX), which yields two terms to be controlled, ∥( ˆCλ −Cλ) ˆC 1 2 X,λ∥ S2(H,Y) ≤∥( ˆCY X −Cλ ˆCX)gλ( ˆCX) ˆC 1 2 X,λ∥ S2(H,Y) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ (I) +∥Cλrλ( ˆCX) ˆC 1 2 X,λ∥ S2(H,Y) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ (II) To control term (I), we use the definition of the filter function gλ (Definition 2) to obtain that ∥ˆCX,λgλ( ˆCX)∥H→H ≲1. Thus it suffices to control the term ∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ S2(H,Y) = ∥1 n ∑n i=1 ξ(xi,yi)∥S2(H,Y), where ξ(x,y) = (y −Cλϕ(x)) ⊗C −1 2 X,λϕ(x). We proceed by bounding E[∥ξ(X,X)∥m S2(H,Y)] for m ≥1, and then use Bernstein’s inequality to derive the upper bound on ∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ S2(H,Y) . To control term (II), Lemma 9 in Appendix C.1 shows that (II) ≲∥ˆC 1 2 X,λrλ( ˆCX)gλ(CX)C β+1 2 X ∥ H→H . The term on the right side is bounded in prior work on scalar-valued spectral method, and we refer the reader to [54, Theorem 16]. The results of Theorem 4 are then obtained by choosing regularization parameter λ = λ(n) to optimally trade off approximation and estimation errors. 5 Conclusion In this work, we have rigorously explored the theoretical properties of vector-valued spectral learning algorithms, focusing on their performance in infinite-dimensional output spaces. 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On the optimality of misspecified kernel ridge regression. In International Conference on Machine Learning, pages 41331–41353. PMLR, 2023. 13 Appendices The appendix is organized as follows. In Section A, we give additional mathematical background and notations. In Section B, we give the proof of Theorem 3 and provide a technical comparison of our proof with [29]. In Section C, we prove Theorem 4. Finally, in Section D, we provide auxiliary results used in the main proofs. A Additional Background A.1 Hilbert spaces and linear operators Definition 3 (Bochner Lq−spaces, [14]). Let H be a separable Hilbert space and π a probability measure on X. For 1 ≤q ≤∞, Lq(X,FX ,π;H), abbreviated Lq(π;H), is the space of strongly FX −FH measurable and Bochner q-integrable functions from X to H, with the norms ∥f∥q Lq(π;H) = ∫X ∥f∥q H dπ, 1 ≤q < ∞, ∥f∥L∞(π;H) = inf {C ≥0 ∶π{∥f∥H > C} = 0}. Definition 4 (p-Schatten class, e.g. [52]). Let H,H′ be separable Hilbert spaces. For 1 ≤q ≤∞, Sp(H,H′), abbreviated Sp(H) if H = H′, is the Banach space of all compact operators C from H to H′ such that ∥C∥Sp(H,H′) ∶= ∥(σi(C))i∈I∥ℓp is finite. Here ∥(σi(C))i∈I ∥ℓp is the ℓp−sequence space norm of the sequence of the strictly positive singular values of C indexed by the at most countable set I. For p = 2, we retrieve the space of Hilbert-Schmidt operators, for p = 1 we retrieve the space of Trace Class operators, and for p = +∞, ∥⋅∥S∞(H,H′) corresponds to the operator norm ∥⋅∥H→H′. Definition 5 (Tensor Product of Hilbert Spaces, [1]). Let H,H′ be Hilbert spaces. The Hilbert space H ⊗H′ is the completion of the algebraic tensor product with respect to the norm induced by the inner product ⟨x1 ⊗x′ 1,x2 ⊗x′ 2⟩H⊗H′ = ⟨x1,x2⟩H⟨x′ 1,x′ 2⟩H′ for x1,x2 ∈H and x′ 1,x′ 2 ∈H′ defined on the elementary tensors of H ⊗H′. This definition extends to span{x ⊗x′∣x ∈H,x′ ∈H′} and finally to its completion. The space H ⊗H′ is separable whenever both H and H′ are separable. If {ei}i∈I and {e′ j}j∈J are orthonormal basis in H and H′, {ei ⊗e′ j}i∈I,j∈J is an orthonormal basis in H ⊗H′. Theorem 5 (Isometric Isomorphism between L2(π;Y) and S2(L2(π),Y), Theorem 12.6.1 [1]). Let H be a separable Hilbert space. The Bochner space L2(π;H) is isometrically isomorphic to S2(L2(π),Y) and the isometric isomorphism is realized by the map Ψ ∶S2(L2(π),Y) →L2(π;H) acting on elementary tensors as Ψ(f ⊗y) = (ω →f(ω)y). A.2 RKHS embbedings into L2 and Well-specifiedness Recall that Iπ ∶H →L2(π) is the embedding that maps every function in H into its π-equivalence class in L2(π) and that we used the shorthand notation [f] = Iπ(f) for all f ∈H. We define similarly Iπ ∶G →L2(π;Y) as the embedding that maps every function in G into its π-equivalence class in L2(π;Y). Definition 6 (Embedding G into L2(π;Y)). Let Iπ ∶= IY ⊗Iπ be the tensor product of the operator IdY with the operator Iπ (see [1, Definition 12.4.1.] for the definition of tensor product of operators). Iπ maps every function in G into its π-equivalence class in L2(π;Y). We then use the shorthand notation [F] = Iπ(F) for all F ∈G. Remark 5. Let {dj}j∈J be an orthonormal basis of Y and recall that {√µi[ei]}i∈I forms an orthonormal basis of [H]1. Let F ∈[G]1. Then F can be represented as the element C ∶= ∑i∈I,j∈J aijdj ⊗√µi[ei] in S2([H]1,Y) by definition of [G]1 with ∥C∥2 1 = ∑i,j a2 ij. Hence defining ¯C ∶= ∑i∈I,j∈J aijdj ⊗√µiei we have C = ¯C π−a.e. and ∥¯C∥2 G = ∑ i∈I,j∈J a2 i,j = ∥C∥2 1 < +∞. Taking the elements identifying ¯C in G gives a representer ¯F of F in G. 14 A.3 Additional Notations In the following, we fix {dj}j∈J an orthonormal basis of Y, where J is at most countable. Recall that {µ1/2 i ei} i∈I is an ONB of (kerIπ)⊥in H, and {[ei]}i∈I is an ONB of ran Iπ in L2(π). Let {˜ei}i∈I′ be an ONB of kerIπ (with I ∩I′ = ∅), then {µ1/2 i ei} i∈I ∪{˜ei}i∈I′ forms an ONB of H, and {dj ⊗µ1/2 i ei} i∈I,j∈J ∪{dj ⊗˜ei}i∈I′,j∈J forms an ONB of Y ⊗H ≃G. For any Hilbert space H, linear operator T ∶H →H and scalar λ > 0, we define Tλ ∶= T + λIH. B Saturation Effect with Tikhonov Regularization - Proof of Theorem 3 In the following proofs a quantity hn ≥0 depending on n ≥1, but independent of τ the confidence level, is equal to o(1) if hn →0 when n →+∞. We will make extensive use of the following notation in the subsequent analysis. Definition 7 (Empirical L2(π)−norm). Denoted by ⟨⋅,⋅⟩2,n, the empirical L2(π)−norm associated to points {xi}n i=1 independently and identically sampled from the distribution of X, is defined as, for any f,g ∈H, ⟨f,g⟩2,n ∶= ⟨ˆCX,f ⊗g⟩S2(H) = ⟨ˆCXf,g⟩H = ⟨ˆC 1 2 Xf, ˆC 1 2 Xg⟩ H = 1 n n ∑ i=1 f(xi)g(xi). This induces an inner product on H, with associated norm, ∥f∥2 2,n = ⟨f,f⟩2,n = 1 n n ∑ i=1 f(xi)2. Definition 8. Fix x ∈X and λ > 0. The regularized canonical feature map is defined as fx,λ(⋅) = C−1 X,λk(x,⋅) ∶X →H. Recall from Eq. (11) that the ridge estimator ˆFλ defined in Eq. (7) can be expressed as ˆCλ = ˆCY Xgλ( ˆCX), ˆFλ(⋅) = ˆCλϕ(⋅) ∈G, where in Theorem 3 we focus on Tikhonov regularization where gλ(x) = (x + λ)−1. In that setting we have rλ (x) ∶= 1 − x x + λ = − λ x + λ. (13) Proof of Theorem 3. Since β ≥2, F∗∈[G]β ⊆[G]1, therefore F∗has a representer ¯F in G such that F∗= ¯F π-a.e. (see Remark 5), and by Theorem 1, ¯F(⋅) = ¯Cϕ(⋅), with ¯C ∈S2(H,Y). Define the errors ϵi ∶= yi −¯Cϕ(xi), i = 1,...,n, that are i.i.d samples with the same distribution as ϵ ∶= Y −¯Cϕ(X). By assumption E[∥ϵ∥2 Y ∣X] ≥σ2 and by definition E[ϵ ∣X] = 0. By Eq. (13), we have rλ ( ˆCX) ∶= I −ˆCX ˆC−1 X,λ = −λ ˆC−1 X,λ. 15 The following bias-variance decomposition is the essence of the proof. In the following derivation we abbreviate S2(L2(π),Y) to S2 L2(π;Y) to L2 and x1,...,xn to xn to save space. E[∥[ ˆFλ] −F∗∥ 2 L2 ∣xn] = E[∥[ ˆCY X ˆC−1 X,λ −¯C]∥ 2 S2 ∣xn] = E ⎡⎢⎢⎢⎢⎣ ∥[( 1 n n ∑ i=1 yi ⊗ϕ(xi)) ˆC−1 X,λ −¯C]∥ 2 S2 ∣xn ⎤⎥⎥⎥⎥⎦ = E ⎡⎢⎢⎢⎢⎣ ∥[ 1 n n ∑ i=1 ( ¯Cϕ(xi) + ϵi) ⊗ϕ(xi) ˆC−1 X,λ −¯C]∥ 2 S2 ∣xn ⎤⎥⎥⎥⎥⎦ = E ⎡⎢⎢⎢⎢⎣ ∥[−¯Crλ ( ˆCX) + 1 n n ∑ i=1 ϵi ⊗( ˆC−1 X,λϕ(xi))]∥ 2 S2 ∣xn ⎤⎥⎥⎥⎥⎦ = ∥[ ¯Crλ ( ˆCX)]∥ 2 S2 + 1 n2 n ∑ i=1 E[∥ϵi∥2 Y ∣xi]∥[ ˆC−1 X,λϕ(xi)]∥ 2 L2(π) ≥λ2 ∥[ ¯C ˆC−1 X,λ]∥ 2 S2 + σ2 n2 n ∑ i=1 ∥[ ˆC−1 X,λϕ(xi)]∥ 2 L2(π) . The second term is a lower bound on the variance while the first term is a lower bound on the bias. Bounding the Bias term. The idea is to first show that the population analogue of ∥[ ¯C ˆC−1 X,λ]∥ 2 S2(L2(π),Y) can be bounded below by a non-zero constant. We can then bound the difference between the empirical and population version of ∥[ ¯C ˆC−1 X,λ]∥ 2 S2(L2(π),Y) using a concentration inequality. By Lemma 1, for λ > 0, there is a constant c > 0 (see Lemma 1 for the exact value of c) such that ∥[ ¯CC−1 X,λ]∥ 2 S2(L2(π),Y) ≥c > 0. Furthermore by Lemma 2, there is a constant c0 > 0 (see Lemma 2 for the exact value of c0) such that for any τ ≥log(4), with probability at least 1 −4e−τ, for n ≥(c0τ)(4+2p) and 1 ≥λ ≥n− 1 2+p , we have ∣∥[ ¯CC−1 X,λ]∥ 2 S2(L2(π),Y) −∥[ ¯C ˆC−1 X,λ]∥ 2 S2(L2(π),Y)∣= τ 2o(1). Therefore, under the same high probability, ∥[ ¯C ˆC−1 X,λ]∥2 S2(L2(π),Y) ≥c −τ 2o(1). It leads to our final bound on the bias term, for a constant ρ2 ≥0 and for sufficiently large n ≥1, where the hidden index bound depends on τ, we have λ2 ∥[ ¯C ˆC−1 X,λ]∥ 2 S2(L2(π),Y) ≥ρ1λ2. (14) Bounding the Variance Term. Using the norm from Definition 7, we have the following chain of identities. σ2 n2 n ∑ i=1 ∥[ ˆC−1 X,λϕ(xi)]∥ 2 L2(π) = σ2 n2 n ∑ i=1∫X ⟨ϕ(X), ˆC−1 X,λϕ(xi)⟩ 2 H dπ(x) = σ2 n ∫X ∥ˆC−1 X,λϕ(X)∥2 2,ndπ(x). (15) Therefore it suffices to consider ∫X ∥ˆC−1 X,λk(x,⋅)∥2 2,ndπ(x). Combining the result of Lemma 4 and Lemma 5 we obtain that for 1 ≥λ ≥n− 1 2+p with probability at least 1 −6e−τ, for n ≥(c0τ)4+2p, the following bounds hold simultaneously for all x ∈X: ∥ˆC 1 2 X ( ˆC−1 X,λ −C−1 X,λ)k(x,⋅)∥ H ≤τo(1) ∥[C−1 X,λk(x,⋅)]∥2 2,n −1 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) ≥−τo(1) ∥[C−1 X,λk(x,⋅)]∥2 2,n −3 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) ≤τo(1). 16 Fix x ∈X. Using the algebraic identity a2 −b2 = (a −b)(2b + (a −b)), and recalling that by Definition 7, ∥f∥2 2,n = ∥ˆC 1 2 Xf∥ 2 H , we deduce ∣∥ˆC 1 2 X ˆC−1 X,λk(x,⋅)∥ 2 H −∥ˆC 1 2 XC−1 X,λk(x,⋅)∥ 2 H ∣ ≤∥ˆC 1 2 X ( ˆC−1 X,λ −C−1 X,λ)k(x,⋅)∥ H ⋅(∥ˆC 1 2 X ( ˆC−1 X,λ −C−1 X,λ)k(x,⋅)∥ H + 2∥C−1 X,λk(x,⋅)∥2,n) ≤τo(1)(τo(1) + 2∥C−1 X,λk(x,⋅)∥2,n). Using Definition 7 again, this reads ∥ˆC−1 X,λk(x,⋅)∥ 2 2,n ≥∥C−1 X,λk(x,⋅)∥ 2 2,n −τo(1)(τo(1) + 2∥C−1 X,λk(x,⋅)∥2,n). We have ∥C−1 X,λk(x,⋅)]∥2 2,n ≤3 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) + τo(1) ≤( √ 1.5∥[C−1 X,λk(x,⋅)]∥L2(π) + √τo(1)) 2 . Hence, ∥ˆC−1 X,λk(x,⋅)∥ 2 2,n ≥∥C−1 X,λk(x,⋅)∥ 2 2,n −τo(1)(∥[C−1 X,λk(x,⋅)]∥L2(π) + √τo(1) + τo(1)) ≥1 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) −τo(1) −τo(1)(∥[C−1 X,λk(x,⋅)]∥L2(π) + √τo(1) + τo(1)) ≥1 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) −τ 2o(1) −τo(1)∥[C−1 X,λk(x,⋅)]∥L2(π). By Lemma 17, ∫X ∥[C−1 X,λk(x,⋅)]∥ 2 L2(π) dπ(x) = N2(λ). Furthermore, by Jensen’s inequality, ∫X ∥[C−1 X,λk(x,⋅)]∥2 L2(π)dπ(x) ≥(∫X ∥[C−1 X,λk(x,⋅)]∥L2(π)dπ(x)) 2 . Recall from Lemma 16 that c1,2λ−p ≤N2(λ) ≤c2,2λ−p. Therefore we have ∫X ∥[C−1 X,λk(x,⋅)]∥L2(π)dπ(x) ≤√c2,2λ−p 2 ∫X ∥[C−1 X,λk(x,⋅)]∥2 L2(π)dπ(x) ≥c1,2λ−p. Hence ∫X ∥ˆC−1 X,λk(x,⋅)∥2 2,ndπ(x) ≥c1,2 2 λ−p −τ 2o(1) −τo(1)√c2,2λ−p 2 ≥(c1,2 2 −τo(1)√c2,2)λ−p −τ 2o(1) Combined with Eq. (15), it leads to our final bound on the variance term, for a constant ρ2 ≥0 and for sufficiently large n ≥1, where the hidden index bound depends on τ, we have σ2 n2 n ∑ i=1 ∥[ ˆC−1 X,λϕ(xi)]∥ 2 L2(π) ≥ρ2 nλp . (16) Putting it together. We are now ready to assemble the lower bounds on the variance and on the bias. For a fixed confidence parameter τ ≥log(10), for sufficiently large n > 0, where the hidden index 17 bound depends on τ, with probability at least 1 −10e−τ, we have by Eq. (14) and Eq. (16), that for λ = λ(n) satisfying 1 ≥λ ≥n− 1 2+p , E[∥[ ˆFλ] −F∗∥ 2 L2(π;Y) ∣x1,...,xn] ≥ρ1λ2 + ρ2n−1λ−p where ρ1,ρ2 have no dependence on n. Recall Young’s inequality, for r,q > 1 satisfying r−1+q−1 = 1, we have for all a,b ≥0, a + b ≥r 1 r q 1 q a 1 r b 1 q . We apply Young’s inequality with r−1 = p/(2 + p) and q−1 = 2/(2 + p), there exists a constant c1 > 0 such that ρ1λ2 + ρ2n−1λ−p ≥c1 (λ2) p 2+p (λ−pn−1) 2 2+p = c1n− 2 2+p . To conclude the proof, let λ = λ(n) be an arbitrary choice of regularization parameter satisfying λ(n) →0. We have just covered the case 1 ≥λ ≥n− 1 2+p and the case 0 < λ ≤n− 1 2+p is covered by [29, Section B.4]. Lemma 1. For any λ ≤1 and C ∈S2(H,Y), with C ⊥̸ S2(ran Sπ,Y)7, we have ∥[CC−1 X,λ]∥ 2 S2(L2(π),Y) ≥ ∑ i∈I,j∈J a2 ij µi (µi + 1)2 > 0, with aij ∶= ⟨dj,C√µiei⟩Y, i ∈I,j ∈J. Proof. Recell the notations of Section A.3. Define {aij}i∈I∩I′,j∈J such that aij ∶= ⟨dj,C√µiei⟩Y for i ∈I,j ∈J and aij ∶= ⟨dj,C˜ei⟩Y for i ∈I′,j ∈J. Then, on one hand, since C ∈S2(H,Y), C = ∑ i∈I,j∈J aijdj ⊗(√µiei) + ∑ i∈I′,j∈J aijdj ⊗˜ei. On the other hand, C−1 X,λ = ∑ i∈I (µi + λ)−1(√µiei) ⊗(√µiei) + λ−1 ∑ i∈I′ ˜ei ⊗˜ei. Therefore, noting that ˜ei = 0 π−a.e. for all i ∈I′, we have, [CC−1 X,λ] = ⎡⎢⎢⎢⎢⎣ ∑ i∈I,j∈J aij(µi + λ)−1dj ⊗(√µiei) + ∑ i∈I′,j∈J aij λ dj ⊗˜ei ⎤⎥⎥⎥⎥⎦ = ∑ i∈I,j∈J aij √µi µi + λdj ⊗[ei]. Therefore the S2(L2(π),Y)-norm can be evaluated in closed form using Parseval’s identity, ∥[CC−1 X,λ]∥ 2 S2(L2(π),Y) = ∑ i∈I,j∈J a2 ij µi (µi + λ)2 ≥ ∑ i∈I,j∈J a2 ij µi (µi + 1)2 , where we used that {dj ⊗[ei]}j∈J,i∈I is orthonormal in Y ⊗L2(π), and λ ≤1. The right hand side has no dependence on λ or n. Furthermore, under assumption (EVD+), µi > 0 for all i ∈I, therefore the right hand side term equals zero if and only if aij = 0 for all i ∈I,j ∈J. Since by assumption C ⊥̸ S2(ran Sπ,Y), the right hand side is strictly positive. Lemma 2. Suppose Assumption (EVD) holds with p ∈(0,1]. Let C ∈S2(H,Y) such that [C] ∈ S2([H]2,Y). There is a constant c0 > 0 such that for any τ ≥log(4), with probability at least 1 −4e−τ, for n ≥(c0τ)(4+2p) and 1 ≥λ ≥n− 1 2+p , we have ∣∥[CC−1 Xλ]∥ 2 S2(L2(π),Y) −∥[C ˆC−1 Xλ]∥ 2 S2(L2(π),Y)∣≤τ 2o(1) We have c0 ∶= 8κmax{√c2,1,1}) where c2,1 is defined in Lemma 16. 7⊥̸ is the notation for “not being orthogonal to”. 18 Proof. Using the identity A−1 −B−1 = A−1(B −A)B−1, we obtain C−1 X,λ −ˆC−1 X,λ = C−1 X,λ( ˆCX −CX) ˆC−1 X,λ We apply Lemma 22 with γ = 0, ∥[C (C−1 X,λ −ˆC−1 X,λ)]∥S2(L2(π),Y) =∥[C ˆC−1 X,λ( ˆCX −CX)C−1 X,λ]∥S2(L2(π),Y) =∥C ˆC−1 X,λ( ˆCX −CX)C−1 X,λC 1 2 X∥ S2(H,Y) ≤∥C ˆC −1 2 X,λ∥ S2(H,Y) ∥ˆC −1 2 X,λC 1 2 X,λ∥ H→H ⋅∥C −1 2 X,λ( ˆCX −CX)C −1 2 X,λ∥ H→H ∥C −1 2 X,λC 1 2 X∥ H→H (17) We consider each of the four terms in line (17). The last term is bounded above by 1 and the first term is bounded above by λ−1 2 ∥C∥S2(H,Y). By Lemma 20 applied with s = 1/2, we have for the second term ∥ˆC −1 2 X,λC 1 2 X,λ∥ H→H ≤∥ˆC−1 X,λCX,λ∥ 1 2 H→H . Then, by Lemma 18, for τ ≥ log(2), with probability at least 1 −2e−τ, for √ nλ ≥ 8τκ √ max{N(λ),1}, we have ∥ˆC−1 X,λCX,λ∥H→H ≤2. Since N(λ) ≤c2,1λ−p by Lemma 16, and λ ≤1, it suffices to verify that λ satisfies √ nλ ≥8τκmax{√c2,1,1}λ−p 2 . Since λ ≥n− 1 2+p by assumption, we deduce the sufficient condition n ≥(τc0)2(2+p), where c0 ∶= 8κmax{√c2,1,1}. We bound the third term using Lemma 16 [33]. For τ ≥log(2), with probability at least 1 −2e−τ, we have λ−1 2 ∥C −1 2 X,λ(CX −ˆCX)C −1 2 X,λ∥ H→H ≤4κ2ξδ 3nλ 3 2 + √ 2κ2ξδ nλ2 , where we define ξδ ∶= log 2κ2(N1(λ) + 1) e−τ∥CX∥H→H . By assumption λ ≥n− 1 2+p . We thus have nλ 3 2 ≥n 1+2p 4+2p and nλ2 ≥n p 2+p . On the other hand, since 1 ≥λ ≥n− 1 2+p , using Lemma 16, we have ξδ ≤log 82(c2,1λ−p + 1) e−τ∥CX∥H→H ≤log 2(c2,1 + 1)n p 2+p e−τ∥CX∥H→H ≤log 2(c2,l + 1) e−τ∥CX∥H→H + p 2 + p log n. The first term does not depend on n, and the second term is logarithmic in n. Putting everything together with a union bound, we get a bound on (17). With probability at least 1 −4e−τ, for n ≥(c0τ)(4+2p), we have ∥[C (C−1 X,λ −ˆC−1 X,λ)]∥S2(L2(π),Y) ≤∥C∥S2(H,Y) √ 2⎛ ⎝ 4ξδ 3n 0.5+p 2+p + √ 2ξδ n p 2+p ⎞ ⎠= τo(1) The derivations in the proof of Lemma 1 show that [CC−1 X,λ] = ∑ i∈I,j∈J aij √µi µi + λdj ⊗[ei], 19 with aij ∶= ⟨dj,C√µiei⟩Y, i ∈I,j ∈J. Note that since [C] ∈S2([H]2,Y), we have ∥[C]∥2 S2([H]2,Y) = XXXXXXXXXXX ∑ i∈I,j∈J aijdj ⊗(√µiei) XXXXXXXXXXX 2 S2([H]2,Y) = ∑ i∈I,j∈J a2 ij µi < +∞. Hence, ∥[CC−1 X,λ]∥ 2 S2(L2(π),Y) = ∑ i∈I,j∈J a2 ij µi (µi + λ)2 ≤ ∑ i∈I,j∈J a2 ij µi = ∥[C]∥2 S2([H]2,Y) < +∞. (18) Using the equality a2 −b2 = (a −b)(a + b) and the reverse triangular inequality, we obtain the following bound, with probability at least 1 −4e−τ, for n ≥(c0τ)(4+2p), ∣∥[CC−1 X,λ]∥ 2 S2(L2(π),Y) −∥[C ˆC−1 X,λ]∥ 2 S2(L2(π),Y)∣ ≤∥[C (C−1 X,λ −ˆC−1 X,λ)]∥S2(L2(π),Y) (∥[CC−1 X,λ]∥S2(L2(π),Y) + ∥[C ˆC−1 X,λ]∥S2(L2(π),Y)) ≤τo(1)(2∥[CC−1 X,λ]∥S2(L2(π),Y) + ∥[C (C−1 X,λ −ˆC−1 X,λ)]∥S2(L2(π),Y)) ≤τo(1)(2∥[C]∥S2([H]2,Y) + τo(1)) =τ 2o(1), where in the second last line we used Equation (18). Lemma 3. Fix x ∈X and fx,λ as in Definition 8. For τ ≥log(2), with probability at least 1 −2e−τ (note that this event depends on x), ∣∥fx,λ∥2 2,n −∥[fx,λ]∥2 L2(π)∣≤1 2∥[fx,λ]∥2 L2(π) + 5τκ2 3λ2n. Proof. We start with ∥fx,λ∥∞≤κ∥fx,λ∥H ≤κ2λ−1. We apply Proposition 3 to f = fx,λ, with M = κ2λ−1. For τ ≥log(2), with probability at least 1 −2e−τ, ∣∥fx,λ∥2 2,n −∥[fx,λ]∥2 L2(π)∣≤1 2∥[fx,λ]∥2 L2(π) + 5τκ2 3λ2n. Lemma 4. Suppose that X is a compact set in Rd and that k ∈Cθ(X × X) for θ ∈(0,1] (Definition 11). Assume that 1 ≥λ ≥n− 1 2+p . With probability at least 1 −2e−τ, it holds for all x ∈X simultaneously that ∥C−1 X,λk(x,⋅)∥2 2,n ≥1 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) −τo(1), ∥C−1 X,λk(x,⋅)∥2 2,n ≤3 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) + τo(1). Proof. The proof follows [29, Lemma C.11]. As we use different notations and tracking of constants, we provide a similar proof in our setting for completeness. By Lemma 24, there exists an ϵ-net F ⊆Kλ ⊆H with respect to ∥⋅∥∞such that there exists a positive constant c with ∣F∣≤c(λϵ)−2d θ , for ϵ to be determined later. Using Lemma 3 and a union bound over the finite set F, with probability at least 1 −2e−τ, it holds simultaneously for all f ∈F that ∣∥f∥2 2,n −∥[f]∥2 L2(π)∣≤1 2∥[f]∥2 L2(π) + 5(τ + log(∣F∣))κ2 3λ2n . (19) 20 We work in the event where Equation (19) holds for all f ∈F. By definition of an ϵ-net and Kλ, for any x ∈X, there exists some f ∈F such that ∥C−1 X,λk(x,⋅) −f∥∞≤ϵ, which in particular implies that ∣∥[C−1 X,λk(x,⋅)]∥L2(π) −∥[f]∥L2(π)∣≤ϵ ∣∥C−1 X,λk(x,⋅)∥2,n −∥f∥2,n∣≤ϵ. Since ∥C−1 X,λk(x,⋅)∥∞≤κ2λ−1, using the algebraic identity a2 −b2 = (a−b)(2b+(a−b)), we obtain ∣∥[C−1 X,λk(x,⋅)]∥2 L2(π) −∥[f]∥2 L2(π)∣≤ϵ(2κ2λ−1 + ϵ) ∣∥C−1 X,λk(x,⋅)∥2 2,n −∥f∥2 2,n∣≤ϵ(2κ2λ−1 + ϵ). We therefore have, ∥C−1 X,λk(x,⋅)∥2 2,n ≤∥f∥2 2,n + ϵ(2κ2λ−1 + ϵ) ≤3 2∥[f]∥2 L2(π) + 5(τ + log(∣F∣)κ2 3λ2n + ϵ(2κ2λ−1 + ϵ) ≤3 2∥[C−1 X,λk(x,⋅)]∥2 L2(π) + 5(τ + log(∣F∣)κ2 3λ2n + 2ϵ(2κ2λ−1 + ϵ). We now choose ϵ = 1 n and bound the error term. Recall that 1 ≥λ ≥n− 1 2+p , therefore, 5(τ + log(∣F∣)κ2 3λ2n + 2ϵ(2κ2λ−1 + ϵ) ≤5(τ + log(∣F∣)κ2 3 n− p 2+p + 2(2κ2n −1−p 2+p + 1 n2 ) ≤5κ2 3 (τ + log(cλ−2d θ n 2d θ ))n− p 2+p + 2(2κ2n −1−p 2+p + 1 n2 ) = τo(1). Lemma 5. For 1 ≥λ ≥n− 1 2+p , with probability at least 1 −4e−τ, for n ≥(c0τ)4+2p, we have for all x ∈X simultaneously ∥ˆC 1 2 X ˆC−1 X,λ(CX −ˆCX)C−1 X,λk(x,⋅)∥ H = τo(1), (20) where c0 is the same constant as in Lemma 2. Proof. ∥ˆC 1 2 X ˆC−1 X,λ(CX −ˆCX)C−1 X,λk(x,⋅)∥ H =∥ˆC 1 2 X ˆC −1 2 X,λ ˆC −1 2 X,λC 1 2 X,λC −1 2 X,λ(CX −ˆCX)C−1 X,λk(x,⋅)∥ H ≤∥ˆC 1 2 X ˆC −1 2 X,λ∥ H→H ∥ˆC −1 2 X,λC 1 2 X,λ∥ H→H ⋅∥C −1 2 X,λ(CX −ˆCX)C −1 2 X,λ∥ H→H ∥C −1 2 X,λk(x,⋅)∥ H We already saw in the proof of Lemma 2 that the first term is bounded by 1 and there is a constant c0 > 0 such that for τ ≥log(2), with probability at least 1 −2e−τ, for n ≥(c0τ)4+2p, the second term is bounded by √ 2. For the third term we also saw in the proof of Lemma 4 that for τ ≥log(2), with probability at least 1 −2e−τ, we have λ−1 2 ∥C −1 2 X,λ(CX −ˆCX)C −1 2 X,λ∥ H→H ≤4κ2ξδ 3nλ 3 2 + √ 2κ2ξδ nλ2 , 21 where we defined ξδ = log 2κ2(N1(λ) + 1) e−τ∥CX∥H→H . Finally, the fourth term is bounded above by λ−1 2 κ. Note that the bound on the fourth term is independent of x, so it holds simultaneously for all x ∈X. This is in contrast with the setting of Lemma 4 where for each fixed x ∈X corresponds an element in the ϵ-net of F for which we have a high probability bound, and therefore we must use a union bound in order for the bound to hold simultaneously for all x ∈X in the proof of Lemma 4. As in the proof of Lemma 4 since 1 ≥λ ≥n− 1 2+p , we have ξδ ≤log 2(c2,l + 1) e−τ∥CX∥H→H + p 2 + p log n. In the bound on ξδ above, the first term does not depend on n, and the second term is logarithmic in n. Putting everything together by union bound, with probability at least 1 −4e−τ, for n ≥(c0τ)4+2p, we have ∥ˆC 1 2 X ˆC−1 X,λ(CX −ˆCX)C−1 X,λk(x,⋅)∥ H = τo(1). Remark 6 (Comparison to [29]). We explicit the differences between our proof strategy and the proof strategy of [29]. • Scalar versus vector-valued: lower bounding the bias in our case require us to accommodate for the vector-valued setting (see Lemma 1). • New proof of the bias: we lower bound the bias through Lemma 2, while [29] obtain the lower bound in Lemma C.7; however the proof of Lemma C.7 implicitly uses the equality ∥A−1∥= ∥A∥−1, with ∥⋅∥the operator norm, see Eq. (69) [29] and the preceding equations. It holds that ∥A−1∥≥∥A∥−1, but ∥A−1∥≤∥A∥−1 may not hold in general. We therefore develop a new proof for this step, leading to Lemma 2. • New proof of the variance: we lower bound the variance in Lemma 5, while [29] lower bound the variance in Lemma C.12; to show Eq. (20), [29] use a covering argument involving N(Kλ,∥⋅∥H,ϵ) (Lemma C.10). However, a close look at the proof of Lemma C.10 (last inequality of the proof) reveals that λi λ+λi was mistaken for λ λ+λi and plugging the correct term in the proof would lead to a vacuous bound. As explained in the proof of Lemma 5, we therefore develop a proof that is free of a covering number argument for this step. C Learning rates for spectral algorithms To upper bound the excess-risk, we use a decomposition involving the approximation error expressed as Fλ −F∗and the estimation error expressed as ˆFλ −Fλ. ∥[ ˆFλ] −F∗∥γ ≤∥[ ˆFλ −Fλ]∥γ + ∥[Fλ] −F∗∥γ , where ˆFλ is the empirical estimator based on general spectral regularization (Eq. (11)) and Fλ is its counterpart in population (Eq. (10)). Note that this is a different decomposition than the bias-variance decomposition used in the proof of Theorem 3. The proof structure is as follows: 1. Fourier expansion C.1. 2. Approximation Error C.2. 3. Estimation error C.3 22 C.1 Fourier expansion Recall the notations defined in Appendix A.3. The family {dj}j∈J is an ONB of Y, the family {µ1/2 i ei}i∈I is an ONB of (kerIπ)⊥and the family {˜ei}i∈I′ is an ONB of kerIπ such that {µ1/2 i ei} i∈I ∪{˜ei}i∈I′ forms an ONB of H. Furthermore, recall that {µβ/2 i [ei]}i∈I is an ONB of [H]β, β ≥0. Lemma 6 (Fourier expansion). Suppose Assumption (SRC) holds with β ≥0. By definition of the vector-valued interpolation space and by Theorem 5, we have F∗= ∑ i∈I,j∈J aijdj[ei], aij = ⟨F∗,dj[ei]⟩L2(π;Y) , ∥F∗∥2 β = ∑ i∈I,j∈J a2 ij µβ i . (21) Then, we have the following equalities with respect to this Fourier decomposition. 1. The Hilbert-Schmidt operator Cλ ∈S2(H,Y), Eq. (10), can be written as Cλ = ∑ i∈I,j∈J aijgλ(µi)√µidj ⊗√µiei. (22) 2. The Hilbert-Schmidt operator (CY X −CλCX)C −1 2 X,λ ∈S2(H,Y) can be written as (CY X −CλCX)C −1 2 X,λ = ∑ i∈I,j∈J aijrλ(µi)(µi + λ)−1 2 √µi (dj ⊗√µiei) (23) 3. The Hilbert-Schmidt operator CY X ∈S2(H,Y) can be written as CY X = ⎛ ⎝∑ i∈I,j∈J aijµ −β 2 i dj ⊗√µiei ⎞ ⎠C β+1 2 X (24) Proof. We first derive the Fourier expansion of CY X, CY X = EX,Y [Y ⊗ϕ(X)] = EX [F∗(X) ⊗ϕ(X)] (25) = EX ⎡⎢⎢⎢⎢⎣ ∑ i∈I,j∈J aijei(X)dj ⊗ϕ(X) ⎤⎥⎥⎥⎥⎦ = EX ⎡⎢⎢⎢⎢⎣ ∑ i∈I,j∈J aijdj ⊗(∑ k∈I √µkek(X)√µkek)ei(X) ⎤⎥⎥⎥⎥⎦ = ∑ ijk aij √µk ⋅EX[ek(X)ei(X)] ⋅dj ⊗(√µkek) = ∑ i∈I,j∈J aij √µidj ⊗(√µiei), (26) where in Eq. (25) we used the tower property of conditional expectation and in Eq. (26) we used the fact that {[ei]}i∈I forms an orthonormal system in L2(π). We can manipulate Eq. (26) to derive Eq. (24), CY X = ∑ i∈I,j∈J aijµ 1 2 −β+1 2 i dj ⊗(C β+1 2 X (√µiei)) = ⎛ ⎝∑ i∈I,j∈J aijµ −β 2 i dj ⊗√µiei ⎞ ⎠C β+1 2 X . By the spectral decomposition of CX Eq. (6) and spectral calculus (Definition 9), we have that gλ(CX) = ∑ i∈I gλ(µi)√µiei ⊗√µiei + gλ(0) ∑ i∈I′ ˜ei ⊗˜ei, (27) rλ(CX) = ∑ i∈I rλ(µi)√µiei ⊗√µiei + ∑ i∈I′ ˜ei ⊗˜ei. (28) 23 where we used rλ(0) = 1. Proof of Eq. (22). Using Eq. (26) and (27), we have Cλ = ⎛ ⎝∑ i∈I,j∈J aij √µidj ⊗(√µiei)⎞ ⎠(∑ k∈I gλ(µk)(√µkek) ⊗(√µkek) + gλ(0) ∑ l∈I′ ˜el ⊗˜el) = ∑ ijk aij √µigλ(µk)δikdj ⊗(√µkek) (29) = ∑ i∈I,j∈J aij √µigλ(µi)dj ⊗(√µiei), where in Eq. (29), we recall the fact that {√µiei}i∈I forms an ONB of (kerIπ)⊥and {˜ei}i∈I′ forms an ONB of kerIπ. Proof of Eq. (23). Using Eq. (26) and (28), we have (CY X−CλCX)C −1 2 X,λ = CY Xrλ(CX)C −1 2 X,λ = ⎛ ⎝∑ i∈I,j∈J aij √µidj ⊗(√µiei)⎞ ⎠(∑ k∈I rλ(µk)√µkek ⊗√µkek + ∑ l∈I′ ˜el ⊗˜el)C −1 2 X,λ = ⎛ ⎝∑ ijk aij √µirλ(µk)dj ⊗(√µkek)δik ⎞ ⎠C −1 2 X,λ = ∑ i∈I,j∈J aij √µirλ(µi)dj ⊗(C −1 2 X,λ(√µiei)) = ∑ i∈I,j∈J aij √µi(µi + λ)−1 2 rλ(µi)dj ⊗(√µiei), Lemma 7. Suppose Assumption (SRC) holds with β ≥0, then the following bound is satisfied, for all λ > 0 and 0 ≤γ ≤1, we have ∥[Fλ]∥2 γ ≤E2∥F∗∥2 min{γ,β}λ−(γ−β)+. For the definition of E, see Eq. (8). Proof. We adopt the notations of Lemma 6. By Parseval’s identity and Eq. (22), we have ∥[Fλ]∥2 γ = ∥Cλ∥2 S2([H]γ,Y) = ∑ i∈I,j∈J a2 ijgλ(µi)2µ2−γ i . In the case of γ ≤β, we bound gλ(µi)µi ≤E using Eq. (8). Then, by Eq. (21), ∥[Fλ]∥2 γ ≤E2 ∑ i∈I,j∈J a2 ij µγ i = E2∥F∗∥2 γ. In the case of γ > β, we apply Eq. (8) to gλ(µi)µ 1−γ−β 2 i ≤Eλ−γ−β 2 to obtain, using Eq. (21) again, ∥[Fλ]∥2 γ = ∑ i∈I,j∈J gλ(µi)2µ2−(γ−β) i µ−β i a2 ij ≤E2λ−(γ−β) ∑ i∈I,j∈J µ−β i a2 ij = E2λ−(γ−β)∥F∗∥2 β. 24 Lemma 8. Suppose Assumption (SRC) holds for 0 ≤β ≤2ρ, with ρ the qualification. Then, the following bound is satisfied, for all λ > 0, we have ∥(CY X −CλCX)C −1 2 X,λ∥ S2(H,Y) ≤ωρ∥F∗∥βλ β 2 . For the definition of ωρ, see Eq. (9). Proof. Recall that in Lemma 6 we used the decomposition F∗= ∑ i∈I,j∈J aijdj[ei], where Assumption (SRC) implies that ∥F∗∥2 β = ∑ij a2 ij µβ i < ∞. Using Eq. (23) in Lemma 6 and Parseval’s identity w.r.t. the ONS {dj ⊗µ1/2 i ei}i∈I,j∈J in S2(H,Y), we have ∥(CY X −CλCX)C −1 2 X,λ∥ S2(H,Y) = ⎛ ⎝∑ i∈I,j∈J a2 ijr2 λ(µi)(µi + λ)−1µi ⎞ ⎠ 1 2 ≤⎛ ⎝∑ i∈I,j∈J a2 ij µβ i r2 λ(µi)µβ i ⎞ ⎠ 1 2 ≤∥F∗∥β sup i∈I rλ(µi)µ β 2 i ≤∥F∗∥βωρλ β 2 . Lemma 9. Suppose Assumption (SRC) holds with β ≥0, then for all λ > 0, we have ∥Cλrλ ( ˆCX) ˆC 1 2 X,λ∥ S2(H,Y) ≤B ∥ˆC 1 2 X,λrλ( ˆCX)gλ(CX)C β+1 2 X ∥ H→H , where ∥F∗∥β = B < ∞. Proof. Recall that Lemma 6 we used the decomposition F∗= ∑ i∈I,j∈j aijdj[ei], where ∥F∗∥2 β = ∑ij a2 ij µβ i = B2 < ∞. Using Eq. (24) in Lemma 6 and Cλ = CY Xgλ(CX), we have ∥Cλrλ ( ˆCX) ˆC 1 2 X,λ∥ S2(H,Y) = XXXXXXXXXXX ⎛ ⎝∑ ij aijµ −β 2 i dj ⊗√µiei ⎞ ⎠C β+1 2 X gλ(CX)rλ( ˆCX) ˆC 1 2 X,λ XXXXXXXXXXXS2(H,Y) ≤B ∥C β+1 2 X gλ(CX)rλ( ˆCX) ˆC 1 2 X,λ∥ H→H , where we notice that the S2(H,Y) norm of the first term is exactly the β norm of F∗, which is given by B. Recalling that CX, ˆCX are self adjoint, we prove the final result by taking the adjoint and using that an operator has the same operator norm as its adjoint. C.2 Approximation Error Lemma 10. Let Fλ be given by Eq. (10) based on a general spectral filter satisfying Definition 2 with qualification ρ ≥0. Suppose Assumption (SRC) holds with parameter β ≥0 and define βρ = min{β,2ρ}, then the following bound is satisfied, for all λ > 0 and 0 ≤γ ≤βρ, ∥[Fλ] −F∗∥2 γ ≤ω2 ρ ∥F∗∥2 βρ λβρ−γ. 25 Proof. In Eq. (10), we defined Fλ(⋅) = Cλϕ(⋅). On the other hand, in Lemma 6 we obtained the Fourier expansion of Cλ leading to Eq. (22). Thus we have for π−almost all x ∈X, Fλ(x) = ∑ i∈I,j∈J aijµigλ(µi)djei(x). Therefore, [Fλ] −F∗= ∑ i∈I,j∈J aij(1 −µigλ(µi))dj[ei] = ∑ i∈I,j∈J aijrλ(µi)dj[ei]. Suppose β ≤2ρ, using Parseval’s identity w.r.t. the ONB {djµγ/2 i [ei]}i∈I,j∈J of [G]γ, we have ∥[Fλ] −F∗∥2 γ = XXXXXXXXXXX ∑ i∈I,j∈J aij µγ/2 i rλ(µi)djµγ/2 i [ei] XXXXXXXXXXX 2 γ = ∑ i∈I,j∈J a2 ij µγ i r2 λ(µi) = ∑ i∈I,j∈J a2 ij µβ i r2 λ(µi)µβ−γ i ≤ω2 ρλβ−γ ∑ i∈I,j∈J a2 ij µβ i = ∥F∗∥2 βω2 ρλβ−γ where we used Eq. (9) in the definition of a filter function, together with 0 ≤β ≤2ρ and 0 ≤γ ≤β, which taken together implies that 0 ≤β−γ 2 ≤ρ. Finally, if β ≥2ρ, then since [G]β ⊆[G]2ρ, we can perform the last derivations again with β = 2ρ to obtain the final result. C.3 Estimation error Before proving the main results we recall two embedding properties for the vector-valued interpolation space [G]β (Definition 1). The first embedding property lifts the property (EMB) defined for the scalar-valued RKHS [H]α to the vector-valued RKHS [G]α. Lemma 11 (L∞-embedding property - Lemma 4 [31]). Under (EMB) the inclusion operator Iα,β π ∶ [G]α ↪L∞(π;Y) is bounded with operator norm A, Theorem 6 (Lq-embedding property - Theorem 3 [31]). Let Assumption (EMB) be satisfied with parameter α ∈(0,1]. For any β ∈[0,α), the inclusion map Iqα,β π ∶[G]β ↪Lqα,β(π;Y) is bounded, where qα,β ∶= 2α α−β . The Lq-embedding property was first introduced in the scalar-valued setting in [55] and then lifted to the vector-valued setting by [31]. Its role is to replace a boundedness condition on the ground truth function F∗. We now explain how the Lq-embedding property can be combined with Assumption (EMB) and a truncation technique. Lemma 12. Recall that π is the marginal measure of X on X. For t ≥0, define the measurable set Ωt as follows Ωt ∶= {x ∈X ∶∥F∗(x)∥Y ≤t} Let q > 0. Assume that F∗∈Lq(π;Y). In other words, there exists some constant cq > 0 such that ∥F∗∥Lq(π;Y) = (∫X ∥F∗(x)∥q Ydπ(x)) 1 q = cq < +∞, Then we have the following conclusions 1. The π-measure of the complement of Ωcan be bounded by π({x ∉Ωt}) ≤cq q tq . 26 2. Recall that {xi}n i=1 are i.i.d. samples distributed according to π. If t = n 1 ˜q for ˜q < q, then we can conclude as follows. For a fixed parameter τ > 0, for all sufficiently large n, where the hidden index bound depends on q˜q−1 and τ, we have π⊗n (∩n i=1{xi ∈Ωt}) ≥1 −e−τ. Proof. The first claim is a straightforward application of Markov’s inequality, as follows π({x ∉Ωt}) = π (∥F∗(x)∥Y > t) ≤Eπ [∥F∗(X)∥q Y] tq = cq q tq . To show the second claim, we first evaluate the probability that there exists some xi’s that lies outside Ωt, π⊗n (∪n i=1{xi ∉Ωt}) = 1 −π⊗n (∩n i=1{xi ∈Ωt}) = 1 −π({xi ∈Ωt})n ≤1 −(1 −cq q tq ) n ≤cq qn tq , where in the last inequality we used Bernoulli’s inequality, which states that for r ≥1 and 0 ≤x ≤1, (1 −x)r ≥1 −rx. By assumption t = n 1 qt for some fixed q > qt > 0. We thus have π⊗n (∪n i=1{xi ∉Ωt}) ≤cq qn1−q qt ≤e−τ, for sufficiently large n, where the hidden index bound depends on q qt and τ. We adapt [31, Lemma 5] to the spectral algorithms setting. Lemma 13. Suppose Assumptions (SRC) and (EMB) hold for some 0 ≤β ≤2ρ, with ρ the qualification, then the following bounds are satisfied, for all 0 < λ ≤1, ∥[Fλ] −F∗∥2 L∞≤(∥F∗∥L∞+ Amax{E,ωρ}∥F∗∥β)2 λβ−α, (30) ∥[Fλ]∥2 L∞≤A2E2∥F∗∥2 min{α,β}λ−(α−β)+, (31) Proof. We use Lemma 11 and Lemma 7 to write: ∥[Fλ]∥2 ∞≤A2∥[Fλ]∥2 α ≤A2E2∥F∗∥2 min{α,β}λ−(α−β)+. This proves Eq. (31). To show Eq. (30), in the case β ≤α we use the triangle inequality, Eq. (31) and λ ≤1 to obtain ∥[Fλ] −F∗∥∞≤∥F∗∥∞+ ∥[Fλ]∥∞ ≤(∥F∗∥∞+ AE∥F∗∥β)λ−α−β 2 . In the case β > α, Eq. (30) is a consequence of Lemma 11 and Lemma 10 with γ = α (here we use the assumption 0 ≤β ≤2ρ), ∥[Fλ] −F∗∥2 ∞≤A2∥[Fλ] −F∗∥2 α ≤A2ω2 ρ∥F∗∥2 βλβ−α ≤(∥F∗∥∞+ Aωρ∥F∗∥β)2λβ−α. We adapt [54, Theorem 13] to the vector-valued setting. 27 Theorem 7. Suppose that Assumptions (EMB), (EVD), (MOM) and (SRC) hold for 0 ≤β ≤2ρ, where ρ is the qualification, and p ≤α ≤1. Denote, for i = 1,...,n, ξi = ξ(xi,yi) = ((yi −Cλϕ(xi)) ⊗ϕ(xi))C −1 2 X,λ, and for t ≥0, Ωt = {x ∈X ∶∥F∗(x)∥Y ≤t} Then for all τ ≥1, with probability at least 1 −2e−τ, we have ∥1 n n ∑ i=1 ξi1{xi ∈Ωt} −E[ξ(X,Y )1{X ∈Ωt}]∥ S2(H,Y) ≤τ ⎛ ⎝c1λ β 2 −αn−1 + c2λ−α 2 n−1(t + R + A) + c3 √ N1(λ) √n + c4 √nλ α−β 2 ⎞ ⎠ where R is the constant from Assumption (MOM), and c1 = 8 √ 2A2 max{E,ωρ}∥F∗∥β c2 = 8 √ 2A c3 = 8 √ 2σ c4 = 8 √ 2A∥F∗∥βωρ where A is the constant from Assumption (EMB), and E,ωρ are defined in Eq. (8) and (9) respectively. Proof. We wish to apply vector-valued Bernstein’s inequality, namely Theorem 10. We thus compute, E[∥ξ(X,Y )1{X ∈Ωt}∥m S2(H,Y)] = E[1{X ∈Ωt}∥(Y −Cλϕ(X)) ⊗(C −1 2 X,λϕ(X))∥ m S2(H,Y) ] = E[1{X ∈Ωt}∥(Y −Cλϕ(X))∥m Y ∥C −1 2 X,λϕ(X)∥ m H ] = ∫Ωt ∥C −1 2 X,λϕ(x)∥ m H ∫Y ∥y −Cλϕ(x)∥m Y dp(x,dy)dπ(x). (32) First we consider the inner integral, by Assumption (MOM), ∫Y ∥(y −Cλϕ(x))∥m Y dp(x,dy) ≤2m−1 (∫Y ∥y −F∗(x)∥m Y + ∥Fλ(x) −F∗(x)∥m Y )dp(x,dy) = m!σ2(2R)m−2 + 2m−1 ∥Fλ(x) −F∗(x)∥m Y . Plugging the above inequality into Eq. (32), as well as introducing the shorthand, hx ∶= C −1 2 X,λϕ(x), we have E[∥ξ(X,Y )1{X ∈Ωt}∥m S2(H,Y)] ≤m!σ2(2R)m−2 ∫Ωt ∥hx∥m Hdπ(x) (33) + 2m−1 ∫Ωt ∥hx∥m H ∥Fλ(x) −F∗(x)∥m Y dπ(x). We bound term (33) using Lemma 15 and Lemma 17 with l = 1. We have, ∫Ωt ∥hx∥m Hdπ(x) ≤(Aλ−α 2 )m−2N1(λ). Therefore we bound term (33) as follows, m!σ2(2R)m−2 ∫Ωt ∥hx∥m Hdπ(x) ≤m!σ2 (2AR λ α 2 ) m−2 N1(λ). 28 If β ≥α, by Assumption (EMB), ∥F∗∥∞≤A∥F∗∥α ≤A∥F∗∥β. Hence by Lemma 13, ∥[Fλ] −F∗∥∞≤(∥F∗∥∞+ Amax{E,ωρ}∥F∗∥β)λ β−α 2 ≤A(1 + max{E,ωρ})∥F∗∥βλ β−α 2 . If β < α, by Lemma 13, we have for π-almost all x ∈Ωt, ∥F∗(x) −Fλ(x)∥Y ≤t + ∥[Fλ]∥L∞(π;Y) ≤t + AE∥F∗∥βλ β−α 2 . Therefore, for all β ∈[0,2ρ], ∥(F∗−[Fλ])1X∈Ωt∥L∞(π;Y) ≤t + A(1 + max{E,ωρ}∥F∗∥βλ β−α 2 ) =∶χ(t,λ). Using Lemma 17 with l = 1, we have, 2m−1 ∫Ωt ∥hx∥m H∥F∗(x) −Fλ(x)∥m Y dπ(x) ≤2m−1χ(t,λ)m−2(Aλ−α 2 )m∥F∗−[Fλ]∥2 L2(π;Y) =(2χ(t,λ)A λ α 2 ) m−2 ∥F∗−[Fλ]∥2 L2(π;Y) 2A2 λα ≤m!(2χ(t,λ)A λ α 2 ) m−2 ∥F∗−[Fλ]∥2 L2(π;Y) 2A2 λα . Putting everything together, E[∥ξ(X,Y )1{X ∈Ωt}∥m S2(H,Y)] ≤m!(2(R + χ(t,λ))A λ α 2 ) m−2 (σ2N1(λ) + ∥F∗−[Fλ]∥2 L2(π;Y) 2A2 λα ). We now apply Theorem 10 with L ←2(R + χ(t,λ))A λ α 2 σ ←2σ √ N1(λ) + ∥F∗−[Fλ]∥L2(π;Y) 2A λ α 2 We bound ∥F∗−[Fλ]∥L2(π;Y) using Lemma 10 with γ = 0, ∥F∗−[Fλ]∥L2(π;Y) ≤ωρ∥F∗∥βλ β 2 . The conclusion is, for all τ ≥1, with probability at least 1 −2e−τ, we have ∥1 n n ∑ i=1 ξi1{xi ∈Ωt} −E[ξ(X,Y )1{X ∈Ωt}]∥ S2(H,Y) ≤4 √ 2τ ⎛ ⎜ ⎝ 2σ √ N1(λ) + ∥F∗−[Fλ]∥L2(π;Y) 2A λ α 2 √n + 2(R + χ(t,λ))A nλ α 2 ⎞ ⎟ ⎠ ≤4 √ 2τ ⎛ ⎝ 2σ √n √ N1(λ) + 2A∥F∗∥βωρ √nλ α−β 2 + 2(R + t + A)A nλ α 2 + 2A2 max{E,ωρ}∥F∗∥β nλα−β 2 ⎞ ⎠. Lemma 14. Suppose that the same assumptions and notations listed in Theorem 7 hold. 1. Suppose β + p > α, and λ ≍n− 1 β+p . For any fixed τ ≥1, with probability at least 1 −2e−τ, suppose that the truncation level t satisfies t ≤n 1 2 (1+ p−α p+β ), then there exists a constant c > 0 such that ∥1 n n ∑ i=1 ξi1{xi ∈Ωt} −E[ξ(X,Y )1{X ∈Ωt}]∥ S2(H,Y) ≤cτn−1 2 β β+p . 29 2. Suppose β +p ≤α, and λ ≍( n logθ(n)) 1 α for some θ > 1. For any fixed τ ≥1, with probability at least 1 −2e−τ, suppose that the truncation level t satisfies t ≤n 1 2 (1−β α ), then there exists a constant c > 0 such that ∥1 n n ∑ i=1 ξi1{xi ∈Ω} −E[ξ(X,Y )1{X ∈Ω}]∥ G ≤cτ ( n logθ(n) ) −β 2α Proof. Note that Theorem 7 yields the same conclusion as in the scalar-valued case proved in [54, Theorem 13]. The Lemma then follows from the analysis for the scalar-valued case in the proof of [54, Theorem 15]. We adapt [55, Theorem 15] to the vector-valued setting. Theorem 8. Suppose that Assumptions (EMB), (EVD), (MOM) and (SRC) hold for 0 ≤β ≤2ρ, where ρ is the qualification, and p ≤α ≤1. 1. In the case of β + p > α, choosing λ ≍n− 1 β+p , for any fixed τ ≥log(4), when n is sufficiently large, with probability at least 1 −4e−τ, we have ∥(( ˆCY X −Cλ ˆCX) −(CY X −CλCX))C −1 2 X,λ∥ S2(H,Y) ≤cτn−1 2 β β+p (34) where c is a constant independent of n,τ,λ. 2. In the case of β + p ≤α, choosing λ ≍( n logθ(n)) −1 α for some θ > 1 . We make the additional assumption that there exists some α′ < α such that Assumption (MOM) is satisfied for α′ < α. Then, for any fixed τ ≥log(4), when n is sufficiently large, where the hidden index bound depends on α −α′, with probability at least 1 −4e−τ, we have ∥(( ˆCY X −Cλ ˆCX) −(CY X −CλCX))C −1 2 X,λ∥ S2(H,Y) ≤cτ ( n logθ(n) ) −β 2α (35) where c is a constant independent of n,τ,λ. Proof. By assumption (EMB) and Theorem 6, if β < α, then the inclusion map Iqα,β π ∶[G]β ↪Lqα,β(π;Y) is bounded, where qα,β ∶= 2α α−β . If β ≥α, then by Lemma 11 the inclusion map I∞ π ∶[G]β ↪L∞(π;Y) is bounded and therefore [G]β is continuously embedded into Lq(π;Y) for any q ≥1. In the rest of the proof, we will use q to denote qα,β, unless otherwise specified. Furthermore, we will use cq = ∥F∗∥Lq(π;Y). We first consider the case β +p > α. We can easily verify using β +p > α that the following inequality holds 1 2 (1 + p −α p + β ) > 1 2 ( p p + β ) > α −β 2α = 1 qα,β . Choose t = n˜q−1, where 1 ˜q = 1 2 (1 2 (1 + p −α p + β ) + 1 q ). We thus have n 1 2 (1+ p−α p+β ) > t = n˜q−1 > n 1 qα,β . Thus the assumptions for both Lemma 12 and Lemma 14 are satisfied. 30 We then consider the case β + p ≤α. We now apply Assumption (EMB) and Theorem 6 to α′ instead of α. We obtain that the inclusion map I qα′,β π is bounded, where we recall that qα′,β is defined to be 2α′ α′−β . Since x ↦ 2x x−β is monotonically decreasing for x > β, we obtain the inequality 2α′ α′ −β > 2α α −β . We choose t = n 1 qα,β . By construction, t satisfies the assumptions in Lemma 14. Furthermore, the assumptions of Lemma 12 are satisfied, with F∗∈Lq′(π,Y), and t = n 1 qα,β . Having established the applicability of Lemma 14 and Lemma 12, let us turn our attention to proving the results of the Theorem. Denote ξ(x,y) = (y −Cλϕ(x)) ⊗(C −1 2 X,λϕ(x)) We compute E[ξ(x,y)] = (CY X −CλCX)C −1 2 X,λ 1. The β + p > α case. Have ∥1 n n ∑ i=1 ξi −E[ξ(x,y)]∥ S2(H,Y) ≤∥1 n n ∑ i=1 ξi1{xi ∈Ωt} −E[ξ(x,y)1{x ∈Ωt}]∥ S2(H,Y) +∥1 n n ∑ i=1 ξi1{xi ∈Ωc t}∥ S2(H,Y) +∥E[ξ(x,y)1{x ∈Ωc t}∥S2(H,Y) We can bound the first term with probability at least 1 −2e−τ by cτ ( n logθ(n)) −β 2α , according to Lemma 14. By Lemma 12, with probability at least 1 −e−τ, for sufficiently large n, xi ∈Ωt for all i ∈[n], whereby the second term is zero. It remains to bound the third term, where our bound will be deterministic. Using Jensen’s inequality, we have, ∥E[ξ(x,y)1{x ∈Ωc t}]∥S2(H,Y) ≤E[∥ξ(x,y)1{x ∈Ωc t}∥S2(H,Y)] = E[∥(y −Cλϕ(x))1{x ∉Ωt}∥Y ⋅∥C −1 2 X,λϕ(x)∥ H ] ≤Aλ−α 2 E[∥(y −Cλϕ(x))1{x ∉Ωt}∥Y] where in the third line we used Lemma 17. We first split the second term into an approximation error and a noise term using triangle inequality. E[∥(y −Cλϕ(x))1{x ∉Ωt}∥Y] ≤E[∥(y −F∗(x))1{x ∉Ωt}∥Y]+E[∥(F∗(x) −Fλ(x))1{x ∉Ωt}∥Y] We bound the first term using the tower property of conditional expectation, E[∥(y −Cλϕ(x))1{x ∉Ωt}∥Y] ≤Eπ [E[∥y −Cλϕ(x)∥Y ∣x]1{x ∉Ωt}] ≤Eπ [E[∥y −Cλϕ(x)∥2 Y ∣x] 1 2 1{x ∉Ωt}] ≤σπ(x ∉Ωt) ≤σcq q tq . where in the third inequality we used Assumption (MOM) with q = 2, and in the fourth inequality we used Lemma 12. We bound the second term using Cauchy-Schwarz inequality and Lemma 10 with γ = 0, E[∥(F∗(x) −Fλ(x))1{x ∉Ωt}∥Y] ≤P(x ∉Ωt) 1 2 ∥F∗∥βωρλ β 2 Therefore, using Lemma 12, we have, ∥E[ξ(x,y)1{x ∈Ωc t}∥S2(H,Y) ≤Aλ−α 2 ⎛ ⎝ σcq q tq + c q 2q t q 2 ∥F∗∥βωρλ β 2 ⎞ ⎠. (36) 31 We now plug in λ ≍n− 1 β+p . Recall that by construction t > n 1 q . Thus, λ−α 2 t−q ≲n−1n α 2(β+p) < n−1n β+p 2(β+p) = n−1 2 ≤n−1 2 β β+p λ β−α 2 t−q 2 ≲n−1 2 n −(β−α) 2(β+p) < n−1 2 n p 2(β+p) = n−1 2 β β+p We’ve therefore proved inequality (34). 2. The β + p ≤α case. We proceed similarly to the β + p > α case. We have, ∥1 n n ∑ i=1 ξi −E[ξ(x,y)]∥ S2(H,Y) ≤∥1 n n ∑ i=1 ξi1{xi ∈Ωt} −E[ξ(x,y)1{x ∈Ωt}]∥ S2(H,Y) + ∥1 n n ∑ i=1 ξi1{xi ∈Ωc t}∥ S2(H,Y) + ∥E[ξ(x,y)1{x ∈Ωc t}]∥S2(H,Y) We can bound the first term with probability at least 1 −2e−τ by cτ ( n logθ(n)) −β 2α , according to Lemma 14. By Lemma 12, with probability at least 1 −e−τ, for sufficiently large n, xi ∈Ωt for all i ∈[n], whereby the second term is zero. We bound the third term by Eq. (36). We now plug in λ ≍( n logθ(n)) −1 α . Recall that by construction t > n 1 q . Thus, λ−α 2 t−q ≲n−1 ( n logθ(n) ) 1 2 < ( n logθ(n) ) −1 2 ≤( n logθ(n) ) −β 2α λ β−α 2 t−q 2 ≲n−1 2 ( n logθ(n) ) α−β 2α < ( n logθ(n) ) −β 2α We have therefore proved inequality (35). We adapt [54, Theorem 16] to the vector-valued setting. Theorem 9 (Bound of estimation error). Suppose that assumptions (EMB), (EVD), (MOM) and (SRC) hold for 0 ≤β ≤2ρ, where ρ is the qualification, and p ≤α < 1. For 0 ≤γ ≤1, with γ ≤β, 1. In the case of β + p > α, by choosing λ ≍n− 1 β+p , for any fixed τ ≥log(4), when n is sufficiently large, with probability at least 1 −4e−τ, we have ∥[ ˆCλ −Cλ]∥2 S2([H]γ,Y) ≤cτ 2n−β−γ β+p , where c is a constant independent of n,τ. 2. In the case of β + p ≤α, by choosing λ ≍( n logθ(n)) −1 α , for any fixed τ ≥log(4), when n is sufficiently large, with probability at least 1 −4e−τ, we have ∥[ ˆCλ −Cλ]∥2 S2([H]γ,Y) ≤cτ 2 ( n logθ(n) ) −β−γ α where c is a constant independent of n,τ. Proof. Firstly, we establish the applicability of Lemma 19. 1. The β + p > α case. Have λ ≍n− 1 β+p , hence nλα ≳n β+p−α β+p whereas using λ ≤∥CX∥H→H for sufficiently large n, as well as Lemma 16, 8A2τ log (2eN(λ)∥CX∥H→H + λ ∥CX∥H→H ) ≤8A2τ log(4ec2,1λ−p) ≲8A2τ (log(4ec2,1) + p β + p log(n)) 32 Therefore, for a fixed τ > 0, for all sufficiently large n, Eq. (40) in Lemma 19 is satisfied. 2. The β + p ≤α case. Have λ ≍( n logθ(n)) −1 α for some θ > 1, hence nλα ≥logθ(n) whereas similar to the β + p > α case, we ahve 8A2τ log (2eN(λ)∥CX∥H→H + λ ∥CX∥H→H ) ≲8A2τ (log(4ec2,1) + p α log ( n logθ(n) )) Therefore, for a fixed τ > 0, for all sufficiently large n, Eq. (40) in Lemma 19 is satisfied. We thus conclude for all α ∈(0,1], with probability ≥1 −2e−τ, Eq. (41) and (42) are satisfied simultaneously. We exploit the following decomposition ∥[ ˆCλ −Cλ]∥S2([H]γ,Y) ≤∥( ˆCλ −Cλ)C 1−γ 2 X ∥ S2(H,Y) ≤∥( ˆCλ −Cλ) ˆC 1 2 X,λ∥ S2(H,Y) ⋅∥ˆC −1 2 X,λC 1 2 X,λ∥ H→H ⋅∥C −1 2 X,λC 1−γ 2 X ∥ H→H ≤∥( ˆCλ −Cλ) ˆC 1 2 X,λ∥ S2(H,Y) ⋅3 ⋅sup i∈N µ 1−γ 2 i √µi + λ ≤∥( ˆCλ −Cλ) ˆC 1 2 X,λ∥ S2(H,Y) ⋅3λ−γ 2 , (37) where in the first inequality we used Lemma 22, in the third inequality we used Eq. (42) and in the last inequality we used Lemma 21. We consider the following decomposition ˆCλ −Cλ = ˆCλ −Cλ ( ˆCXgλ ( ˆCX) + rλ ( ˆCX)) = ( ˆCY X −Cλ ˆCX)gλ ( ˆCX) −Cλrλ ( ˆCX). Hence ∥[ ˆCλ −Cλ]∥ 2 S2([H]γ,Y) ≤18λ−γ ((I)2 + (II)2), where (I) = ∥( ˆCY X −Cλ ˆCX) ˆC 1 2 X,λgλ ( ˆCX)∥ S2(H,Y) (II) = ∥Cλrλ ( ˆCX) ˆC 1 2 X,λ∥ S2(H,Y) . Term (I). The high level idea is to bound (I) by exploiting the first axiom of the filter function (8), where gλ( ˆCX) is intuitively a regularized inverse of ˆCX, by grouping it with ˆCX,λ. (I) ≤∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ S2(H,Y) ⋅∥C 1 2 Xλ ˆC −1 2 X,λ∥ H→H ⋅∥ˆCX,λgλ ( ˆCX)∥H→H ≤∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ S2(H,Y) ⋅ √ 3 sup t∈[0,κ2] (t + λ)gλ(t) ≤∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ S2(H,Y) ⋅2 √ 3E. where the second inequality follows from Eq. (42). We consider the following decomposition ∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ 2 S2(H,Y) ≤2∥(( ˆCY X −Cλ ˆCX) −(CY X −CλCX))C −1 2 X,λ∥ 2 S2(H,Y) + 2∥(CY X −CλCX)C −1 2 X,λ∥ 2 S2(H,Y) 33 We bound the first term by Theorem 8 and the second term by Lemma 8. This yields, for τ ≥log(4), with probability at least 1 −4e−τ, for some constant c > 0 which does not depend on n,τ,λ, ∥( ˆCY X −Cλ ˆCX)C −1 2 X,λ∥ 2 S2(H,Y) ≤2ω2 ρ∥F∗∥2 βλβ + ⎧⎪⎪⎪⎨⎪⎪⎪⎩ cτ 2n− β β+p β + p ≥α cτ 2 ( n logθ(n)) −β α β + p < α ≤ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ τ 2(c + 2∥F∗∥2 βω2 ρλβ)n− β β+p β + p ≥α τ 2(c + 2∥F∗∥2 βω2 ρλβ)( n logθ(n)) −β α β + p < α where we used that τ > 1. So collecting all the relevant constants together, we can write the upper bound of term (I) as follows: with probability at least 1 −4e−τ, for some constant c′ > 0 (different from the c before) which does not depend on n,τ,λ, we have (I) ≤c′τ ⋅ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ n−1 2 β β+p β + p ≥α ( n logθ(n)) −β 2α β + p < α. Term (II). Using Lemma 9, we have (II) ≤B ∥ˆC 1 2 X,λrλ( ˆCX)gλ(CX)C β+1 2 X ∥ H→H The second term is the same as the scalar-valued case, which is bounded in Step 3 of the proof of [54, Theorem 16]. We define ∆1 ∶= 32max{β −1 2 ,1}Eωρκβ−1λ 1 2 n−min(β,3)−1 4 By the proof of [54, Theorem 16], we have, with probability at least 1 −6e−τ (II) ≤6BωρEλ β 2 + ∆1Bτ1{β > 2}. 1. Case β + p > α. In this case λ ≍n− 1 β+p . We note that for β > 2, ∆1 as a function of n can be written as ∆1 ≍n− 1 2(β+p) −min(β,3)−1 4 Note that 1 2(β + p) + min(β,3) −1 4 − β 2(β + p) = 1 2 ( p β + p + min(β,3) −1 2 ) > 0 Hence ∆1 ≲n− β 2(β+p) Therefore we have shown that there exists some constant c′′ > 0, independent of n,λ,τ, such that with probability at least 1 −6e−τ, for sufficiently large n, ∥[ ˆCλ −Cλ]∥S2([H]γ,Y) ≤c′′τn−1 2 β−γ β+p . 2. Case β + p ≤α. In this case β ≤α ≤1, and λ ≍( n logθ(n)) −1 α . We have also shown that there exists some constant c′′ > 0, independent of n,λ,τ, such that with probability at least 1 −6e−τ, for sufficiently large n, ∥[ ˆCλ −Cλ]∥S2([H]γ,Y) ≤c′′τ ( n logθ(n) ) −β−γ 2α Putting together Lemma 10 and Theorem 9, we have proved Theorem 4. 34 D Auxiliary Results D.1 Spectral Calculus, Proof of Proposition 1 and Empirical Solution Definition 9 (Spectral Calculus; see 16, Chapter 2.3). Let H be a Hilbert space. Consider g ∶R →R and a self-adjoint compact operator A ∶H →H admitting a spectral decomposition written as A = ∑ i∈I λihi ⊗hi. We then define g(A) ∶H →H as g(A) ∶= ∑ i∈I g(λi)hi ⊗hi whenever this series converges in operator norm. Proof of Proposition 1. We define the sampling operator S ∶Rn →H and it dual S∗∶H →Rn, S ∶Rn →H, S∗∶H →Rn, α ↦ n ∑ i=1 αiϕ(xi) f ↦(f(xi))n i=1 We can verify that ˆCX = n−1SS∗and K = S∗S. Let Y = (yi)n i=1 ∈Rn. We have, for all x ∈X, ˆFλ(x) = ˆCλϕ(X) = ( 1 n n ∑ i=1 yi ⊗ϕ(xi))gλ( ˆCX)ϕ(X) = n ∑ i=1 yi ⟨ϕ(xi), 1 ngλ(n−1SS∗)ϕ(X)⟩ H = YT S∗( 1 ngλ(n−1SS∗)ϕ(X)) (38) = YT 1 ngλ(n−1S∗S)S∗ϕ(X) (39) = YT 1 ngλ(n−1K)S∗ϕ(X) = YT 1 ngλ(n−1K)kx. To go from (38) to (39), we make the following observation. Consider the singular value decomposition of the compact operator S, there is m ≤n such that S = m ∑ i=1 √σifi ⊗ei where (ei)i, (fi)i are orthonormal sequences in Rn and H respectively. We then have SS∗= m ∑ i=1 σifi ⊗fi, S∗S = m ∑ i=1 σiei ⊗ei. Therefore, we deduce S∗gλ (SS∗ n ) = (∑ i √σiei ⊗fi)⎛ ⎝∑ j gλ (σj n )fj ⊗fj ⎞ ⎠ = ∑ i,j gλ (σj n )√σiei ⊗fj⟨fi,fj⟩H = ∑ i gλ (σi n )√σiei ⊗fi 35 Similarly, gλ (S∗S n )S∗= ⎛ ⎝∑ j gλ (σj n )ej ⊗ej ⎞ ⎠(∑ i √σiei ⊗fi) = ∑ i,j gλ (σj n )√σiej ⊗fi⟨ei,ej⟩Rn = ∑ i gλ (σi n )√σiei ⊗fi Hence we have proved S∗gλ (SS∗ n ) = gλ (S∗S n )S∗ as desired. Proposition 2. Any minimizer F ∈G of En(F) ∶= 1 n n ∑ i=1 ∥yi −F(xi)∥2 Y on G must satisfy ˆCY X = ˆC ˆCX, C ∈S2(H,Y), where F(⋅) = Cϕ(⋅). Proof. By Corollary 1, it is equivalent to solve the following optimization problem on S2(H,Y), min C∈S2(H,Y) 1 n n ∑ i=1 ∥yi −Cϕ(xi)∥2 Y. Recall for a Hilbert-Schmidt operator L ∈S2(H,Y), we have ⟨L,a ⊗b⟩S2(H,Y) = ⟨a,Lb⟩Y. Using this, we re-write the objective as an inner product in S2(H,Y): 1 n n ∑ i=1 ∥yi −Cϕ(xi)∥2 Y = 1 n n ∑ i=1 −2⟨C,yi ⊗ϕ(xi)⟩S2(H,Y) + ⟨C,(Cϕ(xi)) ⊗ϕ(xi)⟩S2(H,Y) + constant = −2⟨C, ˆCY X⟩S2(H,Y) + ⟨C,C ˆCX⟩S2(H,Y) Taking the Fréchet derivative with respect to C and setting in to zero, we obtain the following first order condition ˆCY X = C ˆCX D.2 Properties Related to Assumptions (EMB) and (EVD) Lemma 15 (Lemma 13 [17]). Under (EMB), the following inequality is satisfied, for λ > 0 and π-almost all x ∈X, ∥(CX + λIdH)−1 2 k(x,⋅)∥ H ≤Aλ−α 2 . Definition 10 (l-effective dimension). For l ≥1, the l-effective dimension Nl ∶(0,∞) →[0,∞) is defined by Nl(λ) ∶= Tr[Cl XC−l X,λ] = ∑ i≥1 ( µi µi + λ) l The 1-effective dimension is widely considered in the statistical analysis of kernel ridge regression (see [6], [5], [32], [34], [17]). The following lemma provides upper and lower bounds for the l−effective dimension. 36 Lemma 16. Suppose Assumption (EVD) holds with parameter p ∈(0,1], for any λ ∈(0,1], there exists a constant c2,l > 0 independent of λ such that Nl(λ) ≤c2,lλ−p. If furthermore, Assumption (EVD+) holds with parameter p ∈(0,1), for any λ ∈(0,1], there exists a constant c1,l > 0 independent of λ such that c1,lλ−p ≤Nl(λ) ≤c2,lλ−p. The proof can be found in [29] (Proposition D.1), but as the proof is incomplete we provide a full proof for completeness. This allows us to detect that the value p = 1 in Assumption (EVD+) is not compatible with the assumption of a bounded kernel (see Remark 7 below). Proof. Nl(λ) ≤∑ i≥1 ( D2 D2 + λi 1 p ) l (x ↦ x x + λ is monotonically increasing) ≤∫ +∞ 0 ( D2 D2 + λx 1 p ) l dx (i ↦( D2 D2 + λi1/p ) l is positive and decreasing) = ∫ +∞ 0 ( D2 D2 + y 1 p ) l dy λp (y1/p = λx1/p) ≤λ−p ⎛ ⎝1 + ∫ +∞ 1 ( D2 D2 + y 1 p ) l dy⎞ ⎠ Let us now consider the integral. Let us first consider p ≤1 < l, ∫ ∞ 1 ( D2 D2 + y 1 p ) l dy ≤Dl 2 ∫ ∞ 1 y−l p dy = Dl 2 p l −p Therefore, using λ ≤1, we can take c2,l = 1 + Dl 2 p l−p. The remaining edge case p = 1 = l, is covered by [17, Lemma 11] with c2,1 = ∥CX∥S1(H). For the lower bound, we proceed similarly. For p ∈(0,1) (and therefore p < l), Nl(λ) ≥∑ i≥1 ( D1 D1 + λi 1 p ) l (x ↦ x x + λ is monotonically increasing) ≥∫ ∞ 1 ( D1 D1 + λx 1 p ) l dx (i ↦( D1 D1 + λi1/p ) l is positive and decreasing) = ∫ ∞ 1 ( D1 D1 + y 1 p ) l dy λp (y1/p = λx1/p) ≥λ−p ∫ ∞ 1 ( D1 D1 + 1) l y−l p dy = λ−p ( D1 D1 + 1) l p l −p. Therefore, we can take c1,l = ( D1 D1+1) l p l−p. Remark 7. We note that Assumption (EVD+) with p = 1 is not compatible with the assumption that k is bounded (Assumption 3). Indeed, suppose that µi ≥D1i−1, for all i ≥1. Recall that {[ei]}i≥1 forms an orthonormal set in L2(π). By Mercer’s theorem, κ2 ≥∫X k(x,x)π(dx) = ∑ i≥1 µi ∫X ei(x)2π(dx) = ∑ i≥1 µi ≥D1 ∑ i≥1 i−1 = +∞, which leads to a contradiction. 37 Lemma 17. For any l ∈[1,2], the following equality holds, ∫X ∥[C −l 2 X,λk(x,⋅)]∥ 2 2−l dπ(x) = Nl(λ). In particular for l = 1, ∫X ∥C −1 2 X,λk(x,⋅)∥ 2 H dπ(x) = N1(λ), and for l = 2, ∫X ∥[C−1 X,λk(x,⋅)]∥ 2 L2(π) dπ(x) = N2(λ). Proof. Fix x ∈X. Since k(x,⋅) ∈H, and {µ1/2 i ei} i∈I is an ONB of (kerIπ)⊥, we have that π−almost everywhere k(x,⋅) = ∑ i∈I ⟨k(x,⋅),µ1/2 i ei⟩Hµ1/2 i ei = ∑ i∈I µiei(x)ei. This is Mercer’s Theorem [51]. On the other hand, π−almost everywhere, C−l/2 X,λ = ∑ i∈I (µi + λ)−l/2(√µiei) ⊗(√µiei). Therefore, [C−l/2 X,λ k(x,⋅)] = ∑ i∈I µi (µi + λ)l/2 ei(x)[ei], and by Parseval’s identity, using that {µ(2−l)/2 i [ei]}i∈I is an ONB of [H]2−l, ∥[C−l/2 X,λ k(x,⋅)]∥2 2−l = ∑ i∈I ( µi µi + λ) l ei(x)2 Therefore, ∫X ∥[C−l/2 X,λ k(x,⋅)]∥2 2−ldπ(x) = ∑ i∈I ( µi µi + λ) l ∫X ei(x)2dπ(x) = Nl(λ), where we used that ([ei])i∈I forms an orthonormal set in L2(π). D.3 Concentration Inequalities The following Theorem is from [17, Theorem 26]. Theorem 10 (Bernstein’s inequality). Let (Ω,B,P) be a probability space, H be a separable Hilbert space, and ξ ∶Ω→H be a random variable with E[∥ξ∥m H] ≤1 2m!σ2Lm−2 for all m ≥2. Then, for τ ≥1 and n ≥1, the following concentration inequality is satisfied P n ⎛ ⎝(ω1,...,ωn) ∈Ωn ∶∥1 n n ∑ i=1 ξ (ωi) −EP ξ∥ 2 H ≥32τ 2 n (σ2 + L2 n )⎞ ⎠≤2e−τ. In particular, for τ ≥1 and n ≥1, P n ((ω1,...,ωn) ∈Ωn ∶∥1 n n ∑ i=1 ξ (ωi) −EP ξ∥ H ≥4 √ 2 τ √n (σ + L √n)) ≤2e−τ. Lemma 18. Let τ ≥log(2), with probability at least 1 −2e−τ, for √ nλ ≥8τκ √ max{N(λ),1}, we have ∥ˆC−1 X,λCX,λ∥H→H ≤2. Proof. The proof is identical to [5, Proposition 5.4] with the only difference that in their setting κ = 1. 38 Proposition 3 (Proposition C.9 29). Let π be a probability measure on X,f ∈L2(π) and ∥f∥∞≤M. Suppose we have x1,...,xn sampled i.i.d. from π. Then, for any τ ≥log(2), the following holds with probability at least 1 −2e−τ: 1 2∥f∥2 L2(π) −5τM 2 3n ≤∥f∥2 2,n ≤3 2∥f∥2 L2(π) + 5τM 2 3n , where ∥⋅∥2,n was defined in Definition 7. Lemma 19 (Lemma 12 54). Let Assumptions (EMB), (SRC) and (MOM) be satisfied. For τ ≥1, if λ and n satisfy that n ≥8A2τλ−α log (2eN(λ)∥CX∥H→H + λ ∥CX∥H→H ) (40) then the following operator norm bounds are satisfied with probability not less than 1 −2e−τ ∥C −1 2 X,λ ˆC 1 2 X,λ∥ 2 H→H ≤2, (41) ∥C 1 2 X,λ ˆC −1 2 X,λ∥ 2 H→H ≤3. (42) D.4 Miscellaneous results Lemma 20 (Cordes inequality [18]). Let A,B be two positive bounded linear operators on a separable Hilbert space H and s ∈[0,1]. Then ∥AsBs∥H→H ≤∥A∥s H→H∥B∥s H→H Lemma 21 (Lemma 25 [17]). For λ > 0 and s ∈[0,1], we have sup t≥0 ts t + λ ≤λs−1 We recall the following basic Lemma from [30, Lemma 2]. Lemma 22. For 0 ≤γ ≤1 and F ∈G, the inequality ∥[F]∥γ ≤∥CC 1−γ 2 X ∥ S2(H,Y) holds, where C = ¯Ψ−1(F) ∈S2(H,Y). If, in addition, γ < 1 or C ⊥Y ⊗kerIπ is satisfied, then the result is an equality. Definition 11. Let X ⊆Rd be a compact set and θ ∈(0,1]. For a function f ∶X →R, we introduce the Hölder semi-norm [f]θ,X ∶= sup x,y∈X,x≠y ∣f(x) −f(y)∣ ∥x −y∥θ , where ∥⋅∥represents the usual Euclidean norm. Then, we define the Hölder space Cθ(X) ∶= {f ∶X →R ∣[f]θ,X < +∞}, which is equipped with the norm ∥f∥Cθ(X) ∶= sup x∈X ∣f(x)∣+ [f]θ,X . The next lemma is used to prove Lemma 24 below. It appears in [29, Lemma A.3], albeit the use of an erroneous equality in their proof: ∥k(x,⋅) −k(y,⋅)∥2 H = k(x,x)k(y,y) −k(x,y)2. We therefore provide our own proof of this result. Lemma 23. Assume that H is an RKHS over a compact set X ⊆Rd associated with a kernel k ∈Cθ(X × X) for θ ∈(0,1]. Then, we have H ⊆C θ 2 (X) and [f] θ 2 ,X ≤ √ 2[k]θ,X×X ∥f∥H 39 Proof. For all (x,y) ∈X and f ∈H, by the reproducing property and Cauchy–Schwarz inequality, ∣f(x) −f(y)∣= ∣⟨k(x,⋅) −k(y,⋅),f⟩H∣≤∥f∥H∥k(x,⋅) −k(y,⋅)∥H. Then, using k ∈Cθ(X × X), we obtain ∥k(x,⋅) −k(y,⋅)∥2 H = k(x,x) + k(y,y) −2k(x,y) ≤2[k]θ,X×X ∥x −y∥θ, which concludes the proof. We derive as a corollary a quantitative upper bound on the ϵ-covering number of the the set of (spectral) regularized kernel basis function with respect to the ∥⋅∥∞norm. Lemma 24 (Lemma C.10 by 29). Assume that H is an RKHS over a compact set X ⊆Rd associated with a kernel k ∈Cθ(X × X) for θ ∈(0,1]. Assume that k(x,x) ≤κ2 for all x ∈X. Then, we have that for all ϵ > 0, N(Kλ,∥⋅∥∞,ϵ) ≤c(λϵ)−2d θ where Kλ ∶= {C−1 X,λk(x,⋅)}x∈X , and c is a positive constant which does not depend on λ,ϵ and only depends on κ and [k]θ,X×X . N(Kλ,∥⋅∥∞,ϵ) denotes the ϵ−covering number of the set Kλ in the norm ∥⋅∥∞(see [49, Definition 6.19] for the definition of covering numbers). 40 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We specifically point out the settings and detailed contributions of our work in both abstract and introduction. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,475	Maximum Entropy Inverse Reinforcement Learning of Diffusion Models with Energy-Based Models Sangwoong Yoon1, Himchan Hwang2, Dohyun Kwon1,3†, Yung-Kyun Noh1,4†, Frank C. Park2,5† 1Korea Institute for Advanced Study, 2Seoul National University, 3University of Seoul, 4Hanyang University, 5Saige Research, †Co-corresponding Authors swyoon@kias.re.kr, himchan@robotics.snu.ac.kr, dhkwon@uos.ac.kr, nohyung@hanyang.ac.kr, fcp@snu.ac.kr Abstract We present a maximum entropy inverse reinforcement learning (IRL) approach for improving the sample quality of diffusion generative models, especially when the number of generation time steps is small. Similar to how IRL trains a policy based on the reward function learned from expert demonstrations, we train (or fine-tune) a diffusion model using the log probability density estimated from training data. Since we employ an energy-based model (EBM) to represent the log density, our approach boils down to the joint training of a diffusion model and an EBM. Our IRL formulation, named Diffusion by Maximum Entropy IRL (DxMI), is a minimax problem that reaches equilibrium when both models converge to the data distribution. The entropy maximization plays a key role in DxMI, facilitating the exploration of the diffusion model and ensuring the convergence of the EBM. We also propose Diffusion by Dynamic Programming (DxDP), a novel reinforcement learning algorithm for diffusion models, as a subroutine in DxMI. DxDP makes the diffusion model update in DxMI efficient by transforming the original problem into an optimal control formulation where value functions replace back-propagation in time. Our empirical studies show that diffusion models fine-tuned using DxMI can generate high-quality samples in as few as 4 and 10 steps. Additionally, DxMI enables the training of an EBM without MCMC, stabilizing EBM training dynamics and enhancing anomaly detection performance. 1 Introduction Generative modeling is a form of imitation learning. Just as an imitation learner produces an action that mimics a demonstration from an expert, a generative model synthesizes a sample resembling the training data. In generative modeling, the expert to be imitated corresponds to the underlying data generation process. The intimate connection between generative modeling and imitation learning is already well appreciated in the literature [1, 2]. The connection to imitation learning plays a central role in diffusion models [3, 4], which generate samples by transforming a Gaussian noise through iterative additive refinements. The training of a diffusion model is essentially an instance of behavioral cloning [5], a widely adopted imitation learning algorithm that mimics an expert’s action at each state. During training, a diffusion model is optimized to follow a predefined diffusion trajectory that interpolates between noise and data. The trajectory provides a step-by-step demonstration of how to transform Gaussian noise into a sample, allowing diffusion models to achieve a new state-of-the-art in many generation tasks. Behavioral cloning is also responsible for the diffusion model’s key limitation, the slow generation speed. A behavior-cloned policy is not reliable when the state distribution deviates from the expert demonstration [6, 7]. Likewise, the sample quality from a diffusion model degrades as the gap 38th Conference on Neural Information Processing Systems (NeurIPS 2024). ⋯ 𝐱0 Diffusion Model 𝜋(𝐱) Energy-Based Model Training Data 𝐱𝑇(= 𝐱) ⋯ 𝐱1 Positive Sample Negative Sample Reward 𝑞𝐱 𝑝𝐱 𝐱𝑇−1 Figure 1: (Left) Overview of DxMI. The diffusion model π(x) is trained using the energy of q(x) as a reward. The EBM q(x) is trained using samples from π(x) as negative samples. (Right) ImageNet 64 generation examples from a 10-step diffusion model before DxMI fine-tuning (up) and after fine-tuning (down). Only the last six steps out of ten are shown. between training and generation grows. A diffusion model is typically trained to follow a fine-grained diffusion trajectory of 1,000 or more steps. Since 1,000 neural network evaluations are prohibitively expensive, fewer steps are often used during generation, incurring the distribution shift from the training phase and thus degraded sample quality. Speeding up a diffusion model while maintaining its high sample quality is a problem of great practical value, becoming an active field of research [8, 9, 10, 11, 12, 13]. The slow generation in diffusion models can be addressed by employing inverse reinforcement learning (IRL; [14]), another imitation learning approach. Unlike behavioral cloning, which blindly mimics the expert’s action at each state, the IRL approach first infers the reward function that explains the trajectory. When applied to diffusion models, IRL allows a faster generation trajectory to be found by guiding a sampler using the learned reward [11, 12, 13]. This approach is more frequently referred to as adversarial training because a common choice for the reward function is a discriminator classifying the training data and diffusion model’s samples. However, resembling GAN, this adversarial approach may have similar drawbacks, such as limited exploration. The first contribution of this paper is formulating maximum entropy IRL [15, 16, 2] for a diffusion model. Our formulation, named Diffusion by Maximum Entropy IRL (DxMI, pronounced "diby-me"), is a minimax problem that jointly optimizes a diffusion model and an energy-based model (EBM). In DxMI, the EBM provides the estimated log density as the reward signal for the diffusion model. Then, the diffusion model is trained to maximize the reward from EBM while simultaneously maximizing the entropy of generated samples. The maximization of entropy in DxMI facilitates exploration and stabilizes the training dynamics, as shown in reinforcement learning (RL) [17], IRL [15], and EBM [18] literature. Furthermore, the entropy maximization lets the minimax problem have an equilibrium when both the diffusion model and the EBM become the data distribution. The diffusion model update step in DxMI is equivalent to maximum entropy RL. However, this step is challenging to perform for two reasons. First, the diffusion model update requires the gradient of the marginal entropy of a diffusion model’s samples, which is difficult to estimate for discrete-time diffusion models, such as DDPM [3]. Second, back-propagating the gradient through the whole diffusion model is often infeasible due to memory constraints. Even with sufficient memory, the gradient may explode or vanish during propagation through time, causing training instability [19]. Our second contribution is Diffusion by Dynamic Programming (DxDP), a novel maximum entropy RL algorithm for updating a diffusion model without the above-mentioned difficulties. First, DxDP mitigates the marginal entropy estimation issue by optimizing the upper bound of the original objective. In the upper bound, the marginal entropy is replaced by conditional entropies, which are easier to compute for a diffusion model. Then, DxDP removes the need for back-propagation in time by interpreting the objective as an optimal control problem and applying dynamic programming using value functions. The connection between optimal control and diffusion models is increasingly gaining attention [20, 21, 22], but DxDP is the first instance of applying discrete-time dynamic programming directly to train a diffusion model. Compared to policy gradient methods, we empirically find that DxDP converges faster and provides stronger diffusion models. As an RL algorithm, DxDP may have broader utility, such as fine-tuning a diffusion model from human or AI feedback. 2 We provide experimental results demonstrating the effectiveness of DxMI in training diffusion models and EBMs. On image generation tasks, DxMI can train strong short-run diffusion models that generate samples in 4 or 10 neural network evaluations. Also, DxMI can be used to train strong energy-based anomaly detectors. Notably, DxMI provides a novel way to train EBM without MCMC, which is computationally demanding and sensitive to the choice of hyperparameters. The paper is structured as follows. In Section 2, we introduce the necessary preliminaries. Section 3 presents the DxMI framework, and Section 4 proposes the DxDP algorithm. Experimental results and related work are provided in Sections 5 and 6, respectively. Section 7 concludes the paper. The code for DxMI can be found in https://github.com/swyoon/Diffusion-by-MaxEntIRL.git. 2 Preliminaries Diffusion Models. The diffusion model refers to a range of generative models trained by reversing the trajectory from the data distribution to the noise distribution. Among diffusion models, we focus on discrete-time stochastic samplers, such as DDPM [3], which synthesize a sample through the following iteration producing x0, x1, . . . , xT ∈RD: x0 ∼N(0, I) and xt+1 = atxt + µ(xt, t) + σtϵt for t = 0, 1, . . . , T −1, (1) where ϵt ∼N(0, I) and µ(x,t) is the output of a neural network. The coefficients at ∈R and σt ∈R>0 are constants. Note that we reverse the time direction in a diffusion model from the convention to be consistent with RL. The final state xT is the sample generated by the diffusion model, and its marginal distribution is π(xT ). We will often drop the subscript T and write the distribution as π(x). The conditional distribution of a transition in Eq. (1) is denoted as π(xt+1\|xt). For continuous-time diffusion models [23], Eq. (1) corresponds to using the Euler-Maruyama solver. The generation process in Eq. (1) defines a T-horizon Markov Decision Process (MDP) except for the reward. State st and action at are defined as st = (xt, t) and at = xt+1. The transition dynamics is defined as p(st+1\|st, at) = δ(at,t+1), where δ(xt,t) is a Dirac delta function at (xt, t). With a reward function defined, a diffusion model can be trained with RL [11, 24, 19, 25]. In this paper, we consider a case where the reward is the log data density log p(x), which is unknown. Energy-Based Models. An energy-based model (EBM) q(x) uses a scalar function called an energy E(x) to represent a probability distribution: q(x) = 1 Z exp(−E(x)/τ), E : X →R, (2) where τ > 0 is temperature, X is the compact domain of data, and Z = R X exp(−E(x)/τ)dx < ∞ is the normalization constant. The standard method for training an EBM is by minimizing KL divergence between data p(x) and EBM q(x), i.e., minq KL(p\|\|q), where KL(p\|\|q) := R X p(x) log(p(x)/q(x))dx. Computing the gradient of KL(p\|\|q) with respect to EBM parameters requires MCMC sampling, which is computationally demanding and sensitive to hyperparameters. The algorithm presented in this paper serves as an alternative method for training an EBM without MCMC. EBMs have a profound connection to maximum entropy IRL, where Eq. (2) serves as a model for an expert’s policy [15, 16, 2]. In maximum entropy IRL, x corresponds to an action (or a sequence of actions), and E(x) represents the expert’s cost of the action. The expert is then assumed to generate actions following q(x). This assumption embodies the maximum entropy principle because Eq. (2) is a distribution that minimizes the cost while maximizing the entropy of the action. Here, τ balances cost minimization and entropy maximization. 3 Diffusion by Maximum Entropy Inverse Reinforcement Learning 3.1 Objective: Generalized Contrastive Divergence We aim to minimize the (reverse) KL divergence between a diffusion model π(x) and the data density p(x). This minimization can improve the sample quality of π(x), particularly when T is small. min π∈Π KL(π(x)\|\|p(x)) = max π∈Π Eπ[log p(x)] + H(π(x)), (3) 3 where Π is the set of feasible π(x)’s, and H(π) = − R π(x) log π(x)dx is the differential entropy. This minimization is a maximum entropy RL problem: The log data density log p(x) is the reward, and π(x) is the stochastic policy. However, we cannot solve Eq. (3) directly since log p(x) is unknown in a typical generative modeling setting. Instead, training data from p(x) are available, allowing us to employ an Inverse RL approach. In this paper, we present Diffusion by Maximum Entropy IRL (DxMI) as an IRL approach for solving Eq. (3). We employ an EBM q(x) (Eq. (2)) as a surrogate for p(x) and use log q(x) as the reward for training the diffusion model instead of log p(x). At the same time, we train q(x) to be close to p(x) by minimizing the divergence between p(x) and q(x): min π∈Π KL(π(x)\|\|q(x)) and min q∈Q KL(p(x)\|\|q(x)), (4) where Q is the feasible set of EBMs. When the two KL divergences become zero, p(x) = q(x) = π(x) is achieved. However, minq∈Q KL(p(x)\|\|q(x)) is difficult due to the normalization constant of q(x). Instead, we consider an alternative minimax formulation inspired by Contrastive Divergence (CD; [26]), a celebrated algorithm for training an EBM. Objective. Let p(x) be the data distribution. Suppose that Q and Π are the feasible sets of the EBM q(x) and the diffusion model π(x), respectively. The learning problem of DxMI is formulated as follows: min q∈Q max π∈Π KL(p(x)\|\|q(x)) −KL(π(x)\|\|q(x)). (5) We shall call Eq. (5) Generalized CD (GCD), because Eq. (5) generalizes CD by incorporating a general class of samplers. CD [26] is originally given as minq∈Q KL(p(x)\|\|q(x))−KL(νk p,q(x)\|\|q(x)). Here, νk p,q(x) is a k-step MCMC sample distribution where Markov chains having q(x) as a stationary distribution are initialized from p(x). GCD replaces νk p,q(x) with a general sampler π(x) at the expense of introducing a max operator. When the models are well-specified, i.e., p(x) ∈Q = Π, Nash equilibrium of GCD is p(x) = q(x) = π(x), which is identical to the solution of Eq. (4). Meanwhile, there is no need to compute the normalization constant, as the two KL divergences cancel the normalization constant out. Note that the objective function (Eq. (5)) can be negative, allowing q(x) be closer to p(x) than π(x). Our main contribution is exploring the application of discrete-time diffusion models (Eq. (1)) as π(x) in GCD and not discovering GCD for the first time. GCD is mathematically equivalent to a formulation called variational maximum likelihood or adversarial EBM, which has appeared several times in EBM literature [27, 28, 29, 30, 31, 32, 33]. However, none of them have investigated the use of a diffusion model as π(x), where optimization and entropy estimation are challenging. We discuss the challenges in Section 3.2 and provide a novel algorithm to address them in Section 4. 3.2 Alternating Update of EBM and Diffusion Model In DxMI, we update a diffusion model and an EBM in an alternative manner to find the Nash equilibrium. We write θ and ϕ as the parameters of the energy Eθ(x) and a diffusion model πϕ(x), respectively. While EBM update is straightforward, we require a subroutine described in Section 4 for updating the diffusion model. The entire procedure of DxMI is summarized in Algorithm 1. EBM Update. The optimization with respect to EBM is written as minθ Ep(x)[Eθ(x)] − Eπϕ(x)[Eθ(x)]. During the update, we also regularize the energy by the square of energies γ(Ep(x)[Eθ(x)2] + Eπϕ(x)[Eθ(x)2]) for γ > 0 to ensure the energy is bounded. We set γ = 1 unless stated otherwise. This regularizer is widely adopted in EBM [34, 35]. Difficulty of Diffusion Model Update. Ideally, diffusion model parameter ϕ should be updated by minimizing KL(πϕ\|\|qθ) = Eπϕ(x)[Eθ(x)/τ] −H(πϕ(x)). However, this update is difficult in practice for two reasons. First, marginal entropy H(πϕ(x)) is difficult to estimate. Discrete-time diffusion models (Eq. (1)) do not provide an efficient way to evaluate log πϕ(x) required in the computation of H(πϕ(x)), unlike some continuous-time models, e.g., continuous normalizing flows [36, 23]. Other entropy estimators based on k-nearest neighbors [37] or variational methods [38, 39] do not scale well to 4 Algorithm 1 Diffusion by Maximum Entropy IRL 1: Input: Dataset D, Energy Eθ(x), Value V t ψ(xt), and Sampler πϕ(x0:T ) 2: st ←σt // Initialize Adaptive Velocity Regularization (AVR) 3: for x in D do // Minibatch dimension is omitted for brevity. 4: Sample x0:T ∼πϕ(x0:T ). 5: minθ Eθ(x) −Eθ(xT ) +γ(Eθ(x)2 + Eθ(xT )2) // Energy update 6: for t = T −1, . . . , 0 do // Value update 7: minψ h sg[V t+1 ψ (xt+1)] + τ log π(xt+1\|xt) + τ 2s2 t \|\|xt −xt+1\|\|2 −V t ψ(xt) i2 8: end for 9: for xt randomly chosen among x0:T do // Sampler update 10: Sample one-step: xt+1 ∼πϕ(xt+1\|xt) // Reparametrization trick 11: minϕ V t+1 ψ (xt+1(ϕ)) + τ log π(xt+1\|xt) + τ 2s2 t \|\|xt −xt+1(ϕ)\|\|2 // xt+1 is a function of ϕ. 12: end for 13: s2 t ←αs2 t + (1 −α)\|\|xt −xt+1\|\|2/D // AVR update 14: end for high-dimensional spaces. Second, propagating the gradient through time in a diffusion model may require significant memory. Also, the gradient may explode or vanish during propagation, making the training unstable [19]. 4 Diffusion by Dynamic Programming In this section, we present a novel RL algorithm for a diffusion model, Diffusion by Dynamic Programming (DxDP), which addresses the difficulties in updating a diffusion model for the reward. DxDP leverages optimal control formulation and value functions to perform the diffusion model update in DxMI efficiently. Note that DxDP can be used separately from DxMI to train a diffusion model for an arbitrary reward. 4.1 Optimal Control Formulation Instead of solving minϕ KL(πϕ(xT )\|\|qθ(xT )) directly, we minimize its upper bound obtained from the data processing inequality: KL(πϕ(xT )\|\|qθ(xT )) ≤KL(πϕ(x0:T )\|\|qθ(xT )˜q(x0:T −1\|xT )). (6) Here, we introduce an auxiliary distribution ˜q(x0:T −1\|xT ), and the inequality holds for an arbitrary choice of ˜q(x0:T −1\|xT ). In this paper, we consider a particularly simple case where ˜q(x0:T −1\|xT ) is factorized into conditional Gaussians as follows: ˜q(x0:T −1\|xT ) = T −1 Y t=0 ˜q(xt\|xt+1), where ˜q(xt\|xt+1) = N(xt+1, s2 tI), st > 0. (7) Now we minimize the right-hand side of Eq. (6): minϕ KL(πϕ(x0:T )\|\|qθ(xT )˜q(x0:T −1\|xT )). When we plug in the definitions of each distribution, multiply by τ, and discard all constants, we obtain the following problem: min ϕ E πϕ(x0:T ) " Eθ(xT ) + τ T −1 X t=0 log πϕ(xt+1\|xt) + τ T −1 X t=0 1 2s2 t \|\|xt+1 −xt\|\|2 # , (8) which is an optimal control problem. The controller πϕ(·) is optimized to minimize the terminal cost Eθ(xT ) plus the running costs for each transition (xt, xt+1). The first running cost log πϕ(xt+1\|xt) is responsible for conditional entropy maximization, because Eπ[log πϕ(xt+1\|xt)] = −H(πϕ(xt+1\|xt)). The second running cost regularizes the “velocity" \|\|xt+1 −xt\|\|2. The temperature τ balances between the terminal and running costs. We have circumvented the marginal entropy computation in GCD, as all terms in Eq. (8) are easily computable. For the diffusion model considered in this paper (Eq. (1)), the conditional entropy has a particularly simple expression H(πϕ(xt+1\|xt)) = D log σt + 0.5D log 2π. Therefore, optimizing the entropy running cost amounts to learning σt’s in diffusion, and we treat σt’s as a part of the diffusion model parameter ϕ in DxMI. 5 Figure 2: 2D density estimation on 8 Gaussians. Red shades indicate the energy (white is low), and the dots are generated samples. Table 1: Quantitative results for 8 Gaussians experiment. SW denotes the sliced Wasserstein distance between samples and data. AUC is computed for classification between data and uniform noise using the energy. The standard deviation is computed from 5 independent samplings. The ideal maximum value of AUC is about 0.906. Method T Pretrain τ SW (↓) AUC (↑) DDPM 5 0.967±0.005 DDPM 10 0.824±0.002 DDPM 100 0.241±0.003 DDPM 1000 0.123±0.014 DxMI 5 ⃝ 0 0.074±0.018 0.707 DxMI 5 ⃝ 0.01 0.074±0.017 0.751 DxMI 5 ⃝ 0.1 0.068±0.004 0.898 DxMI 5 ⃝ 1 1.030±0.004 0.842 DxMI 5 × 0.1 0.076±0.011 0.883 4.2 Dynamic Programming We propose a dynamic programming approach for solving Eq. (8). Dynamic programming introduces value functions to break down the problem into smaller problems at each timestep, removing the need for back-propagation in time. Then, a policy, a diffusion model in our case, is optimized through iterative alternating applications of policy evaluation and policy improvement steps. Value Function. A value function, or cost-to-go function V t ψ(xt), is defined as the expected sum of the future costs starting from xt, following π. We write the parameters of a value function as ψ. V t ψ(xt) = Eπ " Eθ(xT ) + τ T −1 X t′=t log πϕ(xt′+1\|xt′) + T −1 X t′=t τ 2s2 t′ \|\|xt′+1 −xt′\|\|2 xt # , (9) for t = 0, . . . , T −1. Note that V T (xT ) = E(xT ). A value function can be implemented with a neural network, but there are multiple design choices, such as whether to share the parameters with π(x) or E(x), and also whether the parameters should be shared across time. We explore the options in our experiments. Policy Evaluation. During policy evaluation, we estimate the value function for the current diffusion model by minimizing the Bellman residual, resulting in the temporal difference update. min ψ Ext,xt+1∼π[(sg[V t+1 ψ (xt+1)] + τ log πϕ(xt+1\|xt) + τ 2s2 t \|\|xt −xt+1\|\|2 −V t ψ(xt))2], (10) where sg[·] denotes a stop-gradient operator indicating that gradient is not computed for the term. Policy Improvement. The estimated value is used to improve the diffusion model. For each xt in a trajectory x0:T sampled from πϕ(x0:T ), the diffusion model is optimized to minimize the next-state value and the running costs. min ϕ Eπϕ(xt+1\|xt) V t+1 ψ (xt+1) + τ log πϕ(xt+1\|xt) + τ 2s2 t \|\|xt −xt+1\|\|2 xt . (11) In practice, each iteration of policy evaluation and improvement involves a single gradient step. Adaptive Velocity Regularization (AVR). We additionally propose a method for systematically determining the hyperparameter st’s of the auxiliary distribution ˜q(x0:T −1\|xT ). We can optimize st such that the inequality Eq. (6) is as tight as possible by solving mins0,...,sT −1 KL(πϕ(x0:T )\|\|qθ(xT )˜q(x0:T −1\|xT )). After calculation (details in Appendix A), the optimal s∗ t can be obtained analytically: (s∗ t )2 = Ext,xt+1∼π[\|\|xt −xt+1\|\|2]/D. In practice, we can use exponential moving average to compute the expectation Ext,xt+1∼π[\|\|xt −xt+1\|\|2] during training: s2 t ←αs2 t + (1 −α)\|\|xt −xt+1\|\|2/D where we set α = 0.99 for all experiment. 6 4.3 Techniques for Image Generation Experiments When using DxDP for image generation, one of the most common applications of diffusion models, we introduce several design choices to DxDP to enhance performance and training stability. The resulting algorithm is summarized in Algorithm 2. Time-Independent Value Function. In image generation experiments (Section 5.2), we let the value function be independent of time, i.e., V t ψ(xt) = Vψ(xt). Removing the time dependence reduces the number of parameters to be trained. More importantly, a time-independent value function can learn better representation because the value function is exposed to diverse inputs, including both noisy and clean images. On the contrary, a time-dependent value function V t ψ(xt) never observes samples having different noise levels than the noise level of xt. Time Cost. Also, in the value update (Eq. (10)) step of image generation experiments, we introduce time cost function R(t) > 0, which replaces the running cost terms τ log πϕ(xt+1\|xt) + τ\|\|xt −xt+1\|\|2/(2s2 t). The time cost R(t) only depends on time t. The modified value update equation is given as follows: min ψ Ext,xt+1∼π[(sg[Vψ(xt+1)] + R(t) −Vψ(xt))2]. (12) Meanwhile, we retain the running cost terms in the diffusion model (policy) update step (Eq. (11)). The time cost R(t) is predetermined and fixed throughout training. The introduction of time cost is motivated by the observation that the running costs can fluctuate during the initial stage of training, posing difficulty in value function learning. The time cost stabilizes the training by reducing this variability. Moreover, the time cost ensures that the value function decreases monotonically over time. Such monotonicity is known to be beneficial in IRL for episodic tasks [16]. We employ two types of R(t): “linear" and “sigmoid". A linear time cost is given as R(t) = c where we use c = 0.05. The linear time cost encourages the value to decrease linearly as time progresses. The sigmoid time cost is R(t) = σ(−t+T/2))−σ(−t−1+T/2)), where σ(x) = (1+exp(−x))−1. With the sigmoid time cost, the value function is trained to follow a sigmoid function centered at T/2 when plotted against the time. Other forms of R(t) are also possible. Separate tuning of τ. In image generation, we assign different values of temperature τ for entropy regularization log πϕ(xt+1\|xt) and velocity regularization \|\|xt −xt+1\|\|2/(2s2 t), such that the resulting running cost becomes τ1 log πϕ(xt+1\|xt) + τ2\|\|xt −xt+1\|\|2/(2s2 t). Typically, we found τ1 > τ2 beneficial, indicating the benefit of exploration. Setting τ1 ̸= τ2 does not violate our maximum entropy formulation, as scaling τ2 is equivalent to scaling s2 t’s, which can be set arbitrarily. 5 Experiments In this section, we provide empirical studies that demonstrate the effectiveness of DxMI in training a diffusion model and an EBM. We first present a 2D example, followed by image generation and anomaly detection experiments. More details on experiments can be found in Appendix C. 5.1 2D Synthetic Data We illustrate how DxMI works on 2D 8 Gaussians data. DxMI is applied to train a five-step diffusion model (T = 5) with a corresponding time-dependent value network, both parametrized by timeconditioned multi-layer perceptron (MLP). The last time step (T = 5) of the value network is treated as the energy. The sample quality is measured with sliced Wasserstein distance (SW) to test data. Also, we quantify the quality of an energy function through the classification performance to uniform noise samples (Table 1). First, we investigate the effect of maximum entropy regularization τ. Setting an appropriate value for τ greatly benefits the quality of both the energy and the samples. When τ = 0.1, the samples from DxMI have smaller SW than the samples from a full-length DDPM do. The energy also accurately captures the data distribution, scoring high AUC against the uniform noise. Without entropy regularization (τ = 0), DxMI becomes similar to GAN [40]. The generated samples align moderately well with the training data, but the energy does not reflect the data distribution. When τ is too large (τ = 1), the generated samples are close to noise. In this regime, DxMI behaves similarly 7 Table 2: CIFAR-10 unconditional image generation. †: the starting point of DxMI fine-tuning. NFE FID (↓) Rec. (↑) Score SDE (VE) [23] 2000 2.20 0.59 PD [10] 8 2.57 Consistency Model [42] 2 2.93 PD [10] 1 8.34 2-Rectified Flow [43] 1 4.85 0.50 Consistency Model [42] 1 3.55 StyleGAN-XL [44] 1 1.85 0.47 Backbone: DDPM DDPM [3] 1000 3.21 0.57 FastDPM† [8] 10 35.85 0.29 DDIM [45] 10 13.36 SFT-PG [11] 10 4.82 0.606 DxMI 10 3.19 0.625 τ = 0 10 3.77 0.613 Linear time cost 10 3.39 0.595 No time cost 10 5.18 0.595 DxMI + Value Guidance 10 3.17 0.623 τ = 0 10 3.72 0.613 Backbone: DDGAN DDGAN† [46] 4 4.15 0.523 DxMI 4 3.65 0.532 Table 3: ImageNet 64×64 conditional image generation. †: the starting point of DxMI fine-tuning. NFE FID (↓) Prec. (↑) Rec. (↑) ADM [47] 250 2.07 0.74 0.63 DFNO [48] 1 8.35 PD [10] 1 15.39 0.59 0.62 BigGAN-deep [49] 1 4.06 0.79 0.48 Backbone: EDM EDM (Heun) [50] 79 2.44 0.71 0.67 EDM (Ancestral)† 10 50.27 0.37 0.35 EDM (Ancestral)† 4 82.95 0.26 0.25 Consistency Model [42] 2 4.70 0.69 0.64 Consistency Model [42] 1 6.20 0.68 0.63 DxMI 10 2.68 0.777 0.574 τ = 0 10 2.72 0.782 0.564 Linear time cost 10 2.81 0.742 0.594 DxMI+Value Guidance 10 2.67 0.780 0.574 τ = 0 10 2.76 0.786 0.560 DxMI 4 3.21 0.758 0.568 τ = 0 4 3.65 0.767 0.552 Linear time cost 4 3.40 0.762 0.554 DxMI+Value Guidance 4 3.18 0.763 0.566 τ = 0 4 3.67 0.770 0.541 to Noise Contrastive Estimation [41], enabling energy function learning to a certain extent. These effects are visualized in Fig. 2. Next, we experiment on whether pre-training a sampler as DDPM helps DxMI. Table 1 suggests that the pre-training is beneficial but not necessary to make DxMI work. We also visualize the value functions in Fig. 3 and find that the time evolution of value interpolates the data distribution and a Gaussian distribution. 5.2 Image Generation: Training Diffusion Models with Small T Table 4: LSUN Bedroom 256 × 256 unconditional image generation. NFE FID (↓) Prec. (↑) Rec. (↑) StyleGAN2 [51] 1 2.35 0.59 0.48 Backbone: EDM EDM [50] 79 2.44 0.71 0.67 Consistency Model [42] 2 5.22 0.68 0.39 DxMI 4 5.93 0.563 0.477 On image generation tasks, we show that DxMI can be used to fine-tune a diffusion model with reduced generation steps, such as T = 4 or 10. We test DxMI on unconditional CIFAR-10 [52] (32 × 32), conditional ImageNet [53] downsampled to 64 × 64, and LSUN Bedroom [54] (256×256), using three diffusion model backbones, DDPM [3], DDGAN [46], and variance exploding version of EDM [50]. The results can be found in Table 2, 3, and 4. Starting from a publicly available checkpoint of each pretrained backbone, we first adjust the noise schedule for the target sampling steps T. When adjusting the noise, for DDPM, we follow the schedule of FastDPM [8], and for EDM, we use Eq. (5) of [50]. No adjustment is made for DDGAN, which was originally built for T = 4. The adjusted models are used as the starting point of DxMI training. A single CIFAR-10 run reaches the best FID in less than 4 hours on four A100 GPUs. We set τ1 = 0.1 and τ2 = 0.01. The sigmoid time cost is used for all image generation experiments. The sample quality is measured by FID [55], Precision (Prec., [56]), and Recall (Rec., [56]). ResNet is used as our value function and is trained from scratch. More experimental details are in Appendix C.2. Short-run diffusion models fine-tuned by DxMI display competitive sample quality. Unlike distillation methods, which are often limited by their teacher model’s performance, DxMI can surpass the pretrained starting point. Although DxMI does not support single-step generation, DxMI offers a principled approach to training a high-quality generative model with a moderate computation burden 8 (Appendix D). Note that DDGAN does not fit the formulation of DxMI, as π(xt+1\|xt) in DDGAN is not Gaussian. Nevertheless, DxMI can still enhance sample quality, showing its robustness. Furthermore, DxMI outperforms SFT-PG [11], another IRL approach implemented with a policy gradient. For a fair comparison, we have ensured that the backbone and the initial checkpoint of SFT-PG and DxMI are identical. Thus, the performance gap can be attributed to the two differences between SFT-PG and DxMI. First, DxMI uses dynamic programming instead of policy gradient. In DxDP, the value function is more directly utilized to guide the learning of the diffusion model. Meanwhile, in policy gradient, the role of the value function is variance reduction. As SFT-PG also requires a value function during training, the computational overhead of DxMI is nearly identical to SFT-PG. Second, DxMI incorporates the maximum entropy principle, which facilitates exploration. We also conduct ablation studies for the components of DxMI and append the results in Table 2 and 3. First, the temperature parameters are set to zero τ1 = τ2 = 0 in the sampler update (11). Then, we compare the linear time cost to the sigmoid time cost. In both cases, We observe the increase in FID and the decrease in Recall. To investigate whether the trained value function captures useful information, we implement value guidance, where we shift the trajectory of generation slightly along the value function gradient, similarly to classifier guidance [47] and discriminator guidance [57]. When sampling the next step xt+1, we add a small drift with coefficient λ, i.e., xt+1 ←xt+1 −λσt∇xt+1Vψ(xt+1). We observe sample quality metric improvement until λ is 0.5. This observation suggests that the value function gradient is aligned well with the data density gradient. 5.3 Energy-Based Anomaly Detection and Localization Table 5: MVTec-AD multi-class anomaly detection and localization experiment. Anomaly detection (DET) and localization (LOC) performance are measured in AUC. Due to the space constraint, only the average AUC over 15 classes is presented. The full results are provided in Table 6. Model DET LOC DRAEM [58] 88.1 87.2 MPDR [59] 96.0 96.7 UniAD [60] 96.5±0.08 96.8±0.02 DxMI 97.0 ±0.11 97.1±0.02 τ = 0 67.9±5.90 84.6±4.02 We demonstrate the ability of DxMI to train an accurate energy function on an anomaly detection task using the MVTec-AD dataset [61], which contains 224×224 RGB images of 15 object categories. We follow the multi-class problem setup proposed by [60]. The training dataset contains normal object images from 15 categories without any labels. The test set consists of both normal and defective object images, each provided with an anomaly label and a mask indicating the defect location. The goal is to detect and localize anomalies, with performance measured by AUC computed per object category. This setting is challenging because the energy function should reflect the multi-modal data distribution. Following the preprocessing protocol in [60, 59], each image is transformed into a 272×14×14 vector using a pre-trained EfficientNet-b4 [62]. DxMI is conducted in a 272-dimensional space, treating each spatial coordinate independently. With the trained energy function, we can evaluate the energy value of 14x14 spatial features and use max pooling and bilinear interpolation for anomaly detection and localization, respectively. We use separate networks for the energy function and the value function in this experiment, as the primary goal is to obtain an accurate energy function. We employ an autoencoder architecture for the energy function, treating the reconstruction error of a sample as its energy [63]. The diffusion model and the value function are five-step time-conditioned MLPs. Unlike conventional diffusion models, DxMI allows for a flexible choice of π(x0). We set the initial distribution for the sampler to the data distribution applied with noise, aiming to identify the energy value more precisely near the data distribution. More experimental details can be found in Appendix C.3. DxMI demonstrates strong anomaly classification and localization performance, as shown in Table 5. This result indicates that the trained energy function effectively captures the boundary of normal data. When entropy maximization is disabled by τ = 0, the diffusion model fails to explore and only exploits regions of minimum energy, resulting in poor performance. We observe that a moderate level of τ = 0.1 benefits both the sampler and the energy function, as it encourages exploration and provides a suitable level of diversity in negative samples. 9 6 Related Work Faster Diffusion Models. Significant effort has been dedicated to reducing the number of generation steps in diffusion models during sampling while preserving sample quality. One popular approach is to keep the trained diffusion model unchanged and improve the sampling phase independently by tuning the noise schedule [8, 64, 65, 9], improving differential equation solvers [50, 66, 67, 68], and utilizing non-Markovian formulations [45, 69, 70]. While these methods are training-free, the sample quality can be further improved when the neural network is directly tuned for short-run sampling. Distillation methods train a faster diffusion sampler using training signal from a longer-run diffusion model, showing strong performance [10, 71, 72, 48, 73, 43, 42, 74]. A distilled model usually cannot outperform the teacher model, but adversarial or IRL methods may exceed full-length diffusion models. Hybrid methods [13, 12] combine distillation with adversarial loss, while other methods [46, 75] apply adversarial training to each denoising step. DxMI and SFT-PG [11] rely fully on adversarial training for final samples, allowing beneficial deviations from the diffusion path and reducing statistical distance from the data. RL for Diffusion Model. RL is often employed to fine-tune diffusion models for a reward function. The source of the reward signal can be a computer program [24, 76, 20, 77, 78], or a human evaluator [19, 79, 25]. DxMI focuses on a setting where the estimated log data density is the reward. When RL is applied to diffusion models, the policy gradient [80] is the dominant choice [24, 76, 11, 77]. DxMI offers a value function-based approach as an alternative to the policy gradient. Maximum entropy RL for diffusion models is investigated in [20, 81, 82] but only in the continuous-time setting. DxDP investigates the discrete-time setting, which is more suitable for accelerating generation speed. Energy-Based Models. DxMI provides a method of utilizing a diffusion model to eliminate the need for MCMC. Many existing EBM training algorithms rely on MCMC, which is computationally expensive and difficult to optimize for hyperparameters [34, 83, 18, 84, 85, 35, 63, 59]. Joint training of an EBM with a separate generative model is a widely employed strategy to avoid MCMC. EBMs can be trained jointly with a normalizing flow [86, 87], a generator [30, 88, 89], or a diffusion model [90, 33]. DxMI shares the objective function with several prior works in EBM [27, 28, 29, 30, 31, 33]. However, none of the works use a diffusion model directly as a sampler. Related Theoretical Analyses. The convergence guarantees of entropy-regularized IRL are provided in [91, 92] under the assumption of a linear reward and the infinite time horizon. Their guarantees are not directly applicable to a practical instance of DxMI, mainly due to the nonlinearity of the reward function, the continuous state and action spaces, and the finite-horizon setting. Establishing the convergence guarantee for DxMI could be an important future research direction. On the other hand, theoretical analyses have been conducted on MaxEnt RL under finite state and action spaces [93], which is relevant for the discrete version of DxDP. More focused analysis on entropy regularized RL for diffusion models is provided in [94]. 7 Conclusion In this paper, we leverage techniques from sequential decision making to tackle challenges in generative modeling, revealing a significant connection between these two fields. We anticipate that this connection will spur a variety of algorithmic innovations and find numerous practical applications. Broader Impacts. DxMI may facilitate deep fakes or fake news. However, trained on relatively low-resolution academic datasets, the models created during our experiments are not capable enough to cause realistic harm. Generative models trained solely using DxMI may possess fairness issues. Limitations. Training multiple components simultaneously, DxMI introduces several hyperparameters. To reduce the overhead of practitioners, we provide a hyperparameter exploration guideline in Appendix B. DxMI is not directly applicable to training a single-step generator. However, a diffusion model fine-tuned by DxMI can be distilled to a single-step generator. DxMI does not offer the flexibility of using a different value of generation steps T during the test time. Direct theoretical analysis of DxMI is challenging since the models are built on deep neural networks. Theoretical analysis that rationalizes the empirical results will be an important direction for future work. 10 Acknowledgments and Disclosure of Funding S. Yoon is supported by a KIAS Individual Grant (AP095701) via the Center for AI and Natural Sciences at Korea Institute for Advanced Study and IITP/MSIT (RS-2023-00220628). H. Hwang and F. Park are supported in part by IITP-MSIT grant RS-2021-II212068 (SNU AI Innovation Hub), IITP-MSIT grant 2022-220480, RS-2022-II220480 (Training and Inference Methods for Goal Oriented AI Agents), MSIT(Ministry of Science, ICT), Korea, under the Global Research Support Program in the Digital Field program(RS-2024-00436680) supervised by the IITP(Institute for Information & Communications Technology Planning & Evaluation), KIAT grant P0020536 (HRD Program for Industrial Innovation), SRRC NRF grant RS-2023-00208052, SNU-AIIS, SNU-IPAI, SNU-IAMD, SNU BK21+ Program in Mechanical Engineering, SNU Institute for Engineering Research, Microsoft Research Asia, and SNU Interdisciplinary Program in Artificial Intelligence. D. Kwon is partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00252516 and No. RS-2024-00408003) and the POSCO Science Fellowship of POSCO TJ Park Foundation. Y.-K. 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Fine-tuning of diffusion models via stochastic control: entropy regularization and beyond. arXiv preprint arXiv:2403.06279, 2024. 16 A Adaptive Velocity Regularization We are interested in the following optimization problem: min s0,...,sT −1 KL(πϕ(x0:T )\|\|qθ(xT )˜q(x0:T −1\|xT )). (13) Plugging in our choice of log ˜q(xt\|xt+1) = −\|\|xt−xt+1\|\|2 2s2 t −D log st, we can rewrite the optimization problem as follows. min s0,...,sT −1 T −1 X t=0 Ext,xt+1∼π \|\|xt −xt+1\|\|2 2s2 t + D 2 log s2 t , (14) where constant term with respect to st is omitted. Since the object function is separable, we can solve the optimization for each st independently. min st Ext,xt+1∼π \|\|xt −xt+1\|\|2 2s2 t + D log st, t = 0, ..., T −1. (15) This optimization has an analytic solution: (s∗ t )2 = Ext,xt+1∼π \|\|xt −xt+1\|\|2 /D. B Guideline for Hyperparameters Value function. The most crucial design decision in DxMI is how to construct the value function. This design revolves around two primary axes. The first axis is whether the value function should be time-dependent or time-independent. The second axis is whether the value function should share model parameters with other networks, such as the sampler or the energy network. Our experiments demonstrate various combinations of these design choices. In a 2D experiment, the time-dependent value function shares parameters with the EBM, which is a recommended approach for smaller problems. In an image experiment, we employ a time-independent value function that shares parameters with the energy network, effectively making the value function and the energy function identical. This design choice promotes monotonicity and efficient sample usage, as a single value function learns from all intermediate samples. In an anomaly detection experiment, we use a time-dependent value function that does not share parameters with any other network. This design is suitable when a specific structure needs to be enforced on the energy function, such as with an autoencoder, and when making the structure time-dependent is not straightforward. While these are the options we have explored, there are likely other viable possibilities for designing the value function. Coefficient τ. Although the coefficient τ plays an important role in DxMI, we recommend running DxMI with τ = 0 when implementing the algorithm for the first time. If everything is in order, DxMI should function to some extent. During normal training progression, the energy values of positive and negative samples should converge as iterations proceed. After confirming that the training is progressing smoothly, you can start experimenting with increasing the value of τ. Since γ determines the absolute scale of the energy, the magnitude of τ should be adjusted accordingly. We set γ = 1 in all our experiments and recommend this value. In such a case, the optimal τ is likely to be less than 0.5. Learning rate. As done in two time-scale optimization [55], we use a larger learning rate for the value function than for the sampler. When training the noise parameters σt in the diffusion model, we also assign a learning rate 100 times larger than that of the sampler. C Details on Implementations and Additional Results C.1 2D Experiment Sample quality is quantified using sliced Wasserstein-2 distance (SW) with 1,000 projections of 10k samples. The standard deviation is computed from 5 independent samplings. Density estimation 17 Algorithm 2 Diffusion by Maximum Entropy IRL for Image Generation 1: Input: Dataset D, Energy Eθ(x), Value Vψ(xt), and Sampler πϕ(x0:T ) 2: st ←σt // AVR initialization 3: for x in D do // Minibatch dimension is omitted for brevity. 4: Sample x0:T ∼πϕ(x). 5: minθ Eθ(x) −Eθ(xT ) +γ(Eθ(x)2 + Eθ(xT )2) // Energy update 6: for t = T −1, . . . , 0 do // Value update 7: minψ[sg[Vψ(xt+1)] + R(t) −Vψ(xt)]2 8: end for 9: for xt randomly chosen among x0:T do // Sampler update 10: Sample one-step: xt+1 ∼πϕ(xt+1\|xt) // Reparametrization trick 11: minϕ V t+1 ψ (xt+1(ϕ)) −τ1D log σt + τ2 2s2 t \|\|xt −xt+1(ϕ)\|\|2 // xt+1 is a function of ϕ. 12: end for 13: s2 t ←αs2 t + (1 −α)\|\|xt −xt+1\|\|2/D // AVR update 14: end for performance is measured by AUC on discriminating test data and uniform samples over the domain. AUC is also computed with 10k samples. V(x,t = 0) V(x,t = 1) V(x,t = 2) V(x,t = 3) V(x,t = 4) E(x) (= V(x,t = 5)) Figure 3: Value functions at each time step (τ = 0.1 case). Blue indicates a low value. C.2 Image Generation Datasets. CIFAR-10 is retrieved through torchvision API. ImageNet is downloaded from Kaggle and downsampled to 64 × 64 following https://github.com/openai/guided-diffusion. We apply random horizontal flips on all images. When computing FID, the whole 50,000 training images of CIFAR-10 are used. We make sure that no JPEG compression is used during processing. For ImageNet, we use the batch stat file provided by https://github.com/openai/guided-diffusion. Models. For DDPM on CIFAR-10, we use the checkpoint provided by the SFT-PG repository [11]. For DDGAN, we utilize the official checkpoint. For EDM on ImageNet 64, we use the checkpoint from the consistency model repository [42]. In all experiments, we employ the same ResNet architecture, which does not include any normalization layers, such as batch normalization. Training. For all runs, we use a batch size of 128. In the CIFAR-10 experiments, we use the Adam optimizer with a learning rate of 10−7 for the sampler weights, 10−5 for the value weights, and 10−5 for the σt’s. In the ImageNet 64 experiments, we use RAdam with a learning rate of 10−8 for the sampler. Additionally, we utilize a mixed precision trainer to handle FP16 weights. The value weights are updated with a learning rate of 10−5 using Adam. The σt’s are updated with a learning rate of 10−6. To select the best model, we periodically generate 10,000 images for CIFAR-10 and 5,000 images for ImageNet. The checkpoint with the best FID score is selected as the final model. Evaluation For computing FID, Precision, and Recall scores, we used the TensorFlow-based evaluation script provided by Consistency Models [42] repository, which is based on the codebase of [47]. All the images are saved in a PNG format when fed to the evaluation script. Additional Details. For optimal performance in image generation, we often tune the coefficients of two running costs separately. Let us denote the coefficient of log π(xt+1\|xt) as τ1 and the coefficient of 1 2s2 t \|\|xt −xt+1\|\|2 as τ2. In the CIFAR-10 experiments with T = 10 and T = 4, we set τ1 = 0.1 18 and τ2 = 0.01. In the ImageNet experiments with T = 10, we set τ1 = τ2 = 0.01, and for T = 4, we set τ1 = 0.1 and τ2 = 0.01. We believe the optimal τ values vary for each setting due to differences in noise schedules and magnitudes. For example, the DDPM backbone is a variance-preserving formulation, while EDM is a variance-exploding formulation. Exploring a unified method for selecting the entropy regularization parameter is an interesting research direction. C.3 Anomaly Detection Dataset and Feature Extraction. MVTec AD is a dataset designed to evaluate anomaly detection techniques, particularly for industrial inspection. It contains over 5000 high-resolution images divided into 15 different categories of objects and textures. Each category includes a set of defect-free training images and a test set with both defective and non-defective images. The original dataset consists of high-resolution RGB images sized 224×224. Following the methods used in similar studies [59, 60], we extract a 272×14×14 full feature representation of each image using a pre-trained EfficientNet-b4 [62]. We then train the energy function using DxMI in 272dimensional space, treating every spatial feature from the training images as normal training data. Each data point x in R272 is projected to S272 through normalization. This projection is effective because the direction of the feature often represents the original image better than its magnitude. Anomaly Detection and Localization. For anomaly detection, the energy of a single image is calculated by applying max pooling to the energy value of each spatial feature. For anomaly localization, the energy values of the 14×14 spatial features are upsampled to match the original 224×224 image resolution using bilinear interpolation. Model Design. We utilize an autoencoder architecture for the energy function, as described in [63], using the reconstruction error of a sample as its energy value. Considering that the data distribution lies on S272, we appropriately narrow the function classes for the energy and sampler. Specifically, we constrain the decoder manifold of the energy function and the mean prediction of the sampler to remain on S272. We pretrain the energy function (i.e., the autoencoder) to minimize the reconstruction error of the training data. Both the sampler and the value function are trained from scratch. Choice of π(x0). Unlike traditional diffusion models, DxMI permits a flexible choice of π(x0). To train the energy function effectively near the data distribution, we set the initial distribution for the sampler as the data distribution corrupted with noise. To apply meaningful noise to the data in S272, we use the pretrained autoencoder that is also used for the initial energy function to project the samples from the data distribution to the latent space, apply perturbations, and then restore the data to produce initial samples, as suggested in [59]. To maintain a consistent initial distribution, we fix the autoencoder used for generating the initial samples. Additional Details. We use autoencoder with latent dimension 128 as the energy function. Encoder and decoder each consist of an MLP with 3 hidden layers and 1024 hidden dimensions. We use a time-conditional MLP for the sampler and value function, encoding the time information into a 128-dimensional vector using sinusoidal positional encoding. The input xt is concatenated with the time embedding vector. We use MLPs with 3 hidden layers of 2048 and 1024 hidden dimensions for the sampler and value function, respectively. The model is trained for 100 epochs with a batch size of 784 (=4×14×14) using the Adam optimizer. We use a learning rate of 10−5 for the sampler and value function and 10−4 for the energy function. D Implementation and Computational Complexity of DxMI DxMI (Algorithm 1) may seem complicated, but it largely mirrors the procedure of Max Ent IRL, with the exception of the AVR update. Notably, DxDP, the Max Ent RL subroutine within DxMI, is significantly simpler than standard actor-critic RL algorithms. Unlike these algorithms—such as Soft Actor Critic (SAC) [17], which requires training both a value function and two Q-functions—DxMI only trains a single value function and no Q-functions. 19 Table 6: MVTec-AD detection and localization task in the unified setting. AUROC scores (percent) are computed for each class. UniAD and DRAEM results are adopted from [60]. The largest value in a task is marked as boldface. Detection Localization DxMI UniAD MPDR DRAEM DxMI UniAD MPDR DRAEM Bottle 100.0 ±0.00 99.7±0.04 100.0 97.5 98.5±0.03 98.1±0.04 98.5 87.6 Cable 97.1± 0.37 95.2±0.84 95.5 57.8 96.6±0.10 97.3±0.10 95.6 71.3 Capsule 89.8± 0.61 86.9±0.73 86.4 65.3 98.5±0.03 98.5±0.01 98.2 50.5 Hazelnut 100.0± 0.04 99.8±0.10 99.9 93.7 98.4±0.04 98.1±0.10 98.4 96.9 Metal Nut 99.9± 0.11 99.2±0.09 99.9 72.8 95.5±0.03 94.8±0.09 94.5 62.2 Pill 95.4± 0.66 93.7±0.65 94.0 82.2 95.6±0.07 95.0±0.16 94.9 94.4 Screw 88.9± 0.51 87.5±0.57 85.9 92.0 98.6±0.08 98.3±0.08 98.1 95.5 Toothbrush 92.2±1.46 94.2±0.20 89.6 90.6 98.8±0.04 98.4±0.03 98.7 97.7 Transistor 99.2±0.28 99.8±0.09 98.3 74.8 96.0±0.13 97.9±0.19 95.4 65.5 Zipper 96.3±0.50 95.8±0.51 95.3 98.8 96.7±0.08 96.8±0.24 96.2 98.3 Carpet 99.9±0.04 99.8±0.02 99.9 98.0 98.8±0.02 98.5±0.01 98.8 98.6 Grid 98.6±0.28 98.2±0.26 97.9 99.3 97.0±0.07 96.5±0.04 96.9 98.7 Leather 100.0±0.00 100.0±0.00 100.0 98.7 98.5±0.03 98.8±0.03 98.5 97.3 Tile 100.0±0.00 99.3±0.14 100.0 95.2 95.2 ±0.14 91.8±0.10 94.6 98.0 Wood 98.3±0.33 98.6±0.08 97.9 99.8 93.8±0.07 93.2±0.08 93.8 96.0 Mean 97.0±0.11 96.5±0.08 96.0 88.1 97.1±0.02 96.8±0.02 96.7 87.2 DxMI does not demand excessive computation, a major concern in image generation experiments. In these experiments, the only additional component beyond the diffusion model is the EBM, which shares the same network as the value function. Additionally, the EBM used in DxMI is typically much smaller than the diffusion model, imposing minimal computational overhead. For instance, in our CIFAR-10 experiment (T=10), the EBM consists of 5M parameters, compared to 36M in the diffusion model. This is also significantly smaller than the critic networks used in GANs, such as the 158.3M parameters in BigGAN [49]. In practice, our CIFAR-10 experiment completes in under 24 hours on two A100 GPUs, while the ImageNet 64 experiment takes approximately 48 hours on four A100 GPUs. E Interpretation of the policy improvement method In this section, we show that our policy improvement method introduced in Eq. (11) can effectively minimize the KL divergence between the joint distributions, KL(πϕ(x0:T )\|\|qθ(xT )˜q(x0:T −1\|xT )). The policy improvement at each timestep t can be expressed as min π(xt+1\|xt) Ext,xt+1∼π V t+1(xt+1) + τ log π(xt+1\|xt) −τ log ˜q(xt\|xt+1) + const. (16) Here omitted the parameters ϕ and ψ to avoid confusion. Using the definition of value function, above minimization can be expressed as min π(xt+1\|xt) Eπ(xt:T ) " E(xT ) + τ T −1 X t′=t log π(xt′+1\|xt′) −τ T −1 X t′=t log ˜q(xt′\|xt′+1) # , (17) where Eπ(xt:T ) " E(xT ) + τ T −1 X t′=t log π(xt′+1\|xt′) −τ T −1 X t′=t log ˜q(xt′\|xt′+1) # (18) = τEπ(xt:T ) [−log q(xT ) −log ˜q(xt:T −1\|xT ) + log π(xt:T ) −log Z −log π(xt)] (19) = τKL(π(xt:T )\|\|q(xT )˜q(xt:T −1\|xT )) −τ log Z + τH(xt) (20) Note that xt is fixed in this optimization problem, and the optimization variable is xt+1. Therefore the policy improvement step at time t is equivalent to min π(xt+1\|xt) KL(π(xt:T )\|\|q(xT )˜q(xt:T −1\|xT )) (21) Therefore, the policy improvement step at each time step t results in the minimization of the KL divergence between the joint distribution, π(x0:T ) and ˜q(x0:T ). 20 F Additional Image Generation Results Additional samples from the image generation models are presented in Fig. 4 and Fig. 5. Figure 4: Randomly selected samples from CIFAR-10 training data, SFT-PG (T = 10, FID: 4.32), and DxMI (T = 10, FID: 3.19). Figure 5: Randomly selected samples from ImageNet 64×64 training data, Consistency Model (T = 1, FID: 6.20), DxMI (T = 4, FID: 3.21), and DxMI (T = 10, FID: 2.68). Note that the Consistency Model samples distort human faces, while the DxMI samples depict them in correct proportions. 21 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: Our contributions are the maximum entropy IRL formulation for diffusion models and a learning algorithm inspired by dynamic programming. These contributions are explicitly stated in the abstract and the introduction. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We have dedicated a paragraph to the discussion of limitations in Section 7. 3. Theory Assumptions and Proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [NA] Justification: No formal theorem or proposition is provided in the main manuscript. 4. Experimental Result Reproducibility Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: We have put our best effort into making the proposed method reproducible. The concrete description of the proposed algorithm is provided in Algorithm 1. The detailed information on the model and implementation are described in Section 5 and Appendix C. We release our source code and model checkpoints at https://github.com/swyoon/ Diffusion-by-MaxEntIRL. Our public codebase will include instructions for preparing data and evaluating performance metrics. 5. Open access to data and code Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: We provide our code at https://github.com/swyoon/ Diffusion-by-MaxEntIRL. 6. Experimental Setting/Details Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: Section 5 and Appendix C describe experimental details including hyperparameters, model architecture, data preparation, and the choice of optimizers. We also provide a guideline for tuning hyperparameters in Appendix B. 7. Experiment Statistical Significance Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: We report error bars for 2D experiments where repetitive experiments can be readily conducted. We did not do statistical significance testing. 22 8. Experiments Compute Resources Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: We provide information on compute resource in Section 5 and Appendix C. 9. Code Of Ethics Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? Answer: [Yes] Justification: We have reviewed and followed the NeurIPS Code of Ethics. 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: We have included a paragraph discussing the potential negative impact of our work in Section 7. 11. Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: Although we have image generation models, they are not equipped with safeguards. Our models are trained on relatively low-resolution images, such as CIFAR-10 and ImageNet 64×64, which reduces the likelihood of causing harm in the real world. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: We have cited all datasets and baseline models used in our experiment. We also provide the license information in Appendix C. 13. New Assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: Our work does not provide new assets. 14. Crowdsourcing and Research with Human Subjects Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justification: Our work does not involve any crowd sourcing or experiments with human subjects. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? 23 Answer: [NA] Justification: Our work does not involve experiments with human subjects. 24	2024	776
4,476	Towards Heterogeneous Long-tailed Learning: Benchmarking, Metrics, and Toolbox Haohui Wang, Weijie Guan, Jianpeng Chen, Zi Wang, Dawei Zhou Computer Science, Virginia Tech {haohuiw,skjguan,jianpengc,ziwang,zhoud}@vt.edu Abstract Long-tailed data distributions pose challenges for a variety of domains like ecommerce, finance, biomedical science, and cyber security, where the performance of machine learning models is often dominated by head categories while tail categories are inadequately learned. This work aims to provide a systematic view of long-tailed learning with regard to three pivotal angles: (A1) the characterization of data long-tailedness, (A2) the data complexity of various domains, and (A3) the heterogeneity of emerging tasks. We develop HEROLT, a comprehensive longtailed learning benchmark integrating 18 state-of-the-art algorithms, 10 evaluation metrics, and 17 real-world datasets across 6 tasks and 4 data modalities. HEROLT with novel angles and extensive experiments (315 in total) enables effective and fair evaluation of newly proposed methods compared with existing baselines on varying dataset types. Finally, we conclude by highlighting the significant applications of long-tailed learning and identifying several promising future directions. For accessibility and reproducibility, we open-source our benchmark HEROLT and corresponding results at https://github.com/SSSKJ/HeroLT. 1 Introduction In the era of big data, many high-impact domains, such as e-commerce [1, 2], finance [3], biomedical science [4, 5], and cyber security [6, 7], naturally exhibit long-tailed data distributions, where a few head categories1 are well-studied with abundant data, while massive tail categories are under-explored with scarce data. To name a few, in financial transaction networks, the majority of transactions fall into a few head classes that are considered normal, like wire transfers and credit card payments. However, a large number of tail classes correspond to various fraudulent transaction types, like synthetic identity transactions and money laundering. Although fraudulent transactions rarely occur, detecting them is essential for preventing unexpected financial loss [8, 9]. Another example is antibiotic resistance genes, which can be classified based on the antibiotic class they confer to and their transferable ability. Genes with mobility and strong human pathogenicity may not be detected frequently and are viewed as tail classes, but these resistance genes have the potential to be transmitted from the environment to bacteria in humans, thereby posing an increasing global threat to human health [10, 11]. Massive long-tailed learning studies have been conducted in recent years, proposing methods like the cost-sensitive focal loss to effectively address the data imbalance by reshaping the standard crossentropy loss [12], and graph augmentation by interpolating tail node embeddings and generating new edges [13]. The advancements in tackling long-tailed problems drove the publication of several studies on the survey of long-tailed problems [14, 15, 16, 17, 18], which generally categorize existing 1Long-tailed problems occur in labels or input data (like degrees of nodes), collectively named as category. 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. A1. LongTailedness A3. Task Heterogeneity A2. Data Complexity Entity Alignment Node Classification Classification Image Classification Instance Segmentation Multi-Label Text Classification Information Extraction Recommendation Sequential Data Grid Data Relational Data Extreme # of Categories Data Imbalance Tabular Data Heterogeneous Long-Tailed Learning [13, 30, 31, 32] [33, 34] [1, 35, 36, 37, 38] [39] [40, 41, 42, 43, 26, 44, 45] [46, 47, 48] [49, 50, 51] [52] [53, 54, 55] [56, 57] [39, 58, 59] [60, 61] [41, 62, 63, 64] [65, 66, 67] [68, 69] Figure 1: The systematic view of heterogeneous long-tailed learning concerning three pivotal angles, including long-tailedness (colored in red), data complexity (green), and task heterogeneity (blue). works by different types of learning algorithms (e.g., re-sampling [19, 20], cost-sensitive [21, 22], transfer learning [23, 24], decoupled training [25, 26], and meta learning [27, 28, 29]). Despite tremendous progress, some pivotal questions largely remain unresolved, e.g., how can we characterize the extent of long-tailedness of given data? how do long-tailed algorithms perform with regard to different tasks on different domains? To fill the gap, this work aims to provide a systematic view of long-tailed learning with regard to three pivotal angles in Figure 1: (A1) the characterization of data long-tailedness: long-tailed data exhibits a highly skewed data distribution and an extensive number of categories; (A2) the data complexity of various domains: a wide range of complex domains may naturally encounter long-tailed distribution, e.g., tabular data, sequential data, grid data, and relational data; and (A3) the heterogeneity of emerging tasks: it highlights the need to consider the applicability and limitations of existing methods on heterogeneous tasks. With three major angles in long-tailed learning, we design extensive experiments that conduct 18 state-of-the-art algorithms and 10 evaluation metrics on 17 real-world benchmark datasets across 6 tasks and 4 data modalities. Key Takeaways: Through extensive experiments (see Section 3), we find (1) most works mainly focus on data imbalance while paying less attention to the extreme number of categories in the long-tailed distribution; (2) surprisingly none of the algorithms statistically outperforms others across all tasks and domains, emphasizing the importance of algorithm selection in terms of scenarios. In general, we summarize the main contributions of HEROLT as below: • Comprehensive Benchmark. We conduct a comprehensive review and examine long-tailed learning concerning three pivotal angles: (A1) the characterization of data long-tailedness, (A2) the data complexity of various domains, and (A3) the heterogeneity of emerging tasks. • Insights and Future Directions. With comprehensive results, our study highlights the importance of characterizing the extent of long-tailedness and algorithm selection while identifying open problems and opportunities to facilitate future research. • The Open-Sourced Toolbox. We provide a fair and accessible performance evaluation of 18 state-of-the-art methods on multiple benchmark datasets using accuracy-based and ranking-based evaluation metrics at https://github.com/SSSKJ/HeroLT. 2 HeroLT: Benchmarking Heterogeneous Long-Tailed Learning 2.1 Preliminaries and Problem Definition Here we provide a general definition of long-tailed learning as follows. Given a long-tailed dataset D = {x1, . . . , xn} of n samples from C categories, let Y denote the label set of category, where a 2 0 10 20 30 Category 0 200 400 600 800 Frequency (a) Longtailed distribution Data Pareto 0 5 10 0 5 10 (b) Obey Assumption 1, 2 Head Tail 0 5 10 0 5 10 (c) Violate Assumption 1 Head Tail 0 5 10 0 5 10 (d) Violate Assumption 2 Head Tail 0 5 10 0 5 10 (e) Violate Assumption 1, 2 Head Tail 0.0 0.5 1.0 1.5 2.0 2.5 Figure 2: Illustrative figures of a synthetic long-tailed distributed data. (a) long-tailed distribution of categories. (b) Visualization of obeying of Assumption 1, 2. (c) Visualization of violating of Assumption 1. (d) Visualization of violating of Assumption 2. (e) Visualization of violating of Assumption 1, 2. sample xi may belong to one or more categories. For example, in image classification, one sample belongs to one category; in document classification, one document is often associated with multiple categories (i.e., topics). For simplicity, we represent D = {D1, . . . , DC}, where Dc, c = 1, . . . , C denotes the subset of samples belong to category c, and the size of each category nc = \|Dc\| following a descending order. As an instantiation, a synthetic long-tailed dataset is given in Figure 2(a). Categories with massive samples refer to head categories, while categories with only a few samples refer to tail categories. Without the loss of generality, we make the following assumptions in long-tailed learning regarding the decision region of head and tail categories on the embedding space. Assumption 1 (Smoothness Assumption for Head Category). Given a long-tailed dataset, the distribution of the decision region of each head category is sufficiently smooth. Assumption 2 (Compactness Assumption for Tail Category). Given a long-tailed dataset, the tail category samples can be represented as a compact cluster in the feature space. These assumptions ensure that tail categories are identifiable and meaningful. If Assumption 1 of smoothness is violated, then it is challenging to identify the head category clearly. For example in Figure 2(c), it is difficult to depict the decision region of the head category. Similarly, if Assumption 2 of compactness is violated, as seen in Figure 2(d), it is difficult to determine whether it is noise data or data from tail category. Based on the assumptions, we give the definition of long-tailed learning. Problem 1. Long-Tailed Learning. Given: a training set D of n samples from C distinct categories following long-tailed distribution, and the label set of category Y. The data follows long-tailed distribution, i.e., the frequency of the categories is approximated as limy→∞etyP(Y > y) = ∞, ∀t > 0, where Y is a random variable. Find: a function f : X →Y that gives accurate label predictions on both head and tail categories. 2.2 Benchmark Angles in HEROLT Angle 1: Long-Tailedness in Terms of Data Imbalance and Extreme Number of Categories. While extensive public datasets and algorithms are available for benchmarking, these datasets or algorithms often exhibit varying extents of long-tailedness or focus on specific characteristics of long-tailed problems, making it challenging to select appropriate datasets and baselines to evaluate new algorithms. Two key properties of long-tailed data lie in highly skewed data distribution and an extreme number of categories. The former introduces a significant difference in sample sizes between head and tail categories, resulting in a bias towards the head categories [17], while the latter poses challenges in learning classification boundaries due to the increasing number of categories [70, 71]. To measure the long-tailedness of a dataset, we introduce three metrics in Table 1. Firstly, a commonly used metric is imbalance factor (IF) [72], and a value closer to 1 means a more balanced dataset. Secondly, Gini coefficient can be used as a metric to quantify long-tailedness [18]. It ranges from 0 to 1, where a smaller value indicates a more balanced dataset. IF quantifies data imbalance between the most majority and the most minority categories, while Gini coefficient measures overall imbalance, unaffected by extreme samples or absolute data size. However, the two metrics pay more attention to data imbalance and may not reflect the number of categories. Therefore, Pareto-LT Ratio is proposed to jointly evaluate the two aspects of the data long-tailedness. Intuitively, the higher the skewness of the data distribution, the higher Pareto-LT Ratio may be; the more number of categories, the higher 3 Table 1: Three metrics for measuring long-tailedness of datasets. Q(p) = min{y : Pr(Y ≤y) = p, 1 ≤y ≤C} is the quantile function of order p ∈(0, 1) for Y. Metric name Computation Description Imbalance Factor [72] n1/nC Ratio to the size of largest and smallest categories. Gini Coefficient [18] PC i=1 PC j=1 \|ni−nj\| 2nC Relative mean absolute differences between category sizes. Pareto-LT Ratio C−Q(0.8) Q(0.8) The number of categories of the last 20% samples to the number of categories of the rest 80% samples. the value of Pareto-LT Ratio. In light of three long-tailedness metrics, we gain a better understanding of which datasets and baselines to consider when evaluating a newly proposed algorithm. • When all three metrics have large values, it indicates a dataset with a severe long-tailed distribution, necessitating an algorithm that addresses both data imbalance and the extreme number of categories. • If Gini coefficient and IF are relatively small, but Pareto-LT Ratio is large, the main challenges of the dataset may lie in massive categories, as exemplified by the experiments in Section 3.4. In such cases, methods like extreme classification [73, 74] and meta learning [27, 28, 29] would be preferred. • When Gini coefficient and IF are large, but Pareto-LT Ratio is relatively small, it suggests that the challenges of data imbalance may be more significant than the number of categories. Algorithms employing techniques like re-sampling [19, 20] or re-weighting [21, 22] may already effectively handle data imbalance, as exemplified by the experiments in Section 3.5. • If all three metrics are small, the extent of the long-tailedness of the dataset may be small, and ordinary machine learning methods may achieve decent performance. In addition to the above metrics, we integrate the Bayes Imbalance Impact Index (BI3 [75]) and the Complementary Cumulative Distribution Function (CCDF) into our toolbox. While BI3 and CCDF are useful for certain application domains, i.e., BI3 for binary classification and CCDF for visualizing label distributions, they are not specifically designed for long-tailed learning. Therefore, we mention them here briefly without providing a detailed analysis. We give a detailed evaluation of long-tailedness metrics (AppendixB.1), and show the long-tailedness of benchmark datasets (Table 2). We analyze whether long-tailed algorithms are specifically designed to address data imbalance and an extreme number of categories (AppendixB.2), and provide a comprehensive experimental analysis (Section3). Angle 2: Data Complexity with 17 Datasets across 4 Data Modalities. Most existing long-tailed benchmarks mainly focus on image datasets [14, 15, 16, 17, 18]. However, in real-world applications, various types of data including tabular, grid, sequential, and relational data, face long-tailed problems. To fill this gap, we consider data complexity based on data types with long-tailed distribution. • Tabular data comprises samples (rows) with the same set of features (columns) and is used in practical applications (e.g., medicine, finance, and e-commerce). Specifically, Glass [76] is a tabular dataset with feature attributes, where the number of samples in different categories follows a long-tailed distribution. In addition to node connections, graph data such as Amazon [55] often include node attributes, which can be regarded as tabular data. • Sequential data denotes that data points in the dataset are dependent on the other points. A common example is a time series such as S&P 500 Daily Changes [39], where the input (price over date) shows a long-tailed distribution. Another example of sequential data is text composed of manuscripts, messages, and reviews, such as Wikipedia [68] containing Wikipedia articles with the long-tailed distribution of Wikipedia categories. • Grid data records regularly spaced samples over an area. Images can be viewed as grid data by mapping grid cells onto pixels one for one. The labels of images often exhibit long-tailed distribution, as observed in commonly used datasets (e.g., ImageNet [41], iNatural [62], and LVIS [63]), remote sensing datasets [77], and 3D point cloud datasets [78]. Furthermore, HOI categories in HICO-DET [66] and V_COCO [67], which are designed for human-object interaction detection in images, follow the long-tailed distribution. Videos can be regarded as a combination of sequential and grid data. In INSVIDEO dataset[64] for micro-video, hashtags exhibit long-tailed distribution. While the NYU-Depth dataset [65], which is composed of video sequences as recorded by depth cameras, exhibits a long-tailed per-pixel depth. 4 Table 2: Datasets available in HEROLT benchmark. The values in this table correspond to input data. Data Statistics Long-Tailedness Dataset Data # of Categories Size # of Edges IF Gini Pareto Glass [76] Tabular 6 214 8 0.380 0.500 Abalone [79] Tabular 28 4177 689 0.172 0.333 Housing [80] Tabular 1,460 EURLEX-4K [68] Sequential 3,956 15,499 1,024 0.342 3.968 AMAZONCat-13K [68] Sequential 13,330 1,186,239 355,211 0.327 20.000 Wiki10-31K [68] Sequential 30,938 14,146 11,411 0.312 4.115 ImageNet-LT [41] Grid 1,000 115,846 256 0.517 1.339 Places-LT [41] Grid 365 62,500 996 0.610 2.387 iNatural 2018 [62] Grid 8,142 437,513 500 0.647 1.658 CIFAR 10-LT (100) [72] Grid 10 12,406 100 0.617 1.751 CIFAR 10-LT (50) [72] Grid 10 13,996 50 0.593 1.751 CIFAR 10-LT (10) [72] Grid 10 20,431 10 0.520 0.833 CIFAR 100-LT (100) [72] Grid 100 10,847 100 0.498 1.972 CIFAR 100-LT (50) [72] Grid 100 12,608 50 0.488 1.590 CIFAR 100-LT (10) [72] Grid 100 19,573 10 0.447 0.836 LVIS v0.5 [63] Grid 1,231 693,958 26,148 0.381 6.250 Cora-Full [53] Relational&Tabular 70 19,793 146,635 62 0.321 0.919 Wiki [81] Relational 17 2,405 25,597 45 0.414 1.000 Email [82] Relational&Tabular 42 1,005 25,934 109 0.413 1.263 Amazon-Clothing [55] Relational&Tabular 77 24,919 208,279 10 0.343 0.814 Amazon-Electronics [55] Relational&Tabular 167 42,318 129,430 9 0.329 0.600 • Relational data organizes data points with defined relationships. One specific type of relational data is represented by graphs, which consist of nodes connected by edges. Therefore, in addition to long-tailed distributions in node classes, node degrees, referring to the number of edges connected to a node, may exhibit a long-tailed distribution [31]. It is frequently observed that the majority of nodes have only a few connected edges, while only a small number of nodes have a large number of connected edges, as seen in datasets like Cora [53], CiteSeer [54], and Amazon [55]. Moreover, in knowledge graphs like YAGO [56] and DBpedia [57], the distribution of entities may exhibit long-tailed distribution, with only a few entities densely connected to others. In HEROLT, we collect 17 datasets across 4 data modalities (tabular, sequential, grid, and relational data) as discussed in Appendix B.3. Table 2 shows data statistics (e.g., size, #categories) and long-tailedness of these datasets. Angle 3: Task Heterogeneity with 18 Algorithms on 6 Tasks. While visual recognition has long been recognized as a significant aspect of long-tailed problems, real-world applications involve different tasks with unique learning objectives, presenting unique challenges for long-tailed algorithms. Despite its importance, this crucial angle has not been well explored in existing benchmarks. We aim to benchmark long-tailed algorithms based on various tasks they are designed to solve to fill the gap. • Object recognition [83, 84] assigns each sample to one label. Data imbalance usually occurs in binary classification or multi-class classification, while long-tailed problems focus on multi-class classification with a large number of categories. • Multi-label text classification [49, 50, 51, 85] involves assigning the most relevant subset of class labels from an extremely large label set to each document. However, the extremely large label space often leads to significant data scarcity, particularly for rare labels in the tail. Consequently, many long-tailed algorithms are specifically designed to address the long-tailed problem in this task. Additionally, tasks such as sentence-level few-shot relation classification [86] and information extraction (containing three sub-tasks named relation extraction, entity recognition, and event detection) [52] are also frequently addressed by long-tailed algorithms. • Image classification [26, 40, 42, 43, 44, 45], which involves assigning a label to an entire image, is a widely studied task in long-tailed learning that received extensive research attention. Furthermore, there are some long-tailed algorithms focusing on similar tasks, including out-ofdistribution detection [87, 88], image retrieval [89, 90], image generation [91, 92], visual relationship recognition [93, 94, 95, 96] and video classification [64, 97]. • Instance segmentation [46, 47] is a common and crucial task that gained significant attention in the development of long-tailed algorithms aimed at enhancing the performance of tail classes. It involves the identification and separation of individual objects within an image, including the 5 Table 3: Algorithms available in the HEROLT benchmark. Algorithm Venue Long-tailedness Task SMOTE [19] 02JAIR Data imbalance Object recognition NearMiss [104] 03ICML Data imbalance Object recognition X-Transformer [49] 20KDD Data imbalance, extreme # of categories Multi-label text classification XR-Transformer [50] 21NeurIPS Data imbalance, extreme # of categories Multi-label text classification XR-Linear [51] 22KDD Data imbalance, extreme # of categories Multi-label text classification OLTR [41] 19CVPR Data imbalance, extreme # of categories Image classification BALMS [46] 20NeurIPS Data imbalance Image classification, Instance segmentation TDE [42] 20NeurIPS Data imbalance Image classification Decoupling [26] 20ICLR Data imbalance Image classification BBN [40] 20CVPR Data imbalance Image classification MiSLAS [43] 21CVPR Data imbalance Image classification PaCo [105] 21ICCV Data imbalance, extreme # of categories Image classification GraphSMOTE [13] 21WSDM Data imbalance Node classification ImGAGN [30] 21KDD Data imbalance Node classification TailGNN [31] 21KDD Data imbalance, extreme # of categories Node classification LTE4G [32] 22CIKM Data imbalance, extreme # of categories Node classification SmoteR [102] 13EPIA Data imbalance Regression SMOGN [103] 17PKDD/ECML Data imbalance Regression detection of object boundaries and the assignment of a unique label to each object. It contains several parts: object detection [98, 99] and semantic segmentation [25]. • Node classification [100, 13, 30, 31, 32] involves assigning labels to nodes in a graph based on node features and connections between them. The emergence of long-tailed algorithms for this task is relatively recent, but the development is flourishing. Additionally, there are some long-tailed learning studies addressing entity alignment tasks [33, 34] in knowledge graphs. • Regression [101, 102, 103] involves learning from long-tailed data with continuous (potentially infinite) target values. For example, the goal in [101] is to infer a person’s age from their visual appearance, where age is a continuous target that can be highly imbalanced. Unlike classification, continuous targets lack clear category boundaries, causing difficulty when directly utilizing traditional methods like re-sampling and re-weighting. Additionally, continuous labels inherently possess a meaningful distance between targets. In HEROLT, we have a comprehensive collection of algorithms for object recognition, multi-label text classification, image classification, instance segmentation, node classification, and regression tasks (Table 3). We discuss what technologies the algorithms use and how they solve long-tailed problems in Appendix B.2. 3 Experiment Results and Analyses We conduct extensive experiments to further answer the question: how do long-tailed learning algorithms perform with regard to different tasks on different domains? In this section, we present the performance of 18 state-of-the-art algorithms on 6 typical long-tailed learning tasks and 17 real-world datasets across 4 data modalities. 3.1 Experiment Setting Hyperparameter Settings. For all the 18 algorithms in HEROLT, we use the same hyperparameter settings on the same task for a fair comparison. Refer to the Appendix C.1 for more information. Evaluation Metrics. We evaluate different long-tailed algorithms by several basic metrics, which are divided into accuracy-based metrics including accuracy (Acc) [106], precision [107], recall [107], and balanced accuracy (bAcc) [107]; ranking-based metrics such as mean average precision (MAP) [108]; regression metrics such as mean-average-error (MAE), mean-squared-error (MSE), Pearson correlation and error geometric mean (GM); and running time. The computation formula and description of the metrics are in Table 10 in Appendix C.2. 3.2 Algorithm Performance on Object Recognition Recently, there has been very limited work considering object recognition tasks on pure tabular long-tailed data. We compare the performance of two methods in Table 4. We find: (1) SMOTE 6 Table 4: Comparing the methods on long-tailed tabular datasets. Each point is the mean and standard deviation of 10 runs. "Time" means training time plus inference time. Dataset Method Acc (%) bAcc (%) Precision (%) Recall (%) mAP (%) Time (s) Glass SMOTE 97.0±1.1 98.5±0.5 95.1±1.1 98.5±0.5 99.9±0.0 0.6 NearMiss 76.7±0.0 88.1±0.0 92.1±0.0 88.1±0.0 98.9±0.0 0.4 Abalone SMOTE 20.1±1.1 19.1±1.1 11.5±0.6 19.1±1.1 17.4±0.2 12.4 NearMiss 23.4±0.0 10.3±0.0 7.0±0.0 10.3±0.0 18.8±0.0 2.8 Table 5: Comparison of different methods on long-tailed sequential datasets for multi-label text classification tasks. "Time" refers to inference time. The results of bAcc@k are not reported since no relevant literature discusses how to use this metric in the multi-label setting. Dataset Method Acc (%) Precision (%) Recall (%) MAP (%) Time (s) @1 @3 @5 @1 @3 @5 @1 @3 @5 @1 @3 @5 Eurlex4K XR-Transformer 88.2 75.8 62.8 88.2 75.8 62.8 17.9 45.2 61.1 88.2 82.0 75.6 66.9 X-Transformer 87.0 75.2 62.9 87.0 75.2 62.9 17.7 44.8 61.2 87.0 81.1 75.1 433.9 XR-Linear 82.1 69.6 58.2 82.1 69.6 58.2 16.6 41.4 56.6 82.1 75.9 69.9 0.2 AmazonCat13K XR-Transformer 96.7 83.6 67.9 96.7 83.6 67.9 27.7 63.3 79.0 96.7 90.5 83.1 78.3 X-Transformer 96.7 83.9 68.6 96.7 83.9 68.6 27.6 63.4 79.7 96.7 90.6 83.4 428.6 XR-Linear 93.0 78.9 64.3 93.0 78.9 64.3 26.3 59.7 75.2 93.0 86.1 78.9 0.3 Wiki1031K XR-Transformer 88.0 79.5 69.7 88.0 79.5 69.7 5.3 14.0 20.1 88.0 83.9 79.2 117.3 X-Transformer 88.5 78.5 69.1 88.5 78.5 69.1 5.3 13.8 19.8 88.5 83.6 78.7 433.2 XR-Linear 84.6 73.0 64.3 84.6 73.0 64.3 5.0 12.7 18.4 84.6 78.7 73.7 1.1 (using an upsampling technique) shows superior performance to NearMiss (using a downsampling technique). Specifically, SMOTE is 85.43% highly balanced accuracy than NearMiss on the Abalone dataset. (2) While Acc treats all samples equally, bAcc considers data imbalance by averaging across classes. Although NearMiss achieves higher accuracy on the imbalanced Abalone dataset, it tends to exhibit a bias toward majority classes and does not sufficiently improve the minority classes, resulting in a lower bAcc score. Precision provides an insight into how much we can trust the model when it identifies a sample as Positive. Precision of NearMiss is lower as it may wrongly classify minority samples into other classes. Recall assesses the model’s ability to identify all the Positive samples. NearMiss fails to characterize minority classes, it typically scores lower on Recall. Similarly, MAP is calculated by averaging across all classes and exhibits a similar trend with the other metrics. 3.3 Algorithm Performance on Multi-Label Text Classification In Table 5, we provide a performance comparison of three methods for multi-label text classification tasks on sequential datasets. We find: (1) Among these methods, XR-Transformer and X-Transformer demonstrate comparably superior performance on multiple metrics. On Eurlex-4K, XR-Transformer achieves a 1.37% improvement in ACC@1 compared to the second-best method. On AmazonCat13K, X-Transformer exhibits a 1.03% improvement in Acc@5 than the suboptimal method. (2) Conversely, XR-Linear consistently exhibits the poorest performance across all three datasets, which verifies the effectiveness of recursive learning of transformer encoders than linear methods. However, XR-Linear is more than 100x faster than XR-Transformer, making it suitable for scenarios with large dataset sizes and strict requirements on time-consuming. (3) Notably, all methods exhibit a limitation in effectively recognizing certain classes, as indicated by the observed low recalls across all datasets. 3.4 Algorithm Performance on Image Classification and Instance Segmentation Among the considered algorithms, OLTR and BALMS utilize meta-learning; BBN and MisLAS employ mixup techniques; TDE uses causal inference; BALMS, Decoupling, and MisLAS decouple the learning process; and PaCo utilizes contrastive learning. We have the following observations from the results in Table 6: (1) No single technique (e.g., meta-learning, decoupled training, mixup, or contrastive learning) can consistently perform the best across all tasks and datasets, and there are limited algorithms that consider both classification and segmentation tasks. Therefore, in contrast to taxonomy based on techniques in methods, the three novel angles we propose (data long-tailedness, data complexity, and task heterogeneity) may be more suitable for benchmarking. (2) PaCo and MisLAS consistently perform well in accuracy on three natural long-tailed datasets. In particular, Paco exhibits a remarkable overall accuracy of 58.3% on ImageNet-LT dataset, surpassing the 7 Table 6: Comparison of different methods on natural long-tailed grid datasets. "Time" refers to inference time. Recall equals to bAcc for multi-class classification. Dataset Task Method Acc (%) Precision Recall/bAcc MAP Time Many Medium Few Overall (%) (%) (%) (s) ImageNet-LT Image classification OLTR 37.9 36.1 30.8 36.1 52.4 36.1 20.5 69.8 BALMS 50.1 39.6 25.3 41.6 41.2 41.6 20.6 29.0 TDE 60.5 47.2 30.4 50.1 50.1 51.0 28.7 30.9 Decoupling 64.0 33.8 5.8 41.6 41.6 49.9 21.1 21.8 BBN 59.4 45.4 16.3 46.6 47.5 46.6 24.9 43.0 MiSLAS 60.9 46.8 32.5 50.0 39.0 38.0 21.0 18.9 PaCo 67.8 56.5 37.8 58.3 58.3 58.3 37.5 58.9 Places-LT Image classification OLTR 44.0 40.6 28.5 39.3 39.5 39.3 17.9 30.6 BALMS 41.0 39.9 30.2 38.3 38.1 38.3 17.1 29.0 TDE 30.5 29.3 19.5 27.8 27.8 29.1 9.8 18.6 Decoupling 40.6 39.1 28.6 37.6 37.6 38.2 16.7 28.9 BBN 34.8 32.0 5.8 27.5 30.7 27.5 9.4 15.4 MiSLAS 42.4 41.8 34.7 40.5 41.0 40.5 19.2 16.2 PaCo 34.8 48.1 38.4 41.4 41.2 41.4 20.1 56.1 iNatural 2018 Image classification OLTR 62.5 52.2 42.2 48.8 50.8 48.8 33.3 33.9 BALMS 37.1 31.9 7.9 28.7 32.5 28.7 10.4 52.9 TDE 63.1 62.1 54.8 59.3 59.3 65.6 43.7 24.8 Decoupling 69.0 65.8 63.1 65.1 65.1 71.0 50.6 23.9 BBN 61.8 73.5 67.7 69.7 72.8 69.7 55.9 29.3 MiSLAS 72.3 72.3 70.7 71.6 74.6 71.6 58.0 19.6 PaCo 66.7 68.0 69.4 68.4 71.0 68.4 53.9 53.8 LVIS v0.5 Instance segmentation BALMS 62.9 34.7 16.1 60.0 37.1 46.8 37.1 1436.1 Table 7: Comparing the methods on semi-synthetic long-tailed datasets with three imbalance factors (100, 50, 10). "Time" refers to inference time. Recall equals to bAcc for multi-class classification. Dataset Method Acc (%) Precision (%) Recall/bAcc (%) MAP (%) Time (s) 100 50 10 100 50 10 100 50 10 100 50 10 CIFAR-10-LT OLTR 28.1 29.1 33.7 21.9 20.9 33.7 28.1 29.1 33.7 15.5 15.1 18.7 10.1 BALMS 84.2 86.7 91.5 84.3 86.7 91.5 84.2 86.7 91.5 72.8 76.9 84.8 1.7 TDE 80.9 82.9 88.3 80.9 82.9 88.3 81.4 83 88.2 68.0 70.7 79.2 1.8 Decoupling 53.5 61.2 73.4 53.5 61.2 73.4 55.4 62.3 73.5 34.0 42.0 57.3 1.8 BBN 80.7 83.4 88.7 81.0 84.0 88.8 80.7 83.4 88.7 67.5 71.8 80.0 5.2 MiSLAS 82.5 85.7 90.0 83.4 86.1 90.1 82.5 85.7 90.1 70.5 75.2 82.3 8.2 PaCo 85.9 88.3 91.7 86.0 88.3 91.7 85.9 88.3 91.7 75.5 79.4 85.1 4.6 CIFAR-100-LT OLTR 7.9 9.2 12.5 5.4 6.5 12.0 7.9 9.2 12.5 1.9 2.2 3.2 12.9 BALMS 51.0 56.2 55.2 50.0 56.0 54.8 51.0 56.2 55.2 29.5 34.9 33.6 2.2 TDE 43.5 48.9 58.9 43.5 48.9 58.9 42.8 49.3 59.7 21.0 26.0 36.8 1.7 Decoupling 34.0 36.4 51.5 33.9 36.4 51.5 32.2 35.7 51.4 14.0 15.8 29.2 1.8 BBN 40.9 46.7 59.7 43.3 48.0 59.9 40.9 46.7 59.7 19.9 24.5 38.2 4.3 MiSLAS 47.0 52.3 63.3 46.5 52.5 63.4 47.0 52.3 63.2 25.4 30.5 42.3 7.2 PaCo 51.2 55.4 66.0 51.2 55.9 66.2 51.2 55.4 66.0 34.9 34.2 46.0 2.5 suboptimal method by 16.37%, MisLAS exhibits a remarkable overall accuracy of 71.6% on iNatural 2018 dataset. The superiority of PaCo and MisLAS is further evident in the tail category, where their few-shot accuracy surpasses other methods by 27.15% and 14.90% on Places-LT. (3) The few-shot accuracy is often lower than the many-shot accuracy for all methods. For example, on ImageNet-LT, Decoupling exhibits particularly poor performance in terms of few-shot accuracy, with a decrease of 90.94% compared to many-shot accuracy. But on iNatural 2018, PaCo achieves a few-shot accuracy of 69.4%, surpassing many-shot accuracy of 66.7% and medium-shot accuracy of 68.0%. CIFAR-10-LT and CIFAR-100-LT are semi-synthetic datasets, where the number of samples in each class is determined by a controllable imbalance factor (typically 10, 50, 100). Similarly, we have the observations as shown in Table 7: (1) PaCo and BALMS show the top-two best accuracy performance on the synthetic CIFAR-10-LT and CIFAR-100-LT with varying degrees of IF. In conjunction with experimental results on natural datasets, the overall performances of decoupled learning (BALMS and MiSLAS) and contrastive learning (Paco) are generally superior. In addition, decoupled learning has the potential to be applied to multiple tasks and therefore has the potential to deal with data long-tailedness and task heterogeneity. but the stable performances in the few-shot categories still need to be considered. (2) As we control IF of the synthetic datasets to be larger, the long-tailed 8 Table 8: Comparing the methods on long-tailed relational datasets. Each point is the mean and standard deviation of 10 runs. "Time" means training time plus inference time. Dataset Method Acc (%) bAcc (%) Precision (%) Recall (%) MAP (%) Time (s) Cora-Full GraphSmote 60.5±0.8 51.9±0.6 60.2±0.8 51.9±0.6 54.9±0.4 718.8 ImGAGN 4.2±0.8 1.5±0.1 0.2±0.1 1.5±0.1 2.7±0.1 69.6 Tail-GNN 63.8±0.3 54.6±0.6 63.8±0.4 54.6±0.6 66.8±0.6 906.2 LTE4G 60.8±0.5 54.6±0.5 61.5±0.6 54.6±0.5 51.1±0.9 281.6 Wiki GraphSmote 66.0±0.9 51.1±2.0 66.3±1.1 51.1±2.0 64.4±1.9 52.3 ImGAGN 45.4±6.7 24.5±4.1 44.8±5.0 24.5±4.1 64.1±6.2 14.3 Tail-GNN 63.6±0.7 47.7±1.1 64.0±1.4 47.7±1.1 65.0±2.1 22.5 LTE4G 58.2±18.5 48.9±12.5 60.2±20.1 48.9±12.5 59.3±16.5 53.1 Email GraphSmote 58.3±1.2 34.2±1.4 54.4±2.0 34.2±1.4 44.6±2.4 126.2 ImGAGN 42.7±1.9 23.0±1.4 38.4±2.5 23.0±1.4 35.7±2.2 19.7 Tail-GNN 56.5±1.7 34.5±1.6 55.6±0.4 34.5±1.6 58.0±3.0 8.6 LTE4G 58.7±2.0 34.3±2.1 54.1±2.5 34.3±2.1 47.0±2.1 72.1 Amazon_Clothing GraphSmote 66.3±0.4 63.2±0.4 64.9±0.3 63.2±0.4 57.6±0.9 919.1 ImGAGN 30.3±1.1 12.8±0.7 23.3±1.2 12.8±0.7 32.0±0.7 89.6 Tail-GNN 69.2±0.6 65.9±0.5 67.7±0.5 65.9±0.5 68.7±0.1 768.7 LTE4G 65.6±0.6 64.2±0.6 64.8±0.5 64.2±0.6 54.3±2.7 349.7 Amazon_Eletronics GraphSmote 56.2±0.5 51.7±0.5 55.6±0.4 51.7±0.5 36.2±1.9 3406.2 ImGAGN 17.1±1.1 7.4±0.6 12.3±0.7 7.4±0.6 16.8±0.7 218.5 Tail-GNN OOM OOM OOM OOM OOM OOM LTE4G 55.8±0.4 53.0±0.5 56.1±0.3 53.0±0.5 32.7±2.2 1112.8 Table 9: Comparing the methods on long-tailed tabular regression dataset. "Time" means training time plus inference time. Dataset Method MAE MSE Pearson (%) GM Time Many Med. Few All Many Med. Few All Many Med. Few All Many Med. Few All (s) Housing SmoteR 0.12 0.12 0.20 0.12 0.02 0.02 0.07 0.02 50.2 98.2 96.9 97.3 0.08 0.08 0.14 0.08 56.6 SMOGN 0.40 0.35 0.43 0.37 0.17 0.13 0.24 0.15 53.2 97.3 91.3 95.4 0.38 0.33 0.36 0.35 35.0 phenomenon is more severe (showing higher values of Gini coefficient, IF, and Pareto-LT Ratio), and the performance of nearly all methods exhibits a decline. (3) CIFAR-100-LT dataset appears more affected than CIFAR-10-LT under the different settings of IF, possibly because it has a more severe long-tailed phenomenon with a larger number of categories. 3.5 Algorithm Performance on Node Classification From the results of long-tailed algorithms for node classification on relational datasets in Table 8, we have: (1) No method consistently outperforms all others across all datasets. The performance of different runs on the same dataset may have large variances, e.g., LTE4G on Wiki. (2) GraphSmote exhibits promising performance on Wiki dataset. Wiki presents high IF and Gini coefficient yet a low Pareto-LT Ratio, hinting that the main challenge may stem from data imbalance as discussed in Angle 1. Therefore, employing the simple augmentation method may yield great results. Amazon_Clothing exhibits relatively low IFs and Gini coefficients but high Pareto-LT Ratios, which may indicate the need for increased focus on addressing the challenge caused by the number of categories (but it does not mean it necessarily contains more categories). The insight may elucidate why Tail-GNN and LTE4G, which can characterize the extensive number of categories, exhibit more significant performance. Although these metrics can give some understanding of a dataset, none can comprehensively and accurately depict all of the characteristics, and the analysis may exhibit limitations in specific datasets. (3) Tail-GNN exhibits superior performance on Cora-full and Amazon_clothing. Especially on Cora-full, Tail-GNN achieves a 4.93% higher accuracy than the second-best method. However, the scalability of Tail-GNN shows limitations. It faces out-of-memory problems on Amazon_Eletronics with 42,318 nodes and 129,430 edges. (4) The performance of ImGAGN is relatively weak since it considers only one class as the tail class by default. This limitation becomes apparent in datasets with a large number of classes. Nonetheless, ImGAGN shows a performance improvement by adjusting the number of classes considered as tail. In addition, ImGAGN is less time-consuming and is 5x faster than LTE4G on the largest Amazon_Eletronics dataset. 9 3.6 Algorithm Performance on Regression There has been limited work considering regression tasks on long-tailed data. In Table 9, we present a comparison of the performance of two methods. It is observed that SmoteR outperforms SMOGN in terms of MAE, MSE, Pearson correlation, and GM metrics across many-shot, medium-shot, and few-shot regions, as well as overall performance. However, SmoteR requires a longer time compared to SMOGN. Additionally, SMOGN appears to overfit to the many-shot regions during training. 4 Related Work Recently, several papers that review long-tailed problems have been published. One of the earliest works [16] aims to improve the performance of long-tailed algorithms through a reasonable combination of existing tricks. Yang et al. [18] conduct a survey on long-tailed visual recognition and shows a taxonomy of existing methods. Fu et al. [15] divide methods into three categories, namely, training, fine-tuning, and inference stage. Fang et al. [14] group methods based on balanced data, balanced feature representation, balanced loss, and balanced prediction. Zhang et al. [17] categorize them into class re-balancing, information augmentation, and module improvement. Although these papers summarize studies on long-tailed visual recognition, they pay less attention to the extent of long-tailedness and fail to consider data complexity and task heterogeneity. Next, we highlight the similarities and differences of long-tailed learning with related areas, e.g., imbalanced learning [20] and few-shot learning [109, 110]. Imbalanced Learning focuses on learning from imbalanced data, where minority classes contain significantly fewer training samples than majority classes. In imbalance learning, the class numbers may be small, while the number of minority samples is not necessarily small. But in long-tailed learning, the number of classes is larger and samples in tail classes are often very scarce. Few-shot Learning aims to train well-performing models from limited supervised samples. Long-tailed datasets, like few-shot datasets, have limited labeled samples in tail classes but with an imbalanced distribution. In contrast, few-shot datasets tend to be more balanced. 5 Conclusions and Future Work In this paper, we introduce HEROLT, the most comprehensive heterogeneous long-tailed learning benchmark with 18 state-of-the-art algorithms and 10 evaluation metrics on 17 real-world benchmark datasets across 6 tasks and 4 data modalities. Based on the analyses of three pivotal angles, we gain valuable insights into the characterization of data long-tailedness, the data complexity of various domains, and the heterogeneity of emerging tasks. Our benchmark and evaluations are released at https://github.com/SSSKJ/HeroLT. On top of them, we suggest intellectual challenges and promising research directions in long-tailed learning: (C1) Theoretic Challenge: Current work lacks sufficient theoretical tools for analyzing long-tailed models like their generalization performance. (C2) Algorithmic Challenge: Existing research typically focuses on one task in one domain, while there is a trend to consider multiple forms of input data (e.g., text and images) by multi-modal learning [111, 112, 113], or to solve multiple learning tasks (e.g., segmentation and classification) by multi-task learning [114, 115]. (C3) Application Challenge: In open environments, many datasets exhibit long-tailed distributions. However, long-tailed problems in domains like antibiotic resistance genes [10, 11] receive insufficient attention. Acknowledgments and Disclosure of Funding We thank the anonymous reviewers for their constructive comments. 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[N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] 3. If you ran experiments (e.g. for benchmarks)... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Please see the supplementary materials for the implementation details. The released toolbox is available at https://github.com/SSSKJ/HeroLT (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Please see the supplementary materials. (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We report the average value and standard deviations over 10 runs. (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please see the supplementary materials. 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes] (b) Did you mention the license of the assets? [Yes] All the datasets we use are publicly available. (c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our toolbox. (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We use publicly available datasets. (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] The dataset has no personally identifiable information or offensive content. 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] 19 A Long-Tailed Data Distributions In this section, we will give some real-world examples with long-tailed distributions. Cora-Full dataset consists of 19,793 scientific publications classified into one of seventy categories [53]. As shown in Figure 3(a), this dataset exhibits a prominent long-tailed distribution, wherein the number of instances belonging to the head categories far surpasses that of the tail categories. Similarly, Amazon_Eletronics dataset [55] also exhibits long-tailed distribution (see Figure 3(b)), where each product is considered as a node belonging to a product category in "Electronics." Despite the emergence of machine learning methods aimed at facilitating accurate classification, further solutions are called due to the challenges of long-tailed distribution. 0 20 40 60 Category (a) Cora-Full 0 200 400 600 800 Frequency 0 50 100 150 Category (b) Amazon_Eletronics 0 200 400 600 800 Figure 3: The data distributions on two commonly used datasets exhibit prominent long-tailed distributions. In addition, we present a motivative application of the recommendation system [116] as shown in Figure 4, which naturally exhibits long-tailed data distributions coupled with data complexity [2] (e.g., tabular data and relational data) and task heterogeneity (e.g., user profiling [1] and recommendation [2]). Additionally, heterogeneous long-tailed learning has various real-world applications, such as financial fraud detection [117, 118] and ARG prediction [119]. Figure 4: The data distributions on two commonly used datasets exhibit prominent long-tailed distributions. B More Details on HEROLT B.1 Long-tailedness Metrics To measure the long-tailedness of a dataset, a commonly used metric is the imbalance factor, and Gini coefficient is used as a measurement in [18]. In this section, we analyze the strengths and weaknesses of each metric and introduce Pareto-LT Ratio to evaluate the long-tailedness of a dataset. 20 Imbalance Factor. To measure the skewness of long-tailed distribution, [72] first introduces the imbalance factor as the size of the largest majority class to the size of the smallest minority class: IF = n1/nC (1) where nc, c = 1, 2, . . . , C, represents the size of each category following descending order. The range of imbalance factor is [1, ∞). Intuitively, the larger the value of IF indicates a more imbalanced dataset. Gini Coefficient. Gini coefficient is a measure of income inequality used to quantify the extent to which the distribution of income among a population deviates from perfect equality. [18] propose to use Gini coefficient as a long-tailedness metric since long-tailedness is similar to inequality between each category. Gini = PC i=1 PC j=1 \|ni −nj\| 2nC (2) Gini coefficient ranges from 0 to 1, where a larger value indicates that the dataset is more imbalanced. Pareto-LT Ratio. We propose a new metric named Pareto-LT Ratio to measure the long-tailedness. The design of this metric is inspired by the Pareto distribution, which is defined as: Pareto −LT = C −Q(0.8) Q(0.8) (3) where Q(p) = min{y : Pr(Y ≤y) = p, 1 ≤y ≤T} is the quantile function of order p ∈(0, 1) for Y. The numerator represents the number of categories to which the last 20% instances belongs, and the denominator represents the number of categories to which the other 80% instances belongs in the dataset. Intuitively, the higher the skewness of the data distribution, the larger the ratio will be; the more classes, the larger the long-tailedness ratio. Figure 5: An example to compare the three long-tailedness metrics. Imbalance factor is intuitive and easy to calculate, but it only considers the imbalance between the largest and smallest classes. Gini coefficient indicates the overall degree of category imbalance, unaffected by extreme samples or absolute data size. However, the two metrics pay more attention to data imbalance and may not reflect the number of categories. Pareto-LT Ratio is proposed to characterize two properties of long-tailed datasets: (1) data imbalance and (2) extreme # of categories. For better understanding, we give a specific example of the three long-tailedness metrics (as shown in Figure 5). As the number of categories increases, the difficulty of classifying a long-tailed dataset therefore increases. For example, we down-sampled 7 categories from the original Cora-Full dataset. Although the two datasets are clearly different, the imbalance factor remains the same (62 for both the original and down-sampled datasets) as the number of samples in the most majority and most minority categories does not change. Gini coefficient indicates the overall degree of category imbalance and thus shows a large increase of value in the down-sampled dataset (0.321 for original and 0.441 for down-sampled). However, the data complexity also includes the number of categories, i.e., 70 classes in Fig(a) v.s 7 classes in Fig(b), which is not reflected in imbalance factor and Gini coefficient. As the number of categories of the down-sampled dataset decreases dramatically, Pareto-LT Ratio better 21 characterizes the differences between the original Cora dataset (0.919) and its down-sampled dataset (0.750) by a small decrease of the metric value. B.2 HEROLT Algorithm List In this section, we introduce 18 popular and recent methods for solving long-tailed problems in our benchmark according to whether they can solve the problems of data imbalance and an extreme number of categories. The baseline algorithms are selected due to the following reasons: (1) Data Complexity: The selected methods provide comprehensive coverage of heterogeneous data (i.e, tabular, sequential, grid, and relational data) in long-tailed learning problems. (2) Task Heterogeneity: we explore a range of techniques for long-tailed learning approaches (i.e., data augmentation, metalearning, decoupled training, mixup), designing for various tasks (i.e, object recognition, multi-label text classification, image classification, instance segmentation, node classification, and regression). (3) SOTA Performance: Most of the selected methods are the SOTAs, which are recently published and highly cited. (4) Open Source: All of the selected methods are open-sourced in GitHub. In addition, our toolbox is still being updated, and we will include more algorithms in the future. SMOTE [19] generates synthetic samples by feature interpolation minority samples with their nearest. NearMiss [104] is a downsampling method that selects samples based on the distance of majority class samples to minority class samples. X-Transformer [49] consists of three components: semantic label indexing, deep neural matching, and ensemble ranking. Semantic label indexing decomposes extreme multi-label classification (XMC) problems into a feasible set of subproblems with much smaller output space by label clustering; deep neural matching fine-tunes a Transformer model for each subproblem; ensemble ranking is conditionally trained based on the instance-cluster assignment and embeddings from the Transformer and is used to aggregate scores from subproblems. X-Transformer solves data imbalance and an extreme number of categories. XR-Transformer [50] is a transformer-based XMC framework that fine-tunes pre-trained transformers recursively leveraging multi-resolution objectives and cost-sensitive learning. The embedding generated at the previous task is used to bootstrap the non pre-trained part for the current task. XR-Transformer solves data imbalance and an extreme number of categories. XR-Linear [51] is designed for XMC problem. It consists of recursive linear models that traverse an input from the root of a hierarchical label tree to a few leaf node clusters and return top-k relevant labels within the clusters as predictions. XR-Linear solves data imbalance and an extreme number of categories. OLTR [41] learns from natural long-tailed open-end distributed data. It consists of two main modules: dynamic meta embedding and modulated attention. The former combines a direct image feature and an associated memory feature to transfer knowledge between head and tail classes, and the latter maintains discrimination between them. Therefore, OLTR solves the challenges of data imbalance and an extreme number of categories. BALMS [46] presents Balanced Softmax, an unbiased extension of Softmax, to accommodate the label distribution shift between training and testing. In addition, it applies a Meta Sampler to learn the optimal class re-sample rate by meta learning. Therefore, BALMS can address data imbalance. TDE [42] is a framework for considering long-tailed problems using causal inference. It constructs a casual graph with four variables: momentum, object feature, projection of head direction, and model prediction, using casual intervention in training and counterfactual reasoning in inference to preserve "good" feature while cut off "bad" confounding effect. TDE solves data imbalance. Decoupling [26] decouples the learning procedure into representation learning and classification. The authors find that instance-balance sampling gives more generalizable representations and can achieve state-of-art performance after properly adjusting the classifiers. Decoupling can address the challenge of data imbalance. BBN [40] consists of two branches: the conventional learning branch with a uniform sampler for learning universal patterns for recognition and the re-balancing branch with a reversed sampler for modeling the tail data. Then the predicted outputs of these bilateral branches are aggregated in 22 the cumulative learning part using an adaptive trade-off parameter, which first learns the universal features from the original distribution and then gradually pays attention to the tail data. By different data samplers (such as mixup which is popular for solving long-tailed problems) and cumulative learning strategy, BBN can address data imbalance. MiSLAS [43] decouples representation learning and classifier learning. It uses mixup and designs label-aware smoothing to handle different degrees of over-confidence for classes and improve classifier learning. It also uses shift learning on the batch normalization layer to reduce dataset bias in the decoupling framework. MiSLAS can solve data imbalance. PaCo [105] presents parametric contrastive learning. To mitigate the problem of contrastive loss bias on head categories from an optimization perspective, the authors propose a set of parametric class-wise learnable centers, which adaptively change the intensity of pushing samples belonging to the same category close to each other. The follow-up work GPaCo [120] removes the momentum encoder, which achieves better model performance and robustness. PaCo can solve data imbalance and the extreme number of categories. GraphSMOTE [13] is the first work considering node class imbalance on graphs proposed in 2021. It first uses a GNN-based feature extractor to learn node representations and then applies SMOTE to generate synthetic nodes for the minority classes. Then an edge generator pre-trained on the original graph is introduced to model the existence of edges among nodes. The augmented graph is used for classification with a GNN classifier. By generating nodes for minority classes, GraphSMOTE increases the number of labeled samples for these classes, thus addressing the data imbalance. ImGAGN [30] is an adversarial learning-based approach. It uses a generator to generate a set of synthetic minority nodes and topological structures, then uses a discriminator to discriminate between real and fake (i.e., generated) nodes and between minority and majority nodes, which can solve the challenges of data imbalance. However, ImGAGN sets the smallest class as the minority class and the residual classes as the majority class, which fails when the dataset contains the large number of categories. Tail-GNN [31] Due to the specificity of rational data, the long-tailed problem on graphs includes the long-tailed of category and the long-tailed of node degree. Tail-GNN [31] focuses on solving the long-tailed problem of degree by introducing transferable neighborhood translation to capture the relational tie between a node and its neighboring nodes. Then it complements missing information of the tail nodes for neighborhood aggregation. Tail-GNN learns robust node embeddings by narrowing the gap between head and tail nodes in terms of degree and addresses the challenges of data imbalance and an extreme number of categories in degree level. LTE4G [32] splits the nodes into four balanced subsets considering class and degree long-tailed distributions. Then, it trains an expert for each balanced subset and employs knowledge distillation to obtain the head student and tail student for further classification. Lastly, LTE4G devises a class prototype-based inference. Because LTE4G uses knowledge distillation across the head and tail and considers the tail classes together, it can solve data imbalance and an extreme number of categories. SmoteR [102] modifies the well-known SMOTE algorithm to handle regression tasks, where the target variable is continuous. It generates synthetic samples and applies over-sample and under-sample, helping to balance the distribution of the training data. SMOGN [103] generates synthetic samples by combining an under-sampling strategy with two over-sampling strategies to address the challenges of imbalanced regression. It adjusts the training data distribution to handle rare and extreme values of a continuous target variable. B.3 HEROLT Dataset List Long-tailed challenges exist for various real-world data, such as tabular data, sequential data, grid data, and rational data. In this section, we briefly describe the collection of datasets selected for the initial version of our benchmark. In addition, we will show more datasets in past long-tailed studies on our toolbox page, including the statistics information (e.g., size, #categories) and the long-tailedness (e.g., imbalance factor, Gini coefficient, and Pareto-LT Ratio). Glass [76] is a dataset from USA Forensic Science Service. Motivated by criminological investigation, the glass left at the scene of the crime can be classified into 6 types based on its oxide content. 23 Abalone [79] is a dataset for predicting the age of abalone from physical measurements, such as length, diameter, height, and weight. EURLEX-4K [68] consists of legal documents from the European Union, and the number of instances in the training and test sets are 15,499 and 3,865. AMAZONCat-13K [68] contains product descriptions from Amazon, and the number of instances in the training and test sets are 1,186,239 and 306,782. Wiki10-31K [68] is a collection of Wikipedia articles, and the number of instances in the training and test sets are 14,146 and 6,616. ImageNet-LT [41] is a long-tailed version sampled from the original ImageNet-2012, which is a large-scale image dataset constructed based on the WorldNet structure. The train set has 115,846 images from 1000 categories with maximally 1280 images per class and minimally 5 images per class. The test and valid sets have 50,000 and 20,000 samples and are balanced. Places-LT [41] is a long-tailed version of scene classification dataset Places-2. The train set contains 62,500 images from 365 categories with maximally 4980 images per class and minimally 5 images per class. The test and valid sets are balanced with 100 and 20 images per class accordingly. iNatural 2018 [62] is a species classification dataset with a train set with 437,513 images for 8142 classes. The class frequencies follow a natural power law distribution with a maximum number of 4,980 images per class and a minimum number of 5 images per class. The test and valid sets contain 149,394 and 24,426 images, respectively. CIFAR 10-LT and CIFAR 100-LT [72]. The original CIFAR dataset has two versions: CIFAR 10 and CIFAR 100, where the former has 10 classes, 6,000 images in every class, while the latter has 100 classes with 600 samples in each class. CIFAR 10-LT and CIFAR 100-LT are two long-tailed versions of CIFAR dataset (named semi-synthetic long-tailed datasets in this paper), where the number of samples in each class is determined by a controllable imbalance factor. The commonly used imbalance factors are 10, 50, 100. The test set remains unchanged with even distribution. LVIS v0.5 [63] is a large vocabulary instance segmentation dataset with 1231 classes. It contains a 693,958 train set, and a relatively balanced test/ valid set. Housing [80] is a dataset designed to predict the selling price. It contains 79 explanatory variables that describe various aspects of residential homes in Ames. Cora-Full [53] is a citation network dataset. Each node represents a paper with a sparse bag-of-words vector as the node attribute. The edge represents the citation relationships between two corresponding papers, and the node category represents the research topic. Email [82] is a network constructed from email exchanges in a research institution, where each node represents a member, and each edge represents the email communication between institution members. Wiki [81] is a network dataset of Wikipedia pages, with each node representing a page and each edge denoting the hyperlink between pages. Amazon-Clothing [55] is a product network that contains products in "Clothing, Shoes and Jewelry" on Amazon, where each node represents a product and is labeled with low-level product categories. The node attributes are constructed based on the product’s description, and the edges are established based on their substitutable relationship ("also viewed"). Amazon-Electronics [55] is another product network constructed from products in "Electronics". The edges are created with the complementary relationship ("bought together") between products. C Details on Experiment Setting C.1 Hyperparameter Settings Here we provide the details of parameter settings. We implement X-Transformer, XR-Transformer, and XR-Linear by applying the best hyperparameter settings from their original papers. In particular, for XR-Linear, we set the number of clusters (i.e., beam size) predicted by the matcher to 10 and chose teacher-forced negation as the hard negation sampling scheme. We implement all experiments in 24 PyTorch and use ResNet-50 for ImageNet-LT dataset, ResNet-152 for Places-LT dataset, ResNet-50 for iNatural 2018 dataset, and ResNet-32 for CIFAR 10-LT and CIFAR 100-LT as the backbones for all methods. For method-related hyperparameters, we use the default settings for all methods on all datasets following the original papers. For GraphSMOTE, we set the weight of edge reconstruction loss to 1e−6 as in the original paper. For LTE4G, we adopt the best hyperparameter settings reported in the paper. For Tail-GNN, we set the degree threshold to 5 (i.e., nodes having a degree of no more than 5 are regarded as tail nodes), which is the default value in the original paper. C.2 Evaluation Metrics Considering the long-tailed distribution, accuracy, precision, recall, balanced accuracy, mean average precision, mean-average-error, mean-squared-error, Pearson correlation, error geometric mean, and time are used as the evaluation metrics. Table 10: Ten metrics for evaluating long-tailed algorithms, where TP, TN, FP, FN stand for true positive, true negative, false positive, false negative, APi is average precision for class i, T is the total number of classes, yi and ˆyi are the actual and predicted values of the i-th data point, ¯yi and ¯ˆyi are the means of the actual and predicted values, n is the total number of data points. For classification tasks (e.g., object recognition, multi-label text classification, image classification, instance segmentation, and node classification), we give computations of two-class classification, which are slightly different for different tasks in our benchmark. Metric name Task Computation Description Acc Classification T P +F N T P +T N+F P +F N Correct predictions of the algorithm on the dataset. Precision Classification T P T P +F P Fraction of correctly predicted positive instances against total positively predicted instances. Recall Classification T P T P +F N Fraction of correctly predicted positive instances against total positively classified instances. bAcc Classification T P/(T P +F N)+T N/(T N+F P ) 2 Arithmetic mean of the recalls for all classes. MAP Classification 1 T PT i=1 APi Average over APs for all classes. MAE Regression 1 n Pn i=1 \|ei\| Average of absolute prediction errors. MSE Regression 1 n Pn i=1 e2 i Average of squared prediction errors. Pearson Regression Pn i=1(yi−¯y)(ˆyi−¯ˆy) √Pn i=1(yi−¯y)2√Pn i=1(ˆyi−¯ˆy)2 Linear relationship between actual and predicted values. GM Regression Qn i=1 \|ei\| 1 n Geometric mean of absolute prediction errors. Time All Time (training time/inference time) of the algorithm. Accuracy (Acc) [106] provides an overall measure of the model’s correctness in predicting the entire test set. Acc favours the majority class as each instance has the same weight and contributes equally to the value of accuracy. In image classification and instance segmentation tasks, besides the overall accuracy across all classes, we follow previous work to comprehensively assess the performance of each method. We calculate the accuracy of three distinct subsets: many-shot classes (classes with over 100 training samples), medium-shot classes (classes with 20 to 100 training samples), and few-shot classes (classes with under 20 training samples). In the task of multi-label learning, each instance can be assigned to multiple classes, making predictions fully correct, partially correct, or fully incorrect. To capture the quality of these predictions, we utilize ACC@k(k = 1, 3, 5) as evaluation metrics, which measure the top-k accuracy. Precision [107] measures the number of correctly predicted positive instances against all the instances that were predicted as positive. In this paper, we use macro-precision, which is calculated by averaging the precision scores for each predicted class. The Macro approach considers all the classes equally, regardless of their size, ensuring the effect of majority classes is considered as important as that of minority classes. Recall [107] is a metric that measures the proportion of true positive instances out of all the positive instances. In our evaluation, we utilize macro-recall, which is calculated by averaging recall for each actual class. Notably, recall equals accuracy when the test set is balanced. Balanced accuracy (bAcc) [107] is the arithmetic mean of recall values calculated for each individual class. It is insensitive to imbalanced class distribution, as it treats every class with equal weight and importance and ensures minority classes have a more than proportional influence. 25 Mean average precision (MAP) [108] is a ranking-based metric that is the average of Average Precision (AP) over different classes, where AP is the area under the precision-recall curve. Mean absolute error (MAE) [101] measures the average magnitude of errors between predicted and actual values without considering their direction. Mean squared error (MSE) [101] calculates the average of the squared differences between predicted and actual values. It gives a higher weight to larger errors due to the squaring operation, making it particularly useful for identifying models that make fewer large mistakes. Pearson correlation coefficient [101] quantifies the linear relationship between the actual and predicted values. It ranges from -1 to 1, where values closer to 1 or -1 indicate a strong positive or negative correlation, respectively, while a value near 0 indicates no linear relationship. Error geometric mean (GM) [101] represents the geometric mean of the errors for better prediction fairness. It provides an indication of the central tendency of multiplicative differences. 26	2024	2327
4,477	Simplified and Generalized Masked Diffusion for Discrete Data Jiaxin Shi∗, Kehang Han∗, Zhe Wang, Arnaud Doucet, Michalis K. Titsias Google DeepMind Abstract Masked (or absorbing) diffusion is actively explored as an alternative to autoregressive models for generative modeling of discrete data. However, existing work in this area has been hindered by unnecessarily complex model formulations and unclear relationships between different perspectives, leading to suboptimal parameterization, training objectives, and ad hoc adjustments to counteract these issues. In this work, we aim to provide a simple and general framework that unlocks the full potential of masked diffusion models. We show that the continuous-time variational objective of masked diffusion models is a simple weighted integral of cross-entropy losses. Our framework also enables training generalized masked diffusion models with state-dependent masking schedules. When evaluated by perplexity, our models trained on OpenWebText surpass prior diffusion language models at GPT-2 scale and demonstrate superior performance on 4 out of 5 zero-shot language modeling tasks. Furthermore, our models vastly outperform previous discrete diffusion models on pixel-level image modeling, achieving 2.75 (CIFAR-10) and 3.40 (ImageNet 64×64) bits per dimension that are better than autoregressive models of similar sizes. Our code is available at https://github.com/google-deepmind/md4. 1 Introduction Since their inception [1, 2, 3], diffusion models have emerged as the workhorse for generative media, achieving state-of-the-art in tasks such as image synthesis [4, 5, 6], audio [7, 8] and video generation [9, 10, 11, 12, 13]. The majority of existing successes are for continuous state space diffusions. While diffusion models have been extended to discrete state spaces [1, 14, 15] and have been successfully applied to applications ranging from graph generation [16], text-to-sound generation [17] or protein design [18], they remain not as widely used as their continuous counterparts as they are not competitive with autoregressive models in important domains such as text modeling. This has motivated the development of continuous space diffusion models where the discrete data are embedded in the Euclidean space [19, 20, 21, 22, 23] or the simplex [24, 25, 26, 27, 28]. We believe that one of the reasons for the limited success of discrete diffusions is that they have been hindered by fairly complex formulations and training objectives. This paper is a step towards closing this gap. In this work, we focus on “masked” (or “absorbing”) diffusions, a discrete diffusion formulation first presented by Austin et al. [14], and later explored by the literature from various perspectives [29, 30, 31, 32]. We follow here a continuous-time framework which has proven very useful to improve the training and understanding of continuous state space diffusions [see e.g., 3, 33, 34]. We make several technical contributions which simplify the training of these models and improve significantly their performance. Our contributions are as follows: • Using elementary arguments, we establish several properties for the forward process induced by this model and its corresponding time reversal, improving our understanding of this model class. ∗Equal contribution. Correspondence to: jiaxins@google.com. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). • We provide a remarkably simple expression of the Evidence Lower Bound (ELBO) for masked diffusion models, showing that it corresponds to a weighted integral over time of cross-entropy losses. Similarly to continuous space diffusions [33], this objective can be rewritten in terms of signal-to-noise ratio and exhibits invariance properties. • We develop a unifying understanding of previously proposed continuous-time discrete diffusion models [29, 32, 35], revealing the changes they made to our ELBO objective and/or model parameterization. We show that these changes either lead to expensive model evaluations, or large variance in training, or breaking the consistency between forward and reverse processes. • On GPT-2 scale text modeling and pixel-level image modeling tasks, masked diffusions trained using our simple ELBO objective outperform previous proposals, leading to the best likelihood and zero-shot transfer performance among discrete diffusion models. • Finally, based on our simplified masked diffusion formulation, we propose a generalized masked diffusion model that allows state-dependent masking schedules. This generalized masked diffusion model further improves predictive performance measured by test likelihoods. Concurrent work by Ou et al. [36] and Sahoo et al. [37] derives a similar simplified expression of the ELBO. Ou et al. [36]’s derivation relies on an observation similar to the one we made in Proposition 1. 2 Masked Diffusion Consider a sentence where we progressively replace each word with a special mask token, transforming the sentence into a sequence of masks. Our goal is to train a generative model that reverses this process, effectively turning a sentence of masks back into meaningful text. More formally, assume our data consists of tokens from a finite discrete state space with m possible states, represented by integers 0, 1, . . . , m −1 and their corresponding one-hot vectors e0, e1, . . . , em−1. To accommodate the masking process, we augment this space with an additional mask state, denoted by the index m. The masking process transitions each token to the mask state at a random time. This process, known as the forward process, is applied independently to each token (e.g., each word), progressively converting the data into a sequence of mask tokens. By learning to reverse this masking process, we create a generative model capable of producing coherent discrete data. Discrete-time forward process. We start with the case of a single token and later expand to multiple dimensions. We define the forward process as a Markovian sequence of discrete random variables xt indexed by time t, where t runs from 0 to 1. Throughout the work, we abuse the notation such that xt can be either an integer or its corresponding one-hot vector, whenever it is clear from the context. We divide [0, 1] into T intervals, and let s(i) = (i −1)/T, t(i) = i/T. Following Austin et al. [14], the state transition between [s(i), t(i)] is determined by a transition matrix of size (m + 1) × (m + 1): Qi = (1 −βi)I + βi1e⊤ m, where 1 is an all-one vector of size m + 1, em represents a one-hot vector where element at index m is 1. Each entry [Qi]jk denotes the probability of transition from the state j to the state k: [Qi]jk = q(xt(i) = k\|xs(i) = j) = (1 −βi)δjk + βiδkm. This means that, with probability 1 −βi, xt(i) = xs(i), otherwise it jumps to the mask state. Given the above transition matrix, the marginal distribution at time t(i) given x0 is q(xt(i)\|x0) = Cat(xt(i); ¯Q⊤ i x0) = x⊤ 0 ¯Qixt(i). Here, we use Cat(x; p) to denote a Categorical distribution where p is the vector of probabilities of being in each category, and ¯Qi ≜Qi j=1 Qj = αiI + 1 −αi 1e⊤ m for αi = Qi j=1(1 −βj). We expect αT to become very small or zero for a sufficiently large T such that q(x1\|x0) for any x0 will become a delta mass at the mask state. Continuous-time limit. We can define a continuous-time forward process by taking a limit of the above discrete-time process. We first specify a continuous function β(t) such that βi = β(t(i))/T. We then let T →∞in the discrete-time process and compute the limit of ¯Qi (proved in Austin et al. 14, Appendix A.6, see also App. A) as ¯Q(t) ≜lim T →∞ ¯Qi = αtI + (1 −αt)1e⊤ m, where αt ≜exp − Z t 0 β(s)ds , (1) 2 0.00 0.25 0.50 0.75 1.00 t 0.0 0.5 1.0 t 0.00 0.25 0.50 0.75 1.00 t 20 15 10 5 0 t t/(1 t) linear geometric cosine poly2 poly0.5 Figure 1: Masking schedules in the literature: (Left) αt; (Right) weight of the cross-entropy loss w.r.t. t; Equations for these schedules are given in Tab. 4 in Appendix. so that q(xt\|x0) = Cat(xt; ¯Q(t)⊤x0). For two arbitrary times, 0 ≤s < t ≤1, the transition distribution that is compatible with the above marginal (i.e., q(xt\|x0) = P xs q(xt\|xs)q(xs\|x0)) is q(xt\|xs) = Cat(xt; ¯Q(s, t)⊤xs), where ¯Q(s, t) ≜¯Q(s)−1 ¯Q(t) = αt αs I + 1 −αt αs 1e⊤ m. Note that Austin et al. [14] did not derive this explicit form of transition matrix between two arbitrary time s and t, which appeared later in Zhao et al. [38] concurrently with our work. Masking schedules. From the definition of αt, we have that α0 = 1. And similar to the discretetime formulation, we would like α1 be zero or very close to zero. We provide a summary of masking schedules from literature that satisfy these properties in Fig. 1. The linear schedule was proposed in Sohl-Dickstein et al. [1] for binary variables and then re-derived by Austin et al. [14] from mutual information for discrete-time models. The geometric schedule αt is plotted for ¯βmin = 10−5 and ¯βmax = 20. It was first used for continuous diffusions [3] and then for discrete by Lou et al. [32]. The cosine schedule was originally proposed in MaskGIT [39], an iterative unmasking generative model inspired by diffusion. This schedule has the property of slowing down the unmasking process at the beginning of the reverse generation. Aligning with their observation, we find that this results in a lower chance of conflicting tokens being unmasked simultaneously at the start of generation, thereby enhancing the overall generation quality. Time reversal of the forward process given x0. The analytic property of our forward process allows to compute many quantities of interest in closed form. One such quantity frequently used in diffusion models is the time reversal of the forward process given x0: q(xs\|xt, x0) for s ≤t. We derive it in App. C as q(xs\|xt, x0) = Cat(xs; ¯Rx0(t, s)⊤xt), where ¯Rx0(t, s) = I + αs −αt 1 −αt em(x0 −em)⊤. From the transition matrix ¯Rx0(t, s) ∈R(m+1)×(m+1) we can see the reverse process conditioned on x0 has a very simple logic—if xt is a mask, with probability αs−αt 1−αt , it will jump to the state x0 at time s, otherwise it will stay masked. Once xt is unmasked, it remains in the same state until the end. 3 Model and Objective For a discrete-time masked diffusion process, we define our generative model by approximately reversing the forward transitions using a reverse model pθ(xs\|xt). One way to define this model is pθ(xs\|xt) ≜q(xs\|xt, µθ(xt, t)), (2) where µθ(xt, t) ∈∆m+1 is a probability vector parametrized by a neural network fθ with a softmax applied to the output logits (note the m-th output is forced to 0 since the clean data cannot be masks): µθ(xt, t) = softmax(fθ(xt, t)) xt = m, xt xt ̸= m. (3) This is known as mean-parameterization since it leverages a prediction model for the mean of x0. A matrix-form depiction of pθ(xs\|xt) is shown in Fig. 7 (right). In fact, we can select a time-invariant parametrization µθ(xt, t) = µθ(xt) as [36] showed that p(x0\|xt) given xt = x is identical for any t. 3 Besides pθ(xs\|xt), we also need to specify p(x0\|xt(1)) and the prior distribution p(xt(T )) = p(x1). Following the practice in continuous diffusion models [33], we choose p(x0\|xt(1)) ∝q(xt(1)\|x0). And since q(x1\|x0) ≈δx1,m for any x0 as α1 ≈0, we set p(x1) ≈δx1,m, see App. E. We then write out the discrete-time diffusion model objective [1, 2], which is a lower bound of the log marginal likelihood of data x0 under the model p (known as the Evidence Lower Bound, or ELBO): log p(x0) ≥Eq(xt(1)\|x0)[log p(x0\|xt(1))] −KL(q(x1\|x0)∥p(x1)) −LT , where LT = PT i=2 Eq(xt(i)\|x0)[KL(q(xs(i)\|xt(i), x0)∥pθ(xs(i)\|xt(i)))]. For the above choices of the prior distribution, the term KL(q(x1\|x0)∥p(x1)) becomes zero. Under the reverse model (2), the KL divergence terms in LT becomes (proof in App. D) KL(q(xs\|xt, x0)∥pθ(xs\|xt)) = −αs −αt 1 −αt δxt,m · x⊤ 0 log µθ(xt, t), which is a simple cross-entropy loss between the predicted logits and the clean data. In App. D, we show that LT is a Riemann sum and is lower bounded by the corresponding continuous integral: L∞≜lim T →∞LT = Z 1 t(1) α′ t 1 −αt Eq(xt\|x0) δxt,m · x⊤ 0 log µθ(xt, t) dt, (4) where α′ t denotes the derivative of αt with respect to t. Therefore, we can obtain an ELBO that is tighter than that of any finite T by pushing T →∞. This ELBO can be further simplified by letting t(1) →0. As a result, Eq(xt(1)\|x0)[log p(x0\|xt(1))] goes to 0 and the ELBO becomes −L∞. For continuous state-space diffusions, the ELBO depends on the signal-to-noise ratio (SNR) at its endpoints but is otherwise invariant to the noise schedule [33]. We establish here a similar result for discrete diffusions. Consider choosing αt = σ(λt), where σ represents the sigmoid function σ(x) = 1 1+e−x . In this context, the log-SNR is defined by λt = log αt 1−αt = log-SNR(t). By making a change of variables in (4) to make everything a function of the log-SNR, we obtain L∞= Z λ1 λt(1) σ(λ)E˜q(xλ\|x0) δxλ,m · x⊤ 0 log ˜µθ(xλ, λ) dλ. where ˜µθ(x, λ) := µθ(x, t) and ˜q(xλ\|x0) := q(xt\|x0) for t = log-SNR−1(λ). This shows that the only effect αt has on the loss is through the values of the SNR at the endpoints. Still, because we draw uniform samples of t to estimate the integral, the choice of masking schedule affects the variance. Multidimensional data. In the previous sections, xt was assumed to be a single discrete token. To extend the method to multidimensional data, let xt be now a sequence (x(1) t , x(2) t , . . . , x(N) t ), where each element x(n) t represents a discrete token. We select a forward process which factorizes across all N tokens: q(xt\|xs) = QN n=1 q(x(n) t \|x(n) s ). As a result, the forward marginals q(xt\|x0) and reversal q(xs\|xt, x0) also factorize. In this case, we define the reverse model as pθ(xs\|xt) ≜ QN n=1 q(x(n) s \|x(n) t , µ(n) θ (xt, t)), where µθ(xt, t) is a neural network that takes the full N tokens as input and outputs N probability vectors.2 The n-th output µ(n) θ (xt, t) is a prediction model for E[x(n) 0 \|xt], the mean value of the n-th token. Repeating above derivations gives L(N) ∞ ≜ Z 1 0 α′ t 1 −αt Eq(xt\|x0) hP n:x(n) t =m(x(n) 0 )⊤log µ(n) θ (xt, t) i dt. (5) We term our simple masked diffusion model trained with loss (5) MD4 (Masked Discrete Diffusion for Discrete Data). A single step of MD4 training algorithm is described in Alg. 1 in Appendix. 4 Sampling We use ancestral sampling from our discrete-time reverse process for generation. We have found this yields slightly higher sample quality compared to other methods such as Euler discretization [29, 32]. For conditional generation tasks such as infilling, we find that the simple approach works best — we keep the conditioning tokens unmasked throughout the generation process. A complete description of the sampling algorithm can be found in Alg. 2 in Appendix. 2We intentionally choose the reverse model to factorize across dimensions because the true reverse transition q(xs\|xt) factorizes in the continuous-time limit (as s approaches t). 4 128 256 512 Total Steps T 0 20 40 60 80 100 120 FID Linear t Linear t, cosine grid Cosine t Cosine t (class cond) 0 50 100 150 200 250 Steps 0 20 40 60 80 Tokens Revealed per Step Linear t Cosine t Figure 2: Left: FID evaluation for 50k samples randomly generated from MD4 on pixel-level modeling of ImageNet 64×64 (numbers in Tab. 6). Right: Number of tokens revealed per generation step (T = 256). Each image consists of 64 × 64 × 3 = 12288 tokens. Impact of schedules and discretization. For comparing different sampling configurations, we primarily use the FID score [40] on image datasets as our evaluation metric. We favor it over text generative perplexity3 used in prior work [32], as the latter can be misleadingly reduced by lowering sample diversity [41]. We initially trained our model using the linear schedule, which achieves the best final ELBO overall; however, we found that sampling did not perform well with a standard uniform discretization grid t(i) = i T . We hypothesize that time discretization can lead to conflicts by generating multiple tokens in a single step. We then switched to the cosine schedule (Tab. 4) that slows down unmasking at the beginning of reverse process. This drastically improves the FID on ImageNet 64×64 from 70 to 17 for T = 256 steps (Fig. 2, left). Building on this observation, we suggest using a “cosine” discretization grid for sampling in models trained with a linear schedule: t(i) = cos π 2 1 −i T . (6) This induces the same discretization in αt as the cosine schedule with a uniform grid, leading to comparable sample quality, as shown in Fig. 2 (left). In Fig. 2 (right), we plot the number of tokens unmasked per step for linear and cosine schedules with a uniform grid. We believe the cosine schedule performs better because it leverages information redundancy: with more tokens revealed, the remaining tokens become more predictable, reducing conflicts when unmasking them in a single step. Although these findings were originally developed on images, we find them translate well to text (see Fig. 10). we expect other techniques such as top-p sampling [41], classifier-free guidance [42, 43], and predictor-correctors [29, 44] to further improve sample quality of our models. While we reserve these for future work, we note that the JAX [45] implementation of categorical sampling implicitly truncates small probabilities, creating a similar effect to top-p sampling. See App. G for details. 5 Relation to Existing Work We discuss how to unify several existing masked diffusion models using our framework. Continuous-Time Markov Chains (CTMC). To show the connection with the CTMC view presented in Austin et al. [14], Campbell et al. [29], we can write out the forward and reverse masked diffusion using CTMC machinery. To see this, for a short time ∆t, given x0, the Taylor expansions of our forward and reverse transition matrices at t are ¯Q(t, t + ∆t) = I + Q(t)∆t + o(∆t) for Q(t) ≜β(t)(1e⊤ m −I), (7) ¯Rx0(t, t −∆t) = I + Rx0(t)∆t + o(∆t) for Rx0(t) ≜− α′ t 1 −αt em(x0 −em)⊤, (8) where Q(t) and Rx0(t) are known as the transition rate matrices. Austin et al. [14] derived the same Q(t) in App. A.6 of their paper. However, they did not explore the reverse process or a 3Perplexity of generated samples scored by a large language model such as GPT-2. 5 continuous-time objective. Campbell et al. [29] derived an alternative ELBO expression using rate matrices, which Kitouni et al. [46] further simplified for absorbing diffusion. In App. H.1, we show how to recover their expression by separating out a constant from our ELBO expression (4) and applying a discrete “integration-by-part”. A key limitation of their expression is that it needs N evaluations of the prediction model µθ(·, t) to compute an inner summation. To circumvent this computational burden, they used a doubly stochastic estimate. However, this leads to significantly higher variance compared to the analytic cross-entropy (4) which only requires one pass of µθ(·, t). Please refer to App. H.2 for more details. Score parameterization. While so far we used a prediction model µθ(xt, t) for the mean of clean data given xt (i.e., mean parameterization), one can choose other ways of parameterizing the reverse model. Lou et al. [32], Benton et al. [35] proposed to parameterize the discrete “score” s(xt, t)j ≜ qt(j) qt(xt) and introduced a score-based loss for discrete diffusions. In App. H.3, we provide an alternative derivation of their loss which is simpler. We show the link between score and mean parameterizations through the following proposition. Proposition 1 (Score Parameterization vs. Mean Parameterization). Let qt be the marginal distribution of the masked diffusion defined in Sec. 2 at time t. The discrete score s(xt, t)j = qt(j) qt(xt) for a mask state xt = m and j ̸= m can be expressed as s(m, t)j = αt 1 −αt E[x0\|xt = m]⊤ej, which satisfies X j̸=m s(m, t)j = αt 1 −αt . (9) Proposition 1 (proved in App. H.3) implies that a reasonable score model for a mask state is sθ(m, t)j = αt 1 −αt µθ(m, t)j. (10) Indeed, substituting (10) into the score-based loss of Lou et al. [32], Benton et al. [35] recovers our objective (4). In Lou et al. [32], the score is parameterized as a neural network without enforcing the constraint in (9). This means the learned reverse model can be incompatible with the forward process. We find that our parameterization, which enforces the constraint, leads to more stable training and better results. Any-order autoregressive models. The continuous-time reverse process of our masked diffusion model can be viewed as an any-order autoregressive model (AO-ARM) [47]. To see this, we reorder the tokens according to the timing of their unmasking events in the reverse process. For all tokens, the cumulative distribution functions (CDFs) of unmasking times {τn}N n=1 are identical and satisfy P(τn ≤t) = P(x(n) t = m) = 1 −αt. As a result, the ordering is uniformly random across all possible arrangements, and the token prediction during each unmasking event represents a prediction step in AO-ARMs. This connection was initially pointed out in Hoogeboom et al. [48, App. C]. The relation between our simplified ELBO (5) and the AO-ARM objective is independently clarified by Ou et al. [36]. Despite this equivalence, our work demonstrates that the masking schedule αt introduces a new degree of freedom in the design of such models. Variations in αt can lead to different distributions of unmasking times, significantly impacting performance in diffusion-style parallel sampling under time discretization, as shown in Fig. 2. Other related work. Due to space constraint, we defer the discussion on other related work, including MaskGIT [39], discrete flow matching [49], SDDM [30], Blackout diffusion [50] and SUNDAE [51], to App. H.4. 6 Generalization to State-dependent Masking Schedules Consider a scenario where some tokens hold more significance than others and we would like to unmask them earlier in the process. To achieve this, we introduce state-dependent masking schedules, where the probability of unmasking a token depends not only on time, but also on the token’s value. We first define the forward process for a single token xt. Let αt be a m + 1 dimensional vector function, i.e., there is a different function αt,i for each possible value i of the token xt. Also, by 6 Mayor Muriel Bowser said after meetings with Commissioner Busby on Thursday that the new plan will be on board in December. Mayor ә Bowser said ә meetings ә Commissioner ә on Thursday that ә new plan will be ә board in ә ә Mayor Muriel Bowser said after meetings ә Commissioner ә on Thursday that ә new plan will be ә board in December ә Mayor ә ә said ә ә ә ә ә ә ә that ә new plan ә ә ә ә ә ә ә 500 steps 700 steps 850 steps 1000 steps Figure 3: Iterative unmasking process for an unconditionally generated sample by MD4. This visualization only includes a subsequence from a generated sequence of 1024 tokens. "?" represents masks. Masked tokens are revealed sequentially: green (steps 500-700), yellow (700-850), and red (850-1000). Additional unconditional generation from MD4 can be found in App. K.5. vector αt αs we denote the element-wise division of the two vectors. We define the forward transition as q(xt\|xs) = Cat(xt; ¯Q(s, t)⊤xs) where ¯Q(s, t) = diag αt αs + I −diag αt αs 1e⊤ m and diag αt αs is a diagonal matrix with the vector αt αs in its diagonal. The probability of moving from current state xs to a future state xt (either the same as xs or mask) is determined by a state-dependent rate αt αs ⊤xs, while the marginal at time s given x0 is q(xs\|x0) = Cat(xs; ¯Q(s)⊤x0) for ¯Q(s) = diag(αs) + (I −diag(αs))1e⊤ m. Further, for any time 0 ≤s < t ≤1 it holds that q(xt\|x0) = P xs q(xt\|xs)q(xs\|x0) so the above is a valid continuous-time Markov chain. Given the forward conditionals and marginals, we can now compute the time reversal conditioned on x0. The full form of q(xs\|xt, x0) is derived in App. I.1. For xt = m, we have q(xs\|xt = m, x0) = q(xs\|xt = m, x0, x0x⊤ 0 ) = 1−αs 1−αt ⊤ x0e⊤ mxs + αs−αt 1−αt ⊤ x0x⊤ 0 xs. (11) This suggests that the reverse model given xt = m can be chosen as pθ(xs\|xt = m) ≜q(xs\|xt = m, µθ(xt, t), diag(µθ(xt, t))) where µθ(xt, t) is a neural network that approximates E[x0\|xt] while diag(µθ(xt, t)) approximates E[x0x⊤ 0 \|xt] = diag(E[x0\|xt]). We show in App. I.1 that the negative continuous-time ELBO for the state-dependent rate case is L∞= Z 1 0 α′ t 1 −αt ⊤ Eq(xt\|x0) δxt,m · (x0 −µθ(xt, t) + x0x⊤ 0 log µθ(xt, t)) dt. (12) Here, α′ t is the elementwise derivative of αt. This generalizes the MD4 loss (4), which is recovered when αt is a scalar schedule times a vector of ones. For N tokens, the model further generalize similarly to Sec. 3 and the loss is given in (32). We call this generalized model GenMD4. To learn the token dependent masking schedule using ELBO optimization, we parametrize the m + 1 dimensional function αt using the polynomial schedule (see Fig. 1) as αt,i = 1 −twi and optimize each parameter wi > 0.4 The value of wi, through the masking probability 1 −αt,i, determines how fast the token with value i jumps to the mask state. Since in the loss (12) the distribution q(xt\|x0) depends on αt and thus the vector w, optimizing w poses a discrete gradient estimation problem [see, e.g., 52]. Naive autodiff leads to biased gradients and pushes w towards zero because the gradients cannot propagate through the (discrete) samples drawn from q(xt\|x0). To fix this, we used the REINFORCE leave-one-out estimator [53, 54] to compute low-variance unbiased gradients for optimizing w. Details are given in App. I.2. 4We only need m learnable parameters wi, for i = 0, . . . , m −1, since x0 can never be the mask token. For the final mask dimension we can choose an arbitrary fixed value such as wm = 0. 7 Table 1: Zero-shot unconditional perplexity on five benchmark datasets from Radford et al. [57]. The numbers for other methods are from Lou et al. [32] except our reimplementation of SEDD Absorb. Our MD4 model achieves the best result on all benchmarks except LAMBADA where it is the second best. ∗The GPT-2 numbers are reported for the GPT-2 checkpoint pretrained on WebText instead of OWT thus is not a direct comparison. Size Method LAMBADA WikiText2 PTB WikiText103 IBW Small GPT-2 (WebText)∗ 45.04 42.43 138.43 41.60 75.20 D3PM ≤93.47 ≤77.28 ≤200.82 ≤75.16 ≤138.92 Plaid ≤57.28 ≤51.80 ≤142.60 ≤50.86 ≤91.12 SEDD Absorb ≤50.92 ≤41.84 ≤114.24 ≤40.62 ≤79.29 SEDD Absorb (reimpl.) ≤49.73 ≤38.94 ≤107.54 ≤39.15 ≤72.96 MD4 (Ours) ≤48.43 ≤34.94 ≤102.26 ≤35.90 ≤68.10 Medium GPT-2 (WebText)∗ 35.66 31.80 123.14 31.39 55.72 SEDD Absorb ≤42.77 ≤31.04 ≤87.12 ≤29.98 ≤61.19 MD4 (Ours) ≤44.12 ≤25.84 ≤66.07 ≤25.84 ≤51.45 7 Experiments 7.1 Text Text is natural discrete data with rich structures. For comparison with prior work, we evaluate likelihood on two datasets: text8 [55], a character-level text modeling benchmark, and OpenWebText [56], an open clone of the unreleased WebText dataset used to train GPT-2 [57]. We also assess our model’s performance on downstream tasks by training on FineWeb-Edu [58], a high-quality dataset of fine educational text commonly used by the open-source community for comparing LLMs. Unless otherwise specified, a linear schedule and a cosine sampling grid are employed. 0 200 400 600 800 1000 Training steps (1 unit = 1000 steps) 102 2 × 101 3 × 101 4 × 101 6 × 101 OpenWebText Eval Perplexity Gaussian Diffusion-S SEDD-S MD4-S GenMD4-S MD4-M Figure 4: Perplexity on OpenWebText (OWT) validation set during training. The final numbers are reported in Tab. 5 in Appendix. OpenWebText. We train MD4 of GPT-2 small (S) and GPT-2 medium (M) sizes on OpenWebText and evaluate zero-shot perplexity on five benchmark datasets used in Radford et al. [57]. We keep our evaluation setup the same as SEDD [32]. To ensure fair comparison, we reimplemented SEDD in our codebase. Our implementation led to slightly better results than those reported in their paper. As seen in Tab. 1, our small model outperforms previous best discrete diffusion models on all five tasks. We are also better than GPT-2 on all tasks except LAMBADA where we are the second best method. When scaling up to medium size, MD4 similarly beats SEDD and GPT-2 on 4 out of 5 tasks. To confirm that the strong zero-shot performance stems from improved training, we plot perplexity on 2% OpenWebText validation set in Fig. 4. Our models converge faster and have better final likelihoods than prior methods. We also observed that SEDD [32] has training instabilities, likely due to score parameterization breaking consistency between forward and reverse processes (Sec. 5). Although GenMD4 achieves lower perplexity than MD4, we observed that the learned ws can overfit to dataset statistics, making it less effective on zero-shot transfer tasks. We also assess our models’ generation quality. Fig. 3 shows a randomly selected, notably coherent sample from MD4-medium and its denoising process. Fig. 10 demonstrates MD4’s text infilling ability and highlights a substantial quality gain when transitioning from uniform to cosine discretization (see Sec. 4). Despite MD4’s strong performance on quantitative metrics like generative perplexity, we have placed these results in Appendix Fig. 8 due to the metric’s inherent unreliability, as noted in Sec. 4. We emphasize the more reliable FID-based assessments found in our image experiments. 8 Table 2: Bits Per Character (BPC) on Text8 test set. All models use standard 12-layer transformers similar to GPT-2 small [57] except Discrete Flow which uses 8 × 3 layers. Method BPC (↓) Continuous Diffusion Plaid [22] (Our impl.) ≤1.48 BFN [26] ≤1.41 Any-order Autoregressive ARDM [48] ≤1.43 MAC [61] ≤1.40 Autoregressive IAF/SCF [62] 1.88 AR Argmax Flow [15] 1.39 Discrete Flow [59] 1.23 Transformer AR [14] 1.23 Discrete Diffusion Mult. Diffusion [15] ≤1.72 D3PM Uniform [14] ≤1.61 D3PM Absorb [14] ≤1.45 SEDD Absorb [32] ≤1.39 MD4 (Ours) ≤1.37 GenMD4 (Ours) ≤1.34 Table 3: Bits Per Dimension (BPD) on CIFAR-10 test set and Downsampled ImageNet 64×64 [63] validation set. All models in the table are trained without data augmentation. Method #Params BPD (↓) CIFAR-10 Autoregressive PixelRNN [63] 3.00 Gated PixelCNN [64] 3.03 PixelCNN++ [65] 53M 2.92 PixelSNAIL [66] 46M 2.85 Image Transformer [67] 2.90 Sparse Transformer [68] 59M 2.80 Discrete Diffusion D3PM Absorb [14] 37M ≤4.40 D3PM Gauss [14] 36M ≤3.44 Campbell et al. [29] 36M ≤3.59 Campbell et al. [29] Absorb 28M ≤3.52 MD4 (Ours) 28M ≤2.75 ImageNet 64×64 Autoregressive PixelRNN [63] 3.63 Gated PixelCNN [64] 3.57 Sparse Transformer [68] 152M 3.44 Routing Transformer [69] 3.43 Perceiver AR [68] 770M 3.40 Discrete Diffusion MD4 (Ours) 198M ≤3.40 Text8. Following prior work [14, 32], we trained masked diffusion models on text8 and evaluate the bits-per-character on the test set (details in App. J.1). As seen in Tab. 2, our models outperform previous discrete and continuous diffusion models, as well as state-of-the-art AO-ARMs which are closely related to discrete diffusion [48]. Our model is only beaten by an autoregressive (AR) transformer and the AR-backbone Discrete Flow [59]. We believe this is because AR models only require learning a fixed generation order thus better utilize model capacity. Text8’s small vocabulary (26 letters and a space) led us to expect limited flexibility from our state-dependent formulation. However, using the generalized objective in (12), GenMD4 achieved significantly better BPC than MD4, demonstrating the potential of state-dependent diffusion for discrete data. 101 102 103 steps / 1000 25 30 35 40 45 hellaswag accuracy (%) MD4-S MD4-M MD4-L AR-S AR-M AR-L Figure 5: Hellaswag accuracy vs. training steps for MD4 and AR models at GPT-2 small, medium, and large scales. FineWeb-Edu. We train MD4 on FineWebEdu and evaluate its zero-shot accuracy on the Hellaswag dataset [60], a popular common sense inference benchmark for LLMs. We directly compared MD4 to its AR counterparts – transformers with identical configurations (except for causal masking) trained on the same data. Results are summarized in Fig. 5. MD4 demonstrates steady performance growth with increasing scale. While outperformed by AR models of the same size, the performance gap does not widen as model size increases. For example, AR-small reaches 30% accuracy in 50k steps, while MD4-small takes 200k steps (4x data efficiency difference). At the medium scale, AR achieves 37% in 270k steps, compared to MD4’s 1 million steps. 7.2 Pixel-level image modeling Unlike continuous diffusion which struggles with discrete data, we show that MD4, a discrete diffusion model, performs well on inherently continuous data, suggesting its potential for unifying 9 Figure 6: Non cherry-picked unconditional samples from MD4 trained on ImageNet 64x64, treating pixels as discrete tokens. More samples can be found in Fig. 9 in Appendix. The model is optimized for likelihood instead of visual quality—see e.g., Kingma et al. [33] for samples from a continuous diffusion model optimized similarly for likelihood. modalities. We follow Austin et al. [14] and train MD4 on order-agnostic image data from CIFAR-10 and downsampled ImageNet 64×64 [63]. Each image is treated as a set of 256-valued discrete tokens, making the model agnostic to pixel proximity. We compare to other discrete diffusion and AR models with reported likelihood results on these datasets, although to our knowledge there are no published result on discrete diffusion for ImageNet 64 × 64 that directly model raw pixel space. Tab. 3 summarizes our results. We establish a new state-of-the-art for discrete diffusion models, outperforming previous work [14, 29] by a significant margin. Our CIFAR-10 result surpasses the best reported AR result. On ImageNet 64 × 64, our results are competitive with Transformer AR models that are 4× larger, as well as a strong continuous diffusion model VDM [33]. Notably, despite lacking knowledge of the ordinal structure of pixel values, MD4 outperforms models trained with this inductive bias, including D3PM Gauss and Campbell et al. [29] where the noising distribution is a discrete Gaussian that assigns larger probabilities to near pixel values. To isolate the differences caused by training objectives, we also implemented the Campbell et al. [29] objective with the absorbing process, showing its high variance hinders learning even with our architecture. We provide a random sample from our ImageNet 64×64 model in Fig. 6. More results can be found in App. K. In Fig. 2, we plot the FID values of samples generated under different choices of schedules and discretization grids. We can see that the model with the linear schedule plus a cosine grid achieves an FID close to the model with cosine schedule, both significantly outperform the linear schedule with a uniform grid. We further trained a class-conditional model on ImageNet 64×64 that boosts the FID to around 7. Although these are not state-of-the-art FIDs on ImageNet 64×64, we emphasize our models are optimized for likelihood instead of sample quality. 8 Conclusion In this work, we revisit masked diffusion models, focusing on a flexible continuous-time formulation. Existing works in this area are not easily accessible to non-specialists and present ELBOs that are difficult to optimize, often resulting in performance that is not competitive with continuous diffusions and AR models. The framework we propose provides a very simple expression of the ELBO as a weighted integral of cross-entropy losses. Additionally, we propose a generalized masked diffusion formulation (GenMD4), where the masking schedule depends on the current state of the process, and derive its corresponding ELBO. On text data, our MD4 models outperform existing discrete and continuous diffusion models. For pixel-level image modeling, we significantly improve discrete diffusion results, outperforming similar-sized AR models and achieving comparable likelihoods to continuous diffusion models such as VDM. GenMD4 provides further improvements in terms of likelihoods over the state-independent case. Although we have improved masked diffusion models, they still suffer from limitations. First, in some tasks such as text8, masked diffusions are not yet competitive with AR models. We conjecture that this is because AR models can better leverage model capacity since they only require learning one order. It would be interesting to develop better architectures for discrete diffusions. Moreover, GenMD4 is promising, but it can easily overfit to the dataset, making it less effective for zero-shot transfer compared to simpler versions. 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The forward transition matrix Qi ∈R(m+1)×(m+1) (m is vocabulary size) at time t(i) is [Qi]jk =        1 j = k = m 1 −βi j = k ̸= m βi k = m, j ̸= m 0 otherwise or more compactly written as Qi = (1 −βi)I + βi1e⊤ m, where 1 denotes an all-one vector of size m + 1, and em is an one-hot vector of size m + 1 with the m-th element (recall that counting starts from 0) being one. We use an one-hot vector xt of length m + 1 to denote the discrete state. The forward conditionals are defined as q(xt(i)\|xs(i)) = Cat(xt(i); Q⊤ i xs(i)) = x⊤ s(i)Qixt(i), (13) where Q⊤ i xs(i) is the probabilities for each of the m + 1 categories that xt(i) can take. The marginal forward distribution at time t(i) given x0 is q(xt(i)\|x0) = Cat(xt(i); ¯Q⊤ i x0) = x⊤ 0 ¯Qixt(i), where ¯Qi = Qi j=1 Qj = Qi j=1(1 −βj)I + 1 −Qi j=1(1 −βj) 1e⊤ m. To see what this leads to in continuous time, we let βi = β(t(i)) T and T →∞: iY j=1 (1 −βj) = exp i X j=1 log(1 −βj) = exp i X j=1 −β(t(j)) T + o(1/T) T →∞ → exp − Z t(i) 0 β(s)ds . We let ¯Q(t) denote the limit of ¯Qi in this case: ¯Q(t) = exp − Z t 0 β(s)ds I + 1 −exp − Z t 0 β(s)ds 1e⊤ m ≜αtI + (1 −αt)1e⊤ m. Here we define αt ≜exp(− R t 0 β(s)ds). And the marginal forward transition is q(xt\|x0) = Cat(xt; ¯Q(t)⊤x0) = x⊤ 0 ¯Q(t)xt = αtx⊤ 0 xt + (1 −αt)e⊤ mxt. (14) 15 B Continuous-time derivation We consider a continuous-time Markov chain with transition rates Q(t) = (Qi −I)/(1/T) = β(t)(1e⊤ m −I). (15) For simplicity, we let Q = 1e⊤ m −I. The marginal forward distribution at time t given x0 is q(xt\|x0) = Cat(xt; ¯Q(t)⊤x0), where ¯Q(t) = exp Z t 0 Q(s)ds = exp Q Z t 0 β(s)ds = exp(¯β(t)Q). Here we define ¯β(t) ≜ R t 0 β(s)ds. The matrix exponential can be computed via eigendecomposition: ¯β(t)Q = UΛU −1, where U = I −eme⊤ m + 1 √n + 11e⊤ m, U −1 = I + √ n + 1eme⊤ m −1e⊤ m, Λ = ¯β(t)(eme⊤ m −I), and thus exp(Λ) = αtI + (1 −αt)eme⊤ m, ¯Q(t) = U exp(Λ)U −1 = αtI + (1 −αt)1e⊤ m. A simpler derivation uses the following property: Q2 = −Q. Therefore, ¯Q(t) = exp(¯β(t)Q) = I + ¯β(t)Q + 1 2 ¯β(t)2Q2 + 1 3 ¯β(t)3Q3 + . . . = I + Q −(1 −¯β(t) + 1 2 ¯β(t)2 −1 3 ¯β(t)3 + . . . )Q = I + Q −exp(−¯β(t))Q = αtI + (1 −αt)1e⊤ m. This marginal forward transition matrix at time t coincides with the result (1) we get by taking the limit of discrete-time derivation. Arbitrary discretization of the continuous-time forward process. For the discrete-time process we have defined the per-step transition in (13). For the continuous-time process, we can derive the transition matrix ¯Q(s, t)ij ≜q(xt = j\|xs = i) between two arbitrary time s and t as the solution to the following differential equation (known as Kolmogorov forward equation) d dt ¯Q(s, t) = ¯Q(s, t)Q(t) where Q(t) = β(t)Q with initial condition ¯Q(s, s) = I. The solution is given by ¯Q(s, t) = exp (¯β(t) −¯β(s))Q = ¯Q(s)−1 ¯Q(t). Routine work (using the Woodbury matrix inversion lemma) shows that ¯Q(t)−1 = α−1 t I + (1 −α−1 t )1e⊤ m. Plugging the result back, we get the forward transition distribution from s to t: q(xt\|xs) = Cat(xt; ¯Q(s, t)⊤xs) = x⊤ s ¯Q(s, t)xt, (16) where ¯Q(s, t) ≜¯Q(s)−1 ¯Q(t) = αt αs I + 1 −αt αs 1e⊤ m. 16 Figure 7: The reverse transition probability and our generative model. Left: q(xs = ·\|xt = ·, x0) in matrix form where first index is xt and second index is xs. Right: pθ(xs = ·\|xt = ·) ≜q(xs = ·\|xt = ·, µθ(xt, t)) also in matrix form. C Time reversal of the forward process given x0 The analytic property of our forward process allows to compute many quantities of interest in closed form. One such quantity frequently used in diffusion models is the time reversal of the forward process given x0: q(xs\|xt, x0). We can compute it using (14) and (16) as q(xs\|xt, x0) = q(xt\|xs)q(xs\|x0) q(xt\|x0) =      αs−αt 1−αt x⊤ s x0 xs ̸= m, xt = m 1−αs 1−αt xs = m, xt = m x⊤ s xt xt ̸= m. (17) Visually, eqn (17) is a R(m+1)×(m+1) matrix (Fig. 7, left) whose first index is xt and the second is xs. The matrix is almost an identity matrix except the last row corresponding to xt is the mask token. The last row means with probability of αs−αt 1−αt the mask token gets unmasked to become x0, and with probability of 1−αs 1−αt it remains masked. Alternatively, we can rewrite the above using reverse transition matrix ¯Rx0(t, s) ∈R(m+1)×(m+1) as q(xs\|xt, x0) = Cat(xs; ¯Rx0(t, s)⊤xt), where ¯Rx0(t, s) = I + αs −αt 1 −αt em(x0 −em)⊤. We are also interested in what would happen in the infinitesimal time limit, i.e., when s = t −∆t and ∆t →0. Note that αt−∆t −αt = −α′ t∆t + o(∆t). Plugging it into the original formula, we get ¯Rx0(t, t −∆t) = I − α′ t 1 −αt em(x0 −em)⊤∆t + o(∆t). Comparing the above with the transition rate matrix Rx0(t) definition ¯Rx0(t, t −∆t) = I + Rx0(t)∆t + o(∆t), we have determined the transition rate matrix for the reverse process conditioned on x0: Rx0(t) = − α′ t 1 −αt em(x0 −em)⊤. (18) 17 D Details of the ELBO Using (17) and (3), we compute the KL divergences between forward and reverse transitions KL(q(xs\|xt, x0)∥pθ(xs\|xt)) = KL(q(xs\|xt, x0)∥q(xs\|xt, µθ(xt, t))) (19) = (Pm xs=0 q(xs\|xt, x0) log q(xs\|xt,x0) q(xs\|xt,µθ(xt,t)) xt = m 0 xt ̸= m = δxt=m X k̸=m αs −αt 1 −αt x⊤ 0 ek log x⊤ 0 ek µθ(xt, t)⊤ek = −δxt=m αs −αt 1 −αt x⊤ 0 log µθ(xt, t). Note that 0 log 0 = 0. Alternatively, this result can be easily obtained from the visual depictions of q(xs\|xt, x0) and pθ(xs\|xt) shown in Fig. 7. In this case, the reconstruction term becomes Eq(xt(1)\|x0)[log p(x0\|xt(1))] = m X k=0 qt(1)\|0(k\|x0) log qt(1)\|0(k\|x0) P j̸=m qt(1)\|0(k\|j) = αt(1) · log αt(1) αt(1) + (1 −αt(1)) log 1 m = −(1 −αt(1)) log m. The prior KL term can be computed as KL(q(x1\|x0)∥p(x1)) = KL(δx1,m∥δx1,m) = 0. As usual, we take the continuous-time limit by letting T →∞: L∞≜lim T →∞LT = lim T →∞ T X i=2 −αs(i) −αt(i) s(i) −t(i) s(i) −t(i) 1 −αt(i) x⊤ 0 Eq(xt(i)\|x0) δxt(i),m log µθ(xt(i), t(i)) = Z 1 t(1) α′ t 1 −αt x⊤ 0 Eq(xt\|x0) [δxt,m log µθ(xt, t)] dt. E Avoiding undefined KL divergence When defining the forward process, we often do not want α1 to be exactly 0, or equivalently, λ1 to be ∞for numerical stability reasons. Instead, we set λ1 to be a finite value, and thereby α1 has a small positive value. This has a problem that the support of q(x1\|x0) is no longer {m} and instead becomes {m, x0}. As a result, the KL divergence between q(x1\|x0) and p(x1) is undefined because q(x1\|x0) is not absolutely continuous with respect to p(x1) = δx1,m. To resolve the issue, we modify the prior distribution p(x1) such that it has support over all m + 1 values. One such choice is letting p(x1) = α1 m X j̸=m δx1,j + (1 −α1)δx1,m. Then, the prior KL divergence term becomes KL(q(x1\|x0)∥p(x1)) = m X x1=0 q(x1\|x0) log q(x1\|x0) p(x1) = m X x1=0 (α1δx1,x0 + (1 −α1)δx1,m) log α1δx1,x0 + (1 −α1)δx1=m p(x1) = α1 log α1 α1/m + (1 −α1) log 1 −α1 1 −α1 = α1 log m. 18 F Details of Training and Sampling with MD4 F.1 Training Algorithm 1 A single step of training with MD4. Input: data minibatch {xi t}B i=1, network µθ(·, t), masking schedule αt for i = 1, . . . , B do (in parallel): ti ←mod(u + i/B, 1), u ∼U[0, 1] for n ∈[N], mask out each token xi,(n) 0 independently with probability 1 −αti to obtain xi ti for n ∈[N], if x(n) ti =m, compute weighted cross entropy loss α′ ti 1−αti (xi,(n) 0 )⊤log µ(n) θ (xi ti, ti) Sum over all weighted cross entropy losses for mask positions and optimize via autodiff F.2 Sampling Algorithm 2 Unconditional and conditional generation (e.g., infilling) with MD4. Input: Context sequence xc of length N, with masks indicating the target areas for generation Init: {t(i)}T i=0 ←discretize([0, 1]), xt(T ) ←xc for i = T, T −1, . . . , 1 do t ←t(i), s ←t(i −1) for n ∈[N], if x(n) t = m, draw x(n) s ∼Cat( αs−αt 1−αt µ(n) θ (xt, t) + 1−αs 1−αt em) else x(n) s ←x(n) t return x0. G JAX Categorical Sampling and Implicit Top-p We noticed that the following equivalent implementation of Alg. 2 leads to significantly worse sample quality in JAX: Algorithm 3 Variant of Alg. 2 that yields lower sample quality when implemented in JAX. Input: Token sequence xc of length N, with masks indicating the target areas for generation Init: {t(i)}T i=0 ←discretize([0, 1]), xt(T ) ←xc for i = T, T −1, . . . , 1 do t ←t(i), s ←t(i −1) for n ∈[N] do (in parallel) draw u ∼U[0, 1] if x(n) t = m and u < αs−αt 1−αt then draw x(n) s ∼Cat(µ(n) θ (xt, t)) else x(n) s ←x(n) t return x0. However, mathetically it is equivalent to Alg. 2 and should produce identical results. Our investigation revealed that the issue arises because Alg. 2 scales the output probabilities of µθ by a small factor αs−αt 1−αt as s is close to t, causing some categories to have very low probabilities. JAX, however, implements categorical sampling using Gumbel argmax, which is less numerically stable than methods like binary search. As a result, categories with low probabilities are rarely sampled, even when their cumulative probability is significant. In our experiment, we found that categories with probabilities below 1e-8 are rarely sampled out of a total of 50K categories. Thus, Alg. 2 implicitly performs top-p sampling (with a dynamic p) under JAX’s categorical sampling, yielding better sample quality than Alg. 3 where µθ is not scaled by a small factor and has fewer categories truncated. 19 H Unifying Existing Masked Diffusion Models H.1 The CTMC point of view We first prove a lemma that connects the forward and reverse transition rate matrices. This follows from the results in [29] but we give a proof for completeness. Lemma 2. The forward transition rate matrix Q(t) and the reverse transition rate matrix (given x0) Rx0(t) satisfy: Rx0(t)kj = Q(t)jk qt\|0(j\|x0) qt\|0(k\|x0) for j ̸= k. (20) Proof Consider the reverse transition from time t + τ to t. For j ̸= k, Bayes’ rule yields q(xt = j\|xt+τ = k, x0) = q(xt = j\|x0)q(xt+τ = k\|xt = j) q(xt+τ = k\|x0) = q(xt = j\|x0)(δjk + Q(t)jkτ + o(τ)) q(xt+τ = k\|x0) τ→0 = δkj + q(xt = j\|x0) q(xt = k\|x0)Q(t)jkτ + o(τ). Then, it follows from the definition of the transition rate matrix that Rx0(t)kj = Q(t)jk qt\|0(j\|x0) qt\|0(k\|x0). Proposition 3. We use the shorthand Rθ(t)kj to denote the approximate reverse transition rate from the state k to j obtained by substituting our prediction model µθ(k) for x0 in Rx0(t)kj. Then, the continuous-time objective (4) can be equivalently expressed as L∞= − Z 1 t(1) Eqt\|0(k\|x0) h Rθ(t)kk + X j̸=k Q(t)kj log Rθ(t)jk i dt + C, (21) where C is a constant independent of θ. Proof To rewrite our objective L∞with the transition rate matrices, we first go back to (19). There, instead of plugging in the explicit form of ¯Rx0(t, s), we substitute it with (8) which leverages the transition rate Rx0(t). To simplify the notation, we assume xt = k and use the shorthand Rθ(t)kj ≜Rµθ(k)(t)kj. We then have KL(q(xt−∆t\|xt, x0)∥pθ(xt−∆t\|xt)) = KL(Cat(xs; ¯Rx0(t, t −∆t)⊤ek)∥Cat(xs; ¯Rµθ(k)(t, t −∆t)⊤ek)) = m X j=0 e⊤ k (I + Rx0(t)∆t + o(∆t))ej log e⊤ k (I + Rx0(t)∆t + o(∆t))ej e⊤ k (I + Rθ(t)∆t + o(∆t))ej = (1 + Rx0(t)kk∆t) log 1 + Rx0(t)kk∆t + o(∆t) 1 + Rθ(t)kk∆t + o(∆t) + X j̸=k (Rx0(t)kj∆t) log Rx0(t)kj∆t + o(∆t) Rθ(t)kj∆t + o(∆t) + o(∆t) = (Rx0(t)kk −Rθ(t)kk)∆t + X j̸=k (Rx0(t)kj∆t) log Rx0(t)kj∆t + o(∆t) Rθ(t)kj∆t + o(∆t) + o(∆t). 20 For the last identity, we have used the fact that log(1 + x) = x + o(x). To obtain L∞, we take the limit of LT as T →∞, which is equivalent to letting ∆t = 1/T →0. We obtain L∞= lim T →∞ T X i=2 Eq(xt(i)\|x0)[KL(q(xs(i)\|xt(i), x0)∥pθ(xs(i)\|xt(i)))] = lim T →∞ T X i=2 Eq(xt(i)\|x0) h Rx0(t(i))kk −Rθ(t(i))kk + X j̸=k Rx0(t(i))kj log Rx0(t(i))kj∆t + o(∆t) Rθ(t(i))kj∆t + o(∆t) ∆t + o(∆t) i = Z 1 t(1) Eqt\|0(k\|x0) h Rx0(t)kk −Rθ(t)kk + X j̸=k Rx0(t)kj log Rx0(t)kj Rθ(t)kj i dt. Note that Rx0(t) is a constant matrix independent of θ. Absorbing all constant terms into C, we have L∞= − Z 1 t(1) Eqt\|0(k\|x0) h Rθ(t)kk + X j̸=k Rx0(t)kj log Rθ(t)kj i dt + C. Next, we subtitute Rx0(t) with the forward transition rate using Lemma 2: L∞= − Z 1 t(1) Eqt\|0(k\|x0) h Rθ(t)kk + X j̸=k Q(t)jk qt\|0(j\|x0) qt\|0(k\|x0) log Rθ(t)kj i dt + C = − Z 1 t(1) h m X k=0 qt\|0(k\|x0)Rθ(t)kk + m X k=0 X j̸=k Q(t)jkqt\|0(j\|x0) log Rθ(t)kj i dt + C = − Z 1 t(1) h m X k=0 qt\|0(k\|x0)Rθ(t)kk + m X k=0 X j̸=k Q(t)kjqt\|0(k\|x0) log Rθ(t)jk i dt + C, where the last identity used the discrete analog to integration-by-part (or summation-by-part): P k=0 P j̸=k f(j, k) = P k=0 P j̸=k f(k, j). Rearranging the terms then gives (21). H.2 Differences from Campbell et al. [29] Campbell et al. [29] used the first term of (21) as the training loss. A key limitation of this loss function is from the inner summation term X j̸=k Q(t)kj log Rθ(t)jk. For single dimension case, the sum is analytically computable due to the sparse structure of Rθ(t)—if xt = k is mask, the second term disappears; otherwise the only possible neighbor j is a mask. However, for multidimensional data, j will represent all N −1 neighbors in the forward process, i.e., the states we get from mask out a single unmasked dimension of xt = k. Recall that Rθ(t)jk is computed as substituting our neural network prediction model µθ(j) for x0 in Rx0(t)jk. Therefore, the summation together with Rθ(t)kk requires N evaluations of µθ(·). This is prohibitive since the neural network model is usually expensive. To resolve this issue, Campbell et al. [29] proposed to rewrite the sum as Ej∼˜q(·\|k) [Zk log Rθ(t)jk] where ˜q(j\|k) = Q(t)kj Zk , Zk ≜ X j′̸=k Q(t)kj′ and estimate it through Monte Carlo. Taking into account the outer expectation under qt\|0(k\|x0), the computation of the loss then becomes a doubly stochastic estimate (using k ∼qt\|0(k\|x0) and j ∼˜q(j\|k)) which suffers from large variance. In contrast, the form of our loss (4) only requires evaluating µθ once for a single stochastic estimation of the expectation w.r.t. q(xt\|x0). 21 H.3 Score parameterization We provide a simpler derivation of the score-based loss [32, 35] below. We start from the form of the ELBO in (21) and rewrite it as L∞= Z 1 t(1) Eqt\|0(k\|x0) h X j̸=k Rµθ(t)kj −Rx0(t)kj + Rx0(t)kj log Rx0(t)kj Rµθ(t)kj i dt. (22) For the last identity we used the zero-row-sum property of transition rate matrix: Rx0(t)kk = − X j̸=k Rx0(t)kj. If we plug (20) into (22) and reparameterize with a score model sθ(xt)j ≜qt\|0(j\|µθ(xt)) q(xt\|µθ(xt)) , (23) we recover the score entropy loss function from Lou et al. [32], Benton et al. [35]: L∞= Z 1 t(1) Eqt\|0(k\|x0) h X j̸=k Q(t)jk sθ(k)j −qt\|0(j\|x0) qt\|0(k\|x0) log sθ(k)j + ψ qt\|0(j\|x0) qt\|0(k\|x0) i dt, (24) where ψ(y) ≜y log y −y. Note that our derivation above is different and simpler than that of Campbell et al. [29] (which Lou et al. [32] is based on) since we leverage the conditional reverse transition rate given x0 instead of the transition rate matrix of the reverse process. We can further simplify the loss with the following relationship between the conditional score and x0: qt\|0(j\|x0) qt\|0(k\|x0) = x⊤ 0 ¯Q(t)ej x⊤ 0 ¯Q(t)ek = αt 1 −αt x⊤ 0 ej for k = m, j ̸= k. (25) Note that only the result under the case k = m is needed. This is because when xt is unmasked, at any time between 0 and t, the state must stay unchanged and remain x0. As a result, KL(q(xt−∆t\|xt, x0)∥pθ(xt−∆t\|xt)) = 0 for xt ̸= m. From (15), we know Q(t)jk = β(t)(δmk −δjk). Combining (25) and (24), we get L∞= Z 1 t(1) β(t) Eqt\|0(k\|x0) δmk X j̸=k sθ(k)j − αt 1 −αt x⊤ 0 log sθ(k) + ψ αt 1 −αt dt. (26) Further, we can show the connection between (26) and (4) by reverting the score parameterization to a mean parameterization using (23), or equivalently sθ(xt)j = αt 1−αt µθ(xt)⊤ej. By doing so, we obtain L∞= Z 1 t(1) β(t) Eqt\|0(k\|x0) δmk X j̸=k sθ(k)j − αt 1 −αt x⊤ 0 log µθ(k) + αt 1 −αt dt. Observing that X j̸=m sθ(m)j = αt 1 −αt , (27) we conclude that this recovers the objective in (4). Interestingly, in Lou et al. [32] the score parameterization is not constrained to satisfy (27). That means the learned reverse model might be incompatible with the forward process. Below, we prove Proposition 1 using the result from Eq. (25). Proof of Proposition 1 qt(j) qt(m) = P x0 qt\|0(j\|x0)q(x0) qt(m) = P x0 qt\|0(j\|x0)q0\|t(x0\|m) qt\|0(m\|x0) = Ex0\|xt=m qt\|0(j\|x0) qt\|0(m\|x0) = Ex0\|xt=m αt 1 −αt x⊤ 0 ej = αt 1 −αt E[x0\|xt = m]⊤ej. 22 H.4 Other related work. MaskGIT [39]. MaskGIT is a diffusion-inspired iterative denoising model for discrete image tokens obtained through models such as VQ-VAE [70]. Training of MaskGIT follows the steps: (a) Sample t ∈[0, 1]. (b) Given a mask scheduling function γ(t), sample γ(t)N tokens to place masks. (c) For data x0 of size (m + 1) × N and the partially masked state xt, minimize the negative log-likelihood LMaskGIT = − Z 1 0 Ext h P n:x(n) t =m(x(n) 0 )⊤log µ(n) θ (xt, t) i dt. (28) Our forward process satisfies qt\|0(m\|x0) = 1 −αt. Therefore, when we set the mask scheduling function as γ(t) = 1 −αt we obtain a loss similar to (5) without the α′ t 1−αt weighting. Note that there remains a difference in the sampling distribution of xt: in the masked diffusion forward process, tokens are sampled independently and do not necessarily yield exactly (1 −αt)N mask tokens at time t, though the expected number is (1 −αt)N. One might be interested in whether the uniform weighting can be recovered by selecting an appropriate schedule αt. However, solving αt such that α′ t = αt −1 yields αt = cet + 1 and there is no c that satisfies both α0 = 1 and α1 = 0. This shows that training with the MaskGIT loss (28) may not be faithfully optimizing the model likelihood. Discrete flow matching [49]. For the linear schedule αt = 1 −t, our reverse transition rate matrix (8) conditioned on x0 is: Rx0(t) = − α′ t 1 −αt em(x0 −em)⊤= 1 t em(x0 −em)⊤. This is the same as the conditional reverse transition rate used in Campbell et al. [49, Eq. (22)]—note that their time t is reversed, and the rate matrix was therefore in the form Rx0(t) = 1 1−tem(x0−em)⊤. SDDM [30]. Sun et al. [30] proposed a pseudo-likelihood-like objective for training discrete diffusion models that can also be applied to masked diffusion. However, their objective encounters the same challenge as Campbell et al. [29] — requiring N passes of the mask prediction model. To mitigate this, they introduced a new transformer architecture, which unfortunately leads to some performance degradation. Blackout diffusion [50]. Santos et al. [50] proposed a “blackout” diffusion process that gradually diffuses images to a black state. While this approach is similar to masked diffusion on binary data, key differences emerge when dealing with larger state spaces. In their method, image pixel intensities gradually fade out, whereas ours directly transition to a mask state. Our method offers more flexibility, being applicable to general discrete state spaces without requiring predefined structural relationships. It also demonstrates competitive performance in image generation, achieving this without knowing pixel value proximity. SUNDAE [51, 71]. Unlike masked diffusion, SUNDAE uniformly corrupts data with random tokens in the vocab (known as uniform discrete diffusion [14]). Additionally, it uses a second loss term from cross entropy between clean data and 1-step unrolled model prediction. Similar ideas have been proposed in [72]. I Details for state-dependent rates I.1 Derivations and time continuous limit All derivations in this section assume that xt is a single token, while for N tokens the masked diffusion with state-dependent rates factorises across the N tokens. Learning from data of N tokens using variational inference is discussed in App. I.2. 23 Given the forward transition q(xt\|xs) and marginal q(xs\|x0) derived in main text (Sec. 6) The reversal given x0 is q(xs\|xt, x0) = Cat(xs; ¯Rx0(t, s)⊤xt) for ¯Rx0(t, s)jk =      αs−αt 1−αt ⊤x0x⊤ 0 ek j = m, k ̸= m 1−αs 1−αt ⊤x0 j = m, k = m δjk j ̸= m. or alternatively can be written as q(xs\|xt, x0) = q(xt\|xs)q(xs\|x0) q(xt\|x0) = h α⊤ t xs α⊤ s xs x⊤ s xt + (1 −α⊤ t xs α⊤ s xs )e⊤ mxt i α⊤ s x0x⊤ 0 xs + (1 −α⊤ s x0)e⊤ mxs α⊤ t x0x⊤ 0 xt + (1 −α⊤ t x0)e⊤ mxt . (29) To simplify this expression we consider the two cases: either xt = m (i.e. xt is mask) or xt ̸= m where in the second case xt = x0. For the case xt = m, the denominator in (29) simplifies as q(xt = m\|x0) = 1 −α⊤ t x0 due to x⊤ 0 xt = 0 since x0 ̸= m, i.e. the observed token x0 cannot be a mask. Then given that xt = m the probability that xs = xt = m is 1 −α⊤ s x0 1 −α⊤ t x0 = (1 −αs)⊤x0 (1 −αt)⊤x0 = 1 −αs 1 −αt ⊤ x0 (30) while the remaining probability for xs = x0 ̸= m is (αs −αt)⊤x0 1 −α⊤ t x0 = (αs −αt)⊤x0 (1 −αt)⊤x0 = αs −αt 1 −αt ⊤ x0. (31) Then, combining (30) and (31) to write q(xs\|xt = m, x0) in an unified way yields the expression (11) in the main Sec. 6. In the second case, when xt = x0 ̸= m, q(xs\|xt ̸= m, x0) from (29) simplifies dramatically and it becomes q(xs\|xt ̸= m, x0) = x⊤ t xs which is a point mass that sets xs = xt. Derivation of the continuous-time limit of the loss in (12). To simplify the notation, we let ξs,t ≜αs−αt 1−αt . We first compute the KL divergence terms in the discrete-time ELBO as KL(q(xs\|xt, x0)∥pθ(xs\|xt)) = (Pm xs=0 q(xs\|xt, x0) log q(xs\|xt,x0) pθ(xs\|xt) xt = m 0 xt ̸= m = δxt,m h X k̸=m ξ⊤ s,tx0x⊤ 0 ek log ξ⊤ s,tx0x⊤ 0 ek ξ⊤ s,tdiag(µθ(xt, t))ek + (1 −ξs,t)⊤x0 log (1 −ξs,t)⊤x0 (1 −ξs,t)⊤µθ(xt, t) i = δxt,m h −ξ⊤ s,tx0x⊤ 0 log µθ(xt, t) + (1 −ξs,t)⊤x0 log (1 −ξs,t)⊤x0 (1 −ξs,t)⊤µθ(xt, t) i . Let ∆t ≜1 T = t(i) −s(i) for all i. Plugging αt−∆t = αt −α′ t∆t + o(∆t) into the above formula and letting γt = α′ t 1−αt , we get KL(q(xs\|xt, x0)∥pθ(xs\|xt)) = δxt,m γ⊤ t x0x⊤ 0 log µθ(xt, t)∆t + 1 + γ⊤ t x0∆t · log 1 + γ⊤ t x0∆t + o(∆t) 1 + γ⊤ t µθ(xt, t)∆t + o(∆t) + o(∆t) = δxt,m γ⊤ t x0x⊤ 0 log µθ(xt, t)∆t + 1 + γ⊤ t x0∆t γ⊤ t x0∆t −γ⊤ t µθ(xt, t)∆t + o(∆t) + o(∆t) = δxt,m γ⊤ t x0x⊤ 0 log µθ(xt, t)∆t + γ⊤ t x0∆t −γ⊤ t µθ(xt, t)∆t + o(∆t) = δxt,m · γ⊤ t (x0x⊤ 0 log µθ(xt, t) + x0 −µθ(xt, t))∆t + o(∆t). 24 Therefore, lim T →∞ T X i=2 Eq(xt(i)\|x0)[KL(q(xs(i)\|xt(i), x0)∥pθ(xs(i)\|xt(i)))] = lim T →∞ T X i=2 Eq(xt(i)\|x0)[δxt(i),m · γ⊤ t (x0x⊤ 0 log µθ(xt(i), t(i)) + x0 −µθ(xt(i), t(i)))∆t + o(∆t)] = Z 1 t(1) γ⊤ t Eq(xt(i)\|x0)[δxt,m · (x0x⊤ 0 log µθ(xt, t) + x0 −µθ(xt, t))]dt. Letting t(1) →0 proves the result. I.2 Training and gradient estimation The model is applied to data consisted of N tokens where x0 = (x1 0, . . . , x(N) 0 ) and where each state in the masked diffusion is xt = (x1 t, . . . , x(N) t ). The reverse generated model has a factorizing transition conditional of the form QN n=1 pθ(x(n) s \|xt) where pθ(x(n) s \|xt) = q(x(n) s \|x(n) t , µ(n) θ (xt, t)) has a form that depends on whether x(n) t = m or x(n) t ̸= m. For the first case: pθ(x(n) s \|x(n) t = m, {x(k) t }k̸=n) = 1 −αs 1 −αt ⊤ µ(n) θ (xt, t)e⊤ mx(n) s + αs −αt 1 −αt ⊤ diag(µ(n) θ (xt, t))x(n) s , where µ(n) θ (xt, t) = softmax(fθ(xt)) is a m + 1 dimensional probability vector modelled by a NN (where the final value is constrained to be zero since µ(n) θ (xt, t) is a reconstruction of x(n) 0 which cannot be mask, so in practice the NN classifier needs to have a softmax output only over the m actual token classes). Crucially, note that the NN classifier receives as input the full state xt of all tokens, while additional time features to encode t are also included. When x(n) t ̸= m the reverse transition model is set to be pθ(xs\|x(n) t ̸= m, {x(k) t }k̸=n) = (x(n) t )⊤x(n) s which matches precisely q(x(n) s \|x(n) t = m, x(n) 0 ) = (x(n) t )⊤x(n) s from the forward process. The full negative lower bound for state-dependent rates and assuming N tokens is given by L(N) ∞ = Z 1 0 α′ t 1 −αt ⊤ Eq(xt\|x0) hP n:x(n) t =m(x(n) 0 −µ(n) θ (xt, t) + x(n) 0 (x(n) 0 )⊤log µ(n) θ (xt, t)) i dt. (32) Given that each αt,i = 1 −twi, the reverse model becomes pθ(x(n) s \|x(n) t ̸= m, {x(k) t }k̸=n) = ew log s t ⊤µ(n) θ (xt, t)e⊤ mx(n) s + 1 −ew log s t ⊤diag(µ(n) θ (xt, t))x(n) s , where w is the m + 1 dimensional vector of all wis. Note that the probability of x(n) s staying in the mask state, i.e., x(n) s = m depends on the full xt and it is given by ew log s t ⊤µ(n) θ (xt, t) = Pm−1 i=0 ewi log s t µ(n) θ (xt, t)i while the probability for x(n) s to take a certain non-mask token value i is 1 −ewi log s t µ(n) θ (xt, t)i. The gradient wrt t is α′ t,i = −witwi−1 and α′ t,i 1−αt,i = −wi t the above loss is written as L(N) ∞ = − Z 1 0 1 t w⊤Eq(xt\|x0) hP n:x(n) t =m(x(n) 0 −µ(n) θ (xt, t) + x(n) 0 (x(n) 0 )⊤log µ(n) θ (xt, t)) i dt, where w is the vector of all wi’s. An unbiased gradient over the NN parameters θ is straightforward to obtain since we just need to sample one time point t and an xt ∼q(xt\|x0) to approximate the integral and expectation and then use the gradient: −∇θ X n:x(n) t =m 1 t w⊤ x(n) 0 −µ(n) θ (xt, t) + x(n) 0 (x(n) 0 )⊤log µ(n) θ (xt, t) . The gradient wrt the w parameters is more complex since these parameters appear also in the discrete distribution q(xt\|x0) which is not reparametrizable. To deal with this we need REINFORCE 25 unbiased gradients [73, 74], and in our implementation we consider REINFORCE leave-one-out (RLOO) [53, 54] with two samples. Firstly, the exact gradient wrt w of the exact loss is written as − Z 1 0 1 t Eq(xt\|x0) [g(xt, x0)] dt − Z 1 0 1 t Eq(xt\|x0) [f(xt, x0)∇w log q(xt\|x0)] dt. (33) where g(xt, x0) = X n:x(n) t =m (x(n) 0 −µ(n) θ (xt, t)+x(n) 0 (x(n) 0 )⊤log µ(n) θ (xt, t)), f(xt, x0) = w⊤g(xt, x0). Note that g(xt, x0) is a vector while f(xt, x0) is a scalar. The left term in (33) is easy since it just requires sampling t and xt ∼q(xt\|x0), while the right term is the REINFORCE term which could have high variance. For this second term we use RLOO with two samples x1 t, x2 t and construct the unbiased estimate −1 2t ∇w log q(x1 t\|x0) −∇w log q(x2 t\|x0) f(x1 t, x0) −f(x2 t, x0) . Thus, the overall unbiased gradient for w we use is −1 2t g(x1 t, x0) + g(x2 t, x0) + ∇w log q(x1 t\|x0) −∇w log q(x2 t\|x0) f(x1 t, x0) −f(x2 t, x0) . J Experimental Details In all experiments, the model is trained with a continuous-time loss while samples are drawn from the discrete-time reverse model of 1000 timesteps unless otherwise noted. We used an exponential moving average factor 0.9999 for all evaluation including sample generation. J.1 text8 We followed the standard dataset split as in Austin et al. [14], Lou et al. [32] and trained our models on text chunks of length 256 for 1 million steps with batch size 512. All models in the table used a standard 12-layer transformer architecture unless otherwise noted. Our transformer has also the same number of heads (12) and hidden dimension (784) as in Austin et al. [14], Lou et al. [32]. We used the continuous-time ELBO and drew one sample of t for each data to estimate the integral. To reduce the variance of training, we used the same antithetic sampling trick described in Kingma et al. [33] for continuous diffusion models. We used the linear masking schedule αt = 1 −t and added a small shift ϵ = 10−4 when t is close to 0 and 1 to ensure numerical stability. The shifted schedule is αt = (1 −2ϵ)(1 −t) + ϵ. The shift leads to a support mismatch between q(x1\|x0) and the prior p(x1), leading to an undefined KL divergence term. We explain in app. E how to modify the prior distribution to allow small uniform probabilities in non-mask states to mitigate this problem. The shift leads to a non-zero reconstruction term and KL divergence term for the prior distribution but both are of negligible scale so we can safely ignore them when reporting the ELBO. We used a cosine learning rate schedule with a linear warm up of 2000 steps. We applied channel-wise dropout of rate 0.05 and used AdamW optimizer with learning rate 0.0003 and a weight decay factor of 0.03. Our model is trained on 16 TPU-v5 lite for less than a day. J.2 OpenWebText We kept 2% of the original training set for validation. Our small and medium transformer model have the same number of layers, heads, and hidden dimensions as in Lou et al. [32] and our tokenizer was also kept the same with a vocabulary size of around 50k. The training objective, masking schedule and other architectural choices were kept the same with the text8 experiment. We kept the training hyperparameters the same as text8 experiment except that we reduced the dropout rate to 0.02. J.3 FineWeb-Edu We kept the same training setup as the OpenWebText experiments. Our transformer models have the same number of layers, heads, and hidden dimensions as those of GPT-2 models. We use the same GPT-2 tokenizer. 26 For Hellaswag evaluation, we concatenate question with each answer option, tokenize the concatenated string, pad to the length of 1024. The padded token sequence gets fed to our MD4 model’s loss function for likelihood evaluation. We average 32 Monte Carlo samples to reduce variance. The answer with the highest likelihood estimate is the model’s prediction. J.4 Images We used the same linear masking schedule as in previous experiments in all likelihood results. We used the same U-Net plus self-attention architectures from the continuous diffusion model described in Kingma et al. [33] for CIFAR-10, except that we did not use Fourier feature inputs and added an additional input embedding layer with embedding size the same as the hidden dimension of the model. For ImageNet 64 × 64, we reduced the number of residual blocks (in one side of the U-Net structure) from 64 to 48 and added a 12-layer diffusion transformer [75] with 768 hidden dimension and 12 heads in the middle. For both datasets we used AdamW optimizer and trained for 2M iterations. We used learning rate 0.0004, batch size 256, weight decay factor 0.01 for CIFAR-10 and learning rate 0.0002, batch size 512, weight decay factor 0.03 for ImageNet 64×64. The learning rate follows a cosine annealing after 100 warm up steps. Our CIFAR-10 model is trained on 32 TPU-v5 lite for 24 hours. Our ImageNet-64 × 64 model is trained on 256 TPU-v5 lite for 3.5 days. As explained in Sec. 4, we have observed that the cosine schedule leads to better sample quality so we used it to train a cheaper model for sample visualization. This model differs from the one that achieves best likelihood in that we used 8 residual blocks (in one side of the UNet structure) and a 20-layer diffusion transformer in the middle. All other configurations are kept the same. K Additional Results K.1 Sample quality evaluation by GPT-2 We use the GPT-2 large model to evaluate the perplexity of samples generated by our model, following Lou et al. [32]. Results are shown in Fig. 8. Figure 8: Generative perplexity evaluated by GPT-2 Large following Lou et al. [32]. We compare MD4 against the GPT-2 checkpoint (autoregressive baseline) and SEDD (the previous best discrete diffusion model on this task) in generating 1024-token text sequences. We investigate the effects of two orthogonal factors on sample quality: model size and decoding steps. The numbers for GPT-2 and SEDD are from Lou et al. [32]. 27 K.2 Perplexity on OpenWebText validation set Tab. 5 reports the final perplexity number achieved on OpenWebText validation set, corresponding to Fig. 4. Table 5: Perplexity on OpenWebText validation set. Size Method Perplexity (↓) Small Gaussian Diffusion ≤27.28 SEDD Absorb (reimpl.) ≤24.10 MD4 (Ours) ≤22.13 GenMD4 (Ours) ≤21.80 Medium MD4 (Ours) ≤16.64 K.3 FID evaluation of MD4 trained on ImageNet 64×64 We provide the FID numbers corresponding to Fig. 2 in Tab. 6. Table 6: FID of 50k samples generated by MD4 trained on ImageNet 64× 64, corresponding to Fig. 2. Top three rows show results from an unconditional model, while the bottom row is from a model conditioned on class labels. Uniform discretization grid is used in Alg. 2 unless otherwise noted. Method Timesteps T 64 128 256 512 Linear αt 193.81 128.18 72.94 50.21 Linear αt, cosine grid 42.07 25.16 18.31 18.22 Cosine αt 47.46 23.84 17.8 18.74 Cosine αt, class conditional 30.75 11.39 7.13 7.8 K.4 Additional unconditional generation from MD4 trained on ImageNet 64×64 We provide more unconditional generation results from our pixel-level modeling experiments on ImageNet 64×64 in Fig. 9. K.5 Additional unconditional generation from MD4 trained on OpenWebText Below we include two unconditioned text samples generated by our MD4 Medium model trained on OpenWebText. K.5.1 MD4-M unconditional sample 1: 1024 tokens like, I don’t have to be alive? Sometimes there are things that are too real and you’re really supposed to experience them. So that’s a good feeling. That is the scary thing. Not actually, being able to experience things, being able to do these things, when you’re doing them, which, for most people having to wake in a dream is something that seems the most significant, and then you think about it the next day. It’s like the hope of the future, and you wake up right now thinking about it. What happens is,, then you have to stop and think about it and then all of a sudden, somebody always says, "You’re dreaming." And sometimes I wonder if this is a good time to teach your gut instincts to your actors when you’re doing a show like this. Because even on this particular show, it feels like everyone’s been through this all the time before, if even a few years ago. I mean, if you’re doing a show together, at least not on continuous development, you you’re a vet. I mean, you should really be along. 28 Figure 9: More unconditional samples from MD4 trained on ImageNet 64×64. If you’re not sure, well -VS: I’m working on that one. Did any of you guys feel that an instinct could work? I thought, "Well, because you didn’t do ’Deadwood’ you should stop doing this." But when I read the story for the first time, I thought, "I think this is going to work." What I can’t picture is a way to hold this apart. 29 VS: That’s me. It’s what we have to do. So do we. When we wrote the first episode, we wrote a script that we felt like me and myself would want to see. I knew that I wanted to be able to be in something -- and I wanted to be able to take refuge in something that was real, that you could see and just really step out of yourself. And then I saw it. Then, you get rehearsing it and doing it. And then I actually started shooting. I think I knew I didn’t think it was going to be good. But, I know it was good. And now people are talked about because it’s not good enough. Growing up, you say that you just completely hated the show, "Lost." Isn’t that what you wish for at the end of the day? VS: I don’t like the concept. And so there’s a lot that you don’t know about that, so I think for me to have had these ideas, if you didn’t understand even that it was coming out of this world that doesn’t exist, we might never get together. It’s so weird. This happened to happen at the same time? VS: Yes. It happened to happen at basically the same time. Nobody’s even had a show or had a movie/come out of the movie, but ... VS: If I’m going to pretend I’m definitely not you and have to live through that stuff, I don’t think I’m going to swallow that. I didn’t expect it to do quite that long. There are always things now that happen with ’Deadwood’ where you don’t know where it’s going to end up next time, but I think there are occasions now where we have to keep the fight, even if ’Lost’ was pretty consistent in the mindset and the form. VS: I’m glad that we did fight the odds, because we should have understood that there was a direct link. But there was almost a sense of not that we had showed up on the same day, we know we work in the same pieces, but a lot of stuff we don’t know about. Some of it, we need to deal with. We also just have to accept the language, and there are a lot of things where we take from them and we do this what they did because we want to K.5.2 MD4-M unconditional sample 2: 1024 tokens the groups let recreational vehicles use the three roads that will stay open in the meantime of fighting off the permit. "The purpose of the permit is to make sure that we work with the NPS and made roadways and rest areas. We’re not just scaring guys kind of messing around." Community plans to build an urban bike facility marched forward at the ongoing staff meeting of the King County Commission. Trail will be finished just south of the Greenview 5. Instead of continuing with a pedestrian and bike trail to the MBTA’s campus, these two trails could bridle the areas from Market to 14 and carry communities closer. "This project will provide a car-free path to King County," said Andrew Weed. It’s been put the brakes on in the past several months, but there are those residents still skeptical. "I’ve addressed some of the community concerns that’ve been raised. They’ve expressed some of their concerns. I don’t think it’s terribly reasonable from a 30 transportation standpoint." The trail had been set up to meet on for more than a year when the council approved funding for a different proposal. Mayor Muriel Bowser said after meetings with Commissioner Bushell on Thursday that the new plan will be on board in December. "There’s enough of a finish for this project to roll out on time, and we’re going to get it done," Bowser said. For the public, the campaign appears over. “There was one meeting that I feel like I lost at last night’s meeting," said Shelley Potts, a local resident. Local resident Joel Grimy, who lives on Uman Road, met residents there as well. And in other groups that rode through Mayor assistant Stacey Land and even her son held fliers saying to look for light sign, and also met with Bowser’s son, Deion Bowser, about a future plan to also have a dog park on the transit corridor. Advocates at Brickley’s event, many one waited at least 11 minutes in during the start of the public meeting, said they expect at least another month from the Board of Commissioners, even after a public hearing on Nov. 13. "We’ve been trying to be a talkative board where we are meeting in advance, being respectful of folks," Bowser said. He considered that the proposal for the section of trail between the Greenview 5 and 3 “has to move on a schedule. We have other historic preservation projects that would take over that.” But Chad Routledge, a local advocate of the project, spoke out against the mayor’s plan. “The mayor has sent a new meeting to the public using the same route that resulted from the loud criticism and onslaught of complaints from the community committee back during the public hearing,” Routledge said. The BDC doesn’t have a particular plan-turns around for the end of the planned path, and says “nothing practical can happen right now.” But, she said the agency still "looking to make investments in facilities along the route." And still there is another part of the trail that might be just as much a wish for the dogs, as cars: the district wants to go west foot a couple blocks south, to make the trail safer for dogs. “I feel that the accessibility of the trail is pretty important. I think the education of the trail, and the uses along different routes are very important pieces of a balanced outcome,” said Bushell. Trams coming off Route 1 K.6 Conditional generation from MD4 trained on OpenWebText We share conditionally generated text samples by MD4 Medium in Fig. 10 and observe that slow unmasking near t = 1, enabled by the cosine schedule, tends to help produce more consist and meaningful samples than uniform unmasking counterpart. 31 Figure 10: Conditionally generated text samples from MD4-M. Top: MD4-M trained with the linear schedule, sampled with a uniform grid; Middle: MD4-M trained with the linear schedule, sampled with the cosine grid; Bottom: MD4-M trained with the cosine schedule, sampled with a uniform grid. Context text shown in blue, model-generated text in black. K.7 Effect of discretization on zero-shot perplexity We carried out ablation study on the effect of discretization on zero-shot perplexity. Results are included in Tab. 7. Note that this is an inference ablation with the same trained model (MD4-S trained with the continuou-time objective). Table 7: Effect of discretization on zero-shot perplexity. Size Timesteps LAMBADA WikiText2 PTB WikiText103 IBW Small T = 100 ≤49.8 ≤36.1 ≤105.2 ≤36.1 ≤70.3 T = 1000 ≤48.5 ≤35.0 ≤102.5 ≤35.0 ≤68.4 T = 10000 ≤48.4 ≤34.9 ≤102.4 ≤34.9 ≤68.2 T = ∞(continuous) ≤48.4 ≤34.9 ≤102.3 ≤35.9 ≤68.1 32 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: Our main theoretical and experimental contributions are claimed in the abstract and demonstrated in the paper. They reflect the paper’s contributions and scope. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] . Justification: The limitations of our work are detailed in the very last paragraph of the paper (see Section 7). 3. 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4,478	Stabilize the Latent Space for Image Autoregressive Modeling: A Unified Perspective Yongxin Zhu1,3, Bocheng Li2,3, Hang Zhang4, Xin Li, Linli Xu∗2,3, Lidong Bing 1School of Data Science, University of Science and Technology of China 2School of Computer Science and Technology, University of Science and Technology of China 3State Key Laboratory of Cognitive Intelligence, 4Zhejiang University zyx2016@mail.ustc.edu.cn,bcli@mail.ustc.edu.cn,hangzhang_scu@foxmail.com lixin4ever@gmail.com,linlixu@ustc.edu.cn,binglidong@gmail.com Abstract Latent-based image generative models, such as Latent Diffusion Models (LDMs) and Mask Image Models (MIMs), have achieved notable success in image generation tasks. These models typically leverage reconstructive autoencoders like VQGAN or VAE to encode pixels into a more compact latent space and learn the data distribution in the latent space instead of directly from pixels. However, this practice raises a pertinent question: Is it truly the optimal choice? In response, we begin with an intriguing observation: despite sharing the same latent space, autoregressive models significantly lag behind LDMs and MIMs in image generation. This finding contrasts sharply with the field of NLP, where the autoregressive model GPT has established a commanding presence. To address this discrepancy, we introduce a unified perspective on the relationship between latent space and generative models, emphasizing the stability of latent space in image generative modeling. Furthermore, we propose a simple but effective discrete image tokenizer to stabilize the latent space for image generative modeling by applying K-Means on the latent features of self-supervised learning models. Experimental results show that image autoregressive modeling with our tokenizer (DiGIT) benefits both image understanding and image generation with the next token prediction principle, which is inherently straightforward for GPT models but challenging for other generative models. Remarkably, for the first time, a GPT-style autoregressive model for images outperforms LDMs, which also exhibits substantial improvement akin to GPT when scaling up model size. Our findings underscore the potential of an optimized latent space and the integration of discrete tokenization in advancing the capabilities of image generative models. The code is available at https://github.com/DAMO-NLP-SG/DiGIT. 1 Introduction In recent years, remarkable advancements have been achieved in the field of image generation, principally propelled by the development of latent-based generative models, such as Latent Diffusion Models (LDMs) [34, 30] and Mask Image Models (MIMs) [7, 26]. By employing reconstructive autoencoders such as VQGAN [15] or VAE [23] to compress images into a manageable low dimensional latent space, these models can generate highly realistic and imaginative samples. Concurrently, in light of the transformative impact of autoregressive (AR) generative models, such as Large Language Models [31, 32, 5, 27] in NLP, it becomes compelling to investigate the feasibility of similar paradigms to images. Despite the advances in image autoregressive pre-training, exemplified by ∗Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Model Performance: IS vs. FID Fréchet Inception Distance (FID) ↓ Model Performance: FID vs. Top-1 Accuracy Top-1 Accuracy (%) ↑ 84.5 82.0 79.5 77.0 74.5 72.0 69.5 67.0 64.5 62.0 59.5 57.0 N/A 87.0 DINOv2 iBOT MoCo V3 MAE AIM VIM+ViT-VQGAN iGPT-XL VIM+dVAE VIM+ViT-VQGAN VIM+VQGAN iGPT-L MaskGIT LDM ADM VQGAN DiGIT (+VQGAN) DiGIT (+MaskGIT) DiGIT (+VQGAN) MAGE Fréchet Inception Distance (FID) ↓ N/A 36 32 28 24 20 16 12 8 4 SSL Models Generative Models GIVT Inception Score (IS) ↑ MAGVIT-v2 DiGIT (+VQGAN) BigGAN RQ-Transformer VQGAN VIT-VQGAN ADM LDM-4 DiT-XL/2 RQ-Transformer ViT-VQGAN DiGIT(+VQGAN) L-DiT-7B DiGIT(+MaskGIT) Contextual RQTransformer MaskGIT 4 6 8 10 12 14 16 18 80 100 120 140 GAN Diffusion Models Mask Image Models Autoregressive Models 160 180 200 (a) (b) Figure 1: (a): Linear probe and class-unconditional generation performance of different methods trained and evaluated on ImageNet-1K. (b): Class-conditional generation performance of different methods on ImageNet-1k. The size of the bubbles indicates the number of parameters in the models. DiGIT achieves SOTA performance in linear probing and establishes a new SOTA in image generation within a single model. models such as iGPT [8], AIM [14] and GIVT [36] in the pixel space, VIM [41] and LVM [1] in the latent space, their performance are still inferior to the leading models [34, 12, 7] in image generation, or self-supervised learning models [18, 10, 6, 28] in image understanding tasks. An intriguing observation emerges regarding the presumed optimality of current practices in latent space: as illustrated in Figure 1(b), though sharing the same latent space, autoregressive models significantly lag behind LDMs and MIMs in the image generation task. This discrepancy prompts reevaluating our understanding of latent spaces and their interaction with generative models, suggesting unexplored avenues worth investigating. A central premise in learning theory [20] is that the latent distribution should retain as much critical information of the data distribution as possible, akin to a compression goal. This leads to a common misconception that an optimal latent space for reconstruction equates to optimal generative performance. Nevertheless, in the investigation regarding the reconstruction and generation ability with the popular VQGAN model [42], it is observed that the generation FID will deteriorate when the reconstruction FID becomes lower (where a lower value indicates better performance), challenging the above assumption. To address these intriguing discrepancies, we introduce a unified perspective on the relationship between latent spaces and generative models to analyze what constitutes an optimal latent space for generative models. Our findings reveal that, beyond the compression techniques employed by prevalent latent generative models, an optimal latent space should also aim to minimize the distance between distributions under the condition of incorporating a generative model, which is an aspect often overlooked. We critically assess prevalent methodologies and reveal that the stability of latent space is important for generative models. We argue that the reason why autoregressive models underperform iterative models such as LDMs and MIMs is that iterative models can correct errors brought by the instability of latent space. Drawing from this insight, we propose a straightforward method to stabilize the existing latent space methods for image autoregressive generative models. Unlike conventional autoencoder-style approaches, our approach disentangles the concurrent training of encoders and decoders and commences with encoder-only training through a discriminative self-supervised model [28]. This phase does not necessitate a decoder for pixel reconstruction, enabling the encoder to discern the intrinsic and distinguishable features present within the data. Subsequently, a separate decoder of the autoencoder [15] is trained and tasked solely with the pixel reconstruction process, conditioned on the features identified by the encoder. By focusing initially on the encoder’s capability to extract meaningful data features independently of pixel reconstruction, we lay a foundation for a more stable and feature-rich latent 2 space. The subsequent independent training phase of the decoder ensures that these captured features can be accurately translated back to pixels. In support of the autoregressive generative model, which requires discrete tokens for next token prediction, we employ a strategy inspired by VQGAN [15] to discretize the encoder’s latent feature space with the K-Means clustering algorithm. With this novel image tokenizer induced from the stabilized latent space, the performance of autoregressive generative models in images is enhanced significantly in both image understanding and image generation tasks. We refer to this approach as call Discriminative Generative Image Transformer (DiGIT). Notably, when scaling up the model size, substantial improvements can be achieved. To the best of our knowledge, this is the first evidence that image autoregressive generative models behave analogously to GPT. In essence, this work endeavors to redefine the boundaries of what is possible in image autoregressive modeling through a unified perspective of latent space. In summary, our contributions to the field of image generative models include: • We introduce a unified perspective on the relationship between latent space and generative models, emphasizing the stability of latent space in image generative modeling. • We propose a novel method to stabilize latent space by disentangling the encoder and decoder training processes. Furthermore, a simple yet effective discrete image tokenizer is proposed to improve the image autoregressive generative model’s performance under the philosophy of next token prediction. • The experimental results show that the image autoregressive modeling with our tokenizer leads to SOTA performance in image understanding and generation, with further improvements witnessed when scaling up the model size. 2 Problem Formulation In section 2.1, we formalize the latent space requirements for generative models and categorize current latent-based generative models. Furthermore, in section 2.2, we analyze the stability of different induced latent spaces and propose to stabilize the latent space for autoregressive generative models instead of stabilizing the generation process with an iterative decoding strategy like LDMs. 2.1 Latent Space for Generative Models Drawing inspiration from the complexity perspective of latent space induced from autoencoders in Hu et al. [21], we delve into the latent space for generative models. Generative models aim at learning a distribution to approximate the data distribution PX. Formerly, given a tractable prior distribution PZ and a distance metric D(·, ·) between distributions, the purpose of a generative model g ∈G is to minimize the distance between data distribution PX and the distribution generated by g(Z): min g D(Pg(Z), PX). (1) For example, GANs [16] employ the Gaussian distribution as their prior and utilize a discriminator network as the distance metric. However, the optimal strategy for data representation in generative models is still under-explored. Recent studies on latent diffusion models [34] have identified that direct learning in the pixel space of images is suboptimal. They propose to learn in a latent space induced by a constrained autoencoder model such as VAE [23] or VQGAN [15], which has been demonstrated to improve the perceptual quality. A simple method to construct the latent space is using an encoder f ∈F : Rd →Rdz to map raw data samples x ∈Rd into a latent space f(X) of dimension dz. Consequently, the goal of latent-based generative models is to learn the distribution as per the following formula: min g D(Pg(Z), Pf(X)), (2) where Pf(X) denotes the data distribution in the latent space induced by the encoder f. Despite these advances, determining the optimal latent space configuration for generative models remains unresolved. 3 Distance between distributions in different spaces. Given the ultimate goal of the generative model is to produce image pixels, a decoder model h ∈H : Rdz →Rd, paired with the encoder model f, is necessary to convert latent representations back into pixels. We define a generalized distance between different spaces associated with a decoder h ∈H as: DH(Pf(X), PX) := inf h∈H D(Ph(f(X)), PX). (3) By employing DH to compare distributions across different spaces, we can define the ideal latent space f(X) as the one that minimizes DH(Pf(X), PX). This implies selecting a latent representation that minimizes the empirical objective L(h\|Pf(X)) with the same family H of decoder models. Such a latent configuration depends on both the data and the decoder training methodology. We can formalize it with an autoencoder framework by looking at the encoder and decoder together, and the primary goal becomes the reconstruction of the input sample x, formulated as minf,h L(h(f(x)), x). Once a generative model g successfully approximates the latent distribution Pf(X), the generated sample can be efficiently transformed back into pixels using the decoder h. Similarly, we can define the distance between distributions in the latent space f(X) ∈Rdz and data space X ∈Rd conditioned on the latent-based generative model g ∈G, DG(Pf(X), PX) := inf g∈G D(Pg(Z), Pf(X)), (4) which aims to minimize the empirical objective with the generative model family G. Optimal Latent Space for Generative Models. Now that we have characterized the ideal latent distribution given the family of generative models and the data, the next step is to determine how to find the optimal latent space. At the population level, the objective for the latent-based generative models with a decoder is: min h∈H,g∈G D(Ph(g(Z)), PX) = min h∈H,g∈G D(Ph(f(X)), PX) + D(Pg(Z), Pf(X)), (5) where the first term focuses on optimizing the decoder to enhance the reconstruction quality and the second one is directed towards optimizing the generative model to more accurately approximate the latent space distribution. Inspired by this observation, we can characterize the optimal latent distribution P ∗ f(X) for a given PX from the perspective of minimizing the distance between distributions in different spaces by defining f ∗as argmin f∈F Dlatent(Pf(X), PX) := f ∗(X) = argmin f∈F DG(Pf(X), PX) + DH(Pf(X), PX) (6) Notice that Pf ∗(X) depends on multiple factors, including PX, the distance metric D, and the constructed model families G and H. By integrating DG and DH, we arrive at a comprehensive measure of the distance between the distribution in the latent space and the original data distribution as Dlatent(Pf(X), PX). The second term, DH(Pf(X), PX), is exactly the objective of reconstructive autoencoders. Ultimately, from examining the learning objective pertinent to identifying the optimal latent space for generative models, it becomes evident that: A reconstructive autoencoder does not necessarily establish an advantageous latent space for generative models. Two Pathways of the Latent Space Construction. Although we theoretically analyze the optimization of the optimal latent space for generative models, it is challenging to implement in practice because optimizing (f, g, h) simultaneously is computationally complex. A practical solution is to optimize DG and DH separately, allowing for tractable training. • When DH(Pf(X), PX) is not a target for optimization, it implies that the optimization of the decoder within the generative model framework is bypassed. The encoder independently forms a latent space, aligning with self-supervised learning (SSL) strategies aimed at uncovering lower-dimensional features from unlabeled data. However, learning the generative models in the latent space induced by SSL models remains relatively unexplored. • On the other hand, when DG(Pf(X), PX) remains fixed, the primary objective becomes optimizing the encoder and decoder to effectively learn and represent the latent space, 4 where Dlatent(Pf(X), PX) degrades into an autoencoder learning objective. This approach is evident in recent latent-space-oriented generative models, such as LDM [34, 30], VQGAN (AR)[15], and MaskGIT (MIM) [7], all of which concentrate on learning g in the latent space with the encoder and decoder frozen. 2 While latent generative models such as LDM [34], MaskGIT [7], and VQGAN [15] share the same latent space induced by a reconstructive autoencoder [15] to minimize DH(Pf(X), PX), their image generation performances differ significantly. In the next section, we analyze the reason behind it from the perspective of the latent space. 2.2 Stability of the Latent space We first describe the decoding mechanism of various latent generative models. Both LDM [34] and MaskGIT [7] can be depicted as an iterative sampling procedure given by: p(xT ) = T Y i=1 p(xi\|xi−1). (7) The intermediate states xi in LDMs represent images infused with Gaussian noise of varying variance, whereas for MaskGIT, they denote discretely tokenized images augmented with masks. In contrast, the autoregressive framework of VQGAN (AR) is described as: p(x) = p(x1, . . . , xN) = N Y j=1 p(xj\|x<j), (8) where xi represents the i-th patch in the image sequence. Notice that xi denotes the entire image while xj means the local patch tokens. In the autoregressive decoding process, if the previously sampled tokens are incorrect, the accuracy of subsequent tokens would be affected due to error aggregation. In contrast, the iterative decoding approach allows for the revision of earlier misjudged tokens. When the latent space is unstable that small perturbation in pixels can change the latent distribution significantly, the iterative decoding mechanism employed by LDM and MaskGIT can alleviate the error aggregation problem by allowing for revision of earlier misjudged tokens while autoregressive models cannot. Consequently, a stable latent space is required to reduce errors introduced in the generation process of autoregressive models. This principle forms the foundation of our methodology for developing a metric to evaluate latent spaces with an emphasis on the stability of the latent representations. We examine two primary types of latent spaces: (1) autoencoder induced by minimizing DH(Pf(X), PX) and (2) self-supervised learning (SSL) model induced by minimizing DG(Pf(X), PX). By analyzing network parameters of these models in a linear regime, we derive the following propositions. Proposition 2.1. The latent space spanned by a linear autoencoder is congruent with that spanned by the principal component loading vectors derived in Principal Component Analysis (PCA). Furthermore, the principal component loading vectors can be elucidated from the autoencoder’s weights. Proposition 2.2. The discriminative self-supervised model learns to separate data distributions in the latent space as Linear Discriminant Analysis in principle. Motivated by these theoretical insights, we introduce a metric to assess the stability of the latent space induced by different encoder models. To exemplify these concepts, we refer to an example consisting of two Gaussian distributions in a two-dimensional space, as depicted in Figure 6(a). The results attained from applying the PCA and LDA algorithms are visually depicted in Figure 6(b) and (c) respectively. The distribution embedded by the LDA model exhibits greater separability than that by the PCA model. To quantitatively evaluate stability, we add Gaussian noise of different variances to the original 2D data and subsequently train a linear classifier on the latent space. As Figure 6(d) illustrates, the accuracy of the LDA model consistently surpasses that of PCA. 2DG and DH can achieve zero simultaneously. For example, in the well-known posterior collapse phenomenon in the VAE literature, the latent space f(X) is a tractable Gaussian distribution and DG(Pf(X), PX) can be zero by simply setting the generative models as a Gaussian sampler. If the decoder is strong enough and directly generates samples without conditioning on the encoder output, DH(Pf(X), PX) can be zero as well. 5 Table 1: The stability of latent spaces induced from VQ Token and Discriminative Token (introduced in Section 3), assessed across different Signal-to-Noise Ratio (SNR) levels to evaluate performance under varying signal and noise conditions. SNR 30 25 20 15 10 5 1 0.01 VQ Token change ↓ 0.187 0.317 0.487 0.663 0.805 0.901 0.948 0.956 Disc Token change ↓ 0.114 0.178 0.260 0.355 0.457 0.570 0.687 0.721 VQ Token cos-sim ↑ 0.972 0.949 0.910 0.853 0.777 0.682 0.594 0.571 Disc Token cos-sim ↑ 0.975 0.960 0.940 0.916 0.888 0.855 0.816 0.803 Image Patches Discriminative SSL Model K-Means Discriminative Tokens Visual Features Frozen Trainable DiGIT Tokenize … eos c1 c2 … cL−2 cL−1 Transformer Block Transformer Block Transformer Block Transformer Block c1 c2 … cL−2 cL−1 cL Visual Features Figure 2: The architecture of DiGIT. To evaluate the stability of latent space induced from autoencoders and SSL models, we add Gaussian noise to image pixels and then feed the noisy images to a VQGAN encoder and an SSL encoder DINOv2 [28]. This experiment aims to examine the resilience of the latent spaces induced by these encoders to such disturbances. We measure the rate of change in discrete tokens, specifically VQ tokens for the latent space induced from the VQGAN encoder and discriminative tokens for the latent space induced from the SSL model, and the cosine similarity in conjunction with the strength of the noise introduced. The experimental results in Table 1 demonstrate that the latent space induced from the SSL model exhibits heightened stability compared to that derived from the VQGAN autoencoder. Therefore, we propose to replace the unstable latent space induced by the reconstructive model with a stable latent space induced by the discriminative self-supervised model for autoregressive models. 3 Stabilize the Latent Space with Self-supervised Learning Model In this section, we present a simple but effective image tokenizer that discretizes the feature representations of discriminative SSL models to form discrete tokens for autoregressive models. The architecture of our model is illustrated in Figure 2. Discrete Image Discriminative Tokenizer Drawing inspiration from the VQGAN tokenizer [15], which employs an implicit K-Means clustering algorithm within the latent space to generate discrete tokens for autoregressive modeling, we propose a straightforward approach to perform K-Means clustering on the feature space of discriminative SSL models to obtain discrete tokens. To process a given dataset, our initial step involves gathering the features of image patches, akin to the hidden states produced by SSL models. Then we employ a clustering algorithm to group these patches, resulting in a collection of K clustering centers. These centers constitute the codebook for the discrete tokenizer. To determine the discrete tokens for an image patch at inference, we identify its nearest neighbor in the codebook, which is then designated as the discrete token for the respective patch. Image Autoregressive Modeling After converting images into discrete tokens with the discriminative tokenizer, we treat each image as a sequence by flattening the discrete tokens from images into a 1D sequence in raster order. We train a causal Transformer [38] model with the next token prediction objective, which is the same as the standard approach for language models. 6 Table 2: Linear-probe accuracy of image autoregressive generative models on ImageNet [11]. Methods # Tokens Features # Params Top-1 Acc.↑ iGPT-L [8] 32 × 32 1536 1362M 60.3 iGPT-XL [8] 64 × 64 3072 6801M 68.7 VIM+VQGAN [41] 32 × 32 1024 650M 61.8 VIM+dVAE [41] 32 × 32 1024 650M 63.8 GIVT [36] 16 × 16 1024 304M 65.1 VIM+ViT-VQGAN [41] 32 × 32 1024 650M 65.1 VIM+ViT-VQGAN [41] 32 × 32 2048 1697M 73.2 AIM [14] 16 × 16 1536 0.6B 70.5 DiGIT (Ours) 16 × 16 1024 219M 71.7 DiGIT (Ours) 16 × 16 1536 732M 80.3 4 Experiments 4.1 Implementation Details We take the discriminative SSL model DINOv2 [28] as the encoder for all experiments. The K-Means model is trained on the randomly selected 10% subset of the ImageNet [11] training set. We use the autoregressive model with the same architecture as GPT-2 [32]. We train the DiGIT models with the base and large sizes. The vocabulary size of the tokenizer for the base is 8192 and 16000 for the large size. More implementation details and hyper-parameters are provided in Appendix A.3. 4.2 Image Understanding The GPT model is famous for learning semantic features by a generative training objective of next token prediction. We compare the image understanding ability of different image autoregressive models with linear-probe as described in iGPT [8]. We train a linear classifier on top of the frozen features average from each layer on the ImageNet training set. We report the Top-1 accuracy compared with other image autoregressive models in Table 2. Remarkably, with only 219M parameters, DiGIT achieves a Top-1 accuracy of 71.7%, surpassing both iGPT and VIM-Base, which have a greater number of parameters and operate at more visual tokens. Despite representing images with a smaller token grid size (16 × 16 as opposed to 32 × 32), DiGIT still delivers superior top-1 accuracy, demonstrating the effectiveness of our tokenizer. Moreover, when we scale DiGIT’s parameters from 219M to 732M, the Top-1 accuracy shows an additional increase of 8.6% and reaches 80% for the first time. The improvement indicates that DiGIT with the proposed discriminative tokenizer has the potential for the development of large vision models. 4.3 Image Generation Since the SSL models do not have a paired decoder to recover pixels from latent space, the generative models trained with our discriminative tokenizer require an auxiliary image decoder to render pixels. The discriminative tokenizer can be seamlessly integrated with any existing image generative models trained with a tokenizer induced from a reconstructive autoencoder. In our experiment, we train an autoregressive model VQGAN, and an MIM model MaskGIT as the pixel decoder respectively. The results are presented in Table 3 and Table 4. The autoregressive model equipped with our discriminative tokenizer achieves the SOTA performance with FID reaching 3 for the first time. Furthermore, the performance significantly improves as the model size increases, demonstrating the potential of a large vision model with next token prediction. Interestingly, when utilizing the DiGIT as the conditioning factor, the performance of both the autoregressive and MaskGIT decoders becomes close (4.62 and 4.79). This observation suggests that stabilizing the latent space produces effects analogous to the iterative stabilization decoding mechanism. 4.4 Ablation Study We conduct the ablation study to present a comprehensive analysis of the proposed discriminative tokenizer in image generation and understanding. The results are illustrated in Table 3(a) and 7 Table 3: Class-unconditional image generation on ImageNet with resolution 256 × 256. DiGIT + VQ indicates that we utilize golden discriminative tokens alongside VQ generated by autoregressive models. Type Methods #Param #Epoch FID↓ IS↑ GAN BigGAN [4] 70M 38.6 24.70 Diff. LDM [34] 395M 39.1 22.83 Diff. ADM [12] 554M 26.2 39.70 MIM MAGE [26] 200M 1600 11.1 81.17 MIM MAGE [26] 463M 1600 9.10 105.1 MIM MaskGIT [7] 227M 300 20.7 42.08 MIM DiGIT (+MaskGIT) 219M 200 9.04 75.04 AR VQGAN [15] 214M 200 24.38 30.93 AR GIVT [36] 304M 500 17.70 AR GIVT [36] 1.67B 500 11.02 AR DiGIT (+VQGAN) 219M 400 9.13 73.85 AR DiGIT (+VQGAN) 732M 200 4.59 141.29 validation data DiGIT + VQ 1.92 184.40 validation data VQ only 1.67 175.56 Table 4: Class-conditional image generation on ImageNet with resolution 256 × 256. † denotes the model is trained with classifier-free guidance while all the other models are not. Type Methods #Param #Epoch FID↓ IS↑ GAN BigGAN [4] 160M 6.95 198.2 Diff. ADM [12] 554M 10.94 101.0 Diff. LDM-4 [34] 400M 10.56 103.5 Diff. DiT-XL/2 [30] 675M 9.62 121.50 Diff. L-DiT-7B [30] 7B 6.09 153.32 MIM Contextual RQ-Trans [25] 371M 300 5.45 172.6 MIM+AR VAR [35] 310M 200 4.64 MIM+AR VAR [35] 310M 200 3.60† 257.5† MIM+AR VAR [35] 600M 250 2.95† 306.1† MIM MAGVIT-v2 [42] 307M 1080 3.65 200.5 AR VQVAE-2 [33] 13.5B 31.11 45 AR RQ-Trans [24] 480M 15.72 86.8 AR RQ-Trans [24] 3.8B 7.55 134.0 AR ViTVQGAN [41] 650M 360 11.20 97.2 AR ViTVQGAN [41] 1.7B 360 5.3 149.9 AR GIVT [36] 304M 500 5.67 AR GIVT [36] 1.67B 500 3.46 MIM MaskGIT [7] 227M 300 6.18 182.1 MIM DiGIT (+MaskGIT) 219M 200 4.62 146.19 AR VQGAN [15] 227M 300 18.65 80.4 AR DiGIT (+VQGAN) 219M 400 4.79 142.87 AR DiGIT (+VQGAN) 732M 200 3.39 205.96 validation data DiGIT + VQ 1.92 184.40 validation data VQ only 1.67 175.56 Figure 3(b). For image generation tasks, we take the autoregressive model trained with the VQGAN tokenizer as the baseline. Introducing discriminative tokens leads to a significant improvement, reducing FID to 9.66 and increasing IS to 69.15, underscoring the effectiveness of stabilizing latent space for autoregressive models. Further extending the training duration to 400 epochs yielded additional improvements of 0.53. A substantial advancement is observed when scaling up the model size to 732M, resulting in FID dropping dramatically to 4.59 and IS more than doubling to 141.29. This indicates that increasing the model’s capacity significantly enhances its ability to model complex relationships within the data, which is a similar phenomenon in GPT models. Overall, the study highlights the latent space stabilization and the potential of large-scale training of autoregressive modeling in images with our discriminative tokenizer. 8 FID↓ IS↑ VQ Token 24.38 30.93 + Discriminative Token 9.66 69.15 + Longer Training (400 epoch) 9.13 73.85 + Scale up (732M) 4.59 141.29 (a) 0 2 4 6 8 10 12 14 16 Layer Index 55.0 57.5 60.0 62.5 65.0 67.5 70.0 72.5 Top-1 Accuracy K-Means 8192 K-Means 4096 K-Means 1024 (b) Figure 3: Ablation study of DiGIT. (a) The comparison of tokenizer, training steps, and model size in the image generation task. (b) Linear-probe accuracy from different layers in the pre-trained DiGIT-base with different number of K-Means clusters. For the image understanding task, we investigate the effect of K-Means clusters and features learned in the different layers of DiGIT. We can see that increasing the cluster number can further improve the accuracy of the linear probe, which means the image autoregressive model can benefit from a larger vocabulary. The linear probe accuracy increases quickly from the first transformer block, reaches its peak at the middle layers, and finally decreases a little bit for the last few blocks. This observation connects the image autoregressive model to the text language model where the semantic information is learned in the middle layers of the transformer. 4.5 Comparison of Discrete Tokenizers We conduct an experiment to investigate the effect of different SSL models on latent space. We generally categorize the SSL models into two types according to their pre-training objectives: (1) Global level (MoCo) and patch level (MAE,iBOT), (2) reconstructive (MAE) and discriminative (MoCo, iBOT). At the global level, the loss function is computed using an aggregate output such as [CLS] token or mean pooling. In contrast, patch-level models involve patches directly in loss computation. Reconstructive models, such as MAE, aim to recover image pixels in a manner akin to autoencoders, while discriminative models are optimized to learn the distinguishable features. As demonstrated in Table 4(a), the discriminative objective plays a pivotal role in image generation in that it can stabilize the latent space. Furthermore, because generative models need to predict patches, the inclusion of a patch-level loss function can enhance performance. To assess the stability of latent space induced by our discriminative tokenizer and reconstructive tokenizer. We pre-train two auto-regressive generative models on the ImageNet dataset [11], employing the proposed discriminative tokenizer and VQGAN tokenizer respectively. We provide each model with the upper half of the target image as a conditional prompt for generation, challenging them to complete the lower half of the image. A stable latent space should be able to help the autoregressive model generate the lower half more robustly, maintaining thematic and aesthetic coherence. As shown in Figure 4(b), the FID decreases for both models when given a longer prefix context. However, when the prefix length is reduced from 75% to only 12.5% of the image, the model trained with the VQGAN tokenizer encounters difficulties in producing images that adhere to the specified prompt. In contrast, the model utilizing the discriminative tokenizer effectively continues to produce congruent visual tokens, maintaining low FID scores even with a significantly truncated prefix. 5 Related Work Image Tokenizer The image tokenizer [37, 33, 15] is essential in converting pixels into discrete tokens for autoregressive generative modeling. VQVAE [37] first proposes to assign latent features learned by an encoder to the nearest entry in the learnable codebook embeddings, followed by a decoder to reconstruct the original image pixels. VQGAN [15] further incorporates adversarial loss and perceptual loss to improve the image synthesis quality. RQ-Transformer [24] extends the single-layer quantizer to the multi-layer residual quantizer to augment the visual tokenizer’s ability to capture fine-grained details. ViT-VQGAN [41] incorporates the modern Vision-Transformer [13] into VQGAN to enhance the reconstruction quality. MAGVIT-v2 [42] substitute the online update codebook in VQGAN with a lookup-free quantizer to enable a larger vocabulary for generative language models. 9 SSL Type FID↓ IS↑ Acc@LP Acc@OL† MAE P+R 45.51 18.39 31.40 75.8 MoCo G+D 20.38 45.02 59.22 76.7 iBOT P+D 16.81 57.88 61.10 76.0 VQGAN 24.38 30.93 (a) 10 20 30 40 50 60 70 80 Prefix Length / % 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 FID Disc Tokenizer VQGAN Tokenizer (b) Figure 4: (a): The comparison of tokenizers induced from different SSL models. Acc@LP is obtained by linear probing on the autoregressive model (model size of 39M for 100 epochs) trained with tokenizers. Acc@OL is the linear probe score of the SSL model. “P”: patch, “D”: discriminative, “R”: reconstructive. (b): Generation quality curves in FID on ImageNet 256 × 256 valid set when scaling the prefix length with discriminative tokenizer and reconstructive VQGAN tokenizer. Both are autoregressive models with 219M parameters. Image Autoregressive Modeling Inspired by the success of autoregressive Transformer [38] in text generation tasks, there have been several efforts to replicate it in image generation tasks. One of the pioneering works is iGPT [8], which pre-trains an autoregressive model on pixels with the same architecture as GPT2 [32], achieving promising results in unsupervised visual representation learning. LVM [1] proposes a large-scale vision dataset composed of images and videos, based on which a large vision model is trained. Empirical observations indicate that the model scales effectively across various tasks with in-context learning [40]. Similarly, AIM [14] follows ViT [13] and represents images in the patch. It is observed that the performance of image recognition continues to increase as the model size scales up. VAR [35] proposes a next-scale prediction to generate images from coarse to fine in a hybrid of autoregressive and nonautoregressive manner. Self-supervised Learning Models Self-supervised learning (SSL) [9, 17] plays an important role in learning fundamental visual representations for downstream tasks. Among them, SimCLR [9], MoCo [9, 18, 10] compute losses at the image level through [CLS] token aggregation or pooling operations with contrastive learning. iBOT-style models [43, 6, 28] extend the loss to the patch level, achieving improved performance in dense prediction tasks. BEiT [3] uses VQGAN tokenized sequences as the training target. MAE [19] randomly masks some patches of images and reconstructs the pixels with unmasked patches as the condition. 6 Conclusion In this paper, we make an exploration in the latent space for generative modeling. We introduce a unified perspective on the relationship between latent space and generative models, emphasizing the stability of latent space in image generative modeling. Subsequently, we propose a simple but effective discriminative image tokenizer, followed by an image autoregressive generative model DiGIT. Empirical results indicate that our tokenizer achieves superior performance across both image understanding and image generation tasks. Notably, when DiGIT is scaled up in model size, it exhibits even greater enhancements, indicating the potential for the development of large vision models. Our findings challenge the conventional wisdom that proficiency in reconstruction equates to an effective latent space for auto-regressive generation. Through this work, we aim to rekindle interest in the generative pre-training of image auto-regressive models and encourage a reevaluation of the fundamental components that define latent space for generative models. 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[43] Jinghao Zhou, Chen Wei, Huiyu Wang, Wei Shen, Cihang Xie, Alan Yuille, and Tao Kong. Image BERT pre-training with online tokenizer. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=ydopy-e6Dg. 14 A Appendix A.1 Limitation and Broader Impact Our proposed discriminative tokenizer exhibits remarkable capabilities in both the image understanding and the image generation tasks, which is challenging for a single model. However, the discriminative tokenizer induced by the SSL model cannot directly render pixels. Consequently, we need to train another decoder model to convert tokens from discriminative tokenizers to VQGAN tokens. The potential for a direct generation of RGB pixels from the tokens produced by the discriminative tokenizer remains an uncharted avenue, which we leave for future work. Image generative models possess a dichotomous nature, particularly within the realm of visual media. On the positive side, they foster a myriad of creative endeavors, and methodologies aim to minimize the costs of training and inference, holding the potential to broaden access and democratize the use of this technology. On the negative side, the simplicity with which manipulated content can be crafted and propagated raises serious concerns. This includes the proliferation of misinformation and spam. Furthermore, generative models may inadvertently unveil aspects of their training data, a particularly troubling issue when that data includes sensitive or personal information collected without explicit consent. A.2 Proof of claims in Section 2 Proposition A.1. Consider the optimization problem for X ∈Rn: min W1∈Rm×n,W2∈Rn×m ∥X −W2W1X∥2 F , (9) which is a linear autoencoder. W2 is a minimizer of the problem if and only if its column space is spanned by the first m loading vectors of X. Proof. First, we derive the condition for the optimal W1 in the context of this optimization problem. Setting the gradient of the objective function with respect to W1 to zero leads to W1 being the left Moore-Penrose pseudoinverse of W2 [2]. Similarly, setting the gradient with respect to W2 to zero would identify W2 as the right pseudoinverse of W1: W1 = W † 2 = (W T 2 W2)−1W T 2 . (10) This finding indicates that the optimization can be simplified to focus on a single matrix (either W1 or W2), hence removing the redundancy in the parameters: min W2∈Rn×m ∥X −W2W † 2 X∥2 F . (11) The term W2W † 2 = W2(W T 2 W2)−1W T 2 is recognized as the matrix form of the orthogonal projection operator onto the column space of W2. This property holds true even when the column vectors of W2 are not orthonormal. By performing QR decomposition on W2, W2 = QR where Q is an orthogonal matrix (QT Q = I) and R is an upper triangular matrix, we effectively transform the problem into optimizing over orthogonal matrices. The objective can thus be restated as: min W ∈Rm×n ∥X −W T WX∥2 F , subject to WW T = In×n. (12) This revelation explicitly demonstrates that minimizing the reconstruction error in the space Rm×n demands that W (equivalent to W2 in our context) projects X onto a space spanned by its most significant structural components (in terms of variance), which are precisely the first m loading vectors, or principal components, of X. Proposition A.2. The discriminative self-supervised model learns to separate data distributions in latent space as LDA in principle. Proof. We first consider the objective of Fisher LDA for two-class classification. LDA seeks to find a linear projection that maximizes the separation between the two classes: 15 J(w) = wT SBw wT SW w, (13) where SB is the between-class scatter matrix, SW is the within-class scatter matrix, and w is the projection vector. We take contrastive learning with InfoNCE for analysis because it is the most popular learning objective in discriminative SSL models. For contrastive learning using the asymptotic form of the InfoNCE objective [39], we have two components: L = −1 τ E(x,x+)∼ppos f(x)⊤f(x+) + Ex∼pdata h log Ex−∼pdata h ef(x)⊤f(x−)/τii , (14) where the first term encourages similarity between positive pairs. In LDA, this is analogous to minimizing SW because we want points from the same class to be close to each other when projected onto w. The second term, when expanded using Jensen’s inequality, represents an upper bound on the regularized sum of all pairwise inner products between different embeddings, effectively encouraging dissimilarity between all samples: Ex∼pdata h log Ex−∼pdata h ef(x)⊤f(x−)/τii = 1 m m X i=1 log  1 m m X j=1 eh⊤ i hj/τ  ≥ 1 τm2 m X i=1 m X j=1 h⊤ i hj. (15) When hi = f(xi) are normalized, optimizing this term aims to decrease Sum(WW⊤) = Pm i=1 Pm j=1 h⊤ i hj. This term, being an upper bound for the largest eigenvalue of WW⊤, when minimized, encourages a "flatter" singular value distribution of the embedding space. This flattening makes the embedding space more isotropic, which in turn increases the between-class scatter SB. To make the connection explicit, consider that in LDA, we want to maximize SB, the between-class scatter, which is typically represented as: SB = (µ1 −µ2)(µ1 −µ2)T , (16) where µ1 and µ2 are the means of the two classes. Analogously, in the contrastive learning framework, minimizing the second term scatters the embeddings more uniformly in the high-dimensional space, akin to maximizing between-class scatter. The uniform distribution of the embeddings across the space increases the separation between classes or clusters of points, akin to the effect of maximizing SB. A.3 Implementation Details We extract the hidden states from the third-to-last layer of DINOv2 for discriminative tokenizer training, as it suggests more generalized representations in the intermediate layers compared to the last layer. All DiGIT models are trained from scratch with a batch size of 2048 for the base model and 1024 for the large model, over a duration of 200 epochs. For the image generation task, the decoder model for pixel rendering is trained with a batch size of 2048 for 400 epochs, while the large model uses a batch size of 1024 for 200 epochs. The base model consists of 16 layers with a dimension of 1024 and a hidden dimension of 4096. The large model consists of 24 layers with a dimension of 1536 and a hidden dimension of 6144. Both configurations feature 16 attention heads. For all models, we utilize the Adam [22] optimizer with β = (0.9, 0.98). We employ an inverse square root decay schedule for the learning rate with a peak value of 3e −4 for 10 epochs warm-up. We use the same VQGAN model as MAGE Li et al. [26]. During training, we apply data augmentations including random crops resized to the short edge and horizontal flips. For the decoding strategy, we apply top-k sampling with K = 400 for the large model and K = 200 for the base model. In the autoregressive decoder for pixel rendering, we use a probability of 0.8 with a temperature of 1.0 for top-p sampling. All experiments are conducted using the FAIRSEQ [29] library. 16 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 top-p value 0.0 50.0 100.0 150.0 200.0 FID FID DiGIT FID stage2 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 Inception Score Inception Score DiGIT Inception Score stage2 (a) 0 100 200 300 400 500 600 700 800 900 1000 top-k value 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 FID FID DiGIT FID stage2 40.0 60.0 80.0 100.0 120.0 140.0 Inception Score Inception Score DiGIT Inception Score stage2 (b) Figure 5: FID and Inception Score as a function of top-k, top-p sampling on the image generation task with DiGIT-base. The decoding temperature is fixed to 1.0. The “stage2” denotes the autoregressive model for pixel rendering. 2 0 2 4 Feature 1 2 0 2 4 6 Feature 2 Original Data 4 2 0 2 4 Principal Component 0 10 20 30 40 50 60 70 80 Frequency PCA Reduction Class 1 Class 2 4 2 0 2 4 6 LDA Component 0 10 20 30 40 50 60 70 80 Frequency LDA Reduction Class 1 Class 2 0.0 0.2 0.4 0.6 0.8 1.0 Noise Level 0.6 0.7 0.8 0.9 1.0 Classification Accuracy PCA vs LDA Acc. PCA LDA Figure 6: Toy example of PCA and LDA. 17 A.4 Qualitative Cases Figure 7: Class-unconditional image generation results on ImageNet 256×256 by DiGIT. 18 Figure 8: Class-conditional image generation results on ImageNet 256×256 by DiGIT, where the images in the same row share the same class label. 19 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: Please refer to Section 6. 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4,479	AdaPKC: PeakConv with Adaptive Peak Receptive Field for Radar Semantic Segmentation Teng Li2∗ Liwen Zhang1∗† Youcheng Zhang1 Zijun Hu1 Pengcheng Pi1 Zongqing Lu2 Qingmin Liao2† Zhe Ma1 1Intelligent Science and Technology Academy of CASIC 2Shenzhen International Graduate School, Tsinghua University liteng21@mails.tsinghua.edu.cn∗ lwzhang9161@126.com∗† liaoqm@tsinghua.edu.cn† Abstract Deep learning-based radar detection technology is receiving increasing attention in areas such as autonomous driving, UAV surveillance, and marine monitoring. Among recent efforts, PeakConv (PKC) provides a solution that can retain the peak response characteristics of radar signals and play the characteristics of deep convolution, thereby improving the effect of radar semantic segmentation (RSS). However, due to the use of a pre-set fixed peak receptive field sampling rule, PKC still has limitations in dealing with problems such as inconsistency of target frequency domain response broadening, non-homogeneous and time-varying characteristic of noise/clutter distribution. Therefore, this paper proposes an idea of adaptive peak receptive field, and upgrades PKC to AdaPKC based on this idea. Beyond that, a novel fine-tuning technology to further boost the performance of AdaPKC-based RSS networks is presented. Through experimental verification using various real-measured radar data (including publicly available low-cost millimeter-wave radar dataset for autonomous driving and self-collected Ku-band surveillance radar dataset), we found that the performance of AdaPKC-based models surpasses other SoTA methods in RSS tasks. The code is available at https://github.com/lihua199710/AdaPKC. 1 Introduction As a common remote sensing device, radar exhibits superior robustness in complex environments (e.g., varying weather and lighting conditions) compared to cameras, and it is more cost-effective and resilient in extreme weather scenarios compared to LiDARs. Benefiting from the physical advantages of radar sensors and the powerful capabilities of deep learning techniques, modern deep learning-based radar signal interpretation has become a hot research topic in the field of radio frequency detection technology. It has been extensively explored in autonomous driving [30, 19, 34, 6], UAV surveillance [9, 17], sea monitoring [27, 21, 28], etc. Considering the similar dense representations between radar frequency maps and optical images, most of these works directly transfer convolution networks or modules developed for optical signals to radar perception tasks, such as radar object detection (ROD) and radar semantic segmentation (RSS), and they have achieved impressive performance. Nevertheless, without specific design for the inherent characteristics of radar signals, these approaches fail to fully liberate the potential of deep learning techniques. Recently, PKCIn-Net [32] introduced an innovative convolution operator named PeakConv (PKC), tailored for the efficient analysis of radar signals, and this operator seamlessly integrates the advan∗Equal contribution. †Corresponding author. This research is supported by Young Science Foundation of National Natural Science Foundation of China (No.62206258). 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Car Car Cyclist Car Car Cyclist Car Cyclist Car (a) Synchronized camera image of the first frame (b) Radar frequence map Figure 1: The illustration of variations in target signature and interfering signals in radar frequency map. The first row illustrates the variations of the same target across temporally consecutive frames in the range-Doppler (RD)-amplitude 3D representation. The second row demonstrates the disparities of different targets in the same frame, as well as the same target across different frames, in the range-angle (RA) 2D representation. Cyan and yellow rectangles represent target areas, illustrating the variations of target signature with dimensions, categories, and time, etc. Red ellipses indicate prominent interfering clutter, while purple ellipses represent clutter undergoing significant changes. tages of classic radar detectors [22] with common convolution networks [19, 6]. For radar signals, the frequency responses of objects comprise target echoes and interference, and share a distinct peak-shaped pattern, thus most classic radar detection methods [22, 10, 25, 23] build peak detection algorithms upon constant false alarm rate (CFAR) criteria. Extending from cell averaging-CFAR (CA-CFAR) [22], PKC explicitly embeds a similar band-pass peak enhancement mechanism in a standard convolution operator for better characterising target signatures in radar signals. Concretely, following the guard-reference policy of CA-CFAR, it first estimates interfering signals with the center unit/cell and reference units outside predefined guard bands. Then, with estimated interference, it finishes noise suppression for each cell under test (CUT) in feature space and enhances peak frequency response associated with objects of interest. Despite its superior suitability for radar data than alternative convolution operators [31, 5, 33], there exists even greater potential for PKC to learn peak frequency response of radar signals. Research on CFAR detectors [10, 25, 23, 11] reveals that there exist significant variations in target signature and associated interference within radar signals, rendering the predefined reference cells in CA-CFAR inadequate for precisely locating interfering signals, and this limitation can also be observed in PKC. To provide a clearer depiction, let us delve deeper into these variations present within radar signals, as illustrated in Fig. 1. On the target side, since multi-dimensional radar tensors are generated through a sequence of cascading fast Fourier transformations (FFTs), target signatures along different dimensions exhibit distinct degrees of frequency response tailing (broadening). Additionally, the broadening degrees of different instances would also be influenced by target categories or states, i.e., the relative distances, azimuth or velocity from the radar. On the interference side, the noise or clutter distribution commonly exhibits non-homogeneous and time-varying characteristics. However, since the PKC kernel always gathers the reference units at fixed locations for noise estimation, i.e., the predefined peak receptive field (PRF), the dynamic variations in both targets and interference degrade its performance. In short, the fixed PRF essentially limits the learning ability of PKC, thus, it hinders the RSS model from obtaining better performance. Motivated by the adaptive selection of reference cells in classic CFAR detectors [10, 25, 23], in this work we introduce two novel data-adaptive band-pass filtering mechanisms aimed at upgrading PKC to adaptively adjust its PRF for each CUT in a data-driven manner, namely adaptive PeakConv (AdaPKC). Concretely, both versions of AdaPKC first measure the correlation between CUT and its 2 alternative reference units in high-dimensional feature representations, then select proper reference units and integrate them seamlessly with PKC to effectively take care of the fluctuating dynamics of radar signals. The main contributions of our work are: • We present the first attempt specially tailored for radar signal processing to dynamically adjust the receptive field for convolution operators. Concretely, we propose a novel updated version of PKC, termed as AdaPKC, which can adaptively adjust the PRF (or reference units) at cell-level granularity. And two different implementation versions are provided, which both exhibit enhanced flexibility and robustness in handling fluctuating radar signals compared to original PKC. • To better release the learning ability of AdaPKC, a fine-tuning technology with a thresholding on-line switch is presented. With such technology, the same AdaPKC-based model can even achieve better performance with less computational cost. • To verify the effectiveness of AdaPKCs, quantitative and qualitative experiments are conducted on various real-measured large scale radar datasets including CARRADA [20] collected from a low-cost FMCW (≈77GHz) radar in autonomous driving scenario and self-collected dataset recorded from a Kurz-under (Ku) band (≈17GHz) radar for UAV surveillance and sea monitoring. Results show that AdaPKC-based models achieve SoTA RSS performance and our fine-tuning strategy further brings visible improvements, verifying the scope of application of AdaPKCs. 2 Related Work Receptive field (RF) adjustment. RF is crucial for modern deep convolution models, affecting the granularity of modeling primitives, computation architecture and their representation capabilities, etc. The rational use of RF can directly improve the representation ability of the models, e.g. enlarging the scope [31], multi-scale modeling [2], and dynamically changing shapes [5, 33]. Beyond that, the concept of RF has also been applied to Transformers [7, 16]. These methods are proposed for vision tasks and have achieved significant results. However, compared with conventional convolution, the improvement is not satisfactory enough on radar signals. Recently, by fully considering the characteristics of radar signals, the concept of PRF has been proposed, which makes the original convolution have the ability of band-pass filtering and noise suppression [32]. However, the bandwidth of filtering is pre-set, which hinders the adaptive ability of PKC to radar data. To this end, this paper attempts to study a data-driven PRF adjustment method, and introduces the concept of adaptive PRF (AdaPRF), so as to further improve the RSS performance. Considering that the adaptive adjustment of the suppression (guard) bandwidth and the non-suppression (reference) units is essentially a dynamic adjustment of RF, and from this point of view, the content studied in this paper is related to deformable convolution (DefConv) [5, 33]. Unfortunately, the dynamic RF technology of DefConvs cannot solve the problem in hand, for the following reasons: i) Differences of signaling mechanism. DefConv uses the visual prior information that the target is visually deformed geometrically, which cannot be directly corresponded to the radar signal. ii) Mismatched prediction method. DefConv generates new RF through prior prediction, i.e., the regular RF is still used to infer the sampling point outside regular RF. However, in radar signal, the interference (noise/clutter) with large entropy, is often difficult or even impossible to predict. At present, a better way is under the premise of observation, i.e., posterior measurement or statistics. iii) Different mechanisms of representation. Noise suppression is not required to be considered by DefConv, thus it does not need to distinguish between the center unit and surroundings during calculation. To this end, a novel adaptive RF adjustment method is required. Radar semantic segmentation. Benefiting from the reliable perceptual capabilities, convolutional neural networks (CNNs) play an indispensable role in existing RSS networks for radar frequency maps processing, whether in pure CNN models [13, 8, 19, 32] or transformer-assisted CNN models [34, 12, 6]. RSS-Net [13] utilizes a fully convolutional neural network with encoder-decoder structure to recognize targets in radar scans, and it incorporates an atrous spatial pyramid pooling (ASPP) [2] module to gather multi-scale spatial information. RAMP-CNN [8] employs parallel branches to extract features from multiple views and adopts 3D convolutions to better capture temporal information. TMVA-Net [19] leverages these techniques to develop a multi-view RSS model, which is capable of making semantic predictions across multiple views simultaneously. T-RODNet [12] integrates Swin Transformer [16] modules into a CNN-based RSS model to strengthen its modeling capability. TransRSS [34] and TransRadar [6] introduce attention blocks into the multi-view feature fusion stage to enhance the fusion effectiveness. Recently, as the first fundamental convolution operator tailored for radar signal processing, PKC [32] is proposed. Compared to Dilated Convolution [31] 3 and Deformable ones [5, 33], PKC demonstrates superior RSS performance. However, it is inherently constrained by its fixed peak receptive field, posing challenges in achieving consistent interference (noise/clutter) suppression under the dynamic and time-varying nature of radar signals. By contrast, our AdaPKCs overcome this limitation well by implementing novel data-adaptive band-pass filtering mechanisms. 3 Method In this section, the proposed two versions of AdaPKC with different AdaPRF mechanisms are first introduced in § 3.1. Then, the proposed fine-tuning strategy to further uncover the potential of AdaPKC-based RSS models is presented in § 3.2. We design these models using both multiview [19, 32] and single-view frameworks, which are comprehensively described in Appendix A.2 and A.3 for saving space. 3.1 AdaPKC 3.1.1 AdaPKCξ: PKC w/ Metric-based AdaPRF To ensure the reliability of estimating the proper unit-level PRF, i.e., AdaPRF, for AdaPKC in radar signals, we establish the estimation process in a posterior way: we first define a set of candidate PRFs within the neighbourhood of the center unit, aligning with the local peak response of targets and the local scanning process of convolution, and then design a measuring criterion to evaluate these PRFs, estimating AdaPRF primarily occupied by interfering signals. Motivated by classic CFAR detectors [10, 25, 23], we first demonstrate how to estimate AdaPRF in an explicitly measuring way, referred to as metric-based AdaPRF (AdaPKCξ). To illustrate the mechanism of AdaPKCξ, we begin by defining the search space of alternative PRFs. Reviewing the PRF definition in previous work [32] we can see that, the PRF for center unit xc encompasses xc itself and a set of sampled reference units {x(i) r }Nr i=1, and the area of reference units is governed by horizontal- and verticalsymmetry guard bandwidth bG ≜{bG x, bG y } and reference bandwidth bR ≜{bR x, bR y}, as illustrated in Fig. 2-(a). Following this definition, the PRF adjustment corresponds precisely to the adjustment of the reference unit set, thus we can define the PRF search space by defining the candidate sets of reference units with the adjustment ranges for the guard bandwidth and reference bandwidth. Given that adjusting the reference bandwidth leads to a drastic change in the number of sampled reference units compared to adjusting the guard bandwidth, in AdaPKCξ we keep anytime-fixed bR ≜{bR x = 1, bR y = 1} and denote the set of K guard bandwidth candidates as ΩG ≜{bG k}K k=1 = {bG \| bG min \|x ≤bG x ≤bG max \|x, bG min \|y ≤bG y ≤bG max \|y}, generating K sets of reference units as {{x(i) r\|k}Nk i=1}K k=1 correspondingly. As a result, AdaPRF estimation in AdaPKCξ is equivalent to selecting an appropriate reference unit set from these candidate sets for each CUT (or center unit), as illustrated in Fig. 2-(b). For better explanation, we divide the observed radar signals into three subsets: i) signals reflected directly from a target, St; ii) the target-interfering noise, i.e., the noise coupled with the signal that partially leaks out of the target, St-n; and iii) the target-independent noise, Sn. In practice, it is the part that from St-n really causes misjudgment. Therefore, classic CFAR detectors focus on filtering out such noise either by the extreme value [10, 25] or the median value [23] in amplitude domain of signals. Motivated by this idea, our AdaPKCξ centers its attention on collecting reference units predominantly occupied by such noise. However, as a learnable module, AdaPKCξ needs to process the representation tensors of radar signals, implying that the measurement of the reference units should be conducted on feature space. For some CUT xc = ψ(s; W) and its candidate reference unit xr = ψ(s′; W), where s ∈St, s′ ∈St ∪St-n ∪Sn, and ψ(·; W) ∈RC denotes a convolution layer with shared weights, W. Then, AdaPKCξ should be responsible for transforming these features into a metric space that explicitly delineates their correlation with the target. This transformation is achieved by utilizing the inner product of representations between CUT and its candidate reference unit, i.e., xcx⊤ r , sharing a similar spirit with the matched filter concept in conventional radar signal processing [22] and attention in [26]. 4 (b) AdaPRF estimation in 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃 𝑔𝑔(sort Ξ ) K candidate PRFs 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦\|𝒚𝒚 𝐆𝐆 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦\|𝒚𝒚 𝐆𝐆 Input feature map K×H×W K = 𝑏𝑏max \|𝑥𝑥 G −𝑏𝑏min \|𝑥𝑥 G + 1 ⋅𝑏𝑏max \|𝑦𝑦 G −𝑏𝑏min \|𝑦𝑦 G + 1 Estimation criterion AdaPRF Reference unit under confirmation 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦\|𝒙𝒙 𝐆𝐆 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦\|𝒙𝒙 𝐆𝐆 Metric 𝑘𝑘 Selection ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 𝑏𝑏𝑥𝑥G 𝑏𝑏𝑦𝑦G 𝑏𝑏𝑦𝑦R 𝑏𝑏𝑥𝑥R 𝑘𝑘† (a) Fixed PRF in PKC • ★ Cell Under Test (CUT), 𝐱𝐱𝒄𝒄 Guard unit Sampled reference unit, 𝐱𝐱𝒓𝒓 𝜉𝜉1, 𝜉𝜉2, … , 𝜉𝜉K ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 𝑏𝑏𝑥𝑥G 𝑏𝑏𝑦𝑦G 𝑏𝑏𝑦𝑦R 𝑏𝑏𝑥𝑥R 𝐱𝐱𝑟𝑟↑ Reference bandwidth: 𝐛𝐛R ≜𝑏𝑏𝑥𝑥R, 𝑏𝑏𝑦𝑦R Guard bandwidth: 𝐛𝐛G ≜𝑏𝑏𝑥𝑥G, 𝑏𝑏𝑦𝑦G 𝐱𝐱𝑟𝑟↓ 𝐱𝐱𝑟𝑟← 𝐱𝐱𝑟𝑟→ Figure 2: The illustration of AdaPRF in AdaPKCξ. (a) illustrates the definition of PRF in PKC, whose area is governed by the reference bandwidth bR and guard bandwidth bG; (b) describes the estimation process of AdaPRF in AdaPKCξ, including denoting K candidate PRFs for each CUT, translating these PRFs into metric scores {ξk}K k=1, and finally selecting an appropriate PRF as the AdaPRF with these metric scores. Under this definition, these measures exhibit the following statistical properties, E xcx⊤ r =    E ∥ψ(s; W)∥2 2 , if s′ ∈St E ∥ψ(s′; W)∥2 2 , if s′ ∈St-n 0, if s′ ∈Sn , (1) where, E(·) is the expectation and ∥· ∥2 denotes the L2 norm. From Eq. (1), we can see that the inner product transformation assigns three statistical boundaries to xr from St, St-n and Sn: for s′ ∈St-n, the expectation E xcx⊤ r consistently exhibits smaller value than case for s′ ∈St and larger value than s′ ∈Sn, with a notable separation between their respective magnitudes. This attribute significantly serves to facilitate the subsequent localization of reference units from target-interfering noise. Then we elucidate the process of translating the K available sets of reference units (or PRFs) into the previously discussed metric space. Since different sets may comprise varying numbers of units, we uniformly sample N (16 by default) units as representatives, as illustrated in Fig. 2-(b). For the center unit xc and its kth reference unit set {x(i) r\|k}N i=1, let Rk = {xc, {x(i) r\|k}N i=1} denote its corresponding PRF. Then the correlation value (or metric score), ξk for the kth PRF w.r.t. xc is formulated as ξk = 1 N N X i=1 σ xcx(i)⊤ r\|k /C , (2) where, C is the feature dimension; σ(·) is the sigmoid function which normalizes ξk to (0, 1). With the correlation values Ξ = {ξk}K k=1 for all alternative PRFs, we can select the appropriate PRF R†, which effectively encompasses target-interfering noise. In view of the attribute presented in Eq. 1, we employ the maximum value of the first-order gradient of Ξ as the selection criterion and please refer to Appendix B.1 for the detailed analysis. Then, we have final selection strategy as follows, R† ≜Rk† k=k† ←−{xc} ∪{x(i) r\|k}N i=1, s.t., k† = arg max k {g [sort (Ξ)]} , (3) where arg max operator retrieves the index corresponding to the maximum value in g [sort (Ξ)]; g is the difference function; sort is the descending sort operator. After obtaining R†, AdaPKC performs a convolution operation similar to PeakConv, detailed in Appendix A.1. 3.1.2 AdaPKCθ: PKC w/ Learning-based AdaPRF In AdaPKCξ, a measuring criterion is established in Eq. 3 based on prior knowledge of radar signals, providing a non-parametric way to achieve AdaPRF. Differing from AdaPKCξ, AdaPKCθ learns to 5 7× 1 Conv 1× 7 Conv BN BN 2×H×W Input feature map Guard unit under confirmation Optimal guard bandwidth 2×H×W 𝒃𝒃↑ 𝐆𝐆 𝑏𝑏↑ G 𝑏𝑏↓ G 𝑏𝑏←G 𝑏𝑏→G Modulated sigmoid function 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦 𝐆𝐆 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦 𝐆𝐆 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦 𝐆𝐆 𝒃𝒃𝐦𝐦𝐦𝐦𝐦𝐦 𝐆𝐆 𝑏𝑏min G 𝑏𝑏max G 𝑏𝑏min G 𝑏𝑏max G 𝒃𝒃↓ 𝐆𝐆 𝑏𝑏max G 𝑏𝑏min G 𝑏𝑏max G 𝑏𝑏min G 𝒃𝒃← 𝐆𝐆 𝒃𝒃→ 𝐆𝐆 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 𝑏𝑏→G 𝑏𝑏↓ G 𝑏𝑏𝑦𝑦R 𝑏𝑏𝑥𝑥R 𝑏𝑏←G 𝑏𝑏↑ G (a) Example of candidate PRF Guard bandwidth: 𝐛𝐛G ≜𝑏𝑏↑ G, 𝑏𝑏↓ G, 𝑏𝑏←G, 𝑏𝑏→G 𝐱𝐱𝑟𝑟← 𝐱𝐱𝑟𝑟→ ℛ= 𝐱𝐱𝑐𝑐, 𝐱𝐱𝑟𝑟↑}, 𝐱𝐱𝑟𝑟↓, 𝐱𝐱𝑟𝑟←, {𝐱𝐱𝑟𝑟→ Candidate PRF: 𝐱𝐱𝑟𝑟↑ 𝐱𝐱𝑟𝑟↓ (b) Estimation network for optimal guard bandwidth • ★ Cell Under Test (CUT), 𝐱𝐱𝒄𝒄 Guard unit Sampled reference unit, 𝐱𝐱𝒓𝒓 Figure 3: The illustration of AdaPRF in AdaPKCθ. (a) illustrates an example of candidate PRFs in AdaPKCθ, where the guard bandwidth bG is in a quadruple form; (b) describes the flowchart of the optimal guard bandwidth estimation network, which consists of two parallel branches that sample representative points in their corresponding directions and then automatically measure and select the optimal guard bandwidth. build the criterion in a task-driven manner, employing a small network to estimate AdaPRF, i.e., a parametric way. Thus, given center unit xc and its K candidate PRFs, {Rk}K k=1, the natural way to locate AdaPRF R†, is to define a function f(·; θ) with learnable parameters θ, which is used to produce the likelihood of each Rk being R†, then we can have R† ≜Rk†, where k† = arg max k {f(Rk; θ)}K k=1 . (4) At a rough glance, there is no obvious difference between Eq. 4 and Eq. 3. However, AdaPKCθ involves joint optimization of estimation network parameters (i.e., θ) and segmentation model parameters. The use of arg max operation will lead to the loss of gradient information of θ, as a result, the optimization of the estimation network cannot be driven by the segmentation task. To this end, we transform the discrete estimation problem into a continuous form. Firstly, following the setup of anytime-fixed bR ≜{bR x = 1, bR y = 1} in AdaPKCξ, estimating R† can be equivalently translated into estimating the optimal guard bandwidth bG†. Subsequently, to ensure that the estimation of bG† retains gradient information, the estimation network is designed to generate the continuousvalued bG†, instead of the likelihoods for alternative guard bandwidths. Finally, the derivable linear interpolation is used to associate the continuous-valued bG† with discrete spatial coordinates. Concretely, given an input feature map X ∈RC×H×W , our AdaPKCθ is responsible for obtaining an appropriate guard bandwidth BG† = {bG† h,w}H,W h=1,w=1 ∈R4×H×W . For enhancing the expressive capability of AdaPKC, the horizontal- and vertical-symmetry bG design in the original PRF is extended to a quadruple form, bG = {bG ↑, bG ↓, bG ←, bG →}, so that the shape of PRF would enjoy free change in four directions, i.e., top, bottom, left, right, as illustrated in Fig. 3-(a), resulting in more diverse band-pass filters. As shown in Fig. 3-(b), we use a small network with two paralleled conv blocks, gHrz(·) : RC×H×W →R2×H×W and gVtc(·) : RC×H×W →R2×H×W to estimate bG ∈R4 for each unit of X in horizontal (←→) and vertical (↑↓) directions, respectively. To ensure the directional consistency, the horizontal branch possesses kernels with a size of 1 × (2bG max + 1), and the kernel size of the vertical branch is (2bG max + 1) × 1, correspondingly. All four directions have the same lower bound, bG min = 1 and the same upper bound, bG max = 3 by default. Then, for each xc, its AdaPRF R† = {xc, {x(i)† r }N i=1} (N = 16 by default) can be obtained by the following two steps: Step 1. Guard bandwidth estimation: BG† = (bG max −bG min) · σ β gVtc(X) ⊕gHrz(X) + bG min · 1, (5) 6 where, each bG† h,w in BG† is the learned guard bandwidth for xc of the spatial coordinate (h, w); together with a constant bG max −bG min, the sigmoid activation σ(·) can modulate the input values to (0, bG max −bG min); β(·) and ⊕denotes the BatchNorm and Concat operator, respectively; 1 ∈ Z4×H×W is an all-one cube. Step 2. Reference unit sampling: for better illustration, we first define the default PRF with guard bandwidths equal to 1 in [32] as R0 = {xc, {x(i) r }N i=1}, which is shown in Fig. 2-(a). let pc = (hc, wc) and p(i) r = (h(i) r , w(i) r ) denote the spatial coordinate of the current center unit xc and one of its reference unit x(i) r in R0, respectively. Then we split {x(i) r }N i=1 into four subsets as {x(j) r↑}M j=1, {x(j) r↓}M j=1, {x(j) r←}M j=1 and {x(j) r→}M j=1, according to the four directions, where M = N/4, as indicated by the braces drawn in Fig. 2-(a). By using bG† obtained from Step 1, we can sample R† = {xc, {x(i)† r }N i=1} through linear interpolation, as exemplified in Fig. 3-(a). Taking the top direction as an example, each x(j)† r↑ can be obtained by x(j)† r↑ = h 1 − bG† ↑−⌊bG† ↑⌋ i · xh(j) r↑−⌊bG† ↑−1⌋,w(j) r↑+ bG† ↑−⌊bG† ↑⌋ · xh(j) r↑−⌈bG† ↑−1⌉,w(j) r↑. (6) Sampling values for the rest directions and back-propagation of gradients in linear interpolation are demonstrated in Appendix B.2. 3.2 Fine-tuning AdaPKC with Thresholding On-line Switch To further release the learning ability of AdaPKC, a fine-tuning strategy is proposed. To improve interpretability, we focus mainly on optimizations for the explicit AdaPRF estimation version, i.e., AdaPKCξ. The motivation comes from two perspectives. i) Model confidence: the representation ability and confidence of the model gradually improve with training, hence, during early training phase the less representative features may result in unreliable metric scores calculated in Eq. 2, consequently affecting the selection of AdaPRF and misleading the subsequent learning process. ii) Data sparsity: AdaPKC is responsible for the PRF adaptation for both background units (xc /∈St) and target units (xc ∈St). However, due to the sparsity of target occupation w.r.t. the radar detection range, most units in the input feature map are background units, thus the heavy class-imbalance hinders the model’s concentration on target points and their AdaPRF optimization. To address the above issues, we propose to fine-tune AdaPKCξ with a thresholding on-line switch (FiTOS), and please refer to Appendix C for a visual illustration. Firstly, a pre-trained PKC model with pre-defined PRFs is used to initialize the AdaPKC-based model to be optimized, so that AdaPKC can have a warm-start before PRF adjustment. Thus, the risk of obtaining unreliable metric scores is greatly reduced. Secondly, considering that the majority of background units are occupied by locally similar noise, i.e., their monotonic correlation/metric curves, sort(Ξ), are relatively flat, FiTOS introduces a confidence threshold τ to filter out these background units in spatial dimension on-the-fly. Specifically, if the steepness of the metric curve, i.e., max{g [sort (Ξ)]}, is below the threshold τ, then the corresponding unit is considered a background unit, and we retain the initial PRF, i.e., switch off PRF adjustment. Otherwise, we adopt the newly estimated AdaPRF, i.e., switch on PRF adjustment. As a result, the PRF selection criterion in Eq. (3) is modified as follows, R† ≜Rk† k=k† ←−{xc} ∪{x(i) r\|k}N i=1, s.t., k† = ( arg max k {g [sort (Ξ)]} , if max{g [sort (Ξ)]} > τ k0 , otherwise , (7) where k0 denotes the index of pre-defined PRF in the pre-trained model. 4 Experiments To verify the effectiveness of our methods, we conduct quantitative and qualitative experiments on two public multi-view radar datasets and our self-collected single-view radar dataset. For simplicity of notations, AdaPKCξ-based and AdaPKCθ-based multi-view RSS model is denoted as AdaPKCξ-Net and AdaPKCθ-Net, respectively. The proposed single-view baseline model is named KuRALS-Net. Additionally, we append “FiT” in the upper right corner of the models to indicate the use of FiTOS. 7 4.1 Datasets and Training Setups CARRADA [20] dataset is recorded by a low-cost FMCW radar in millimeter wave band (≈77GHz). It comprises camera-radar synchronised multi-view radar recordings in various scenarios, and contains four categories of objects: pedestrian, cyclist, car and background. The dimensions of provided range-angle-Doppler (RAD) tensors are 256 × 256 × 64 and support multi-view (RA and RD views) RSS task. The dataset splits are the same as in [32, 19]. CARRADA-RAC [32] dataset is derived from CARRADA and mainly calibrates the original RA annotations, see Appendix D.1 for details. KuRALS dataset is self-collected by a Kurz-under band (≈17GHz) surveillance Radar, which is recorded in multiple scenarios, including Aerial vehicles, Land targets and ships on the Sea surface. Different from other public datasets, e.g., CARRADA [20] and CRUW [29], KuRALS aims at exploring the performance of deep models in the field of monitoring radar, hence offering a greater range field (≦6.4km) and higher Doppler resolution (≈0.198m/s). The comprised RD tensors are stored as 2D matrices of size 2048 (range) by 128 (Doppler). This dataset contains 9 sequences of radar recordings and there exist four moving object categories: UAV, pedestrian, vehicle and ship. Training Setups. Following previous works, all models are evaluated with Intersection over Union (IoU) and Dice scores. These metrics are averaged across all classes on the test subset for model performance comparison, yielding mean IoU (mIoU) and mean Dice (mDice). Implementation details are presented in Appendix D.2. 4.2 Investigation of AdaPKC Mechanism To investigate the working mechanism of AdaPKCs, a series of comparison analysis is conducted on CARRADA benchmark. We first compare AdaPKCs with PeakConv and a manual PRF adjustment method. Results in Tab. 1 demonstrate that, both versions of AdaPKC exhibit significant enhancements in RSS performance compared to PeakConv, especially in the RA view, and they incur affordable additional computational complexity and inference speed overhead. Considering the severer signal tailing effect in RA view, this suggests that AdaPKC can better handle situations with frequency response ambiguity. For better illustration, we analyze the distribution of guard bandwidths in Appendix E.1. Additionally, since manually adjusting PRFs directly affects the RSS performance of PeakConv-based models, as discussed in [32], in this work we undertake a similar exploration experiment within a broader range of guard bandwidths. Specifically, different guard bandwidths in range dimension, bG R ∈{1, 2, 3, 4, 5}, are tested, while the guard bandwidths in angle and Doppler dimensions are fixed at 1 for controlling variables. To ensure consistent parameter counts under different bandwidth settings, we adopt the same strategy of uniform sampling as in AdaPKCξ. As shown in Tab. 2, presetting a proper PRF globally in a hyper-parameter way can indeed help the PeakConv-based model achieve better performance. However, the manual adjustment way cannot cater to each unit, and the computational cost of traversing the guard bandwidth in all dimensions and directions is quite large. In contrast, AdaPKC completely automates the PRF adjustment, and the adjustment granularity reaches the unit-level. With this unit-level PRF adaptation capability, AdaPKCs demonstrate superior RSS performance compared to the manual adjustment way. Table 1: Comparison between AdaPKCs and PeakConv. The best and secondary results are marked with bold and underline, correspondingly. Frame rate is calculated on a workstation with an Intel(R) Xeon(R) Platinum 8255C CPU and a Tesla V100-SXM2 GPU. Conv Type #Params @Frames RD View RA View GMACs FPS mIoU mDice mIoU mDice PeakConv 6.3M@5 60.7% 72.6% 43.1% 53.7% 109.8 21.1 AdaPKCξ 6.3M@5 61.2% 73.1% 44.1% 55.1% 109.8 20.7 AdaPKCθ 6.3M@5 61.5% 73.6% 43.6% 54.5% 110.1 18.8 Furthermore, a comparative analysis between AdaPKCs and DefConvs (DefConv and DefConvV2) [5, 33] is conducted under the same RSS framework. Results in Tab. 3 show that, sharing the similar unitlevel dynamic RF adjustment spirit with AdaPKC, DefConvs demonstrate better RSS performance than regular convolution (Conv). However, due to the three task-mismatched reasons discussed in § 2, DefConvs exhibit inferior applicability compared to AdaPKC, highlighting AdaPKC’s suitability for 8 Table 2: The effectiveness of manual PRF adjustment in PKC. bG R represents guard bandwidth in range dimension. bG R RD View RA View mIoU mDice mIoU mDice 1 60.7% 72.6% 43.1% 53.7% 2 61.0% 73.0% 43.3% 54.1% 3 60.7% 72.7% 42.5% 53.2% 4 59.7% 71.5% 43.0% 53.8% 5 60.4% 72.5% 42.7% 53.3% Table 3: Comparison between AdaPKCs and DefConvs (DefConv and DefConvV2). Conv Type #Params @Frames RD View RA View mIoU mDice mIoU mDice Conv 5.6M@5 56.1% 68.0% 37.7% 46.2% DefConv 5.7M@5 58.0% 69.8% 39.1% 48.1% DefConvV2 5.8M@5 58.8% 70.6% 39.3% 48.6% AdaPKCξ 6.3M@5 61.2% 73.1% 44.1% 55.1% AdaPKCθ 6.3M@5 61.5% 73.6% 43.6% 54.5% achieving adaptive receptive fields in radar signals. For supplementary purposes, we also show the comparison results between AdaPKC and dynamic CFAR detectors [10, 25, 23] in Appendix E.2. As further explorations, we present an investigation into adaptive sampling strategies for reference band in Appendix E.3. Additionally, in Appendix E.4 we demonstrate a training strategy to enlarge the search space of alternative PRFs under fixed resource constraints for AdaPKCξ. 4.3 Comparison with State-of-The-Art (SoTA) Our methods are further compared with fashionable visual segmentation models and existing SoTA RSS solutions. The quantitative results on the CARRADA benchmark are illustrated in Tab. 4 and the qualitative comparisons are presented in Appendix F.3. Both AdaPKCξ-Net and AdaPKCθ-Net outperform previous RSS models, including pure CNN models [13, 8, 19, 32] and transformerassisted CNN models [12, 6]. Compared to the baseline model PKCIn-Net, AdaPKCξ-Net exhibits improvements in both RD and RA views, with a particularly notable enhancement in the RA view. AdaPKCθ-Net purely relies on task-driven PRF adjustment, achieving a better performance balance between two views. Additionally, without consuming extra training resources, our proposed FiTOS strategy further enhances RSS performance of AdaPKCξ-Net, achieving an overall superiority over AdaPKCθ-Net. Table 4: SoTA comparison on CARRADA benchmark. ’-’ denotes an unreported and not replicable value. Detailed results by category are presented in Appendix F.1, and the trade-off between performance and complexity of these models is illustrated in Appendix F.2. Frameworks #Params @Frames RD View RA View mIoU mDice mIoU mDice FCN[18] 134.3M@3 54.7% 66.3% 34.5% 40.9% U-Net[24] 17.3M@3 55.4% 68.0% 32.8% 38.2% DeepLabv3+[1] 59.3M@3 50.8% 61.6% 32.7% 38.3% RSS-Net[13] 10.1M@3 32.1% 36.9% 32.1% 37.8% RAMP-CNN[8] 106.4M@9 56.6% 68.5% 27.9% 30.5% TMVA-Net[19] 5.6M@5 56.1% 68.0% 37.7% 46.2% TransRadar[6] 4.9M@5 57.2% 69.1% 39.9% 49.5% T-RODNet[12] 162.0M@4 43.5% 53.6% PKCIn-Net[32] 6.3M@5 60.7% 72.6% 43.1% 53.7% AdaPKCξ-Net 6.3M@5 61.2% 73.1% 44.1% 55.1% AdaPKCθ-Net 6.3M@5 61.5% 73.6% 43.6% 54.5% AdaPKCξ-NetF iT 6.3M@5 62.1% 74.0% 44.3% 55.5% Table 5: Performance comparison on CARRADARAC. Frameworks #Params @Frames RD View RA View mIoU mDice mIoU mDice TMVA-Net 5.6M@5 59.7% 69.9% 46.6% 57.9% PKCIn-Net 6.3M@5 60.6% 72.4% 47.3% 58.7% AdaPKCξ-Net 6.3M@5 61.6% 73.6% 47.9% 59.3% AdaPKCθ-Net 6.3M@5 60.7% 72.7% 48.1% 59.6% AdaPKCξ-NetF iT 6.3M@5 60.8% 72.7% 48.8% 60.5% Table 6: RSS performance comparison on KuRALS. Detailed results by category are presented in Appendix F.4. Frameworks #Params@Frames mIoU mDice FCN 134.3M@5 50.4% 59.4% U-Net 17.3M@5 52.4% 60.1% DeepLabv3+ 59.3M@5 52.6% 61.8% KuRALS-Net 1.2M@5 56.0% 65.5% KuRALS-Net w/ PKC 1.2M@5 56.7% 65.9% KuRALS-Net w/ AdaPKCξ 1.2M@5 57.3% 67.2% KuRALS-Net w/ AdaPKCθ 1.2M@5 57.8% 67.6% KuRALS-Net w/ AdaPKCξ F iT 1.2M@5 58.2% 67.6% Results on CARRADA-RAC dataset are shown in Tab. 5. Similar to the trend on CARRADA, the evaluation results exhibit the superior performance of our methods than these RSS baseline models. AdaPKCθ-Net still shows a relatively balanced performance improvement in both views. AdaPKCξ-Net demonstrates a more pronounced improvement in the RD view, while 9 AdaPKCξ-NetF iT shows a greater enhancement in the RA view. We speculate that this might be attributed to the collaborative training of different views in the multi-view segmentation task. 4.4 RSS Performance on KuRALS To verify the effectiveness of AdaPKCs in other application scenarios, we conduct comparative experiments on KuRALS dataset. Quantitative results are shown in Tab. 6 and the qualitative comparisons are presented in Appendix F.5. Compared to conventional visual segmentation models, our proposed KuRALS-Net offers a stronger baseline and AdaPKCs further boost the RSS performance of KuRALS-Net, validating their application potential in surveillance radar detection scenarios. 4.5 Ablation Study for FiTOS Since the threshold τ plays a vital role in FiTOS, we study the impact by testing different values of τ from 0.1 to 0.9. The summary results on mDice are shown in Fig. 4 and the corresponding mIoU results are presented in Appendix F.6. The optimal outcome for AdaPKCξ-NetF iT in RD and RA view is obtained with τ = 0.6 and 0.7, respectively. It is observed that the performance of AdaPKCξ-NetF iT declines when τ is either too large or too small. When τ goes larger, fewer target units adjust their PRFs during the fine-tuning stage, rendering AdaPKCξ less effective. Conversely, when τ is small, AdaPKCξ tends to focus on background units with random fluctuations, resulting in training confusion. Compared to PKCIn-Net, AdaPKCξ-NetF iT consistently demonstrates superiority under different τs, highlighting the essentiality of PRF adaptation for the original PKC. When compared to AdaPKCξ-Net, AdaPKCξ-NetF iT exhibits superior performance with most values of τ, demonstrating the efficacy of the fine-tuning strategy. Especially, in RD view, the RSS performance shows significant improvement within a broad range of τ. Taking mDice as an example, it consistently surpasses 73.5% for τ ∈[0.5, 0.8], indicating a strong level of robustness w.r.t. τ. 53.5% 54.0% 54.5% 55.0% 55.5% 56.0% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 mDice RA View 72.5% 73.0% 73.5% 74.0% 74.5% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 mDice RD View TAPKCIn-Net TAPKCIn-Net PKCIn-Net −𝐍𝐍𝐍𝐍𝐍𝐍𝑭𝑭𝑭𝑭𝑭𝑭 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃-Net PKCIn-Net 𝜏𝜏 𝜏𝜏 Figure 4: RSS performance of AdaPKCξ-NetF iT with different values of τ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. AdaPKCξ-Net and PKCIn-Net actually corresponds to the case where τ of AdaPKCξ-NetF iT equals to 0 in whole training process and 1 in the fine-tuning stage, respectively. 5 Conclusion This work delves deeply into the convolution operator for radar signals, PKC, and improves upon it. Due to the design of PRF, PKC filters out the reference units corresponding to interfering signals in a band-pass filtering manner. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019. [34] Hao Zou, Zhen Xie, Jiarong Ou, and Yutao Gao. Transrss: Transformer-based radar semantic segmentation. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pages 6965–6972, 2023. 12 Supplementary Material A AdaPKC-based RSS Frameworks A.1 AdaPKC We formulate the detailed process of how AdaPKC conducts PeakConv on R†. For the simplicity of notation, we uniformly represent the R† acquired by the two AdaPKCs as R† = {xc, {x(i)† r }N i=1}. R† can be readily incorporated into both PeakConv operators proposed in [32], including vanillaPKC and ReDA-PKC. Given ReDA-PKC’s enhanced capability in utilizing interference and target information for noise suppression compared to vanilla-PKC, we choose ReDA-PKC as the baseline operator. Therefore, both AdaPKCs share the same structure as ReDA-PKC and differ only in the generation of R†. Analogous to the formalization of ReDA-PKC, we can formulate AdaPKC R†; W ∈RCin×N×Cout : RCin →RCout for each xc as follows, AdaPKC R†; W = V ec nPN i=1 w(i) j · (xc −x(i)† r )⊤oCout j=1 , where, w(i) j ∈RCin (j = 1, · · · , Cout). (S1) where, W is the general learning weights of AdaPKCs; V ec(·) is the vectorization operator. A.2 Multi-view RSS Framework For multi-view RSS tasks on public radar datasets, in alignment with the design of PKCIn-Net [32], we replace the ReDA-PKC in PKCIn-Net with AdaPKCξ or AdaPKCθ. And this brings two multiview RSS models, denoted as AdaPKCξ-Net and AdaPKCθ-Net correspondingly, whose overall frameworks are depicted in Fig. S1. The remaining components of both AdaPKC-Nets keep consistent with PKCIn-Net and TMVA-Net [19]: i) the designed networks take RD, RA and AD tensors as inputs and conduct semantic segmentation on both RD and RA views concurrently; ii) each encoding branch for a single view comprises two 3D convolution blocks, allowing for the comprehensive utilization of temporal information; iii) an ASPP [2] module is used as a component of the encoding branch to aggregate multi-scale spatial information; iv) after the encoding phase, a Latent Space Encoder (LSE) aids in integrating the encoding features from multiple views and transmits them to the decoding section for the final prediction. A.3 Single-view RSS Framework For single-view RSS tasks on our self-collected radar dataset, we establish a lightweight baseline model called KuRALS-Net, the architecture of which is illustrated in Fig. S2. This model takes temporally contiguous multi-frame RD tensors as inputs, utilizing an Encoder with 3D convolutions to simultaneously gather temporal and spatial information. Then, the encoding features pass through 2D convolution blocks followed by an ASPP module for further information fusion. Finally, the fused features are fed into the Decoder to obtain pixel-wise semantic predictions in the RD view. For the validation of AdaPKC, we replace the 2D convolution blocks in KuRALS-Net with different AdaPKC blocks. B Supplementary Details of AdaPKC B.1 Analysis of Selection Criterion in AdaPKCξ We provide a detailed analysis of the selection criterion adopted in Eq. 3. In Eq. 2, the metric score ξk for the kth PRF is calculated by averaging the metric scores of related reference units {x(i) r\|k}N i=1. For reference units mainly occupied by s′ ∈St-n, the corresponding ξk is statistically smaller than those primarily influenced by s′ ∈St and larger than those controlled by s′ ∈Sn. When it is reflected on the curve of sort(Ξ), ξk is likely to lie near the quickly descending position. In view of this 13 Max-Pool 1ⅹ1 2D Conv Concatenation Multi-View Encoding Flow Latent Space Encoding Flow RD Encoder LSE AdaPKC Block RA Decoder RD Decoder ASPP AD Encoding Branch RA Encoding Branch RD Encoding Branch Multi-frames RD input RD output prediction RA output prediction Multi-frames AD input Multi-frames RA input Figure S1: The overall frameworks of AdaPKCξ-Net and AdaPKCθ-Net, which only differ in the AdaPKC Block. Please note that the AdaPKC Block consists of two AdaPKC layers. Max-Pool 1ⅹ1 2D Conv Concatenation RD Encoder Conv Block RD Decoder ASPP Multi-frames RD input RD output prediction Figure S2: The overall framework of KuRALS-Net. The Conv Block can be readily replaced with different AdaPKC blocks to validate the effectiveness of AdaPKC. attribute, we might adopt the median value of Ξ as the selection criterion. However, while the use of median value can locate ξk where it is desired in some case, it is clear that the maximum value of the first order gradient is more robust for a variety of scenarios, as illustrated in Fig. S3. Therefore, to promote the robustness of this selection, we finally employ the maximum value of the first-order gradient. B.2 Reference Units Sampling and Back-propagation in AdaPKCθ The sampling values of reference units in down, left and right directions are obtained by the following forms of linear interpolation, accordingly: x(j)† r↓ = h 1 − bG† ↓−⌊bG† ↓⌋ i · xh(j) r↓+⌊bG† ↓−1⌋,w(j) r↓ + bG† ↓−⌊bG† ↓⌋ · xh(j) r↓+⌈bG† ↓−1⌉,w(j) r↓, (S2) x(j)† r←= 1 − bG† ←−⌊bG† ←⌋ · xh(j) r←,w(j) r←−⌊bG† ←−1⌋ + bG† ←−⌊bG† ←⌋ · xh(j) r←,w(j) r←−⌈bG† ←−1⌉, (S3) 14 sort(Ξ) sort(Ξ) sort(Ξ) Figure S3: The comparison between different selection criteria across three representative scenarios. The median value fails to capture target-interfering noise in the first and last scenarios, whereas our introduced selection criterion demonstrates robustness in locating it. x(j)† r→= 1 − bG† →−⌊bG† →⌋ · xh(j) r→,w(j) r→+⌊bG† →−1⌋ + bG† →−⌊bG† →⌋ · xh(j) r→,w(j) r→+⌈bG† →−1⌉. (S4) For back-propagation of gradients in these linear interpolations, we compute the gradient of Eq. S1 w.r.t. bG† = {bG† ↑, bG† ↓, bG† ←, bG† →} as ∂V ec nPN i=1 w(i) j · (xc −x(i)† r )⊤oCout j=1 ∂bG† = −V ec       N X i=1 w(i) j ∂x(i)† r ∂bG† !⊤   Cout j=1   . (S5) Since {x(i)† r }N i=1 have been split into four subsets based on their directions, and each x(i)† r is solely determined by the guard bandwidth offset in its corresponding direction, there only remain four nonzero items in n ∂x(i)† r ∂bG† oN i=1, i.e., ∂x(j)† r↑ ∂bG† ↑ , ∂x(j)† r↓ ∂bG† ↓ , ∂x(j)† r← ∂bG† ←and ∂x(j)† r→ ∂bG† →, where j ∈[1, M]. For simplicity, we demonstrate the calculation procedure with ∂x(j)† r→ ∂bG† →as an example, and the remaining items follow a similar approach. From Eq. S4, we can get ∂x(j)† r→ ∂bG† → = −xh(j) r→,w(j) r→+⌊bG† →−1⌋+ xh(j) r→,w(j) r→+⌈bG† →−1⌉. (S6) With the combination of Eq. S5 and S6, we can obtain the desired gradients. 15 C Visual Illustration of FiTOS Load weights Confidence threshold, 𝝉𝝉 PeakConv PeakConv Feature Map Fixed PRF Pretraining Flow Finetuning Flow AdaPRF Thresholding On-line Switch Figure S4: The illustration of FiTOS strategy. The Thresholding On-line Switch is detailed in Eq. 7. The detailed process of FiTOS strategy is illustrated in Fig. S4. D Supplementary Details about Datasets and Training Setups D.1 Detailed Description of CARRADA-RAC Dataset The CARRADA-RAC [32] dataset is derived from CARRADA and mainly calibrates the original RA annotations. For the generation of both RD and RA annotations, CARRADA adopts a semiautomatic method relying on the Mean-Shift clustering [3]. However, in CARRADA, the clustering performance is seriously degraded by unreliable centroid initialization from optical images and inaccurate candidate search space in RA representation. To alleviate these issues, CARRADA-RAC proposes an RD association strategy to refine the initial centroid, and introduces a regionalized CFAR for readjusting the search space. D.2 Implementation Details Multi-view RSS. The input sizes of RA, AD and RD views are 256 × 256, 256 × 64 and 256 × 64, respectively. Both AdaPKC-Nets leverage a sequence of 5 input frames for temporal information aggregation, consistent with PKCIn-Net [32]. We train all these models on two NVIDIA-3090 GPUs and use the Adam optimizer [14] for training. The initial learning rate is 1e −4, and decays in a cosine manner by default. We train these models for 300 epochs with a batch size of 6. For FiTOS, the 300 epochs are evenly distributed between the pre-training and fine-tuning stages, and we set τ = 0.6 for the fine-tuning stage by default. We train these models using a combination of weighted cross-entropy loss, Dice loss and coherence loss, configured with the recommended parameters outlined in [19]. 16 Single-view RSS. In contrast to the multi-view RSS task, the experimental configurations for KuRALS dataset maintain consistency with the following two exceptions: i) the input includes only a single view of RD tensor after 0-Doppler frequency elimination, with shape of 2048 × 124; ii) the weighted cross-entropy loss in multi-view RSS is substituted with weighted focal loss [15] to tackle the severer target-background imbalance in KuRALS. E More Exploration of AdaPKC mechanism E.1 Analysis of Guard Bandwidth Distribution in AdaPKCξ-Net We analyze the guard bandwidth distribution in AdaPKCξ-Net from two perspectives: different categories and different views. The distribution histogram of guard bandwidths in AdaPKCξ-Net is illustrated in Fig. S5. From the perspective of different categories, we observe that the guard bandwidth of the background class tends toward a uniform distribution, while that of the foreground class exhibits a more distinctly concentrated trend. For a better illustration, we present the variance values of different categories in Tab. S1, and the remarkable difference in variance between foreground and background classes confirms our observation. This observation aligns with the characteristics of our proposed metric score, indicating that AdaPKCξ consistently enhances features of foregroundclass targets, thus improving their discriminability from the background class. From the perspective of different views, this concentrated trend appears more apparent in the RA view compared to the RD view. Concretely, RA view shows a clear concentration on the guard bandwidth of (2, 1), while RD view exhibits considerable proportions on all guard bandwidths except (1, 1). Moreover, this concentrated guard bandwidth in the RA view remains consistent across different AdaPKC layers, while that of the RD view transitions from (2, 3) in the first layer to (1, 2) in the second layer. This comparative analysis of different views indicates that the guard bandwidth selection in RA view demonstrates more confidence and ease for AdaPKC, which explains the better performance improvement of AdaPKC on the RA view. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) Probability (R, D): bandwidth in the 1st layer of RD view 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) Probability (R, A): bandwidth in the 1st layer of RA view 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) Probability (R, D): bandwidth in the 2nd layer of RD view 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) Probability (R, A): bandwidth in the 2nd layer of RA view background pedestrian cyclist car Figure S5: The category-wise probability distribution histogram of guard bandwidths in AdaPKCξ-Net. 17 Table S1: Comparison of the variance values for different categories. Var1st and Var2nd represents the variance value of the first and second AdaPKC layer, respectively. Category RD View RA View Var1st(%) Var2nd(%) Var1st(%) Var2nd(%) Bkg. 0.19 0.19 0.67 0.33 Ped. 0.58 0.49 2.39 1.47 Cyc. 0.69 0.77 4.38 3.43 Car 0.42 0.72 4.33 3.90 E.2 Comparison between AdaPKC and Dynamic CFAR Detectors We compare the performance of AdaPKC and dynamic CFAR detectors, and the results are shown in Tab. S2. The comparison reveals several significant limitations of CFAR methods: i) They can only detect foreground targets without the ability to categorize them; ii) They show poor target identification performance, struggling with complex target and interference scenarios; iii) They rely on manual parameter tuning and lack adaptive learning capabilities. Therefore, it is evident that enhancing radar target perception paradigms through deep learning is both necessary and practical, serving as one of the key motivations for PKC and AdaPKC. Table S2: Performance comparison between AdaPKC and dynamic CFAR detectors on the CARRADA dataset. Bkg., Ped., Cyc., Car and Frg. represents Background, Pedestrian, Cyclist, Car and Foreground class, respectively, with Foreground class including Pedestrian, Cyclist and Car classes. FPS is calculated by the total inference time of the detection algorithm on both RA and RD views, using a workstation with an Intel(R) Xeon(R) Platinum 8255C CPU and a Tesla V100-SXM2 GPU. View Methods FPS IoU(%) Dice(%) Bkg. Ped. Cyc. Car Frg. Bkg. Ped. Cyc. Car Frg. RD CA-CFAR 47.5 82.9 0.9 90.6 1.7 SO-CFAR 36.0 77.8 1.2 87.5 2.4 GO-CFAR 36.6 97.7 2.9 98.8 5.6 OS-CFAR 42.8 93.8 3.1 96.8 6.1 AdaPKCξ-Net 20.7 99.7 55.1 31.9 58.2 99.8 71.0 48.2 73.4 AdaPKCθ-Net 18.8 99.7 54.6 33.7 58.1 99.8 70.6 50.4 73.5 RA CA-CFAR 47.5 89.2 0.4 94.3 0.8 SO-CFAR 36.0 75.8 0.4 86.2 0.8 GO-CFAR 36.6 97.2 0.6 98.6 1.2 OS-CFAR 42.8 91.6 0.5 95.6 1.0 AdaPKCξ-Net 20.7 99.8 24.4 17.1 35.0 99.9 39.3 29.2 51.9 AdaPKCθ-Net 18.8 99.8 26.5 15.8 32.3 99.9 41.9 27.3 48.9 E.3 Exploration of Adaptive Sampling Strategies for Reference Band We demonstrate the exploration of adaptive sampling strategies for reference band. Sampling for reference band plays an indispensable role in multiple applications of peak convolutions: i) for PKC with different guard bandwidth settings, sampling fixed N reference points ensures consistent parameter counts; ii) for AdaPKCs, different xc within the feature map might claim diverse PRFs, necessitating reference points sampling to meet the requirement of convolution operations; iii) for future research on reference bandwidth adjustment, this sampling mechanism operates in a manner akin to that in AdaPKCs. For simplicity of exploration, we take the mentioned application in PKC as the subject of our experiment. In addition to the common uniform sampling, we explore two adaptive sampling strategies based on the similar principle of AdaPKCξ: For the candidates of reference points {x(i) r }Nr i=1 in a fixed reference band, we first compute their inner product (similarity) with xc, i.e., xcx⊤ r . Then, the first version (v1) of adaptive sampling selects the N least similar reference points to locate target-interfering noise, while the second version (v2) of adaptive sampling chooses the N reference points with intermediate similarity instead. To evaluate the effectiveness 18 of these sampling strategies, we conduct comparative experiments on the CARRADA dataset, with the guard bandwidth of PKC fixed at 2 for all dimensions. The evaluation results are shown in Tab. S3. Compared with uniform sampling, adaptive sampling v1 consistently shows improvement across two views, further affirming the effectiveness of our introduced metric. In the case of adaptive sampling v2, we observe a similar trend as with AdaPKCξ, which exhibits a more pronounced performance enhancement for RA view. Nevertheless, in comparison to the adaptive guard bandwidth adjustment in AdaPKCs, these adaptive samplings of reference band encompass a more limited range of adjustable receptive field, thus demonstrating less improvements on RSS performance. Table S3: Comparison of different sampling strategies for reference band on the CARRADA dataset. The best and secondary results are marked with bold and underline, correspondingly. Sampling Strategy #Params @Frames RD View RA View mIoU mDice mIoU mDice Uniform sampling 6.3M@5 60.9% 72.9% 42.8% 53.6% Adaptive sampling v1 6.3M@5 61.3% 73.1% 43.2% 53.7% Adaptive sampling v2 6.3M@5 61.0% 72.7% 43.5% 54.1% E.4 Exploration of Enlarging PRF Search Space in AdaPKCξ To achieve AdaPRF estimation in a wider range of PRF search space under fixed resource constraints, we propose a novel training strategy named Voting-driven Multi-round Training (Vot-MRT). Due to limitations in GPU memory resources, AdaPKCξ restricts the selection of AdaPRF within a local range. To determine the AdaPRF in a wider range of PRF search space, Vot-MRT employs a step-wise local optimization strategy, inspired by greedy algorithms [4]. Concretely, Vot-MRT involves multiple rounds of training, and each successive round provides a better initialization for the guard bandwidth search space, ΩG, of the latter round. Given that different pixels may pick distinct guard bandwidths, Vot-MRT implements a voting mechanism to gather the most concentrated guard bandwidth for the subsequent round. The detailed algorithm of Vot-MRT strategy is presented in Algo. 1. To evaluate the effectiveness of Vot-MRT, we conduct a quantitative experiment on CARRADA dataset. We keep ΩG 0 consistent with ΩG denoted in § 3.1.1, and set the value of N to 3 by default, indicating that the original training process of AdaPKCξ-Net will be repeated for three rounds in Vot-MRT. The results are shown in Tab. S4 and it can be observed that Vot-MRT brings visible segmentation performance improvements to AdaPKCξ-Net by gradually enlarging the search space of alternative PRFs. F Supplementary Experiment Results F.1 Semantic Segmentation Results by Category on CARRADA The category-wise semantic segmentation results on the CARRADA-Test dataset are shown in Tab. S5. Compared with TMVA-Net, PKCIn-Net employs PeakConv to replace the regular convolution, resulting in impressive performance improvements. Our proposed methods further enhance the capabilities of PeakConv, leading to significant enhancements across nearly all classes and views, with a minor decrease observed in the car class of RA view. Consequently, the proposed methods achieve superior RSS performance and a better trade-off among different classes, which still holds true when compared to other SoTA RSS models. F.2 Performance vs. Complexity The trade-off between performance and complexity of different RSS models is presented in Fig. S6. F.3 Qualitative Comparisons on CARRADA We present in Fig. S7 the qualitative comparisons of different methods on three frames from the CARRADA test split, with each frame exhibiting varying levels of interference with targets. In the 19 Algorithm 1: Voting-driven Multi-round Training Input: Initial guard bandwidth search space: ΩG 0 = {{bG x, bG y } \| bG x ∈ΩG x\|0, bG y ∈ΩG y\|0}; Number of training rounds: N. Initialization: The maximum selection frequency of bG x and bG y : {C∗ x = 0, C∗ y = 0}. for i ←0 to N −1 do AdaPKCξ-Net* ←Getting AdaPKCξ-Net trained from scratch with ΩG i ; Fi ←Getting the feature map calculated by AdaPKCξ-Net; for d in {x, y} do for bG d in ΩG d\|i do Cd ←Getting the cumulative frequency of bG d being selected across all units in Fi; if Cd ≥C∗ d then C∗ d = Cd; bG d = bG d; end end C∗ d = 0; end bG∗= {bG* x , bG* y }; ΩG i+1 ←Getting a new local search space extending from bG∗; end Output: Return AdaPKCξ-Net*. Table S4: Results of AdaPKCξ-Net with Vot-MRT strategy on CARRADA dataset. The number of training rounds is set to 3 by default. Round RD View RA View mIoU mDice mIoU mDice 1 61.2% 73.1% 44.1% 55.1% 2 61.6% 73.6% 43.7% 54.5% 3 61.8% 73.8% 44.6% 55.7% first frame, where the target is interfered with minor noise/clutter, results indicate that all methods can accurately locate and classify targets in relatively clean RD view. However, in the RA view, methods without peak convolution struggle to identify target regions completely in the presence of stronger interference. The interference suppression capability equips PKCIn-Net with superior identification performance, while AdaPKCξ-NetF iT further strengthens this capability and recognizes complete target regions. In the second frame, as clutter interference on targets intensifies, PKCIn-Net can still accurately locate targets in both RD and RA views but faces challenges in differentiating target categories in the highly cluttered RA view. Nevertheless, AdaPKCξ-NetF iT correctly classifies targets in both views. In the third frame, where the signal of the distant car target is weak and the clutter interference is strong, existing methods miss the car target whereas AdaPKCξ-NetF iT can still identify it in the RD view. These results suggest that AdaPKCξ-NetF iT exhibits stronger interference suppression and target recognition capabilities compared to PKCIn-Net and other RSS methods. F.4 Semantic Segmentation Results by Category on KuRALS Considering the limited samples of both ship and land vehicle classes, we merge the two classes into a single vehicle class to mitigate the negative impact of class-imbalance on model training. This class merging is primarily based on two observations: i) both ships and land vehicles belong to rigid transportation vehicles, whose reflected radar signals share considerable similarities; ii) ships and land vehicles are primarily used for marine monitoring tasks and land detection tasks, respectively, and there exist nearly no overlapping application scenarios between them, which minimizes the risk 20 Table S5: RSS performance comparison by category on the CARRADA benchmark. View Frameworks #Params @Frames IoU(%) Dice(%) Bkg. Ped. Cyc. Car mIoU Bkg. Ped. Cyc. Car mDice RD FCN 134.3M@3 99.7 47.7 18.7 52.9 54.7 99.8 24.8 16.5 26.9 66.3 U-Net 17.3M@3 99.7 51.0 33.4 37.7 55.4 99.8 67.5 50.0 54.7 68.0 DeepLabv3+ 59.3M@3 99.7 43.2 11.2 49.2 50.8 99.9 60.3 20.2 66.0 61.6 RSS-Net 10.1M@3 99.3 0.1 4.1 25.0 32.1 99.7 0.2 7.9 40.0 36.9 RAMP-CNN 106.4M@9 99.7 48.8 23.2 54.7 56.6 99.9 65.6 37.7 70.8 68.5 TMVA-Net 5.6M@5 99.7 49.5 22.8 52.5 56.1 99.8 66.2 37.1 68.9 68.0 TransRadar 4.9M@5 99.6 49.5 24.7 54.8 57.2 99.8 66.2 39.7 70.8 69.1 PKCIn-Net 6.3M@5 99.7 54.0 30.4 58.5 60.7 99.8 70.1 46.7 73.9 72.6 AdaPKCθ-Net(ours) 6.3M@5 99.7 54.6 33.7 58.1 61.5 99.8 70.6 50.4 73.5 73.6 AdaPKCξ-NetF iT (ours) 6.3M@5 99.7 56.3 32.8 59.6 62.1 99.9 72.0 49.4 74.7 74.0 RA FCN 134.3M@3 99.8 14.8 0.0 23.3 34.5 99.9 25.8 0.0 37.8 40.9 U-Net 17.3M@3 99.8 22.4 8.8 0.0 32.8 99.9 36.6 16.1 0.0 38.2 DeepLabv3+ 59.3M@3 99.9 3.4 5.9 21.8 32.7 99.9 6.5 11.1 35.7 38.3 RSS-Net 10.1M@3 99.5 7.3 5.6 15.8 32.1 99.8 13.7 10.5 27.4 37.8 RAMP-CNN 106.4M@9 99.8 1.7 2.6 7.2 27.9 99.9 3.4 5.1 13.5 30.5 TMVA-Net 5.6M@5 99.8 25.6 7.5 17.8 37.7 99.9 40.7 13.9 30.2 46.2 TransRadar 4.9M@5 99.7 20.7 11.1 28.2 39.9 99.8 34.3 20.0 44.0 49.5 T-RODNet 162.0M@4 99.9 25.4 9.5 39.4 43.5 99.9 40.5 17.4 56.6 53.6 PKCIn-Net 6.3M@5 99.8 24.0 14.8 33.7 43.1 99.9 38.7 25.8 50.4 53.7 AdaPKCθ-Net(ours) 6.3M@5 99.8 26.5 15.8 32.3 43.6 99.9 41.9 27.3 48.9 54.5 AdaPKCξ-NetF iT (ours) 6.3M@5 99.8 26.2 19.2 32.0 44.3 99.9 41.5 32.2 48.4 55.5 25 30 35 40 45 50 55 60 0 50 100 150 mDice (%) Num. of Model Params. (Millions) RA Performance-VS.-Complexity (mDice) 25 27 29 31 33 35 37 39 41 43 45 0 50 100 150 mIoU (%) Num. of Model Params. (Millions) RA Performance-VS.-Complexity (mIoU) 35 40 45 50 55 60 65 70 75 80 0 50 100 150 mDice (%) Num. of Model Params. (Millions) RD Performance-VS.-Complexity (mDice) 30 35 40 45 50 55 60 65 0 20 40 60 80 100 120 140 mIoU (%) Num. of Model Params. (Millions) RD Performance-VS.-Complexity (mIoU) RD Performance-VS.-Complexity (mDice) FCN U-Net DeepLabv3+ RSS-Net RAMP-CNN TMVA-Net TransRadar T-RODNet PKCIn-Net TAPKCIn-Net (Ours) 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃−𝐍𝐍𝐍𝐍𝐍𝐍𝑭𝑭𝑭𝑭𝑭𝑭(𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎) Figure S6: Performance vs. Complexity. 21 (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) Figure S7: Qualitative comparison of different methods on CARRADA. The three frames from CARRADA test split are with varying levels of interference on targets. For each frame, the top row shows the camera image and the results in RD view, while the bottom row shows the results in RA view. (a) RD/RA tensor, (b) the label of mask, (c) TMVA-Net, (d) TransRadar, (e) PKCIn-Net and (f) AdaPKCξ-NetF iT (ours). Different colors represent different object categories. Black: background, Red: pedestrian, Yellow: cyclist, Cyan: car. 22 of practical application issues due to recognition confusion. As a result, we perform the segmentation task with four categories: background, UAV, pedestrian and vehicle. The semantic segmentation results by category on the KuRALS-Test dataset are presented in Tab. S6. Table S6: RSS performance comparison by category on the KuRALS-Test dataset. Frameworks #Params @Frames IoU(%) Dice(%) Bkg. UAV Ped. Veh. mIoU Bkg. UAV Ped. Veh. mDice FCN 134.3M@5 99.9 68.1 21.9 11.5 50.4 99.9 81.0 35.9 20.6 59.4 U-Net 17.3M@5 99.9 74.8 30.9 4.0 52.4 99.9 85.6 47.2 7.7 60.1 DeepLabv3+ 59.3M@5 99.9 73.1 23.0 14.4 52.6 99.9 84.5 37.4 25.2 61.8 KuRALS-Net(ours-baseline) 1.2M@5 99.9 72.8 39.0 12.2 56.0 99.9 84.3 56.1 21.8 65.5 KuRALS-Net w/ PKC(ours-baseline) 1.2M@5 99.9 79.7 31.9 15.3 56.7 99.9 88.7 48.4 26.5 65.9 KuRALS-Net w/ AdaPKCθ(ours) 1.2M@5 99.9 78.4 33.2 19.4 57.8 99.9 87.9 49.9 32.6 67.6 KuRALS-Net w/ AdaPKCξ F iT (ours) 1.2M@5 99.9 81.5 31.6 19.6 58.2 99.9 89.8 48.0 32.7 67.6 F.5 Qualitative Comparison on KuRALS The qualitative results on KuRALS dataset are illustrated in Fig. S8. Compared to our baseline methods and other segmentation models, our AdaPKCs demonstrate more accurate localization and classification of targets. (a) (b) (c) (d) (e) (f) (g) (h) Figure S8: Qualitative comparison on KuRALS. (a) RD tensor, (b) the label of mask, (c) FCN, (d) U-Net, (e) DeepLabv3+, (f) KuRALS-Net (ours-baseline), (g) KuRALS-Net w/ PKC (ours-baseline), (h) KuRALS-Net w/ AdaPKCξ F iT (ours). Different colors represent different object categories. Black: background, Red: UAV, Yellow: Pedestrian, Cyan: Vehicle. F.6 More Results in Ablation Study for FiTOS Fig. S9 illustrates the segmentation results of mIoU for AdaPKCξ-NetF iT with different values of τ ∈{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. 60.5% 61.0% 61.5% 62.0% 62.5% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 mIoU RD View 43.0% 43.5% 44.0% 44.5% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 mIoU RA View TAPKCIn-Net TAPKCIn-Net PKCIn-Net −𝐍𝐍𝐍𝐍𝐍𝐍𝑭𝑭𝑭𝑭𝑭𝑭 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝝃𝝃-Net PKCIn-Net 𝜏𝜏 𝜏𝜏 Figure S9: RSS performance of mIoU for AdaPKCξ-NetF iT with different values of τ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. 23 G Broader Discussions Limitations. We have collected a Ku-band continuous wave radar dataset to validate the effectiveness of our proposed method in surveillance radar detection scenarios. Nevertheless, pulse-Doppler radar is also commonly used in these scenarios and investigation on such radar datasets would further uncover the potentials and issues of our methods in practical applications. However, there is currently a lack of publicly available datasets for pulse-Doppler monitoring radar. To alleviate this issue, we will try to collect and release a pulse-Doppler radar dataset to enable more comprehensive validation and analysis of our methods and other works. Societal Impacts. Our approach is applicable to various practical applications, such as perceptions for autonomous driving, UAV surveillance and marine monitoring. However, inappropriate usage may lead to decreased reliability, potentially resulting in safety and other issues. 24 NeurIPS Paper Checklist 1. 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4,480	Arctique: An artificial histopathological dataset unifying realism and controllability for uncertainty quantification Jannik Franzen1,2,6,∗,, Claudia Winklmayr1,∗, Vanessa E. Guarino1,5,∗, Christoph Karg1,∗, Xiaoyan Yu1,5, Nora Koreuber1, Jan P. Albrecht1,5, Philip Bischoff2,3,4, Dagmar Kainmueller1,6,1 Max-Delbrück-Center for Molecular Medicine in the Helmholtz Association and Helmholtz Imaging, Berlin, Germany 2 Charité—Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Institute of Pathology, Berlin, Germany 3 Berlin Institute of Health at Charité—Universitätsmedizin Berlin 4 German Cancer Consortium, German Cancer Research Center, Partner Site Berlin 5 Humboldt-Universität zu Berlin, Faculty of Mathematics and Natural Sciences, Berlin, Germany 6 Digital Engineering Faculty of the University of Potsdam {firstname.lastname}@mdc-berlin.de ∗equal contribution Abstract Uncertainty Quantification (UQ) is crucial for reliable image segmentation. Yet, while the field sees continual development of novel methods, a lack of agreedupon benchmarks limits their systematic comparison and evaluation: Current UQ methods are typically tested either on overly simplistic toy datasets or on complex real-world datasets that do not allow to discern true uncertainty. To unify both controllability and complexity, we introduce Arctique, a procedurally generated dataset modeled after histopathological colon images. We chose histopathological images for two reasons: 1) their complexity in terms of intricate object structures and highly variable appearance, which yields challenging segmentation problems, and 2) their broad prevalence for medical diagnosis and respective relevance of high-quality UQ. To generate Arctique, we established a Blender-based framework for 3D scene creation with intrinsic noise manipulation. Arctique contains up to 50,000 rendered images with precise masks as well as noisy label simulations. We show that by independently controlling the uncertainty in both images and labels, we can effectively study the performance of several commonly used UQ methods. Hence, Arctique serves as a critical resource for benchmarking and advancing UQ techniques and other methodologies in complex, multi-object environments, bridging the gap between realism and controllability. All code is publicly available, allowing re-creation and controlled manipulations of our shipped images as well as creation and rendering of new scenes. 1 Introduction The crucial importance of reliable UQ for the deployment of segmentation algorithms to safetycritical real-world settings has long been recognized by the machine learning community, and the field has seen substantial development of methodology over past years (see e.g. [12, 1, 31]). However, there is a glaring lack of comprehensive evaluation of UQ methods, which makes it difficult to contextualize new methods within the existing paradigms, and renders the choice of suitable UQ 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. methods burdensome for practitioners. One reason for the lack of comparative insight is that often UQ methods are developed from theoretical considerations and tested on hand-crafted toy datasets but fail to provide meaningful, interpretable results on complex real-world datasets [20, 6]. Towards more insightful benchmarking of UQ methods, it is desirable to establish benchmark datasets with ground-truth uncertainty. However, in real-world settings, ground truth uncertainty is usually unattainable. Thus related works have resorted to empirically obtained (and therefore not fully quantifiable and/or controllable) distribution shifts and label noise [20, 3], which has greatly advanced the field, albeit by construction still does not facilitate comprehensive insight into method behavior. Synthetic data generation offers a promising avenue towards improved insight by providing clearly defined data properties and annotations (see [17] for an example from the realm of Explainable AI). However, previous synthetic data generation methodologies proposed in the context of challenging image segmentation problems either excel in controllability but fall short in complexity [30, 38], or vice-versa aim at improved complexity and realism but at the cost of falling short in controllability [8, 40], the latter because learnt image generation, while able to offer some level of conditioning on sought image properties, neither provides full control nor full insight into the image generation process. To address this gap, we introduce Arctique (ARtificial Colon Tissue Images for Quantitative Uncertainty Evaluation), a procedurally generated histopathological dataset designed to mirror the properties of images derived from H&E stained colonic tissue biopsies, as acquired routinely for safety-critical medical diagnoses in clinical practice [35]. Histopathological images offer a rich and challenging landscape for the application of advanced machine learning methodology, particularly in segmentation [2, 25]. The essential task of accurately delineating and classifying cellular structures is challenging even for trained professionals, due to many sources of uncertainty, e.g. overlapping structures, partial information from the underlying physical tissue-slicing process, and the inherent variability of biological tissues. The demanding nature of this task is reflected in the relative scarcity of fully annotated real-world data sets and high inter-annotator variability (see e.g [14]). Arctique offers the creation of realistic synthetic histopathological images at full controllability, allowing users to manipulate a range of easily interpretable parameters that effectively serve as "sliders" for imageas well as label uncertainties. Arctique provides 50,000 pre-rendered 512x512-sized images for training and evaluation of segmentation tasks, shipped with exact masks (2D and 3D), metadata storing characteristics of cellular objects, and rendering parameters to re-generate scenes. Furthermore, Arctique provides two main avenues for the controlled study of uncertainty: (1) a blender-based generation framework, which allows to re-generate and manipulate scenes, and (2) a data loader for post-processing images and emulating noisy labels. To assess Arctique’s degree of realism, we show that segmentation networks trained exclusively on Arctique can achieve promising zero-shot performance on real H&E images, proving its ability to learn meaningful attributes. To showcase how Arctique can be used for insightful benchmarking of UQ methods, we assess foreground-background segmentation and semantic segmentation and measure the effect of uncertainty in the images and the labels separately. We benchmark the performance of four widely used UQ-methods (Maximum Softmax Response (MSR), Test Time Augmentation (TTA [37]), MonteCarlo Dropout (MCD [11]) and Deep Ensembles (DE [22])). For each uncertainty scenario we measure model performance, predictive uncertainty, epistemic uncertainty and aleatoric uncertainty. Overall, we find that our manipulations generally increase predictive uncertainty in all four benchmarked UQ methods. In particular, we find that their aleatoric uncertainty components mostly track our devised label-level manipulations while their epistemic components mostly track our devised image-level manipulations. This serves as proof-of-concept that Arctique facilitates meaningful and comprehensive UQ benchmarking. Arctique was rendered and assessed on an internal resource of Nvidia A40 GPUs. Our work amounted to an estimated total of 150 GPU-hours. Dataset access The current version of our dataset, as well as the complete version history, can be accessed via https://doi.org/10.5281/zenodo.11635056. We provide access to up to 50,000 training and 1,000 test images along with their corresponding instance and semantic masks, including 400 additional exemplary variations corresponding to 50 of the test images. We also provide the dataset used for the experimental results presented in this paper as well as the respective noisy variations. The complete codebase containing scripts for dataset generation, model training and uncertainty estimation is available on GitHub: https://github.com/Kainmueller-Lab/arctique 2 epithelial lymphocytes eosinophils fibroblasts plasma cells image 3D labels sectioning plane focus plane orthografic camera parameters b a c 1 2 3 4 1 2 3 4 1 2 3 4 Figure 1: Generation Process: (a) To generate complex microscopic images, Arctique artificially replicates the H&E colon image creation protocol. From left to right: Initially, the colonic macrostructure (i.e., the outer epithelial layer) is constructed. This geometry is then artificially sliced, cell nuclei and other objects are placed, and the resulting scene is rendered along with its corresponding 3D stack of instance and semantic masks. (b) The result is a synthetic image (top) with corresponding semantic and instance mask (bottom) featuring numerous cell nuclei that (1) overlap, (2) lie outside the focal plane, (3) exhibit distinct characteristics, and (4) can be confused with perturbing elements. (c) A typical image of a natural H&E stained slice of colonic tissue (top) and the corresponding segmentation (bottom).The epithelium is characterized by its distinctive flower-like structures, known as crypts. The stroma, located between the crypts, forms the supportive connective tissue framework. 2 The Dataset Histopathological Hematoxylin & Eosin (H&E) stained tissue slices captured under light microscopy pose a significant challenge for segmentation models. This is due to their inherent complexity, manifested by overlapping cells, varying staining intensities, limitations introduced by the physical tissue-slicing process, and general heterogeneity of biological tissues. The scarcity of exact annotated data further aggravates the problem, hindering the development of robust segmentation models. Our synthetic dataset Arctique addresses these challenges by mimicking the complexity of real H&E stained colon tissue images akin to those in the Lizard dataset [14]. To this end, we devised a Python-based image generation pipeline on top of the 3D ray-tracing software Blender. This approach, as opposed to the alternative of relying on generative models, ensures controllability (and reproducibility) while allowing for the creation of realistic scenes. Consequently, each of our 3 pre-rendered images and labels can be easily recreated and subjected to controlled modifications. We generate each H&E image along with its corresponding masks via the following procedural data generation pipeline, as shown in Figure 1a: 1. Macro-structure: We begin by generating a 3D model representing the characteristic topology of the colon tissue architecture. Specifically, we focus on the epithelial crypts, which do not only follow an intriguing pattern (see Figure 1a left) but are also of pathological relevance. For example, colon cancer typically originates at this outer layer of the colon. To model the outer tissue topology, we arrange rod-shaped crypts in a densely packed hexagonal pattern, as depicted in Figure 1a (for details see Appendix). 2. Placing of cells: Next, we populate our scene with five predominant cell types. Within the stroma, the connective tissue between the crypts, we randomly distribute plasma cells, lymphocytes, eosinophils, and fibroblasts. The cells of the epithelial crypts, i.e. epithelial cells and corresponding goblet cells (white bubbles, see Figure 1), are placed according to a 3D adaptation of the Voronoi cell generation algorithm (cf. [36, 23] and Appendix). Each cell model includes the nucleus and, when highlighted by staining, the surrounding cytoplasm. For instance, Figure 1b3 illustrates the peanut-shaped nucleus of an eosinophil within its reddish-stained cytoplasm. The individual cell types are characterized by controllable parameters such as typical diameter, elongation, and nucleus shape. A comprehensive description of the parameter sampling methodology is provided in the Appendix. 3. Sectioning: A significant source of complexity arises from the fact that histopathological images are 2D slices of an underlying 3D architecture. To replicate this, we digitally slice through our 3D macro structure and cells. This approach ensures that the visible features of cells vary depending on their location and orientation relative to the sectioning plane. For example, in Figure 1b2, we can faintly observe two fibroblasts: one cut along its major axis and another cut vertically. 4. Staining: In real-world histopathological images, the staining colors are derived from Hematoxylin & Eosin (H&E) staining. Hematoxylin binds to DNA in the nucleus, giving it a purple appearance, while eosin stains the surrounding tissue architecture in a reddish-purple hue (see Figure 1c). To replicate this, we model the staining of the cytoplasm, cell nuclei, and surrounding tissue using controllable parameters such as staining hue, staining intensity, and inherent staining intensity noise. This is achieved using Blender-specific shaders, as detailed in the Appendix. 5. Rendering: The final scene is rendered using ray tracing from a virtual camera positioned above the light source and tissue slice (see Appendix). By adjusting the camera’s focal plane, we achieve a depth-blurring effect characteristic of histopathological light microscopy images. This workflow enables the generation of both 2D images and 3D image stacks. Moreover, the synthetic image generation provides precise 3D pixel-wise semantic- and instance masks and their corresponding 2D projection, serving as exact ground truth. Parameters: Various parameters can be gradually adjusted to control the rendering process and allow for precise customization of the generated images: Cell/Nuclei shapes: Adjustments include cell diameter, elongation, bending, and shape noise for linear interpolation between cell types. Cell distribution: Parameters cover cell locations, occurrence ratios of cell types, and cell density in the stroma. Tissue parameters: Configurations include tissue thickness and degrees of tearing. Staining parameters: Settings include staining colors and intensities for cells, nuclei, and tissue. By consulting with a pathologist, we fine-tuned these parameters to align with the images from the Lizard dataset. 2.1 Assessment of Realism Mimicking the generation protocol of real histopathological images, we were able to qualitatively incorporate many of their defining properties. Figure 1b demonstrates the fidelity of our dataset in capturing these characteristic nuances: 1b1) depicts a ring of densely packed and overlapping epithelial cell nuclei. Note that with Arctique, we enable a thorough investigation of such overlaps by offering precise 3D masks alongside their 2D projections, surpassing the limitations of standard 2D annotations typically used in real H&E slices; 1b2) illustrates the "blurring" effect of a fibroblast (characterized by its elongated shape) located outside the focus plane; and 1b3) showcases an artificial eosinophil cell, characterized by its distinctive peanut-shaped nucleus. Our dataset accurately models this characteristic feature, including the reddish staining hue of the surrounding cytoplasm, which contrasts with other cytoplasmic staining patterns. We also incorporated realistic sources of noise, 4 0 0.2 0.4 0.6 0.8 0 1 2 3 0 0.2 0.4 0.6 0.8 Inference on real data depth map real data noisy ARC. ARCTIQUE 3 b c HoverNext (Baumann et al.) pred. noisy ARCTIQUE groundtruth pred. noisy ARCTIQUE groundtruth a real data noisy ARC. depth map ARCTIQUE real data noisy ARC. depth map ARCTIQUE F1 semantic F1 AP Hausdorff epithelial connective eosinophil lymphocyte plasma Figure 2: Inference on the Lizard dataset using HoVer-NeXt (HN) models trained on Arctique: (a) Graphical illustration of the Arctique variants used for zero-shot learning, arranged on the left by complexity level (from most to least complex and noisy). Each variant aims to enhance the model’s generalization across diverse structural and textural details. On the right, a schematic representation depicts the post-processed raw class- and instance map outputs from the HN model during inference. (b) and (c) show visual and quantitative results for instance- and semantic segmentation, respectively, with bar plots comparing the baseline HN model trained on Lizard data (black) to the three HN models trained on simulated datasets of varying complexity. All metrics and predictions are averaged across 5 inference rounds, each with 16 Test-Time Augmentations. Note that the colors of the bars in (c) correspond to the colors of celltypes in the example. such as red blood cells, which exhibit a red staining hue similar to the cytoplasm of eosinophils (see Figure 1b4). Zero-shot segmentation of real data: To support these qualitative efforts, we quantitatively assess the applicability of Arctique in a segmentation context by training a HoVer-NeXt (HN) model [4] on Arctique. This panoptic segmentation architecture has been shown to yield state-of-the-art results when trained on real H&E data. After training, we conduct zero-shot inference on real H&E data [14]. To validate Arctique’s ability to infer semantically meaningful intermediate attributes, we compare baseline results with a model trained a) on a "noisy" Arctique version containing randomly injected anomalies, and b) on a simplified version consisting of depth maps only. As shown in Figure 2, considering the instance segmentation task, both F1 score and Hausdorff distance metric (weighted for true positives per class) support Arctique’s realism and value. Qualitative tile inspections further reveal that Arctique can detect previously discordant nuclei. Even regarding the semantic segmentation task, the metrics per class indicate a positive correlation between predictions and observations for the most abundant cell type, i.e. epithelial cells, without any fine-tuning. Overall (except for the default and noisy version for F1 and AP score), the metrics exhibit a clear trend between increasing heterogeneity 5 in the synthetic data and better segmentation performance. In summary, these findings indicate that segmentation models trained on Arctique not only learn features pertinent to classical segmentation tasks, but also suggest that Arctique may serve as a promising surrogate training dataset. For further details on the training process, datasets description, and metrics comparison, see the Appendix. 3 Benchmarking uncertainty quantification methods To showcase the capabilities of the Arctique dataset for benchmarking uncertainty quantification methods, we study the effect of image-level and label-level uncertainties on foreground-background segmentation (FG-BG-Seg) and semantic segmentation (Sem-Seg) [24]. To serve as a proof-ofconcept, we evaluate the performance of established algorithms on our data, namely segmentation with a UNet backbone [32, 18], and uncertainty estimation with four popular methods, two ensemblebased, namely Monte-Carlo Dropout (MCD, [11]) and Deep Ensembles (DE, [22]), one heuristic, namely Test Time Augmentation (TTA, [37]), and, for comparative purposes, one deterministic model known as Maximum Softmax Response (MSR).1 (See Appendix for implementation details). In accordance with [20], we use the predictive entropy H[Y \|x, D], conditional on the training set D = {xi, yi}n i=1, as the uncertainty measure of our predictive distribution p(y\|x, D), called predictive uncertainty (PU). For all UQ models except MSR we estimate p(y\|x, D), by sampling from the models and averaging the softmax outputs over the samples. Following [7, 21], we can then perform an information-theoretic decomposition to disentangle, respectively, the epistemic and the aleatoric components. In this setting, the epistemic uncertainty is defined as the mutual information between the output y and model parameters ω: H[Y \|x, D] \| {z } Predictive Unc. (PU) = I[Y ; ω\|x, D] \| {z } Epistemic Unc. (EU) + Ep(ω\|D)[H[Y \|x, ω]] \| {z } Aleatoric Unc. (AU) . (1) Eq. (1) shows that the aleatoric component correlates with the ambiguity inherent to the data and we should expect high values when there is a mismatch between image and label [21]. In particular, this implies that the aleatoric component is only meaningful for in-distribution data. The epistemic component, on the other hand, correlates with the model’s lack of knowledge. It is sensitive to out-of-distribution (OOD) data and can be compensated for by the addition of new training data. While the UQ measures yield pixel-wise results, we want to relate the uncertainty measures to our image- and label-level manipulations and are thus interested to aggregate pixel-level results to obtain uncertainty scores per image. It has been shown in [20] that the specific type of aggregation employed can hugely influence the behavior of uncertainty metrics. To account for this we tested the three aggregation strategies discussed in [20]: image-level aggregation, where uncertainty scores for all pixels are summed for each image and averaged over the dataset; patch-level aggregation, where uncertainty scores are aggregated within a sliding window and the maximum of the patch scores is taken as the image-level score; and threshold-level aggregation, which considers only uncertainty scores above an empirical quantile bQu(by)(p) for a chosen uncertainty measure u(by), then calculates their mean. All results presented in the main text are generated using threshold-level aggregation and normalization based on the image size. In the Appendix, we provide results from alternative aggregation strategies for comparison. In our experiments, we validate uncertainty measures using the variables defined in Eq. (1). Our approach differs from previous studies, such as [20, 29, 10], by focusing on how well informationtheoretic definitions of aleatoric and epistemic uncertainty capture true uncertainty within our dataset, especially for complex tasks like semantic segmentation. While some studies, like [15, 26], examine epistemic uncertainty for segmentation, they are generally limited to in-distribution data. Additionally, we track prediction accuracy across varying noise levels, expecting accuracy to decrease as overall uncertainty increases. 1Where a deterministic model predicts a categorical distribution p(y\|x, ω), we define 1 −MRS as 1 − max c p(y = c\|x, ω), a metric employed as computationally cheap alternative to the predictive entropy [16] despite depending only on a single model realization (see also [27]). 6 b a c Accuracy Predictive Uncertainty (PU) AU noisy label prediction softmax dropout tta noise increase AU EU Accuracy EU Figure 3: Illustration of two types of label uncertainty and their effect on model performance and uncertainty measure. (a) Effect of noisy class labels on Sem-Seg: illustrations on the left show an example of possible label confusion. The two large panels in the middle show model performance across noise levels (x-axis) as measured by accuracy and predictive uncertainty for all four UQ methods. The two smaller panels on the right show aleatoric and epistemic uncertainty for DE, TTA and MCD. (Note that MSR does not permit decomposition, therefore not shown.) (b) Effect of noisy label shapes on FG-BG-Seg: subpanels analogous to (a). (c) Qualitative example of the impact of noisy labels for FG-BG-Seg on prediction performance and how this is captured in the PU maps. 3.1 Label Noise In the first step, we look at the effect of uncertain labels. In biomedical data, this is a common issue as complex and ambiguous images yield high disagreement even among expert annotators. In real-world images, we should expect some correlation between uncertainty in images and uncertainty in the labels. For example, cells with lower contrast staining might be harder to identify for human annotators leading to more missing labels. However, we believe, that there is a benefit to studying label-noise in isolation as it can give us valuable insights into model calibration and the sensitivity of UQ methods [6, 13, 19]. 7 We devise two types of label-noise tailored to different segmentation tasks: Sem-Seg: Class labels are randomly switched. The noise level reflects the probability for each single-cell label to be switched to another class. FG-BG-Seg: Labels of single cells are manipulated by shifting, scaling, elastic transform or completely removed (missing label). The noise-level reflects the probability that any single cell is affected by any of these modifications. Both types of label noise are illustrated in the top row of Figure 3. Figure 3 summarizes the results of uncertainty evaluation in the presence of label noise: for both segmentation tasks, we find that performance decreases with increasing label noise. This is to be expected as unreliable labels make it harder for the model to learn generalizable patterns. Predictive uncertainty (PU) increases as a function of label noise across both tasks. This confirms that what we would intuitively consider as "making the segmentation task harder" will actually decrease performance and increase uncertainty. Further, we find that across both segmentation tasks the majority of uncertainty stems from the aleatoric component. This is in keeping with the theoretical claim that aleatoric uncertainty mainly captures data-inherent uncertainty. While aggregate measures offer a convenient way to assess how training with noisy labels impact a model’s uncertainty, Arctique also provides exact labels, enabling a more detailed study of the label manipulation at pixel-level. Figure 3 shows in detail how models trained with and without label-noise learn about training images. Specifically, we observe that when some cell labels are consistently missing, the model still predicts the corresponding cells. Despite this, the uncertainty maps still highlight high uncertainty in regions with corrupted masks, suggesting that uncertainty quantification may address the common challenge of sparse annotations in biomedical images. Conversely, in densely packed regions, the model tends to interpolate across missing labels, as seen in the epithelial crypt in the bottom left of 3(c). Here, the uncertainty maps effectively capture the increased uncertainty in areas where cells are incorrectly merged. This duality underscores the value of evaluating UQ methods for their ability to handle both sparse and ambiguous label regions effectively. 3.2 Image Noise The greatest advantage of having full control over the dataset creation is that it allows us to perform targeted manipulations to certain aspects of the image while leaving all others unchanged. We are thus able to create samples that gradually transform from near-OOD, where outlier and inlier classes are quite similar, to far-OOD, where an outlier is more distinct. This method of generating data is fundamentally different from other common strategies, such as applying augmentations like color shifts or blur [5, 32], where we achieve global manipulations which do not correlate with input features; or testing on images from a different domain [34] (e.g. data from a different organ) where the exact impact on image components is unknown and uncontrolled. For the Nuclei-Intensity manipulation, we progressively reduce the staining of the cells’ nuclei until they become less distinct from their surroundings, while preserving details in other regions and simulating real-world inconsistencies in staining. In contrast, an image-level reduction uniformly lowers contrast across the entire image, a simpler manipulation achievable with basic augmentation techniques and far from the intended use of Arctique. For the Blood-Stain manipulation, we gradually increase both the red stains and the number of blood cells, simulating realistic, extreme variations in blood cell abundance. This adjustment reflects a possible scenario in histology where red-stained artifacts may be mistaken for cell types like eosinophils, which naturally exhibit red staining. Figure 4 shows examples for both types of image-level manipulations and their effects on model performance and uncertainty. Subfigure 4 (a) shows the impact of manipulating the nuclei-intensity on the FG-BG-Seg task. As might be expected, we observe a decrease in accuracy and an increase in the uncertainty measures as staining intensity decreases. While the aggregated effects may appear subtle, the error maps reveal a clearly visible decline in prediction performance: as staining weakens, the model starts to hallucinate cells in the tissue of the crypt-structures. In Subfigure 4 (b) we illustrate the effect of manipulating the the blood-cells and -stains on the Sem-Seg task. Even with perfect masks, the model already tends to misclassify blood cells as eosinophils. As the abundance of blood cells increases, the number of false positives rises, leading to 8 a FG/BG Accuracy Predictive Uncertainty (PU) Aleatoric Uncertainty (AU) Epistemic Uncertainty (EU) prediction dropout b Semantic certain prediction error dropout eosinophil (FP) Accuracy EU AU Figure 4: Illustration of Image-level noise: (a) Illustration of an image undergoing decreasing intensity of nuclei staining. The small image patches on the top illustrate qualitatively how FG-BG prediction performance and PU (for the example of MCD) are affected as staining is removed. The four panels on the bottom summarize for all four uncertainty methods how accuracy, PU, AU and EU react to the gradual change in staining. (b) illustrates the effect of the increasing prevalence of blood-cells. Similar as in (a) the small image patches on the top show the qualitative changes in semantic prediction performance and uncertainty. Here we additionally show the error maps next to the PU maps to highlight how blood cells are incorrectly identified as eosinophil cells, however the model remains confident in its prediction. The four panels on the bottom are arranged analogous to (a) and further illustrate the decrease in performance while uncertainty remains relatively unchanged. a significant drop in accuracy, as shown in both the qualitative error maps and accuracy results in 4 (b). Consequently, this phenomenon leads to the miscalibration of uncertainty methods. In fact, the error maps reveal that regions affected by blood cells exhibit particularly low uncertainty values, indicating that high error rates do not correlate with high uncertainty. For DE we even observe a slight decrease in the uncertainty as the prevalence of blood cells increases, further emphasizing the weak correlation between error rates and uncertainty values. We conclude that both experiments demonstrate Arctique’s capabilities for in-depth analysis of uncertainty behavior. The two manipulations highlight common challenges encountered in realworld H&E images: without perfect labels, it becomes nearly impossible to ascertain whether high uncertainty values indicate subtle features in the data or stem from a miscalibrated uncertainty model. In the case of the Nuclei-Intensity alteration, the uncertainty method effectively identifies a genuine issue. In contrast, with the Blood-Stain manipulation, the uncertainty quantification (UQ) models demonstrate inadequacies in correctly calibrating the model. 9 4 Discussion UQ carries the promise to increase the reliability of machine learning models so that these models can be more widely deployed even in safety-critical domains. To this end, we must be confident that the UQ methods we develop follow through on their claims. We believe that Arctique constitutes a valuable first step for a more thorough and interpretable evaluation of UQ metrics. While the domain of histopathology may represent a specialized domain it serves as a valuable testbed due to its versatility and the presence of common uncertainty sources, such as missing or incorrect labels and overlapping instances. Moreover, this domain is particularly relevant for UQ, as it is critical for medical diagnosis and often suffers from incomplete and inaccurate annotations. Our main goal in this publication is to introduce the Arctique dataset and illustrate its utility for evaluating UQ methods, yet it also opens numerous promising avenues for further research. One important follow-up would be to expand the range of studied UQ methods, particularly in Active Learning (AL), where uncertainty plays a central role in sampling strategies. Recent studies suggest that prioritizing high epistemic uncertainty can improve AL performance, while focusing on aleatoric uncertainty may be less effective ([6], [28]). Arctique’s controlled uncertainty levels make it suitable for evaluating AL sampling, integrating uncertainty into optimization, and exploring domain adaptation strategies ([9], [39], [33]). In particular, Arctique allows to straightforwardly combine multiple sources of uncertainty at any level, thus constituting a unique testbed for methodology that seeks to disentangle AU and EU. Our dataset can also be applied in the context of Explainable AI (XAI) evaluations, where transparent decision-making is crucial for trustworthiness. In contrast to simpler datasets like FunnyBirds [17], which focus on single-class tasks, Arctique offers a realistic multi-object environment. This complexity allows XAI methods to be benchmarked on relevant concepts that reflect the characteristics of real data, and to analyze interpretability and predictiveness across complex co-occurrence patterns, as for example cell nuclei and cytoplasm. Finally, future research could extend our framework with image- and label modifications, encompassing imaging modalities, tissue types, and staining variations. We encourage users to devise their own modifications suitable to their specific evaluation needs. A direct next step could be studying uncertainty for related tasks such as panoptic segmentation or 3D models. To conclude, our work contributes Arctique, a complex, realistic yet fully controllable dataset of synthetic images, together with a broad range of "sliders" for targeted manipulation. As a proof-ofconcept, we show that we can tailor label- and image manipulations such that they are selectively picked up by the aleatoric and epistemic components of established UQ methods, which suggests that Arctique is a valuable resource for UQ methods development and benchmarking, with clear potential for extensions into orthogonal methodological realms like XAI. Acknowledgments We wish to thank Aasa Feragen, Kilian Zepf, Paul Jäger and Lorenz Rumberger for inspiring discussions. Funding: German Research Foundation (DFG) Research Training Group CompCancer (RTG2424), DFG Research Unit DeSBi (KI-FOR 5363, project no. 459422098), DFG Collaborative Research Center FONDA (SFB 1404, project no. 414984028), DFG Individual Research Grant UMDISTO (project no. 498181230), Synergy Unit of the Helmholtz Foundation Model Initiative, Helmholtz Einstein International Berlin Research School In Data Science (HEIBRiDS). References [1] Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U Rajendra Acharya, et al. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information fusion, 76:243–297, 2021. [2] Mohammed M Abdelsamea, Usama Zidan, Zakaria Senousy, Mohamed Medhat Gaber, Emad Rakha, and Mohammad Ilyas. A survey on artificial intelligence in histopathology image analysis. 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4,481	Rethinking the Membrane Dynamics and Optimization Objectives of Spiking Neural Networks Hangchi Shen1,2, Qian Zheng3,4, Huamin Wang1,2,∗, Gang Pan3,4 1College of Artificial Intelligence, Southwest University 2Chongqing Key Laboratory of Brain Inspired Computing and Intelligent Chips 3The State Key Lab of Brain-Machine Intelligence, Zhejiang University 4College of Computer Science and Technology, Zhejiang University stephen1998@email.swu.edu.cn, hmwang@swu.edu.cn, {qianzheng, gpan}@zju.edu.cn Abstract Despite spiking neural networks (SNNs) have demonstrated notable energy efficiency across various fields, the limited firing patterns of spiking neurons within fixed time steps restrict the expression of information, which impedes further improvement of SNN performance. In addition, current implementations of SNNs typically consider the firing rate or average membrane potential of the last layer as the output, lacking exploration of other possibilities. In this paper, we identify that the limited spike patterns of spiking neurons stem from the initial membrane potential (IMP), which is set to 0. By adjusting the IMP, the spiking neurons can generate additional firing patterns and pattern mappings. Furthermore, we find that in static tasks, the accuracy of SNNs at each time step increases as the membrane potential evolves from zero. This observation inspires us to propose a learnable IMP, which can accelerate the evolution of membrane potential and enables higher performance within a limited number of time steps. Additionally, we introduce the last time step (LTS) approach to accelerate convergence in static tasks, and we propose a label smooth temporal efficient training (TET) loss to mitigate the conflicts between optimization objective and regularization term in the vanilla TET. Our methods improve the accuracy by 4.05% on ImageNet compared to baseline and achieve state-of-theart performance of 87.80% on CIFAR10-DVS and 87.86% on N-Caltech101. The code is available at https://github.com/StephenTaylor1998/IMP-SNN. 1 Introduction In recent years, deep learning technology based on artificial neural networks (ANNs) has made significant breakthroughs in many fields [1, 2, 3, 4, 5]. However, as we all known, its high training and inference costs have become a major obstacle to restrict its further widespread applications. To overcome these challenges, a new generation of neural network architectures named spiking neural networks (SNNs) [6] have been developed, which may be a feasible path to refer to the efficient dynamics mechanism of biological nervous systems [7]. SNNs leverage the dynamic mechanisms of membrane potential integration and fire from biological neural networks, which can process time-varying input data by using a single model [8] as biological neurons, maintaining linearly increasing computational complexity. Therefore, the advantages of energy efficiency and biologically plausible dynamical mechanisms [7, 9, 10] make them a bridge between the fields of brain science and artificial intelligence, which is widely regarded as the next generation of ANNs [6]. In addition, it is worth mentioning that SNNs can cleverly avoid multiplication operations on neuromorphic chips ∗Corresponding author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). [11, 12, 13] by spike-based computing, achieving synaptic computation and membrane potential accumulation solely through addition operations, which can significantly enhance their computational efficiency. It should be noted that training SNNs to achieve the comparable performance with the same ANN architecture remains a formidable challenge at present. The conversion method of ANNs to SNNs (ANN2SNN) [14, 15, 16, 17, 18] has proved to be an effective approach for obtaining high-performance SNNs. However, these methods mainly incorporate knowledge from the ANN’s learning of static inputs into the converted SNNs, which motivates us to concern about the biological plausibility of these SNNs as they do not require the acquisition of spatiotemporal information. Another alternative is to leverage surrogate gradient [19, 20, 21] and backpropagation [22] through time to train high-performing SNNs. This method only requires four time steps to achieve an accuracy surpassing that of conversion methods requiring hundreds of time steps, making it the most promising training approach for currently available energy-efficient SNNs. Therefore, due to the development of direct training methods, SNNs have been extended to various tasks, including image classification[23, 24, 25], image reconstruction [26, 27, 28], object detection [29], natural language processing [30, 31], etc, and have demonstrated significant energy efficiency advantages in these fields. It is a pity that these training methods and architecture designs typically consider the firing rate or average membrane potential of the last layer as the output, instead of fully exploring the impacts of the membrane potential on the model performance and the training process [32, 33]. (a) Adjusting IMP. (b) Adjusting input intensity. Figure 1: Membrane potentials and spikes (yellow) generated by adjusting IMP and input intensity. In this paper, we find that the initial membrane potential (IMP) has different effects on the membrane potential integration and spike firing compared to input intensity(Figure 1). Then, we discover that the novel spike patterns can be generated by adjusting the IMP of spiking neurons and additional mappings between firing patterns can be established, which motivates us to propose a learnable IMP. Furthermore, we find that in static tasks, the variation of the output accuracy at each time step for SNNs is directly related to the membrane potential, as it is the only time-varying term. Then we can explain why the convergence speed of temporal efficient training (TET) [34] in static tasks is significantly lower than standard direct training (SDT) [20]. To address this problem, we propose a method called last time steps (LTS) to achieve faster convergence in static tasks. And then, we propose a label-smoothed TET loss for neuromorphic tasks, which can outperform vanilla TET on neuromorphic datasets [35, 36]. It is worth noting that the proposed novel IMP method can obtain significant performance without any additional adjustments for the network structure and training method. By simply setting the IMP of the original LIF neurons to a learnable version, we achieve SOTA accuracies of up to 87.8% and 86.32% on CIFAR10DVS [35] and NCaltech101 [36], respectively. Furthermore, our LTS method improves the accuracy of SEW-ResNet50 [37] on the ImageNet1k [38] dataset to 71.83%, surpassing the vanilla SEW-ResNet152, 69.26%. The main contributions are as follows: 1) We discover that SNNs can generate new spike patterns by adjusting the IMP values, and prove that on the static tasks, the variation of SNN accuracy at each time step is only caused by the change of membrane potential. In addition, we innovatively introduce the learnable IMP in SNNs to accelerate the evolution of membrane potential. 2) To alleviate the slow convergence of TET on the static tasks, we propose the LTS method, which can accelerate the rate of convergence on the static tasks. Additionally, we construct a label-smoothed TET loss to further enhance the performances of SNNs on the neuromorphic tasks. 2 3) Compared with the baselines on the neuromorphic datasets and the large-scale static dataset ImageNet1k, our methods can achieve significant improvements. Moreover, there is almost no difference in the computational overhead and inference speed compared to the original models. 2 Related Works 2.1 Neuron Dynamics Modeling The leaky integrate and analog fire spiking neuron [39] was proposed to replace binary spike with analog values for transmission, alleviating the issue of decreased performance in SNNs. The parametric leaky integrate-and-fire spiking neuron [40] was introduced to design a learnable dynamic model, enabling each neuron to learn optimal membrane time constants, thus increasing neuronal diversity. The gated leaky integrate-and-fire neuron [41] was employed for a channel-wise parameterization approach to fully parameterize the spiking neuron, including learnable decay mechanisms, potential thresholds, reset voltages, input conductance, and gating factors. The multi-level firing method [42] was used to enhance the expressive ability and achieve more efficient gradient propagation by integrating neurons with different thresholds to realize multi-level firing. Parallel spiking neurons [43] were discussed to remove the membrane potential reset process and redefine the dynamic mechanism in a non-iterative manner, addressing the difficulty of ordinary spiking neurons in learning long-term dependencies. 2.2 Direct Training Methods The binary spikes emitted by spiking neurons during the forward phase are generated by a step function, which is a non-differentiable activation. In the backward propagation phase, this step function can be replaced with a surrogate gradient [19] to achieve direct training. The most common direct training method currently is backpropagation through time (BPTT) [22], which treats spiking neural networks (SNNs) as a special type of recurrent neural network (RNN). In this approach, gradients are propagated backward along the temporal dimension, which requires more computational resources and memory compared to their corresponding ANNs [44]. tdBN [21] explored normalization methods for spiking neural networks (SNNs) and achieved direct training of large-scale SNNs on the ImageNet dataset for the first time. Based on this, a more effective normalization method called TEBN [45] was proposed, which rescales the presynaptic inputs at each time step using distinct weights. TET [34] enabling spiking neural networks (SNNs) to converge to flatter minima compared to SDT [20], which enhances generalization capabilities. OTTT [46] and SLTT [47] simplified the gradient calculations along the temporal dimension in BPTT, significantly reducing memory and computational costs. 3 Analysis of Membrane Dynamics In this section, we first investigate how the initial membrane potential affects neuronal spike patterns, and then analyze how the dynamic evolution of membrane potential drives improved SNN performance. These analyses underscore the critical role of membrane dynamics in SNNs and provide new insights into its impact on the SNN’s representational capacity and convergence. 3.1 Preliminary of Spiking Neurons and Loss Functions Spiking neurons are the fundamental unit of SNNs, used to simulate the dynamic behavior of brain neurons. Their operation is described by dynamical equations that are related to membrane potential and input current. The leaky integrate-and-fire (LIF) [7] model is one of the commonly used spiking neuron models, and its iterative form of the dynamical equations can be represented as follows: h[t] = (1 −τ)s[t] + I[t], h ∈RT ×N, I ∈RT ×N (1) o[t] = h[t] > Vth, o ∈{0, 1}T ×N, Vth ∈R (2) s[t + 1] = h[t] −o[t], s ∈RT ×N, s[0] ∈{0}N (3) where I[t] represents the neural input current at time t, τ represents the membrane potential decay coefficient. When τ = 0, it reduces to the Integrate-and-Fire (IF) [7] neuron. s[t] represents the state of the membrane potential at time step t, and s[0] is the state of the IMP, which is typically set 3 to a constant 0. h[t] represents the change in membrane potential during time step t, o[t] indicates whether the neuron fires a spike at time t, and Vth represents the firing threshold of the neuron. The most commonly used loss functions in direct training SNNs are SDT [20] and TET [34]. The SDT loss function LSDT is defined as: LSDT = LCE( 1 T × T X t y[t], ygt), (4) here, T represents the total number of time steps, y[t] represents the raw output of the model at each time step, ygt represents the ground truth label, and LCE denotes the cross-entropy loss. SDT aggregate the outputs of the SNN by taking the mean of the outputs from all time steps, then calculate the loss based on the voting result. In TET, the averaging step is placed after the calculation of cross entropy loss: LTET = 1 T × T X t LCE(y[t], ygt), (5) the LTET calculates the loss for each time step individually and then aggregates the losses from each time step to obtain the final loss. This approach can effectively improve the performance of SNN on neuromorphic datasets. 3.2 Membrane Dynamics Related to IMP The membrane potential is commonly reset to zero before the next task in the current implementations of SNNs. However, through some experiments and the analysis of experimental results, we find that novel firing patterns and pattern mappings can be generated by adjusting IMP. Firing Pattern Input Intensity 0 0 0 1 0 0 1 0 0 1 0 1 [0.25,0.33) [0.33,0.50) [0.50,0.75) 1 1 1 1 0 0 0 0 ( - ∞ ,0.25) [1.00, + ∞) IMP=0.0 IF Firing Pattern 0 1 0 0 1 0 1 0 IMP=0.5 Input Intensity 0 0 0 1 0 0 1 0 [0.125,0.166) [0.166,0.250) [0.250,0.500) 0 0 0 0 ( - ∞ ,0.125) [0.500,1.000) [1.000, + ∞ ) 1 1 1 1 IF IMP=1.0 Firing Pattern Input Intensity 1 0 0 1 1 0 1 0 [0.333,0.500) [0.500,0.100) 1 0 0 0 ( - ∞ ,0.333) [1.000, + ∞ ) 1 1 1 1 IF Figure 2: All firing patterns that IF neurons can generate under constant intensity input and 4 time steps. The red box highlights the disappearing firing patterns, while the green and yellow boxes denote the additional firing patterns due to the IMP change. Observation 1: Novel firing patterns under constant intensity input can be generated by adjusting IMP. Figure 2 displays the firing patterns (spike sequences) generated by an IF neuron under constant input intensity. The number of patterns varies depending on the IMP. When the value of IMP increases from 0.0 to 0.5, the pattern ’0101’ disappears, while the new patterns ’0100’ and ’1010’ emerge, resulting in increased pattern diversity. Conversely, setting the IMP to 1.0 leads to the emergence of new patterns ’1000’, ’1001’, and ’1010’, but reduces the overall number of patterns. These additional patterns that cannot be generated if IMP is set to 0, can benefit static tasks by enabling SNNs to encode more information during the processing of static inputs. Observation 2: New mappings of firing patterns can be generated by adjusting IMP. Apart from its impact on input encoding, our primary focus is whether the model’s capabilities can be enhanced by modifying IMP. In ANNs, artificial neurons can map any single input variable to any value by adjusting the weights. Similarly, we hope that spiking neurons can also map input sequences to as many firing patterns as possible, thereby enabling the network to have better representation capabilities. From figure 3, we observe that every output pattern has at least one available pattern mapping. However, we can notice that black areas present in the figure, which means that no matter how the synaptic weights are adjusted, mapping among these spike patterns still cannot be established. In other words, spiking neurons can theoretically generate all firing patterns, but they cannot map any 4 (a) IMP=0.0 (b) IMP=0.25 (c) Learnable IMP Figure 3: Pattern mapping of IF neuron over 4 time steps. The horizontal and vertical axes in the figure represent all possible spike patterns (16 total) that IF neurons may receive and emit. The white squares indicate that IF neuron can receive the spike pattern from the horizontal axis and emit a spike pattern on the vertical axis, known as pattern mapping. specific pattern to all patterns. Figure 3b illustrates that through adjusting the IMP value from 0 to 0.25 can generate additional pattern mappings. Furthermore, Figure 3c demonstrates that when IMP is learnable, it exhibits a greater potential for establishing mappings among spike patterns. Therefore, we believe that the learnable IMP can effectively improve the expression capacity of spiking neurons. 3.3 Membrane Potential Evolution in Static Tasks We further explore the effect of membrane potential variations on static tasks through a image classification task. We define SNN as follows to focus on its performance at each time step, (s[t + 1], y[t]) ←−f(x[t], s[t], θ), (6) where f(·) represents the network computation, θ represents the network weights, s[t] represents the set of membrane potentials of all neurons in SNN with time step t, x[t] represents the input at time step t, and y[t] is the corresponding output. Assuming x = x(t) is a constant input intensity for t = 1, 2, ..., T, then we can simplify Eq. 6 as: y[t] = f(x, s[t], θ). (7) On the static tasks, the temporal variations are determined solely by the state of the membrane potential s[t], resulting in corresponding changes for the output y[t]. Observation 3: In static tasks, the accuracy of SNNs at the each time step is sensitive to the current MP. Figure 4 demonstrates the test accuracy of SNNs at each time step on cifar10 dataset [48], which shows that the accuracy is extremely low at T=1, only 10.76%. 10.76 90.44 92.04 92.05 92.36 T=1 T=2 T=3 T=4 mean 0 20 40 60 91 92 93 94 Accuracy Constant IMP Figure 4: The test accuracy at each time step on the CIFAR10 dataset. However, as the time step T increases, the model accuracy exhibits an upward trend, exceeding 90%. It is worth noting that on the static tasks, since the input x and weight θ are fixed at each time step, the model’s output at each time step is entirely determined by the current state of the membrane potential s[t]. For instance, when the membrane potential evolves to a "sufficiently good" state, such as at t = 4, the SNNs only requires the current membrane potential s[4] and input x to achieve an accuracy of 92.05%, which is close to the model’s final performance of 92.36%. Therefore, these findings prompt us to reconsider how to accelerate the evolution of the membrane potential to enhance the SNNs performances within a limited number of time steps. Observation 4: TET performs well on the neuromorphic tasks but exhibits slow convergence on the static tasks. We compare the SDT loss and the TET loss on the static datasets and the neuromorphic 5 Table 1: Test accuracy of TET and SDT on the static and neuromorphic datasets. Loss Static Dataset(SEW-R18) Neuromorphic Dataset(VGG11) Function CIFAR10/100 ImageNet100 ImageNet1k CIFAR10DVS DVSG128 NCaltech101 SDT Loss 94.56/76.58 78.42 63.21 84.3 98.26 85.78 TET Loss 94.33/76.40 77.80 62.92 85.6 98.61 86.32 (a) T=1 (b) T=2 (c) T=4 (d) T=6 Figure 5: The convergence speed of TET and SDT on the static data. datasets, and find that TET loss has a significant advantage on the neuromorphic datasets, but it is not superior to SDT loss on the static datasets, as shown in Table 1. We believe that this phenomenon arises from the constant intensity input on the static datasets. By applying the TET loss Eq. 5 to the SNNs with the static input Eq. 7, we have: LTET = 1 T × T X t LCE(y[t], ygt), where y[t] = f(s[t], x, θ). (8) It can be observed that on static tasks, the membrane potential s[t], as the only time-varying term in the dynamic system, evolves gradually as the time step progresses. However, according to Observation 3, we know SNNs are sensitive to s[t], which means that it is difficult to output the same result for different s[t]. The optimization goal of the TET loss is to make the network output the correct results for every time step, i.e. y[t] = ygt. In this case, TET requires more iterations to build a flat "landscape", slowing down the convergence, as shown in Figure 5. Additionally, if the SNNs are insensitive to the value of s[t] and even tends to the same value for all time steps, i.e. y[1] = y[2] = ... = y[T] = 1 T PT t y[t], then the output of the first time step is sufficient to represent the results of all T time steps. In this case, the computations for the subsequent time steps become redundant and meaningless. 4 Methods 4.1 Learnable IMP The membrane potential in the current dynamic model of SNNs is typically initialized to a uniform constant value (usually 0). Based on Observation 1 and Observation 2, we find that the learnable IMP can enhance the expressive power of SNNs. Based on Observation 3, the learnable IMP can allow the membrane potential to start from a non-zero value, which may help improve the performances of SNNs on the static tasks. We can assign an independent learnable IMP for each spiking neuron. According to Eq. 1, the membrane potential accumulation can be represented as follows, h[t] = ˆs[t] + x[t], x ∈RT ×N, h ∈RT ×N, ˆs ∈RT ×N, ˆs[0] ∈RN, (9) where ˆs[0] represents the state of the IMP that is extended from zero to the real number. As a counterpart to the method of setting IMP to 0, in the initialization process, we set IMP to a uniform distribution with an expected value of 0, ˆs[0] = Uniform(−λ, λ)N, (10) where λ is a hyper-parameter used to control the boundaries of the uniform distribution, ensuring the IMP remains within an appropriate range. Since the current implementation of SNNs requires memory allocation to store membrane potential states, replacing 0 with a trained IMP during the inference process will not incur additional computational overhead. 6 4.2 LTS Method for the Static Tasks We propose a new post-processing representation method called last time step (LTS) to alleviate the convergence difficulties of TET on the static tasks, based on Observation 3 and Observation 4. This approach masks all outputs of the SNNs before the LTS and only retains the output of the LTS as the result of the entire model, which ensures that the SNNs can generate the most "high-quality" membrane potential without interference before the LTS T, y[T] = f(s[T], x, θ). (11) where y[T] represents the LTS’s output of the SNNs. When using only the LTS as the output, both SDT and TET losses yield the same representation, LLTS = LCE(y[T], ygt). (12) 4.3 Label Smooth TET Loss for the Neuromorphic Tasks Based on the results in Table 1, we recommend using TET loss to achieve better performances when dealing with the neuromorphic tasks. In the original version of TET [34], an additional mean squared error (MSE) regularization term was added to control the firing level of the last layer of the model, given by LREG = 1 T PT t=1 LMSE(y[t], ϕ), where ϕ denotes the target firing level. The coefficient λ controls the proportion of the two losses, and the complete loss is defined as LTotal = (1 −λ)LTET + λLREG. We think that this setup will prevent the training loss of the model from converging to zero, because when LREG = 0, the model will output a constant value 1 at every time steps, rendering the model unable to perform the classification task. On the other hand, when LTET = 0, LMSE > 0. Considering that LMSE plays a role similar to a smoothing process in the loss function, we propose removing LREG and replacing the cross-entropy loss with a label smooth cross-entropy loss, as shown by the following equation: LTET-S = 1 T × T X t LCE(f(s[t], x, θ), ˆygt), where ˆygt = (1 −ϵ)ygt + ϵ K , (13) here, ygt represents the ground truth, ˆygt represents the smoothed label, ϵ represents the smoothing factor, and K represents the number of classes. It can be observed that LTET-S can effectively avoid the trade-off between firing level and classification accuracy for model training, and can theoretically allow the training loss to converge to zero. 5 Experiments In this section, we demonstrate the effectiveness of our proposed method by extensive experiments. We compare the results of our method with other methods on both the neuromorphic dataset and the static dataset. Additional training procedures and other hyperparameter settings are provided in the appendix A. 5.1 Execution Speed Benchmark of IMP T=2 T=4 T=8 T=16 T=32 0 5 10 15 20 IF 28 IF+IMP 28 IF 212 IF+IMP 212 IF 216 IF+IMP 216 IF 220 IF+IMP 220 Figure 6: Execution time (ms) for the forward and backward pass of IF neurons, w/wo IMP. We compare the execution speed and the memory consumption between the vanilla IF neurons and IF+IMP neurons in Figures 6. The number of neurons are set to 28, 212, 216, and 220, with time steps of 2, 4, 8, 16, and 32. All neurons are implemented by using spikingjelly and PyTorch, and the computations are performed on GPU. It can be observed that there is almost no difference (about ±1.03%) in the execution time between the IF neurons with IMP and the vanilla IF neurons, including forward and backward propagation. In addition, since the computational consumption of SNNs is mainly caused by synaptic computation, the additional overhead caused by adding IMP can be neglected. 7 5.2 Convergence Speed of LTS on the Static Data (a) T=1 (b) T=2 (c) T=4 (d) T=6 Figure 7: Convergence speed of LTS, TET, and SDT on the static data. We have conducted a validation of the convergence speed of SDT, TET, and LTS on the commonly used time steps (1, 2, 4, and 6), as shown in Figure 7. The application of LTS post-processing has resulted in an improvement in the convergence speed of SNNs. 5.3 Performances on the Neuromorphic Data Classification Table 2: Comparison of our methods and other SOTA methods on the neuromorphic datasets. Size refers to the input resolution of SNNs. Dataset Method SNN Architecture Size Time Steps Accuracy(%) CIFAR10-DVS GLIF[41] Wide 7B Net 48 16 78.10 NDA[49] VGG 48 10 79.60 TET[34] VGG 48 10 83.17 TEBN[45] VGG 48 10 84.90 PSN[43] VGG 48 10 85.90 IMP(ours) VGG 48 10 85.90 IMP+TET-S(ours) VGG 48 10 87.10 IMP+TET-S(ours) VGG 48 8 87.80 PLIF[40] PLIF Net 128 20 74.80 TDBN[21] ResNet-19 128 10 67.80 Dspike[50] ResNet-18 128 10 75.40 KLIF[51] PLIF Net 128 15 70.90 SEW ResNet[52] Wide 7B Net 128 16 74.40 Spikformer[23] Spikformer 128 10 78.90 Spikformer[23] Spikformer 128 16 80.90 NDA[49] VGG 128 10 81.70 IMP(ours) VGG 128 16 86.30 IMP+TET-S(ours) VGG 128 16 87.00 N-Caltech101 NDA[49] VGG 48 10 78.20 EventMix[53] ResNet18 48 10 79.47 ESP[54] SNN7-LIFB 48 10 81.74 TCJA[55] TCJA-SNN 48 10 82.50 TKS[56] VGG-TKS 48 10 84.10 IMP(ours) VGG 48 10 84.68 IMP+TET-S(ours) VGG 48 10 85.01 EventDrop[57] VGG 128 10 74.04 NDA[49] VGG 128 16 83.70 EventRPG[58] VGG 128 10 85.62 STR[59] VGG 128 10 85.91 IMP(ours) VGG 128 16 86.12 IMP+TET-S(ours) VGG 128 16 87.86 We have applied our methods to a simple spiking VGG model and compare them with the SOTA SNNs on the neuromorphic datasets. Since the CIFAR10DVS and NCaltech101 datasets are not pre-divided into training and testing sets, we split these datasets in a 9:1 ratio. To ensure a fair 8 comparison with other existed methods, we adopt the two configurations with resolutions of 48 and 128, respectively. The data preprocessing and training settings refer to the appendix A.8. Table 2 reports the experimental results on the CIFAR10DVS and NCaltech101 datasets. On the CIFAR10DVS dataset, when setting the resolution to 48, the IMP method achieves the SOTA accuracy of 85.9%, which is 2.73% higher than the baseline [34] and is on par with the current SOTA method PSN. It should be noted that the removal of the reset process in PSN means that the spiking activities in the previous time steps will not affect the membrane potential values in the subsequent time steps. When we set the resolution to 128, the IMP method once again demonstrates its superiority, achieving the SOTA accuracy of 86.3%, which exceeds all other methods, including the current SOTA data augmentation method EventRPG[58]. In addition, we further explore the impact of using smoothed TET loss on model performance. The experimental results show that the performance of the model has been significantly improved under both configurations, achieving accuracies of 87.00% and 87.10% respectively. In addition, the accuracy can be further improved to 87.8% by setting the time step to 8. On the NCaltech101 dataset, our method can also demonstrate excellent performance. Specifically, when we set the resolution to 48, our method achieves the current SOTA accuracy of 85.01%. When switching to resolution of 128, our method further demonstrates its advantages, achieving the SOTA accuracy of 87.86%. 5.4 Performances on the Static Data Classification Table 3: Comparison of our methods and other methods on the ImageNet1k dataset. Method Network Architecture Reset Params Time Steps Accuracy(%) PSN[43] SEW ResNet-18 11.69 4 67.63 SEW ResNet-34 21.79 4 70.54 Dspike[50] ResNet-34 ! 21.79 6 68.19 VGG-16 ! 138.42 5 71.24 TET[34] SEW ResNet-34 ! 21.79 4 68.00 TDBN[21] ResNet-34 ! 21.79 6 67.05 TEBN[45] SEW ResNet-34 ! 21.79 4 68.28 GLIF[41] ResNet-34 ! 21.79 4 67.52 Spikformer[23] Spikformer-6-512 ! 23.37 4 72.64 Spikformer-8-512 ! 29.68 4 73.38 SEW ResNet[52] SEW ResNet-18 ! 11.69 4 63.18 SEW ResNet-34 ! 21.79 4 67.04 SEW ResNet-50 ! 25.56 4 67.78 SEW ResNet-101 ! 44.55 4 68.76 SEW ResNet-152 ! 60.19 4 69.26 LTS SEW ResNet-18 ! 11.69 4 64.33(+1.15) SEW ResNet-34 ! 21.79 4 68.10(+1.06) SEW ResNet-50 ! 25.56 4 71.24(+3.46) IMP+LTS SEW ResNet-18 ! 14.17 4 65.38(+2.20) SEW ResNet-34 ! 25.54 4 68.90(+1.86) SEW ResNet-50 ! 36.67 4 71.83(+4.05) We apply our proposed IMP and LTS post-processing methods to the standard SEW-ResNet[52] architecture and compare them with the SOTA spiking neurons and the directly training SNN methods on the large-scale static dataset ImageNet1k [38]. Table 3 presents the detailed experimental results on the large-scale static dataset ImageNet1k. Specifically, by applying the LTS post-processing method to the SEW-ResNet18/34/50 models, we can obtain the accuracy improvements of 1.15%, 1.06%, and 3.46% compared to the baselines, respectively. These results demonstrate the effectiveness of LTS on the large-scale datasets. Furthermore, with the introduction of the learnable IMP, the accuracy can be further increased by 2.2%/1.86%/4.05%. With 9 the LTS post-processing and learnable IMP, our SEW-ResNet50 achieves an accuracy of 71.83%, surpassing the accuracy of the vanilla SEW-ResNet152, 69.26%. 5.5 Further Ablation Studies Table 4: Ablation Study on CIFAR10DVS and Imagenet100. Dataset Method Spiking Network Time-steps Accuracy(%) CIFAR10-DVS SDT(ϵ = 0.0) VGG 10 83.70 TET(ϵ = 0.0) VGG 10 84.90 TET-S(ϵ = 0.1) VGG 10 85.60 TET-S(ϵ = 0.01) VGG 10 86.10 TET-S(ϵ = 0.001) VGG 10 85.40 IMP+SDT(λ = 0.0) VGG 10 83.70 IMP+TET(λ = 0.0) VGG 10 85.90 IMP+TET-S(λ = 0.0) VGG 10 86.20 IMP+TET-S(λ = 0.2) VGG 10 87.10 IMP+TET-S(λ = 0.4) VGG 10 86.40 ImageNet100 TET SEW-ResNet18 4 78.50 SDT SEW-ResNet18 4 79.10 LTS SEW-ResNet18 4 80.20 IMP+TET SEW-ResNet18 4 78.70 IMP+SDT SEW-ResNet18 4 79.90 IMP+LTS SEW-ResNet18 4 80.80 Table 4 presents the results of a series of ablation studies conducted on the CIFAR10-DVS and ImageNet100 datasets, aimed at analyzing the impact of various factors on model performance. This helps understand the role of different components and parameters in the overall model, and aids in optimizing the model design. For the CIFAR10-DVS dataset, we explored methods including SDT, TET, and their variants TET-S and versions combined with IMP. VGG was used as the spiking neural network structure. The results show that for different ϵ values, the TET-S+IMP method achieved the best accuracy, with IMP+TET-S (λ = 0.2) reaching 87.10%, the highest among all methods on the CIFAR10-DVS dataset. For the ImageNet100 dataset, we tried TET, SDT, LTS, and their versions combined with IMP, using SEW-ResNet18 as the spiking neural network structure. On this dataset, the LTS method and its combined version with IMP, IMP+LTS, performed the best, reaching 80.80% accuracy. 6 Conclusions We have proposed a learnable IMP by rethinking the membrane dynamics of SNNs to enhance the dynamics mechanism of spiking neurons. Additionally, we have presented a LTS post-processing method for the static tasks and a label-smoothed TET loss for the neuromorphic tasks. It is worth mentioning that our methods only require very minor modifications to the settings and loss functions of spiking neurons to effectively improve the performance of SNNs on the static tasks and the neuromorphic tasks. At the same time, almost no additional computational cost is required. Since our proposed method has broad compatibility with existing model structures and training methods, it can be widely applied on the existed methods to further improve their network performances. Acknowledgement This work was supported by Fundamental Research Funds for the Central Universities (Grant No. SWU021002), Project of Science and Technology Research Program of Chongqing Education Commission (Grant No. KJZD-K202100203), Key R&D Program of Zhejiang (2022C01048), and National Natural Science Foundation of China (Grant Nos. U1804158, 62376247, U20A20220, and 62334014). 10 References [1] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H. 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[59] Dengyu Wu, Yi Qi, Kaiwen Cai, Gaojie Jin, Xinping Yi, and Xiaowei Huang. Direct training needs regularisation: Anytime optimal inference spiking neural network. arXiv preprint arXiv:2405.00699, 2024. 14 A Appendix / supplemental material A.1 Broader Impacts This paper focuses on the fundamental research of spiking neural networks, with the goal of revealing the impact of membrane dynamics on the network and optimizing its performance. Generally, there are no negative societal impacts in this work. A.2 Limitations IMP has a small gradient during training, which makes it sensitive to initialization (Figure 4). In addition, learnable IMP may lead to an excessive number of parameters, as it assigns initial states to each neuron, although this has the same computational cost as setting the IMP to 0. The advantage of LTS may reduce when the time step is set too large, due to the supervision is only applied at the last time step. Therefore, we only recommend using LTS on static tasks with time steps less than 8 (Table 8), which should be able to handle most situations. Additionally, the performance of the combination of LTS and the latest spike transformer technology is not yet clear. Furthermore, we have not found a unified loss function that can achieve superior performance on both static tasks and neuromorphic tasks, which remains a challenge in the current research. A.3 Convergence Speed We compared the convergence speed of TET and SDT at different time steps (T=1,2,4,6,8,12,16,24,32). For static tasks, TET’s convergence speed was lower than SDT’s, and the difference in convergence speed diminished as the time step increased. (a) T=1 (b) T=2 (c) T=4 (d) T=6 (e) T=8 (f) T=16 (g) T=24 (h) T=32 Figure 8: The convergence speed of SDT, TET and LTS on static data. A.4 Compared with Transformer-based SNNs The performance of SEW-ResNet-LTS can be close to some Transformer-based SNNs (Table 5). Table 5: Accuracy and theoretical energy consumption compared with Transformer-based SNNs. Model Param (M) SOPs (G) Power (mJ) Accuracy Spikformer-8-384 16.81 6.82 7.73 70.24 Spikformer-6-512 23.37 8.69 9.42 72.46 Spike-driven 8-384 16.81 3.90 72.28 Meta-SpikeFormer 15.10 16.70 74.10 SEW-R50-LTS (ours) 25.56 3.10 2.79 71.24 SEW-R50-LTS+IMP (ours) 36.67 3.45 3.11 71.83 15 A.5 Energy Consumption IMP does not incur significant additional theoretical power consumption, but can effectively improve the performance of SNNs (Table 6). Table 6: Accuracy and theoretical energy consumption on ImageNet1k dataset. Model Training Method Accuracy SOPs(G) Power(mJ) SEW-ResNet18 TET 62.92 1.36055 1.22449 SEW-ResNet18 SDT 63.21 1.37418 1.23676 SEW-ResNet18 LTS 64.33 1.21427 1.09285 SEW-ResNet18 LTS+IMP 65.38 1.31371 1.18234 SEW-ResNet34 TET 67.98 3.59539 3.23585 SEW-ResNet34 SDT 68.10 3.52732 3.17459 SEW-ResNet34 LTS 68.10 3.11694 2.80525 SEW-ResNet34 LTS+IMP 68.90 3.12180 2.80962 SEW-ResNet50 TET 69.87 3.40181 3.06163 SEW-ResNet50 SDT 70.33 3.20071 2.88064 SEW-ResNet50 LTS 71.24 3.10432 2.79389 SEW-ResNet50 LTS+IMP 71.83 3.45325 3.10792 A.6 Performance of LTS on DVS Tasks The LTS method can lead to information loss, especially on DVS tasks with a large number of time steps (Table 7). Therefore, we suggest considering the use of LTS only in static tasks, as the effectiveness of LTS relies on the assumption of having the same input at each time step. Table 7: Accuracy on CIFAR10DVS dataset with different time-steps. Model Training Method T=4 T=8 T=10 T=16 VGG TET 83.8 85.0 85.8 86.4 VGG SDT 83.4 84.3 84.4 85.1 VGG LTS 83.7 83.0 82.9 82.3 A.7 Enhancing Performance by Learnable IMP The learnable IMP can significantly improve the accuracy of the first time step and lead to better overall performance (Figure 9). 10.76 90.44 92.04 92.05 92.36 T=1 T=2 T=3 T=4 mean 0 20 40 60 91 92 93 94 Accuracy Constant IMP (a) ConstantIMP 72.35 91.97 92.32 92.8 93.19 T=1 T=2 T=3 T=4 mean 0 20 40 60 91 92 93 94 Accuracy Learnable IMP (b) LearnableIMP Figure 9: The test accuracy at each time step on the CIFAR10 dataset. 16 A.8 Experimental Configurations and Hyperparameter Settings Table 8 lists the key parameters required for training on the static datasets ImageNet1k, ImageNet100, CIFAR10, and CIFAR100. Table 9 outlines the key parameters used for training on the neuromorphic datasets CIFAR10-DVS-128, CIFAR10-DVS-48, N-Caltech101-128, and N-Caltech101-48. Table 8: Experimental configurations on static task. hyper-parameter ImageNet1K ImageNet100 CIFAR10 CIFAR100 architecture SEW-ResNet SEW-ResNet SEW-ResNet SEW-ResNet time steps 4 4 4 4 enable TEBN No No No No detach reset Yes Yes Yes Yes spiking neuron IF+IMP IF+IMP IF+IMP IF+IMP sg function Atan Atan Atan Atan membrane decay optimizer AdamW AdamW AdamW AdamW learning rate 0.001 0.001 0.001 0.001 weight decay 5e-4 5e-4 5e-4 5e-4 momentum epoch 320 200 200 200 warm up 10 10 10 10 lr schedule cosine cosine cosine cosine loss function SDT/TET SDT/TET SDT/TET SDT/TET label smooth 0.1 0.1 0.1 data augment standard standard standard standard enable cutmix No Yes Yes Yes enable mixup No Yes Yes Yes GPUs 4 1 1 1 Table 9: Experimental configurations on neuromorphic task. hyper-parameter CIFAR10DVS-48 CIFAR10DVS-128 NC101-48 NC101-128 architecture VGG11 VGG11 VGG11 VGG11 time steps 10 16 10 16 enable TEBN Yes No Yes No detach reset Yes Yes Yes Yes spiking neuron LIF+IMP LIF+IMP LIF+IMP LIF+IMP sg function ZIF sigmoid ZIF sigmoid membrane decay 0.25 0.5 0.25 0.5 optimizer SGD AdamW SGD AdamW learning rate 0.1 0.001 0.1 0.001 weight decay 5e-4 0.06 5e-4 0.06 momentum 0.9 0.9 epoch 200 200 150 150 warm up 0 30 0 30 lr schedule cosine cosine cosine cosine loss function SDT/TET SDT/TET SDT/TET SDT/TET label smooth 0.01 0.01 0.01 0.01 event augment standard NDA standard NDA enable cutmix No Yes No Yes enable mixup No Yes No Yes GPUs 1 1 1 1 A.9 On-chip Learning The following approach can be useful for implementing on-chip learning IMP: (1) Use an auxiliary neuron to distribute IMP (by firing a spike) to the other neurons at the initial time step. 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4,482	Read-ME: Refactorizing LLMs as Router-Decoupled Mixture of Experts with System Co-Design Ruisi Cai∗1, Yeonju Ro∗1, Geon-Woo Kim1, Peihao Wang1, Babak Ehteshami Bejnordi2, Aditya Akella1, Zhangyang Wang1 1The University of Texas at Austin, 2Qualcomm AI Research {ruisi.cai, gwkim, peihaowang, atlaswang}@utexas.edu, {yro, akella}@cs.utexas.edu, behtesha@qti.qualcomm.com Abstract The proliferation of large language models (LLMs) has led to the adoption of Mixture-of-Experts (MoE) architectures that dynamically leverage specialized subnetworks for improved efficiency and performance. Despite their benefits, MoE models face significant challenges during inference, including inefficient memory management and suboptimal batching, due to misaligned design choices between the model architecture and the system policies. Furthermore, the conventional approach of training MoEs from scratch is increasingly prohibitive in terms of cost. In this paper, we propose a novel framework Read-ME that transforms pre-trained dense LLMs into smaller MoE models (in contrast to “upcycling" generalist MoEs), avoiding the high costs of ground-up training. Our approach employs activation sparsity to extract experts. To compose experts, we examine the widely-adopted layer-wise router design and show its redundancy, and thus we introduce the pre-gating router decoupled from the MoE backbone that facilitates system-friendly pre-computing and lookahead scheduling, enhancing expert-aware batching and caching. Our codesign therefore addresses critical gaps on both the algorithmic and system fronts, establishing a scalable and efficient alternative for LLM inference in resource-constrained settings. Read-ME outperforms other popular open-source dense models of similar scales, achieving improvements of up to 10.1% on MMLU, and improving mean end-to-end latency up to 6.1%. Codes are available at: https://github.com/VITA-Group/READ-ME. 1 Introduction The success of Mixture-of-Experts (MoE) [1, 2] - such as recently exemplified by the Mixtral model [3] in the era of large language models (LLMs) - lies in its remarkable ability to leverage the collective expertise of specialized sub-networks, or "experts," each proficient in handling specific subsets or aspects of the data. By dynamically routing data through these experts, MoE models effectively capture complex patterns, adapt to diverse data distributions, and offer superior predictive accuracy compared to traditional single-model approaches. In addition to performance promise, MoEs also have a natural appeal for resource-limited devices due to their high sparsity, and therefore reduced activated parameters per token, which can potentially save inference costs [4, 5, 6, 7]. However, MoE inference presents significant challenges for key system-level objectives: • Memory Management: Although MoEs activate only a subnetwork during inference, expert selection is determined on the fly by a layerwise router, complicating efficient prefetching. This often forces reliance on naive prefetching algorithms. For example, prior work has *Equal contribution: authors are listed alphabetically. A. Akella and Z. Wang also advised this work equally. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Router Router Router Request Queue Pre-gated Inputs ? ? ? ? ? ? ? Request Queue ? Pre-gated Inputs Inference Serving at time=0 Inference Serving at time=1 Deployment Pre-trained Dense Model Refactoring Expert 1 Expert 2 Full Pre-trained Weight Router Router Token routered to Expert 1 Token routered to Expert 2 ? Tokens to be routered Figure 1: Overview of Read-ME. This figure shows the refactoring of a pre-trained dense model (in yellow) into two experts (in red and green). After refactoring, the model is deployed, and the serving timeline is depicted. At time t = 0, multiple inference requests (each a sequence of tokens) are queued, with expert assignment for each token undecided ("?") until processed by the router. Our router pre-gates tokens before inference, enabling expert-aware batching. Tokens are routed to their respective experts and batched accordingly: at t = 0 for Expert 1 (red) and at t = 1 for Expert 2 (green). New tokens enter the queue at each time step, with routing computed only for incoming tokens marked "?". predicted the next expert using hidden states from the previous layer and applied an LRU cache replacement for recently used experts [8]. While effective under certain conditions, such strategies depend on assumptions about expert locality and token predictability, which can become sub-optimal if those assumptions are violated (as shown in Table 4). • Token Batching: Token batching techniques critical for efficient inference (e.g., [9]) are poorly suited to MoE architectures, where each batch contains tokens invoking different experts across layers, rendering batching strategies ineffective (§ 4.2). Moreover, traditional MoEs are typically trained from scratch, which becomes prohibitively expensive as model scales increase. To mitigate this, some approaches, such as “upcycling" [10], reuse pretrained dense LLMs to initialize experts in an MoE. While that efficiently scales MoEs by leveraging smaller, pre-trained models, it does not address the inference-related challenges mentioned earlier. In this work, we tackle the opposite challenge: how to create a smaller MoE model from larger pre-trained models that enables resource-efficient inference while minimizing training costs? Despite existing efforts [11, 12, 13, 14], this problem remains underexplored. Approaches like [11, 12, 13] attempt MoE refactorization but still adopt systems-unfriendly layer-wise structures for inference. Similarly, [14] focuses on dynamically selecting "important" neurons during pre-filling and pruning others during generation, but this is limited to long-content generation and requires neuron importance identification for each sequence. To address both training and inference challenges, we introduce a holistic MoE framework dubbed Read-ME. To minimize training costs, we “refactorize” a pre-trained dense LLM into specialized experts through activation sparsity and optimize the routing policy (§ 3). For efficient inference, we examine the redundancy of layer-wise routers (§ 2.1, § 2.2) and propose decoupling the router from the MoE backbone (§ 2.3). This allows us to pre-gate all requests (token sequences) before inference and apply lookahead scheduling based on the experts to which tokens will be dispatched. Consequently, we propose an expert-aware batching algorithm (§ 4.2) and an optimal expert caching strategy inspired by Belady’s offline caching algorithm [15] (§ 4.1). Figure 1 illustrates our framework. Our key contributions are: • We transform large pre-trained LLMs into Mixture-of-Experts (MoE) models with fewer activated parameters and small additional training cost (1 billion tokens). Our approach outperforms popular open-source models and compression techniques of similar scale on downstream tasks like MMLU [16]. • We analyze the widely adopted layer-wise routers in existing MoEs and reveal design redundancies. Current caching policies and batching algorithms are poorly suited to layerwise MoEs. We propose a novel pre-gating router, decoupled from the MoE backbone, enabling better system-level optimization. 2 • Our system achieves a 6.1% reduction in mean latency and a 10% improvement in tail latency compared to state-of-the-art systems. Our caching algorithm is both provably and empirically optimal, thanks to our algorithm-system co-design. 2 Pre-gating Sparse Mixture of Experts In this section, we introduce our motivation and design of pre-gating MoE which enables system-level acceleration by sharing and precomputing expert selection for each layer. 2.1 System Drawbacks of Conventional Sparse MoE Design An Mixture-of-Expert (MoE) [1, 2, 17, 3] layer consists of a routing network G and a set of N expert networks {F1, ..., FN}. In the forward pass, the routing network will first process input sequences and generate the gating weights. Then a size-K subset of experts will be dynamically activated and their outputs will be combined as final outputs according to the gating weights. In LLMs, MoE is typically adopted in the Feed-Forward Networks (FFN) within each transformer block [1, 2, 3]. Suppose an LLM has L layers, the output of the l-th layer can be formulated as: y = N X i=1 I(\|{j ∈[N] : G(l)(x)j ≥G(l)(x)i}\| ≤K)G(l)(x)iF (l) i (x), (1) where the superscripts indicate the layer indices, G(l), F (l) are point-wise functions operating on tokens individually, and I(·) is the indicator function which filters experts with top-K gating weights. For shorthand, we denote I(l) i = I(\|{j ∈[N] : G(l)(x)j ≥G(l)(x)i}\| ≤K). As shown in Eq. 1, conventional MoEs assign a separate router to each layer. While this is commonly used by open-source MoEs like Mixtral [3] and OpenMoE [18], we highlight its system inefficiency. Layer-wise gating makes it difficult to predict which expert to load until runtime (§ 4.1), and complicating request batching (§ 4.2). Specifically, layer-wise routers select the l-th layer expert i : I(l) i = 1 based on the (l −1)-th layer outputs, which prevents pre-scheduling and pre-loading of data or model weights. This issue is especially problematic for billion-level parameter MoEs, where experts are usually distributed across devices (GPUs and CPUs in a machine) or even machines; in such situations, layer-wise selection accentuates high overheads of data I/O and communication among servers in the critical path of inference. 2.2 Redundancy of Layer-wise Router In this section, we demonstrate that layer-wise gating patterns are redundant in an MoE. In particular, we empirically find that expert selections between two adjacent layers are highly correlated. Distance w. E1 Expert 1 Expert 2 Distance w. E2 Pre-trained Model (Frozen) Token j Router 🔥 Loss Expert Choice ❄️ (a) (b) (c) Figure 2: (a) Visualization of transition matrix between the (l-1)-th layer and the l-th layer, where each coordinate [{s, t}, {i, j}] represents P(S(l) = {i, j}\|S(l−1) = {s, t}). The row-wise sparse pattern suggests that the router decision becomes almost deterministic given the previous layer’s decision. (b) Mutual information I(S(l); S(l−1)), which indicates the learned knowledge shared by two neighboring layers is high. (c) Overview figure of router tuning and router distillation loss. We use Mixtral-8×7B (N = 8, K = 2) [3] as a study case and analyze router decisions among its layers. Define the random variable S(l) = {i ∈[N] : I(l) i = 1} as the pair of experts selected 3 for each layer (\|S(l)\| = 2). We are interested in the conditional probability of S(l) between two consecutive layers: P(S(l) = {i, j}\|S(l−1) = {s, t}). The transition matrix of the last two layers from Mixtral-8×7B is depicted in Figure 2 (a). The row-wise sparse pattern implies that the expert selection is almost deterministic given the previous layer’ choices. For example, for tokens choosing expert-3 and expert-5 in the 30th layer, over 70% will select expert-1 and expert-5 at the 31st layer. To further validate this observation, we plot the mutual information between expert choices of every two neighboring layers: I(S(l); S(l−1)). As reflected in the right sub-figure of Figure 2, knowing expert pairs used in the last layer significantly reduces the uncertainty of the next layer. Thus, the implicit knowledge learned by each router is extensively shared across layers. 2.3 Pre-Computed Routing Policy The above observations suggest that among the many N K L routing paths, only a few are used during the inference. Therefore, layer-wise routing decisions are unnecessary for MoEs. Instead, we can separate the routerfrom the MoE backbone and pre-compute the routing path all at once. First of all, we assume the indices of experts handling one domain of tokens are aligned, i.e. {F (1) i , · · · , F (l) i } always forms a routing path. We defer our approach to the construction of aligned experts to §3. Next, we let a singleton network G generate gating weights for all layers. In particular, we adopt one transformer block with causal attention as the model architecture of G. Gating weights computed in this way not only leverage the states of the current token but also take the information from the past tokens into consideration. Thus, tokens will have expert selections similar to the recent tokens, which ensures cache-friendly inference (see more details in § 5.3). Suppose the input sequence is (xt)t=1,··· ,T , the output for the t-th token at the l-th layer is: yt = N X i=1 I(\|{j ∈[N] : G(x≤t)j ≥G(x≤t)i}\| ≤K)G(x≤t)iF (l) i (xt), (2) where x≤t = (x1, · · · , xt) represents all the tokens before the t-th token. We note that G is independent of layer index l. Despite a subtle change, it brings profound benefits to enable systemlevel optimization. In brief, by separating the gating network from the transformer layers, expert selection can be determined at the outset and used to schedule the data-loading procedure for each layer. We defer more details on system co-design to §4. 3 Re-factoring Language Model with Pre-Gating MoE In this section, we introduce the main technique to re-use a dense pre-trained model to construct our pre-gating MoE proposed in §2. In short, our approach first initializes each expert by structured pruning of a dense model on the corresponding data domains. Afterward, we instantiate a gating network shared across layers and continue joint training of the router and experts. Domain-Aware Expert Construction. We construct a set of small experts by pruning the dense model with different data domains. To begin with, we point out that public language corpora often contain metadata indicating the domain of each subset. For example, the training dataset of LLaMA family [19] can be split into scientific articles [20], novels [21], and QAs [22], etc. We utilize this metadata to group data entries in the training corpus into N sub-domains {D1, · · · , DN}. Observing that feature channels on each subset are sparsely activated [23], we compute the average magnitude of a channel on each subset and keep top activated neurons to form the domain expert. Formally, let the number of experts equal to the number of sub-domains, and assume the dense model is a two-layer FFN with hidden size D: F0(x) = W 2σ(W 1x), then the i-th experts with hidden size d are initialized as: Fi(x) = W 2M ⊤ i σ(M iW 1x), ∀i ∈[N], in which M i is obtained by: arg max M∈{0,1}d×D Ex∼Di∥MW 1x∥1 s.t. M1D = 1, M ⊤1d ≤1, (3) where M is constrained to be a selection matrix without replacement. The mask for each layer is jointly optimized so that the resultant experts are aligned layerwise and dedicated to the same data distribution. In our experiments, we set d ≈D/2. In addition, we observe that a certain 4 subset of channels is essential for all data, potentially due to the system prompt and the presence of commonsense knowledge. Therefore, we isolate the corresponding neurons as the permanent expert, which will be activated for all tokens, similar to previous designs [18, 24]. Continual Training Objective. After initializing experts via structured pruning, we perform joint training of randomly initialized gating networks and expert subnetworks via causal language modeling. In addition, we propose routing distillation loss to enhance the alignment between expert choice in pre-gating MoE and the activation sparsity in the original dense model. We illustrate the training of our router in Fig. 2 (c). Suppose the predicted token has embedding xt+1. We feed xt+1 into the original dense model F0 and get a sparse selection matrix M 0 that indicates neurons with top 50% magnitude similar to Eq. 3. Then we penalize this loss function: LRD = DKL softmax(G(x≤t+1))∥softmax([∥M 0M ⊤ 1 ∥2 F , · · · , ∥M 0M ⊤ N∥2 F ]) . (4) Here, DKL(·∥·) represents Kullback–Leibler divergence. ∥(M 0M ⊤ j ∥2 F = 1⊤ d M 0M ⊤ j 1d computes the Hamming distance between two masks induced by M 0, M j. We apply softmax to normalize these scores as the estimated selection probability of each expert for predicted token xt+1. 4 Expert-aware Inference System We demonstrate how our refactoring and pregating concepts enable a novel, high-performance, and efficient MoE inference method. We address two key challenges in existing MoE models’ inference: inadequate memory management and limited support for batched inference. Our problem setting is broad, aiming to serve multiple requests using an MoE model, each comprising a sequence of tokens. This differs from previous systems, which focused on optimizing performance for individual requests. 4.1 Pre-gating Optimized Expert Prefetching and Caching MoE models promise reduced memory usage during inference by loading only the parameters of required experts, skipping the rest. However, traditional layer-wise gating imposes significant loading costs. Previous approaches, such as on-demand loading [25], prefetching [26], and expert caching [8, 27], attempt to address this. However, on-demand loading adds overhead to the critical inference path, and prefetching often loads unnecessary experts due to incomplete routing information, leading to suboptimal memory usage and performance [28]. Additionally, caching strategies, based on request characteristics like temporal locality or activation sparsity, have mostly been evaluated in isolated single-request scenarios. In practice, expert caches are shared across multiple requests, making cache policies relying on per-request traits suboptimal. A global view across all requests is necessary for effective caching (see Table 4). Our work leverages pre-gating to develop more informed prefetching and caching strategies, resulting in significant system-level improvements. Fine-grained Prefetching. By design, our pre-gating MoE architecture enables us to prefetch the exact expert layers needed for a token or a request, avoiding guesswork. To further hide the latency in prefetching, we pipeline and thus overlap loading of experts and experts’ computation at layer-wise granularity: specifically, while computing the ith layer’s forward path in the compute stream, we load the i + 1st layer’s experts in a separate loading stream. Belady-inspired Caching. Prefetching can hide the loading latency of all but the first layer, which incurs significant cost. To mitigate this, we need a cache that stores relevant initial layers, and we argue that pre-gating enables an optimal caching strategy. The classical Belady algorithm is known to be the optimal offline cache replacement algorithm, replacing the object that will be accessed farthest in the future. While impractical in real-world systems (due to unknown future accesses), our pre-gating architecture allows us to approximate it. By decoupling the router from the backbone MoE, we can compute future expert references across requests in advance, enabling near-optimal cache replacement. Suppose that the cache at time step t −1 is as follows: C(t −1) = {e1, e2, ..., ek}, where the cache is of size k and is filled with k experts e1...k. F(e, t) represents the next time after t when expert e will be requested. Then, our policy chooses the expert eevict = argmaxe∈C(t−1)F(e, t) for eviction. 5 Router 1 Router 2 Router 3 ? ? ? ? ? ? ? Request Queue Uniq(Activated Experts) 2 3 3 Batched! Requests Arrival Traditional Layer-wise MoE Route Expert Route Expert Route Expert Time Route Expert Expert Expert Pre-gating MoE (Read-ME) Batching Happens Batching Happens Figure 3: Challenges of MoE serving in current serving systems and Read-ME’s batching pipeline. 4.2 Expert-aware Batching Current serving systems heavily rely on batching to improve inference efficiency, but effective batching for MoE models remains challenging. As shown in Figure 3 (a), each token in MoE models may invoke a different set of experts per layer, leading to multiple expert activations for a batch of requests. For example, in a toy model with 4 experts per layer and a batch of 3 tokens (one per request), 2/3/3 experts would be activated across the layers. In the Mixtral8x7B model [3] applied to the chatbot arena dataset [29], we observed an average activation of 7.63 out of 8 experts, even with a modest batch size of 56.8. The core challenge is that while each token requires computation from only one expert per layer, it must wait for all other tokens in the batch to complete their expert computations in the same layer [30]. This bottleneck repeats at each layer, reducing the efficiency of batching. Ideally, a single loaded expert would serve multiple tokens in a batch, but this is rarely achieved, affecting both performance and efficiency. For example, we observe a linear increase in average per-token processing latency as the number of unique experts per batch grows (see Figure 3 (b)). In contrast, pre-gating enhances inference performance by enabling the delayed creation of an optimal batch based on required experts. For a given set of tokens, we pre-gate each one and select a subset for batching, depending on their identified expert requirements. The goal is to minimize the number of unique experts across all layers while maximizing the number of tokens in the batch. Moreover, as discussed in § 2.3, our expert selection remains consistent across layers—if a token is assigned to Expert 1, it will be routed to Expert 1 in every layer. This approach, combined with our batching strategy, ensures optimal efficiency. Algorithm 1 provides our batching pseudocode. We note that in other MoEs, such batching isn’t feasible because, as shown in Figure 3, their expert selection at each layer remains unknown until the request reaches the router. In Read-ME, experts are determined first, which allows batches to be created and submitted to MoE layers efficiently. 5 Evaluation Table 1: Details of router design. Following the standard Transformer architecture [31], the inserted router adds only 18 million additional parameters. # Layers 1 # Heads 4 Vocab size 32000 Embedding Dim. 512 Feature Dim. 512 MLP Intermediate Dim. 512 Activation Function SwiGLU [32] Positional Embedding RoPE [33] Normalization RMSNorm [34] # Params 18.0 M In this section, we start by describing the experimental details in § 5.1. Then we validate the refactorization effectiveness on downstream tasks in § 5.2. In § 5.3, we evaluate the effectiveness of pre-gating and batching. § 5.4 analyzes memory optimization techniques. In addition, we provide experimental details in § 5.1, and more experimental results in §. A. 5.1 Experimental Details Model and Dataset We perform the MoE refactorization based on Llama2-7B-chat [19] model, a popular open-source model pre-trained on 2 trillion tokens. The training corpus [35] involves the data collected from 7 different resources: Arxiv [20], Books [21], Common Crawl, C4 [36], Github, Wikipedia [37], and StackExchange [22]. To generate experts, we collect 16 samples from each data domain, with each sample consisting of 4096 consecutive tokens. During router tuning, we use the subset of RedPajama dataet [35], with the same curation strategy. We present our detailed router design in Table 1. We use the standard Transformer [31] architecture with a 1-layer, 4-head 6 Algorithm 1 Read-ME Expert-aware Batching Algorithm (pseudocode) Input NumExperts, ReqQueueByExpert, MaxTokenLen Output ScheduledReq 1: for k ←0 to NumExperts −1 do 2: len_reqs_per_experts[k] ←len(ReqQueByExpert[k]) 3: end for 4: while true do 5: E ←argmax(len_reqs_per_experts) 6: if len_reqs_per_experts[E] < (MaxTokenLen −len(ScheduledReq)) then 7: ScheduledReq ←ScheduledReq ∪ReqQueueByExpert[k] 8: ReqQueueByExpert[E] ←[ ] 9: len_reqs_per_experts[E] ←0 10: else if MaxTokenLen −len(ScheduledReq) ≥0 then 11: n_available ←MaxTokenLen −len(ScheduledReq) 12: ScheduledReq ←ScheduledReq ∪ReqQueueByExpert[k][: n_available] 13: ReqQueueByExpert[E] ←ReqQueueByExpert[k][n_available :] 14: len_reqs_per_experts[E] ←len(ReqQueueByExpert[E]) 15: break 16: else 17: break 18: end if 19: end while design. The router is lightweight, consisting of 18 million additional parameters, and incurs negligible computational overhead. We use 8 A100 GPUs with 80GB of memory for all tuning experiments. Table 2: Hyper-parameter choice during the training. Stage Router Tuning Expert Tuning # Iteration per Round 100 200 # Rounds 8 8 Initial LR at Round 0 5e−4 5e−5 LR Decay within Round Cosine Cosine LR Decay type across Rounds Exponential Exponential LR Decay rate across Rounds 0.8 0.8 Weight Decay 0.01 0.01 Batch Size 64 128 Sequence Length 4096 4096 # Tokens per Round 26 M 105 M # Tokens in Total 1.04 B Continual-Tuning Details To co-optimize the router and expert networks, we iteratively tune each model component. Specifically, we first optimized the router by LRD, as detailed in § 3, for 100 steps. We use the batch size of 64 in this router tuning stage. During this router tuning stage, we freeze the expert weights and solely tune the router weights. Then, during the expert tuning stage, we fix the router weights and modify the expert weights via language modeling loss, for 200 steps, with a batch size of 128. We set sequence length to 4096 for all stages, following the choice in the pre-training stage of Llama2 model [19]. This iterative training schedule is conducted 8 times. Detailed visualizations of the training dynamics are provided in Section A.1. For each round, the router tuning and expert tuning stages will cost 26 million and 105 million tokens, respectively. The whole continual-tuning process merely uses 1.04 billion tokens, negligible compared to the pre-training cost (2 trillion tokens). During each round of tuning, we use the cosine learning rate decay. At round 0, the initial learning rates are 5e−4 for router tuning and 5e−5 for expert tuning. The initial learning rate decays exponentially with a decay rate of 0.8 as the number of rounds increases. Inference System Evaluation For our workload, we utilize the Chatbot Arena Conversation Dataset [29] to generate inference requests and replay conversation traces. Our setup employs a single A100 GPU with 80GB of memory. The implementation is built on top of DeepSpeed inference engine [38]. We use normalized latency as our primary metric, defined as the end-to-end latency divided by the generated token length, in line with previous works [9, 39, 38]. 5.2 Downstream Task Evaluations We first validate the refactorization effectiveness on downstream tasks, as shown in Table 3, comparing it to other models of similar scales, including the open-source models that trained from scratch, and 7 OpenMoE Llama2-7b Read-ME 0 10 20 30 40 50 Latency[ms] Router Attention Expert/MLP Others 0 20 40 60 80 100 120 0 2 4 p95 latency (proposed) proposed 0 20 40 60 80 100 120 0 2 4 6 p95 latency (prefil) prefil 0 20 40 60 80 100 120 Latency [ms] 0 2 4 6 p95 latency (decoding) decoding Distribution 0 10 20 30 40 50 60 Temporal Distance 21 23 25 27 29 211 # Tokens #Token=2921 #Token=850 Read ME Mixtral-8x7B Figure 5: Latency evaluation and Temporal locality analysis. (Left) Single inference latency measured on a 124 token generation task. (Center) Latency distribution measured on synthetic workload replaying Chatbot Arena Dataset [29] (§ 5.1). (Right) Temporal distance measured on Arxiv dataset [20], and a subset of Redpajama [35]. the dense models pruned from larger pre-trained LLMs. We achieve the best average performance, outperforming all model variants from the Pythia [40] and Open-Llama-v2 [41] families, as well as Sheared-Llama [42]. We use just 1 billion training tokens, considerably less than other models. Table 3: Downstream task evaluation of our proposed method (Read-ME) compared to open-source models, including dense models Pythia and Open-Llama-v2, the MoE model OpenMoE, and the compression method Sheared-Llama. The evaluation includes zero-shot performance on WinoGrande, ARC-Easy, LogiQA, CoQA; 5-shot performance on MMLU; 10-shot on Hellaswag; and 25-shot on ARC-Challenge. The “#Param” column presents in the form of (# Activated-Parameters - # Total-Parameters). Training cost is measured by the number of tokens used. For compression methods like ours and Sheared-Llama, only tokens used for conversion are counted, excluding Llama-2 pre-training costs. Method #Param Cost MMLU Hell. Wino. ARC-E ARC-C LogiQA CoQA avg. Sheared-Llama 2.7B 50B 26.4% 70.8% 67.0% 67.0% 41.2% 28.3% 71.7% 53.2% Pythia 2.8B 300B 26.9% 60.8% 59.7% 64.4% 36.4% 27.7% 61.9% 48.3% Open-Llama-v2 3.4B 1T 25.7% 67.6% 63.5% 66.5% 39.0% 28.1% 54.4% 49.3% OpenMoE 2.1B-8B 1.1T 26.2% 45.5% 60.3% 64.1% 30.3% Read-ME 4.7B-17B 1B 38.9% 68.5% 67.7% 66.6% 42.3% 29.7% 74.8% 55.5% Pythia 6.9B 300B 25.5% 67.1% 64.1% 67.3% 31.3% 25.3% 63.6% 49.2% Open-Llama-v2 6.9B 1T 40.2% 66.7% 66.0% 63.0% 36.0% 27.6% 64.5% 52.0% Llama-2 6.9B 2T 45.3% 78.6% 69.3% 76.4% 53.0% 31.0% 75.9% 61.4% Read-ME Pythia Open-Llama Llama2 SliceGPT LaCo Compresso LLM-Pruner Open-Llama Pythia Sheared-Llama OpenMoE Figure 4: Evaluation of Read-MEon MMLU [16] benchmark, compared to other open-source models and compression techniques ( performance numbers are collected from their respective papers. In Fig. 4, we further provide a direct comparison with other compression methods, which converts a large LLM to a small dense variant, on MMLU [16] benchmarks. Besides open-source models and Sheared-Llama [42] which are mentioned in the previous table, we additionally include recent compression techniques, including LLM-Pruner [43], SliceGPT [44], LaCo [45], and Compresso [46], as our baselines. ReadMEachieves the best performance among the models with the number of activation parameters less than 5 billion, and shows comparable performance with Open-Llama-v2-7B [41]. More analysis is included in § A.2. 5.3 Pre-gating and Expert-aware Batching Inference Latency Breakdown. We evaluate the impact of the auto-regressive router introduced by our refactoring of the dense MoE on per-request inference latency. Unlike conventional layer-wise routers, usually linear layers, our auto-regressive router comprises a multi-head attention layer and an MLP layer (see § 2.3), potentially raising its computational cost. Fig. 5 (left) illustrates the average per-token latency breakdown of a single isolated inference request measured in OpenMoE [18] with conventional layerwise routers, our refactored model with pregating router, and the original dense Llama2-7b model [19] we refactored. We find that the computational 8 overhead of our auto-regressive router is minimal – its contribution of 0.4% is much less compared to the router’s net contribution in other MoE models (3.95%). This is because we use a single router unlike other models with gating for each MoE layer; also, our router design is compact with only 18M parameters (Table 1). Compared to the dense model, we achieve a net 19% reduction in latency via refactoring the MLP to MoE. Batched Inference. We now evaluate the efficacy of our expert-aware batching. Fig 5 (center) shows the latency distribution and the 95-th percentile latency (p95) during batched inference. We compare with two widely used techniques – Decoding-prioritized batching [38], and Prefill-prioritized batching [39, 47]. These methods utilize distinct queues for decoding requests and prefill requests, prioritizing batching of tokens from decoding and prefill requests, respectively. Prioritizing either decoding or prefill requests yields comparable performance. In contrast, our method of constructing batches based on activated experts enhances the mean latency by 5.0-6.1% and reduces the p95 latency by 9.5-10.0% compared to these approaches. The primary reason for this improvement is that our batching approach directly reduces the average number of unique experts invoked per batch by leveraging pre-gated information. Specifically, for decoding-prioritized and prefill-prioritized batching, the average number of unique experts per batch was 5.08 and 5.21, respectively, whereas our method reduces this to 3.51. We observed a significant performance impact as prefill requests invoke more experts per batch compared to decoding requests. Prefill requests require tokens to be dispatched to different experts, making it impractical to batch tokens by shared experts due to attention operations. As a result, a substantial number of experts are invoked for each batch, negatively affecting performance. Fortunately, our auto-regressive router design improves temporal locality in prefill requests, often allowing tokens within the same request to select the same or a small number of experts. We explore this locality in greater detail in the following section. High Temporal Locality. To analyze the locality, we measure the temporal distance of the tokens in a sequence (Fig. 5 (c)). We define temporal distance as the distance between two tokens selecting the same expert within a sequence [48]. Our result shows that our router leads to a smaller distance, indicating a high degree of temporal locality. Specifically, out of 4096 tokens, 2921 tokens follow the choice of the last token, compared to 850 tokens in Mixtral-8×7B. The locality is attributed to the auto-regressive design of our router, where the router’s decision is based on the current and all previous tokens. As a result, a given token is likely to have similar expert selections with its recent predecessor tokens. However, note that this temporal locality appears only within the token sequence of a single request and does not appear across different requests. 5.4 Memory-Efficient Inference 2 3 4 5 Cache Capacity 0 50 100 150 200 250 300 350 Latency [ms] Prefetching On-demand Loading Figure 6: Latency impact of prefetching: We measured end-to-end latency on a synthetic workload generated by replaying Chatbot Arena Dataset [29]. (Appendix 5.1) We evaluate how well our approach can ensure good performance while improving memory efficiency. In particular, we constrain the expert cache capacity to k (that is, up to k experts can reside in accelerator memory). In this setup, if a requested expert is not in memory, it must be loaded from host memory, potentially increasing loading latency. As explained in § 4.1, this loading overhead can be mitigated with prefetching, provided that we know which expert will be needed in ReadME. We compare the end-to-end latency of requests from the prefetching our approach enables (Prefetching) versus not leveraging prefetching (On-demand Loading) [25]. Figure 6 shows that for varying cache capacities, we consistently outperform On-demand Loading, with up to 30% better latency. In addition to proactively loading experts into memory, our approach also retains experts in a cache to further use memory optimally. Table 4 compares three representative caching policies’ hit ratios across varying cache capacities, including the Beladyinspired approach that our architecture enables. As noted earlier, our approach accommodates multiple requests where each request has a token sequence, in contrast with prior works focusing on a single request/token-sequence [8, 27]. 9 Cache Policy Cache Capacity Random LRU Belady 2 34.19% 33.90% 44.16% 3 50.14% 52.42% 61.82% 4 67.52% 66.95% 77.21% 5 82.91% 83.48% 88.03% Table 4: Cache hit ratio measured in batched inference setup. When multiple requests share the expert cache, temporal locality within a single request cannot be leveraged across requests, limiting its effectiveness. This explains why LRU, which works well in single-request scenarios, underperforms in our setup. In contrast, our Belady-based algorithm excels at all cache capacities by utilizing future expert information across requests, thanks to the pre-gating router. When cache capacity is constrained by system memory, latency can be significantly reduced with an optimized cache policy. Our Belady approach notably improves latency, particularly under limited cache sizes, though we omit detailed results for brevity. 6 Related Work MoE Refactorization. Recent “MoE-fication" methods [11, 12, 13, 49] optimize or group channels using graph-based techniques but still rely on system-inefficient layer-wise routers. In contrast, we are the first to identify the redundancy in layer-wise routers and propose a pre-gating router that enables expert pre-fetching. Similar to [50, 14, 51], we leverage activation sparsity [23] to construct experts, adaptively identifying important neurons and evicting less-important ones during inference. Efficient Inference Serving. To deal with the limited memory in resource-constrained settings, prior LLM inference works focused on optimizations such as offloading parameters to host memory [52, 53, 25], quantization [54, 55, 56], sparsity [57, 58] and MoE architectures [4, 59, 26]. However, while token batching [9] has garnered significant attention for dense models [39, 47, 38, 60], it remains problematic and underexplored in the context of MoE models. Pre-gated MoE [28] is related to Read-MEas they too fine-tune a router to pre-gate using the ith layer’s hidden states to compute the i + 1th layer’s routing; but they still maintain a layer-wise architecture which constrains batching. SiDA-MoE [61] separates the router from the inference path. However, tokens cannot be batched together because they do not share routing decisions across all layers. In addition, the offline routing function of SiDA is an approximation that may incorrectly guess expert selection, especially when the model scales. In contrast, Read-MEhas exact routing, ensuring no performance drop during inference. Mixtral-offloading [8] introduces speculation to “guess" routing decisions, resorting to costly ondemand loading if speculation fails. Caching is commonly used [62, 52, 63, 53, 64], including in MoE systems [8, 27], which typically focus on single requests. Prior caching methods are limited by layer-wise routing and lack of foresight into future requests. 7 Conclusions and Limitations We address the under-explored challenge of reusing a pre-trained LLM to create a smaller MoE model that enables efficient inference with minimal training cost. By leveraging activation sparsity, we construct specialized experts and integrate them via a router. Upon analyzing the layer-wise router design used in all open-source MoEs, we identify its inefficiency and redundancy. To overcome this, we propose a pre-gating router, decoupled from the MoE backbone, enabling system-level optimizations that were previously unattainable. Limitations. Our serving system is designed for a single accelerator, and extending it to distributed serving remains a non-trivial task for future work. 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We visualize the validation loss during the first 4 rounds out of the total 8 rounds of training. In Fig. 7, the router tuning stages are marked in gray, while the expert tuning stages are marked in orange. Two observations can be drawn from Figure 7: (1) The validation loss decreases during both router tuning and expert tuning stages. (2) The validation loss reduction from router tuning saturates after two rounds, while the validation loss continues to decrease during expert tuning. A.2 MoE Achieves Better Efficiency-Accuracy Trade-off than Dense Models. Prior compression-based works [42, 43, 44, 45, 46] focus on converting a large dense pre-trained model into a smaller dense model. However, we argue that a smaller MoE model (i.e. the MoE model with the smaller number of activation parameters) is a better target architecture. To ensure a fair comparison, we (1) derive a small dense model with 4.7B parameters, matching the size of a single expert network, using the same amount of data, and (2) fine-tune the obtained dense model for an equivalent number of steps. As shown in Table 5, refactorizing the pre-trained model into an MoE structure, rather than a smaller dense variant, leads to significant performance improvement. The models are evaluated based on performance on the MMLU [16], and perplexity across seven data domains included in RedPajama [35]. Table 5: We compare the Read-MEperformance with dense model, and report the MMLU performance and perplexity on 7 data domains. By adopting an MoE as the target structure instead of dense model, our model achieve significantly better overall performance. Evaluation Arxiv Books C4 Common Crawl Github StackExchange Wikipedia MMLU Dense 5.63 1.94 11.78 9.68 3.75 13.42 6.24 27.1% Read-ME 4.18 1.31 10.57 7.72 2.39 12.52 3.94 38.9% A.3 Read-ME Remains Effective without Prior Knowledge of the Training Domain We additionally use the Mistral [65] model as the pre-trained dense model, and convert it to the MoE structure, with the proposed method. The task is challenging because we do not have prior knowledge on the Mistral original training data, and our experiment in Table 6 shows that our method remains effective without the prior knowledge of the original training domain. Table 6: Ablation study on Mistral [65] pre-trained model. Method Pre-trained Fine-tune #Param MMLU Hell. Wino. ARC-E ARC-C LogiQA CoQA avg. Domain Domain Read-ME-Llama-2 Red-pajama Red-pajama 4.7B-17B 38.9% 68.5% 67.7% 66.6% 42.3% 29.7% 74.8% 55.5% Llama-2 Red-pajama 6.9B 45.3% 78.6% 69.3% 76.4% 53.0% 31.0% 75.9% 61.4% Read-ME-Mistral N/A Red-pajama 4.7B-17B 39.2% 79.1% 68.2% 77.1% 49.3% 30.9% 76.2% 60.0% Mistral N/A 6.9B 62.1% 84.5% 79.3% 82.7% 63.7% 33.5% 80.3% 69.4% A.4 Computational Cost of Auto-regressive Router For a detailed cost analysis of auto-regressive router that we introduced, we added: (1) FLOPs comparison, (2) latency, and (3) latency breakdown with a larger batch size (high-throughput scenarios) of a Traditional Router (TR) and an Autoregressive Router (AR). To focus solely on the router’s impact on latency, we controlled other variables (e.g., the number of activated parameters) to be the same. Note that the computational cost of both the traditional router and the autoregressive router is theoretically linear to batch size. Therefore, when the batch size is high (in high-throughput 15 Table 7: Flops comparison between Traditional router and Auto-regressive router Traditional Router Auto-regressive Router Flops/sample 4.7 KFLOPs 3 KFLOPs Table 8: Latency [ms] comparison between Traditional router and Auto-regressive router bsz=5 bsz=5 bsz=10 bsz=10 bsz=20 bsz=20 bsz=30 bsz=30 TR AR TR AR TR AR TR AR Router 1.76 0.61 1.80 0.61 1.78 0.61 1.93 0.61 Attention 18.13 18.18 18.28 18.13 18.49 18.36 19.59 19.66 Expert/MLP 22.43 21.75 24.59 22.53 24.97 22.99 30.17 28.31 Sum 42.31 40.55 44.67 41.27 45.23 41.96 51.69 48.59 Table 9: Latency breakdown comparison between Traditional router and Auto-regressive router bsz=5 bsz=5 bsz=10 bsz=10 bsz=20 bsz=20 bsz=30 bsz=30 TR AR TR AR TR AR TR AR Router 4.15% 1.50% 4.02% 1.48% 3.93% 1.46% 3.74% 1.26% Attention 42.85% 44.85% 40.92% 43.92% 40.87% 43.75% 37.90% 40.47% Expert/MLP 53.01% 53.65% 55.06% 54.59% 55.20% 54.80% 58.36% 58.27% Sum 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% scenarios), the cost increases linearly. In both cases, the computation can be parallelized, so this remains negligible in end-to-end latency even in high-throughput scenarios. In fact, we would like to clarify that the bottleneck in high-throughput scenarios is actually the expert layers, as seen in Table 9 – Expert/MLP row. This issue can be addressed by the methods discussed in Section 4. Traditional layerwise routers do not allow for efficient system design, which underscores the need for a careful co-design of routers. 16 NeurIPS Paper Checklist The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and precede the (optional) supplemental material. The checklist does NOT count towards the page limit. 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4,483	Understanding the Expressive Power and Mechanisms of Transformer for Sequence Modeling Mingze Wang School of Mathematical Sciences, Peking University, Beijing, China mingzewang@stu.pku.edu.cn Weinan E † Center for Machine Learning Research and School of Mathematical Sciences, Peking University, Beijing, China AI for Science Institute, Beijing, China weinan@math.pku.edu.cn Abstract We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the dot-product self-attention, positional encoding and feed-forward layer, affect its expressive power, and we study their combined effects through establishing explicit approximation rates. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads. These theoretical insights are validated experimentally and offer natural suggestions for alternative architectures. 1 Introduction In recent years, Transformer networks (Vaswani et al., 2017) have emerged as foundational models, setting new benchmarks across various domains, including natural language processing (NLP), computer vision (CV), and protein folding. Despite their impressive practical achievements, the underlying mechanisms and theoretical foundations of Transformer networks remain largely elusive. Transformer networks encompass various components, posing challenges to their comprehensive understanding. A typical Transformer comprises multiple layers, each consisting of a multi-head self-attention (Attn) sub-layer and a feed-forward network (FFN) sub-layer, integrated with residual blocks. FFN is a two-layer nonlinear network, while Attn includes dot-product (DP) and positional encoding (PE). To get a better understanding of how Transformer works in practice, we need to study several key issues. These include: (i) How do the key hyper-parameters, for example, the number of layers, the number of Attn heads and the with of FFN layers, affect the performance of the Transformer network? (ii) How do the Attn and FFN layers contribute differently to the overall performance? (iii) How does DP attention work, and is the DP structure necessary? (iv) How efficient is PE in modeling long-range correlations? Extensive empirical research on Transformer components has led to the proposal of numerous alternatives to the current structure of Transformer. For example, several relative positional encodings (RPE) (Shaw et al., 2018; Raffel et al., 2020; Su et al., 2024; Press et al., 2022) have been proposed 38th Conference on Neural Information Processing Systems (NeurIPS 2024). to substitute the original absolute positional encoding (APE), yielding superior performance in challenging tasks like length generalization (Ontanón et al., 2022; Csordás et al., 2021; Anil et al., 2022). Additionally, the necessity of the computationally expensive DP in Attn layers has been widely questioned, and researchers proposed numerous alternatives of DP that show considerable efficacy in specific tasks (Kitaev et al., 2020; Wang et al., 2020; Choromanski et al., 2020; Tay et al., 2021; Allen-Zhu and Li, 2023). Nonetheless, these explorations have not yielded a satisfactory theoretical understanding of the mechanisms of these components. In this work, we investigate the expressive power of Transformer and the underlying mechanisms of its components for sequence modeling. Our contributions are summarized as follows: We categorize three types of sequence modeling tasks with varying complexity, which are relevant to a broad spectrum of application areas. Task I: Modeling fixed, long but sparse memories. This is relevant to sparse Boolean functions and the traditional n-gram model in NLP. Task II: Modeling adaptive, long but sparse memories. This is relevant to multi-step reasoning tasks as well as various NLP tasks such as dependency parsing, sentiment analysis, and continuation writing. Task III: Modeling essentially sparse memories. Examples include feature representation in CV and wavelet analysis in classical signal processing. For these sequence modeling tasks, we theoretically investigate the expressive power of Transformer and its variants, establishing explicit approximation rates. Our meticulous analysis provides theoretical insights into the underlying mechanisms of Transformer components. Specifically, • The distinct roles of the number of layers, the number of Attn heads, and the width of FFN layers. Deeper Transformer are capable of handling memories with more intricate interrelationships, such as nested relationships (Thm 4.4). In contrast, for memories lacking such interrelationships, single-layer Transformer with sufficient number of Attn heads and FFN width should suffice (Thm 4.1). This is quite intuitive: If the content of the next token relies on a few previous tokens in an independent way, we can treat each such dependence by a separate attention head. There is no need for many layers. Additionally, increasing the depth can also alleviate the reliance on the number of heads and width (Prop 4.5). • The different roles of Attn layers and FFN layers. Our results consistently suggest that: FFN layers are tasked with approximating nonlinear memory functions and the readout function, while Attn layers are responsible for extracting the tokens from these memory locations. • The functionality and necessity of DP. For the relatively simple Task I, DP is not necessary and can be omitted (Thm 3.1). However, for the more complex Task II, the cooperation between DP and RPE provides the needed interaction between the temporal space and the token space, crucial for the extraction of adaptive memories (Thm 4.1 and 4.4). Additionally, for Task II, while the nonlinearity provided by DP is necessary (Prop 4.2), a computationally efficient alternative to DP exists, as we show in Prop 4.3. • The efficiency of RPE in modeling long-range correlations. Our results consistently suggest that the primary role of RPE is to approximate the memory kernels. Specifically, for Task III, we demonstrate that Transformer with suitable RPE can handle heavy-tailed memories, thus overcoming the Curse of Memory faced by recurrent neural networks (Thm 5.1). Moreover, our findings give theoretical support to the choice of RPE in practice. Finally, we conduct experiments to validate our theoretical insights. 2 Preliminaries Basic notations. We use bold-faced letters for vectors or matrices and lowercase letters for scalars, e.g. x = (x1, · · · , xd)⊤∈Rd and W = (Wij)m×n ∈Rm×n. The standard Euclidean inner product between two vectors is denoted by ⟨·, ·⟩, and the lp norm of a vector is represented by ∥·∥p. We employ standard big-O notations O, Ω, Θ to hide absolute positive constants and use ˜O, ˜Ω, ˜Θ to further hide logarithmic constants. For any positive integer n, let [n] = {1, · · · , n}. Denote by I{E} the indicator function for an event E. Denote by a ∨b = max{a, b} for real number a, b. 2 2.1 Sequence modeling with long but sparse memories Sequence modeling. For convenience, we consider input sequences of infinite length (t ∈Z). It is important to note, however, that our theoretical framework can be adapted to finite-length input sequences by masking distant tokens. Formally, the output sequence Y = (yt)t∈Z ∈Rc×Z is generated from the input sequence X = (xt)t∈Z ∈X ⊂Rd×Z via an unknown mapping H·(·) dependent on the input sequence up to the prediction time, and this can be expressed as: yt = Ht(X) = f(xt, xt−1, xt−2, · · · ), t ∈Z. (1) Our objective is to learn the mapping H·(·). Additionally, we define the norm \|\|\|H\|\|\| := supt∈Z supX∈X ∥Ht(X)∥. Without loss of generality, we assume ∥xt∥2 ≤1 for any X ∈X and set the output dimension c = 1 for simplicity. Long but sparse memories. To model such sequences, we define three types of memories: fixed, long but sparse memories; adaptive, long but sparse memories; and essentially sparse memories. These memory types are prevalent in sequence modeling tasks across diverse domains such as NLP, CV, signal processing, and sparse function representation. In Section 3, 4, and 5, we will formally define these different types and investigate Transformer’s capacity to model them. 2.2 Transformer architecture Transformer network. Transformer (Vaswani et al., 2017) is a network architecture designed for processing sequences and generating predictions. Given an input sequence X, Transformer executes the following steps. Initially, each d-dimensional (dim) input token is transformed into a D-dim vector through an embedding mapping such as x(0) t = WExt + bE, where WE ∈RD×d, bE ∈RD. Subsequently, a typical L-layer Transformer with residual block operates according to the formulation: X(l−1 2 ) = X(l−1) + Attn(l)(X(l−1)), l ∈[L]; X(l) = X(l−1 2 ) + FFN(l)(X(l−1 2 )), l ∈[L]. (2) At the l-th layer, FFN(l)(·) denotes a standard (point-wise) two-layer ReLU networks with m neurons: for a given input x ∈RD, FFN(l)(x) = Pm k=1 a(l) k σ b(l)⊤ k x + c(l) k , where σ(·) is the activation function such as ReLU. Additionally, in the final (L-th) FFN layer, the residual block is omitted, commonly referred to as the readout function. Moreover, Attn(l)(·) refers to a multi-head self-attention, as elaborated below. Multi-head self-attention. Our focus lies on standard dot-product Attn, denoted as Attn(l)(·) and consisting of H heads. When applied to an input sequence X, Attn operates as follows: Attn(l)(X) = W (l) O H X h=1 W (l,h) V Xsoftmaxc D W (l,h) Q X, W (l,h) K X E + R(l,h) . (3) Here, the parameters W (l,h) Q , W (l,h) K , W (l,h) V , W (l,h) O correspond to the query, key, value, output matrices of the (l, h)-th head, respectively. softmaxc represents taking softmax normalization across column. Furthermore, R(l,h) ∈RZ×Z denotes the relative positional encoding matrix, which satisfies R(l,h) t,s = −∞for t < s in the next-token prediction paradigm. Consequently, the t-th output of Attn is expressed as: Attn(l) t (X) = W (l) O H X h=1 +∞ X s=0 W (l,h) V xt−s exp D W (l,h) Q xt, W (l,h) K xt−s E + R(l,h) t,t−s P+∞ j=0 exp D W (l,h) Q xt, W (l,h) K xt−j E + R(l,h) t,t−j . Logarithmic and Power relative positional encoding. As highlighted in Section A, among various types of RPEs, the RPEs used in T5 and KERPLE(log) demonstrate superior performance over Alibi, significantly outperforming other RPEs and APEs in the length generalization task (Kazemnejad et al., 2023; Chi et al., 2022). This finding motivates our focus on the T5-type, KERPLE(log), and Alibi-type RPEs throughout this paper. All of these RPE matrices are Toeplitz, with the form of 3 Rt,s = r(t −s). Notably, for T5 and KERPLE(log), r(t −s) undergoes an initial linear decrease followed by a logarithmic decrease as the relative distance t −s increases (Please refer to Section G.1 for more details). In contrast, for Alibi, r(t −s) decreases linearly. Inspired by these discussions, we examine the following RPEs with different decay rates: ϕlog(z) = −log z, z ≥1 −∞, otherwise ; ϕlin(z) = −z, z ≥0 −∞, otherwise . We will study Transformer with ϕtype RPE (type ∈{log, lin}). Specifically, the RPE in the (l, h)-th head (3) is as follows: R(l,h) t,s := p(l,h)ϕtype(t −s), (4) where p(l,h) ∈R+ is a trainable parameter. Remark 2.1. For standard Transformer (2) incorporating Attn (3) with RPE (4), the parameters are: the embedding matrix WE; a(l) k , b(l) k , c(l) k in the FFN layers; W (l,h) Q , W (l,h) K , W (l,h) V , p(l,h), W (l) O in the Attn layers. Notably, the number of parameters is independent of the sequence length, thus enabling the model to handle input sequences of arbitrary length. Remark 2.2. In the subsequent sections, we will analyze Transformer and its variants. For the sake of brevity, some shorthand notations are introduced here. For examples, Transformer (2) using ϕlog/ϕlin RPE (4) is referred to as “Transformer with log/lin-RPE”; Transformer with W (l,h) Q , W (l,h) K = 0 is called “dot-product-free Transformer”. 2.3 Expressive power via approximation theory This paper delves into the expressive power of Transformer through the lens of approximation theory, with a specific focus on establishing explicit approximation rates for Transformers in modeling long but sparse memories. Approximation rates v.s. universal approximation. In approximation theory, results are generally categorized into two types: universal approximation (density-type) and approximation rates (Jacksontype) (Jackson, 1930). Universal approximation investigates whether the hypothesis class is dense in the target class. Although this property is fundamental, it does not offer detailed insights into approximation efficiency. In contrast, approximation rates go deeper, emphasizing the efficiency of the approximation. A typical example within this framework is the approximation theory of two-layer neural networks (2NNs). Barron space of 2NNs. The well-known universal approximation result for 2NNs asserts that 2NNs can approximate any continuous function (Barron, 1992; 1993; 1994). Nonetheless, this result lacks a characterization of the approximation efficiency, i.e., how many neurons are needed to achieve a certain approximation accuracy? This gap was addressed by the Barron space theory (E et al., 2019; 2021; Ma et al., 2020). It is established that for any function within Barron space f ∈B (Appendix G.2), 2NNs with m neurons (denoted by Hm) can approximate them efficiently, at a rate of inffm∈Hm ∥f −fm∥≤O(∥f∥B /√m), remarkably independent of the input dimension d, thus avoiding the Curse of Dimensionality (Bellman, 1966; Bach, 2017). 3 Fixed, long but M-sparse memories 3.1 Problem formulation Fixed, long but M-sparse memories. In this section, we investigate a fundamental category of long but sparse memories. Our focus is on scenarios where the positions of the sparse memories remain fixed and are independent of the tokens. The target function is represented by: yt = f(xt, xt−T1, · · · , xt−TM ), (5) where 1 ≤T1 < · · · < TM < +∞signify the fixed positions of the memories. Despite the memories being fixed (token-independent) and sparse (finite M), the task can still be complex due to the potentially long-range memories (T1, · · · , TM can be large enough). 4 Examples. (I) For Boolean inputs, (5) aligns with sparse Boolean functions, also studied in (Edelman et al., 2022; Bhattamishra et al., 2022). Notably, Bhattamishra et al. (2022) observed that Transformers outperform LSTMs in learning sparse parities. (II) Selecting the simplest case of Ti = i in (5) corresponds to the traditional n-gram model, which consists of short and sparse memories. Target class. We focus on target functions described in (5). The readout function f is considered within the standard Barron space B, i.e., which can be effectively approximated by 2NNs. Moreover, we assume that f is Lipschitz, denoted by f ∈L. Thus, we can focus more on investigating the memory extraction power of Transformer. Formally, we define the target class for modeling fixed, long but M-sparse memories as: HFix := H : Ht(X) = (5), where 1 ≤T1 < · · · < TM < +∞, f ∈B ∩L . (6) Transformer hypothesis class. As mentioned in Section 1, one of our main aims is to study the necessity and roles of different components in Transformer, such as DP and RPE. This section focuses on the “simplest” one-layer Transformer and investigates whether it can effectively model this task. Formally, our hypothesis class includes all one-layer DP-free Transformers, configured with H Attn heads and FFN width m: T FDPF,type (1,H,m) := TF : TF is a 1-layer, H-head, m-width dot-product-free Transformer with type-RPE . (7) 3.2 Theoretical results and insights Theorem 3.1 (Approximation rate). For any target H ∈HFix (6), rate n ∈N+, and H, m ∈N+, there exists a 1-layer Transformer TF ∈T FDPF,type (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =    O C(n) Hn PM i=1 e0.01Ti n+1 , type = lin O C(n) Hn PM i=1 T 1.01 i n+1 , type = log . Theorem 3.1 establishes the approximation rate of one-layer DP-free Transformer for modeling fixed, long but sparse memories. Here, the model complexity is governed by the number of Attn heads H and the width of FFN layers m, while the target complexity arises from the lengths of the memories T1, · · · , TM and the complexity of the readout function f. The approximation error comprises two components: the error in the FFN component EFFN and the error in the Attn component EAttn(type). The error EFFN aligns with classical results, showcasing its effectiveness in approximating Barron functions. On the other hand, EAttn(type) hinges on the capacity of the Attn block for modeling long-range memories. Specifically, with increasing memory length, the necessary number of Attn heads grows at a small exponential rate for lin-RPE and at a polynomial rate for log-RPE. The proof of Theorem 3.1 is deferred to Appendix B. We can draw some insights from Theorem 3.1 and its proof. Different roles of the Attn layer and the FFN layer. The Attn and FFN layers fulfill distinct roles in this task. Specifically, the FFN layer efficiently approximates the nonlinear readout function f, while the Attn layer is responsible for extracting the token xt−Ti by approximating the memory kernel I{· = Ti}. These components together enable effective modeling of fixed, long, but sparse memories. Non-necessity of DP. Theorem 3.1 suggests that the DP component in Attn is not necessary and can be omitted for modeling fixed, long but sparse memories. This is due to the relative simplicity of modeling fixed memory kernels. In a more complex scenario in Section 4, the role of the dot-product becomes important. In contrast to Edelman et al. (2022), which utilizes the property of DP to prove that Transformer can model sparse Boolean functions, our result reveals that one-layer Transformer can successfully tackle the same task even without the dot product in the attention layer. Effect of RPE types on expressivity. Our result indicates that the type of the RPE used in the Attn layer subtly influences the Transformer’s ability to model long-range memories. As the range of the memory increases, the required head number grows at a slightly exponential rate for lin-RPE 5 and at a polynomial rate for log-RPE. The subtle difference is attributed to the relative simplicity of approximating the memory kernel I{· = Ti}. We will explore a more complex task in Section 5, where the impact of different types of RPE becomes even more pronounced. 4 K-Adaptive, long but M-sparse memories 4.1 Problem formulation In this section, we delve into a more complex modeling scenario closely aligned with typical language processing tasks. K-Adaptive, long but M-sparse memories. This section investigates the scenario where the positions of the sparse memories are “adaptive”, meaning they depend on the input tokens. The target function is formulated as: yt = f(xt, xt−t1, · · · , xt−tM ), (8) where the positions of the memory tokens t1, · · · , tM follow a nested relationship: t1 = g1(xt); t2 = g2(xt, xt−t1); · · · ; tK+1 = gK+1(xt, xt−t1, · · · , xt−tK); · · · ; tM = gM(xt, xt−t1, · · · , xt−tK). Here, M denotes the number of memory tokens, and K measures the nesting complexity in the memory structure. We assume that memory functions gi generate positive integers for the input tokens, and there exist maximum values Ti such that gi ≤Ti. In this adaptive framework, each position of the memory token depends on multiple input tokens and is nested within other memory structures, leading to potential influence of later memory tokens by the earlier ones. To facilitate understanding, we first consider a warm-up case, i.e., K = 0 in (8). In this case, the positions of memories only depend on the current token, without interaction with each other. It can be represented as: yt = f(xt, xt−t1, · · · , xt−tM ), (9) where ti = g(xi), i ∈[M]. Target class. The target classes for modeling adaptive, long but sparse memories in both warm-up and general cases are as follows: HAdap (1,M) := H : Ht(X) = (9), where gi ∈B, 1 ≤gi ≤Ti, i ∈[M]; f ∈B ∩L . (10) HAdap (K,M) := H : Ht(X) = (8), where gi ∈B, 1 ≤gi ≤Ti, i ∈[M]; f ∈B ∩L . (11) Examples. Adaptive memories are commonly encountered in practical scenarios. (I) Adaptive sparse Boolean functions, e.g., yt = xt · xt−g(xt) · xt−g(xt−g(xt)), where X ∈{±1}Z, g(x) = 1 for x = 1 and g(x) = 2 for x = −1. This fits within our framework (8) with K = M = 2. (II) Multi-step reasoning, e.g., modeling the K-adaptive, long, but K-sparse memories contains a complicated K-step reasoning task, which require the sequential search following the rule ((· · · ((xt 7→xt−t1) 7→ xt−t2 · · · ) 7→xt−tK−1) 7→xt−tK. (III) In NLP tasks like dependency parsing, part-of-speech tagging, sentiment analysis, or continuation writing, the positions of relevant prefix tokens usually depend on the context itself, and can vary depending the content. Additionally, the nested structure is a fundamental characteristic of natural language (Hawkins, 2021). Transformer hypothesis class. Some previous works Yun et al. (2019); Kim et al. (2022) treated the softmax with normalization as an approximation of hardmax, suggesting the potential importance of the normalization. In contrast, in this section, we remove the normalization in the denominator of softmax and investigate its ability for sequence modeling. Additionally, to address the discreteness of time and memory values, we consider Transformer with specific precision, as detailed in Appendix C. The precision technique is widely used in LLM training (Kalamkar et al., 2019), such as BFloat16. Formally, the hypothesis class is defined as follows, encompassing all normalization-free L-layer Transformer, configured with H Attn heads and FFN width m and using type-RPE and specific precision. T Ftype (L,H,m) := TF :TF is an L-layer, H-head, m-width Transformer with type-RPE and specific precision . (12) 6 4.2 Theoretical results and insights: The warm-up case Theorem 4.1 (Approximation rate, warm-up case). For any target H ∈HAdap (1,M) (8), rate n ∈N+, and H, m ∈N+, there exists a two-layer Transformer TF ∈T Ftype (2,H,m) (12) and a constant C(n) such that: if the width satisfies m ≥ ˜Ω PM i=1 ∥gi∥2 B , type = lin ˜Ω PM i=1 ∥log gi∥2 B T 2 i , type = log, then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =    O C(n) Hn PM i=1 e0.01Ti n+1 , type = lin O C(n) Hn PM i=1 T 1.01 i n+1 , type = log . In Theorem 4.1, we present the approximation rate of two-layer Transformer for the warm-up case: modeling 1-adaptive, long but M-sparse memories. This theorem reveals that the approximation error comprises two distinct components: the error in the FFN component EFFN and the error in the Attn component EAttn(type). A critical difference from 3.1 is the presence of the condition related to the width m of FFN layers. This term arises from using the FFN layer to approximate the memory function gi. Owing to the discreteness of memory gi and the implementation of rounding operations, the approximation within rounding accuracy all achieves zero error after rounding, while it can not get correct rounding beyond this accuracy. In contrast, the error EFFN is caused by using FFN to approximate the readout function f, the same as EFFN in Theorem 3.1. The proof of Theorem 4.1 can be found in Appendix C.1. Theorem 4.1 and its proof offer several critical insights into the underlying mechanism of Transformer. Distinct roles of Attn layers and FFN layers. Our proof elucidates that the FFN layers are tasked with approximating the readout function f and memory functions gi, while the Attn layers are responsible for the extraction of the adaptive memories. It is essential to clarify the difference between “approximating memory functions” and “memory extraction”. The former refers to utilizing some function to estimate the memory function gi, whereas the latter pertains to extracting the token xt−gi(xt) from the memory location. Cooperation between DP and RPE. In the 2-nd Attn layer, the extraction of the memory functions is achieved through an interplay between DP and RPE. Specifically, this is done through a nice interaction between the temporal space (provided by RPE) and the token space (provided by DP). Please refer to Appendix C.1 for more details. Rethinking DP in Attn. Our proof highlights that the core mechanism of Attn is to provide a nice interaction between the temporal space and the token space through the cooperation of DP and RPE. This leads us to the following question: Is DP in Attn necessary and replaceable? The following two propositions provide some hints. Proposition 4.2 (DP vs. DP-free (informal)). There exists a target H ∈HAdap (1,1) (10) such that: (A) For any ϵ > 0, there exists a 1-layer Attn AttnDP such that H −AttnDP ≤ϵ. (B) For any 1-layer DP-free Attn AttnDPF, a uniform lower bound holds: H −AttnDPF ≥2 3. Proposition 4.2 reveal a significant distinction in the expressiveness of two network types for modeling adaptive, long, but sparse memories. Specifically, 1-layer Attn with DP can effectively model this task, while 1-layer DP-free Attn provably fails. This finding underscores the essential role of DP in providing the necessary nonlinearity for Attn to model adaptive memories. The formal version of Proposition 4.2 and its proof can be found in Appendix C.2. Proposition 4.3 (Substitute for DP (informal)). There exists a substitute structure for DP, requiring only O(D) parameters (compared to O(D2) in standard DP) that can effectively model H ∈ Hadap (1,M) (10). Specifically, if we substitute DP with this structure, 1-layer Transformer can achieve the same approximation rate as stated in Section 4.1. 7 Proposition 4.3 demonstrates the existence of a structurally simpler yet effective alternative to traditional DP for modeling (10). This alternative is proposed based on our insights into the role of Attn in facilitating the interaction between the temporal space and the token space. Specifically, we propose a more direct structure to achieve this interaction. More details are deferred to Appendix C.3. 4.3 Theoretical results and insights: The general case Theorem 4.4 (Approximation rate, general case). For any target H ∈HAdap (K,M), rate n ∈N+, and H, m ∈N+, there exists an L-layer (L = K + 1 + I{M ≥K + 1}) Transformer TF ∈T Ftype (L,H,m) (12) and a constant C(n) such that: if the width satisfies if the width satisfies m ≥ ˜Ω maxi∈[K] ∨PM i=K+1 ∥gi∥2 B , type = lin, ˜Ω maxi∈[K] ∨PM i=K+1 ∥log gi∥2 B T 2 i , type = log , then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN= ˜O ∥f∥B √m ,EAttn(type)=    O C(n) Hn qPK l=1e0.02(n+1)Tl+ PM l=K+1e0.01Tl2n+2 ,type=lin O C(n) Hn qPK l=1T 2.02(n+1) l + PM l=K+1T 1.01 l 2n+2 ,type=log . In Theorem 4.4, we establish the approximation rate of deep Transformer for modeling K-adaptive, long but M-sparse memories. Similar to that in Theorem 4.1, the approximation error divides into two distinct terms. A key difference from Theorem 4.1 is the impact of the nested relationships among the memory functions on the required number of layers, Attn heads, and the width of FFN layers. The nested structure within the initial K memories mandates sequential processing in the first K layers one by one. If M ≥K + 1, then in the K + 1-th layer, the remaining M −K non-nested memory functions tK+1, · · · , tM are concurrently processed. The proof of Theorem 4.4 is deferred to Appendix D.1. Distinct roles of the number of layers L, the number of Attn heads H, and the width of FFN layers m. Theorem 4.4 and its proof highlight the distinct roles of three key hyper-parameters of Transformer: L, H, and m. Deeper Transformer are capable of handling the memories with more intricate nested relationships, requiring a K + 1 layer network for a nesting complexity of K. In contrast, the number of heads and width needed is dictated by the individual complexity of memory functions themselves (∥gi∥B,∥log gi∥B, Ti for memory gi), necessitating that each layer’s Attn heads and FFN width are sufficient to capture the memory functions extracted in that layer. This understanding is quite intuitive: If the content of the next token relies on a few previous tokens in an independent way, we can treat each such dependence with a separate attention head. There is no need for many layers. Mitigating required head and width with depth. Recalling Theorem 4.1, the memories lacking nested relationships can be efficiently approximated by 2-layer Transformer with a sufficient number of heads and width. The subsequent proposition further explores how increasing the depth of Transformer can influence its efficiency for modeling memories without nested relationships. Proposition 4.5 (Deep network, warm-up case). For any target H ∈HAdap (1,M) (8), rate n ∈N+, and H, m ∈N+, there exists an M + 1-layer Transformer TF ∈T Ftype (M+1,H,m) (12) and a constant C(n) such that: if the width satisfies m ≥ ˜Ω maxi∈[K] ∥gi∥2 B , type = lin, ˜Ω maxi∈[K] ∥log gi∥2 B T 2 i , type = log , then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =    O C(n) Hn qPK l=1 e0.02(n+1)Tl , type = lin O C(n) Hn qPK l=1 T 2.02(n+1) l , type = log . Upon comparing Proposition 4.5 with Theorem 4.1, a notable distinction becomes evident between 2-layer and M +1-layer Transformer in terms of the requirement of the number of Attn heads and the width of FFN layers. Specifically, for 2-layer Transformer, the required width is proportionally linked 8 to the sum of all the memory functions’ complexity (∥gi∥B , ∥log gi∥B , Ti for memory function gi). In contrast, for M + 1-layer Transformer, the required width correlates with the maximum complexity of the memory functions, much lower than that for 2-layer Transformer. Similarly, the required number of heads for M +1-layer Transformer is much fewer than that for 2-layer Transformer. Please refer to Appendix D.2 for a detailed comparison. The observation suggests that increased depth can significantly reduce the demands on the number of heads and the width. The underlying reason is that deep networks can distribute the memories across different layers for processing, with each layer focusing on approximating only a single memory function. 5 Essentially M-sparse memories 5.1 Problem formulation In language tasks, each token possesses clear semantic meaning. As a result, the structure of the memory is sparse in the original space. This aligns well with our modeling assumptions discussed in Section 3 and 4. However, in other machine learning tasks, we may encounter situations where the input tokens lack distinct semantic meaning. This might happen in image processing or classical signal processing. In these situations, the memory structure could potentially be dense in the original space. Nonetheless, the memory structure might exhibit sparsity in some transformed domain. We call such memory structure “essentially sparse”. In this section, we study the situation in which the memory structure in long-ranged but essentially sparse. For simplicity, we consider the situation in which the positions of the memory kernels are fixed. The analysis can be easily extended to the situation with an adaptive memory structure. Fixed, essentially M-sparse memory. Consider the following situation: yt = f ((X ∗ρ1) (t), · · · , (X ∗ρM) (t)) , (13) where ρ1(·), · · · , ρM(·) ∈ℓ1(N) serve as memory kernels, and (X ∗ρk)(t) = P+∞ s=0 xt−sρk(s) denotes the convolution of the inputs with kernel ρk. Target class and Transformer hypothesis class. The target class for modeling essentially sparse memories is defined as: HEss := H : Ht(X) = (13), where ρ1, · · · , ρM ∈ℓ1(N), f ∈B ∩L . (14) For the hypothesis class, we consider one-layer dot-product-free Transformer with Attn head number H and FFN width m, as defined in (7). Examples. Essentially sparse memories are prevalent in real-world scenarios: (I) Image Tasks. In CV, a fundamental objective is identifying and representing meaningful “features”, such as ears, nose, etc. These features can often be modeled using convolution kernels, leading to a task in the form y = f (X ∗ρeye, X ∗ρnose, X ∗ρear). This is an extension of the task we discussed above, in which the kernel functions {ρj} are data-dependent (“adaptive” in the terminology used in the previous section). (II) Signal processing. In signal processing, it is commonly the case that the signals are highly sparse under Wavelet or Fourier transforms. For instance, let ψ(·) be a wavelet function and define ψa,b(t) := ψ( t−b a )/ p \|a\|. Then we have y = f (X ∗ψa1,b1, · · · , X ∗ψaM,bM ) where (a1, b1), · · · , (aM, bM) might be data-dependent. (III) Mathematical calculation. Consider algebraic operations where memory exhibits sparsity under specific linear transformations. For example, yt = 10xt + xt−4/(P100 s=0 wsxt−10−s) − P+∞ s=0 vsxt−100−s can be represented in our framework as y = f (X ∗ρ1, · · · , X ∗ρ4), where each ρi represents a specific linear transformation. 5.2 Theoretical results and insights Theorem 5.1 (Approximation rates). (A) Consider HEss (14) with exponentially decayed memory kernels, i.e., there exists β > 0 such that 9 ρ1(t), · · · , ρM(t) = O(e−βt). Then for any target H ∈HEss, rate n ∈[⌊99β⌋], and H, m ∈N+, there exists a 1-layer DP-free Transformer TF ∈T FDPF,lin (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip · EAttn; (B) Consider HEss (14) with polynomially decayed memory kernels, i.e., there exists β > 1 such that ρ1(t), · · · , ρM(t) = O(t−β). Then for any target H ∈HEss, rate n ∈[⌊0.99β⌋−1], and H, m ∈N+, there exists a 1-layer DP-free Transformer TF ∈T FDPF,log (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip · EAttn; where EFFN = ˜O ∥f∥B √m , EAttn = O C(n)M n+1 Hn . Theorem 5.1 illustrates that one-layer DP-free Transformer with lin-RPE is effective in modeling essentially sparse memories with exponentially decayed kernels, and one-layer DP-free Transformer with log-RPE can efficiently model the memories with polynomially decayed kernels. A key difference between Theorem 5.1 and Theorem 3.1 lies in the memory kernels they address. In Theorem 5.1, the Attn layer should approximate general memory kernels ρi(·), instead of approximating indicator kernels I{· = Ti} in Theorem 3.1. The proof of Theorem 5.1 can be found in Appendix E. Overcoming the Curse of Memory (CoM). For recurrent neural networks (RNN), it was discovered (Li et al., 2021; 2022) that both approximation and optimization become exceedingly difficult when the target has long-term memory. This phenomenon is referred as the “curse of memory”, or “CoM”. It was shown in (Li et al., 2021; 2022) that RNN requires an exponentially large number of neurons to approximate targets with heavy-tailed memory kernels, such as the ones that exhibit polynomial decay. In contrast, Theorem 5.1 reveals that Transformer with log-RPE efficiently handles polynomial decaying memory kernels, requiring only a polynomial number of neurons for effective approximation. This finding theoretically elucidates the superior performance of T5’s RPE and KERPLE(log) in length generalization task in practice (Section G.1). 6 Experimental Validation As summarized in Section 1, our theoretical analysis reveals novel insights into the expressive power and mechanisms of Transformer. To validate these insights, we conduct experiments ranging from simple toy models to more complex language model pre-training. Due to space constraints, detailed experimental validation and practical implications of our insights are presented in Appendix H. 7 Conclusion and Future Work In this work, we investigate theoretically the expressive power and the mechanisms of Transformer for modeling long but sparse memories. Our analysis establishes explicit approximation rates and offers much-needed insights into the functionalities of the various components of Transformer. However, we still have a long way to go for a full theoretical understanding of Transformer. For instance, although we have investigated the mechanisms of Transformer in terms of expressive power, the evolution of the mechanisms during the training process remains elusive. Recent studies revealed that Transformer exhibits multi-phase learning dynamics (Boix-Adsera et al., 2023) and undergoes phase transitions (Olsson et al., 2022) during training, akin to the phenomenon of learning with increasing complexity in classical neural networks (Kalimeris et al., 2019; Xu et al., 2019; Rahaman et al., 2019; Abbe et al., 2023a; Wang and Ma, 2023). These and other issues will be studied in future work. Acknowledgments This work is supported in part by the National Key Basic Research Program of China (No. 2015CB856000). We thank Prof. Qianxiao Li, Prof. Lei Wu, Dr. Zhong Li, and Dr. Hongkang Yang for helpful discussions and anonymous reviewers for their valuable suggestions. 10 References Emmanuel Abbe, Enric Boix Adsera, and Theodor Misiakiewicz. 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Anchor function: a type of benchmark functions for studying language models. arXiv preprint arXiv:2401.08309, 2024. 17 15 Appendix A Detailed Related Works 16 B Proof of Section 3 18 B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 C Proof of Section 4.2 23 C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 C.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 C.3 Proof of Proposition 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 D Proof of Section 4.3 38 D.1 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 D.2 Proof of Proposition 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 E Proof of Section 5 50 E.1 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 F Key Lemmas about Approximation 55 F.1 Approximation by the sum of exponential decay . . . . . . . . . . . . . . . . . . . 55 F.2 Approximation by the sum of polynomial decay . . . . . . . . . . . . . . . . . . . 59 G Some Background and Proof Preparation 64 G.1 T5’s relative positional encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 G.2 Barron space theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 G.3 Useful approximation lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 H Experiments 66 H.1 Restatement of our theoretical insights . . . . . . . . . . . . . . . . . . . . . . . . 66 H.2 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 H.3 Practical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A Detailed Related Works Theoretical results of Transformer. We first review the expressive power results of Transformer. Yun et al. (2019) first proved the universal approximation property (UAP) of Transformer, highlighting the crucial role of PE in breaking permutation invariance. Edelman et al. (2022) demonstrated that Transformer can approximate fixed sparse functions. Dehghani et al. (2019); Pérez et al. (2021); Wei et al. (2022a) explored the Turing-completeness of infinite-precision and finite-precision Transformer. Giannou et al. (2023) showed that looped Transformer can implement practical computer programs. 16 Jiang and Li (2023) provided explicit approximation rates for Transformer in sequences modeling with inherent graph structures. Liu et al. (2022) found that Transformer can execute finite-state automata. Ma and Ying (2022) asserted the natural suitability of Attn for achieving permutation equivariance. Besides these affirmative results, several studies characterized the expressivity limitation of Transformers, particularly in modeling formal languages or simulating circuits (Hahn, 2020; Weiss et al., 2021; Bhattamishra et al., 2020; Merrill and Sabharwal, 2023b; Merrill et al., 2022). Additionally Feng et al. (2023); Merrill and Sabharwal (2023a) examined the expressivity of Transformer using Chain of Thought prompting (Wei et al., 2022b). Moreover, some studies showed that the in-context learning ability of Transformer is attainable by simulating gradient-based iterations across various layers (Garg et al., 2022; Akyürek et al., 2022; von Oswald et al., 2023; Von Oswald et al., 2023; Mahankali et al., 2023; Bai et al., 2023; Shen et al., 2023). Besides, experimental studies also provide insights into the mechanisms of Transformer through induction head (Elhage et al., 2021; Olsson et al., 2022), information flow (Wang et al., 2023), anchor functions (Zhang et al., 2024), etc. Positional encoding. One core component of Transformer is the PE, which facilitates the representation of input sequence order. Theoretically, Transformer without PE lacks UAP and is restricted to representing permutation-invariant functions. PE was first introduced in Vaswani et al. (2017). It has limitations in encoding unseen positions. To overcome this difficulty, Shaw et al. (2018) introduced RPE. Subsequent studies proposed various different RPE types. Notable examples include T5’s RPE (Raffel et al., 2020), Rotary RPE (Su et al., 2024) (utilized in PaLM (Chowdhery et al., 2023) and LlaMA (Touvron et al., 2023)), Alibi RPE (Press et al., 2022) (employed in BLOOM (Workshop et al., 2022)), and KERPLE (Chi et al., 2022). A prevailing belief is that RPEs can outperform APEs in the “length generalization task” (Ontanón et al., 2022; Csordás et al., 2021)– the ability to generalize from smaller training contexts to larger ones, a critical challenge for Large Language Models (Anil et al., 2022; Abbe et al., 2023b). However, Press et al. (2022) revealed that the commonly used Rotary RPE may exhibit suboptimal performance in this task. The recent work (Kazemnejad et al., 2023; Chi et al., 2022) conducted systematic experiments comparing the length generalization capabilities of Transformers with various RPEs and APEs, suggesting that the RPEs used in T5 and KERPLE(log) demonstrate superior performance over other types. Rethinking dot-product. Another critical component of Transformer is the DP structure. Due to its quadratic cost as a function of the sequence length, the necessity of DP has always been questioned. Numerous variants of DP have been proposed, demonstrating competitive performance across diverse tasks. Representative examples include Longformer (Beltagy et al., 2020), Big Bird (Zaheer et al., 2020), Reformer (Kitaev et al., 2020), Linformer (Wang et al., 2020), Performer (Choromanski et al., 2020), Synthesizer (Tay et al., 2021), etc. In particular, a recent study (Allen-Zhu and Li, 2023) compared standard and DP-free Transformers in modeling "context-free grammar". Their findings suggested that the presence of DP has a marginal impact on performance. These evidences motivate us to rethink the necessity of DP in Transformer. Sparsity (Donoho, 2006; Candès and Wakin, 2008) has gained considerable attention in sequence modeling. In classical signal processing, there is a prevailing notion that valuable signals are extremely sparse. For example, when representing an image, one often finds that only a few wavelet coefficients hold significant values in wavelet space (Meyer, 1992). In NLP, the starting point off the traditional n-gram model (Shannon, 1948) is that the next token only relies on a few previous tokens. Such models, however, overlook long-range information, often resulting in suboptimal performance. For NLP tasks such as dependency parsing (Nivre and Scholz, 2004), sentiment analysis (Nasukawa and Yi, 2003), part-of-speech tagging (Francis and Kucera, 1979), and continuation writing (Brown et al., 2020; OpenAI, 2023), it is indeed often the case that only a limited subset of preceding information is crucial for accurate prediction. However, these relevant information can be quite distant. For instance, the resolution of a mystery novel may hinge on elements introduced at the outset. Moreover, for Transformer networks, extensive research into the visualization and interpretability has revealed that (i) the learned activation maps of FFN layers are extremely sparse (Li et al., 2023); (ii) the learned self-attention matrices exhibit notable sparsity, yet it does not closely resemble a diagonal configuration (Elhage et al., 2021). These observations suggest that the prediction of the next token is influenced by a small number of previous tokens which might be far away. Therefore, being able to represent sparse but long-range dependence is important for sequence modeling. 17 B Proof of Section 3 B.1 Proof of Theorem 3.1 In this subsection, we give the detailed proofs of fixed, long but sparse memory: yt = f(xt, xt−T1, · · · , xt−TM ), where 1 ≤T1 < · · · < TM < +∞signify the fixed positions of the memories. Theorem B.1 (Restatement of Theorem 3.1). For any target H ∈HFix (6), rate n ∈N+, and H, m ∈N+, there exists a 1-layer Transformer TF ∈T FDPF,type (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =        O C(n) Hn PM i=1 e0.01Ti n+1 , type = lin O C(n) Hn PM i=1 T 1.01 i n+1 , type = log . Proof of Theorem B.1. First, we choose the embedding dimension D = (M + 1)d, and select the simple embedding WE = (Id×d, 0)⊤∈RD×d, bE = 0 ∈RD. Then for any input sequence X = (xt)t∈Z, the token after embedding satisfies: xE t = WExt + bE = (x⊤ t , 0⊤)⊤∈RD. For one-layer Dot-product-free Transformer TF ∈T FDPF,type (1,H,m) with ϕtype, the output token TFt(X) of t-th input token xt satisfies: x(1/2) t = x(0) t + W (1) O H X h=1 Attn(1,h) t (X(0)), x(1) t = FFN(1)(x(1/2) t ) where Attn(1,h) t (X) = W (1,h) V +∞ X s=0 xt−s exp p(1,h)ϕtype(s) P+∞ j=0 exp p(1,h)ϕtype(j) . This proof can be summarized as the following process: · · · xE t · · · Step I. Attn layer ↓ · · · x(1/2) t ≈(x⊤ t , x⊤ t−T1, · · · , x⊤ t−TM )⊤ · · · Step II. FFN layer ↓ · · · x(1) t ≈f(xt, xt−T1, · · · , xt−TM ) · · · Now we give the formal proof. Step I. Extract the memory locations by (Dot-product-free) Attn layer. We consider to use Hk attention heads (from Pk−1 i=1 Hi + 1-th head to Pk i=1 Hi-th head) to extract it, and it satisfies to PM k=1 Hk = H. For simplicity, we denote the following projection matrices: 18 P (k) := (0d×kd Id×d 0) ∈Rd×D, 1 ≤k ≤M. P (k) ⊥ := Ikd×kd 0d×d 0 0 0d×d I(M−k)d×(M−k)d ∈RMd×D, 1 ≤k ≤M. Now we consider the extraction of k-th memory xt−Tk (1 ≤k ≤M). • Case type = lin. By Lemma F.1, for any rate n ∈N+, there exists an constant C(n) and a function ϕexp k (t) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αhe−βht such that βh > 0 and ∥I{· = Tk} −ϕexp k (·)∥ℓ1(N) = +∞ X s=0 \|I{s = Tk} −ϕexp k (s)\| ≤C(n)e0.01(n+1)Tk Hn k . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose p(1,h) = βh, W (1,h) V = αh   +∞ X j=0 exp(−βhj)  δd×d (k+1,1), where δ(k+1,1) ∈RD×D means that: it equals to Id×d for the (k + 1, 1)-th d × d blocks, and 0d×d for the other d × d blocks. Then it holds that: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhs 0kd xt−s 0 ! ∈RD. This implies: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhsxt−s, P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = 0, moreover, the following estimate holds: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) −xt−Tk 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhsxt−s −xt−Tk 2 19 = +∞ X s=0   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βhs −I{s = Tk}  xt−s 2 ≤ +∞ X s=0 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βhs −I{s = Tk} = ∥ϕexp k (·) −I{· = Tk}∥ℓ1(N) ≤C(n)e0.01(n+1)Tk Hn k . • Case type = log. By Lemma F.4, for any rate n ∈N+, there exists an constant C(n) and a function ϕpoly k (t) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αht−βh, such that βh > 1 and I{· = Tk} −ϕpoly k (·) ℓ1(N+) = +∞ X s=1 I{s = Tk} −ϕpoly k (s) ≤C(n)T 1.01(n+1) k Hn k . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose p(1,h) = βh, W (1,h) V = αh   +∞ X j=1 j−βh  δ(k+1,1), where δ(k+1,1) ∈RD×D means that: it equals to Id×d for the (k + 1, 1)-th d × d blocks, and 0d×d for the other d × d blocks. Then it holds that: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=1 s−βhxt−s, P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = 0, moreover, the following estimate holds: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(1,h) t (X(0)) −xt−Tk 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=1 s−βhxt−s −xt−Tk 2 = +∞ X s=1   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhs−βh −I{s = Tk}  xt−s 2 20 ≤ +∞ X s=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhs−βh −I{s = Tk} = ϕpoly k (·) −I{· = Tk} ℓ1(N+) ≤O C(n)T 1.01(n+1) k Hn k ! . Then we combine the results for all k ∈[M] for these two cases. By choose WO = ID, we have: x(1/2) t −     xt xt−t1 ... xt−tM     2 =     xt 0d ... 0d    + M X h=1 Attn(1,h) t (X) −     xt xt−t1 ... xt−tM     2 = M X h=1 Attn(1,h) t (X) −     0d xt−t1 ... xt−tM     2 = M X k=1   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X) − 0kd xt−Tk 0d !  2 ≤ M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X) − 0kd xt−Tk 0d ! 2 = M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(1,h) t (X) −xt−Tk 2 ≤EAttn(type) :=    C(n) PM k=1 e0.01(n+1)Tk Hn k , type = lin C(n) PM k=1 T 1.01(n+1) k Hn k , type = log . Consequently, one detail is to assign the head number {Hk}M k=1 such that the error’s sum EAttn(type) is as small as possible. Our way is solving the minimization problem: min H1,··· ,HM : EAttn(type) s.t. M X k=1 Hk = H, which suggests that we should choose the head number: Hk = e0.01Tk PM j=1 e0.01Tj H, k ∈[M], type = lin; Hk = T 1.01 k PM j=1 T 1.01 j H, k ∈[M], type = log . Thus, we obtain the bound in Step I: EAttn(type) ≤            C(n) Hn M P k=1 e0.01Tk n+1 , type = lin C(n) Hn M P k=1 T 1.01 k !n+1 , type = log . 21 Furthermore, by choosing EAttn(type) ≤1, it holds that x(1/2) t ∞≤ x(1/2) t −     xt xt−t1 ... xt−tM     ∞ +     xt xt−t1 ... xt−tM     ∞ ≤EAttn(type) + 1 ≤2. Step II. Approximate the readout function by FFN layer. In this step, we aim to approximate the function f using two-layer network. By Lemma G.6, there exists a two layer neural network with m neurons defined on RD FFN(1)(y) = m X k=1 akσ(b⊤ k y + ck) such that EFFN := FFN(1) −f L∞([−2,2]D) ≤˜O ∥f∥B √m . The final bound. For any t and X ∈X, it holds that Ht(X) −x(1) t = f (xt, xt−t1, · · · xt−tM ) −FFN(1) x(1/2) t = f (xt, xt−t1, · · · xt−tM ) −f x(1/2) t + f x(1/2) t −FFN(1) x(1/2) t ≤ f (xt, xt−t1, · · · xt−tM ) −f x(1/2) t + f x(1/2) t −FFN(1) x(1/2) t ≤∥f∥Lip x⊤ t , x⊤ t−t1, · · · x⊤ t−tM −x(1/2) t 2 + f −FFN(1) L∞([−2,2]D) ≤∥f∥Lip · EAttn(type) + EFFN, where EFFN = ˜O ∥f∥B √m ; EAttn(type) =            O C(n) Hn PM k=1 e0.01Tk n+1 ! , type = lin O C(n) Hn PM k=1 T 1.01 k n+1 ! , type = log . Due to the arbitrariness of t and X, the proof is completed. 22 C Proof of Section 4.2 In this section, we give the detailed proofs of the approximation theory of Transformer for modeling the warm-up case of adaptive, long but sparse memory: yt = f(xt, xt−t1, · · · , xt−tM ), where the adaptive memory satisfies to: tk = gk(xt), k ∈[M]. Moreover, gk(·) generate positive integers for the input tokens, and there exist maximum values Tk such that 1 ≤gk(xt) ≤Tk holds for any xt and k ∈[M]. To tackle the discrete values of the time and the memory values gk(xt), a modified version of standard FFN, termed “FFN with precision”, us cibsudered. This approach ensures that the output of FFN undergoes a simple rounding operation. Notably, the precision technique is widely used in LLM training (Kalamkar et al., 2019), such as BFloat16. Specifically, for Transformer using RPE with type, we use the following FFN with precision: ] FFN(x) := [FFN(x)], type = lin; ] FFN(x) := log [exp (FFN(x))] , type = log, (15) where [·] signifies rounding to the nearest integer, i.e., [x] = arg min n∈Z \|n −x\| (x ∈R). It is important to note that the rounding obtained by using the operator log[exp(z)], used in (15), is quite fine, which is much finer than the vanilla rounding obtained by [z]. To elaborate, the following proposition is presented: Proposition C.1. For any z ≥1, the following holds: (i) \| log[exp(z)] −z\| ≤ 1 2 min{ez, [ez]}; (ii) \|[z] −z\| ≤1 2. C.1 Proof of Theorem 4.1 Theorem C.2 (Restatement of Theorem 4.1). For any target H ∈HAdap (1,M) (8), rate n ∈N+, and H, m ∈N+, there exists a two-layer Transformer TF ∈T FNF,type (2,H,m) (12) and a constant C(n) such that: if the width satisfies m ≥    ˜Ω PM i=1 ∥gi∥2 B , type = lin ˜Ω PM i=1 ∥log gi∥2 B T 2 i , type = log , then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =        O C(n) Hn PM i=1 e0.01Ti n+1 , type = lin O C(n) Hn PM i=1 T 1.01 i n+1 , type = log . Proof of Theorem C.2. First, we choose the embedding dimension D = (M + 1)(d + 1), and select a simple embedding WE = (Id×d, 0)⊤∈RD×d, bE = 0 ∈RD. Then for any input sequence X = (xt)t∈Z, the token after embedding satisfies: x(0) t = WExt + bE = (x⊤ t , 0⊤)⊤∈RD. 23 To tackle the discrete values of gm(xt), we utilize ] FFN, FFN with precision (15). It ensures that the output of FFN undergoes a simple rounding operation. Thus, for two-layer normalization-free Transformer TF ∈T FNF,type (2,H,m) with ϕtype, the output token x(2) t of t-th input token satisfies: x(1/2) t = x(0) t + W (1) O H X h=1 Attn(1,h) t (X(0)), x(1) t = x(1/2) t + ^ FFN (1) (x(1/2) t ), x(3/2) t = x(1) t + W (2) O H X h=1 Attn(2,h) t (X(1)), x(2) t = FFN(2)(x(3/2) t ), where Attn(l,h) t (X) = W (l,h) V +∞ X s=0 xt−s exp D W (l,h) Q xt, W (l,h) K xt−s E + p(l,h)ϕtype(s) . This proof can be summarized as the following process: • Case type = lin. x(0) t Step I. 1-st Attn ↓ x(1/2) t = x(0) t Step II. 1-st FFN ↓ x(1) t = (x⊤ t , 0⊤, g1(xt), · · · , gM(xt), 1)⊤ Step III. 2-st Attn ↓ x(3/2) t ≈(x⊤ t , x⊤ t−g1(xt), · · · , x⊤ t−gM(xt), g1(xt), · · · , gM(xt), 1)⊤ Step IV. 2-st FFN ↓ x(2) t ≈f(xt, xt−g1(xt), · · · , xt−gM(xt)) • Case type = log. x(0) t Step I. 1-st Attn ↓ x(1/2) t = x(0) t Step II. 1-st FFN ↓ x(1) t = (x⊤ t , 0⊤, log g1(xt), · · · , log gM(xt), log 2)⊤ Step III. 2-st Attn ↓ x(3/2) t ≈(x⊤ t , x⊤ t−g1(xt), · · · , x⊤ t−gM(xt), log g1(xt), · · · , log gM(xt), log 2)⊤ Step IV. 2-st FFN ↓ x(2) t ≈f(xt, xt−g1(xt), · · · , xt−gM(xt)) Now we give the formal proof. Step I. Identity map. For the first Attn layer, we only need to do the identity map by taking W (1) 0 = 0. Then x(1/2) t = x(0) t . Step II. Approximate the adaptive memory function by the first FFN layer. 24 • Case type = lin. Our main idea is that using the first FFN layer to express (x⊤ t , 0⊤, g1(xt), · · · , gM(xt), 1)⊤exactly. First, we consider to approximate the r-th memory function gr(x) by standard FFN. For any r ∈[M], by Lemma G.6, there exists a two-layer neural network with mr neurons f 2NN (1,r)(x) = mr X k=1 a(1,r) k σ b(1,r) k ⊤x + c(1,r) k defined on Rd such that gr −f 2NN (1,r) L∞([−1,1]D) ≤˜O ∥gr∥B √mr . Therefore, if we choose ˜O ∥gr∥B √mr < 1 2, the following holds: gr(xt) −f 2NN (1,r)(xt) ≤ gr −f 2NN (1,r) L∞([−1,1]d) < 1 2, Noticing gr(xt) ∈N+, we have h f 2NN (1,r)(xt) i = gr(xt), which implies: ] f 2NN (1,r)(xt) = h f 2NN (1,r)(xt) i = gr(xt). Consequently, in order to construct the form (0⊤, g1(xt), · · · , gM(xt), 1)⊤∈RD, we need to arrange the parameters a(1,r) k , b(1,r) k , and c(1,r) k (k ∈[mr], r ∈[M]) appropriately. Denote ¯b(1,r) k = (b(1,r) k ⊤, 0⊤)⊤∈RD for k ∈[mr], r ∈[M]. Consider the following two-layer neural network with 1 + PM r=1 mr neurons defined on RD: FFN(1)(x) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ ¯b(1,r) k ⊤x + c(1,r) k + eD · 1 · σ(0 + 1). It is easy to verify that for any x(1/2) t , it holds that FFN(1)(x(1/2) t ) = FFN(1)(x(0) t ) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ ¯b(1,r) k ⊤x(0) t + c(1,r) k + eD · 1 · σ(0 + 1) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ b(1,r) k ⊤xt + c(1,r) k + eD · 1 · σ(0 + 1) = M X r=1 eD−M+r−1f 2NN r (xt) + eD =(0⊤ d , f 2NN (1,1)(xt), · · · , f 2NN (1,M)(xt), 1)⊤∈RD. Moreover, it satisfies that ^ FFN (1) (x(1/2) t ) = h FFN(1)(x(1/2) t ) i 25 =(0⊤ d , h f 2NN (1,1)(xt) i , · · · , h f 2NN (1,M)(xt) i , 1)⊤ =(0⊤ d , g1(xt), · · · , gM(xt), 1)⊤∈RD. Thus, we have achieved our goal in this step: x(1) t = x(1/2) t + ^ FFN (1) (x(1/2) t ) = (x⊤ t , 0⊤, g1(xt), · · · , gM(xt), 1)⊤. • Case type = log. Our main idea is that using the first FFN layer to express (x⊤ t , 0⊤, log g1(xt), · · · , log gM(xt), log 2)⊤exactly. First, we consider to approximate the r-th memory function log gr(x) by standard FFN. For any r ∈[M], by Lemma G.6, there exists a two-layer neural network with mr neurons f 2NN (1,r)(x) = mr X k=1 a(1,r) k σ b(1,r) k ⊤x + c(1,r) k defined on Rd such that log gr −f 2NN (1,r) L∞([−1,1]D) ≤˜O ∥log gr∥B √mr . Therefore, if we choose ˜O ∥log gr∥B √mr < 1 4Tr , the following holds: log gr(xt) −f 2NN (1,r)(xt) ≤ gr −f 2NN (1,r) L∞([−1,1]d) < 1 4Tr , which ensures exp f 2NN (1,r)(xt) −gr(xt) = exp f 2NN (1,r)(xt) −exp (log (gr(xt))) ≤exp max n f 2NN (1,r)(xt), log (gr(xt)) o f 2NN (1,r)(xt) −log (gr(xt)) ≤exp log gr(xt) + 1 4 1 4Tr ≤e1/4 · Tr · 1 4Tr < 1 2. Noticing gr(xt) ∈N+, we have h exp f 2NN (1,r)(xt) i = gr(xt), which implies: ] f 2NN (1,r)(xt) = log h exp f 2NN (1,r) i = log gr(xt). Consequently, in order to construct the form (0⊤, log g1(xt), · · · , log gM(xt), log 2)⊤, we need to arrange the parameters a(1,r) k , b(1,r) k , and c(1,r) k (k ∈[mr], r ∈[M]) appropriately. Denote ¯b(1,r) k = (b(1,r) k ⊤, 0⊤)⊤∈RD for k ∈[mr], r ∈[M]. Consider the following two-layer neural network with 1 + PM r=1 mr neurons defined on RD: FFN(1)(x) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ ¯b(1,r) k ⊤x + c(1,r) k + eD · 1 · σ(0 + log 2). 26 It is easy to verify that for any x(1/2) t , it holds that FFN(1)(x(1/2) t ) = FFN(1)(x(0) t ) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ ¯b(1,r) k ⊤x(0) t + c(1,r) k + eD · 1 · σ(0 + log 2) = M X r=1 X 1+Pr−1 j=0 mj≤k≤Pr j=0 mj eD−M+r−1a(1,r) k σ b(1,r) k ⊤xt + c(1,r) k + eD · 1 · σ(0 + log 2) = M X r=1 eD−M+r−1f 2NN r (xt) + eD log 2 =(0⊤ d , f 2NN (1,1)(xt), · · · , f 2NN (1,M)(xt), log 2)⊤∈RD. Moreover, it satisfies that ^ FFN (1) (x(1/2) t ) = log h exp FFN(1)(x(1/2) t ) i =(0⊤ d , log h exp f 2NN (1,1)(xt) i , · · · , log h exp f 2NN (1,M)(xt) i , log 2, 0⊤)⊤ =(0⊤ d , log g1(xt), · · · , log gM(xt), log 2)⊤. Thus, we have achieved our goal in this step: x(1) t = x(1/2) t + ^ FFN (1) (x(1/2) t ) = (x⊤ t , 0⊤, log g1(xt), · · · , log gM(xt), log 2)⊤. As established in the proof above, the width m must satisfy: m ≥1 + M X r=1 mr =    ˜Ω PM r=1 ∥gr∥2 B , type = lin ˜Ω PM r=1 ∥log gr∥2 B T 2 r , type = log . Step III. Extract the adaptive memories by the second Attn layer. We consider to use Hk attention heads (from Pk−1 i=1 Hi + 1-th head to Pk i=1 Hi-th head) to extract it, and it satisfies to PM k=1 Hk = H. For simplicity, we denote the following projection matrices in RD×D: P (k) := (0d×kd Id×d 0) ∈Rd×D, 1 ≤k ≤M; P (k) ⊥ := Ikd×kd 0d×d 0 0 0d×d I(D−(k+1)d)×(D−(k+1)d) ∈R(D−d)×D, 1 ≤k ≤M; Q(M) := I(M+1)d×(M+1)d 0 ∈R(M+1)d×D. Now we consider the extraction of k-th adaptive memory xt−gk(xt) (1 ≤k ≤M). • Case type = lin. By Lemma F.2, for any rate n ∈N+, there exists a constant C(n) and a function ϕexp k (t; B) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αh exp(−βh(t −B)) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αh exp βhB −βht 27 such that βh > 0 and sup 1≤B≤Tk ∥I{· = B} −ϕexp k (·; B)∥ℓ1(N) ≤C(n)e0.01(n+1)Tk Hn k . Moreover, Noticing that 1 ≤gk(xt) ≤Tk holds for any X = (xt)t∈Z ∈X, the following holds: sup X ∥I{· = gk(xt)} −ϕexp k (·; gk(xt))∥ℓ1(N) ≤ sup 1≤B≤Tk ∥I{· = B} −ϕexp k (·; B)∥ℓ1(N) ≤C(n)e0.01(n+1)Tk Hn k . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose: p(2,h) = βh, W (1) O = ID×D, W (2,h) V = αhδ(d×d) (k+1,1) ∈RD×D, W (2,h) Q = p βhδ(1×1) (D−M+k−1,1) ∈RD×D, W (2,h) K = p βhδ(1×1) (D,1) ∈RD×D, where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. Then it is easy to verify: D W (2,h) Q x(1) t , W (2,h) K x(1) t−s E + p(2,h)ϕlin(s) = −βh s −gk(xt) , which implies: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(1)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βh(s−gk(xt)) 0kd xt−s 0 ! ∈RD, Then it holds that: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βh(s−gk(xt))xt−s, P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(0)) = 0, moreover, the following estimate holds: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(2,h) t (X) −xt−gk(xt) 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βh(s−gk(xt))xt−s −xt−gk(xt) 2 28 = +∞ X s=0   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βh(s−gk(xt)) −I{s = gk(xt)}  xt−s 2 ≤ +∞ X s=0 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βh(s−gk(xt)) −I{s = gk(xt)} = ∥ϕexp k (·; gk(xt)) −I{· = gk(xt)}∥ℓ1(N) ≤C(n)e0.01(n+1)Tk Hn k . • Case type = log. By Lemma F.5, for any rate n ∈N+, there exists a constant C(n) and a function ϕpoly k (t; B) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αh(t/B)−βh = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αh exp βh log B −βh log t such that βh > 1 and sup 1≤B≤Tk I{· = B} −ϕpoly k (·; B) ℓ1(N+) ≤C(n)T 1.01(n+1) k Hn k . Moreover, Noticing that 1 ≤gk(xt) ≤Tk holds for any X = (xt)t∈Z ∈X, the following holds: sup X I{· = gk(xt)} −ϕpoly k (·; gk(xt)) ℓ1(N+) ≤ sup 1≤B≤Tk I{· = B} −ϕpoly k (·; B) ℓ1(N+) ≤C(n)T 1.01(n+1) k Hn k . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose: p(2,h) = βh, W (1) O = ID×D, W (2,h) V = αhδ(d×d) (k+1,1) ∈RD×D, W (2,h) Q = p βhδ(1×1) (D−M+k−1,1) ∈RD×D, W (2,h) K = √βh log 2δ(1×1) (D,1) ∈RD×D, where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. Then it is easy to verify: D W (2,h) Q x(1) t , W (2,h) K x(1) t−s E + p(2,h)ϕlog(s) = −βh log s/gk(xt) , which implies: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(1)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 (s/gk(xt))−βh 0kd xt−s 0 ! ∈RD, 29 Then it holds that: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 (s/gk(xt))−βhxt−s, P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(2,h) t (X(0)) = 0, moreover, the following estimate holds: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(2,h) t (X) −xt−gk(xt) 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 (s/gk(xt))−βhxt−s −xt−gk(xt) 2 = +∞ X s=0   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh(s/gk(xt))−βh −I{s = gk(xt)}  xt−s 2 ≤ +∞ X s=0 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh(s/gk(xt))−βh −I{s = gk(xt)} = ϕpoly k (·; gk(xt)) −I{· = gk(xt)} ℓ1(N+) ≤C(n)T 1.01(n+1) k Hn k . Then we combine the estimate for all k ∈[M] for these two cases. It holds that Q(M)x(3/2) t −     xt xt−g1(xt) ... xt−gM(xt)     2 = xt 0Md + M X h=1 Q(M)Attn(2,h) t (X(1)) −     xt xt−g1(xt) ... xt−gM(xt)     2 = M X h=1 Q(M)Attn(2,h) t (X) −     0d xt−g1(xt) ... xt−gM(xt)     2 = M X k=1   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Q(M)Attn(2,h) t (X) − 0kd xt−gk(xt) 0 !  2 ≤ M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Q(M)Attn(2,h) t (X) − 0kd xt−gk(xt) 0 ! 2 30 = M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(2,h) t (X) −xt−gk(xt) 2 ≤EAttn(type) :=    C(n)e0.01(n+1)Tk Hn k , type = lin C(n)T 1.01(n+1) k Hn k , type = log . Similar to the proof of Theorem B.1, we choose the head number: Hk = e0.01Tk PM j=1 e0.01Tj H, k ∈[M], type = lin; Hk = T 1.01 k PM j=1 T 1.01 j H, k ∈[M], type = log . Thus, we obtain the final bound in Step III: ESoft Attn(type) ≤            C(n) Hn M P k=1 e0.01Tk n+1 , type = lin C(n) Hn M P k=1 T 1.01 k !n+1 , type = log . Furthermore, by choosing EAttn(type) ≤1, it holds that Q(M)x(3/2) t ∞≤ Q(M)x(3/2) t −     xt xt−g1(xt) ... xt−gM(xt)     ∞ +     xt xt−g1(xt) ... xt−gM(xt)     ∞ ≤EAttn(type) + 1 ≤2. Step IV. Approximate the nonlinear function by 2-nd FFN layer. In this step, we aim to approximate the function f using two-layer network. By Lemma G.6, there exists a two-layer neural network with m neurons f 2NN (2) (x) = m X k=1 a(2) k σ b(2) k ⊤x + c(2) k defined on R(M+1)d such that f −f 2NN (2) L∞([−2,2](M+1)d) ≤˜O ∥f∥B √m . Denote ¯b(2) k = (b(2) k ⊤, 0⊤)⊤∈RD for k ∈[m]. And we consider the following two-layer neural network with m neurons defined on RD: FFN(2)(x) := m X k=1 a(2) k σ ¯b(2) k ⊤x + c(2) k . It is easy to verify: 31 FFN(2) x(3/2) t = f 2NN (2) Q(M)x(3/2) t . Moreover, E(2) FFN := f −FFN(2) L∞([−2,2](M+1)d) ≤˜O ∥f∥B √m . The final bound. For any t and ∥X∥∈X, it holds that Ht(X) −x(2) t = f xt, xt−g1(xt), · · · xt−gM(xt) −FFN(2) x(3/2) t = f (xt, xt−t1, · · · xt−tM ) −f 2NN (2) Q(M)x(3/2) t = f xt, xt−g1(xt), · · · xt−gM(xt) −f Q(M)x(3/2) t + f Q(M)x(3/2) t −f 2NN (2) Q(M)x(3/2) t ≤ f (xt, xt−t1, · · · xt−tM ) −f Q(M)x(3/2) t + f Q(M)x(3/2) t −f 2NN (2) Q(M)x(3/2) t ≤∥f∥Lip x⊤ t , x⊤ t−t1, · · · x⊤ t−tM ⊤−Q(M)x(3/2) t 2 + f −f 2NN (2) L∞([−2,2](M+1)D) ≤∥f∥Lip · EAttn + EFFN, where EFFN = ˜O ∥f∥B √m ; EAttn(type) =            O C(n) Hn PM k=1 e0.01Tk n+1 ! , type = lin O C(n) Hn PM k=1 T 1.01 k n+1 ! , type = log . Moreover, recalling our proof in Step II, we further need the hard condition: m ≥    ˜Ω PM r=1 ∥gr∥2 B , type = lin ˜Ω PM r=1 ∥log gr∥2 B T 2 r , type = log . Due to the arbitrariness of t and X, the proof is completed. Remark C.3. The core step in this proof is Step III, where the extraction of the memory functions is achieved through a a nice interaction between the temporal space (provided by RPE) and the token space (provided by DP). Specifically, the memory functions gi(xt) (in token space) are mapped into the temporal space s, resulting in the form of xs−gi(xt) for DP Attn with lin-RPE or log(s/gi(xt)) for DP Attn with log-RPE. 32 C.2 Proof of Proposition 4.2 For Proposition 16, we denote the following one-layer Attn hypothesis class: AT T N type (1,H) := Attn : TF is a 1-layer, H-head (normalization-free) Attn with type-RPE ; AT T N DPF,type (1,H) := TF : Attn is a 1-layer, H-head DP-free Attn with type-RPE . (16) Proposition C.4 (The formal version of Proposition 4.2). Consider 1-layer Attn. Then, there exists a target H ∈Hadap (1,1) (10) such that: (A) (Attn with DP) For any ϵ > 0, there exists a 1-layer Attn AttnDP ∈AT T N type (1,H) such that H −AttnDP ≤ϵ. (B) (Attn without DP) For any 1-layer DP-free Attn AttnDPF ∈AT T N DPF,type (1,H) , a uniform lower bound holds: H −AttnDPF ≥2 3. Proof of Proposition C.4. Consider the following target function H ∈HAdap 1,1 . Let the input sequence X ∈X = {−1, 0, 1}Z, and we consider the target yt = Ht(X) := xt−g(xt), where the adaptive memory is g(x) =    0, x = −1 1, x = 0 2, x = 1 . Part (A). The Efficiency of Attn with DP. First, we choose the embedding dimension D = 2, and select simple embedding WE = bE = (1, 0)⊤∈R2×1. Then for any input sequence X = (xt)t∈Z, the token after embedding satisfies: x(0) t = WExt + bE = (xt, 1)⊤. We consider one-layer normalization-free Self-attention with ϕexp, which has the form: AttnDP t (X) = WO H X h=1 W (1,h) V +∞ X s=0 x(0) t−s exp D W (l,h) Q x(0) t , W (1,h) K x(0) t−s E + p(1,h)s . By Lemma F.2 (for n = 1), there exists a constant C > 0 and a function ϕexp(t; B) = H X h=1 αh exp(−βh(t −B)) = H X h=1 αh exp βhB −βht such that βh > 0 and sup 0≤B≤2 ∥I{· = B} −ϕexp(·; B)∥ℓ1(N) ≤Ce0.01·2·2 H = O 1 H . 33 Moreover, Noticing that 0 ≤g(xt) ≤2 holds for any X = (xt)t∈Z ∈X, the following holds: sup X ∥I{· = g(xt)} −ϕexp(·; g(xt))∥ℓ1(N) ≤sup 0≤B≤2 ∥I{· = B} −ϕexp(·; B)∥ℓ1(N) ≤O 1 H . Therefore, for attention heads (1 ≤h ≤H), we can choose: p(1,h) = βh, W (1) O = (1, 0)⊤∈R2×1, W (1,h) V = αhδ(1×1) (1,1) ∈R2×2, W (1,h) Q = p βh(1, 1)⊤∈R2×1, W (1,h) K = p βh(0, 1)⊤∈R2×1, where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. Then it is easy to verify: D W (1,h) Q x(0) t , W (1,h) K x(0) t−s E + p(2,h)s = −p(1,h) s −(xt + 1) = −p(1,h) s −g(xt) . Thus, the following estimate holds: AttnDP t (X) −xt−g(xt) 2 = H X h=1 αh +∞ X s=0 e−βh(s−g(xt))xt−s −xt−g(xt) 2 = +∞ X s=0 H X h=1 αhe−βh(s−g(xt)) −I{s = g(xt)} ! xt−s 2 ≤ +∞ X s=0 H X h=1 αhe−βh(s−gk(xt)) −I{s = g(xt)} = ∥ϕexp(·; g(xt)) −I{· = g(xt)}∥ℓ1(N) ≤O 1 H . Due to the arbitrariness of t and X, the proof is completed: for any ϵ > 0, we only need to use H = Ω(1/ϵ) heads to approximate it. Part (B). The Hardness Result of Attn without DP. We consider one-layer Dot-product-free Self-attention with ϕexp or ϕlog. For any input X, the corresponding output can be written as: AttnDPF t (X) = +∞ X s=0 ρs(WExt−s + bE). For simplicity, we denote: x(0) t = WExt + bE, then we have the following estimate: \|\|\|H −AttnDPF\|\|\| = sup t sup X \|HDPF t (X) −Attnt(X)\| ≥sup X \|H0(X) −AttnDPF 0 (X)\| 34 = sup (··· ,x−2,x−1,x0) x−g(x0) − +∞ X s=0 ρsx(0) −s fixing x−s = 0 for s ≥3 ≥ sup (x−2,x−1,x0) x−g(x0) − 2 X s=0 ρsx(0) −s x0∈{−1,0,1} = max ( max (x−2,x−1) −1 − ρ0(−WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 , max (x−2,x−1) x−g(0) − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 , max (x−2,x−1) x−g(1) − ρ0(WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 ) = max (x−2,x−1) max n x−1 − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 , −1 − ρ0(−WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 , x−2 − ρ0(WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 o max{a,b,c}≥1 3 (a+b+c) ≥ 1 3 max (x−2,x−1) x−1 − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 + −1 − ρ0(−WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 + x−2 − ρ0(WE + bE) + ρ1x(0) −1 + ρ2x(0) −2 \|a\|+\|b\|≥\|a+b\| ≥ 1 3 max (x−2,x−1) x−1 − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 + −1 + x−2 −2 ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 ≥1 3 max (x−2,x−1) x−1 − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 + −1 + x−2 2 − ρ0bE + ρ1x(0) −1 + ρ2x(0) −2 \|a\|+\|b\|≥\|a−b\| ≥ 1 3 max (x−2,x−1) x−1 + 1 2 −x−2 2 = 2 3. C.3 Proof of Proposition 4.3 In this subsection, we propose a structurally simpler yet effective alternative to traditional Dot-product in Self-attention. This alternative is proposed based on our insights into the role of Attn in facilitating interaction between the temporal-space and the token-space. Specifically, we propose a more direct structure to achieve the interaction ϕtype(s) −ϕtype w⊤xt . Definition C.5 (TMX Transformer). We define the TMX (“t minus x”) Transformer as follows. In standard Transformer (2), we modify the term D W (l,h) Q xt, W (l,h) K xt−s E + p(l,h)ϕtype(s) in Multi-head Dot-product Self-attention to the new formulation: p(l,h) ϕtype (s) −ϕtype w(l,h) ⊤xt , 35 where the parameters w(l,h) ∈RD×1. Notice that in TMX Transformer, the revised term requires only O(D) parameters, much less than O(D2) in standard Dot-product Transformer. Consequently, we define the following TMX Transformer hypothesis class. T FTMX,type (1,H,m) := TF :TF is a 1-layer, H-head, m-width (normalization-free) TMX Transformer with type-RPE . (17) Proposition C.6 (The formal version of Proposition 4.3). Under the same conditions in Theorem 4.1, there exists a two-layer TMX Transformer TF ∈T FTMX,type (2,H,m) (17) such that it can achieve the same approximation rate as standard Transformer presented in Theorem 4.1. Proof of Proposition C.6. It is worth noting that TMX Transformer only replaces D W (l,h) Q xt, W (l,h) K xt−s E + p(l,h)ϕtype(s) in standard Transformer with p(l,h) ϕtype (s) −ϕtype w(l,h) ⊤xt . Therefore, the proof is highly similar to that of Theorem C.2. We only need to prove that TMX Attn can also achieve Step I and Step III in the proof of Theorem C.2. Step I. Step I is trivial due to the same use of the residual block. Step III. Extract the adaptive memories by the second Attn layer. We still consider to use Hk attention heads (from Pk−1 i=1 Hi + 1-th head to Pk i=1 Hi-th head) to extract it, and it satisfies to PM k=1 Hk = H. Now we consider the extraction of k-th adaptive memory xt−gk(xt) (1 ≤k ≤M). • Case type = lin. For the proof of standard Transformer (the proof of Theorem C.2), for the attention heads (Pk−1 i=1 Hi+1 ≤h ≤Pk i=1 Hi), we can construct specific p(2,h), W (2,h) Q , W (2,h) K , W (2,h) V such that D W (2,h) Q x(1) t , W (2,h) K x(1) t−s E + p(2,h)ϕlin(s) = −βh s −gk(xt) . In this proof, we only need to prove that we can also construct specific w(l,h), W (2,h) V such that p(2,h) ϕlin(s) −ϕlin w(2,h) ⊤x(1) t = −βh s −gk(xt) . Recalling the proof of Theorem C.2, x(1) t = (x⊤ t , 0⊤, g1(xt), · · · , gM(xt), 1)⊤∈RD. Therefore, we can choose p(2,h) = βh, w(2,h) = δ(1×1) (D−M+k−1,1) ∈RD, W (2,h) V = αhδ(d×d) (k+1,1) ∈RD×D. where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. The the following holds: p(2,h) ϕlin(s) −ϕlin w(2,h) ⊤x(1) t 36 = −p(2,h) s −w(2,h) ⊤x(1) t = −βh s −gk(xt) . • Case type = log. For the proof of standard Transformer (the proof of Theorem C.2), for the attention heads (Pk−1 i=1 Hi+1 ≤h ≤Pk i=1 Hi), we can construct specific p(2,h), W (2,h) Q , W (2,h) K , W (2,h) V such that D W (2,h) Q x(1) t , W (2,h) K x(1) t−s E + p(2,h)ϕlog(s) = −βh log s/gk(xt) . In this proof, we only need to prove that we can also construct specific w(l,h), W (2,h) V such that p(2,h) ϕlog(s) −ϕlog w(2,h) ⊤x(1) t = −βh log s/gk(xt) . Recalling the proof of Theorem C.2, x(1) t = (x⊤ t , 0⊤, log g1(xt), · · · , log gM(xt), log 2)⊤. Therefore, we can choose p(2,h) = βh, w(2,h) = δ(1×1) (D−M+k−1,1) ∈RD, W (2,h) V = αhδ(d×d) (k+1,1) ∈RD×D. where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. The the following holds: p(2,h) ϕlog(s) −ϕlog w(2,h) ⊤x(1) t = −p(2,h) log s/ w(2,h) ⊤x(1) t = −βh log s/gk(xt) . The rest of the proof is exactly the same as the proof of Theorem C.2, and we do not repeat it. 37 D Proof of Section 4.3 D.1 Proof of Theorem 4.4 In this subsection, we give the detailed proofs for the general case of K-adaptive, long but M-sparse memory: yt = f(xt, xt−t1, · · · , xt−tM ), where the adaptive memories satisfy: t1 = g1(xt); t2 = g2(xt, xt−t1); · · · tK+1 = gK+1(xt, xt−t1, · · · , xt−tK); · · · tK+2 = gK+2(xt, xt−t1, · · · , xt−tK); · · · tM = gK+1(xt, xt−t1, · · · , xt−tK), where 1 ≤tk ≤Tk holds for any k ∈[M]. Theorem D.1 (Restatement of Theorem 4.4). For any target H ∈HAdap (K,M), rate n ∈N+, and H, m ∈ N+, there exists an L-layer (L = K + 1 + I{M ≥K + 1}) Transformer TF ∈T FNF,type (L,H,m) (12) and a constant C(n) such that: if the width satisfies m ≥        ˜Ω max i∈[K] ∨ M P i=K+1 ∥gi∥2 B , type = lin, ˜Ω max i∈[K] ∨ M P i=K+1 ∥log gi∥2 B T 2 i , type = log , then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =            O C(n) Hn r PK l=1 e0.02(n+1)Tl + PM l=K+1 e0.01Tl 2n+2 ! , type = lin O C(n) Hn r PK l=1 T 2.02(n+1) l + PM l=K+1 T 1.01 l 2n+2 ! , type = log . Proof of Theorem D.1. First, we choose the embedding dimension D = (M + 1)(d + 1), and select the same embedding matrix WE = (Id, 0)⊤∈RD×d, bE = 0 =∈RD as the proof of Theorem C.2. Moreover, we still use the network with precision ] FFN defined in Appendix C.1 to tackle the discrete values of memories. Then for any input sequence X = (xt)t∈Z, the token after embedding satisfies: x(0) t = WExt + bE = (x⊤ t , 0⊤)⊤∈RD. Thus, for L-layer (L = K + 1 + I{M ≥K + 1}) normalization-free Transformer TF ∈T FNF,type (L,H,m) with ϕtype, the output token x(K+1) t of t-th input token satisfies: x(l−1/2) t = x(l) t + W (l) O H X h=1 Attn(l,h) t (X(l−1)), 1 ≤l ≤L + 1 38 x(l) t = x(l−1/2) t + ^ FFN (l) (x(l−1/2) t ), 1 ≤l ≤L x(L+1) t = FFN(L+1)(x(L+1/2) t ), where Attn(l,h) t (X) = W (l,h) V +∞ X s=0 xt−s exp D W (l,h) Q xt, W (l,h) K xt−s E + p(l,h)ϕtype(s) . Since the proof of this theorem is similar to the proof of Theorem C.2, we mainly discuss the differences. The proof can be summarized as the following process: • Case type = lin. – Regime M ≥K + 1. x(0) t Step 1. 1-st Attn ↓ x(1/2) t = x(0) t Step 2. 1-st FFN ↓ x(1) t = x(1/2) t + (0⊤, t1, 0⊤ M−1, 1)⊤ Step 3. 2-st Attn ↓ x(3/2) t ≈x(1) t + (0⊤ d , x⊤ t−t1, 0⊤)⊤ Step 4. 2-st FFN ↓ x(2) t = x(3/2) t + (0⊤, t2, 0⊤ M−1)⊤ Step 5. 3-st Attn ↓ x(5/2) t ≈x(2) t + (0⊤ 2d, x⊤ t−t2, 0⊤)⊤ · · · Step 2K + 1. K + 1-st Attn ↓ x(K+1/2) t ≈x(K) t + (0⊤ Kd, x⊤ t−tK, 0⊤)⊤ Step 2K + 2. K + 1-st FFN ↓ x(K+1) t = x(K+1/2) t + (0⊤, tK+1, · · · , tM, 0)⊤ Step 2K + 3. K + 2-st Attn ↓ x(K+3/2) t ≈x(K+1) t + (0⊤ (K+1)d, x⊤ t−tK+1, · · · , x⊤ t−tM , 0)⊤ Step 2K + 4. K + 2-st FFN ↓ x(K+2) t ≈f(xt, xt−t1, · · · , xt−tM ) – Regime M = K. x(0) t Step 1. 1-st Attn ↓ x(1/2) t = x(0) t Step 2. 1-st FFN ↓ x(1) t = x(1/2) t + (0⊤, t1, 0⊤ M−1, 1)⊤ Step 3. 2-st Attn ↓ x(3/2) t ≈x(1) t + (0⊤ d , x⊤ t−t1, 0⊤)⊤ 39 Step 4. 2-st FFN ↓ x(2) t = x(3/2) t + (0⊤, t2, 0⊤ M−1)⊤ Step 5. 3-st Attn ↓ x(5/2) t ≈x(2) t + (0⊤ 2d, x⊤ t−t2, 0⊤)⊤ · · · Step 2K + 1. K + 1-st Attn ↓ x(K+1/2) t ≈x(K) t + (0⊤ Kd, x⊤ t−tK, 0⊤)⊤ Step 2K + 2. K + 1-st FFN ↓ x(K+1) t ≈f(xt, xt−t1, · · · , xt−tM ) • Case type = log. – Regime M ≥K + 1. x(0) t Step 1. 1-st Attn ↓ x(1/2) t = x(0) t Step 2. 1-st FFN ↓ x(1) t = x(1/2) t + (0⊤, log t1, 0⊤ M−1, log 2)⊤ Step 3. 2-st Attn ↓ x(3/2) t ≈x(1) t + (0⊤ d , x⊤ t−t1, 0⊤)⊤ Step 4. 2-st FFN ↓ x(2) t = x(3/2) t + (0⊤, log t2, 0⊤ M−1)⊤ Step 5. 3-st Attn ↓ x(5/2) t ≈x(2) t + (0⊤ 2d, x⊤ t−t2, 0⊤)⊤ · · · Step 2K + 1. K + 1-st Attn ↓ x(K+1/2) t ≈x(K) t + (0⊤ Kd, x⊤ t−log tK, 0⊤)⊤ Step 2K + 2. K + 1-st FFN ↓ x(K+1) t = x(K+1/2) t + (0⊤, log tK+1, · · · , log tM, 0)⊤ Step 2K + 3. K + 2-st Attn ↓ x(K+3/2) t ≈x(K+1) t + (0⊤ (K+1)d, x⊤ t−tK+1, · · · , x⊤ t−tM , 0)⊤ Step 2K + 4. K + 2-st FFN ↓ x(K+2) t ≈f(xt, xt−t1, · · · , xt−tM ) – Regime M = K. x(0) t Step 1. 1-st Attn ↓ x(1/2) t = x(0) t Step 2. 1-st FFN ↓ x(1) t = x(1/2) t + (0⊤, log t1, 0⊤ M−1, log 2)⊤ Step 3. 2-st Attn ↓ x(3/2) t ≈x(1) t + (0⊤ d , x⊤ t−t1, 0⊤)⊤ Step 4. 2-st FFN ↓ 40 x(2) t = x(3/2) t + (0⊤, log t2, 0⊤ M−1)⊤ Step 5. 3-st Attn ↓ x(5/2) t ≈x(2) t + (0⊤ 2d, x⊤ t−t2, 0⊤)⊤ · · · Step 2K + 1. K + 1-st Attn ↓ x(K+1/2) t ≈x(K) t + (0⊤ Kd, x⊤ t−tK, 0⊤)⊤ Step 2K + 2. K + 1-st FFN ↓ x(K+1) t ≈f(xt, xt−t1, · · · , xt−tM ) For simplicity, we denote the following projection matrices: P (k) := (0d×kd Id×d 0) ∈Rd×D, 1 ≤k ≤M; P (k) ⊥ := Ikd×kd 0d×d 0 0 0d×d I(D−(k+1)d)×(D−(k+1)d) ∈R(D−d)×D, 1 ≤k ≤M; Q(k) := I(k+1)d×(k+1)d 0 ∈R(k+1)d×D, 1 ≤k ≤M; Q(k) ⊥:= 0 I(D−(k+1)d)×(D−(k+1)d) ∈R(D−(k+1)d)×D, 1 ≤k ≤M; R := 0(M−K)d×(K+1)d I(M−K)d×(M−K)d 0 ∈R(M−K)d×D; R⊥:= I(K+1)d×(K+1)d 0 0 0 0 I(M+1)×(M+1) ∈R(D−(M−K)d)×D. Step 1 is trivial due to the use of the residual block. Step 2. In the same way as Step II in the proof of Theorem C.2, we obtain the conclusion in this step: If the width of FFN satisfies m ≥    ˜Ω ∥g1∥2 B , type = lin ˜Ω ∥log g1∥2 B T 2 1 , type = log , then the following holds: x(1) t = x(1/2) t + (0⊤, t1, 0⊤ M, 1)⊤. Thus, (E1) holds for l = 2. Step 3 ∼Step 2K + 1. • Case type = lin. – FFN layers. We use l-th (2 ≤l ≤K) FFN layer to express l-th memory tl exactly. By Lemma G.6, there exists a two-layer neural network with m neurons defined on Rld f 2NN (l) (x) = m X k=1 a(l) k σ(b(l) k ⊤x + c(l) k ) such that gl −f 2NN (l) L∞([−2,2]ld) ≤˜O ∥gl∥B √m . For l-th FFN layer, we only need to arrange the parameters a(l) k , b(l) k , and c(l) k (k ∈ [m], 2 ≤l ≤M). 41 Denote ¯b(l) k = (b(l) k ⊤, 0⊤)⊤∈RD for k ∈[m], 2 ≤l ≤K −1. Consider the following two-layer neural network with m neurons defined on RD: FFN(l)(x) = m X k=1 eD−M+l−1a(l) k σ ¯b(l) k ⊤x + c(l) k . It is easy to verify FFN(l)(x) = 0⊤, f 2NN (l) Q(l)x , 0⊤ D−M+l−1 ⊤ ∈RD, ∀x ∈RD. Notice that if the width m satisfies ˜O ∥gl∥B √m < 1 4, and the input x(l−1/2) t satisfies ∥gl∥Lip · (x⊤ t , · · · , x⊤ t−tl−1)⊤−Q(l−1)x(l−1/2) t 2 ≤1 4 the following holds: gl(xt, · · · , xt−tl−1) −f 2NN (l) Q(l)x(l−1/2) t ≤ gl(xt, · · · , xt−tl−1) −gl(Q(l−1)x(l−1/2) t ) + gl(Q(l−1)x(l−1/2) t ) −f 2NN (l) (Q(l−1)x(l−1/2) t ) ≤∥gl∥Lip (x⊤ t , · · · , x⊤ t−tl−1)⊤−Q(l−1)x(l−1/2) t 2 + gl −f 2NN (l) L∞ <1 4 + 1 4 = 1 2, Noticing tl = gl(xt, · · · , xt−tl−1) ∈N+, we have ] f 2NN (l) (Q(l−1)x(l−1/2) t ) = tl. Thus, it holds that: ^ FFN (l) (x(l−1/2) t ) = (0⊤, tl, 0⊤ D−M+l−1)⊤. – Attn layers. By Lemma F.2, for any rate n ∈N+, there exists a constant C(n) and K functions ϕexp l (t; B) = X h=1 αl,h exp(−βl,h(t −B)) = H X h=1 αl,h exp βl,hB −βl,ht , 1 ≤l ≤K such that βl,h > 0 and sup 1≤B≤Tl ∥I{· = B} −ϕexp l (·; B)∥ℓ1(N) ≤C(n)e0.01(n+1)Tl Hn , 1 ≤l ≤K. Moreover, Noticing that 1 ≤gl(·) ≤Tl holds for any X = (xt)t∈Z ∈X and 2 ≤l ≤K, the following holds: sup X ∥I{· = tl} −ϕexp l (·; tl)∥ℓ1(N) ≤ sup 1≤B≤Tl ∥I{· = B} −ϕexp l (·; B)∥ℓ1(N) ≤C(n)e0.01(n+1)Tl Hn . 42 Therefore, for the attention heads h (h ∈[H]) in each layer l (1 ≤l ≤K), we can choose: p(l+1,h) = βl,h, W (l+1) O = I, W (l+1,h) V = αl,hδ(d×d) (l+1,1) ∈RD×D, W (l+1,h) Q = p βl,hδ(1×1) (D−M+l,1) ∈RD×(D/H), W (l+1,h) K = p βl,hδ(1×1) (D,1) ∈RD×(D/H), where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. • Case type = log. – FFN layers. We use l-th (2 ≤l ≤K) FFN layer to express l-th memory tl exactly. By Lemma G.6, there exists a two-layer neural network with m neurons defined on Rld f 2NN (l) (x) = m X k=1 a(l) k σ(b(l) k ⊤x + c(l) k ) such that log gl −f 2NN (l) L∞≤˜O ∥log gl∥B √m . For l-th FFN layer, we only need to arrange the parameters a(l) k , b(l) k , and c(l) k (k ∈ [m], 2 ≤l ≤M). Denote ¯b(l) k = (b(l) k ⊤, 0⊤)⊤∈RD for k ∈[m], 2 ≤l ≤K −1. We consider the following l-th layer 2NN with m neurons defined on RD: FFN(l)(x) = m X k=1 eD−M+l−1a(r) k σ ¯b(r) k ⊤x + c(r) k . It is easy to verify FFN(l)(x) = 0⊤, f 2NN (l) Q(l)x , 0⊤ D−M+l−1 ⊤ ∈RD, ∀x ∈RD. Notice that if the width m satisfies ˜O ∥log gl∥B √m < 1 8Tl , and the input x(l−1/2) t satisfies ∥log gl∥Lip · (x⊤ t , · · · , x⊤ t−tl−1)⊤−Q(l)x(l−1/2) t 2 ≤ 1 8Tl , the following holds: log gl(xt, · · · , xt−tl−1) −f 2NN (l) (Q(l)x(l−1/2) t ) ≤ log gl(xt, · · · , xt−tl−1) −log gl(Q(l)x(l−1/2) t ) + gl(Q(l)x(l−1/2) t ) −f 2NN (l) (Q(l)x(l−1/2) t ) 43 ≤∥log gl∥Lip (x⊤ t , · · · , x⊤ t−tl−1)⊤−Q(l)x(l−1/2) t 2 + log gl −f 2NN (l) L∞ < 1 8Tl + 1 8Tl = 1 4Tl , which ensures exp f 2NN (l) (Q(l)x(l−1/2) t ) −gl(xt, · · · , xt−tl−1) = exp f 2NN (l) (Q(l)x(l−1/2) t ) −exp log gl(xt, · · · , xt−tl−1) ≤exp max n f 2NN (l) (Q(l)x(l−1/2) t ), log gl(xt, · · · , xt−tl−1) o · f 2NN (l) (Q(l)x(l−1/2) t ) −log gl(xt, · · · , xt−tl−1) ≤exp log gl(xt, · · · , xt−tl−1) + 1 8 1 4Tr ≤e1/8 · Tr · 1 4Tr < 1 2. Noticing tl = gl(xt, · · · , xt−tl−1) ∈N+, we have ] f 2NN (l) (Qlx(l−1/2) t ) = log exp h FFN(l)(xt, · · · , xt−tl−1) i = log tl. Thus, it holds that: ^ FFN (l) (x(l−1/2) t ) = (0⊤, log tl, 0⊤ D−M+l−1)⊤. – Attn layers. By Lemma F.5, for any rate n ∈N+, there exists a constant C(n) and K functions ϕpoly l (t; B) = X h=1 αl,h(t/B)−βl,h = H X h=1 αl,h exp −βl,h log(t/B) , 1 ≤l ≤K such that βl,h > 0 and sup 1≤B≤Tl I{· = B} −ϕpoly l (·; B) ℓ1(N+) ≤C(n)T 1.01(n+1) l Hn , 1 ≤l ≤K. Moreover, Noticing that 1 ≤gl(·) ≤Tl holds for any X = (xt)t∈Z ∈X and 1 ≤l ≤K, the following holds: sup X I{· = tl} −ϕpoly l (·; tl) ℓ1(N+) ≤ sup 1≤B≤Tl I{· = B} −ϕpoly l (·; B) ℓ1(N+) ≤C(n)T 1.01(n+1) l Hn . Therefore, for the attention heads h (h ∈[H]) in each layer l (1 ≤l ≤K), we can choose: p(l+1,h) = βl,h, W (l+1) O = I, W (l+1,h) V = αl,hδ(d×d) (l+1,1) ∈RD×D, W (l+1,h) Q = p βl,hδ(1×1) (D−M+l,1) ∈RD×(D/H), W (l+1,h) K = p βl,hδ(1×1) (D,1) ∈RD×(D/H), where δ(r×r) (p1,p2) means that: it equals to Ir×r for the (p1, p2)-th r × r blocks, and 0r×r for the other r × r blocks. 44 Similar to the estimate in Step II and Step III in the proof of Theorem C.2, it is easy to prove the following estimates by induction. If the width satisfies m ≥        ˜Ω max l∈[K] ∥gl∥2 B = ˜Ω ∥gK∥2 B , type = lin ˜Ω max l∈[K] ∥log gl∥2 B T 2 l = ˜Ω ∥log gK∥2 B T 2 K , type = log , and the head number satisfies        C(n) Hn qPK−1 l=1 e0.02(n+1)Tl ≤ 1 4 max l∈[K−1]∥gl∥Lip , type = lin C(n) Hn qPK−1 l=1 T 2.02(n+1) l ≤ 1 4 max l∈[K−1]∥log gl∥Lip , type = log , then the following estimates hold: • (E1) for any 2 ≤l ≤K, x(l) t = x(l−1/2) t + (0⊤, tl, 0⊤ M−l+1)⊤; • (E2) for any 1 ≤l ≤K, P (l) ⊥x(l+1/2) t = P (l) ⊥x(l) t ; • (E3) for any 1 ≤l ≤K, P (l) x(l+1/2) t − x(l) t + (0⊤ ld, x⊤ t−tl, 0⊤)⊤ 2 ≤ ( C(n)e0.01(n+1)Tl Hn , type = lin C(n)T 1.01(n+1) l Hn , type = log ; Q(l) x(l+1/2) t −x(0) t 2 ≤    C(n) Hn qPl j=1 e0.02(n+1)Tl, type = lin C(n) Hn qPl j=1 T 2.02(n+1) j , type = log . The Remained Steps. • Regime M ≥K + 1. Step 2K + 2 and 2K + 3. In the similar way as Step 3 ∼Step 2K −1 in this proof and Step II, Step III in the proof of Theorem C.2, it is easy to verify the following estimate. If the width satisifes m ≥    ˜Ω PM l=K+1 ∥gl∥2 B , type = lin ˜Ω PM l=K+1 ∥log gl∥2 B T 2 l , type = log , and the head number satisfies        C(n) Hn qPK l=1 e0.02(n+1)Tl ≤ 1 4 max l∈[K]∥gl∥Lip , type = lin C(n) Hn qPK l=1 T 2.02(n+1) l ≤ 1 4 max l∈[K]∥log gl∥Lip , type = log , then the following estimates hold: x(K+1) t = x(K+1/2) t + (0⊤, tK+1, · · · , tM, 0)⊤; 45 R⊥x(K+3/2) t = R⊥x(K+1) t ; R x(K+3/2) t − x(K+1) t + (0⊤ (K+1)D, x⊤ t−tK+1, · · · , x⊤ t−tM , 0⊤)⊤ 2 ≤          C(n) PM l=K+1 e0.01Tl H n n+1 n+1 , type = lin C(n) PM l=K+1 T 1.01 l H n n+1 !n+1 , type = log , Q(M) x(K+3/2) t −x(0) t 2 ≤        C(n) Hn r PK l=1 e0.02(n+1)Tl + PM l=K+1 e0.01Tl 2n+2 , type = lin C(n) Hn r PK l=1 T 2.02(n+1) l + PM l=K+1 T 1.01 l 2n+2 , type = log . Step 2K + 4 and the final bound. In the same way as Step IV in the proof of Theorem C.2, there exists FFN, such that the following estimate holds for any t and X: Ht(X) −x(K+2) t ≤∥f∥Lip · Q(M) x(K+3/2) t −x(0) t 2 + EFFN = ∥f∥Lip · EAttn(type) + EFFN, where EFFN = O ∥f∥B √m , EAttn(type) = Q(M) x(K+3/2) t −x(0) t 2 =        C(n) Hn r PK l=1 e0.02(n+1)Tl + PM l=K+1 e0.01Tl 2n+2 , type = lin C(n) Hn r PK l=1 T 2.02(n+1) l + PM l=K+1 T 1.01 l 2n+2 , type = log . Recalling our analysis, we need the head number satisfies        C(n) Hn qPK l=1 e0.02(n+1)Tl ≤ 1 4 max l∈[K]∥gl∥Lip , type = lin C(n) Hn qPK l=1 T 2.02(n+1) l ≤ 1 4 max l∈[K]∥log gl∥Lip , type = log . Due to    C(n) Hn qPK l=1 e0.02(n+1)Tl ≤EAttn(type), type = lin C(n) Hn qPK l=1 T 2.02(n+1) l ≤EAttn(type), type = log , when we is large enough, this condition holds naturally and do not affect the approximation rate. Moreover, we need the following condition on the width: 46 m ≥        ˜Ω max l∈[K] ∨PM l=K+1 ∥gl∥2 B , type = lin ˜Ω max l∈[K] ∨PM l=K+1 ∥log gl∥2 B T 2 l , type = log , • Regime M = K. Step 2K + 2 and the final bound. In the same way as Step IV in the proof of Theorem C.2, there exists FFN, such that the following estimate holds for any t and X: Ht(X) −x(K+1) t ≤∥f∥Lip · Q(M) x(K+1/2) t −x(0) t 2 + EFFN = ∥f∥Lip · EAttn(type) + EFFN, where EFFN = O ∥f∥B √m , EAttn(type) = QM x(K+1/2) t −x(0) t 2 =    C(n) Hn qPK l=1 e0.02(n+1)Tl, type = lin C(n) Hn qPK l=1 T 2.02(n+1) l , type = log . Recalling our analysis, we need the head number satisfies        C(n) Hn qPK−1 l=1 e0.02(n+1)Tl ≤ 1 4 max l∈[K−1]∥gl∥Lip , type = lin C(n) Hn qPK−1 l=1 T 2.02(n+1) l ≤ 1 4 max l∈[K−1]∥log gl∥Lip , type = log . Due to    C(n) Hn qPK−1 l=1 e0.02(n+1)Tl ≤EAttn(type), type = lin C(n) Hn qPK−1 l=1 T 2.02(n+1) l ≤EAttn(type), type = log , when H is large enough, this condition holds naturally and do not affect the approximation rate. Moreover, we need the following condition on the width: m ≥        ˜Ω max l∈[K] ∥gl∥2 B , type = lin ˜Ω max l∈[K] ∥log gl∥2 B T 2 l , type = log , Combining these two regimes, we complete our proof. 47 D.2 Proof of Proposition 4.5 Proof of Proposition 4.5. This proposition is a direct corollary of Theorem 4.4. It can be seen as a special case of M = K in Theorem 4.4. Therefore, under the same conditions, there exists a K + 1-layer Transformer TF ∈ T FNF,type (K+1,H,m) (12) and a constant C(n) such that: if the width satisfies m ≥      ˜Ω max i∈[K] ∥gi∥2 B , type = lin, ˜Ω max i∈[K] ∥log gi∥2 B T 2 i , type = log , then the following approximation rate holds: \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type), where EFFN = ˜O ∥f∥B √m and EAttn(type) =        O C(n) Hn qPK l=1 e0.02(n+1)Tl , type = lin O C(n) Hn qPK l=1 T 2.02(n+1) l , type = log . Comparison between Proposition 4.5 and Theorem 4.1. We compare 2-layer Transformer and M + 1-layer Transformer regarding the requirement of the number of heads and width. • The required width of FFN layers. – For 2-layer Transformer, the required width of FFN layers m(2) need is proportionally linked to the sum of all the memory functions’ complexity: m(2) need =        ˜Ω P i∈[K] ∥gi∥2 B , type = lin, ˜Ω P i∈[K] ∥log gi∥2 B T 2 i , type = log . – For M + 1-layer Transformer, the required width of FFN layers m(M+1) need correlates with the maximum complexity of the memory functions: m(M+1) need =      ˜Ω max i∈[K] ∥gi∥2 B , type = lin, ˜Ω max i∈[K] ∥log gi∥2 B T 2 i , type = log . It is easy to see that: m(M+1) need m(2) need = max{a1, · · · , aM} PM k=1 ak , max{a1, · · · , aM} ≤ M X k=1 ak • The required number of Attn heads. To achieve the same EAttn(type) = ϵ, 48 – for 2-layer Transformer, the required number of Attn heads H(2) need satisfies: ϵ =        O C(n) H(2) need n qPK l=1 e0.02(n+1)Tl , type = lin O C(n) H(2) need n qPK l=1 T 2.02(n+1) l , type = log . – for M + 1-layer Transformer, the required number of Attn heads H(M+1) need satisfies: ϵ =        O C(n) H(M+1) need n PM i=1 e0.01Ti n+1 , type = lin O C(n) H(M+1) need n PM i=1 T 1.01 i n+1 , type = log . It is easy to see that: H(M+1) need H(2) need !2n = b2 1 + · · · + b2 M (b1 + · · · bM)2 , b2 1 + · · · + b2 M ≤(b1 + · · · bM)2. This finding suggests that increased depth can significantly reduce the demands on the number of heads and the width. The underlying reason is that deep networks can distribute memories across different layers for processing, with each layer focusing on approximating only a single memory function. 49 E Proof of Section 5 E.1 Proof of Theorem 5.1 In this subsection, we give the detailed proofs of the warm-up case of (fixed) essentially sparse memories as follows: yt = f ((X ∗ρ1) (t), · · · , (X ∗ρM) (t)) , where ρ1(·), · · · , ρM(·) ∈ℓ1(N) serve as memory kernels, and (X ∗ρk)(t) = P+∞ s=0 xt−sρk(s) denotes the convolution of the inputs with kernel ρk. Theorem E.1 (Restatement of Theorem 5.1). (A) Consider HEss (14) with exponentially decayed memory kernels, i.e., there exists β > 0 such that ρ1(t), · · · , ρM(t) = O(e−βt). Then for any target H ∈HEss, rate n ∈[⌊99β⌋], and H, m ∈N+, there exists a 1-layer DP-free Transformer TF ∈T FDPF,exp (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type); (B) Consider HEss (14) with polynomially decayed memory kernels, i.e., there exists β > 1 such that ρ1(t), · · · , ρM(t) = O(t−β). Then for any target H ∈HEss, rate n ∈[⌊0.99β⌋−1], and H, m ∈N+, there exists a 1-layer DP-free Transformer TF ∈T FDPF,poly (1,H,m) (7) and a constant C(n) such that \|\|\|H −TF\|\|\| ≤EFFN + ∥f∥Lip EAttn(type); where EFFN = ˜O ∥f∥B √m and EAttn(type) = O C(n)M n+1 Hn . Proof of Theorem E.1. The proof of this theorem is highly similar to the proof of Theorem B.1. The only difference is that the Attn layer needs to be used to approximate general memory kernel ρk(·) instead of simple I{· = Tk}. But for the completeness of the proof in this section, we still provide the detailed proof. First, we choose the embedding dimension D = Md, and select the simple embedding WE = (Id×d, 0)⊤∈RD×d, bE = 0 ∈RD. For any input sequence X = (xt)t∈Z, the token after embedding satisfies: xE t = WExt + bE = (x⊤ t , 0⊤)⊤∈RD. Then for one-layer Dot-product-free Transformer TF ∈T FDPF,type (1,H,m) without residual blocks, the output token TFt(X) of t-th input token xt satisfies: x(1/2) t = W (1) O H X h=1 Attn(1,h) t (X(0)), x(1) t = FFN(1)(x(1/2) t ) where Attn(1,h) t (X) = W (1,h) V +∞ X s=0 xt−s exp p(1,h)ϕtype(s) P+∞ j=0 exp p(1,h)ϕtype(j) . This proof can be summarized as the following process: · · · xE t · · · 50 Step I. Attn layer ↓ · · · x(1/2) t ≈((X ∗ρ1)(t), · · · , (X ∗ρM)(t))⊤ · · · Step II. FFN layer ↓ · · · x(1) t ≈f ((X ∗ρ1)(t), · · · , (X ∗ρM)(t)) · · · Now we give the formal proof. Step I. Extract the memory locations by (Dot-product-free) Attn layer. We consider to use Hk attention heads (from Pk−1 i=1 Hi + 1-th head to Pk i=1 Hi-th head) to extract it, and it satisfies to PM k=1 Hk = H. P (k) := 0d×(k−1)d Id×d 0 ∈Rd×D, 1 ≤k ≤M. P (k) ⊥ := I(k−1)d×(k−1)d 0d×d 0 0 0d×d I(M−k−1)d×(M−k−1)d ∈R(M−1)d×D, 1 ≤k ≤M. Now we consider the extraction of k-th memory (X ∗ρk)(t) (1 ≤k ≤M). • Case (A). Approximating exponentially decayed memories by type = lin. Because there exists β > 0 such that ρk(t) = O(e−βt), by Lemma F.3, for any n ∈[⌊99β⌋] and m ∈N+, there exists an absolute constant C(n) only depending on n and a function ϕexp k (t) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αhe−βht such that βh > 0 and ∥ρk(·) −ϕexp k (·)∥ℓ1(N) = +∞ X s=0 \|ρk(s) −ϕexp k (s)\| ≤C(n) mn . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose p(1,h) = βh, W (1,h) V = αh   +∞ X j=0 exp(−βhj)  δd×d (k,1), where δ(k,1) ∈RD×D means that: it equals to Id×d for the (k, 1)-th d × d blocks, and 0d×d for the other d × d blocks. Then it holds that: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhs 0(k−1)d xt−s 0 ! ∈RD, This implies: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhsxt−s, 51 P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = 0, moreover, the following estimate holds: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(1,h) t (X(0)) −(X ∗ρk)(t) 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=0 e−βhsxt−s − +∞ X s=0 xt−sρk(s) 2 = +∞ X s=0   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βhs −I{s = Tk}  xt−s 2 ≤ +∞ X s=0 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhe−βhs −ρk(s) = ∥ϕexp k (·) −ρk(·)∥ℓ1(N) ≤C(n) Hn k . • Case (B). Approximating polynomially decayed memories by type = log. Because there exists β > 0 such that ρk(t) = O(t−β), by Lemma F.6, for any n ∈ [⌊0.99β⌋−1] and m ∈N+, there exists an absolute constant C(n) only depending on n and a function ϕpoly k (t) = X Pk−1 i=1 Hi+1≤h≤Pk i=1 Hi αht−βh such that βh > 1 and ρk(·) −ϕpoly k (·) ℓ1(N+) = +∞ X s=1 ρk(s) −ϕpoly k (s) ≤C(n) mn . Therefore, for these attention heads (Pk−1 i=1 Hi + 1 ≤h ≤Pk i=1 Hi), we can choose p(1,h) = βh, W (1,h) V = αh   +∞ X j=1 j−βh  δ(k,1), where δ(k,1) ∈RD×D means that: it equals to Id×d for the (k, 1)-th d × d blocks, and 0d×d for the other d × d blocks. Then it holds that: Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=1 s−βh 0(k−1)d xt−s 0 ! , This implies: P (k) Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=1 s−βhxt−s, 52 P (k) ⊥ Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X(0)) = 0, Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(1,h) t (X(0)) −(X ∗ρk)(t) 2 = Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αh +∞ X s=1 s−βhxt−s − +∞ X s=0 xt−sρk(s) 2 = +∞ X s=1   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhs−βh −ρk(s)  xt−s 2 ≤ +∞ X s=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 αhs−βh −ρk(s) = ϕpoly Hk (·) −ρk(·) ℓ1(N+) ≤C(n) Hn k . Then we combine the estimate for all k ∈[M] for these two cases. By choose WO = ID, we have: x(1/2) t −    (X ∗ρ1)(t) ... (X ∗ρM)(t)    2 = M X k=1   Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X) − 0(k−1)d (X ∗ρk)(t) 0d !  2 ≤ M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 Attn(1,h) t (X) − 0(k−1)d (X ∗ρk)(t) 0d ! 2 = M X k=1 Pk i=1 Hi X h=Pk−1 i=1 Hi+1 P (k)Attn(1,h) t (X(0)) −(X ∗ρk)(t) 2 ≤EAttn := M X k=1 C(n) Hn k , for both Case (A) and Case (B). Consequently, one detail is to assign the head number {Hk}M k=1 such that the error’s sum EAttn(type) is as small as possible. Here, we simply choose the same Hk: Hk = H M , k ∈[M]. Thus, we obtain the bound in Step I: EAttn = M X k=1 C(n) Hn k = C(n)M n+1 Hn . 53 Furthermore, by choosing EAttn ≤1, it holds that x(1/2) t ∞≤ x(1/2) t −    (X ∗ρ1)(t) ... (X ∗ρM)(t)    ∞ +    (X ∗ρ1)(t) ... (X ∗ρM)(t)    ∞ ≤EAttn + 1 ≤2. Step II. Approximate the readout function by FFN layer. In this step, we aim to approximate the function f using two-layer network. By Lemma G.6, there exists a two-layer neural network with m neurons defined on RD FFN(1)(y) = m X k=1 akσ(b⊤ k y + ck) such that EFFN := FFN(1) −f L∞([−2,2]D) ≤˜O ∥f∥B √m . The final bound. For any t and X ∈X, it holds that Ht(X) −x(1) t = f((X ∗ρ1)(t), · · · , (X ∗ρM)(t)) −FFN(1) x(1/2) t = f((X ∗ρ1)(t), · · · , (X ∗ρM)(t)) −f x(1/2) t + f x(1/2) t −FFN(1) x(1/2) t ≤ f((X ∗ρ1)(t), · · · , (X ∗ρM)(t)) −f x(1/2) t + f x(1/2) t −FFN(1) x(1/2) t ≤∥f∥Lip ((X ∗ρ1)(t)⊤, · · · , (X ∗ρM)(t)⊤)⊤−x(1/2) t 2 + f −FFN(1) L∞([−2,2]D) ≤∥f∥Lip · EAttn + EFFN, where EFFN = ∥f∥B √m ; EAttn = C(n)M n+1 Hn , for both Case (A) and Case (B). Due to the arbitrariness of t and X, the proof is completed. 54 F Key Lemmas about Approximation F.1 Approximation by the sum of exponential decay Lemma F.1 (Exp decay, fixed Delta function). For any T ∈N+, n, m ∈N+, there exists and absolute constant C(n) only depending on n and a ϕexp m (t) = m P k=1 αke−βkt such that ∥I(· = T) −ϕexp m (·)∥ℓ1(N) ≤C(n)e0.01(n+1)T mn . where βk > 0 holds for any k ∈[m]. Proof of Lemma F.1. Let α, γ > 0 be constants, and they will take specific values at the end of the proof. First, recall the standard bump function on [−1, 1]: Ψ(x) := ( exp − 1 1−x2 , x ∈(−1, 1) 0, otherwise , and we can define the following constants for T ≥1: µT = e−αT , σT = e−αT −e−α(T +1). Then we consider the following bump function ΨT ∈C∞([0, 1]): ΨT (x) = ( VT Ψ x−µT σT , x ∈(µT −σT , µT + σT ) 0, otherwise , where VT is a scaling constant such that ΨT (e−αT ) = eγT . First, we consider the approximation of ΨT on [0, 1]. Notice that ΨT ∈C∞([0, 1]), and Ψ(k) T (0) = 0 for any k ∈N. For the standard bump function Ψ, for any n ∈N+, there exists an absolute constant M(n) > 0 only depending on n, such that max 0≤k≤10 sup x∈[−1,1] Ψ(k)(x) ≤M(n). Notice that for any k ∈N and x ∈[0, 1], Ψ(k) T (x) = VT σk T Ψ(k) x −µT σT . Therefore, the following upper bound holds: MT (n) = max 0≤k≤n VT σk T M(n) = VT σn T M(n) = eγT · e e−αT −e−α(T +1)n M(n) = M(n)e (1 −1/e)n e(γ+nα)T := C(n, α)e(γ+nα)T . By Lemma G.5, for any m ∈N+, there exists a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|ΨT (x) −Qm(x)\| ≤MT (n) mn ≤C(n, α)e(γ+nα)T mn . Now we use the transform x = e−αt on the function Ψ and consider ΦT (t) := e−γtΨT (e−αt), t ∈[0, +∞). 55 It is easy to verify that ΦT satisfies that ΦT (t) N = I(t = T). Moreover, we consider the function Pm(t) := e−γtQm(e−αt), t ∈[0, +∞). Then by choosing α = γ = 0.01, the following error estimate holds: ∥Pm(·) −I(· = T)∥ℓ1(N) = +∞ X t=0 \|Pm(t) −ΦT (t)\| = +∞ X t=0 e−γt\|Qm(e−αt) −ΨT (e−αt)\| ≤ +∞ X t=0 e−γt MT (n) mn ≤C(n, α)e(γ+nα)T mn +∞ X t=0 e−γt ≤C(n)e0.01(n+1)T mn 1 1 −e−γ = ˜C(n)e0.01(n+1)T mn . Finally, notice that Pm(t) = e−γtQm (e−αt) = m−1 P k=0 αke−(0.01+0.01k), so we can select ϕexp m (t) := Pm(t). Lemma F.2 (Exp decay, adaptive Delta function). For any T ∈N, n, m ∈N+, there exists an absolute constant C(n) only depending on n and a ϕexp m (t; B) = m P k=1 αke−βk(t−B) such that max 1≤B≤T ∥I(· = B) −ϕexp m (·; B)∥ℓ1(N) ≤C(n)e0.01(n+1)T mn . where βk > 0 holds for any k ∈[m]. Proof of Lemma F.2. The key point of the proof is to note that the adaptability of B can be eliminated by the translation operator t −B. First, recall our proof of Lemma F.1. For the same ΨT (·), for any n, m ∈N+, there exists an absolute constant C(n) only depending on n and a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|ΨT (x) −Qm(x)\| ≤C(n)e0.01(n+1)T mn . Moreover, using the transform x = e−0.01(t−B+T ) (t ≥0) on the function Ψ and consider ΦT (t; B) := e−0.01(t−B+T )ΨT e−0.01(t−B+T ) , t ∈[0, +∞). It is easy to verify that ΦT (·; ·) satisfies that ΦT (t; B) N = I(t = B). And we consider the function Pm(t; B) := e−0.01(t−B+T )Qm e−0.01(t−B+T ) , t ∈[0, +∞). 56 Then, for any 1 ≤B ≤T, the following error estimate holds: ∥Pm(·; B) −I(· = B)∥ℓ1(N) = +∞ X t=0 \|Pm(t; B) −ΦT (t; B)\| = +∞ X t=0 e−0.01(t−B+T ) Qm e−0.01(t−B+T ) −ΨT e−0.01(t−B+T ) ≤ +∞ X t=0 e−0.01t sup x∈[0,1] \|Qm(x) −ΨT (x)\| ≤C(n)e0.01(n+1)T mn +∞ X t=0 e−0.01t = ˜C(n)e0.01(n+1)T mn . Due to the arbitrariness of B, the proof is completed. Lemma F.3 (Exp decay, fixed Delta function). Consider a exponentially decayed memory ρ(·): there exists β > 0 such that ρ(t) = O(e−βt). Then for any n ∈[⌊99β⌋] and m ∈N+, there exists an absolute constant C(n) only depending on n and a ϕexp m (t) = m P k=1 αke−βkt such that ∥ρ(·) −ϕexp m (·)∥ℓ1(N) ≤C(n) mn , where βk > 0 holds for any k ∈[m]. Proof of Lemma F.3. There exists C > 0 such that \|ρ(t)\| ≤Ce−βt. Let α, γ > 0 be constants, and they will take specific values at the end of the proof. First, recall the standard bump function on [−1, 1]: Ψ(x) := ( exp − 1 1−x2 , x ∈(−1, 1) 0, otherwise , and we can define the following constants for T ≥1: µT = e−αT , σT = 1 2 e−αT −e−α(T +1) , and we consider the following bump function ΨT ∈C∞([0, 1]): ΨT (x) = ( VT Ψ x−µT σT , x ∈(µT −σT , µT + σT ) 0, otherwise , where VT is a scaling constant such that ΨT (e−αT ) = eγT ρ(T). Consequently, we consider the sum of bump functions on [0, 1]: φ(x) := +∞ X T =1 ΨT (x). It is easy to verify that (µT1 −σT1, µT1 + σT1) ∩(µT2 −σT2, µT2 + σT2) = ∅for any T1 ̸= T2 and φ(x) = ΨT (x), µT −σT ≤x ≤µT + σT 0, otherwise . 57 First, we study the property of φ(·). We denote the absolute constants Mk = supx \|φ(k)(x)\|. Notice that for any k ∈N, Ψ(k) T (x) = VT σk T Ψ(k) x −µT σT . Therefore, it holds that sup x∈(µT −σT ,µT +σT ) \|φ(k)(x)\| = sup x∈(µT −σT ,µT +σT ) \|Ψ(k) T (x)\| ≤VT σk T Mk = eγT ρ(T) e−αT −e−α(T +1)k Mke ≤ CMke (1 −e−α)k e(γ+kα−β)T . Therefore, if β ≥γ + kα, then the following uniform bounds holds: sup x∈(0,1] \|φ(k)(x)\| = sup T ≥1 sup x∈(µT −σT ,µT +σT ) \|φ(k)(x)\| ≤sup T ≥1 CMke (1 −e−α)k e(γ+kα−β)T ≤ CMke (1 −e−α)k := C(k, α). Consequently, we consider the smoothness of Φ at x = 0. Recalling the previous results, for any x ∈(0, 1], we have \|φ(k)(x)\| x ≤C(k, α)e(γ+kα−β)T µT −σT = 2C(k, α) 1 −e−α e(γ+(k+1)α−β)T , x ∈(µT −σT , µT + σT ); \|φ(k)(x)\| x = 0, otherwise Thus, by induction, it is easy to verify that for any i < β−γ α (i ∈N), φ(i)(0) = 0. Therefore, for any n < β−γ α (n ∈N), φ(k)(0) = 0 holds for any 0 ≤k ≤n. Moreover, there exists absolute constant C(n, α) such that: max 0≤k≤n sup x∈[0,1] \|φ(k)(x)\| ≤C(n, α). By Lemma G.5, for any m ∈N+, there exists a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|φ(x) −Qm(x)\| ≤C(n, α) mn . Now we use the transform x = e−αt (t ≥0) on the function φ and consider Φ(t) := 1 eγt φ 1 eαt , t ∈[0, +∞). It is easy to verify that Φ satisfies that Φ(t) N = ρ(t) N. Moreover, we consider the function Pm(t) := 1 eγt Qm 1 eαt , t ∈[0, +∞). 58 Then for any n < β−γ α (n ∈N), the following error estimate holds: ∥Pm(·) −ρ(·)∥ℓ1(N) = +∞ X t=0 \|Pm(t) −Φ(t)\| = +∞ X t=0 e−γt Qm e−αt −ΨT e−αt ≤C(n, α) mn +∞ X t=0 e−γt. By choosing α = 5 · 10−3 and γ = 10−2β, it holds that 99β < β−γ 2α = β−γ α . Thus, we obtain our result: for any n ∈[⌊99β⌋] (β ≥1/99), the following error estimate holds: ∥Pm(·) −ρ(·)∥ℓ1(N) ≤C(n) mn +∞ X t=0 e−γt = C(n) mn 1 1 −e−10−2β = ˜C(n) mn . F.2 Approximation by the sum of polynomial decay Lemma F.4 (Poly decay, fixed Delta function). For any T, n, m ∈N+, there exists an absolute constant C(n) only depending on n and a ϕpoly m (t) = m P k=1 αkt−βk such that I(· = T) −ϕpoly m (·) ℓ1(N+) ≤C(n)T 1.01(n+1) mn , where βk > 1 holds for any k ∈[m]. Proof of Lemma F.4. Let α, γ > 0 be constants, and they will take specific values at the end of the proof First, recall the standard bump function on [−1, 1]: Ψ(x) := ( exp − 1 1−x2 , x ∈(−1, 1) 0, otherwise , and we can define the following constants for T ≥1: µT = 1 T α , σT = 1 T α − 1 (T + 1)α . Then we consider the following bump function ΨT ∈C∞([0, 1]): ΨT (x) = ( VT Ψ x−µT σT , x ∈(µT −σT , µT + σT ) 0, otherwise , where VT is a scaling constant such that ΨT ( 1 T α ) = T 1+γ. First, we consider the approximation of ΨT on [0, 1]. Notice that ΨT ∈C∞([0, 1]), and Ψ(k) T (0) = 0 for any k ∈N. For the standard bump function Ψ, for any n ∈N+, there exists an absolute constant M(n) > 0 only depending on n, such that max 0≤k≤n sup x∈[−1,1] Ψ(k)(x) ≤M(n). Notice that for any k ∈N and x ∈[0, 1], Ψ(k) T (x) = VT σk T Ψ(k) x −µT σT . 59 Therefore, the following upper bound holds: MT (n) = max 0≤k≤n VT σk T M(n) = VT σn T M(n) = T 1+γe (1/T α −1/(T + 1)α)n M(n) ≤T 1+γ(T + 1)n(1+α)M(n)e αn ≤2nM(n)e αn T 1+γ+n(1+α) := C(n, α)T 1+γ+n(1+α). By Lemma G.5, for any m ∈N+, there exists a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|ΨT (x) −Qm(x)\| ≤MT (n) mn ≤C(n, α)T 1+γ+n(1+α) mn . Now we use the transform x = 1 tα (t ≥1) on the function Ψ and consider ΦT (t) := 1 t1+γ ΨT 1 tα , t ∈[1, +∞). It is easy to verify that ΦT satisfies that ΦT (t) N+ = I(t = T). Moreover, we consider the function Pm(t) := 1 t1+γ Qm 1 tα , t ∈[1, +∞). Then by choosing α = γ = 0.01, the following error estimate holds: ∥Pm(·) −I(· = T)∥ℓ1(N+) = +∞ X t=1 \|Pm(t) −ΦT (t)\| = +∞ X t=1 1 t1+γ Qm 1 tα −ΨT 1 tα ≤ +∞ X t=1 1 t1+γ MT (n) mn ≤C(n, α)T 1+γ+n(1+α) mn +∞ X t=1 1 t1+γ = C(n)T 1.01(n+1) mn +∞ X t=1 1 t1+0.01 = ˜C(n)T 1.01(n+1) mn . Finally, notice that Pm(·) satisfies to Pm(t) = 1 t1+γ Qm 1 tα = m−1 P k=0 αkt−(1.01+0.01k), so we can select ϕpoly m (t) := Pm(t). Lemma F.5 (Poly decay, adaptive Delta function). For any T, n, m ∈N+, there exists an absolute constant C(n) only depending on n and a ϕpoly m (t; B) = m P k=1 αk(t/B)−βk such that max 1≤B≤T I(· = B) −ϕpoly m (·; B) ℓ1(N+) ≤C(n)T 1.01(n+1) mn , where βk > 1 holds for any k ∈[m]. Proof of Lemma F.5. The key point of the proof is to note that the adaptability of B can be eliminated by the rescaling operator t/B. 60 First, recall our proof of Lemma F.4. For the same ΨT (·), for any n, m ∈N+, there exists an absolute constant C(n) only depending on n and a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|ΨT (x) −Qm(x)\| ≤C(n)T 1.01(n+1) mn . We use the transform x = 1 t0.01 (t ≥1) on the function Ψ and consider ΦT (t; B) := B tT 1.01 ΨT B tT 0.01! , t ∈[1, +∞). It is easy to verify that ΦT (·; ·) satisfies that ΦT (t; B) N+ = I(t = B). And we consider the function Pm(t; B) := B tT 1.01 Qm B tT 0.01! , t ∈[1, +∞). Then, for any 1 ≤B ≤T, the following error estimate holds: ∥Pm(·; B) −I(· = B)∥ℓ1(N+) = +∞ X t=1 \|Pm(t; B) −ΦT (t; B)\| = +∞ X t=1 B tT 1.01 Qm B tT 0.01! −ΨT B tT 0.01! ≤ +∞ X t=1 1 t1.01 sup x∈[0,1] \|Qm(x) −ΨT (x)\| ≤C(n)T 1.01(n+1) mn +∞ X t=1 1 t1.01 = ˜C(n)T 1.01(n+1) mn . Due to the arbitrariness of B, the proof is completed. Lemma F.6 (Poly decay, fixed Delta function). Consider a polynomially decayed memory ρ(·): there exists β > 1 such that ρ(t) = O(t−β). Then for any n ∈[⌊0.99β⌋−1] and m ∈N+, there exists an absolute constant C(n) only depending on n and a ϕpoly m (t) = m P k=1 αkt−βk such that ρ(·) −ϕpoly m (·) ℓ1(N+) ≤C(n) mn , where βk > 1 holds for any k ∈[m]. Proof of Lemma F.6. There exists C > 0 such that \|ρ(t)\| ≤C/tβ. Let α, γ > 0 be constants, and they will take specific values at the end of the proof First, recall the standard bump function on [−1, 1]: Ψ(x) := ( exp − 1 1−x2 , x ∈(−1, 1) 0, otherwise , 61 and we can define the following constants for T ≥1: µT = 1 T α , σT = 1 2 1 T α − 1 (T + 1)α , and we consider the following bump function ΨT ∈C∞([0, 1]): ΨT (x) = ( VT Ψ x−µT σT , x ∈(µT −σT , µT + σT ) 0, otherwise , where VT is a scaling constant such that ΨT ( 1 T α ) = T 1+γρ(T). Consequently, we consider the sum of bump functions on [0, 1]: φ(x) := +∞ X T =1 ΨT (x). It is easy to verify that (µT1 −σT1, µT1 + σT1) ∩(µT2 −σT2, µT2 + σT2) = ∅for any T1 ̸= T2 and φ(x) = ΨT (x), µT −σT ≤x ≤µT + σT 0, otherwise . First, we study the property of φ(·). We denote the absolute constants Mk = supx \|φ(k)(x)\|. Notice that for any k ∈N, Ψ(k) T (x) = VT σk T Ψ(k) x −µT σT . Therefore, it holds that sup x∈(µT −σT ,µT +σT ) \|φ(k)(x)\| = sup x∈(µT −σT ,µT +σT ) \|Ψ(k) T (x)\| ≤VT σk T Mk = T 1+γρ(T) 1 T α − 1 (T +1)α k 2kMke ≤(T + 1)k(1+α)T 1+γ−βC2kMke αk ≤2k(2+α)CMke αk T 1+γ+k(1+α)−β. Therefore, if k ≤β−(1+γ) 1+α , the following uniform bounds hold: sup x∈(0,1] \|φ(k)(x)\| = sup T ≥1 sup x∈(µT −σT ,µT +σT ) \|φ(k)(x)\| ≤sup T ≥1 2k(2+α)CMke αk T 1+γ+k(1+α)−β ≤2k(2+α)CMke αk := C(k, α). Consequently, we consider the smoothness of Φ at x = 0. Recalling the previous results, for any x ∈(0, 1], we have \|φ(k)(x)\| x ≤C(k, α)T 1+γ+k(1+α)−β µT −σT ≤C(k, α)22+α α T 1+γ+(k+1)(1+α)−β, x ∈(µT −σT , µT + σT ); \|φ(k)(x)\| x = 0, otherwise Thus, by induction, it is easy to verify that for any i < β−(1+γ) 1+α (i ∈N), φ(i)(0) = 0. 62 Therefore, for any n < β−(1+γ) 1+α (n ∈N+), φ(k)(0) = 0 holds for any 0 ≤k ≤n. Moreover, the following uniform bound holds: max 0≤k≤n sup x∈[0,1] \|φ(k)(x)\| ≤C(n, α). By Lemma G.5, for any m ∈N+, there exists a polynomial Qm(x) = m−1 P k=0 αkxk such that sup x∈[0,1] \|φ(x) −Qm(x)\| ≤C(n, α) mn . Now we use the transform x = 1 tα (t ≥1) on the function φ and consider Φ(t) := 1 t1+γ φ 1 tα , t ∈[1, +∞). It is easy to verify that Φ satisfies that Φ(t) N+ = ρ(t) N+. Moreover, we consider the function Pm(t) := 1 t1+γ Qm 1 tα , t ∈[1, +∞). Then for any n < β−(1+γ) 1+α (n ∈N), the following error estimate holds: ∥Pm(·) −ρ(·)∥ℓ1(N+) = +∞ X t=1 \|Pm(t) −Φ(t)\| = +∞ X t=1 1 t1+γ Qm 1 tα −ΨT 1 tα ≤C(n, α) mn +∞ X t=1 1 t1+γ . By choosing α = 10−2 and γ = 10−4β, we have 0.99β −1 = β−γ 1+α −1 = β−(1+γ+α) 1+α < β−(1+γ) 1+α . Thus, we obtain our result: for any n ∈[⌊0.99β⌋−1] (β ≥2/0.99), the following error estimate holds: ∥Pm(·) −ρ(·)∥ℓ1(N+) ≤C(n) mn +∞ X t=1 1 t1+γ ≤C(n) mn +∞ X t=1 1 t1+10−4 = ˜C(n) mn . 63 G Some Background and Proof Preparation G.1 T5’s relative positional encoding The T5’s Relative Positional Encoding is primary focus of this study. Its standard form in practical applications (Raffel et al., 2020) adheres to Rt,s = r(t −s), where −r(n) =      n, if n < B B + ⌊B · log(n/B) log(D/B)⌋, if B ≤n < D 2B −1, if n ≥D . Here, D is a large integer, signifying the longest distance of concern, while B is a small integer. One can see that for n < B, r(·) exhibits polynomial decay, whereas for B < n < D , r(·) demonstrates logarithmic decay. Consequently, the overall decay rate of r(·) is logarithmic. The following Table further provides an example of standard T5’s Relative Positional Encoding. Table 1: An example of standard T5’s Relative Positional Encoding t −s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 −r(t −s) 0 1 2 3 4 5 6 7 8 8 8 8 9 9 9 9 t −s 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 · · · −r(t −s) 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 · · · G.2 Barron space theory The well-known universal approximation result for 2NNs asserts that 2NNs can approximate any continuous function (Barron, 1992; 1993; 1994). Nonetheless, this result lacks a characterization of the approximation efficiency, i.e., how many neurons are needed to achieve a certain approximation accuracy? This gap was addressed by the Barron space theory (E et al., 2019; 2021). It is established that for any function within Barron space f ∈B, 2NNs with m neurons (denoted by Hm) can approximate them efficiently, at a rate of inffm∈Hm ∥f −fm∥≤O(∥f∥B /√m), remarkably independent of the input dimension d, thus avoiding the Curse of Dimensionality (Bellman, 1966; Bach, 2017). Specifically, the Barron space is defined by: Definition G.1 (Barron space (E et al., 2019; 2021; Ma et al., 2020)). Consider functions f : X →R that admit the following representation: f(x) = Z Ω aσ(b⊤x + c)ρ(da, db, dc), x ∈X. For any p ∈[1, +∞], we define the Barron norm: ∥f∥Bp := inf ρ Eρ [\|a\|p(∥b∥1 + \|c\|)p] 1/p . Then the Barron space are defined as: Bp := {f ∈C : ∥f∥Bp < +∞}. Proposition G.2. For any p ∈[1, +∞], Bp = B∞and ∥f∥Bp = ∥f∥B∞. Remark G.3. From the Proposition above, the Barron spaces Bp are equivalent for any p ∈[1, +∞]. Consequently, in this paper, we use B and ∥·∥B to denote the Barron space and Barron norm. Remark G.4. For Barron space B, both Direct and Inverse Approximation Theorems hold (E et al., 2021). In this paper, we mainly utilize the Direct Approximation Theorem, stated in Lemma G.6. 64 G.3 Useful approximation lemmas Lemma G.5 (Jackson (1930)). Let f ∈Cn([0, 1]) with f(0) = f ′(0) = · · · = f (n)(0) = 0. Then for any m ∈N+, there exists a polynomial Qm(x) = m−1 P k=0 αkxk such that ∥f −Qm∥L∞([0,1]) ≤M(n) mn , where M(n) = max k≤n f (k) L∞([0,1]). Lemma G.6 (Ma et al. (2020)). For any f ∈B and m ∈N, there exists a two-layer ReLU neural network fm(x) = m P k=1 akσ(b⊤ k x + ck) with m neurons such that ∥f −fm∥L∞([0,1]d) ≤˜O ∥f∥B √m . 65 H Experiments H.1 Restatement of our theoretical insights As detailed in Section 1, our theoretical analysis reveals the following novel insights into the expressive power and mechanisms of Transformer: Insight (1). The distinct roles of the number of layers, the number of Attn heads, and the width of FFN layers. (1a) Deeper Transformers can handle tasks with memories with more intricate interrelationships, such as nested relationships (Type II). (1b) In contrast, for tasks with memories lacking such interrelationships (Type I), a single-layer Transformer with sufficient Attn heads and FFN width should suffice. Insight (2). The different roles of Attn layers and FFN layers. (2a) FFN layers are tasked with approximating nonlinear memory functions and the readout function, (2b) while Attn layers are responsible for extracting the tokens from the memory locations. Insight (3). The functionality and necessity of Dot-product (DP). (3a) For the relatively simple Task I, DP is not necessary and can be omitted. (3b) However, for the more complex Task II, DP provides necessary nonlinearity: the cooperation between DP and RPE provides the needed interaction between the temporal space and the token space. Insight (4). The efficiency of Relative Positional Encoding (RPE) in modeling long-range correlations. The primary role of RPE is to approximate the memory kernels. (4a) Transformer with log-type RPE can handle heavy-tailed memories. (4b) Transformer with lin-type RPE can handle light-tailed memories. H.2 Experimental Validation To validation of our theoretical insights (1a)∼(4b), we conduct 8 experiments, from simple toy models to more complex LLM pre-training. The experiments are conducted on 1 A100. H.2.1 Validation of Insight (1a) Objective. As indicated in Section 4, numerous NLP tasks exhibit complex interrelationships among tokens and belong to our Task II. This experiment aims to verify our Insight (1a): for such tasks, increasing the number of layers L is more efficient than increasing the number of Attn heads H. Setup. Specifically, we pretrain decoder-only Transformers (Vaswani et al., 2017) with different L and H on the OpenWebText dataset (Gokaslan and Cohen, 2019) for 10,000 iterations (approximately 1B tokens) on 1 A100, using cross-entropy loss and AdamW with the same hyperparameters. To ensure comparability, we meticulously balance the total number of parameters across both experimental setups. Results and conclusion. The final validation losses are shown in Table 2. By comparing these two subtables, the benefits brought by increasing L are much greater than the benefits brought by increasing H (0.802 > 0.136), thereby corroborating our Insight (1a). Table 2: Results of the experiment supporting Insight (1a). L = 1, H = 8 (26M) L = 1, H = 12 (29M) L = 1, H = 12 (32M) 5.796 (baseline) 5.689 (↓0.107) 5.660 (↓0.136) L = 1, H = 8 (26M) L = 4, H = 8 (29M) L = 8, H = 8 (32M) 5.796 (baseline) 5.374 (↓0.422) 4.994 (↓0.802) H.2.2 Validation of Insight (1b) Objective. As mentioned in Section 3, sparse Boolean functions have no interactions among the memories and belong to our Task I. This experiment aims to verify our Insight (1b): for such tasks, a single-layer Transformer equipped with a sufficient number of Attn heads H and FFN width m suffices, and there is no need to increase the number of layers L. 66 Setup. Specifically, we train single-layer DP-free Transformers with different H and m to learn a sparse Boolean target function f ∗: f ∗(x) := g∗(x48, x56, x99) := P64 k=1 ReLU(⟨w∗ k, (x48, x56, x99)⟩) for input sequence x = (x1, · · · , x1000) ∈{±1}1000, where w∗ k are generated by w∗ k ∼N(0, I3). Training proceeds for 10,000 iterations (1M samples) using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are shown in Table 3. As shown in this table, a single-layer Transformer equipped with a sufficient H (32) and m (256) is adequate for representing this sparse Boolean function. This empirical evidence supports our Insight (1b). Table 3: Results of the experiment supporting Insight (1b). H = 2, m = 16 H = 8, m = 64 H = 32, m = 256 0.21 0.04 0.01 H.2.3 Validation of Insight (2a) Objective. This experiment aims to verify our Insight (2a): to learn a sparse Boolean function with a “complex” readout function and “simple” memories, increasing the FFN width m can significantly improve the performance, whereas increasing the number of Attn heads H brings almost no benefit. Setup. Specifically, we train single-layer DP-free Transformers with different H and m to learn a sparse Boolean function with a "complex" readout function (g∗) and a "simple" single memory (x99): f ∗(x) := g∗(x99) := P64 k=1 ReLU(w∗ k · x99) for any input sequence x = (x1, · · · , x1000) ∈ {±1}1000, where w∗ k are generated by w∗ k ∼N(0, 1). Training proceeds for 10,000 iterations (1M samples) using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are shown in Table 4. The tables indicate that, for learning a sparse Boolean function with a “complex” readout function and “simple” memories, increasing m can significantly improve the performance (0.49 →0.002), almost completing this task perfectly. Conversely, increasing H fails to yield substantial improvement. This empirical evidence supports our Insight (2a). Table 4: Results of the experiment supporting Insight (2a). m = 8 m = 64 m = 512 H = 8 0.49 0.006 0.002 H = 8 H = 64 H = 512 m = 8 0.49 0.49 0.52 H.2.4 Validation of Insight (2b) Objective. Contrasting with Experiment (2a), this experiment aims to verify our Insight (2b): for learning a sparse Boolean function with a “simple” readout function and “complex” memories, increasing the number of Attn headers H can substantially improve the performance while increasing FFN width m will offer almost no benefit. Setup. Specifically, we train single-layer DP-free Transformers with different H and m to learn a sparse Boolean function with a “simple” linear readout function (g∗) and relatively “complex” memories (x48, x56, x99): f ∗(x) := g∗(x48, x56, x99) := x48 + x56 + x99 for any input sequence x = (x1, · · · , x1000) ∈{±1}1000. Training processes for 10,000 iterations (1M samples), using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are presented in Table 5. The tables indicate that, for learning a sparse Boolean function with a “simple” readout function and “complex” memories, increasing m can significantly improve the performance (1.16 →10−6), closely achieving task perfection. In contrast, increasing m brings almost no benefits. This empirical evidence supports our Insight 2(b). H.2.5 Validation of Insight (3a) Objective. As mentioned in Section 3, learning sparse Boolean functions has no interactions among the memories and belongs to our Task I. This experiment aims to verify our insight (3a): for such 67 Table 5: Results of the experiment supporting Insight (2b). m = 2 m = 64 m = 256 H = 2 1.16 0.81 1.23 H = 2 H = 64 H = 256 m = 2 1.16 <1e-6 <1e-6 tasks, a DP-free Transformer equipped with a sufficient number of Attn heads H and FFN width m is sufficiently capable. Moreover, there is no need to use DP structure in Attn. Setup. Specifically, we train single-layer DP-free Transformers with different H and m and with DP or without DP to learn a sparse Boolean target function f ∗: f ∗(x) := g∗(x48, x56, x99) := P64 k=1 ReLU(⟨w∗ k, (x48, x56, x99)⟩) for input sequence x = (x1, · · · , x1000) ∈{±1}1000, where w∗ k are generated by w∗ k ∼N(0, I3). Training proceeds for 10,000 iterations (1M samples) using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are shown in Table 6. The findings illustrate that a DP-free Transformer equipped with a sufficient H (32) and m (256) is adept at accurately representing the given sparse Boolean function. Additionally, the Incorporation of the DP structure into the layers contributes marginally to performance enhancement. This substantiates our Insight (3a). Table 6: Results of the experiment supporting Insight (3a). H = 2, m = 16 H = 8, m = 64 H = 32, m = 256 with DP 0.21 0.04 0.01 without DP 0.17 0.11 0.02 H.2.6 Validation of Insight (3b) Objective. As indicated in Section 4, numerous NLP tasks exhibit complex interrelationships among tokens and belong to our Task II. This experiment aims to verify our Insight (3b): for such tasks, the utilization of DP structure in Attn layers is necessary. Setup. Specifically, we pre-train Transformers with DP or without DP on the OpenWebText dataset for 10,000 iterations (approximately 1B tokens) on 1 A100, using cross-entropy loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are presented in Table 7. As shown in the table, for NLP pre-training tasks, Transformer incorporating DP structure is more efficient than Transformer without DP (5.796 < 5.830, 5.374 < 5.486, 4.994 < 5.274), thereby supporting our Insight 3(b). Table 7: Results of the experiment supporting Insight (3b). L = 1, H = 8 L = 4, H = 8 L = 8, H = 8 with DP 5.796 5.374 4.994 without DP 5.830 5.486 5.274 H.2.7 Validation of Insight (4a) Objective. This experiment aims to verify our Insight (4a): for learning Task III with heavy-tailed memories, Transformers with log-type RPE are efficient, whereas those with lin-type RPE fail. Setup. Specifically, we train single-layer, FFN-free, DP-free Transformers with log-type RPE or lintype RPE and varying numbers of Attn heads H. The target function involves a heavy-tailed memory kernel ρ(t) = t−0.5: f ∗(x) := P1000 s=1 xsρ(1000−s) for any input sequence x = (x1, · · · , x1000) ∈ {±1}1000. Training processes for 10,000 iterations (1M samples) using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are shown in Table 8. As shown in the table, to learn heavy-tailed memories, even single-head Transformer with log-type RRE can complete it 68 perfectly (< 10−5). Conversely, Transformers employing lin-type RRE exhibit limited improvement even with up to 64 heads (0.19). This empirical evidence supports our Insight (4a). Table 8: Results of the experiment supporting Insight (4a). H = 1 H = 4 H = 16 H = 64 type=log <1e-5 <1e-5 <1e-5 <1e-5 type=lin 0.73 0.68 1.16 0.19 H.2.8 Validation of Insight (4b) Objective. In contrast to Experiment (4a), this experiment aims to verify that for learning our Task III with light-tailed memories, Transformers with lin-type RPE are efficient, whereas those with log-type RPE fail. Setup. Specifically, we train single-layer, FFN-free, DP-free Transformers with log-type RPE or lintype RPE and varying numbers of Attn heads H. The target function involves a heavy-tailed memory kernel ρ(t) = e−5t: f ∗(x) := P1000 s=1 xsρ(1000 −s) for any input sequence x = (x1, · · · , x1000) ∈ {±1}1000. Training processes for 10,000 iterations (1M samples) using squared loss and AdamW with the same hyperparameters. Results and conclusion. The final validation losses are shown in Table 9. As shown in the table, to learn light-tailed memories, even single-head Transformer with lin-type RRE can complete it perfectly (< 10−7). Conversely, Transformers employing log-type RRE exhibit limited improvement even with up to 64 heads (5.3 × 10−4). This empirical evidence supports our Insight (4b). Table 9: Results of the experiment supporting Insight (4b). H = 1 H = 4 H = 16 H = 64 type=log 9.1e-4 3.7e-3 2.6e-3 5.3e-4 type=lin <1e-7 <1e-7 <1e-7 <1e-7 H.3 Practical Implications Our theoretical insights and empirical evidence can directly lead to the following 8 practical implications, such as the strategic selection of Transformer hyperparameters for specific tasks. • Implication (1a). (supported by Insight (1a) and Experiment (1a)) For sequence modeling tasks with complex interrelations between memories, such as many NLP applications, enhancing the number of layers L is more beneficial than increasing the number of Attn heads H and FFN width m. • Implication (1b). (supported by Insight (1b) and Experiment (1b)) For simple sequence modeling tasks with almost no memory intercorrelation, such as the learning of sparse Boolean function, improving performance necessitates only sufficient H and m in a single-layer Transformer, without a need to increase L. • Implication (2a). (supported by Insight (2a) and Experiment (2a)) For sequence modeling tasks with complex readout or memory functions, increasing m can significantly improve performance. • Implication (2b). (supported by Insight (2b) and Experiment (2b)) For sequence modeling tasks with multiple memories, increasing H can markedly improve performance. • Implication (3a). (supported by Insight (3a) and Experiment (3a)) For simple sequence modeling tasks with almost no memory correlations, such as learning sparse Boolean functions, omitting the DP structure in Attn layers can still perform well. • Implication (3b). (supported by Insight (3b) and Experiment (3b)) 69 For sequence modeling tasks with complex correlations between memories, such as many NLP tasks, preserving the DP structure in attention layers is crucial for achieving high performance due to its indispensable nonlinearity. • Implication (4a). (supported by Insight (4a) and Experiment (4a)) For sequence modeling tasks with heavy-tailed memories, the employment of log-type RPE (such as T5’s RPE and KERPLE (log)) is recommended over lin-type RPE (such as Alibi). • Implication (4b). (supported by Insight (4b) and Experiment (4b)) For sequence modeling tasks with light-tailed memories, the employment of lin-type RPE (such as Alibi) is recommended over log-type RPE (such as T5’s RPE and KERPLE (log)). 70 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,484	Span-Based Optimal Sample Complexity for Weakly Communicating and General Average Reward MDPs Matthew Zurek Department of Computer Sciences University of Wisconsin-Madison matthew.zurek@wisc.edu Yudong Chen Department of Computer Sciences University of Wisconsin-Madison yudong.chen@wisc.edu Abstract We study the sample complexity of learning an ε-optimal policy in an averagereward Markov decision process (MDP) under a generative model. For weakly communicating MDPs, we establish the complexity bound eO SA H ε2 , where H is the span of the bias function of the optimal policy and SA is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters S, A, H, and ε, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. We also initiate the study of sample complexity in general (multichain) average-reward MDPs. We argue a new transient time parameter B is necessary, establish an eO SA B+H ε2 complexity bound, and prove a matching (up to log factors) minimax lower bound. Both results are based on reducing the average-reward MDP to a discounted MDP, which requires new ideas in the general setting. To optimally analyze this reduction, we develop improved bounds for γ-discounted MDPs, showing that eO SA H (1−γ)2ε2 and eO SA B+H (1−γ)2ε2 samples suffice to learn ε-optimal policies in weakly communicating and in general MDPs, respectively. Both these results circumvent the well-known minimax lower bound of eΩ SA 1 (1−γ)3ε2 for γ-discounted MDPs, and establish a quadratic rather than cubic horizon dependence for a fixed MDP instance. 1 Introduction The paradigm of Reinforcement learning (RL) has demonstrated remarkable successes in various sequential learning and decision-making problems. Empirical successes have motivated extensive theoretical study of RL algorithms and their fundamental limits. The RL environment is commonly modeled as a Markov decision process (MDP), where the objective is to find a policy π that maximizes the expected cumulative rewards. Different reward criteria are considered, such as the finite horizon total reward Eπ PT t=0 Rt and the infinite horizon total discounted reward Eπ [P∞ t=0 γtRt] with a discount factor γ < 1. The finite horizon criterion only measures performance for T steps, and the discounted criterion is dominated by rewards from the first 1 1−γ time steps. In many situations where the long-term performance of the policy π is of interest, we may prefer to evaluate policies by their long-run average reward limT →∞(1/T)EπPT −1 t=0 Rt . A foundational theoretical problem in RL is the sample complexity for learning a near-optimal policy using a generative model of the MDP [10], meaning the ability to obtain independent samples of the next state given any initial state and action. For the finite horizon and discounted reward criteria, the sample complexity of this task has been thoroughly studied (e.g., [2, 3, 15, 19, 1, 12]). However, despite significant effort (reviewed in Section 1.1), the sample complexity of the average reward setting is unresolved in existing literature. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Our contributions In this paper, we resolve the sample complexity of weakly communicating Average-Reward MDPs (AMDP) in terms of H := ∥h⋆∥span, the span of the bias (a.k.a. relative value function) of the optimal policy. We show that eO SAH/ε2 samples suffice to find an ε-optimal policy of a weakly communicating MDP with S states and A actions. This bound, presented in Theorem 2, is the first that matches the minimax lower bound eΩ SAH/ε2 up to log factors. Furthermore, we initiate the study of sample complexity for average-reward general MDPs, which refers to the class of all finite-space MDPs without any restrictions [14]. General MDPs are not necessarily weakly communicating and all their optimal policies may be multichain. In this general setting, we demonstrate the span H alone cannot characterize the sample complexity, as the lower bound in Theorem 4 exhibits instances which require ≫HSA/ε2 samples. This observation motivates our introduction of a new transient time bound parameter B, which in conjunction with H captures the sample complexity of general average-reward MDPs. Specifically, our Theorem 8 shows that eO SA B+H ε2 samples suffice to learn an ε-optimal policy, and Theorem 4 provides a matching minimax lower bound of Ω SA B+H ε2 . We remark that it is trivially impossible to achieve low regret in standard online settings of general MDPs, since the agent may become trapped in a closed class of low reward states [4]. The simulator setting is natural for studying general MDPs since it avoids this fatal issue, although the existence of multiple closed classes with different long-run rewards still plays a fundamental role in the minimax sample complexity, as reflected in the dependence on B. To establish the above upper bounds, we adopt the reduction-to-discounted-MDP approach [9, 20], and improve on prior work by developing enhanced sample complexity bounds for γ-discounted MDPs (DMDPs). We improve the analysis of variance parameters related to DMDPs using a new multistep variance Bellman equation, which is applied in a recursive manner to bound the variance of near-optimal policies. For general (multichain) MDPs, we further utilize law-of-totalvariance ideas to bound the total variance contribution from transient states, which present new challenges significantly different to their behavior in the weakly communicating setting. Our averageto-discounted reduction also requires new techniques, because many structural properties used in earlier reduction arguments no longer hold for general MDPs. Our analysis leads to DMDP sample complexities of eO SA H (1−γ)2ε2 and eO SA B+H (1−γ)2ε2 to learn ε-optimal policies in weakly communicating and general MDPs, respectively. Notably, the latter bound, valid for all MDPs, circumvents the existing lower bound eΩ SA (1−γ)3ε2 [3, 15]. Whereas this minimax lower bound allows the adversary to choose the transition matrix P based on γ with B ≈ 1 1−γ [3, Theorem 3], our result reflects the complexity of a fixed MDP P through its parameters H, B and a quadratic dependence on the effective horizon 1 1−γ . This fixed-P complexity is essential for our particular algorithmic approach, where the reduction discount γ is chosen depending on P. It is also a more relevant framework in general for many RL problems where the discount factor is tuned for best performance on a particular instance. 1.1 Comparison with related work on average-reward MDPs We summarize in Table 1 existing sample complexity results for average reward MDPs. Various parameters have been used to characterize the sample complexity of average reward MDPs, including the diameter D of the MDP, the uniform mixing time bound τunif for all policies, and the span H of the optimal bias; formal definitions are provided in Section 2. All sample complexity upper bounds involving τunif require the strong assumption that all stationary deterministic policies have finite mixing times. Otherwise, τunif = ∞by definition, which for example occurs if some policy induces a periodic Markov chain. It is also possible to have D = ∞, while H and our newly introduced B are always finite for finite state-action spaces. As shown in [20], there is generally no relationship between D and τunif; they can each be arbitrarily larger than the other. On the other hand, it has been shown that H ≤D [4] and that H ≤8τunif [20]. Therefore, either of the first two minimax lower bounds in Table 1 (which both use hard instances that are weakly communicating) imply a lower bound of eΩ SA H ε2 and thus the minimax optimality of our Theorem 2. To the best of our knowledge, no prior work has considered the average-reward sample complexity of general (potentially multichain) MDPs. Existing results make assumptions at least as strong as weakly communicating or uniformly bounded mixing times. 2 Method Sample Complexity Reference Comments Primal-Dual SMD eO SA τ 2 unif ε2 [8] requires uniform mixing Reduction to DMDP eO SA τunif ε3 [9] requires uniform mixing Policy Mirror Descent eO SA τ 3 unif ε2 [13] requires uniform mixing Reduction to DMDP eO SA τunif ε2 [22] requires uniform mixing Reduction to DMDP eO SA H ε3 [20] weakly communicating Refined Q-Learning eO SA H2 ε2 [26] weakly communicating Reduction to DMDP eO SA H ε2 Our Theorem 2 weakly communicating Reduction to DMDP eO SA B+H ε2 Our Theorem 8 general MDPs Lower Bound eΩ SA τunif ε2 [9] implies eΩ SA H ε2 Lower Bound eΩ SA D ε2 [20] implies eΩ SA H ε2 Lower Bound eΩ SA B+H ε2 Our Theorem 4 general MDPs Table 1: Algorithms and sample complexity bounds for average reward MDPs with S states and A actions. The goal is finding an ε-optimal policy under a generative model. Here H := ∥h⋆∥span is the span of the optimal bias, τunif is a uniform upper bound on mixing times of all policies, and D is the MDP diameter, with the relationships H ≤8τunif and H ≤D. B is the transient time parameter. The work [9] was the first to develop an algorithm based on reduction to a discounted MDP with a discount factor of γ = 1 − ε τunif . Their argument was improved in [20], which improved the uniform mixing assumption to only assuming a weakly communicating MDP, and used a smaller discount factor γ = 1 −ε H. These arguments both make essential use of the fact that the optimal gain is independent of the starting state, which does not hold for general MDPs. After analyzing the reductions, both [9] and [20] then solved the discounted MDPs by appealing to the algorithm from [12]. To the best of our knowledge, the algorithm of [12] is the only known algorithm for discounted MDPs which could work with either reduction, as the reductions each require a ε 1−γ -optimal policy from the discounted MDP, and other known algorithms for discounted MDPs do not permit such large suboptimality levels. (We discuss algorithms for discounted MDPs in more detail below.) Other algorithms for average-reward MDPs are considered in [9, 13, 26]. The above results fall short of matching the minimax lower bounds. While preparing this manuscript, we became aware of [22], which considers the uniform mixing setting and obtains a minimax optimal sample complexity eO SA τunif ε2 in terms of τunif. Although developed independently, their work and ours have several similarities. We both utilize discounted reductions and observe that it is possible to improve the sample complexity of the resulting DMDP task by improving the analysis of variance parameters. They accomplish the improvement by leveraging the uniform mixing assumption, whereas we make use of the low span of the optimal policy. Note that H ≤8τunif holds in general and there exist MDPs with H ≪τunif = ∞, so our Theorem 2 is strictly stronger than the result of [22]. 1.2 Comparison with related work on discounted MDPs We discuss a subset of results for discounted MDPs in the generative setting. Several works [15, 19, 1, 12] obtain the minimax optimal sample complexity of eO SA 1 (1−γ)3ε2 for finding an ε-optimal policy w.r.t. the discounted reward. However, only [12] is able to show this bound for the full range of ε ∈(0, 1 1−γ ]. As mentioned, the reduction from average reward MDPs requires a large ε in the resulting discounted MDP, making it unsurprising that all of [9, 20, 22] as well as our Algorithm 1 essentially use their algorithm. The matching lower bound is established in [15, 3]. As mentioned earlier, both we and the authors of [22, 21] independently observed that the eΩ SA 1 (1−γ)3ε2 sample complexity lower bound can be circumvented in the settings that arise 3 under the average-to-discounted reductions. The authors of [22, 21] assume uniform mixing and obtain a discounted MDP sample complexity of eO SA τunif (1−γ)2ε2 , first in [21] by modifying the algorithm of [19], and then in [22] under a wider range of ε by instead modifying the analysis of [12]. The work [21] also proves a matching lower bound. Our Theorem 1 for discounted MDPs attains a sample complexity of eO SA H (1−γ)2ε2 assuming only that the MDP is weakly communicating. Again, in light of the relationship that H ≤8τunif, our results are strictly better (ignoring constants), and their lower bound also establishes the optimality of our Theorem 1. 2 Problem setup and preliminaries A Markov decision process (MDP) is given by a tuple (S, A, P, r), where S is the finite set of states, A is the finite set of actions, P : S × A →∆(S) is the transition kernel with ∆(S) denoting the probability simplex over S, and r : S ×A →[0, 1] is the reward function. Let S := \|S\| and A := \|A\| denote the cardinality of the state and action spaces, respectively. Unless otherwise noted, all policies considered are stationary Markovian policies of the form π : S →∆(A). For any initial state s0 ∈S and policy π, we let Eπ s0 denote the expectation with respect to the probability distribution over trajectories (S0, A0, S1, A1, . . . ) where S0 = s0, At ∼π(St), and St+1 ∼P(· \| St, At). Equivalently, this is the expectation with respect to the Markov chain induced by π starting in state s0, with the transition probability matrix Pπ given by (Pπ)s,s′ := P a∈A π(a\|s)P(s′ \| s, a). We also define (rπ)s := P a∈A π(a\|s)r(s, a). We occasionally treat P as an (S × A)-by-S matrix where Psa,s′ = P(s, a, s′). We also let Psa denote the row vector such that Psa(s′) = P(s, a, s′). For any s ∈S and any bounded function X of the trajectory, we define the variance Vπ s [X] := Eπ s (X −Eπ s [X])2, with its vector version Vπ [X] ∈RS given by (Vπ [X])s = Vπ s [X]. For s ∈S, let es ∈RS be the vector that is all 0 except for a 1 in entry s. Let 1 ∈RS be the all-one vector. For each v ∈RS, define the span semi-norm ∥v∥span := maxs∈S v(s) −mins∈S v(s). Discounted reward criterion A discounted MDP is a tuple (S, A, P, r, γ), where γ ∈(0, 1) is the discount factor. For a stationary policy π, the (discounted) value function V π γ : S →[0, ∞) is defined, for each s ∈S, as V π γ (s) := Eπ s [P∞ t=0 γtRt], where Rt = r(St, At) is the reward received at time t. It is well-known that there exists an optimal policy π⋆ γ that is deterministic and satisfies V π⋆ γ γ (s) = V ⋆ γ (s) := supπ V π γ (s) for all s ∈S [14]. In discounted MDPs the goal is to compute an ε-optimal policy, which we define as a policy π satisfying V π γ −V ⋆ γ ∞≤ε. We define one more variance parameter VPπ V π γ ∈RS, specific to a given policy π, by VPπ V π γ s := P s′∈S (Pπ)s,s′ V π γ (s′) −P s′′ (Pπ)s,s′′ V π γ (s′′) 2. Average-reward criterion In an MDP (S, A, P, r), the average reward per stage or the gain of a policy π starting from state s is defined as ρπ(s) := limT →∞1 T Eπ s PT −1 t=0 Rt . The bias function of any stationary policy π is hπ(s) := C-limT →∞Eπ s PT −1 t=0 (Rt −ρπ(St)) , where C-lim denotes the Cesaro limit. When the Markov chain induced by Pπ is aperiodic, C-lim can be replaced with the usual limit. For any policy π, its ρπ and hπ satisfy ρπ = Pπρπ and ρπ + hπ = rπ + Pπhπ. A policy π⋆is Blackwell-optimal if there exists some discount factor ¯γ ∈(0, 1) such that for all γ ≥¯γ we have V π⋆ γ ≥V π γ for all policies π. Henceforth we let π⋆denote some fixed Blackwell-optimal policy, which is guaranteed to exist when S and A are finite [14]. We define the optimal gain ρ⋆∈RS by ρ⋆(s) = supπ ρπ(s) and note that we have ρ⋆= ρπ⋆. For all s ∈S, ρ⋆(s) ≥maxa∈A Psaρ⋆, or equivalently ρ⋆≥Pπρ⋆for all policies π (and this maximum is achieved by π⋆). We also define h⋆= hπ⋆(and we note that this definition does not depend on which Blackwell-optimal π⋆is used, if there are multiple). For all s ∈S, ρ⋆and h⋆satisfy ρ⋆(s) + h⋆(s) = maxa∈A:Psaρ⋆=ρ⋆(s) rsa + Psah⋆, known as the (unmodified) Bellman equation. A weakly communicating MDP is such that the states can be partitioned into two disjoint subsets S = S1 ∪S2 such that all states in S1 are transient under any stationary policy and within S2, any state is reachable from any other state under some stationary policy. In weakly communicating MDPs ρ⋆is a constant vector (all entries are equal), and thus (ρ⋆, h⋆) are also a solution to the modified Bellman equation ρ⋆(s) + h⋆(s) = maxa∈A rsa + Psah⋆. When discussing weakly communicating MDPs we occasionally abuse notation and treat ρ⋆as a scalar. A stationary policy is multichain if it 4 induces multiple closed irreducible recurrent classes, and an MDP is called multichain if it contains such a policy. Weakly-communicating MDPs always contain some gain-optimal policy which is unichain (not multichain), but in general MDPs, all gain-optimal policies may be multichain and ρ⋆ may not be a constant vector. All uniformly mixing MDPs are weakly communicating. In the average reward setting, our goal is find an ε-optimal policy, defined as a policy π such that ∥ρ⋆−ρπ∥∞≤ε. Complexity parameters Our most important complexity parameter is the span of the optimal bias function H := ∥h⋆∥span. In addition, for general MDPs we introduce a new transient time parameter B, defined as follows. Let Π be the set of deterministic stationary policies. For each π ∈Π, let Rπ be the set of states which are recurrent in the Markov chain Pπ, and let T π = S \ Rπ be the set of transient states. Let TRπ = inf{t : St ∈Rπ} be the first hitting time of a state which is recurrent under π. We say an MDP satisfies the bounded transient time property with parameter B if for all policies π and states s ∈S we have Eπ s [TRπ] ≤B, or in words, the expected time spent in transient states (with respect to the Markov chain induced by π) is bounded by B. We recall several other parameters used in the literature to characterize sample complexity. The diameter is defined as D := maxs1̸=s2 infπ∈Π Eπ s1 [ηs2], where ηs denotes the hitting time of a state s ∈S. For each policy π, if the Markov chain induced by Pπ has a unique stationary distribution νπ, we define the mixing time of π as τπ := inf n t ≥1 : maxs∈S e⊤ s (Pπ)t −ν⊤ π 1 ≤1 2 o . If all policies π ∈Π satisfy this assumption, we define the uniform mixing time τunif := supπ∈Π τπ. Note that D and τunif are generally incomparable [20], while we always have H ≤D [4] and H ≤8τunif [20]. It is possible for τunif = ∞, for instance if there are any policies which induce periodic Markov chains. Also, D = ∞if there are any states which are transient under all policies. However, H and B are finite in any MDP with S, A < ∞. Also if τunif is finite, Lemma 27 shows B ≤4τunif. We assume access to a generative model [10], also known as a simulator. This means we can obtain independent samples from P(· \| s, a) for any given s ∈S, a ∈A, but P itself is unknown. We assume the reward function r is deterministic and known, which is standard in generative settings (e.g., [1, 12]) since otherwise estimating the mean rewards is relatively easy. Specifically, to learn an ε-optimal policy for the discounted MDP, we would need to estimate each entry of r to accuracy O((1 −γ)ε), which requires a lower order number of samples eO SA (1−γ)2ε2 . For this reason we assume (as in [20]) that H ≥1. Using samples from the generative model, our Algorithm 1 constructs an empirical transition kernel bP. For a policy π, we use bV π γ (s) to denote the value function computed with respect to the Markov chain with transition matrix bPπ (as opposed to Pπ). Our Algorithm 1 also utilizes a perturbed reward function er, and we use the notation V π γ,p(s) to denote a value function computed using this reward (and Pπ); more concretely, we replace Rt with eRt = er(St, At) in the definition above of V π γ . We use the notation bV π γ,p when using bP and er simultaneously. 3 Main results for weakly communicating MDPs Our approach is based on reducing the average-reward problem to a discounted problem. We first present our algorithm and guarantees for the discounted MDP setting. As discussed in Subsection 1.1, our algorithm of choice, Algorithm 1, is essentially the same as the one presented in [12], with a slightly different perturbation level ξ. Algorithm 1 constructs an empirical transition kernel bP using n samples per state-action pair from the generative model, and then solves the resulting empirical (perturbed) MDP ( bP, er, γ). As noted in [12], the perturbation ensures bπ⋆ γ,p can be computed exactly in poly( 1 1−γ , S, A, log(1/δε)) time by multiple standard MDP solvers. We remark in passing that the SA-by-S transition matrix bP has at most nSA nonzero entries. Our Theorem 1 provides an improved sample complexity bound for Algorithm 1 under the setting that the MDP is weakly communicating. Theorem 1 (Sample Complexity of Weakly Communicating DMDP). Suppose the discounted MDP (P, r, γ) is weakly communicating, H ≤ 1 1−γ , and ε ≤H. There exists a constant C2 > 0 such that, for any δ ∈(0, 1), if n ≥C2 H (1−γ)2ε2 log SA (1−γ)δε , then with probability at least 1 −δ, the policy bπ⋆ γ,p output by Algorithm 1 satisfies V ⋆ γ −V bπ⋆ γ,p γ ∞≤ε. 5 Algorithm 1 Perturbed Empirical Model-Based Planning input: Sample size per state-action pair n, target accuracy ε, discount factor γ 1: for each state-action pair (s, a) ∈S × A do 2: Collect n samples S1 s,a, . . . , Sn s,a from P(· \| s, a) 3: Form the empirical transition kernel bP(s′ \| s, a) = 1 n Pn i=1 I{Si s,a = s′}, for all s′ ∈S 4: end for 5: Set perturbation level ξ = (1 −γ)ε/6 6: Form perturbed reward er = r + Z where Z(s, a) i.i.d. ∼Unif(0, ξ) 7: Compute a policy bπ⋆ γ,p which is optimal for the perturbed empirical discounted MDP ( bP, er, γ) 8: return bπ⋆ γ,p Since we observe n samples for each state-action pair, Theorem 1 shows that a total number of eO HSA (1−γ)2ε2 samples suffices to learn an ε-optimal policy. This bound improves on the eO SA (1−γ)3ε2 complexity bound from [12] when the span H is no larger than the effective horizon 1 1−γ . This assumption holds in many situations, as can be seen by using the relationships H ≤D or H ≤8τunif. On the other hand, in the regime with H > 1 1−γ , the existing bound eO SA (1−γ)3ε2 , also achieved by Algorithm 1, is superior. In this regime, the discounting effectively truncates the MDP at a short horizon 1 1−γ before the long-run behavior of the optimal policy (as captured by H) kicks in. Proof highlights for Theorem 1. The key to obtaining this improved complexity is a careful analysis of certain instance-specific variance parameters. It suffices to bound bV π⋆ γ γ,p −V π⋆ γ γ ∞and bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞by O(ε). The prior DMDP complexity of SA (1−γ)3ε2 is obtained using the wellknown law-of-total-variance argument [3, 1, 12], which ultimately yields a sample complexity like eO q SA (1−γ)ε2 Vπ⋆γ [P∞ t=0 γtRt] ∞ to bound bV π⋆ γ γ,p −V π⋆ γ γ ∞≤O(ε). From here, the variance of the cumulative discounted reward Vπ⋆ γ [P∞ t=0 γtRt] ∞is bounded by 1 (1−γ)2 , since the total reward in a trajectory is within [0, 1 1−γ ]. We instead seek to bound Vπ⋆ γ [P∞ t=0 γtRt] ∞≤O H 1−γ . Assume H is an integer. The first step is to decompose Vπ⋆ γ [P∞ t=0 γtRt] recursively like Vπ⋆ γ " ∞ X t=0 γtRt # = Vπ⋆ γ "H−1 X t=0 γtRt + γHV π⋆ γ γ (SH) # + γ2H Pπ⋆γ H Vπ⋆ γ " ∞ X t=0 γtRt # (see our Lemma 13). This is a multi-step version of the standard variance Bellman equation (e.g., [16, Theorem 1]). Ordinarily an H-step expansion would not be useful, since the term V π⋆ γ γ (SH) by itself appears to have fluctuations on the order of 1 1−γ in the worst case depending on SH (note SH is the random state encountered at time H). However, in our setting, we should have V π⋆ γ γ (SH) ≈ 1 1−γ ρ⋆+ h⋆(SH), reducing the magnitude of the random fluctuations to order H = ∥h⋆∥span. (See Lemma 11 for a formalization of this approximation which first appeared in [23].) Therefore expansion to H steps achieves the optimal tradeoff between maintaining Vπ⋆ γ hPH−1 t=0 γtRt +γHV π⋆ γ γ (SH) i ≤O H2 and minimizing γ2H. As desired this yields Vπ⋆ γ [P∞ t=0 γtRt] ∞≤O H2 1−γ2H = O H 1−γ , where 1 1−γ2H ≤O 1 H(1−γ) requires 1 1−γ ≥H. See Lemma 15 for the complete argument. We would like to use a similar argument as above to bound the second term bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞, which is the “evaluation error” of the empirically optimal policy bπ⋆ γ,p. However, applying the same argument would give a bound in terms of V bπ⋆ γ,p γ span, which, unlike for the analogous term involving the true optimal policy π⋆ γ, is not a priori bounded in terms of H. (If we instead assumed uniform mixing, we could immediately bound this by O(τunif).) Thus, to control the variance associated with evaluating bπ⋆ γ,p, we are able to recursively bound V bπ⋆ γ,p γ span ≤O H + bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ , which can be shown to yield the desired sample complexity. 6 Now we present our main result for the average-reward problem in the weakly communicating setting. Applied in this setting with a DMDP target accuracy of ε = H, our Algorithm 2 reduces the problem to γ-discounted MDP with γ = 1 − ε 12H and then calls Algorithm 1 with target accuracy H. Algorithm 2 Average-to-Discount Reduction input: Sample size per state-action pair n, target accuracy ε ∈(0, 1], DMDP target accuracy ε 1: Set γ = 1 − ε 12ε 2: Obtain bπ⋆from Algorithm 1 with sample size per state-action pair n, accuracy ε, discount γ 3: return bπ⋆ We have the following sample complexity bound for Algorithm 2. Theorem 2 (Sample Complexity of Weakly Communicating AMDP). Suppose the MDP (P, r) is weakly communicating. There exists a constant C1 > 0 such that for any δ, ε ∈(0, 1), if n ≥C1 H ε2 log SAH δε and we call Algorithm 2 with ε = H, then with probability at least 1 −δ, the output policy bπ⋆satisfies the elementwise inequality ρ⋆−ρbπ⋆≤ε1. Again, since we observe n samples for each state-action pair, this result shows that eO HSA ε2 total samples suffice to learn an ε-optimal policy for the average reward MDP. This bound matches the minimax lower bound in [20] and is superior to existing results for weakly communicating MDPs (see Table 1). We note that the proof of Theorem 1 works so long as H is any upper bound of ∥h⋆∥span, hence Algorithm 2 also only needs an upper bound for ∥h⋆∥span. We show in the following theorem that it is in general impossible to obtain a useful upper bound on ∥h⋆∥span with a sample complexity that is a function of only ∥h⋆∥span. This suggests that it is not easy to remove the need for knowledge of ∥h⋆∥span. Theorem 3. For any given n, T ≥1, there exist two MDPs M0 and M1 with S = 4, A = 1 such that M0 has optimal bias span 1, M1 has optimal bias span T, and it is impossible to distinguish between M0 and M1 with probability ≥3 4 with n samples from each state-action pair. Thus even for an MDP with a small span, there exists another MDP that has an arbitrarily large span and is arbitrarily statistically close (that is, cannot be distinguished even with a large sample size n). We emphasize that all previous algorithms in Table 1 also require knowledge of their respective complexity parameters, and such assumptions are pervasive throughout the literature on averagereward RL. The only exception of which we are aware is the contemporaneous work [7], which achieves a suboptimal eO(SA τ 8 unif ε8 ) sample complexity without knowledge of τunif in the uniformly mixing setting. It is unclear if H-based sample complexities are possible without knowing H. Besides the evidence offered by Theorem 3, in the online setting, it has been conjectured that knowledge of H is necessary to obtain an H-dependent regret bound [6, 5, 25]. Moreover, even with knowledge of H, the only known online algorithm with optimal regret is computationally inefficient [25], making it somewhat surprising that our Theorem 2 uses a simple and efficient algorithm. Nevertheless, when H is unknown, one can replace H with the diameter D (since H ≤D). The diameter is known to be estimable [25, 17] and is often a more refined complexity parameter than τunif. Our Theorem 2 is the first to imply the optimal diameter-based complexity eO( SAD ε2 ), given knowledge of D or using a constant-factor upper bound obtained from some estimation procedure. 4 Main results for general MDPs Our starting point for general MDPs is that unlike the weakly communicating setting, their complexity cannot be captured solely by ∥h⋆∥span. We first argue this point informally using the simple example in Figure 1, which is parameterized by a value T > 1. Only state 1 contains multiple actions, and action 2 is optimal since it leads to state 2 which collects reward 0.5 forever, while taking action 1 will always eventually lead to state 3 where the reward is 0 forever. We thus have ρ⋆= [0.5, 0.5, 0]⊤ and ∥h⋆∥span = 0. However, clearly Ω(T) samples are required to even observe a transition 1 →3, so the sample complexity must depend on T ≫H (without observing a transition 1 →3, we cannot determine that action 1 is not optimal). Taking action 1 leads to a large reward of 1 in the short 7 term (for T steps in expectation), so even if we had perfect knowledge of the environment, the optimal γ-discounted policy would not choose the optimal action a = 2 until the effective horizon 1 1−γ ≥Ω(T). Thus 1 1−γ ≈H is insufficient for the reduction to discounted MDP. Note that this instance has its bounded transient time parameter B = T. This example reflects that transient states play a categorically different role in general MDPs: in the weakly communicating setting, states which are transient under all policies can be completely ignored, whereas in this example our action at state 1 fully determines our reward even though state 1 is transient under all policies. 1 2 3 a = 1, R = 1 P(1 \| 1, 1) = 1 −1 T P(3 \| 1, 1) = 1 T a = 2, R = 0.5 R = 0.5 R = 0 Figure 1: A general MDP where γ-discounted approximation fails unless 1 1−γ = Ω(T) ≫∥h⋆∥span. The statistical hardness is formally captured by the following theorem, which uses improved instances to obtain the correct dependence on ε. Theorem 4 (Lower Bound for General AMDPs). For any ε ∈(0, 1/4), B ≥1, A ≥4 and S ∈8N, for any algorithm Alg which is guaranteed to return an ε/3-optimal policy for any input average-reward MDP with probability at least 3 4, there exists an MDP M = (P, r) such that: 1. M has S states and A actions. 2. Letting h⋆be the bias of the Blackwell-optimal policy for M, we have ∥h⋆∥span = 0. 3. M satisfies the bounded transient time assumption with parameter B. 4. Alg requires Ω B log(SA) ε2 samples per state-action pair on M. A similar minimax lower bound holds for the discounted setting. Theorem 5 (Lower Bound for General DMDP). For any ε ∈(0, 1/4), B ≥1, A ≥4 and S ∈8N for any algorithm Alg which is guaranteed to return an ε/3-optimal policy for any input discounted MDP with probability at least 3 4, there exists a discounted MDP M = (P, r, γ) such that: 1. M has S states and A actions. 2. M satisfies the bounded transient time assumption with parameter B. 3. Alg requires Ω B log(SA) (1−γ)2ε2 samples per state-action pair on M. The lower bounds of eO H ε2 from the weakly communicating setting still apply in the general setting. Together with Theorem 4 they imply a eO H+B ε2 lower bound for general average-reward MDPs. Figure 1 demonstrates that, unlike the weakly communicating setting, discounted reduction with 1 1−γ set in terms of only H cannot succeed for general MDPs. (Contrast with Lemma 9 for the analogous theorem from [20] for weakly communicating MDPs.) We remedy this issue and lay the foundation for our matching upper bound by proving a new reduction theorem in terms of H and B; in particular, B measures how much farther ahead we must look in order to determine which closed communicating class will be reached. By Lemma 27 B ≤4τunif, although B is always finite unlike τunif. Theorem 6 (Average-to-Discount Reduction for General MDP). Suppose (P, r) is a general MDP, has an optimal bias function h⋆satisfying ∥h⋆∥span ≤H, and satisfies the bounded transient time assumption with parameter B. Fix ε ∈(0, 1] and set γ = 1 − ε B+H. For any εγ ∈[0, 1 1−γ ], if π is any εγ-optimal policy for the discounted MDP (P, r, γ), then ρ⋆−ρπ ≤ 3 + 2 εγ B+H ε1. Proof highlights. Letting π⋆ γ be the optimal policy for the γ-discounted MDP, our first key observation is that ρ⋆is constant within any irreducible closed recurrent block of the Markov chain Pπ⋆γ, essentially 8 because all states in this block must be reachable from each other with probability one (see Lemma 17). Leveraging the optimality of π⋆ γ, this enables us to bound both V π⋆ γ γ (s) − 1 1−γ ρ⋆(s) and V π⋆ γ γ (s)− 1 1−γ ρπ⋆ γ(s) by O ∥h⋆∥span for any s which is recurrent under π⋆ γ, which when combined demonstrate that the gain ρπ⋆ γ(s) of π⋆ γ is near-optimal for its recurrent states. See Lemma 21. We then leverage the bounded transient time assumption to guarantee that for transient s, V π⋆ γ γ (s) is dominated by the expected returns from recurrent states, since at most O(B) time is spent in transient states. We complete the proof of Theorem 6 by combining these facts, as well as extending them to accommodate approximately optimal policies. Next we establish an improved sample complexity for the discounted problem in the setting relevant to this reduction. This bound matches the lower bound in Theorem 5 up to log factors. Theorem 7 (Sample Complexity of General DMDP). Suppose B + H ≤ 1 1−γ and ε ≤B + H. There exists a constant C3 > 0 such that, for any δ ∈(0, 1), if n ≥C3 B+H (1−γ)2ε2 log SA (1−γ)δε , then with probability 1 −δ, the policy bπ⋆ γ,p output by Algorithm 1 satisfies V ⋆ γ −V bπ⋆ γ,p γ ∞≤ε. Finally, we present our result for the sample complexity of general average-reward MDPs, matching the lower bound in Theorem 4 up to log factors. We again use the reduction Algorithm 2, this time with the larger DMDP target accuracy ε = B + H, leading to a discount factor of γ = 1 − ε 12(B+H). Theorem 8 (Sample Complexity of General AMDP). There exists a constant C4 > 0 such that for any δ, ε ∈(0, 1), if n ≥C4 B+H ε2 log SA(B+H) δε and we call Algorithm 2 with ε = B + H, then with probability at least 1 −δ, the output policy bπ⋆satisfies the elementwise inequality ρ⋆−ρbπ⋆≤ε1. Proof highlights. Similarly to Theorem 2, we seek to bound certain variance parameters, and this time it would suffice to bound the variance of the cumulative discounted reward starting from any state s like V π⋆ γ s [P∞ t=0 γtRt] ≤O H+B 1−γ . Such a bound indeed holds for states s that are recurrent under π⋆ γ, because ρ⋆(St) will remain constant to ρ⋆(s) for all t, since, as mentioned above, ρ⋆is constant on closed irreducible recurrent blocks, and all (St)t≥0 will stay in the same block as s. Therefore, we can almost reuse our argument from the weakly communicating case. However, if s is transient, it is easy to see that V π⋆ γ s [P∞ t=0 γtRt] = Ω 1 1−γ 2 in general (even under the bounded transient time assumption), as we can consider an example where from s we transition to either an absorbing reward 1 state or an absorbing reward 0 state. Thus, when s is transient, instead of bounding V π⋆ γ s [P∞ t=0 γtRt] , we directly work with the sharper variance parameter e⊤ s (I −γPπ⋆γ)−1q VPπ⋆γ V π⋆γ γ , which is also common to the analysis of DMDPs [3, 1, 12] (and in these previous works is bounded in terms of Vπ⋆ γ [P∞ t=0 γtRt] ∞; see Lemma 12 for this relationship). We instead develop a novel law-of-total-variance-style argument which limits the total contribution of transient states to this sharper variance parameter. See Lemma 26 for details. 5 Conclusion In this paper we obtained optimal sample complexities for weakly communicating and general average reward MDPs by improving the analysis of discounted MDPs, revealing a quadratic rather than cubic dependence on the effective horizon for a fixed instance. A limitation of our results (as well as of all previous results) is that the average-to-discounted reduction requires prior knowledge of parameters for optimal complexity, and an interesting open question is whether it is possible to remove this assumption. In conclusion, we believe our results shed greater light on the relationship between the discounted and average reward settings as well as the fundamental complexity of the discounted setting, and we hope that our technical developments can be useful in future work, such as leading to efficient optimal algorithms in the online setting. 9 Acknowledgments and Disclosure of Funding Y. Chen and M. Zurek were supported in part by National Science Foundation CCF-2233152 and DMS-2023239. References [1] Alekh Agarwal, Sham Kakade, and Lin F. Yang. Model-Based Reinforcement Learning with a Generative Model is Minimax Optimal, April 2020. arXiv:1906.03804 [cs, math, stat] version: 3. [2] Mohammad Gheshlaghi Azar, Remi Munos, and Bert Kappen. On the Sample Complexity of Reinforcement Learning with a Generative Model, June 2012. arXiv:1206.6461 [cs, stat]. [3] Mohammad Gheshlaghi Azar, Rémi Munos, and Hilbert J. Kappen. Minimax PAC bounds on the sample complexity of reinforcement learning with a generative model. Machine Learning, 91(3):325–349, June 2013. [4] Peter L. Bartlett and Ambuj Tewari. REGAL: A Regularization based Algorithm for Reinforcement Learning in Weakly Communicating MDPs, May 2012. arXiv:1205.2661. [5] Ronan Fruit, Matteo Pirotta, and Alessandro Lazaric. 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Sharper Model-free Reinforcement Learning for Average-reward Markov Decision Processes, June 2023. arXiv:2306.16394 [cs]. 11 A Proofs for weakly communicating MDPs In this section, we provide the proofs for our main results in Section 3 for weakly communicating MDPs. Before beginning, we note that given that H ≥1, we may assume that H is an integer by setting H ←⌈H⌉, which only affects the sample complexity by a constant multiple < 2 relative to the original parameter H. Let ∥M∥∞→∞:= supv:∥v∥∞≤1 ∥Mv∥∞denote the ℓ∞operator norm of a matrix M. We record the standard and useful fact that (I −γP ′)−1 ∞→∞≤ 1 1−γ for any transition probability matrix P ′, which follows from the Neumann series (I −γP ′)−1 = P t≥0 (γP ′)t and the elementary fact that ∥P ′∥∞→∞≤1. A.1 Technical lemmas First we formally state the main theorem from [20], which gives a reduction from weakly communicating average-reward problems to discounted problems. Lemma 9. Suppose (P, r) is an MDP which is weakly communicating and has an optimal bias function h⋆satisfying ∥h⋆∥span ≤H. Fix ε ∈(0, 1] and set γ = 1 −ε H. For any εγ ∈[0, 1 1−γ ], if π is any εγ-optimal policy for the discounted MDP (P, r, γ), then ρ⋆−ρπ ≤ 8 + 3εγ H ε1. From here, we will first establish lemmas which are useful for proving Theorem 1 on discounted MDPs, and then we will apply the reduction approach of Lemma 9 to prove Theorem 2 on averagereward MDPs. As mentioned in the introduction, a key technical component of our approach is to establish superior bounds on a certain instance-dependent variance quantity which replace a factor of 1 1−γ with a factor of H. Before reaching this step however, to make use of such a bound, we require an algorithm for discounted MDPs which enjoys a variance-dependent guarantee. The work [12] obtains bounds with variance dependence that suffice for our purposes. However, they do not directly present said variance-dependent bounds, so we must slightly repackage their arguments in the form we require. Lemma 10. There exist absolute constants c1, c2 such that for any δ ∈ (0, 1), if n ≥ c2 1−γ log SA (1−γ)δε , then with probability at least 1 −δ, after running Algorithm 1, we have bV π⋆ γ γ,p −V π⋆ γ γ ∞≤γ v u u tc1 log SA (1−γ)δε n (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 (1) and bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤γ v u u tc1 log SA (1−γ)δε n (I −γPbπ⋆ γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6. (2) Proof. First we establish equation (1). The proof of [12, Lemma 1] shows that when n ≥ 16e2 1−γ 2 log 4S log e 1−γ δ , with probability at least 1 −δ we have bV π⋆ γ γ −V π⋆ γ γ ∞≤4γ v u u t2 log 4S log e 1−γ δ n (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ + γ 2 log 4S log e 1−γ δ (1 −γ)n V π⋆ γ γ ∞. (3) 12 Now since bV π⋆ γ γ,p −bV π⋆ γ γ ∞= (I −γ bPπ⋆γ)−1erπ⋆γ −(I −γ bPπ⋆γ)−1rπ⋆γ ∞ ≤ (I −γ bPπ⋆γ)−1 ∞→∞∥er −r∥∞ ≤ ξ 1 −γ = ε 6, we can obtain equation (1) by triangle inequality (although we will choose the constant c1 below). Next we establish equation (2). Using [12, Lemma 6], with probability at least 1 −δ we have that bQ⋆ γ,p(s, bπ⋆ γ,p(s)) −bQ⋆ γ,p(s, a) > ξδ(1 −γ) 3SA2 = εδ(1 −γ)2 18SA2 (4) uniformly over all s and all a ̸= bπ⋆ γ,p(s). From this separation condition (4), the assumptions of [12, Lemma 5] hold (with ω = εδ(1−γ)2 18SA2 in their notation) for the MDP with the perturbed reward er. The proof of [12, Lemma 5] shows that under the event (4) holds, the conditions for [12, Lemma 2] are satisfied (with, in their notation, β1 = 2 log 32 (1−γ)2ωδSA log e 1−γ = 2 log 576S2A3 (1−γ)4δ2ε log e 1−γ ) with additional failure probability ≤δ. The proof of [12, Lemma 2] then shows that, assuming n > 16e2 1−γ 2 log 576S2A3 (1−γ)4δ2ε log e 1−γ , we have bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ,p ∞≤4γ r β1 n (I −γPbπ⋆γ,p)−1q VPbπ⋆γ,p h V bπ⋆ γ,p γ,p i ∞ + γβ1 (1 −γ)n V bπ⋆ γ,p γ,p ∞ (5) where we abbreviated β1 = 2 log 576S2A3 (1−γ)4δ2ε log e 1−γ for notational convenience. We can again calculate that V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞= (I −γPbπ⋆γ,p)−1erbπ⋆γ,p −(I −γPbπ⋆γ,p)−1rbπ⋆γ,p ∞ ≤ (I −γPbπ⋆γ,p)−1 ∞→∞∥er −r∥∞ ≤ ξ 1 −γ = ε 6, so bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ,p ∞+ ε 6 by triangle inequality, essentially giving (2). Finally, to choose the constants c1 and c2, we first note that 2 log 4S log e 1−γ δ ≤β1 < c′ 1 log SA (1−γ)δε for some absolute constant c′ 1, and therefore also all our requirements on n are fulfilled when n ≥16e2 1−γ c′ 1 log SA (1−γ)δε = c′ 2 1−γ log SA (1−γ)δε for another absolute constant c′ 2. Lastly we note that by the union bound the total failure probability is at most 3δ, so to obtain a failure probability of δ′ we may set δ = δ′/3 and absorb the additional constant when defining c1, c2 in terms of c′ 1, c′ 2, and we also then increase c1 by a factor of 4 to absorb the factor of 4 appearing in the first terms within (3) and (5). Now we can analyze the variance parameters (I −γPπ⋆ γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ and (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ∞ , which appear in the error bounds in Lemma 10. We begin by reproducing the following inequality from [23, Lemma 2]. Lemma 11. In a weakly communicating MDP, for all γ ∈[0, 1), it holds that sup s V π⋆ γ γ (s) − 1 1 −γ ρ⋆ ≤H. 13 The following relates the variance parameter of interest to another parameter, the variance of the total discounted rewards. This result essentially appears in [1, Lemma 4] (which was in turn inspired by [3, Lemma 8]), but since their result pertains to objects slightly different than Pπ and VPπ V π γ , we provide the full argument for completeness. Lemma 12. For any deterministic stationary policy π, we have γ (I −γPπ)−1q VPπ V π γ ∞ ≤ r 2 1 −γ v u u t Vπ " ∞ X t=0 γtRt # ∞ . Proof. First we note the well-known variance Bellman equation (see for instance [16, Theorem 1]): Vπ " ∞ X t=0 γtRt # = γ2VPπ V π γ + γ2PπVπ " ∞ X t=0 γtRt # . (6) Now we can basically identically follow the argument of [1, Lemma 4]. The matrix (1−γ)(I−γPπ)−1 has rows which are each probability distributions (are non-negative and sum to 1). Therefore, by Jensen’s inequality and the concavity of the function x 7→√x, for each row s ∈S we have (1 −γ)e⊤ s (I −γPπ)−1q VPπ V π γ ≤ q(1 −γ)e⊤ s (I −γPπ)−1VPπ V π γ . Using this fact we can calculate that, abbreviating v = VPπ V π γ , γ (I −γPπ)−1√v ∞= γ 1 1 −γ (1 −γ)(I −γPπ)−1√v ∞ ≤γ 1 1 −γ q ∥(1 −γ)(I −γPπ)−1v∥∞ = γ 1 √1 −γ q ∥(I −γPπ)−1v∥∞. In order to relate (I −γPπ)−1v ∞to (I −γ2Pπ)−1v ∞in order to apply the variance Bellman equation (6), we calculate (I −γPπ)−1v ∞= (I −γPπ)−1(I −γ2Pπ)(I −γ2Pπ)−1v ∞ = (I −γPπ)−1 ((1 −γ)I + γ(I −γPπ)) (I −γ2Pπ)−1v ∞ = (1 −γ)(I −γPπ)−1 + γI (I −γ2Pπ)−1v ∞ ≤ (1 −γ)(I −γPπ)−1(I −γ2Pπ)−1v ∞+ γ (I −γ2Pπ)−1v ∞ ≤(1 −γ) (I −γPπ)−1 ∞→∞ (I −γ2Pπ)−1v ∞+ γ (I −γ2Pπ)−1v ∞ ≤(1 + γ) (I −γ2Pπ)−1v ∞ ≤2 (I −γ2Pπ)−1v ∞ Combining these calculations with the variance Bellman equation (6), we conclude that γ (I −γPπ)−1√v ∞≤γ 1 √1 −γ q 2 ∥(I −γ2Pπ)−1v∥∞≤ r 2 1 −γ v u u t Vπ " ∞ X t=0 γtRt # ∞ as desired. The following is a multi-step version of the variance Bellman equation, which we will later apply with T = H but holds for arbitrary T. Lemma 13. For any integer T ≥1, for any deterministic stationary policy π, we have Vπ " ∞ X t=0 γtRt # = Vπ "T −1 X t=0 γtRt + γT V π γ (ST ) # + γ2T P T π Vπ " ∞ X t=0 γtRt # 14 and consequently Vπ " ∞ X t=0 γtRt # ∞ ≤ Vπ hPT −1 t=0 γtRt + γT V π γ (ST ) i ∞ 1 −γ2T . Proof. Fix a state s0 ∈S. Letting FT be the σ-algebra generated by (S1, . . . , ST ), we calculate that Vπ s0 " ∞ X t=0 γtRt # = Eπ s0 ∞ X t=0 γtRt −V π γ (s0) !2 = Eπ s0 T −1 X t=0 γtRt + γT V π γ (ST ) −V π γ (s0) + ∞ X t=T γtRt −γT V π γ (ST ) !2 = Eπ s0 " Eπ s0 " T −1 X t=0 γtRt + γT V π γ (ST ) −V π γ (s0) \| {z } A + ∞ X t=T γtRt −γT V π γ (ST ) \| {z } B !2FT ## Using the above shorthands and opening the square, we obtain Vπ s0 " ∞ X t=0 γtRt # = Eπ s0 Eπ s0 A2 + B2 + 2AB FT = Eπ s0 A2 + Eπ s0 B2FT + 2AEπ s0 [B\|FT ] = Eπ s0 A2 + Eπ ST B2 = Eπ s0   T −1 X t=0 γtRt + γT V π γ (ST ) −V π γ (s0) !2 + Eπ ST   ∞ X t=T γtRt −γT V π γ (ST ) !2    = Eπ s0   T −1 X t=0 γtRt + γT V π γ (ST ) −V π γ (s0) !2 + γ2T Eπ ST   ∞ X t=0 γtRt −V π γ (ST ) !2    = Vπ s0 "T −1 X t=0 γtRt + γT V π γ (ST ) # + γ2T e⊤ s0P T π Vπ " ∞ X t=0 γtRt # , where we used the tower property, the Markov property, and the fact that Eπ s0 [B\|FT ] = 0 (which is immediate from the definition of V π γ ). Since e⊤ s0P T π is a probability distribution, it follows from Holder’s inequality that e⊤ s0P T π Vπ [P∞ t=0 γtRt] ≤∥Vπ [P∞ t=0 γtRt]∥∞. Therefore, it holds that Vπ s0 " ∞ X t=0 γtRt # ∞ ≤ Vπ "T −1 X t=0 γtRt + γT V π γ (ST ) # ∞ + γ2T Vπ s0 " ∞ X t=0 γtRt # ∞ and we can obtain the desired conclusion after rearranging terms. We also need the following elemetary inequality. Lemma 14. If γ ≥1 −1 T for some integer T ≥1, then 1 −γ2T 1 −γ ≥ 1 −1 e2 T ≥4 5T. Proof. Fixing T ≥1, we have 1 −γ2T 1 −γ = 1 + γ + γ2 + · · · + γ2T −1 15 which is increasing in γ, so infγ≥1−1 T 1−γ2T 1−γ is attained at γ = 1 −1 T . Now allowing T ≥1 to be arbitrary, note 1−(1−1 T ) 2T 1−(1−1 T ) = T 1 − 1 −1 T 2T so it suffices to show that 1− 1 −1 T 2T ≥1−e2 for all T ≥1. By computing the derivative, one finds that 1− 1 −1 T 2T is monotonically decreasing, so 1 − 1 −1 T 2T ≥lim T →∞1 − 1 −1 T 2T = 1 −1 e2 . We can now provide a bound on the variance of the total discounted rewards under π⋆ γ. Lemma 15. Letting π⋆ γ be the optimal policy for the weakly communicating discounted MDP (P, r, γ), if γ ≥1 −1 H, we have Vπ⋆ γ " ∞ X t=0 γtRt # ∞ ≤5 H 1 −γ . Proof. By using the multi-step variance Bellman equation in Lemma 13, it suffices to bound the quantity Vπ⋆ γ hPH−1 t=0 γtRt + γHV π⋆ γ γ (SH) i ∞. Fixing a state s0 ∈S, V π⋆ γ s0 "H−1 X t=0 γtRt + γHV π⋆ γ γ (SH) # = V π⋆ γ s0 "H−1 X t=0 γtRt + γH V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ # ≤E π⋆ γ s0 H−1 X t=0 γtRt + γH V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤2E π⋆ γ s0 H−1 X t=0 γtRt 2 + 2E π⋆ γ s0 γH V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤2H2 + 2 sup s V π⋆ γ γ (s) − 1 1 −γ ρ⋆ 2 ≤4H2 where in the final inequality we used Lemma 11. Taking the maximum over all states s and combining with Lemma 13 we obtain Vπ⋆ γ " ∞ X t=0 γtRt # ∞ ≤ 4H2 1 −γ2H . Combining this bound with the elementary inequality in Lemma 14, which can be rearranged to show that 1 1−γ2H ≤5 4 1 (1−γ)H, we complete the proof. We also need to control the variance under bπ⋆ γ,p, which requires additional steps. This is done in the following lemma. Lemma 16. We have Vbπ⋆ γ,p " ∞ X t=0 γt eRt # ∞ ≤15 H2 + V bπ⋆ γ,p γ −bV bπ⋆ γ,p γ,p 2 ∞+ V π⋆ γ γ −bV π⋆ γ γ,p 2 ∞ H(1 −γ) . 16 Proof. In light of the multi-step variance Bellman equation in Lemma 13, it suffices to give a bound on Vbπ⋆ γ,p hPH−1 t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) i ∞. We have for any state s0 that V bπ⋆ γ,p s0 "H−1 X t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) # = V bπ⋆ γ,p s0 "H−1 X t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) −γH 1 1 −γ ρ⋆ # ≤E bπ⋆ γ,p s0 H−1 X t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) −γH 1 1 −γ ρ⋆ !2 = E bπ⋆ γ,p s0 H−1 X t=0 γt eRt + γH V bπ⋆ γ,p γ,p (SH) −V π⋆ γ γ (SH) + γH V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ !2 ≤3E bπ⋆ γ,p s0 H−1 X t=0 γt eRt !2 + 3γ2HE bπ⋆ γ,p s0 V bπ⋆ γ,p γ,p (SH) −V π⋆ γ γ (SH) 2 + 3γ2HE bπ⋆ γ,p s0 V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤3E bπ⋆ γ,p s0 H−1 X t=0 γt eRt !2 + 6γ2HE bπ⋆ γ,p s0 V bπ⋆ γ,p γ (SH) −V π⋆ γ γ (SH) 2 + 6γ2H V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞ + 3γ2HE bπ⋆ γ,p s0 V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 , (7) where we have used triangle inequality and the inequalities (a + b)2 ≤2a2 + 2b2 and (a + b + c)2 ≤ 3a2 + 3b2 + 3c2. Now we bound each term of (7). First, we have 3E bπ⋆ γ,p s0 H−1 X t=0 γt eRt !2 ≤3 (H ∥er∥∞)2 ≤3H2(∥r∥∞+ ξ)2 ≤6H2 1 + (1 −γ)ε 6 2! ≤6H2 7 6 2 , where we had (1−γ)ε 6 ≤ ε 6H ≤1 6 because 1 1−γ ≥H and ε ≤H. Clearly it holds that 6γ2HE bπ⋆ γ,p s0 V bπ⋆ γ,p γ (SH) −V π⋆ γ γ (SH) 2 ≤6 V bπ⋆ γ,p γ −V π⋆ γ γ 2 ∞. By an argument identical to those used in the proof of the error bounds in Lemma 10, we get V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ 1 1 −γ ξ = ε 6, so 6γ2H V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞≤ε2 6 ≤H2 6 since ε ≤H. Finally, using Lemma 11, we obtain 3γ2HE bπ⋆ γ,p s0 V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤3 sup s V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤3H2. 17 Using all these bounds in (7), we have V bπ⋆ γ,p s0 "H−1 X t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) # ≤3E bπ⋆ γ,p s0 H−1 X t=0 γt eRt !2 + 6γ2HE bπ⋆ γ,p s0 V bπ⋆ γ,p γ (SH) −V π⋆ γ γ (SH) 2 + 6γ2H V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞ + 3γ2HE bπ⋆ γ,p s0 V π⋆ γ γ (SH) − 1 1 −γ ρ⋆ 2 ≤ 49 6 + 1 6 + 3 H2 + 6 V bπ⋆ γ,p γ −V π⋆ γ γ 2 ∞ ≤12H2 + 6 V bπ⋆ γ,p γ −V π⋆ γ γ 2 ∞. (8) Finally, we use the elementwise inequality V π⋆ γ γ ≥V bπ⋆ γ,p γ ≥bV bπ⋆ γ,p γ,p − bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞1 ≥bV π⋆ γ γ,p − bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞1 ≥V π⋆ γ γ − bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞1 − bV π⋆ γ γ,p −V π⋆ γ γ ∞1, from which it follows that V bπ⋆ γ,p γ −V π⋆ γ γ ∞≤ bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞. Combining this with (8), we conclude V bπ⋆ γ,p s0 "H−1 X t=0 γt eRt + γHV bπ⋆ γ,p γ,p (SH) # ≤12H2 + 12 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞+ 12 bV π⋆ γ γ,p −V π⋆ γ γ 2 ∞. (9) Now combining with Lemma 13 and then using Lemma 14, we have Vbπ⋆ γ,p " ∞ X t=0 γt eRt # ∞ ≤ Vbπ⋆ γ,p hPH−1 t=0 γt eRt + γHV bπ⋆ γ,p γ (SH) i ∞ 1 −γ2H ≤12 H2 + V bπ⋆ γ,p γ −bV bπ⋆ γ,p γ,p 2 ∞+ V π⋆ γ γ −bV π⋆ γ γ,p 2 ∞ 1 −γ2H ≤125 4 H2 + V bπ⋆ γ,p γ −bV bπ⋆ γ,p γ,p 2 ∞+ V π⋆ γ γ −bV π⋆ γ γ,p 2 ∞ H(1 −γ) = 15 H2 + V bπ⋆ γ,p γ −bV bπ⋆ γ,p γ,p 2 ∞+ V π⋆ γ γ −bV π⋆ γ γ,p 2 ∞ H(1 −γ) as desired. A.2 Proofs of Theorems 1 and 2 With the above lemmas we can complete the proof of Theorem 1 on discounted MDPs. Proof of Theorem 1. Our approach will be to utilize our variance bounds within the error bounds from Lemma 10. We will find a value for n which guarantees that bV π⋆ γ γ,p −V π⋆ γ γ ∞and bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞are both ≤ε/2, which guarantees that V bπ⋆ γ,p γ −V π⋆ γ γ ∞≤ε. 18 First we note that the conclusions of Lemma 10 require n ≥ c2 1−γ log SA (1−γ)δε so we assume n is large enough that this holds. Now we bound bV π⋆ γ γ,p −V π⋆ γ γ ∞. Starting with inequality (1) from Lemma 10 and then applying our variance bounds through Lemma 12 and then Lemma 15, we have bV π⋆ γ γ,p −V π⋆ γ γ ∞ ≤γ v u u tc1 log SA (1−γ)δε n (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n r 2 1 −γ v u u t Vπ⋆γ " ∞ X t=0 γtRt # ∞ + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n r 2 1 −γ s 5 H 1 −γ + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n s 10 H (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 where in the last inequality we used the facts that V π⋆ γ γ ∞≤ 1 1−γ and γ ≤1. Now if we assume n ≥360c1 H (1−γ)2ε2 log SA (1−γ)δε , we have bV π⋆ γ γ,p −V π⋆ γ γ ∞≤ v u u tc1 log SA (1−γ)δε n s 10 H (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 ≤1 6 √ ε2 + 1 6 ε2 H + ε 6 ≤ε/2 due to the fact that ε ≤H. Next, to bound bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞, starting from inequality (2) in Lemma 10 and then analogously applying Lemma 12 and then Lemma 16, we obtain bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≤γ v u u tc1 log SA (1−γ)δε n (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆ γ,p γ,p i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n r 2 1 −γ v u u t Vbπ⋆γ,p " ∞ X t=0 γt eRt # ∞ + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n r 2 1 −γ v u u t 15 H2 + V bπ⋆γ,p γ −bV bπ⋆γ,p γ,p 2 ∞+ V π⋆γ γ −bV π⋆γ γ,p 2 ∞ H(1 −γ) + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6. 19 Combining with the fact from above that bV π⋆ γ γ,p −V π⋆ γ γ ∞≤H 2 , as well as the facts that V bπ⋆ γ,p γ,p ∞≤ 1 1−γ , γ ≤1, and √ a + b ≤√a + √ b, we have bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ v u u tc1 log SA (1−γ)δε n r 2 1 −γ v u u t 15 5 4H2 + V bπ⋆γ,p γ −bV bπ⋆γ,p γ,p 2 ∞ H(1 −γ) + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 ≤ v u u tc1 log SA (1−γ)δε n s 30 H(1 −γ)2 r 5 4H2 + r V bπ⋆γ,p γ −bV bπ⋆γ,p γ,p 2 ∞ ! + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 = v u u tc1 log SA (1−γ)δε n s 30 H(1 −γ)2 r 5 4H + V bπ⋆ γ,p γ −bV bπ⋆ γ,p γ,p ∞ ! + c1 log SA (1−γ)δε (1 −γ)2n + ε 6. Rearranging terms gives    1 − v u u tc1 log SA (1−γ)δε n s 30 H(1 −γ)2     bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≤ v u u tc1 log SA (1−γ)δε n s 75H/2 (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n + ε 6. Assuming n ≥120c1 H (1−γ)2ε2 log SA (1−γ)δε , we have 1 − v u u tc1 log SA (1−γ)δε n s 30 H(1 −γ)2 ≥1 −1 2 s ε2(1 −γ)2 H 1 H(1 −γ)2 = 1 −1 2 ε H ≥1 2 since ε ≤H. Also assuming n ≥(75/2) · 242c1 H (1−γ)2ε2 log SA (1−γ)δε we have similarly to before thatv u u tc1 log SA (1−γ)δε n s 75H/2 (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 ≤1 24 s (1 −γ)2ε2 H H (1 −γ)2 + 1 24 (1 −γ)2ε2 H 1 (1 −γ)2 + ε 6 ≤ε 24 + ε 24 + ε 6 = ε 4. Combining these two calculations, we have 1 2 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ε 4, so bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ε 2 as desired. Since we have established that bV π⋆ γ γ,p −V π⋆ γ γ ∞, bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ε 2, since also bV bπ⋆ γ,p γ,p ≥bV π⋆ γ γ,p, we can conclude that V π⋆ γ γ −V bπ⋆ γ,p γ ≤ bV π⋆ γ γ,p −V π⋆ γ γ ∞1 + bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞1 ≤ε1, 20 that is that bπ⋆ γ,p is ε-optimal for the discounted MDP (P, r, γ). We finally note that all our requirements on the size of n can be satisfied by requiring n ≥C2 H (1 −γ)2ε2 log SA (1 −γ)δε := max c2H (1 −γ)2ε2 , 360c1H (1 −γ)2ε2 , (75/2)242c1H (1 −γ)2ε2 log SA (1 −γ)δε ≥max c2 1 −γ , 360c1H (1 −γ)2ε2 , (75/2)242c1H (1 −γ)2ε2 log SA (1 −γ)δε where we used that H (1−γ)2ε2 ≥ H2 (1−γ)ε2 ≥ 1 1−γ (since 1 1−γ ≥H and H ≥ε). We next use Theorem 1 to prove Theorem 2 on average-reward MDPs. Proof of Theorem 2. Using Theorem 1 with target accuracy H and discount factor γ = 1 − ε 12H, we obtain a H-optimal policy for the discounted MDP (P, r, γ) with probability at least 1 −δ as long as n ≥C2 H (1 −γ)2H2 log SA (1 −γ)δε = 122C2 H H2 H2 ε2 log 12H ε SA δε which is satisfied when n ≥C1 H ε2 log SAH δε for sufficiently large C1. Applying Lemma 9 (with error parameter ε 12 since we have chosen γ = 1 −ε/12 H ), we have that ρ⋆−ρbπ⋆≤ 8 + 3H H ε 12 ≤ε1 as desired. A.3 Proof of Theorem 3 Proof of Theorem 3. Fix T, n ≥1. First we define the instances M0 and M1, which have parameters B and ε which we will choose later, using Figure 2. Note that in both MDPs, all states have only one action. The only difference is in the state transition distribution at state 1: For M0 this is a Cat( 1 2, 1 2) distribution and for M1 this is a Cat( 1 2 + ε, 1 2 −ε) distribution, where Cat(p1, p2) denotes the categorical distribution with event probabilities p1 and p2 = 1 −p1. 21 1 2 3 4 R = 1 2 1 2 1 2 R = 1 R = 0 R = 1 2 1 −1 B 1 B Instance M0 1 2 3 4 R = 1 2 1 2 + ε 1 2 −ε R = 1 R = 0 R = 1 2 1 −1 B 1 B Instance M1 Figure 2: MDPs used in Theorem 3 Now we calculate the bias of instance M1. It is easy to check the stationary distribution is µ = [ 1 2, 1 4 + ε 2, 1 4 −ε 2, 0]. Therefore it has optimal gain ρ⋆= 1 2 1 2 + 1 4 + ε 2 = 1 2 + ε 2. Now we claim that the optimal bias is h⋆=   −ε/2 1 2 −ε/2 −1 2 −ε/2 −(B + 1) ε 2  . We can check this by showing that µh⋆= 0 and that ρ⋆1 + h⋆= r + Ph⋆, where P is the transition matrix of the above MDP (again, note that each state has only one action, so there is only one policy, and we use this policy to induce the markov chain with transition matrix P). First, µh⋆= −ε 4 + 1 8 + ε 4 −ε 8 −ε2 4 −1 8 + ε 4 −ε 8 + ε2 4 = 0. It is also easy to check the first three rows of the equality ρ⋆1 + h⋆= r + Ph⋆. For the fourth row, we have h⋆(4) + 1 2 + ε 2 = 1 2 + 1 B h⋆(1) + 1 −1 B h⋆(4) ⇐⇒1 B h⋆(4) = −ε 2B −ε 2 ⇐⇒h⋆(4) = ε 2(B + 1). Thus ∥h⋆∥span = 1 2−ε/2− −(B + 1) ε 2 = 1 2(Bε+1). If we set B = 2T ε −1 2, we have ∥h⋆∥span = T. Also note that the calculation for h⋆holds for any ε, so the optimal bias span of M0 is [0, 1 2, −1 2, 0]⊤, and thus M0 has optimal bias span 1. Finally, to distinguish between the two MDPs M0 and M1, we must be able to determine the next-state distribution of state 1, that is, to distinguish between the two hypotheses Q1 = Cat( 1 2, 1 2) and Q2 = Cat( 1 2 + ε, 1 2 −ε). Given n i.i.d. observations from the transition distribution of state 1, this is a binary hypothesis testing problem between the product distributions Qn 1 and Qn 2. By Le 22 Cam’s bound [24], the testing failure probability is lower bounded by 1 2 (1 −∥Qn 1 −Qn 2∥TV) ≥1 2 1 − r 1 2DKL(Qn 1\|Qn 2) ! = 1 2 1 − rn 2 DKL(Q1\|Q2) , where ∥Qn 1 −Qn 2∥TV and DKL(Qn 1\|Qn 2) denote the total variation distance and Kullback–Leibler (KL) divergence between Qn 1 and Qn 2, respectively, and the last two (in)equalities follow from Pinsker’s inequality and tensorization of KL divergence. By direct calculation, we have DKL(Q1\|Q2) = 1 2 log 1 1 + 2ε + 1 2 log 1 1 −2ε ≤1 2 · −2ε 1 + 2ε + 1 2 · 2ε 1 −2ε log(1 + x) ≤x, ∀x > −1 = 4ε2 1 −4ε2 ≤8ε2 ε ≤1 4. Combining the last two equations, we see that the testing failure probability is at least 1 2 1 − √ 4nε2 . Thus, if we set ε = 1 4√n, the failure probability is at least 1 4. B Proofs for general MDPs In this section, we provide the proofs for our main results in Section 4 for general MDPs. Again, we can assume that H + B is an integer, which only affects the sample complexity by a constant multiple < 2. First we develop more notation which will be useful in the setting of general MDPs. Recall we defined, for any policy π, that Rπ is the set of states which are recurrent in the Markov chain Pπ, and T π = S \ Rπ is the set of transient states. We now present a standard decomposition of Markov chains [14, Appendix A]. For any policy π, possibly after reordering states so that the recurrent states appear first (and are grouped into disjoint irreducible closed sets), we can decompose Pπ = Xπ 0 Yπ Zπ (10) such that Xπ are probabilities of transitions between states which are recurrent under π, Yπ are probabilities of transitions from T π into Rπ, and Zπ are probabilities of transitions between states within T π. Furthermore, supposing there are k irreducible closed blocks within Rπ, Xπ is blockdiagonal of the form Xπ =   Xπ,1 0 · · · 0 0 Xπ,2 · · · 0 ... ... ... ... 0 0 · · · Xπ,k  . The limiting matrix of the Markov chain induced by policy π is defined as the matrix P ∞ π = C-lim T →∞P T π = lim T →∞ 1 T T −1 X t=0 P t π. P ∞ π is a stochastic matrix (all rows positive and sum to 1) since S is finite. We also have PπP ∞ π = P ∞ π = P ∞ π Pπ. Additionally, ρπ = P ∞ π rπ. In terms of our decomposition, we have P ∞ π = X∞ π 0 Y ∞ π 0 (11) 23 where X∞ π =   X∞ π,1 0 · · · 0 0 X∞ π,2 · · · 0 ... ... ... ... 0 0 · · · X∞ π,k  , each X∞ π,i = 1x⊤ π,i for some stochastic row vector x⊤ π,i, and Y ∞ π = (I −Zπ)−1YπX∞ π . Also we have (I −Zπ)−1 = P∞ t=0 Zt π, and P∞ t=0 Zt πYπ = (I −Zπ)−1Yπ has stochastic rows (each row is a probability distribution, that is all entries are positive and sum to 1). With the same arrangement of states as within the above decomposition of Pπ (10), let V π γ = V π γ V π γ decompose V π γ into recurrent and transient states, and generally we use this same notation for any vector x ∈RS: we let x list the values of xs for recurrent x ∈Rπ, x contain xs for s ∈T π, and we assume the entire x has been rearranged so that x = [x x]⊤. Note that the rearrangement of states depends on the policy π so this notation has potential for confusion if applied to objects relating to multiple policies at once, but the policy determining the rearrangement will always be clear from context in our arguments. The main reason we decompose Pπ into recurrent and transient states is the following key observation. Lemma 17. For any policy π, if s, s′ are in the same recurrent block of the Markov chain with transition matrix Pπ, then ρ⋆(s) = ρ⋆(s′). Proof. Define the history-dependent policy ˜π which follows π until its history first contains s′, after which point it follows π⋆. Since ρ⋆(s) is the optimal gain achievable starting at s by following any history-dependent policy [14], we have ρ⋆(s) ≥ρ˜π(s) := limT →∞1 T E˜π s PT −1 t=0 Rt (where E˜π s is defined in the natural way from the distribution over trajectories (S0, A0, . . . ) where At ∼ ˜π(S0, A0, . . . , St) and St+1 ∼P(· \| St, At)). Let Ts′ = inf{t ≥1 : St = s′} be the hitting time of state s′ and let FTs′ be the stopped σ-algebra (with respect to the filtration where for all nonnegative integers t, Ft is the σ-algebra generated by S0, A0, . . . , St, At). Then lim T →∞ 1 T E˜π s T −1 X t=0 Rt = lim T →∞ 1 T E˜π s " E˜π s "T −1 X t=0 Rt FTs′ ## = lim T →∞ 1 T E˜π s   Ts′−1 X t=0 Rt + E˜π s   T −1 X t=Ts′ Rt FTs′     = lim T →∞ 1 T E˜π s   Ts′−1 X t=0 Rt + g(T, Ts′)   = lim T →∞ 1 T Eπ s   Ts′−1 X t=0 Rt + g(T, Ts′)   ≥lim T →∞ 1 T Eπ s [g(T, Ts′)] where g(T, k) := Eπ⋆ s′ hPT −k−1 t=0 Rt i , and we used the tower property, FTs′-measurability of PTs′−1 t=0 Rt, the strong Markov property, and the definition of ˜π. Now note that Ts′ < ∞almost surely since s and s′ are in the same recurrent block, and on the event {Ts′ = k} for any natural number k, we have that lim T →∞ 1 T g(T, k) = lim T →∞ 1 T Eπ⋆ s′ "T −k−1 X t=0 Rt # = ρ⋆(s′) 24 because we can bound 1 T Eπ⋆ s′ "T −1 X t=0 Rt # −k T ≤1 T Eπ⋆ s′ "T −k−1 X t=0 Rt # ≤1 T Eπ⋆ s′ "T −1 X t=0 Rt # and both sides converge to ρ⋆(s′). Therefore g(T,Ts′) T converges almost surely to the constant ρ⋆(s′), and also this random variable is bounded by 1, so by the dominated convergence theorem we have lim T →∞ 1 T Eπ s [g(T, Ts′)] = Eπ s lim T →∞ 1 T g(T, Ts′) = ρ⋆(s′). Thus we have shown that ρ⋆(s) ≥ρ⋆(s′). Since s and s′ were arbitrary states in the same recurrent block we also have ρ⋆(s′) ≥ρ⋆(s), and thus ρ⋆(s) = ρ⋆(s′) as desired. Lemma 18. For any state s which is transient under a policy π, if the MDP satisfies the bounded transient time assumption with parameter B, we have ∞ X t=0 e⊤ s Zt π 1 ≤B. Proof. Let T = inf{t : St ∈Rπ}. Notice that e⊤ s Zt π 1 = Pπ s (T > t). Therefore, we have ∞ X t=0 e⊤ s Zt π 1 ≤ ∞ X t=0 e⊤ s Zt π 1 = ∞ X t=0 Pπ s (T > t) = Eπ s [T] ≤B, where we used a well-known formula for the expectation of nonnegative-integer-valued random variables, and the bounded transient time assumption. Lemma 19. Let s be a transient state under Pπ. Then e⊤ s (I −γPπ)−1 = es⊤P∞ k=1 γkZk−1 π Yπ(I −γXπ)−1 es⊤P∞ t=0 γtZt π . Proof. Using the decomposition of Pπ, we can calculate for any integer t ≥1 that P t π = Xt π 0 Pt k=1 Zk−1 π YπXt−k π Zt π . Therefore, we have e⊤ s (I −γPπ)−1 = e⊤ s ∞ X t=0 γtP t π = es⊤P∞ t=0 γt Pt k=1 Zk−1 π YπXt−k π es⊤P∞ t=0 γtZt π = es⊤P∞ k=1 P∞ t=k γtZk−1 π YπXt−k π es⊤P∞ t=0 γtZt π = es⊤P∞ k=1 γkZk−1 π Yπ P∞ t=k γt−kXt−k π es⊤P∞ t=0 γtZt π = es⊤P∞ k=1 γkZk−1 π Yπ(I −γXπ)−1 es⊤P∞ t=0 γtZt π . Note that we are able to rearrange the order of the summation in the third equality because all summands are (elementwise) positive. 25 B.1 Proof of Theorem 6 Theorem 6, our result which helps reduce general average reward MDPs to discounted MDPs, is proven as a straightforward consequence of the following sequence of lemmas, some of which will also be needed for the proof of our discounted MDP sample complexity bound Theorem 7. Lemma 20. We have V π⋆ γ − 1 1 −γ ρ⋆ ∞ ≤∥h⋆∥span . Proof. We begin by observing that π⋆satisfies ρ⋆+ h⋆= rπ⋆+ Pπ⋆h⋆. Therefore, it holds that V π⋆ γ = (I −γPπ⋆)−1rπ⋆ = (I −γPπ⋆)−1 (ρ⋆+ h⋆−Pπ⋆h⋆) = (I −γPπ⋆)−1ρ⋆+ (I −γPπ⋆)−1 (I −Pπ⋆) h⋆. Since Pπ⋆ρ⋆= ρ⋆, we can calculate that (I −γPπ⋆)−1ρ⋆= X t≥0 γtP t π⋆ρ⋆= X t≥0 γtρ⋆= 1 1 −γ ρ⋆. It also holds that (I −γPπ⋆)−1 (I −Pπ⋆) = X t≥0 γtP t π⋆(I −Pπ⋆) = X t≥0 γtP t π⋆− X t≥0 γtP t+1 π⋆ = Pπ⋆+ X t≥0 (γt+1 −γt)P t+1 π⋆ (12) and P t≥0 γt+1 −γt = (γ −1) P t≥0 γt = −1. Therefore (12) is the difference of two stochastic matrices, and so it follows that (I −γPπ⋆)−1 (I −Pπ⋆) h⋆ ∞≤∥h⋆∥span . Lemma 21. If π⋆ γ is optimal for the discounted MDP (P, r, γ) and s is recurrent under π⋆ γ, then V π⋆ γ γ (s) − 1 1 −γ ρ⋆(s) ≤∥h⋆∥span and V π⋆ γ γ (s) − 1 1 −γ ρπ⋆ γ(s) ≤2 ∥h⋆∥span . These facts can be written as V π⋆γ γ − 1 1−γ ρ⋆ ∞ ≤∥h⋆∥span and V π⋆γ γ − 1 1−γ ρπ⋆γ ∞ ≤2 ∥h⋆∥span respectively. Proof. First note that if s is recurrent for the Markov chain Pπ⋆γ, then all states in the support of e⊤ s Pπ⋆γ are in the same recurrent block as state s, and ρ⋆is constant (and equal to ρ⋆(s)) within this recurrent block by Lemma 17. The (unmodified) Bellman equation states that ρ⋆(s) + h⋆(s) = max a:Psaρ⋆=ρ⋆(s) rsa + Psah⋆. 26 Since we established that e⊤ s Pπ⋆γρ⋆= ρ⋆(s), all actions a in the support of π⋆ γ(a \| s) satisfy Psaρ⋆= ρ⋆(s), and therefore ρ⋆(s) + h⋆(s) = max a:Psaρ⋆=ρ⋆(s) rsa + Psah⋆ ≥ X a∈A π⋆ γ(a \| s) (rsa + Psah⋆) = e⊤ s rπ⋆γ + Pπ⋆γh⋆ . Since this holds for all s ∈Rπ⋆ γ, we can rearrange to obtain that rπ⋆γ ≤ρ⋆+ h⋆−Pπ⋆γh⋆= ρ⋆+ h⋆−Xπ⋆γh⋆. Now we can follow an argument which is similar to that of [23, Lemma 2]. We have V π⋆γ γ = (I −γPπ⋆γ)−1rπ⋆γ = (I −Xπ⋆γ)−1rπ⋆γ ≤(I −Xπ⋆γ)−1 ρ⋆+ h⋆−Xπ⋆γh⋆ using monotonicity of (I −Xπ⋆γ)−1 in the final inequality. Due to the observation above that for all s ∈Rπ⋆ γ, all actions a in the support of π⋆ γ(a \| s) satisfy Psaρ⋆= ρ⋆(s), we have Xπ⋆ γρ⋆= ρ⋆. Therefore we have (I −Xπ⋆γ)−1ρ⋆= ∞ X t=0 γtXπ⋆γρ⋆= ∞ X t=0 γtρ⋆= 1 1 −γ ρ⋆. For the second term, by using an argument which is completely analogous to that used in Lemma 20 we have (I −Xπ⋆γ)−1 h⋆−Xπ⋆γh⋆ ∞≤∥h⋆∥span. Combining these steps we obtain that V π⋆γ γ − 1 1 −γ ρ⋆≤∥h⋆∥span 1. To obtain a lower bound, we can combine the optimality of π⋆ γ for the γ-discounted problem with Lemma 20 to obtain the bound V π⋆γ γ − 1 1 −γ ρ⋆≥V π⋆ γ − 1 1 −γ ρ⋆≥∥h⋆∥span 1. Therefore we can conclude that V π⋆γ γ − 1 1−γ ρ⋆ ∞ ≤∥h⋆∥span. For the second bound in the lemma statement, we first note that, as observed in [20], P ∞ π⋆ γV π⋆ γ γ = P ∞ π⋆ γ ∞ X t=0 γtP t π⋆ γrπ⋆γ = ∞ X t=0 γtP ∞ π⋆ γrπ⋆γ = 1 1 −γ ρπ⋆ γ. Also, as discussed previously, if s ∈Rπ⋆ γ then e⊤ s Pπ⋆γρ⋆= ρ⋆(s), so then we also have e⊤ s P ∞ π⋆γρ⋆= ρ⋆(s) (which can be seen directly from the definition of the limiting matrix P ∞ π⋆ γ). Equivalently, e⊤ s (I −P ∞ π⋆ γ)ρ⋆= 0. Using both of these two observations, we have V π⋆ γ γ (s) − 1 1 −γ ρπ⋆ γ(s) = e⊤ s (I −P ∞ π⋆ γ)V π⋆ γ γ = e⊤ s (I −P ∞ π⋆ γ)(V π⋆ γ γ − 1 1 −γ ρ⋆) = es ⊤(I −X∞ π⋆ γ)(V π⋆γ γ − 1 1 −γ ρ⋆). 27 Therefore, we obtain V π⋆γ γ − 1 1 −γ ρπ⋆γ ∞ ≤ (I −X∞ π⋆γ)(V π⋆γ γ − 1 1 −γ ρ⋆) ∞ ≤ V π⋆γ γ − 1 1 −γ ρ⋆ span ≤2 V π⋆γ γ − 1 1 −γ ρ⋆ ∞ ≤2 ∥h⋆∥span using the first bound from the lemma statement in the final inequality. Lemma 22. We have V π⋆ γ γ − 1 1 −γ ρ⋆ ∞ ≤B + ∥h⋆∥span and V π⋆ γ γ − 1 1 −γ ρπ⋆ γ ∞ ≤B + 2 ∥h⋆∥span . Proof. Note that by combining with Lemma 21, it suffices to prove for any transient state s ∈T π⋆ γ that V π⋆ γ γ (s) − 1 1 −γ ρ⋆(s) ≤B + ∥h⋆∥span and V π⋆ γ γ (s) − 1 1 −γ ρπ⋆ γ(s) ≤B + 2 ∥h⋆∥span . Let s be transient under π⋆ γ. Then starting by using Lemma 19, we can calculate V π⋆ γ γ (s) = e⊤ s (I −γPπ⋆γ)−1rπ⋆γ = ∞ X t=0 γtes ⊤Zt π⋆γrπ⋆γ + γ ∞ X t=0 γtes ⊤Zt π⋆γYπ⋆γ(I −γXπ⋆γ)−1rπ⋆ γ = ∞ X t=0 γtes ⊤Zt π⋆ γrπ⋆γ + γ ∞ X t=0 γtes ⊤Zt π⋆ γYπ⋆γV π⋆γ γ ≤ ∞ X t=0 es ⊤Zt π⋆γrπ⋆γ + ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! V π⋆γ γ . (13) By Lemma 18 we have that ∞ X t=0 es ⊤Zt π⋆γrπ⋆γ ≤ ∞ X t=0 es ⊤Zt π⋆γ 1 rπ⋆γ ∞≤B. Now we can obtain the two bounds in the lemma statement by bounding the second term of (13) in two different ways. For the first bound in the lemma statement, we can use the first bound in Lemma 28 21 to calculate that ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! V π⋆γ γ ≤ ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! 1 1 −γ ρ⋆+ ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! V π⋆γ γ − 1 1 −γ ρ⋆ ∞ 1 = ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! 1 1 −γ ρ⋆+ V π⋆γ γ − 1 1 −γ ρ⋆ ∞ ≤ ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! 1 1 −γ ρ⋆+ ∥h⋆∥span = ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γ ! 1 1 −γ X∞ π⋆γρ⋆+ ∥h⋆∥span = ∞ X t=0 e⊤ s Zt π⋆γYπ⋆γX∞ π⋆γ ! 1 1 −γ ρ⋆+ ∥h⋆∥span = e⊤ s Y ∞ π⋆γ 1 1 −γ ρ⋆+ ∥h⋆∥span = 1 1 −γ e⊤ s P ∞ π⋆γρ⋆+ ∥h⋆∥span ≤ 1 1 −γ ρ⋆(s) + ∥h⋆∥span where we used the fact that X∞ π⋆γρ⋆= ρ⋆and then that e⊤ s P ∞ π⋆γρ⋆≤ρ⋆(s). This gives an upper bound of V π⋆ γ γ ≤ 1 1 −γ ρ⋆(s) + B + ∥h⋆∥span . Combining with the lower bound V π⋆ γ γ (s) ≥V π⋆ γ (s) ≥ 1 1 −γ ρ⋆(s) −∥h⋆∥span , we obtain that V π⋆ γ γ − 1 1 −γ ρ⋆ ∞ ≤B + ∥h⋆∥span which is the first bound in the lemma statement. 29 To obtain the second bound in the lemma statement, using the second bound from Lemma 21, we can calculate for the second term in (13) that ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! V π⋆γ γ ≤ ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! 1 1 −γ ρπ⋆γ + ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! V π⋆γ γ − 1 1 −γ ρπ⋆γ ∞ 1 = ∞ X t=0 es ⊤Zt π⋆ γYπ⋆γ ! 1 1 −γ ρπ⋆γ + V π⋆γ γ − 1 1 −γ ρπ⋆γ ∞ ≤ ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! 1 1 −γ ρπ⋆γ + 2 ∥h⋆∥span = ∞ X t=0 es ⊤Zt π⋆γYπ⋆ γ ! 1 1 −γ P ∞ π⋆γrπ⋆γ + 2 ∥h⋆∥span = ∞ X t=0 es ⊤Zt π⋆γYπ⋆γ ! 1 1 −γ X∞ π⋆γrπ⋆γ + 2 ∥h⋆∥span = 1 1 −γ es ⊤Y ∞ π⋆γ rπ⋆γ + 2 ∥h⋆∥span = 1 1 −γ e⊤ s P ∞ π⋆γrπ⋆γ + 2 ∥h⋆∥span = 1 1 −γ ρπ⋆ γ(s) + 2 ∥h⋆∥span where in the second equality we used the fact that P∞ t=0 e⊤ s Zt π⋆γYπ⋆γ is a probability distribution, and in the final steps we used the decomposition of P ∞ π⋆γ and the fact that ρπ⋆ γ = P ∞ π⋆γrπ⋆γ. Therefore by combining these steps we obtain that V π⋆ γ γ (s) ≤B + 2 ∥h⋆∥span + 1 1 −γ ρπ⋆ γ(s). Combining with the lower bound V π⋆ γ γ (s) ≥V π⋆ γ (s) ≥ 1 1 −γ ρ⋆(s) −∥h⋆∥span ≥ 1 1 −γ ρπ⋆ γ(s) −∥h⋆∥span , we obtain the desired boundV π⋆ γ γ (s) − 1 1 −γ ρπ⋆ γ(s) ≤B + 2 ∥h⋆∥span . Lemma 23. If π satisfies V π γ ≥V π⋆ γ γ −δ1, then V π γ − 1 1 −γ ρπ ∞ ≤3B + 2 ∥h⋆∥span + δ. Proof. Similar to the proof of Lemmas 21 and 22, we will first establish a bound for the states which are recurrent under π. Specifically, we will first show that if s is recurrent under π we have V π γ (s) − 1 1 −γ ρπ(s) ≤2B + 2 ∥h⋆∥span + δ. (14) Letting s ∈Rπ, following steps which are similar to the proof of the second part of Lemma 21, we have V π γ (s) − 1 1 −γ ρπ(s) = e⊤ s (I −P ∞ π )V π γ = e⊤ s (I −P ∞ π )(V π γ − 1 1 −γ ρ⋆) = e⊤ s (I −P ∞ π )(V π⋆ γ γ − 1 1 −γ ρ⋆) + e⊤ s (I −P ∞ π )(V π γ −V π⋆ γ γ ) 30 using the fact discussed in Lemma 21 that e⊤ s (I −P ∞ π )ρ⋆= 0 since s is recurrent under π. Then by triangle inequality, we obtain V π γ (s) − 1 1 −γ ρπ(s) ≤ e⊤ s (I −P ∞ π )(V π⋆ γ γ − 1 1 −γ ρ⋆) + e⊤ s (I −P ∞ π )(V π γ −V π⋆ γ γ ) ≤ V π⋆ γ γ − 1 1 −γ ρ⋆ span + V π γ −V π⋆ γ γ span ≤2 V π⋆ γ γ − 1 1 −γ ρ⋆ ∞ + δ ≤2B + 2 ∥h⋆∥span + δ, where we used the facts that ∥·∥span ≤2 ∥·∥∞and that V π⋆ γ γ ≥V π γ ≥V π⋆ γ γ −δ1. Having established (14), we now extend to transient states using arguments similar to those for the second bound of Lemma 22. Let s be transient under π. Then starting by using Lemma 19, we can calculate V π γ (s) = e⊤ s (I −γPπ)−1rπ = ∞ X t=0 γtes ⊤Zt πrπ + γ ∞ X t=0 γtes ⊤Zt πYπ(I −γXπ)−1rπ = ∞ X t=0 γtes ⊤Zt πrπ + γ ∞ X t=0 γtes ⊤Zt πYπV π γ ≤ ∞ X t=0 es ⊤Zt πrπ + ∞ X t=0 es ⊤Zt πYπ ! V π γ ≤ ∞ X t=0 es ⊤Zt π 1 rπ ∞+ ∞ X t=0 es ⊤Zt πYπ ! V π γ ≤B + ∞ X t=0 es ⊤Zt πYπ ! V π γ (15) using the bounded transient time assumption via Lemma 18 in the final step. Then we can calculate ∞ X t=0 es ⊤Zt πYπ ! V π γ ≤ ∞ X t=0 es ⊤Zt πYπ ! 1 1 −γ ρπ + ∞ X t=0 es ⊤Zt πYπ ! V π γ − 1 1 −γ ρπ ∞ 1 = ∞ X t=0 es ⊤Zt πYπ ! 1 1 −γ ρπ + V π γ − 1 1 −γ ρπ ∞ ≤ ∞ X t=0 es ⊤Zt πYπ ! 1 1 −γ ρπ + 2B + 2 ∥h⋆∥span + δ = ∞ X t=0 es ⊤Zt πYπ ! 1 1 −γ P ∞ π rπ + 2B + 2 ∥h⋆∥span + δ = ∞ X t=0 es ⊤Zt πYπ ! 1 1 −γ X∞ π rπ + 2B + 2 ∥h⋆∥span + δ = 1 1 −γ es ⊤Y ∞ π rπ + 2B + 2 ∥h⋆∥span + δ = 1 1 −γ e⊤ s P ∞ π rπ + 2B + 2 ∥h⋆∥span + δ = 1 1 −γ ρπ(s) + 2B + 2 ∥h⋆∥span + δ, 31 where in the first equality we used the fact that P∞ t=0 e⊤ s Zt πYπ is a probability distribution, in the second inequality we used the bound (14), and in the final steps we used the decomposition of P ∞ π and the fact that ρπ = P ∞ π rπ. Therefore by combining this last bound with the bound (15), we have V π γ (s) ≤3B + 2 ∥h⋆∥span + δ + 1 1 −γ ρπ(s). Combining with the lower bound V π γ (s) ≥V π⋆ γ γ −δ ≥V π⋆ γ (s) −δ ≥ 1 1 −γ ρ⋆(s) −∥h⋆∥span −δ ≥ 1 1 −γ ρπ(s) −∥h⋆∥span −δ, we conclude that V π γ (s) − 1 1 −γ ρπ(s) ≤3B + 2 ∥h⋆∥span + δ as desired. Proof of Theorem 6. Suppose π is εγ-optimal for the discounted MDP (P, r, γ). We can calculate that 1 1 −γ ρπ ≥V π γ −(3B + 2 ∥h⋆∥span + εγ) ≥V π⋆ γ γ −(3B + 2 ∥h⋆∥span + 2εγ) ≥V π⋆ γ −(3B + 2 ∥h⋆∥span + 2εγ) ≥ 1 1 −γ ρ⋆−(3B + 3 ∥h⋆∥span + 2εγ), where in the first inequality we used Lemma 23, in the second inequality we used the fact that π is εγ-optimal, in the third inequality we used the optimality of π⋆ γ for the discounted MDP, and in the final inequality we used Lemma 20. Therefore by mulitplying both sides by 1 −γ, we have that ρπ ≥ρ⋆− ε B + H(3B + 3 ∥h⋆∥span + 2εγ) ≥ρ⋆− 3ε + 2 εγ B + H ε. B.2 Proof of Theorem 7 (Discounted MDP Bounds) In this section, we provide our main result on the sample complexity of general discounted MDPs. Our proof relies on three lemmas that provide bounds on relevant variance parameters. The first lemma controls the variance for π⋆ γ on recurrent states. Lemma 24. Letting π⋆ γ be the optimal policy for the discounted MDP (P, r, γ), if γ ≥1 − 1 B+H, we have max s∈Rπ⋆γ γ e⊤ s (I −γPπ⋆ γ)−1 r VPπ⋆γ h V π⋆γ γ i ≤ s 32 5 B + H (1 −γ)2 . Proof. First, using the decomposition (10), we can calculate for any s ∈Rπ⋆ γ that e⊤ s (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i = es ⊤(I −γXπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i = es ⊤(I −γXπ⋆γ)−1 s VXπ⋆γ V π⋆γ γ . 32 Also due to the decomposition, notice that set Rπ⋆ γ is a closed set for the Markov chain with transition matrix Pπ⋆γ, and furthermore when restricting to the entries corresponding to this closed set we obtain the transition matrix Xπ⋆γ. Therefore we can apply Lemma 12 to this subchain to obtain that γ (I −γXπ⋆γ)−1 s VXπ⋆γ V π⋆γ γ ∞ ≤ r 2 1 −γ v u u u t Vπ⋆γ " ∞ X t=0 γtRt # ∞ . Abbreviating L = B + H, we can also then apply Lemma 13 to bound Vπ⋆γ " ∞ X t=0 γtRt # ∞ ≤ Vπ⋆γ hPL−1 t=0 γtRt + γLV π⋆γ γ (SL) i ∞ 1 −γ2L . We can repeat a similar argument as within Lemma 15 to bound this term. Fixing an initial state s0 ∈Rπ⋆ γ, the key observation is that ρ⋆is constant on the recurrent block of Xπ⋆γ containing s0, and therefore any state trajectory S0 = s0, S1, S2, . . . under the transition matrix Pπ⋆γ will have ρ⋆(SL) = ρ⋆(s0). Therefore for this fixed s0 we have V π⋆ γ s0 "L−1 X t=0 γtRt + γLV π⋆ γ γ (SL) # = V π⋆ γ s0 "L−1 X t=0 γtRt + γL V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(s0) # ≤E π⋆ γ s0 L−1 X t=0 γtRt + γL V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(s0) 2 ≤2E π⋆ γ s0 L−1 X t=0 γtRt 2 + 2E π⋆ γ s0 γL V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(s0) 2 = 2E π⋆ γ s0 L−1 X t=0 γtRt 2 + 2E π⋆ γ s0 γL V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(SL) 2 ≤2L2 + 2 sup s∈Rπ⋆γ V π⋆ γ γ (s) − 1 1 −γ ρ⋆(s) 2 ≤2L2 + 2H2 ≤4L2 where we used Lemma 21 in the penultimate inequality. Applying this argument to all s0 ∈Rπ⋆ γ we obtain Vπ⋆γ "L−1 X t=0 γtRt + γLV π⋆γ γ (SL) # ∞ ≤4L2. 33 Therefore by combining with our initial bounds we have that max s∈Rπ⋆γ γ e⊤ s (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ≤ r 2 1 −γ v u u u t Vπ⋆γ " ∞ X t=0 γtRt # ∞ ≤ r 2 1 −γ v u u u t Vπ⋆γ hPL−1 t=0 γtRt + γLV π⋆γ γ (SL) i ∞ 1 −γ2L ≤ r 2 1 −γ s 4L2 1 −γ2L ≤ r 2 1 −γ s 16L2 5L(1 −γ) ≤ s 32 5 L (1 −γ)2 , where in the penultimate inequality we used Lemma 14 to bound 1 1−γ2L ≤5 4 1 (1−γ)L. The next lemma controls the variance for bπ⋆ γ,p on recurrent states. Lemma 25. Letting bπ⋆ γ,p be the optimal policy for the discounted MDP ( bP, er, γ), if γ ≥1 − 1 B+H, we have max s∈Rbπ⋆γ,p γ e⊤ s (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ≤ s 29 B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ . Proof. Let L = B + H. By the same arguments as in the beginning of the proof of Lemma 24, we have max s∈Rbπ⋆γ,p γ e⊤ s (I −γPbπ⋆ γ,p)−1 r VPbπ⋆γ,p h V bπ⋆ γ,p γ,p i ≤ r 2 1 −γ v u u u t Vbπ⋆γ,p " ∞ X t=0 γt eRt # ∞ ≤ r 2 1 −γ v u u u t Vbπ⋆ γ,p hPL−1 t=0 γt eRt + γLV bπ⋆γ,p γ,p (SL) i ∞ 1 −γ2L so it again suffices to bound Vbπ⋆γ,p hPL−1 t=0 γt eRt + γLV bπ⋆γ,p γ,p (SL) i . Fix s0 ∈Rbπ⋆ γ,p. Again, as observed in Lemma 24, ρ⋆is constant on the recurrent block of Xbπ⋆γ,p containing s0, so we will have 34 ρ⋆(SL) = ρ⋆(s0) with probability one. Therefore (mostly following the steps of Lemma 16) V bπ⋆ γ,p s0 "L−1 X t=0 γt eRt + γLV bπ⋆ γ,p γ,p (SL) # = V bπ⋆ γ,p s0 "L−1 X t=0 γt eRt + γLV bπ⋆ γ,p γ,p (SL) −γL 1 1 −γ ρ⋆(s0) # ≤E bπ⋆ γ,p s0 L−1 X t=0 γt eRt + γLV bπ⋆ γ,p γ,p (SL) −γL 1 1 −γ ρ⋆(s0) !2 = E bπ⋆ γ,p s0 L−1 X t=0 γt eRt + γL V bπ⋆ γ,p γ,p (SL) −V π⋆ γ γ (SL) + γL V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(SL) !2 ≤3E bπ⋆ γ,p s0 L−1 X t=0 γt eRt !2 + 3γ2LE bπ⋆ γ,p s0 V bπ⋆ γ,p γ,p (SL) −V π⋆ γ γ (SL) 2 + 3γ2LE bπ⋆ γ,p s0 V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(SL) 2 ≤3E bπ⋆ γ,p s0 L−1 X t=0 γt eRt !2 + 6γ2LE bπ⋆ γ,p s0 V bπ⋆ γ,p γ (SL) −V π⋆ γ γ (SL) 2 + 6γ2L V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞ + 3γ2LE bπ⋆ γ,p s0 V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(SL) 2 (16) using the inequalities (a + b + c)2 ≤3a2 + 3b2 + 3c2 and (a + b)2 ≤2a2 + 2b2. Now we bound each term of (16) analogously to the steps of Lemma 16. For the first term of (16), 3E bπ⋆ γ,p s0 L−1 X t=0 γt eRt !2 ≤3 (L ∥er∥∞)2 ≤3L2(∥r∥∞+ ξ)2 ≤6L2 1 + (1 −γ)ε 6 2! ≤6L2 7 6 2 , where we had (1−γ)ε 6 ≤ ε 6L ≤1 6 because 1 1−γ ≥L and ε ≤L. For the second term of (16), 6γ2LE bπ⋆ γ,p s0 V bπ⋆ γ,p γ (SL) −V π⋆ γ γ (SL) 2 ≤6 V bπ⋆ γ,p γ −V π⋆ γ γ 2 ∞ ≤6 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 2 where we used (a + b)2 ≤2a2 + 2b2 and the fact that V bπ⋆ γ,p γ −V π⋆ γ γ ∞≤ bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞which was shown in Lemma 16. For the third term of (16), 6γ2L V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞≤6 V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ 2 ∞≤6 ξ 1 −γ 2 = 6 ε 6 2 ≤L2 6 where the fact that V bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ ξ 1−γ is identical to the arguments used in the proof of Lemma 10, and the final inequality is due to the assumption that ε ≤L. For the fourth term of (16), 3γ2LE bπ⋆ γ,p s0 V π⋆ γ γ (SL) − 1 1 −γ ρ⋆(SL) 2 ≤3 V π⋆ γ γ − 1 1 −γ ρ⋆ 2 ∞ ≤3L2 using Lemma 22 for the second inequality. Using all these bounds in (16), we obtain V bπ⋆ γ,p s0 "L−1 X t=0 γt eRt + γLV bπ⋆ γ,p γ,p (SL) # ≤ 49 6 + 1 6 + 3 L2 + 6 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 2 35 and so (since this holds for arbitrary s0 ∈Rbπ⋆ γ,p), we have Vbπ⋆ γ,p "L−1 X t=0 γt eRt + γLV bπ⋆γ,p γ,p (SL) # ≤68 6 L2 + 6 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 2 . Therefore, combining with our initial arguments, max s∈Rbπ⋆γ,p γ e⊤ s (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ≤ r 2 1 −γ v u u u t Vbπ⋆γ,p hPL−1 t=0 γt eRt + γLV bπ⋆γ,p γ,p (SL) i ∞ 1 −γ2L ≤ r 2 1 −γ r 68 6 L2 + 6 bV bπ⋆γ,p γ,p −V bπ⋆γ,p γ ∞+ bV π⋆γ γ,p −V π⋆γ γ ∞ 2 p 1 −γ2L ≤ r 2 1 −γ q 68 6 L2 + r 6 bV bπ⋆γ,p γ,p −V bπ⋆γ,p γ ∞+ bV π⋆γ γ,p −V π⋆γ γ ∞ 2 p 1 −γ2L ≤ r 2 1 −γ q 68 6 L2 + r 6 bV bπ⋆γ,p γ,p −V bπ⋆γ,p γ ∞+ bV π⋆γ γ,p −V π⋆γ γ ∞ 2 q 4 5(1 −γ)L < s 29 L (1 −γ)2 + r 15 L bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ , where we used Lemma 14 to bound 1 1−γ2L ≤5 4 1 (1−γ)L. The next lemma controls the variance on all states. Lemma 26. Under the settings of Lemmas 24 and 25, we have γ (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ ≤4 s B + H (1 −γ)2 and γ (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ∞ ≤8 s B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ . Proof. First we establish the first bound in the lemma statement. As we have already bounded the entries corresponding to the recurrent states of π⋆ γ by Lemma 24, it remains to bound the transient states. Let s ∈T π⋆ γ be an arbitrary transient state. Using Lemma 19, we have e⊤ s γ(I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆ γ γ i = γes ⊤ ∞ X k=1 γkZk−1 π⋆γ Yπ⋆γ(I −γXπ⋆γ)−1 r VPπ⋆γ h V π⋆ γ γ i + γes ⊤ ∞ X t=0 γtZt π⋆γ r VPπ⋆γ h V π⋆γ γ i . (17) 36 Now we bound each of the terms in (17). For the first term, we can calculate γes ⊤ ∞ X k=1 γkZk−1 π⋆γ Yπ⋆γ(I −γXπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ≤γes ⊤ ∞ X k=1 Zk−1 π⋆γ Yπ⋆γ(I −γXπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ≤ es ⊤ ∞ X k=1 Zk−1 π⋆γ Yπ⋆γ 1 γ (I −γXπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ ≤ s 32 5 B + H (1 −γ)2 where we used the fact that es⊤P∞ k=1 Zk−1 π⋆γ Yπ⋆γ is a probability distribution and Lemma 24. For the second term of (17), we have γes ⊤ ∞ X t=0 γtZt π⋆γ r VPπ⋆γ h V π⋆γ γ i = γ es ⊤ ∞ X t=0 γtZt π⋆γ 1 ∞ X t=0 γtes⊤Zt π⋆γ es⊤P∞ t=0 γtZt π⋆γ 1 r VPπ⋆γ h V π⋆γ γ i ≤γ es ⊤ ∞ X t=0 γtZt π⋆γ 1 v u u u t ∞ X t=0 γtes⊤Zt π⋆γ es⊤P∞ t=0 γtZt π⋆γ 1 VPπ⋆γ h V π⋆γ γ i = v u u t es⊤ ∞ X t=0 γtZt π⋆γ 1 v u u tγ2 ∞ X t=0 γtes⊤Zt π⋆γVPπ⋆γ h V π⋆γ γ i (18) where we used Jensen’s inequality since x 7→√x is concave and P∞ t=0 γtes ⊤Zt π⋆γ es⊤P∞ t=0 γtZt π⋆γ 1 is a probability distribution (all entries of this row vector are positive and they sum to 1 due to our normalization). Now we bound each factor in (18). Using Lemma 18, we have v u u t es⊤ ∞ X t=0 γtZt π⋆γ 1 ≤ v u u t es⊤ ∞ X t=0 Zt π⋆γ 1 ≤ √ B. For the second factor in (18), we have ∞ X t=0 γtes ⊤Zt π⋆γVPπ⋆γ h V π⋆ γ γ i ≤ ∞ X t=0 γtes ⊤Zt π⋆γVPπ⋆γ h V π⋆ γ γ i + es ⊤ ∞ X k=1 γkZk−1 π⋆γ Yπ⋆ γ(I −γXπ⋆ γ)−1VPπ⋆γ h V π⋆γ γ i = e⊤ s (I −γPπ⋆γ)−1VPπ⋆γ h V π⋆ γ γ i where the equality step is due to Lemma 19. Now we can apply two steps which are used within Lemma 12 to obtain the desired bound on this term. Abbreviating v = VPπ⋆γ h V π⋆ γ γ i , it is shown within Lemma 12 that γ2 (I −γPπ⋆ γ)−1v ∞≤2γ2 (I −γ2Pπ⋆ γ)−1v ∞≤2 Vπ⋆ γ " ∞ X t=0 γtRt # ∞ ≤ 2 (1 −γ)2 (where the final inequality is because the total discounted return is within [0, 1 1−γ ]). Therefore we can bound the second factor in (18) as v u u tγ2 ∞ X t=0 γtes⊤Zt π⋆γVPπ⋆γ h V π⋆ γ γ i ≤ s 2 (1 −γ)2 = √ 2 1 −γ . 37 Combining all of these bounds back into (17), we have e⊤ s γ(I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ≤ s 32 5 B + H (1 −γ)2 + √ B √ 2 1 −γ < 4 s B + H (1 −γ)2 . Thus we have established the first inequality from the lemma statement. For the second inequality, the argument is entirely analogous, except that we use Lemma 25 instead of Lemma 24, and also in the MDP with the perturbed reward er we have the bound Vπ⋆ γ " ∞ X t=0 γtRt # ∞ ≤ ∥er∥∞ 1 −γ 2 ≤ ∥r∥∞+ ξ 1 −γ 2 ≤ 1 (1 −γ)2 1 + (1 −γ)ε 6 2 ≤ 1 (1 −γ)2 7 6 2 , where we used the fact that (1−γ)ε 6 ≤ ε 6(B+H) ≤1 6 because 1 1−γ ≥B + H and ε ≤B + H. Thus we can obtain the bound γ (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ∞ ≤ s 29 B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ + √ B 7 √ 2 6(1 −γ) ≤8 s B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ . This completes the proof of the lemma. We are now ready to prove Theorem 7 on the sample complexity of general discounted MDPs. Proof of Theorem 7. To prove Theorem 7 we will combine our bounds of the variance parameters in Lemma 26 with Lemma 10. First, starting with (1) from Lemma 10 and combining with the first bound from Lemma 26, we have that there exist absolute constants c1, c2 such that for any δ ∈(0, 1), 38 if n ≥ c2 1−γ log SA (1−γ)δε , then with probability at least 1 −δ bV π⋆ γ γ,p −V π⋆ γ γ ∞≤γ v u u tc1 log SA (1−γ)δε n (I −γPπ⋆γ)−1 r VPπ⋆γ h V π⋆γ γ i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n 4 s B + H (1 −γ)2 + c1γ log SA (1−γ)δε (1 −γ)n V π⋆ γ γ ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n 4 s B + H (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n + ε 6 ≤ε 6 + 1 16 · 62 ε2 B + H + ε 6 ≤ε 2, where the penultimate inequality is under the assumption that n ≥16 · 62c1 B+H ε2(1−γ)2 log SA (1−γ)δε , and the final inequality makes use of the fact that ε ≤B + H. Next, still using Lemma 10, under the same event, we also have bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≤γ v u u tc1 log SA (1−γ)δε n (I −γPbπ⋆γ,p)−1 r VPbπ⋆γ,p h V bπ⋆γ,p γ,p i ∞ + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n   8 s B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ bV π⋆ γ γ,p −V π⋆ γ γ ∞ 1 −γ    + c1γ log SA (1−γ)δε (1 −γ)n V bπ⋆ γ,p γ,p ∞+ ε 6 ≤ v u u tc1 log SA (1−γ)δε n   8 s B + H (1 −γ)2 + r 15 B + H bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞+ (B + H)/2 1 −γ    + c1 log SA (1−γ)δε (1 −γ)n 7 6 1 1 −γ + ε 6 using the second inequality from Lemma 26 for the second inequality, and then we use the fact that V bπ⋆ γ,p γ,p ∞≤ 7 6 1 1−γ which was argued in Lemma 26, as well as the fact from above that 39 bV π⋆ γ γ,p −V π⋆ γ γ ∞≤ε/2 ≤(B + H)/2. After rearranging, we obtain that    1 − v u u tc1 log SA (1−γ)δε n r 15 B + H 1 1 −γ     bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≤ v u u tc1 log SA (1−γ)δε n 8 s B + H (1 −γ)2 + r 15 B + H (B + H)/2 1 −γ ! + c1 log SA (1−γ)δε (1 −γ)2n 7 6 + ε 6 ≤ v u u tc1 log SA (1−γ)δε n 10 s B + H (1 −γ)2 + c1 log SA (1−γ)δε (1 −γ)2n 7 6 + ε 6. (19) If n ≥62 · 102c1 B+H ε2(1−γ)2 log SA (1−γ)δε , then the RHS of (19) is bounded by ε 6 + 7 6 ε2 B + H 1 62 · 102 + ε 6 ≤ 1 6 + 1 62 · 102 + 1 6 ε ≤0.4ε using the assumption that ε ≤B + H. Under the same condition on n, we also have that    1 − v u u tc1 log SA (1−γ)δε n r 15 B + H 1 1 −γ     bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≥ 1 − s ε2 (B + H)2 r 15 62 · 102 ! bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≥ 1 − r 15 62 · 102 ! bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ ≥0.9 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞ where again we used the assumption that ε ≤B + H. Combining these two bounds with the inequality (19), we obtain that 0.9 bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤0.4ε which implies that bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤0.4 0.9ε < ε 2. Since we have established that bV π⋆ γ γ,p −V π⋆ γ γ ∞≤ε 2 and that bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞≤ε 2, since also bV bπ⋆ γ,p γ,p ≥bV π⋆ γ γ,p, we can conclude that V π⋆ γ γ −V bπ⋆ γ,p γ ≤ bV π⋆ γ γ,p −V π⋆ γ γ ∞1 + bV bπ⋆ γ,p γ,p −V bπ⋆ γ,p γ ∞1 ≤ε1, that is that bπ⋆ γ,p is ε-optimal. Finally, we check that all of our conditions on n can be satisfied if n ≥max 62 · 102c1 B + H ε2(1 −γ)2 , 62 · 16c1 B + H ε2(1 −γ)2 , c2 1 −γ log SA (1 −γ)δε , and since 1 1−γ ≥B + H and B + H ≥ε, we have B+H ε2(1−γ)2 ≥ (B+H)2 ε2(1−γ) ≥ 1 1−γ , so the above is guaranteed if we set C3 = max{62 · 102c1, c2} and require n ≥C3 B+H ε2(1−γ)2 log SA (1−γ)δε . 40 B.3 Proof of Theorem 8 (General Average-Reward MDP Bounds) In this section, we prove our main result on the sample complexity of general average-reward MDPs. Proof of Theorem 8. We can combine our bound for discounted MDPs, Theorem 7, with our reduction from average-reward MDPs to discounted MDPs, Theorem 6. Using Theorem 7 with target accuracy B + H and discount factor γ = 1 − ε 12(B+H), we obtain a (B + H)-optimal policy for the discounted MDP (P, r, γ) with probability at least 1 −δ as long as n ≥C3 B + H (1 −γ)2(B + H)2 log SA (1 −γ)δε = 122C3 B + H (B + H)2 (B + H)2 ε2 log 12(B + H) ε SA δε which is satisfied when n ≥C4 B+H ε2 log SA(B+H) δε for sufficiently large C4. Applying Theorem 6 (with error parameter ε 12), we obtain ρ⋆−ρbπ⋆≤ 3 + 2B + H B + H ε 12 ≤ε1 as desired. B.4 Proof of Theorems 4 and 5 (Lower Bounds) In this section, we prove our minimax lower bounds on the sample complexity of general averagereward MDPs (Theorem 4) and discounted MDPs (Theorem 5). Proof of Theorem 4. First consider the MDP instances Ma⋆indexed by a⋆∈{1, . . . , A} shown in Figure 3. In all instances, states 2, 3 and 4 are absorbing states, and state 1 is a transient state. State 1 has A actions and is the only state with multiple actions. At state 1, taking action a = 1 will take the agent to state 4 deterministically; taking action 2 will take the agent back to state 1 with probability P(1\|1, 2) = 1 −1 T , to state 2 with probability P(2\|1, 2), and to state 3 with probability P(3\|1, 2) = 1 −P(1\|1, 2) −P(2\|1, 2). The instances differ only in the values of P(2\|1, a) and P(3\|1, a), which are shown in Figure 3 along with the reward R for each state-action pair. For the MDP instance M1, the optimal policy is taking action a = 1 at state 1, leading to an average reward of 1/2; taking any other action leads to a sub-optimal average reward of 1−2ε 2 . Similarly, for the instance Ma⋆with a⋆∈{2, . . . , A}, the optimal action is a = a⋆with average reward 1+2ε 2 , the action a = 1 has average reward 1 2, and all other actions have average reward 1−2ε 2 . By direct calculation, we find that the span of the optimal policy is ∥h⋆∥span = 0 in all instances. Moreover, by taking any action a ̸= 1, the agent will stay in state 1 for B steps in expectation before transitioning to state 2 or 3, so the bounded transient time is satisfied with parameter B. We next define (A−1)S/4 master MDPs Ms⋆,a⋆indexed by s⋆∈{1, . . . , S/4} and a⋆∈{2, . . . , A} as follows. Each master MDP Ms⋆,a⋆has S/4 copies of sub-MDPs such that the s⋆th sub-MDP is equal to Ma⋆and all other sub-MDPs are equal to M1. We rename the states so that the states of the sth sub-MDP has states 4s + 1, 4s + 2, 4s + 3, 4s + 4 corresponding to states 1, 2, 3, 4 of the instances shown in Figure 3. Note each of these master MDPs has S states and A actions, satisfies the bounded transient time property with parameter B, and has the span of the bias of its Blackwell optimal policy equal to 0. Note that for a given policy π to be ε/3-average optimal in master MDP Ms⋆,a⋆, it must take action a⋆in state 4s⋆+ 1 with probability at least 2/3, and it must take action 1 in states 4s + 1 for s ∈{1, . . . , S/4} \ {s⋆} with probability at least 2/3. Thus, for an algorithm Alg to output an ε/3-average optimal policy π, it must identify the master MDP instance Ms⋆,a⋆(equivalently, the values of s⋆and a⋆), in the sense that there must be exactly one state 4s + 1 where an action a ̸= 1 is taken with probability ≥2/3. Therefore it suffices to lower bound the failure probability of any algorithm Alg for this (A −1)S/4-way testing problem. By 41 1 2 3 4 a ∈{2, . . . , A}, R = (1 + 2ε)/2 P(1 \| 1, a) = 1 −1 B P(2 \| 1, a) = 1−2ε 2B P(3 \| 1, a) = 1+2ε 2B a = 1, R = 1/2 R = 1 R = 0 R = 1/2 Instance M1 1 2 3 4 a ∈{2, . . . , A} \ {a⋆}, R = (1 + 2ε)/2 P(1 \| 1, a) = 1 −1 B P(2 \| 1, a) = 1−2ε 2B P(3 \| 1, a) = 1+2ε 2B a = 1, R = 1/2 R = 1 R = 0 R = 1/2 a = a⋆, R = (1 + 2ε)/2 P(1 \| 1, a⋆) = 1 −1 B P(2 \| 1, a⋆) = 1+2ε 2B P(3 \| 1, a⋆) = 1−2ε 2B Instance Ma⋆, for a⋆∈{2, . . . , A} Figure 3: MDP Instances Used in the Proof of Lower Bound in Theorem 4 construction, for any two distinct index pairs (s⋆ 1, a⋆ 1) and (s⋆ 2, a⋆ 2), the master MDPs Ms⋆ 1,a⋆ 1 and Ms⋆ 2,a⋆ 2 differ only in the state-action pairs (4s⋆ 1, a⋆ 1) and (4s⋆ 2, a⋆ 2), and we have PMs⋆ 1 ,a⋆ 1 (· \| 4s⋆ 1, a⋆ 1) = Cat 1 −1 B , 1 −2ε 2B , 1 + 2ε 2B =: Q1, PMs⋆ 2 ,a⋆ 2 (· \| 4s⋆ 1, a⋆ 1) = Cat 1 −1 B , 1 + 2ε 2B , 1 −2ε 2B =: Q2, where Cat(p1, p2, p3) denotes the categorical distribution with event probabilities pi’s (and vice versa for the distributions of the state action pair (4s⋆ 2, a⋆ 2)). Now we use Fano’s method [18] to lower bound this failure probability. Choose an index J uniformly at random from the set J := {1, . . . , S/4} × {2, . . . , A} and suppose that we draw n iid samples X = (X1, . . . , Xn) from the master MDP MJ; note that under the generative model, each random variable Xi represents an (S × A)-by-S transition matrix with exactly one nonzero entry in each row. Letting I(J; X) denote the mutual information between J and X, Fano’s inequality yields that the failure probability is lower bounded by 1 −I(J; X) + log 2 log((A −1)S/4). 42 We can calculate using the fact that the Pi’s are i.i.d., the chain rule of mutual information, and the form of the construction that I(J; X) = nI(J; X1) ≤n max (s⋆ 1,a⋆ 1),(s⋆ 2,a⋆ 2)∈J : (s⋆ 1,a⋆ 1)̸=(s⋆ 2,a⋆ 2) DKL PMs⋆ 1 ,a⋆ 1 PMs⋆ 2 ,a⋆ 2 = n DKL(Q1 \| Q2) + DKL(Q2 \| Q1) . By direct calculation, we have DKL(Q1\|Q2) = 1 −2ε 2B log 1 −2ε 1 + 2ε + 1 + 2ε 2B log 1 + 2ε 1 −2ε ≤1 −2ε 2B · −4ε 1 + 2ε + 1 + 2ε 2B · 4ε 1 −2ε log(1 + x) ≤x, ∀x > −1 = 16ε2 B(1 + 2ε)(1 −2ε) ≤32ε2 B ε ≤1 4. Also note that DKL(Q2\|Q1) = DKL(Q1\|Q2) in this case. Therefore the failure probability is at least 1 −I(J; P n) + log 2 log((A −1)S/4) ≥1 − n 64ε2 B + log 2 log((A −1)S/4) ≥1 2 − n 64ε2 B log((A −1)S/4), where in the second inequality we assumed A and S are at least a sufficiently large constant. For the above RHS to be smaller than 1/4, we therefore require n ≥Ω( B log(SA) ε2 ). Proof of Theorem 5. The desired DMDP lower bound follows from combining our AMDP lower bound Theorem 4 with the average-to-discount reduction in Theorem 6. B.5 Relationship between transient time and mixing time Lemma 27. In any uniformly mixing MDP, we have B ≤4τunif. Proof. Fix a deterministic stationary policy π. Notice that since all states in the support of the stationary distribution νπ are recurrent, for any s ∈S we have Pπ s (St is transient) = X s′∈T π Pπ s (St = s′) ≤ X s′∈T π Pπ s (St = s′) + X s′∈Rπ \|Pπ s (St = s′) −νπ(s′)\| = X s′∈S \|Pπ s (St = s′) −νπ(s′)\| ≤2 max s∈S 1 2 e⊤ s P t π −νπ 1 ≤2 · 2−⌊t/τunif⌋ where the final inequality uses standard properties of mixing [11, Chapter 4]. Now define T = inf{t : St ∈Rπ}. Then, using a standard formula for the expectation of nonnegative-integer-values random 43 variables, we have for any s ∈S that Eπ s [T] = ∞ X t=0 Pπ s (T > t) = ∞ X t=0 Pπ s (St is transient) ≤2 ∞ X t=0 2−⌊t/τunif⌋ = 2 ∞ X ℓ=0 τunif2−ℓ = 4τunif. Since this bound holds for all s ∈S and all deterministic stationary policies π, we conclude that B ≤4τunif. 44 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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4,485	Almost Surely Asymptotically Constant Graph Neural Networks Sam Adam-Day∗ Michael Benedikt ˙Ismail ˙Ilkan Ceylan Ben Finkelshtein Department of Computer Science University of Oxford Oxford, UK Abstract We present a new angle on the expressive power of graph neural networks (GNNs) by studying how the predictions of real-valued GNN classifiers, such as those classifying graphs probabilistically, evolve as we apply them on larger graphs drawn from some random graph model. We show that the output converges to a constant function, which upper-bounds what these classifiers can uniformly express. This strong convergence phenomenon applies to a very wide class of GNNs, including state of the art models, with aggregates including mean and the attention-based mechanism of graph transformers. Our results apply to a broad class of random graph models, including sparse and dense variants of the Erd˝os–R´enyi model, the stochastic block model, and the Barab´asi-Albert model. We empirically validate these findings, observing that the convergence phenomenon appears not only on random graphs but also on some real-world graphs. 1 Introduction Figure 1: The output of the considered GNNs eventually become constant as the graph sizes increase. Graph neural networks (GNNs) [44, 21] have become prominent for graph machine learning with applications in domains such as life sciences [47, 11, 25, 55]. Their empirical success motivated work investigating their theoretical properties, pertaining to their expressive power [52, 39, 4, 8, 1, 43, 23, 19, 54], generalization capabilities [18, 33, 38, 45, 40], and convergence properties [28, 27, 38, 32, 2]. We consider GNNs outputting real-valued vectors, such as those which classify graphs probabilistically, and ask: how do the outputs of these GNNs evolve as we apply them on larger graphs drawn from a random graph model? Our study provides a surprising answer to this question: the output of many GNNs eventually become independent of their inputs – each model eventually outputs the same values on all graphs – as graph sizes increase (Figure 1). This “almost sure convergence” to a constant distribution is much stronger than convergence to some limit object [28, 27, 32]. The immediate consequence of this strong convergence phenomenon is to upper bound the uniform expressiveness of the considered model architectures: these architectures can uniformly express only the classifiers that are almost ∗Corresponding author. Email: me@samadamday.com 38th Conference on Neural Information Processing Systems (NeurIPS 2024). surely asymptotically constant. In other words, our results provide impossibility results for what tasks are in principle learnable by GNNs. While our top-level results are for graph classification, in the process we provide strong limitations on what node- and edge-classification can be performed by GNNs on random graphs: see, for example Theorem 5.3. Scope of the result. The core approach in graph machine learning is based on iteratively updating node representations of an input graph by an aggregate of messages flowing from the node’s neighbours [20]. This approach can be extended to global aggregates [5]. Our main result holds for all architectures that use weighted mean as an aggregation function and it is extremely robust in the following two dimensions: 1. Model architectures: Our result is very general and abstracts away from low-level architectural design choices. To achieve this, we introduce an aggregate term language using weighted mean aggregation and provide an “almost sure optimization” result for this language: our result states that every term in the language can be simplified to Lipschitz functions for most inputs. Thus, any architecture that can be expressed in this language follows the same convergence law. This includes graph attention networks (GATs) [49], as well as popular (graph) transformers, such as the General, Powerful, Scalable Graph Transformer (GPS) with random walk encodings [41]. The term language can seamlessly capture common design choices, such as skip and jumping knowledge connections [51], or global aggregation schemes [5]. 2. Random graph models: All results apply to a wide class of random graph models, including Erd˝os-R´enyi models of various sparsity levels, the Barab´asi-Albert preferential attachment model, and the stochastic block model. The sparse models are more realistic than their dense counterparts which makes it typically harder to obtain results for them. This is also reflected in our study, as the results for sparse and dense models require very different proofs. Contributions. The key contributions of this paper are as follows: • We introduce a flexible aggregate term language with attractive closure properties (Section 4) and prove an “almost sure convergence” result for this language relative to a wide class of random graph models (Section 5). This result is of independent interest since it pushes the envelope for convergence results from classical logical languages [15, 36] to include aggregation. • We show that a diverse class of architectures acting as real-valued graph classifiers can be expressed in the term language (Section 4). In Section 5 we present “almost sure convergence” results for our term language, from which we derive results about GNNs (Corollary 5.2). The results are robust to many practical architectural design choices and even hold for architectures using mixtures of layers from different architectures. We also show strong convergence results for real-valued node classifiers in many graph models. • We validate these results empirically, showing the convergence of these graph classifiers in practice (Section 6) on graphs drawn from the random models studied. In addition, we probe the real-world significance of our results by testing for convergence on a dataset with varying size dataset splits. Across all experiments we observe rapid convergence to a constant distribution. Interestingly, we note some distinctions between the convergence of the sparse and non-sparse Erd˝os-R´enyi model, which we can relate to the proof strategies for our convergence laws. 2 Related work Uniform expressiveness. The expressive power of MPNNs is studied from different angles, including their power in terms of graph distinguishability [52, 39]. The seminal results of Xu et al. [52], Morris et al. [39] show that MPNNs are upper bounded by the 1-dimensional Weisfeiler Leman graph isomorphism test (1-WL) in terms of graph distinguishability. WL-style expressiveness results are inherently non-uniform, i.e., the model construction is dependent on the graph size. There are also recent studies that focus on uniform expressiveness [4, 42, 2]. In particular, Adam-Day et al. [2] investigate the uniform expressive power of GNNs with randomized node features, which are known to be more expressive in the non-uniform setting [1, 43]. They show that for classical Erd˝os-R´enyi graphs, GNN binary classifiers display a zero-one law, assuming certain restrictions on GNN weights and the random graph model. We focus on real-valued classifiers, where their results do not apply, while dealing with a wider class of random graph models, subsuming popular architectures such as graph transformers. 2 Convergence laws for languages. Our work situates GNNs within a rich term language built up from graph and node primitives via real-valued functions and aggregates. Thus it relates to convergence laws for logic-based languages on random structures, dating back to the zero-one law of Fagin [15], including [35, 46, 31]. We are not aware of any prior convergence laws for languages with aggregates; the only work on numerical term languages is by Gr¨adel et al. [22], which deals with a variant of first-order logic in general semi-rings. Other notions of convergence on random graphs. The works of Cordonnier et al. [9], Keriven et al. [28, 27], Maskey et al. [38], Levie [32] consider convergence to continuous analogues of GNNs, often working within metrics on a function space. The results often focus on dense random graph models, such as graphons [34]. Our approach is fundamentally different in that we can use the standard notion of asymptotic convergence in Euclidean space, comparable to traditional language-based convergence results outlined above, such as those by Fagin [15] and Lynch [36]. The key point is that a.a.s. constancy is a very strong notion of convergence and it does not follow from convergence in the senses above. In fact, obtaining such a strong convergence result depends heavily on the details of the term language, as well as the parameters that control the random graph: see Section 7 for further discussion of the line between a.a.s. convergence and divergence. Our study gives particular emphasis to sparse random graph models, like Barab´asi-Albert, which are closer to graphs arising in practice. 3 Preliminaries 3.1 Featured random graphs and convergence Random graphs. We consider simple, undirected graphs G = (VG, EG, HG) where each node is associated with a vector of node features given by HG : VG →Rd. We refer to this as a featured graph. We are interested in random graph models, specifying for each number n a distribution µn on graphs with n nodes, along with random graph feature models, where we have a distribution µn on featured graphs with n nodes. Given a random graph model and a distribution ν over Rd, we get a random graph feature model by letting the node features be chosen independently of the graph structure via ν. Erd˝os-R´enyi and the stochastic block model. The most basic random graph model we deal with is the Erd˝os-R´enyi distribution ER(n, p(n)), where an edge is included in the graph with n nodes with probability p(n) [14]. The classical case is when p(n) is independent of the graph size n, which we refer as the dense ER distribution. We also consider the stochastic block model SBM(n1, . . . , nM, P), which contains m communities of sizes n1, . . . , nM and an edge probability matrix between communities P ∈RM×M. A community i is sampled from the Erd˝os-R´enyi distribution ER(n, p(n) = Pi,i) and an edge to a node in another community j is included with probability Pi,j. The Barab´asi-Albert preferential attachment model. Many graphs encountered in the real world obey a power law, in which a few vertices are far more connected than the rest [7]. The Barab´asiAlbert distribution was developed to model this phenomenon [3]. It is parametrised by a single integer m, and the n-vertex graph BA(n, m) is generated sequentially, beginning with a fully connected m-vertex graph. Nodes are added one at a time and get connected via m new edges to previous nodes, where the probability of attaching to a node is proportional to its degree. Almost sure convergence. Given any function F from featured graphs to real vectors and a random featured graph model (µn)n∈N, we say F converges asymptotically almost surely (converges a.a.s.) to a vector z with respect to ¯µ if for all ϵ, θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing featured graphs G from µn, we have that ∥F(G) −z∥< ϵ. 3.2 Graph neural networks and graph transformers We first briefly introduce message passing neural networks (MPNNs) [20], which include the vast majority of graph neural networks, as well as (graph) transformers. Message passing neural networks. Given a featured graph G = (VG, EG, HG), an MPNN sets the initial features h(0) v = HG(v) and iteratively updates the feature h(ℓ) v of each node v, for 3 0 ≤ℓ≤L −1, based on the node’s state and the state of its neighbors N(v) by defining h(ℓ+1) v as: UPD(ℓ) h(ℓ) v , AGG(ℓ) h(ℓ) v , {{h(ℓ) u \| u ∈N(v)}} , where {{·}} denotes a multiset and UPD(ℓ) and AGG(ℓ) are differentiable update and aggregation functions, respectively. The final node embeddings are pooled to form a graph embedding vector z(L) G to predict properties of entire graphs. A MEANGNN is an MPNN where the aggregate is mean. GATs. One class of MPNNs are graph attention networks (GATs) [49], where each node is updated with a weighted average of its neighbours’ representations, letting h(ℓ+1) v be: X u∈N(v) exp score h(ℓ) v , h(ℓ) u P w∈N(v) exp score h(ℓ) v , h(ℓ) w W (ℓ)h(ℓ) u , where score is a certain learnable Lipschitz function. Graph transformers. Beyond traditional MPNNs, graph transformers extend the well-known transformer architecture to the graph domain. The key ingredient in transformers is the self-attention mechanism. Given a featured graph, a single attention head computes a new representation for every (query) node v, in every layer ℓ> 0 as follows: ATT(v) = X u∈VG exp scale h(ℓ) v , h(ℓ) u P w∈VG exp scale h(ℓ) v , h(ℓ) w W (ℓ)h(ℓ) u where scale is another learnable Lipschitz function (the scaled dot-product). The vanilla transformer architecture ignores the graph structure. Graph transformer architectures [53, 41, 37] address this by explicitly encoding graph inductive biases, most typically in the form of positional encodings (PEs). In their simplest form, these encodings are additional features pv for every node v that encode a node property (e.g., node degree) which are concatenated to the node features hv. The random walk positional encoding (RW) [12] of each node v is given by: rwv = [rwv,1, rwv,2, . . . rwv,k], where rwv,i is the probability of an i-length random walk that starts at v to end at v. The GPS architecture [41] is a representative graph transformer, which applies a parallel computation in every layer: a transformer layer (with or without PEs) and an MPNN layer are applied in parallel and their outputs are summed to yield the node representations. By including a standard MPNN in this way, a GPS layer can take advantage of the graph topology even when there is no positional encoding. In the context of this paper, we write GPS to refer to a GPS architecture that uses an MPNN with mean aggregation, and GPS+RW if the architecture additionally uses a random-walk PE. Probabilistic classifiers. We are looking at models that produce a vector of reals on each graph. All of these models can be used as probabilistic graph classifiers. We only need to ensure that the final layer is a softmax or sigmoid applied to pooled representations. 4 Model architectures via term languages We demonstrate the robustness and generality of the convergence phenomenon by defining a term language consisting of compositions of operators on graphs. Terms are formal sequences of symbols which when interpreted in a given graph yield a real-valued function on the graph nodes. Definition 4.1 (Term language). AGG[WMEAN] is a term language which contains node variables x, y, z, . . . and terms defined inductively:2 • The basic terms are of the form H(x), representing the features of the node x, and constants c. 2In the body we reserve x, y, z for free variables and u, v, w for concrete nodes in a graph. 4 • Let h: Rd →(0, ∞)d be a function which is Lipschitz continuous on every compact domain. Given terms τ and π, the local h-weighted mean for node variable x is: X y∈N(x) τ(y) ⋆h(π(y)) The interpretation of ⋆will be defined below. The global h-weighted mean is the term: X y τ(y) ⋆h(π(y)) • Terms are closed under applying a function symbol for each Lipschitz continuous F : Rd×k →Rd for any k ∈N+. The weighted mean operator takes a weighted average of the values returned by τ. It uses π to perform a weighting, normalizing the values of π using h to ensure that we are not dividing by zero (see below for the precise definition). To avoid notational clutter, we keep the dimension of each term fixed at d. It is possible to simulate terms with different dimensions by letting d be the maximum dimension and padding the vectors with zeros, noting that the padding operation is Lipschitz continuous. We make the interpretation of the terms precise as follows. See Figure 2 for a graphical example of evaluating a term on a graph. Definition 4.2. Let G = (VG, EG, HG) be a featured graph. Let τ be a term with free variables x1, . . . , xk and ¯u = (u1, . . . , uk) a tuple of nodes. The interpretation Jτ(¯u)KG of term τ graph G for tuple ¯u is defined recursively: • Jc(¯u)KG = c for any constant c. • JH(xi)(¯u)KG = HG(ui) for the ith node’s features. • JF(τ1, . . . , τk)(¯u)KG = F(Jτ1(¯u)KG, . . . , Jτk(¯u)KG) for any function symbol F. • For any term composed using ⋆: u v X y∈N(xi) τ ⋆h(π)(¯u) } ~ G =      P v∈N (ui) Jτ(¯u,v)KGh(Jπ(¯u,v)KG) P v∈N (ui) h(Jπ(¯u,v)KG) if N(ui) ̸= ∅; 0 if N(ui) = ∅. The semantics of global weighted mean is defined analogously and omitted for brevity. 4.0 2.0 0.0 -3.0 1.0 1.0 1.0 1.0 10.0 6.0 2.0 -4.0 4.0 6.0 4.0 2.0 Input graph Figure 2: Evaluation of the term P x∈N(y)(2H(x) + 2) ⋆1.0 on a small graph with scalar features. The term computes the mean of z 7→2z + 2 on each of a node’s neighbours. As each sub-term has one free variable, we can represent the intermediate results as scalar values for each node. A closed term has all node variables bound by a weighted mean operator: so the implicit input is just a featured graph. Definition 4.3. We augment the term language to AGG[WMEAN, RW] by adding the random walk operator rw(x). The interpretation of rw(xi) given a graph G and a tuple of nodes ¯u, is: Jrw(xi)(¯u)KG = [rwui,1, . . . , rwui,d] 5 4.1 How powerful is the term language? Various architectures can be described using this term language. The core idea is always the same: we show that all basic building blocks of the architecture can be captured in the term language and applying this inductively yields the desired result. Let us first note that all linear functions and all commonly used activation functions are Lipschitz continuous, and therefore included in the language. MPNNs with mean aggregation. Consider an L-layer MPNN with mean aggregation, update functions UPD(ℓ) consisting of an activation function applied to a linear transformation, with mean pooling at the end. First, note that mean aggregation can be expressed as: MEANyπ(y) := X y π(y) ⋆1. For each layer 0 ≤ℓ< L, we define a term τ (ℓ)(x) which will compute the representation of a node at layer ℓof the MPNN: • Initialization. Let τ (0)(x) := H(x). Then the value Jτ (0)(x)(u)KG at a node u is the initial node representation H(u). • Layers. For 1 ≤ℓ< L, define τ (ℓ+1)(x) := UPD(ℓ) x, MEANy∈N(x)τ (ℓ)(y) . Then the value at node u is the following, which conforms with the inductive construction of the MPNN: Jτ (ℓ+1)(x)(u)KG = UPD(ℓ)  Jτ (ℓ)(x)(u)KG, 1 \|N(u)\| X v∈N(u) Jτ (ℓ)(x)(v)KG   • Final mean pooling. The final graph representation is computed as τ := MEANxτ (L)(x). The idea is similar for the other architectures, where the difference lies in the aggregation functions. Thus below we only present how the term language captures their respective aggregation functions. Graph transformers. We can express the self-attention mechanism of transformers using the following aggregator: ATTyπ(y) := X y π(y) ⋆exp(scale(π(x), π(y))) The function scale(π(x), π(y)) is a term in the language, since scaled dot product attention is a Lipschitz function. To see how graph transformer architectures such as GPS can be expressed, it suffices to note that we can express both self-attention layers and MPNN layers with mean aggregation, since the term language is closed under addition. The random walk positional encoding can also be expressed using the rw operator. Graph attention networks. The attention mechanism of GAT is local to a node’s neighbours and can be expressed in our term language using similar ideas, except using the local aggregate terms. Additional architectural features. Because the term language allows the arbitrary combination of graph operations, it can robustly capture many common architectural choices used in graph learning architectures. For example, a skip connection or residual connection from layer ℓ1 to layer ℓ2 can be expressed by including a copy of the term for layer ℓ1 in the term for the layer ℓ2 [51]. Global readout can be captured using a global mean aggregation [5]. Attention conditioned on computed node or node-pair representations can be captured by including the term which computes these representations in the mean weight [37]. Capturing probabilistic classification. Our term language defines bounded vector-valued functions over graphs. Standard normalization functions, like softmax and sigmoid, are easily expressible in our term language, so probabilistic classifiers are subsumed. Graph convolutional networks. In Appendix A we show how to extend the term language to incorporate graph convolutional networks (GCNs) [29] by adding a new aggregator. 5 Convergence theorems We start by presenting the convergence theorem. 6 Theorem 5.1. Consider (µn)n∈N sampling a graph G from any of the following models and node features independently from i.i.d. bounded distributions on d features. 1. The Erd˝os-R´enyi distribution ER(n, p(n)) where p satisfies any of the following properties. • Density. p converges to ˜p > 0. • Root growth. For some K > 0 and 0 < β < 1 we have: p(n) = Kn−β. • Logarithmic growth. For some K > 0 we have: p(n) = K log(n) n . • Sparsity. For some K > 0 we have: p(n) = Kn−1. 2. The Barab´asi-Albert model BA(n, m) for any m ≥1. 3. The stochastic block model SBM(n1(n), . . . , nm(n), P ) where n1, . . . , nm : N →N are such that n1(n) + · · · + nm(n) = n and each ni n converges, and P is any symmetric m × m edge probability matrix. Then every AGG[WMEAN, RW] term converges a.a.s. to a constant with respect to (µn)n∈N.3 Concretely, this result shows that for any probabilistic classifier, or other real-valued classifier, which can be expressed within the term language, when drawing graphs from any of these distributions, eventually the output of the classifier will be the same regardless of the input graph, asymptotically almost surely. Thus, the only probabilistic classifiers which can be expressed by such models are those which are asymptotically constant. Corollary 5.2. For any of the random graph featured models above, for any MeanGNN, GAT, or GPS + RW, there is a distribution ¯p on the classes such that the class probabilities converge asymptotically almost surely to ¯p. We now discuss briefly how the results are proven. The cases divide into two groups: the denser cases (the first three ER distributions and the SBM) and the sparser cases (the fourth ER distribution and the BA model). Each is proved with a different strategy. 5.1 Overview of the technical constructions for the denser cases While the theorem is about closed terms, naturally we need to prove it inductively on the term language, which requires consideration of terms with free variables. We show that each open term in some sense degenerates to a Lipschitz function almost surely. The only caveat is that we may need to distinguish based on the “type” of the node – for example, nodes u1, u2, u3 that form a triangle may require a different function from nodes that do not. Formally, for node variables ¯x, an ¯x graph type is a conjunction of expressions E(xi, xj) and their negations. The graph type of tuple ¯u in a graph, denoted GrTp(¯u) is the set of all edge relations and their negations that hold between elements of ¯u. A (¯x, d) feature-type controller is a Lipschitz function taking as input pairs consisting of d-dimensional real vector and an ¯x graph type. The key theorem below shows that to each term π we can associate a feature-type controller eπ which captures the asymptotic behaviour of π, in the sense that with high probability, for most of the tuples ¯u the value of eπ(HG(¯u), GrTp(¯u)) is close to Jπ(¯u)K. Theorem 5.3 (Aggregate Elimination for Non-Sparse Graphs). For all terms π(¯x) over featured graphs with d features, there is a (¯x, d) feature-type controller eπ such that for every ϵ, δ, θ > 0, there is N ∈N such that for all n ≥N, with probability at least 1 −θ in the space of graphs of size n, out of all the tuples ¯u at least 1 −δ satisfy that ∥eπ(HG(¯u), GrTp(¯u)) −Jπ(¯u)K∥< ϵ. This can be seen as a kind of “almost sure quantifier elimination” (thinking of aggregates as quantifiers), in the spirit of Kaila [24], Keisler and Lotfallah [26]. It is proven by induction on term depth, with the log neighbourhood bound playing a critical role in the induction step for weighted mean. Theorem 5.3 highlights an advantage of working with a term language having nice closure properties, rather than directly with GNNs: it allows us to use induction on term construction, which may be more natural and more powerful than induction on layers. Theorem 5.3 also gives strong limitations on node and link classification using GNNs: on most nodes in (non-sparse) random graphs, GNNs can only classify based on the features of a node, they cannot make use of any graph structure. 3The appendix includes additional results for the GCN aggregator. 7 5.2 Overview of the technical constructions for the sparser cases In the sparser cases, the analysis is a bit more involved. Instead of graph types over ¯u, which only specify graph relations among the ¯u, we require descriptions of local neighbourhoods of ¯u. Definition 5.4. Let G be a graph, ¯u a tuple of nodes in G and ℓ∈N. The ℓ-neighbourhood of ¯u in G, denoted N ℓ(¯u) is the subgraph of G induced by the nodes of distance at most ℓfrom some node in ¯u. The “types” are now graphs T with k distinguished elements ¯w, which we call k-rooted graphs. Two k-rooted graphs (T, ¯w) and (U, ¯z) are isomorphic is there is a structure-preserving bijection T →U which maps ¯w to ¯z. The combinatorial tool here is a fact known in the literature as ‘weak local convergence’: the percentage of local neighbourhoods of any given type converges. Lemma 5.5 (Weak local convergence). Consider sampling a graph G from either the sparse ER or BA distributions. Let (T, ¯w) be a k-rooted graph and take ℓ∈N. There is qT ∈[0, 1] such that for all ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ we have that: \|{¯u \| N ℓ(¯u) ∼= (T, ¯w)}\| n −qT < ϵ Our “aggregate elimination”, analogous to Theorem 5.3, states that every term can be approximated by a Lipschitz function of the features and the neighbourhood type. Theorem 5.6 (Aggregate Elimination for Sparser Graphs). For every π(¯x), letting ℓbe the maximum aggregator nesting depth in π, for all k-rooted graphs (T, ¯w) there is a Lipschitz function eT π : R\|T \|·d →Rd such that for each ϵ, θ > 0, there is N ∈N such that for all n ≥N with probability at least 1 −θ in the space of graphs of size n, for every k-tuple of nodes ¯u in the graph such that N ℓ(¯u) ∼= (T, ¯w) we have that eT π (HG(N ℓ(¯u))) −Jπ(¯u)K < ϵ. Compared to Theorem 5.3 the result is much less limiting in what node-classifying GNNs can express. Although combining the sparse and non-sparse conditions covers many possible growth rates, it is not true that one gets convergence for Erd˝os-R´enyi with arbitrary growth functions: Theorem 5.7. There are functions p(n) converging to zero and a term τ in our language such that τ does not converge even in distribution (and hence does not converge a.a.s.) over ER random graphs with growth rate p. Proof. Let p alternate between 1 2 on even n and 1 n on odd n. Consider τ1(x) that returns 0 if x has a neighbour and 1 otherwise, and let τ be the global average of τ1. So τ is the percentage of isolated nodes in the graph. Then τ clearly goes to zero in the non-sparse case. However the probability that a particular node is not isolated in a random sparse graph of size n is (1 −1 n)n−1, which goes to 1 e < 1. Thus JτK diverges on ER(n, p(n)). 6 Experimental evaluation We first empirically verify our findings on random graphs and then on a real-world graph to answer the following questions: Q1. Do we empirically observe convergence? Q2. What is the impact of the different weighted mean aggregations on the convergence? Q3. What is the impact of the graph distribution on the convergence? Q4. Can these phenomena arise within large real-world graphs? All our experiments were run on a single NVidia GTX V100 GPU. We made our codebase available online at https://github.com/benfinkelshtein/GNN-Asymptotically-Constant. Setup. We report experiments for the architectures MeanGNN, GAT [49], and GPS+RW [41] with random walks of length up to 5. Our setup is carefully designed to eliminate confounding factors: • We consider five models with the same architecture, each having randomly initialized weights, utilizing a ReLU non-linearity, and applying a softmax function to their outputs. Each model uses a hidden dimension of 128, 3 layers and an output dimension of 5. • We experiment with distributions ER(n, p(n) = 0.1), ER(n, p(n) = log n n ), ER(n, p(n) = 1 50n), BA(n, m = 5). We also experiment with an SBM of 10 communities with equal size, where an edge between nodes within the same community is included with probability 0.7 and an edge between nodes of different communities is included with probability 0.1. 8 • We draw graphs of sizes up to 10,000, where we take 100 samples of each graph size. Node features are independently drawn from U[0, 1] and the initial feature dimension is 128. To understand the behaviour of the respective models, we will draw larger and larger graphs from the graph distributions. We use five different models to ensure this is not a model-specific behaviour. Further experimental results are reported in Appendix F. 6.1 Empirical results on random graph models In Figure 3, a single model initialization of the MeanGNN, GAT and GPS+RW architectures is used with ER(n, p(n) = 0.1), ER(n, p(n) = log n n ) and ER(n, p(n) = 1 50n). Each curve in the plots corresponds to a different class probability, depicting the average of 100 samples for each graph size along with the standard deviation shown in lower opacity. The convergence of class probabilities is apparent across all models and graph distributions, as illustrated in Figure 3, in accordance with our main theorems (Q1). The key differences between the plots are the convergence time, the standard deviation and the converged values. One striking feature of Figure 3 is that the eventual constant output of each model is the same for the dense and logarithmic growth distributions, but not for the sparse distribution (Q3). We can relate this to the distinct proof strategy employed in the sparse case, which uses convergence of the proportions of local isomorphism types. There are many local isomorphism types, and the experiments show that what we converge to depends the proportion of these. In all other cases the neighbourhood sizes are unbounded, so there is asymptotically almost surely one ‘local graph type’. We observe that attention-based models such as GAT and GPS+RW exhibit delayed convergence and greater standard deviation in comparison to MeanGNN (Q2). A possible explanation is that because some nodes are weighted more than others, the attention aggregation has a higher variance than regular mean aggregation. For instance, if half of the nodes have weights close to 0, then the (a) Dense ER (b) Logarithmic growth ER (c) Sparse ER Figure 3: Each plot shows the five mean class probabilities (in different colours) with standard deviations of a single model initialization over ER(n, p(n) = 0.1), ER(n, p(n) = log n n ), and ER(n, p(n) = 1 50n), as we draw graphs of increasing size. 9 Figure 4: Each plot depicts the standard deviation of Euclidean distances between class probabilities and their respective means across various samples of each graph size for GPS+RW. attention aggregation effectively takes a mean over half of the available nodes. Eventually, however, the attention weights themselves converge, and thus convergence cannot be postponed indefinitely. Figure 4 depicts the outcomes of the GPS+RW architecture for various graph distributions. The analysis involves calculating the Euclidean distance between the class probabilities of the GPS+RW architecture and the mean class probabilities over the different graph samples. The standard deviation across the different graph samples is then derived for each of the 5 different model initializations and presented in Figure 4. The decrease of standard deviation across the different model initializations in Figure 4 indicates that all class probabilities converge across the different model initializations, empirically verifying the phenomenon for varying model initializations (Q1 and Q3). 6.2 Empirical results on large real-world graphs Figure 5: Standard deviation of distances between class probabilities and their means across TIGER-Alaska graph sizes for MeanGNN. Towards Q4, we investigated a large real-world dataset. Many commonly studied graph datasets (e.g. ZINC [13], QM9 [6]) do not exhibit sufficient graph size variance and provide no obvious means to add scaling. We used the TIGER-Alaska dataset [16] of geographic faces. The original dataset has 93366 nodes, while Dimitrov et al. [10] extracted smaller datasets with graphs having 1K, 5K, 10K, 25K and 90K nodes. We chose the modified dataset as it is split by graph size, and consists of graphs differing from our random models (in particular all graphs are planar). Figure 5 shows the results of applying the same five MeanGNNs to graphs of increasing sizes. Strikingly, we again observe a convergence phenomenon, but at a slower pace. 7 Discussion and limitations We have demonstrated a wide convergence phenomenon for real-valued classifiers expressed even in very advanced GNN architectures, and it applies to a great variety of random graph models. Rather than having separate proof techniques per GNN model, our paper introduces a broad language where such models can be situated, and provides techniques at the level of term languages. Although our top-level theorems deal with graph-level tasks, along the way we provide strong limitative results on what can be achieved on random graphs for node- or edge-level real-valued tasks: see Theorem 5.3. The principal limitations of our work come our assumptions. In particular, we assume that the initial node embeddings are i.i.d. This assumption is used in the application concentration inequalities throughout the proofs, so loosening it would require careful consideration. Our main results show that many GNN architectures cannot distinguish large graphs. To overcome this limitation, one could consider moving beyond our term language. For example, if we add sum aggregation, the term values clearly diverge, and similarly if we allow non-smooth functions, such as linear inequalities. Further we emphasize that a.a.s. convergence is not universal for our term language, and it does not hold even for ER with arbitrary p(n) going to 0: see Theorem 5.7. 10 References [1] R. 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Bioinformatics, 34(13):i457–i466, 2018. 13 A Graph Convolutional Networks In the body of the paper, we mentioned that our results can be extended to graph convolutional networks (GCNs) [29]. In this section we show how to extend our term language to capture them. Later in the appendix when providing proof details for our convergence laws, we will utilize the extended language, thus showing the applicability to GCNs. GCNs are instances of the message passing neural network in which the update function is defined as follows. h(ℓ+1) v = X u∈N(v) 1 p \|N(v)\|\|N(u)\| W (ℓ)h(ℓ) u To allow our term language to capture GCNs, we add the new aggregator: GCNy∈N(x)τ(y) The language AGG[WMEAN, GCN, RW] is the result of closing AGG[WMEAN, RW] under this operator. The semantics are extended as follows. q GCNy∈N(xi)τ(¯u) y G = X v∈N(ui) 1 p \|N(ui)\|\|N(v)\| Jτ(¯u, v)KG With this operator it is possible to capture GCNs in the same way as MeanGNNs are captured (Section 4). Moreover, the extended term language permits arbitrary combinations of the GCN aggregator with other language features. B Concentration Inequalities Throughout the appendix, we make use of a few basic inequalities. Theorem B.1 (Markov’s Inequality). Let X be a positive random variable with finite mean. Then for any λ > 0 we have: P(X ≥λ) ≤E[X] λ Proof. See Proposition 1.2.4, p. 8 of [50]. Theorem B.2 (Chebyshev’s Inequality). Let X be a random variable with finite mean µ and finite variance σ2. Then for any λ > 0 we have: P (\|X −µ\| ≥λ) ≤σ2 λ2 Proof. See Corollary 1.2.5, p. 8 of [50]. Theorem B.3 (Chernoff’s Inequality). Let Xi for i ≤n be i.i.d. Bernoulli random variables with parameter p. (1) For any λ > p we have: P n X i=1 Xi ≥λ ! ≤e−np enp λ λ (2) For any λ < p we have: P n X i=1 Xi ≤λ ! ≤e−np enp λ λ Proof. See Theorem 2.3.1, p. 17 of [50] and Exercise 2.3.2, p. 18 of [50]. 14 Theorem B.4 (Hoeffding’s Inequality for bounded random variables). Let Xi for i ≤n be i.i.d. bounded random variables taking values in [a, b] with common mean µ. Then for any λ > 0 we have: P n X i=1 Xi −nµ ≥λ ! ≤2 exp − 2λ2 n(b −a)2 Proof. See Theorem 2.2.6, p. 16 of [50]. Corollary B.5 (Hoeffding’s Inequality for Bernoulli random variables). Let Xi for i ≤n be i.i.d. Bernoulli random variables with parameter p. Then for any λ > 0 we have: P n X i=1 Xi −np ≥λ ! ≤2 exp −2λ2 n C Proof for Erd˝os-R´enyi distributions, the non-sparse cases We new begin with the proof of convergence for the first three cases of the Erd˝os-R´enyi distribution in Theorem 5.1, which we restate here for convenience: Theorem C.1. Consider (µn)n∈N sampling a graph G from the Erd˝os-R´enyi distribution ER(n, p(n)) and node features independently from i.i.d. bounded distributions on d features, where p satisfies any of the following properties. • Density. p converges to ˜p > 0. • Root growth. For some K > 0 and 0 < β < 1 we have: p(n) = Kn−β. • Logarithmic growth. For some K > 0 we have: p(n) = K log(n) n . Then for every AGG[WMEAN, RW] term converges a.a.s. with respect to (µn)n∈N. Note that throughout the following we use u and v to denote both nodes and node variables. C.1 Combinatorial and growth lemmas Remark C.2. At various points in the following we will make statements concerning particular nodes of an Erd˝os-R´enyi graph without fixing a graph. For example, we state that for any u we have that its degree d(u) is the sum of n −1 Bernoulli random variables. To make sense of such statements we can fix a canonical enumeration of the nodes of every graph and view our node meta-variables as ranging over indices. So we would translate the previous statement as “for any node index u the degree of the uth node in ER(n, p(n)) is the sum of n −1 Bernoulli random variables.” Both the combinatorial and growth lemmas, and the main inductive results themselves require saying that certain events hold with high probability. Inductively, we will need strengthenings of these compared with the statements given in the body. We will need to consider conditional probabilities, where the conditioning is on a conjunction of atomic statements about the nodes. Definition C.3. A ∧-description ϕ(¯u) is a conjunction of statements about the variables ¯u of the form uiEuj, which expresses that there exists an edge between ui and uj. A ∧-description ui1Euj1 ∧· · · ∧uimEujm holds if and only if there is an edge between uiℓand ujℓfor each ℓ. The reason for strengthening the results in this way will become clearer when we are proving the local weighted mean induction step P v∈N(ui) ρ(¯u, v) ⋆h(η(¯u, v)). There we will need our auxiliary results to hold conditioned on vEui, along with any additional requirements coming from previous induction steps. We will first need a few auxiliary lemmas about the behaviour of random graphs in the different ER models. We will need to know that in the non-sparse case, neighborhoods are fairly big: Lemma C.4 (Log neighbourhood bound). Let p: N →[0, 1] satisfy density, root growth or logarithmic growth. There is R > 0 such that for all δ, θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), we have that the proportion of all nodes u such that \|N(u)\| ≥R log(n) is at least 1 −δ. 15 We prove a stronger result, which allows us finer control over the growth in each of the non-sparse cases. The strengthening will involve the ∧-descriptions of Definition C.3. Lemma C.5. Take any ∧-description ϕ on k variables, and choose i ≤k. We consider k-tuples of nodes ¯u satisfying ϕ and the degrees of the ith node ui. (1) Let p satisfy the density condition, converging to ˜p. Then for every ϵ, θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for all tuples ¯u which satisfy ϕ we have that: n(˜p −ϵ) < d(ui) < n(˜p + ϵ) (2) Let p satisfy the root growth condition, with p = Kn−β. Then for every R1 ∈(0, 1) and R2 > 1 and for every θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for all tuples ¯u which satisfy ϕ we have that: R1Kn1−β < d(ui) < R2Kn1−β (3) Let p satisfy the log growth condition, with p = K log(n) n . Then for every R1 ∈(0, 1) and R2 > 1 and for every δ, θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for at least 1 −δ of the tuples ¯u which satisfy ϕ we have that: R1K log(n) < d(ui) < R2K log(n) Moreover, there is R3 > 0 such that for every θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for all tuples ¯u which satisfy ϕ we have that: d(ui) < R3K log(n) Note that in the first two cases, we only have claims about all tuples, while in the third case we make an assertion about most tuples, while also asserting an upper bound on all tuples. The upper bound on all tuples does not subsume the upper bound on most tuples, since the former states the existence of an R3, while the latter gives a bound for an arbitrary R2. We now give the proof of the lemma: Proof. (1) Take ϵ, θ > 0. First, there is N1 such that for all n ≥N1: \|p(n) −˜p\| < ϵ Now, any d(u) is the sum of n −1 i.i.d. Bernoulli random variables with parameter p(n): see Remark C.2 for the formalization. Hence by Hoeffding’s Inequality (Theorem B.4) and a union bound, there is N2 such that for all n ≥N2, with probability at least 1 −θ, for every node u we have: d(u) n −˜p < ϵ Letting N = max(N1, N2), the result follows. (2) Take R1 ∈(0, 1) and R2 > 1 and fix θ > 0. Again, any d(u) is the sum of n −1 i.i.d. Bernoulli random variables with parameter p(n). By Chernoff’s Inequality (Theorem B.3), for any node u we have that: P d(u) ≤R1Kn1−β ≤exp Kn1−β (R1 −1 −R1 log(R1)) Since R1 −1 −R1 log(R1) < 0, by a union bound there is N1 such that for all n ≥N1, with probability at least 1 −θ, for every node u we have: d(u) > R1Kn1−β Similarly there is N2 such that for all n ≥N2, with probability at least 1 −θ, for every node u we have: d(u) < R2Kn1−β Letting N = max(N1, N2), the result follows. 16 (3) Take R1 ∈(0, 1) and R2 > 1 and fix δ, θ > 0. By Chernoff’s Inequality (Theorem B.3), for any node u we have that: P d(u) ≤R1Kn1−β ≤nK(R1−1−R1 log(R1)) Since R1 −1 −R1 log(R1) < 0 this probability tends to 0 as n tends to infinity. Hence there is N1 such that for all n ≥N1, for all nodes u: P (d(u) ≤R1 log(n)) < θδ Let Bn be the proportion, our of tuples ¯u which satisfy ϕ, such that: d(ui) ≤R log(n) (the ‘bad’ tuples). Then for n ≥N1, by linearity of expectation E[Bn] ≤θδ. By Markov’s Inequality (Theorem B.1) we have that: P (Bn ≥δ) ≤θδ δ = θ That is, with probability at least 1 −θ we have that the proportionout of tuples ¯u which satisfy ϕ such that: d(ui) > R1 log(n) is at least 1 −δ. Similarly, there is N2 such that for all n ≥N2, with probability at least 1 −θ we have that the proportion out of tuples ¯u which satisfy ϕ such that: d(ui) < R2 log(n) is at least 1 −δ. Letting N = max(N1, N2) the first statement follows. To show the ‘moreover’ part, note that by Chernoff’s Inequality (Theorem B.3), for any R3 > 0 and for any node u we have that: P d(u) ≥R3Kn1−β ≤nK(R3−1−R3 log(R3)) For R3 large enough we have that K(R3 −1 −R3 log(R3)) ≤−2. Hence taking a union bound for every θ > 0 there is N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for all tuples ¯u which satisfy ϕ we have that: d(ui) < R3K log(n) The basic form of the following “regularity lemma” states that for every k ∈N+ and i ≤k, whenever we have a large subset S of the (k + 1)-tuples (¯u, v) with each ui adjacent to v, there is a large subset of the k-tuples ¯u such that each ui is adjacent to a large set of v’s such that (¯u, v) ∈S. The full regularity lemma states that this is the case also when we restrict the tuples ¯u to a certain ∧-description. Lemma C.6 (Regularity Lemma). Let p: N →[0, 1] satisfy one of the conditions other than sparsity. Take a ∧-description ϕ on k variables and fix i ≤k. Then for every γ, θ > 0 there is N ∈N and δ′ > 0 such that for all n ≥N and δ < δ′, with probability at least 1 −θ we have that, whenever S ⊆{(¯u, v) \| ϕ(¯u) ∧uiEv} is such that: \|S\| \|{(¯u, v) \| ϕ(¯u) ∧uiEv}\| ≥1 −δ then there is S′ ⊆{¯u \| ϕ(¯u) and d(ui) > 0} such that: \|S′\| \|{¯u \| ϕ(¯u) and d(ui) > 0}\| ≥1 −γ and for every ¯u ∈S′ we have: \|{v \| (¯u, v) ∈S}\| d(ui) ≥1 −(δ + δ2) 17 For this we make use of the following purely combinatorial lemma. Lemma C.7. Take δ, σ > 0. Let (B1, . . . , Bm) be a sequence of disjoint finite sets of minimum size a and maximum size b. Take S ⊆Sm i=1 Bi such that: \|S\| Pm i=1\|Bi\| ≥1 −δ Let: S′ := i \|S ∩Bi\| \|Bi\| ≥1 −σ Then: \|S′\| m ≥1 − (σ −δ)a (σ −δ)a + δb Proof. Let ¯B := (B1, . . . , Bm). We look for an upper bound on: γ ¯ B,S := 1 −\|S′\| m To do this, take any ( ¯B, S) with proportion at least 1 −δ which minimises γ ¯ B,S. By performing a series of transformations, we show that we can assume that ( ¯B, S) is of a particular form. • First, ensuring that S ∩Bi = Bi for each i ∈S′ does not increase γ ¯ B,S and does not decrease the proportion \|S\| Pm i=1\|Bi\|. • We can also make sure that \|Bi\| = b for each i ∈S′. • Similarly, we can ensure that: \|S ∩Bi\| \|Bi\| = ⌊1 −σ⌋ for each i /∈S′. • Finally, we can make sure that \|Bi\| = a for each i /∈S′. Now, using this nice form for ( ¯B, S), we can compute: 1 −δ ≤ \|S\| Pm i=1\|Bi\| ≤γ ¯ B,Sb + (1 −γ ¯ B,S)(1 −σ)a γ ¯ B,Sb + (1 −γ ¯ B,S)a Rearranging we get: \|S′\| m = 1 −γ ¯ B,S ≥1 − (σ −δ)a (σ −δ)a + δb as required. Proof of Lemma C.6. We proceed differently depending on which of the non-sparse conditions are satisfied. We begin with the Density condition. Say p converges to some ˜p > 0. Fix γ, θ > 0. Now choose ϵ, δ′ > 0 small enough. We will decide how small later, but we need at least ϵ < ˜p. By Lemma C.5 (1) there is N such that for all n ≥N, with probability at least 1 −θ, for all tuples ¯u which satisfy ϕ(¯u) we have that: n(˜p −ϵ) < d(ui) < n(˜p + ϵ) Condition on the event that this holds. Take any δ < δ′ and S ⊆{(¯u, v) \| ϕ(¯u) ∧uiEv} such that: \|S\| \|{(¯u, v) \| ϕ(¯u) ∧uiEv}\| ≥1 −δ 18 Let: S′ := ¯u ϕ(¯u) and \|{v \| (¯u, v) ∈S}\| d(ui) ≥1 −(δ + δ2) Applying Lemma C.7 with σ = δ + δ2, a = n(˜p −ϵ) and n = n(˜p + ϵ), we get that: \|S′\| \|{¯u \| ϕ(¯u) and d(ui) > 0}\| ≥1 − (δ + δ2 −δ)n(˜p −ϵ) (δ + δ2 −δ)n(˜p −ϵ) + δn(˜p + ϵ) = 1 − δ(˜p −ϵ) δ(˜p −ϵ) + (˜p + ϵ) By making ϵ and δ′ small enough, we can make this greater than 1 −γ. This completes the proof for the dense case. We now give the proof for the case of Root growth. Let p(n) = Kn−β. Fix γ, θ > 0. Choose δ′ > 0 small enough. Also choose R1 ∈(0, 1) and R2 > 1. By Lemma C.5 (2) there is N ∈N such that for all n ≥N, with probability at least 1 −θ, for all tuples ¯u which satisfy ϕ(¯u) we have that: R1Kn1−β < d(ui) < R2Kn1−β Proceeding analogously to the dense case, we have that for all δ < δ′: \|S′\| \|{¯u \| ϕ(¯u) and d(ui) > 0}\| ≥1 − δR1Kn1−β δnR1Kn1−β + R2Kn1−β = 1 − δR1 δR1 + R2 By making δ′ and R2 −R1 small enough, we can make this greater than 1 −γ. This completes the proof for the root growth case. Finally, we give the argument for the case of Logarithmic growth. Let p(n) = K log(n) n . Fix γ, θ > 0. Choose ζ, δ′ > 0 small enough. Also choose R1 ∈(0, 1) and R2 > 1. By Lemma C.5 (3), there is R3 > 0 and N ∈N such that for all n ≥N, with probability at least 1 −θ when drawing graphs from ER(n, p(n)), for at least 1 −δ of the tuples ¯u which satisfy ϕ(¯u) we have that: R1K log(n) < d(ui) < R2K log(n) and moreover for all tuples ¯u which satisfy ϕ(¯u) we have that: d(ui) < R3K log(n) Now, let: Q := {¯u \| R1K log(n) < d(ui) < R2K log(n)} For n large enough we have both: \|Q\| \|{¯u \| ϕ(¯u) and d(ui) > 0}\| ≥1 −ζ and (using the fact that outside of Q all nodes ui have degree at most R3K log(n)): \|{(¯u, v) \| ϕ(¯u) ∧uiEv and ¯u ∈Q}\| \|{(¯u, v) \| ϕ(¯u) ∧uiEv}\| ≥1 −ζ Since we have control over ζ, it suffices to restrict attention to: S ⊆{(¯u, v) \| ϕ(¯u) ∧uiEv and ¯u ∈Q} Then, analogously to the dense case, we have that for the worst case, for every δ < δ′: \|S′\| \|{¯u \| ϕ(¯u) and d(ui) > 0}\| ≥1 − δR1 δR1 + R2 By making δ′ and R2 −R1 small enough, we can make this greater than 1 −γ. This completes the proof of Lemma C.6 for the logarithmic growth case. 19 The following lemma will be needed only in the analysis of random walk embeddings. It gives the asymptotic behaviour of the random walk embeddings in the non-sparse cases. We see that the embeddings are almost surely zero. Lemma C.8. Let p: N →N satisfy density, root growth or logarithmic growth. Then for every k ∈N and every ϵ, δ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ, the proportion of nodes v such that: ∥rwk(v)∥< ϵ is at least 1 −δ. Proof. We start with the Dense case. Let p converge to ˜p > 0. Take ϵ, δ, θ > 0. By Hoeffding’s Inequality (Theorem B.4) and taking a union bound, there is N such that for all n ≥N with probability at least 1 −θ we have that: \|d(v) −˜pn\| < ϵn Condition on this event. For any node v the number of length-k walks from v is at least: ((˜p −ϵ)n)k By removing the last node of the walk, the number of length-k walks from v to itself is at most the number of length-(k −1) walks from v. The number of length-(k −1) walks from v is at most: ((˜p + ϵ)n)k−1 Thus the proportion of length-k walks from v which return to v is at most: ((˜p + ϵ)n)k−1 ((˜p −ϵ)n)k = (˜p + ϵ)k−1 (˜p −ϵ)k n−1 This tends to 0 an n tends to infinity. We now argue this for the Root growth case. Let p = Kn−β. As in the proof of Lemma C.6, by Chernoff’s Inequality (Theorem B.3) and taking a union bound, there are 0 < R1 < 1 < R2 and N such that for all n ≥N with probability at least 1 −θ we have that for all v: R1Kn1−β < d(v) < R2Kn1−β Then, as in the dense case, the proportion of length-k walks from v which return to v is at most: R2Kn1−βk−1 (R1Kn1−β)k = Rk−1 2 Rk 1K nβ−1 which tends to 0 as n tends to infinity. Finally, we argue this for the Logarithmic growth case. Let p = K log(n). Take ϵ, δ, θ > 0. As in the proof of Lemma C.6, by Chernoff’s Inequality (Theorem B.3), there are 0 < R1 < 1 < R2 and N such that for all n ≥N1 with probability at least 1−θ we have that for at least 1−δ proportion of nodes v: R1K log(n) < d(v) < R2K log(n) (⋆) Moreover, by Chernoff’s Inequality and a union bound, there is R3 > 0 such that for all n ≥N2 with probability at least 1 −θ we have that for all v: d(v) < R3K log(n) Let N := max(N1, N2). Take n ≥N. We will condition on the event that the above inequalities hold. 20 Take any node v such that equation (⋆) holds. Then, the number of length-k walks from v is at least: R1K log(n)((1 −δ)(R1K log(n)))k−1 The number of length-k walks from v to itself is at most: (R3K log(n))k−1 Thus the proportion of length-k walks from v which return to v is at most: (R3K log(n))k−1 R1K log(n)((1 −δ)(R1K log(n)))k−1 = Rk−1 3 Rk 1(1 −δ)k−1 · 1 log(n) This tends to 0 as n tends to infinity. Finally, we need a simple lemma about division, which has a straightforward proof: Lemma C.9. Let x, y, z, w ∈R and ζ, ξ, ν, Ω> 0 with ν > ξ be such that \|x −y\| < ζ and \|z −w\| < ξ while \|x\|, \|z\| < Ωand z > ν. Then: x z −y w < Ω(ζ + ξ) ν(ν −ξ) C.2 Proving the inductive invariant for the non-sparse cases, and proving the convergence theorem for these cases With the growth lemmas for the non-sparse cases in hand, we return to presenting the main proof of Theorem C.1, the convergence theorem for non-sparse ER distributions. Throughout the following subsection, for notational convenience we allow empty tuples. For a tuple ¯u in a graph let GrTp(¯u) be the graph type of ¯u. For k ∈N let GrTpk be the set of such types with k free variables. For any t ∈GrTpk let: Ext(t) := {t′(¯u, v) ∈GrTpk+1 \| t′(¯u, v) ⊨t(¯u)} For any ui free in t let: Extui(t) := {t′(¯u, v) ∈GrTpk+1 \| t′(¯u, v) ⊨t(¯u) ∧uiEv} We now are ready to begin the proof of Theorem 5.1 for the non-sparse cases. This will involve defining controllers and proving that they satisfy the inductive invariant. Let F be the probability distribution of the node features, which, for the sake of convenience, we assume to have domain [0, 1]d. The domain [0, 1]d can be replaced by the more general domain [a, b]d in the results and proofs without further modification. But we note that the domain [0, 1]d is already sufficient for the results to hold in the general case. Indeed, suppose τ(¯x) is a term which we apply to a graph distribution D with features in [a, b]d. Modify this distribution to D′ by applying the function ¯z 7→(¯z −a)/(b −a) to the features. This is now a distribution with features in [0, 1]d. Modify τ to τ ′ by replacing each H(x) by F(H(x)), where F is the function ¯z 7→(b −a)¯z + a. Then evaluating τ ′ on D′ is equivalent to evaluating τ on D. Note that in each case the probability function p(n) converges to some ˜p (which is 0 in the root growth and logarithmic growth cases). Recall from the body of the paper that a (¯u, d) feature-type controller is a Lipschitz function taking as input pairs consisting of a d-dimensional real vector and a ¯u graph type. Recall from Theorem 5.3 that for each term π, we need to define a controller eπ that approximates it. We first give the construction of eπ and then recall and verify what it means for eπ to approximate π. Take ¯a ∈([0, 1]d)k and t ∈GrTpk, where k is the number of free variables in π. When π = H(u) define: eπ(¯a, t) := ¯a 21 When π = c, a constant, define: eπ(¯a, t) := c When π = rw(u) define: eπ(¯a, t) := 0 When π = f(ρ1, . . . , ρr) define: eπ(¯a, t) := f(eρ1(¯a, t), . . . , eρr(¯a, t)) We start with the construction for the global weighted mean. For any t(¯u) ∈GrTpk and t′(¯u, v) ∈ Ext(t), define α(t, t′) as follows. As an extension type, t′ specifies some number, say r, of edges between the nodes ¯u and the new node v. Define: α(t, t′) := ˜pr(1 −˜p)k−r Note that in both the root growth and logarithmic growth cases, α(t, t′) is non-zero precisely when t′ specifies no edges between ¯u and v. These relative atomic type weights will play a key role in the construction. Now consider a term π = P v ρ(¯u, v) ⋆h(η(¯u, v)). Recall that the semantics are: P v∈G Jρ(¯u, v)KGh(Jη(¯u, v)KG) P v∈G h(Jη(¯u, v)KG) Define: eπ(¯a, t) := fπ(¯a, t) gπ(¯a, t) where: fπ(¯a, t) := X t′∈Ext(t) α(t, t′) E b∼F [eρ(¯a, b, t′)h(eη(¯a, b, t′))] and: gπ(¯a, t) := X t′∈Ext(t) α(t, t′) E b∼F [h(eη(¯a, b, t′))] Note that when ˜p = 0 (as in the root growth and logarithmic growth cases), we have that: α(t, t′) = 1 if t′ specifies no edges between ¯u and v 0 otherwise Therefore the controller becomes, letting t∅be the type with no edges between ¯u and v: eπ(¯a, t) = Eb∼F [eρ(¯a, b, t∅)h(eη(¯a, b, t∅))] Eb∼F [h(eη(¯a, b, t∅))] For the local weighted mean case, given t(¯u) ∈GrTpk and t′(¯u, v) ∈Extui(t), we define αui(t, t′) as follows. Let r be the number of edges specified by t′ between ¯u \ ui and v. Define: αui(t, t′) := ˜pr(1 −˜p)k−r−1 Consider π = P v∈N(ui) ρ(¯u, v) ⋆h(η(¯u, v)). Define: eπ(¯a, t) := fπ(¯a, t) gπ(¯a, t) where: fπ(¯a, t) := X t′∈Extui(t) αui(t, t′) E b∼F [eρ(¯a, b, t′)h(eη(¯a, b, t′))] and: gπ(¯a, t) := X t′∈Extui(t) αui(t, t′) E b∼F [h(eη(¯a, b, t′))] With the controllers eπ defined, we now prove that they satisfy Theorem (Theorem 5.3). We state the strengthened version of this result here using the notation of ∧-decriptions from Definition C.3: 22 Lemma C.10. For every ϵ, δ, θ > 0 and ∧-description ψ(¯u), there is N ∈N such that for all n ≥N, with probability at least 1 −θ in the space of graphs of size n, out of all the tuples ¯u such that ψ(¯u), at least 1 −δ satisfy: ∥eπ(H(¯u), GrTp(¯u)) −Jπ(¯u)K∥< ϵ Proof. Naturally, we prove the lemma by induction. • Base cases H(v) and constant c. These are immediate from the definition. • Base case rw(v). This follows by Lemma C.8. • Induction step for Lipschitz functions f(ρ1, . . . , ρr). Take ϵ, δ, θ > 0 and ∧-description ψ(¯u). By the induction hypothesis there is N ∈N such that for all n ≥N and for every i ≤r, with probability at least 1 −θ, we have that out of all the tuples ¯u such that ψ(¯u), at least 1 −δ satisfy: ∥eρi(H(¯u), GrTp(¯u)) −Jρi(¯u)K∥< ϵ Hence, with probability at least 1 −rθ, out of all the tuples ¯u such that ψ(¯u), at least 1 −rδ satisfy this for every i ≤r. Condition on this event. Take any such tuple ¯u. Then by Lipschitzness of f we have that the normed distance between: eπ(H(¯u), GrTp(¯u)) = f(eρ1(H(¯u), GrTp(¯u)), . . . , eρr(H(¯u), GrTp(¯u))) and: Jπ(¯u)K = f(Jρ1(¯u)K, . . . , Jρr(¯u)K) is at most Lfϵ, where Lf is the Lipschitz constant of f. Since Lf is a constant, this case follows. • Inductive step for P v ρ(¯u, v) ⋆h(η(¯u, v)). Take ϵ, δ, θ > 0. Take a ∧-description ψ(¯u). By the induction hypothesis, there is N1 such that for all n ≥N1, with probability at least 1 −θ, out of all the tuples (¯u, v) which satisfy ψ(¯u), at least 1 −δ satisfy: ∥eρ(H(¯u, v), GrTp(¯u, v)) −Jρ(¯u, v)K∥< ϵ (†ρ) and: ∥eη(H(¯u, v), GrTp(¯u, v)) −Jη(¯u, v)K∥< ϵ (†η) Take γ > 0. We will choose how small γ is later. By Lemma C.6 there is Nγ > N1 such that for all n ≥Nγ, with probability at least 1 −θ, out of all the tuples ¯u such that ψ(¯u), at least 1 −γ, at least 1 −(δ + δ2) of the v’s satisfy equation (†ρ) and equation (†η). Now consider t(¯u) ∈GrTpk and take any t′(¯u, v) ∈Ext(t). For any tuple ¯u of nodes such that GrTp(¯u) = t, define: Jt′(¯u)K := {v \| t′(¯u, v)} Since p(n) converges to ˜p and Ext(t) is finite, there is N3 such that for all n ≥N3 with probability at least 1 −θ we have that for every pair (t, t′) and tuple ¯u of nodes such that t(¯u): \|Jt′(¯u)K\| n −α(t, t′) < 1 2k+1 ϵ (∗) Next, for any tuple ¯u consider the function: f ◦ π(¯u, v) := eρ(HG(¯u, v), GrTp(¯u, v))h(eη(HG(¯u, v), GrTp(¯u, v))) Note that the function: (¯a, b, t′) 7→eρ(¯a, b, t′)h(eη(¯a, b, t′)) is bounded. Let Λ be the diameter of its range. This also bounds f ◦ π. 23 We will now apply Hoeffding’s Inequality to P v\|t′(¯u,v) f ◦ π(¯u, v). Note that the number of summands is \|Jt′(¯u)K\| and the summands are bounded by Λ. Furthermore, the summands are independent and have expected value: E b∼F [eρ(HG(¯u, b), t′)h(eη(HG(¯u, b), t′))] Hence, by Hoeffding’s Inequality (Theorem B.4) for any ¯u the probability that: 1 n X v\|t′(¯u,v) f ◦ π(¯u, v) −\|Jt′(¯u)K\| n E b∼F [eρ(HG(¯u, b), t′)h(eη(HG(¯u, b), t′))] ≥ 1 2k+1 ϵ (♡) is at most: 2d exp −ϵ2\|Jt′(¯u)K\| 2Λ2 By equation (∗) there is N4 such that for all n ≥N4, with probability at least 1 −θ for all t′ such that α(t, t′) > 0 we have that for every tuple ¯u: 2d exp −ϵ2\|Jt′(¯u)K\| 2Λ2 < θδ (♣) Let Bn be the proportion, out of all tuples ¯u for which ψ(¯u) holds4, such that: 1 n X v\|t′(¯u,v) f ◦ π(¯u, v) −\|Jt′(¯u)K\| n E b∼F [eρ(HG(¯u, b), t′)h(eη(HG(¯u, b), t′))] ≥ 1 2k+1 ϵ Note that the property above is exactly the event whose probability is bounded in equation (♡). We can express Bn as the mean of a set of indicator variables, one for each tuple ¯u such that ψ(¯u) holds. Each indicator variable has expected value at most θδ by equation (♡) and equation (♣). Then by linearity of expectation, E[Bn] ≤θδ for all n ≥N4, and hence by Markov’s Inequality (Theorem B.1): P (Bn ≥δ) ≤θδ δ = θ Therefore, for all n ≥max(N3, N4), with probability at least 1 −θ for at least 1 −δ of the tuples ¯u for which ψ(¯u) holds we have that: 1 n X v\|t′(¯u,v) f ◦ π(¯u, v) −\|Jt′(¯u)K\| n E b∼F [eρ(HG(¯u, b), t′)h(eη(HG(¯u, b), t′))] < 1 2k+1 ϵ and therefore, by equation (∗) and the definition of fπ: 1 n X v f ◦ π(¯u, v) −fπ(HG(¯u), GrTp(¯u)) < ϵ (△f) Similarly, for any tuple ¯u consider the function: g◦ π(¯u, v) := h(eη(HG(¯u, v), GrTp(¯u, v))) As above there is N5 ≥N3 such that for all n ≥N5 with probability at least 1 −θ we have that the proportion out of tuples ¯u for which ψ(¯u) holds such that: 1 n X v g◦ π(¯u, v) −gπ(HG(¯u), GrTp(¯u)) < ϵ (△g) is at least 1 −δ. 4Recall that ψ is the ∧-description on which we condition the tuples ¯u in the inductive invariant. 24 Take n ≥max(N1, Nγ, N3, N4, N5). For such an n, these events above hold with probability at least 1 −5θ. So from now on we will condition on the event that they hold. Take any such tuple ¯u. Let t := GrTp(¯u). It suffices to show, using the definition of the interpretation of the weighted mean operator, that: P vJρ(¯u, v)Kh(Jη(¯u, v)K) P v h(Jη(¯u, v)K) −fπ(HG(¯u), t) gπ(HG(¯u), t) < ι(ϵ, δ, γ) for ι which we can make arbitrarily small by choosing ϵ, δ, γ small enough. By Lemma C.9 it suffices to find ζ(ϵ, δ, γ), ξ(ϵ, δ, γ) > 0 which we can make arbitrarily small and constants ν, Ω> 0 such that: 1 n X v Jρ(¯u, v)Kh(Jη(¯u, v)K) −fπ(HG(¯u), t) < ζ(ϵ, δ, γ) (1) 1 n X v h(Jη(¯u, v)K) −gπ(HG(¯u), t) < ξ(ϵ, δ, γ) (2) X v Jρ(¯u, v)Kh(Jη(¯u, v)K) < Ωn (3) X v h(Jη(¯u, v)K) < Ωn (4) ∀i ≤d: "X v h(Jη(¯u, v)K) # i > νn (5) For equation (3) and equation (4) we can use the fact that Jρ(¯u, v)K and Jη(¯u, v)K are bounded and that h is Lipschitz on bounded domains. For equation (5) we use the fact that Jη(¯u, v)K is bounded and that the codomain of h is (0, ∞)d. The proofs of equation (1) and equation (2) are very similar. We prove equation (2) since it is slightly notationally lighter. Let: g∗ π(¯u) := 1 n X v h(eη(HG(¯u, v), t)) Let κ be a bound on the norms of h(eη(¯a, b, t′)) and h(Jη(¯u, v)K). By equation (†η) we have that: 1 n X v h(Jη(¯u, v)K) −g∗ π(¯u) < 1 −(δ + δ2) ϵ + (δ + δ2)2κ2 We can make the right-hand-side as small as we like by taking ϵ and δ sufficiently small. Now note that: g∗ π(¯u) = 1 n X v g◦ π(¯u, v) Hence by equation (△g) we have that: ∥g∗ π(¯u) −gπ(HG(¯u), t)∥< ϵ Therefore: gπ(HG(¯u), t) −1 n X v h(Jη(¯u, v)K) < 1 −(δ + δ2) ϵ + (δ + δ2)2κ2 + ϵ which we can make as small as we like by taking ϵ and δ small enough. 25 • Inductive step for P v∈N(ui) ρ(¯u, v) ⋆h(η(¯u, v)). We proceed similarly to the global weighted mean case, this time making use of the conditioning ∧-description in the inductive invariant. Indeed, notice that when we apply the inductive invariant below, we add uiEv to our condition. Take ϵ, δ, θ > 0. Take a ∧-description ψ(¯u). By the induction hypothesis, there is N1 such that for all n ≥N1, with probability at least 1 −θ, out of all the tuples (¯u, v) which satisfy ψ(¯u) ∧uiEv, at least 1 −δ satisfy: ∥eρ(HG(¯u, v), GrTp(¯u, v)) −Jρ(¯u, v)K∥< ϵ (†loc ρ ) and: ∥eη(HG(¯u, v), GrTp(¯u, v)) −Jη(¯u, v)K∥< ϵ (†loc η ) Take γ > 0. We will choose how small γ is later. By Lemma C.6 there is Nγ > N1 such that for all n ≥Nγ, with probability at least 1 −θ the following event happens. Out of all the tuples ¯u such that ψ(¯u) ∧∃v: uiEv, a proportion of at least 1 −γ of them satisfy the following. At least 1 −(δ + δ2) of the nodes v for which ψ(¯u) ∧uiEv holds also satisfy equation (†loc ρ ) and equation (†loc η ). Now consider t(¯u) ∈GrTpk and take any t′(¯u, v) ∈Extui(t). For any tuple ¯u of nodes such that GrTp(¯u) = t, define: Jt′(¯u)K := {v ∈N(ui) \| t′(¯u, v)} By Lemma C.5 we have that in all the non-sparse cases there is R > 0 and N3 such that for all n ≥N3 with probability at least 1 −θ, the proportion of all tuples ¯u for which ψ(¯u) holds such that: d(ui) ≥R log(n) is at least 1 −δ. Then, as before, there is N4 ≥N3 such that for all n ≥N4, with probability at least 1 −θ, for all pairs (t, t′) the proportion of all tuples ¯u for which ψ(¯u) and t(¯u) hold such that: \|Jt′(¯u)K\| d(ui) −α(t, t′) < 1 2k+1 ϵ For any tuple ¯u define the function: f ◦ π(¯u, v) := eρ(HG(¯u, v), GrTp(¯u, v))h(eη(HG(¯u, v), GrTp(¯u, v))) Taking Λ as before, by Hoeffding’s Inequality for any ¯u the probability that: 1 d(ui) X v∈N(ui)\|t′(¯u,v) f ◦ π(¯u, v) −\|Jt′(¯u)K\| d(ui) E b∼F [eρ(HG(¯u, b), t′)h(eη(HG(¯u, b), t′))] ≥ 1 2k+1 ϵ is at most: 2d exp −ϵ2\|Jt′(¯u)K\| 2Λ2 Using a similar argument to the global weighted mean case, there is N5 such that for all n ≥N5, with probability at least 1 −θ for all t′ such that α(t, t′) > 0 we have that for at least 1 −δ of the tuples ¯u such that ψ(¯u) and t(¯u) hold: 2d exp −ϵ2\|Jt′(¯u)K\| 2Λ2 < θδ We now proceed as in the global weighted mean case, creating a random variable similar to Bn and making using of linearity of expectation. We get that for all n ≥max(N3, N4, N5), with probability at least 1 −θ for at least 1 −δ of the tuples ¯u for which ψ(¯u) holds we have that: 1 n X v f ◦ π(¯u, v) −fπ(HG(¯u), GrTp(¯u)) < ϵ (△loc f ) 26 Similarly, with g◦ π defined as above, there is N6 such that for all n ≥N6, with probability at least 1 −θ for at least 1 −δ of the tuples ¯u for which ψ(¯u) holds we have that: 1 n X v g◦ π(¯u, v) −gπ(HG(¯u), GrTp(¯u)) < ϵ (△loc g ) With equations (†loc ρ ), (†loc η ), (△loc f ) and (△loc g ) established, we can now proceed as before, making use of Lemma C.9. This completes the proof of Theorem 5.3. Applying the lemma to prove the theorem. To prove Theorem C.1, note that eτ is only a function of the GrTp0 (since τ is a closed term), while GrTp0 consists of a single type, ⊤, which is satisfied by all graphs. So eτ is a constant. D Sparse Erd˝os-R´enyi and Barab´asi-Albert We now give the proof for the sparse Erd˝os-R´enyi and Barab´asi-Albert cases of Theorem 5.1. In addition, we extend the result to the language AGG[WMEAN, GCN, RW] defined in Appendix A. We state the full result here. Theorem D.1. Consider (µn)n∈N sampling a graph G from either of the following models and node features independently from i.i.d. bounded distributions on d features. 1. The Erd˝os-R´enyi distribution ER(n, p(n)) where p satisfies Sparsity: for some K > 0 we have: p(n) = Kn−1. 2. The Barab´asi-Albert model BA(n, m) for any m ≥1. Then for every AGG[WMEAN, GCN, RW] term converges a.a.s. with respect to (µn)n∈N. As discussed in the body, this requires us to analyze neighbourhood isomorphism types, which implicitly quantify over local neighbourhoods, rather than atomic types as in the non-sparse cases. Remark D.2. Formally, a k-rooted graph is a tuple (T, ¯u). At various points below to lighten the notation we drop the ‘¯u’ and refer simply to a k-rooted graph T. D.1 Combinatorial tools for the sparse case As in the dense case, before we turn to analysis of our term language, we prove some results about the underlying random graph model that we can use as tools. The key combinatorial tool we will use is the following, which states that for any tuple of nodes ¯u, the proportion of nodes v such that (¯u, v) has any particular neighbourhood type converges. Definition D.3. Let (T, ¯w) be a k-rooted graph and let (T ′, ¯w, y) be a (k + 1)-rooted graph. Then T ′ extends T if T is a subgraph of T ′. Let Ext(T) be the set of all (k + 1)-rooted graphs extending (T, ¯w). Theorem D.4. Consider sampling a graph G from either the sparse Erd˝os-R´enyi or Barab´asi-Albert distributions. Let (T, ¯w) be a k-rooted graph and take ℓ∈N. Then for every (k + 1)-rooted graph (T ′, ¯w, y) which extends (T, ¯w) there is qT ′\|T ∈[0, 1] such that for all ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ we have that for all k-tuples of nodes ¯u such that N G ℓ(¯u) ∼= (T, ¯w): \|{v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)}\| n −qT < ϵ Moreover, we have that: X T ′∈Ext(T ) qT ′\|T = 1 We refer to the qT ′\|T as relative neighborhood weights. They will play a role in defining the controllers, analogous to that of the relative atomic type weights α(t, t′) in the non-sparse case. 27 Theorem D.4 follows from what is known as “weak local convergence” in the literature. It is essentially a non-parametrised version of Theorem D.4. Definition D.5. A sequence (µn)n∈N of graph distributions has weak local convergence if for every k-rooted graph (T, ¯w) and ℓ∈N there is qT ∈[0, 1] such that for all ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ we have that: \|{¯u \| N G ℓ(¯u) ∼= (T, ¯w)}\| n −qT < ϵ and moreover: X T qT = 1 Theorem D.6. The sparse Erd˝os-R´enyi and Barab´asi-Albert distributions have weak local convergence. Proof. See [48, Theorem 2.18] for the sparse Erd˝os-R´enyi distribution and [17, Theorem 4.2.1] for the Barab´asi-Albert distribution. Proof of Theorem D.4. Take T ′ ∈Ext(T). There are two cases. Case 1. There is a path from y to some node wi of ¯w in T ′. Set qT ′\|T = 0. Note that in this case, for all tuples ¯u such that N G ℓ(¯u) ∼= T we have that: {v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)} ⊆N G ℓ(ui)} Since N G ℓ(ui) is determined by T and is thus of fixed size, the proportion: \|{v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)}\| n tends to 0 as n grows. This completes the proof for Case 1. Case 2. There is no path from y to any node of ¯w in T ′. Then for all tuples ¯u such that N G ℓ(¯u) ∼= T we have that: {v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)} = {v \| N G ℓ(v) ∼= N T ′ ℓ(y)} \ N G ℓ(¯u) Hence: \|{v \| N G ℓ(v) ∼= N T ′ ℓ(y)}\| n −\|N G ℓ(¯u)\| n ≤\|{v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)}\| n Also: \|{v \| N G ℓ(¯u, v) ∼= (T ′, ¯w, y)}\| n ≤\|{v \| N G ℓ(v) ∼= N T ′ ℓ(y)}\| n By Lemma 5.5 (with k = 1) we have that: \|{v \| N G ℓ(v) ∼= N T ′ ℓ(y)}\| n converges to some limit qT \|T ′ := qN T ′ ℓ(y) asymptotically almost surely, while: \|N G ℓ(¯u)\| n = \|T\| n converges to 0. This completes the proof for Case 1. Finally, to show that P T ′∈Ext(T ) qT ′\|T = 1, note the set: {N T ′ ℓ(y) \| (T ′, ¯w, y) ∈Ext(T ′)} contains all 1-rooted graphs up to isomorphism which can be the ℓ-neighbourhood of a node. Therefore, by the “moreover” part of Definition D.5 we have that: X T ′∈Ext(T ) qT ′\|T = X T ′∈Ext(T ) qN T ′ ℓ(y) = 1 28 D.2 Proving the inductive invariant for the sparse cases, and proving the convergence theorem for these cases We now begin the proof of Theorem D.1. Recall from the body that we will do this via Theorem 5.6, which requires us to define some Lipschitz controllers on neighbourhood types that approximate a given term π relative to a neighbourhood type. Throughout the following, it will be convenient to assume for every k-rooted graph isomorphism type (T, ¯u) a canonical ordering nodes T = {s1, . . . , s\|T \|} such that: • the first k nodes in the ordering are ¯u and • for any graph G, tuple of nodes ¯u and ℓ, ℓ′ ∈N with ℓ′ < ℓthe canonical ordering of N ℓ′(¯u) is an initial segment of the canonical ordering of N ℓ(¯u). Again we will use F for the probability distribution of the node features. For each subterm π of τ, its reach, denoted Reach(π) is a natural number defined as follows: Reach(H(u)) = 0 Reach(c) = 0 Reach(rw(v)) = d Reach(f(ρ1, . . . , ρr)) = max i≤r Reach(ρi) Reach  X v∈N(u) ρ ⋆h(η)  = max(Reach(ρ), Reach(η)) + 1 Reach X v ρ ⋆h(η) ! = 0 Take a subterm π of τ. Let k be the number of free variables in π. We now define the controller at π for every k-rooted graph (T, ¯w). The controller will be of the form: eT π : ([0, 1]d)\|T \| →Rd Note that when \|T\| = 0 this is a constant. Recall from Theorem 5.6 that our goal is to ensure the following correctness property for our controllers eT π . Property D.7. Let π be any subterm of τ and let ℓ= Reach(π). Consider sampling a graph G from either the sparse Erd˝os-R´enyi or Barab´asi-Albert distributions. For every k-rooted graph (T, ¯w) and ϵ, θ > 0, there is N ∈N such that for all n ≥N with probability at least 1 −θ we have that for every k-tuple of nodes ¯u in the graph such that N ℓ(¯u) ∼= (T, ¯w), taking the canonical ordering N ℓ(¯u) = {s1, . . . , s\|T \|}: eT π (HG(s1), . . . , HG(s\|T \|)) −Jπ(¯u)K < ϵ For notational convenience, we allow each eT π to take additional arguments as input, which it will ignore. Proof of Theorem 5.6. We give the construction of the controllers in parallel with the proof of correctness. • Base case π = H(ui). Here ℓ= Reach(π) = 0. Let eT π (¯a) := ai, the feature value of the ith node in the tuple. Note that for any k-tuple of nodes ¯u in the graph we have: eT π (HG(s1), . . . , HG(s\|T \|)) = HG(si) = Jπ(¯u)K • Base case π = c. Let eT π (¯a) := c. 29 • Base case π = rw(ui). Then ℓ= Reach(π) = d. Note that the random walk embeddings up to length d are entirely determined by the d-neighbourhood. Therefore, given rooted graph (T, ¯w), there is rT such that: Jrw(ui)K = rT for all tuples ¯u such that N ℓ(¯u) = T. So set: eT π (¯a) = rT for any ¯a. • Inductive step for Lipschitz functions π = F(ρ1, . . . , ρr). Note that ℓ= maxj≤r Reach(ρi). Take a k-rooted graph (T, ¯w). For each i ≤r let Tj := N T Reach(ρj)( ¯w). Define: eT π (¯a) := F(eT1 ρ1 (¯a), . . . , eTr ρr (¯a)) Now take ϵ, θ > 0. By the induction hypothesis, there is N such that for all n ≥N with probability at least 1−θ we have that for every k-tuple of nodes ¯u in the graph, letting T = N ℓ(¯u), for every j ≤r: eTj ρj (HG(s1), . . . , HG(s\|Tj\|)) −Jρj(¯u)K < ϵ Hence, by the Lipschitzness of F we have: eT π (HG(s1), . . . , HG(s\|T \|)) −Jπ(¯u)K < LF ϵ where LF is the Lipschitz constant of F. • Inductive step for local weighted mean π = P v∈N(ui) ρ ⋆h(η). In this step we use that the value of a local aggregator is determined by the values of the terms it is aggregating in the local neighbourhood. Take a k-rooted graph (T, ¯w), where k is the number of free variables in π. Note that ℓ= max(Reach(ρ), Reach(η)) + 1. When wi has no neighbours in T, define: eT π (¯a) := 0 and note that for any k-tuple of nodes ¯u in the graph such that N ℓ(¯u) = T we have: Jπ(¯u)K = 0 = eT π (HG(s1), . . . , HG(s\|T \|)) So suppose that wi has some neighbours in T. Enumerate the neighbours as: N T (wi) = {y1, . . . , yr} Define the Lipschitz function: WMeanT (a1, . . . , ar, b1, . . . , br) := Pr j=0 ajh(bj) Pr j=0 h(bj) Note that whenever N G ℓ(¯u) ∼= (T, ¯w), letting N G(ui) = {v1, . . . , vr} be the enumeration of the neighbourhood of ui given by the isomorphism, we have that: Jπ(¯u)K = WMeanT (Jρ(¯u, v1)K, . . . , Jρ(¯u, vr)K, Jη(¯u, v1)K, . . . , Jη(¯u, vr)K) (□) For any yj ∈N(wi), let: Tj := N T ℓ−1( ¯w, yj) T ρ j := N T Reach(ρ)( ¯w, yj) T η j := N T Reach(η)( ¯w, yj) 30 Further, for any ¯a ∈([0, 1]d)\|T \| let ¯aρ j and ¯aη j be the tuple of elements of ¯a corresponding to the nodes of T ρ j and T η j respectively. Let: eT π (¯a) := WMeanT e T ρ 1 ρ (¯aρ 1), . . . , eT ρ r ρ (¯aρ r), e T η 1 η (¯aη 1), . . . , eT η r η (¯aη r), Take ϵ, θ > 0. Applying the induction hypothesis to the term ρ(¯u, v) and to η(¯u, v), which have Reach at most ℓ−1, and using the fact that N T (wi) is finite, there is N such that for all n ≥N with probability at least 1 −θ, for every yj ∈N T (wi) and every (k + 1)-tuple of nodes (¯u, v) such that N G ℓ−1(¯u, v) ∼= (Tj, ¯w, yj) we have that both: e T ρ j ρ (HG(s1), . . . , HG(s\|T ρ j \|)) −Jρ(¯u, v)K < ϵ and: e T η j η (HG(s1), . . . , HG(s\|T η j \|)) −Jη(¯u, v)K < ϵ Under this event, by equation (□) and the Lipschitzness of WMeanT we have: eT π (HG(s1), . . . , HG(s\|T \|)) −Jπ(¯u)K < LT ϵ where LT is the Lipschitz constant of WMeanT . • Inductive step for the GCN aggregator π = GCNv∈N(ui)ρ. This step is very similar to the previous, where we again use the fact that the value of a local aggregator is determined by the values of the terms it is aggregating in the local neighbourhood. The only difference is that we now have a different Lipschitz function: GCNT (a1, . . . , ar) := r X j=1 1 N T (ui) N T (yj) aj The rest of the proof is as before. • Inductive step for global weighted mean π = P v ρ ⋆h(η). Note that ℓ= 0. Let ℓ′ = max(Reach(ρ), Reach(η)). Take a k-rooted graph (T, ¯w). For any tuple ¯u and (k + 1)-rooted graph (T ′, ¯w, y) extending (T, ¯w) let: JT ′(¯u)K := \|{v \| N G ℓ′(¯u, v) ∼= (T ′, ¯w, y)}\| By the parameterized Weak Local Convergence result, Theorem D.4, for every such T ′ extending there is a relative neighborhood weight qT \|T ′ ∈[0, 1] such that for all ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ, for every k-tuple ¯u such that N ℓ(¯u) = T we have that: JT ′(¯u)K n −qT \|T ′ < ϵ and moreover the relative neighborhood weights sum to one: X T ′∈Ext(T ) qT \|T ′ = 1 Consider any (T ′, ¯w, y) ∈Ext(T). In order to define the controller, we need to identify the nodes in the canonical enumeration corresponding to N T ℓ′−1(¯u). Let: rT := N T ℓ′−1(¯u) Given tuples of Rd-vectors ¯a = (a1, . . . , arT ) and ¯b = (brT +1, . . . , b\|T ′\|), let ArrangeT ′(¯a, ¯b) be the \|T ′\|-tuple of vectors obtained by assigning the ai’s to the nodes in N T ℓ′−1(¯u) and the bi’s to the remaining nodes in T ′, according to the canonical enumeration of T ′. Define the controller: eT π (¯a) := f T π (¯a) gTπ (¯a) 31 where: f T π (¯a) := X T ′∈Ext(T ) E ¯b∼F h eT ′ ρ (ArrangeT ′(¯a, ¯b)) · h(eT ′ η (ArrangeT ′(¯a, ¯b))) i · qT \|T ′ and: gT π (a1, . . . , ak) := X T ′∈Ext(T ) E ¯b∼F h h(eT ′ η (ArrangeT ′(¯a, ¯b))) i · qT \|T ′ where in both cases the expectation is taken over bi’s sampled independently from node feature distribution F. Since: eT ′ ρ (a1, . . . , a\|T ′\|)h(eT ′ η (a1, . . . , a\|T ′\|)) and: h(eT ′ η (a1, . . . , a\|T ′\|)) are bounded, and P T ′∈Ext(T ) qT \|T ′ converges, these sums converges absolutely, and hence the controller is well-defined. This completes the definition of the controller. We now present the proof that it is correct. Take ϵ, θ > 0. Take a finite S ⊆Ext(T) such that: X T ′∈S qT \|T ′ > 1 −ϵ (⃝) and each qT \|T ′ > 0 for T ′ ∈S. The guarantee given when we applied the parameterized version of Weak Local Convergence, Theorem D.4, is that there is N1 such that for all n ≥N1 with probability at least 1 −θ, for all T ′ ∈S and every k-tuple ¯u such that N ℓ(¯u) = T we have that: JT ′(¯u)K n −qT \|T ′ < ϵ (‡) Define the function: f ◦ π(¯a, ¯b) := eT ′ ρ (ArrangeT ′(¯a, ¯b)) · h(eT ′ η (ArrangeT ′(¯a, ¯b))) and similarly: g◦ π(¯a, ¯b) := h(eT ′ η (ArrangeT ′(¯a, ¯b))) Note that these are just the integrands used in f T π (¯a) and gT π (¯a). Now, for any (k + 1)-tuple of nodes (¯u, v) let ¯a¯u be the tuple of node features of N G ℓ′−1(¯u) and ¯b(¯u,v) be the tuple of node features of the remaining nodes in N G ℓ′(¯u, v), ordered as in the canonical enumeration of N G ℓ′(¯u, v). Note that f ◦ π is bounded, say by Λ. Then for every γ > 0, by Hoeffding’s Inequality (Theorem B.4) for any tuple ¯u such that N ℓ(¯u) = T and T ′ ∈S we have that: P     X v N G ℓ′(¯u,v)∼ =T ′ f ◦ π(¯a¯u, ¯b(¯u,v)) −\|JT ′(¯u)K\| · E ¯b∼F f ◦ π(¯a¯u, ¯b) ≥\|JT ′(¯u)K\|γ     ≤2 exp −2JT ′(¯u)Kγ2 Λ2 ≤2 exp −2n(q′ T −ϵ)γ2 Λ2 32 where in the last inequality we use equation (‡). Hence, taking a union bound, there is N2 such that for all n ≥N2 with probability at least 1−θ, for all ¯u such that N ℓ(¯u) = T and T ′ ∈Ext(T) we have: X v N G ℓ′(¯u,v)∼ =T ′ f ◦ π(¯a¯u, ¯b(¯u,v)) −\|JT ′(¯u)K\| · E ¯b∼F f ◦ π(¯a¯u, ¯b) < \|JT ′(¯u)K\|γ If we divide both sides of the inequality through by n, take a γ below 1, and apply equation (‡) again, we conclude that there is N4 such that for all n ≥N4 with probability at least 1 −θ, for all ¯u such that N ℓ(¯u) = T and T ′ ∈Ext(T) we have: 1 n X v N G ℓ′(¯u,v)∼ =T ′ f ◦ π(¯a¯u, ¯b(¯u,v)) −qT \|T ′ E ¯b∼F f ◦ π(¯a¯u, ¯b) < ϵ (△f) Similarly, there is N5 such that for all n ≥N5 with probability at least 1 −θ, for all ¯u such that N ℓ(¯u) = T and T ′ ∈Ext(T) we have: 1 n X v N G ℓ′(¯u,v)∼ =T ′ g◦ π(¯a¯u, ¯b(¯u,v)) −qT \|T ′ E ¯b∼F g◦ π(¯a¯u, ¯b) < ϵ (△g) By the induction hypothesis, there is N6 such that for all n ≥N6 with probability at least 1 −θ, for every T ′ ∈S and every (k + 1)-tuple of nodes (¯u, v) such that N G ℓ′(¯u, v) ∼= T ′ we have that both: eT ′ ρ (HG(s1), . . . , HG(s\|T ′\|)) −Jρ(¯u, v)K < ϵ (†ρ) and: eT ′ η (HG(s1), . . . , HG(s\|T ′\|)) −Jη(¯u, v)K < ϵ (†η) Let N := max(N1, N2, N3, N4, N5, N6) and take n ≥N. Then for such n the probability that these six events happen is at least 1 −6θ. We will condition on this. Take any k-tuple of nodes ¯u such that N ℓ(¯u) = T. It suffices to show, using the definition of the interpretation of the weighted mean operator, that: P vJρ(¯u, v)Kh(Jη(¯u, v)K) P v h(Jη(¯u, v)K) −f T π (HG(u1), . . . , HG(uk)) gTπ (HG(u1), . . . , HG(uk)) < ι for some ι which we can make arbitrarily small by choosing ϵ small enough. As above it suffices to find ζ, ξ > 0 which we can make arbitrarily small and constants ν, Ω> 0 such that: 1 n X v Jρ(¯u, v)Kh(Jη(¯u, v)K) −f T π (HG(s1), . . . , HG(s\|T \|)) < ζ (6) 1 n X v h(Jη(¯u, v)K) −gT π (HG(s1), . . . , HG(s\|T \|)) < ξ (7) X v Jρ(¯u, v)Kh(Jη(¯u, v)K) < Ωn (8) X v h(Jη(¯u, v)K) < Ωn (9) ∀i ≤d: "X v h(Jη(¯u, v)K) # i > νn (10) 33 Equation (8), equation (9) and equation (10) follow as before. We show equation (7), the proof of equation (6) being similar. By equation (†η) we have that for every v: h(Jη(¯u, v)K) −g◦ π(¯a¯u, ¯b(¯u,v)) < ϵ Hence: 1 n X v h(Jη(¯u, v)K) −1 n X v g◦ π(¯a¯u, ¯b(¯u,v)) < ϵ Now: 1 n X v g◦ π(¯a¯u, ¯b(¯u,v)) = 1 n X T ′∈Ext(T ) X v N G ℓ′(¯u,v)∼ =T ′ g◦ π(¯a¯u, ¯b(¯u,v)) Letting Λ be a bound on the norm of g◦ π(¯a¯u, ¯b(¯u,v)), by equation (⃝) we have that: 1 n X v g◦ π(¯a¯u, ¯b(¯u,v)) −1 n X T ′∈S X v N G ℓ′(¯u,v)∼ =T ′ g◦ π(¯a¯u, ¯b(¯u,v)) < ϵΛ By equation (△g) we have that: 1 n X T ′∈S X v N G ℓ′(¯u,v)∼ =T ′ g◦ π(¯a¯u, ¯b(¯u,v)) −1 n X T ′∈S qT \|T ′ E ¯b∼F g◦ π(¯a¯u, ¯b) ≤ϵ Finally, by equation (⃝) again we have that: 1 n X T ′∈S qT \|T ′ E ¯b∼F g◦ π(¯a¯u, ¯b) −gT π (¯a¯u) < ϵΛ This completes the proof of the inductive construction of the controllers, thus proving Theorem 5.6. Application to prove the final theorem, Theorem 5.1 for the sparse Erd˝os-R´enyi and Barab´asiAlbert cases. To complete the proof, we note that the term τ has no free variables, and as a subterm of itself has reach 0. The controller e∅ τ of is a constant z, and hence by the induction hypothesis applied to it, for every ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ we have that: ∥JτK −z∥< ϵ E Proof for the stochastic block model We now prove the convergence result for the stochastic block model. The proof follows the same structure as the density case in Theorem 5.1, except that the notion of graph type is augmented slightly. We therefore only indicate the differences in the proof. For this is it helpful to be able to remember the community to which each node belongs. Given any graph G generated by the stochastic block model and node v, let C(v) ∈{1, . . . , m} be the community to which v belongs. To prove the result, we first need that the random work positional encodings converge, as in Lemma C.8. In fact they converge to 0. Lemma E.1. Let n1, . . . , nm : N →N and P be as in Theorem 5.1. Then for every k ∈N and ϵ, θ > 0 there is N ∈N such that for all n ≥N with probability at least 1 −θ, for all nodes v we have that: ∥rwk(v)∥< ϵ 34 Proof. For each j ∈{1, . . . , m}, let nj n converge to qj. Let: rj := q1P1,j + · · · + qmPm,j Note that nrj is the expected degree of a node in community j. By Hoeffding’s Inequality (Theorem B.4) and a union bound, there is N such that for all n ≥N with probability at least 1 −θ, for all v we have that: d(v) n −rC(v) < ϵ Take n ≥N and condition on this event. Take any node v. When rC(v) = 0, the node v has no neighbours, and so rwk(v) = 0. So assume rC(v) > 0. Let j = C(v). Then as in the proof of Lemma C.8 we can show that the proportion of random walks of length k starting at v is at most: (˜p + ϵ)k−1 (˜p −ϵ)k n−1 which converges to 0 as n →∞. We now follow the structure of the proof for the Erd˝os-R´enyi dense cases. This time we augment our vocabulary for atomic types with a predicate Pj for each community j, so that Pj(v) holds if and only if v belongs to community j. With this we define the community atomic type of a tuple of nodes ¯u in graph, notation ComTp(¯u) to be the atomic formulas satisfied by ¯u in the language augmented with the Pj predicates. For k ∈N let ComTpk be the set of all complete community atomic types with k free variables. For each j ∈{1, . . . , m}, let nj n converge to qj. For any type t(¯u) and t′(¯u, v) ∈Ext(t), let: α(t, t′) := qC(v) Further, for any type t(¯u), free variable ui in t and t′(¯u, v) ∈Extui(t), let: αui(t, t′) := PC(ui),C(v)qC(v) For any term π with k free variables, the feature-type controller: eπ : ([0, 1]d)k × ComTpk →[0, 1]d is defined exactly as as in the proof of the density case of Theorem 5.1, using the extension proportions α(t, t′) and αui(t, t′) just defined. We show by induction that every ϵ, δ, θ > 0 and ∧-desc ψ(¯u), there is N ∈N such that for all n ≥N, with probability at least 1 −θ in the space of graphs of size n, out of all the tuples ¯u such that ψ(¯u), at least 1 −δ satisfy: ∥eπ(HG(¯u), GrTp(¯u)) −Jπ(¯u)K∥< ϵ We then proceed as in the proof of the non-sparse Erd˝os-R´enyi cases of Theorem 5.1. The only difference is that when showing equation (△f) and equation (△g) we use the fact that the expected proportion of type extensions t′(¯u, v) of a type t(¯u) at a node uj is α(t, t′). F Additional experiments In this section we provide additional experiments using MeanGNN, GCN [30], GAT [49], and GPS+RW with random walks of length up to 5, over the distributions ER(n, p(n) = 0.1), ER(n, p(n) = log n n ), ER(n, p(n) = 1 50n) and BA(n, m = 5). We also experiment with an SBM of 10 communities with equal size, where an edge between nodes within the same community is included with probability 0.7 and an edge between nodes of different communities is included with probability 0.1. Setup. Our setup is carefully designed to eliminate confounding factors: 35 Figure 6: The 5 class probabilities (in different colours) of a MEANGNN, GCN, GAT and GPS+RW model initialization over the ER(n, p(n) = 0.1) graph distributions, as we draw increasingly larger graphs. • We consider 5 models with the same architecture, each having randomly initialized weights, utilizing a ReLU non-linearity, and applying a softmax function to their outputs. Each model uses a hidden dimension of 128, 3 layers and an output dimension of 5. • We draw graphs of sizes up to 10,000, where we take 100 samples of each graph size. Node features are independently drawn from U[0, 1] and the initial feature dimension is 128. Further details are available in the experiments repository at https://github.com/ benfinkelshtein/GNN-Asymptotically-Constant. Much like in Section 6, the convergence of class probabilities is apparent across all models and graph distributions, in accordance with our main theorems (Q1). We again observe that attention-based models such as GAT and GPS+RW exhibit delayed convergence and greater standard deviation in comparison to MeanGNN and GCN, which further strengthening our previous conclusions. G Acknowledgements This research was funded in whole or in part by EPSRC grant EP/T022124/1. For the purpose of Open Access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission. NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We thoroughly investigate the phenomenon of a.a.s. convergence of GNNs, both from a theoretical and experimental point of view, as claimed in the abstract. Guidelines: 36 Figure 7: The 5 class probabilities (in different colours) of a MEANGNN, GCN, GAT and GPS+RW model initialization over the ER(n, p(n) = log n n ) graph distributions, as we draw increasingly larger graphs. Figure 8: The 5 class probabilities (in different colours) of a MEANGNN, GCN, GAT and GPS+RW model initialization over the ER(n, p(n) = 1 50n) graph distributions, as we draw increasingly larger graphs. 37 Figure 9: The 5 class probabilities (in different colours) of a MEANGNN, GCN, GAT and GPS+RW model initialization over the SBM graph distributions, as we draw increasingly larger graphs. Figure 10: The 5 class probabilities (in different colours) of a MEANGNN, GCN, GAT and GPS+RW model initialization over the BA(n, m = 5) graph distributions, as we draw increasingly larger graphs. 38 Figure 11: A three-layer GCN with hidden dimension 128 is trained on the ENZYMES dataset with one class removed so that there are five output classes. This model is then run on graphs drawn from ER(n, p(n) = 0.1) for increasing sizes n, and the mean output probabilities are recorded, along with standard deviation • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,486	M3LEO: A Multi-Modal, Multi-Label Earth Observation Dataset Integrating Interferometric SAR and Multispectral Data Matt Allen University of Cambridge, UK mja78@cam.ac.uk Francisco Dorr Independent, Argentina fran.dorr@gmail.com Joseph A. Gallego-Mejia Drexel University, USA jagallegom@unal.edu.co Laura Martínez-Ferrer Universitat de València, Spain laura.martinez-ferrer@uv.es Anna Jungbluth European Space Agency, Climate Office, UK anna.jungbluth@esa.int Freddie Kalaitzis University of Oxford, UK freddie.kalaitzis@cs.ox.ac.uk Raúl Ramos-Pollán Universidad de Antioquia, Colombia raul.ramos@udea.edu.co Abstract Satellite-based remote sensing has revolutionised the way we address global challenges in a rapidly evolving world. Huge quantities of Earth Observation (EO) data are generated by satellite sensors daily, but processing these large datasets for use in ML pipelines is technically and computationally challenging. Specifically, different types of EO data are often hosted on a variety of platforms, with differing degrees of availability for Python preprocessing tools. In addition, spatial alignment across data sources and data tiling for easier handling can present significant technical hurdles for novice users. While some preprocessed Earth observation datasets exist, their content is often limited to optical or near-optical wavelength data, which is ineffective at night or in adverse weather conditions. Synthetic Aperture Radar (SAR), an active sensing technique based on microwave length radiation, offers a viable alternative. However, the application of machine learning to SAR has been limited due to a lack of ML-ready data and pipelines, particularly for the full diversity of SAR data, including polarimetry, coherence and interferometry. In this work, we introduce M3LEO, a multi-modal, multi-label Earth observation dataset that includes polarimetric, interferometric, and coherence SAR data derived from Sentinel-1, alongside multispectral Sentinel-2 imagery and a suite of auxiliary data describing terrain properties such as land use. M3LEO spans approximately 17M data chips, each measuring 4x4 km, across six diverse geographic regions. The dataset is complemented by a flexible PyTorch Lightning framework, with configuration management using Hydra, to accommodate its use across diverse ML applications in Earth observation. Additionally, we provide tools to process any dataset available on popular platforms such as Google Earth Engine for seamless integration with our framework. We show that the distribution shift in self-supervised embeddings is substantial across geographic regions, even when controlling for terrain properties. Data is available at huggingface.co/M3LEO, and code at github.com/spaceml-org/M3LEO. 38th Conference on Neural Information Processing Systems (NeurIPS 2024) Track on Datasets and Benchmarks. 1 Introduction Satellite-based Earth observation data is fundamental in addressing global problems in a rapidly changing world, offering large-scale, high resolution, high frequency data for applications from tracking wildfires [1] and deforestation [2] to refugee settlement mapping [3] and war zone damage assessment [4, 5]. Information from these tasks is critical in crafting responses to man-made [6] and environmental crises [7], but is constrained by the use of optical (wavelengths from visible to near-infrared) sensing data. Such sensors are unable to operate in adverse weather or cloudy conditions [8], or at night, limiting their usefulness for time-critical tasks such as natural disaster management [7], environmental protection [9] or maritime surveillance [10]. Synthetic Aperture Radar (SAR) sensing presents an alternative that is able to overcome these limitations. Unlike optical sensors, SAR instruments actively illuminate terrain using microwave pulses, ensuring visibility without the need for daylight. These long-wavelength pulses can penetrate cloud cover and other adverse atmospheric conditions such as dust, making SAR-based sensing a valuable alternative to optical sensors for robust day-night coverage. Additionally, microwave radiation can penetrate some solid objects such as small-scale vegetation or soils — allowing, for example, measurements of properties relating to soil moisture under vegetation [11] or the identification of archaeological features hidden below ground [12]. In addition to exploiting the wavelength and active illumination of SAR data, the complex nature of SAR signals — returning both amplitude and phase — can also be leveraged to provide insights beyond what is possible using optical data. Coherence, the complex correlation between pairs of SAR acquisitions, has been used successfully for tasks including flood detection [13, 14], detection of urban damage [15] and forest canopy height measurement [16]. The phase difference between co-registered SAR acquisitions, measured through interferometry, enables the detection of surface height changes with millimetre accuracy, independently of the horizontal resolution of the sensor. This capability is critical for monitoring geological phenomena such as earthquakes [17], landslides [18], glacial movement [19, 20], magma chamber growth [21] and infrastructure deformation [22]. While SAR offers opportunities to overcome the limitations of optical sensors, and to give insights that are impossible to provide using visible wavelengths, it is associated with substantial additional complexity. The automated analysis of optical data, sometimes used in fusion with SAR data, has seen great success in recent years [23–25] — including the development of large foundation models able to make use of planetary-scale datasets [26, 27]. The application of large-scale deep learning to SAR data without simultaneous use of optical data, however, is more limited [28]. The complexities of processing SAR data, particularly estimating coherence and performing interferometry — which require processing phase information as using complex numbers — mean that the full diversity of SAR data types is not available at scale in formats compatible with machine learning (ML) pipelines. To address these challenges we introduce M3LEO, a large-scale multi-modal Earth Observation dataset comprising polarimetric, interferometric and coherence data for SAR as well as Sentinel-2 data and auxiliary datasets describing surface properties such as land use, biomass measurements and elevation. We provide this data pre-processed as ML-readable, tiled images to abstract complex parameter choices typically made by domain experts. We also include a flexible PyTorch Lightning framework, parameterised using Hydra [29], to further minimise barriers to usage. We include a preliminary analysis on distribution shift for terrain properties and the appearance of SAR data across geographic regions. Finally, in addition to the pre-formatted data we offer for download, we provide tools enabling ML practitioners to process any data retrievable from Google Earth Engine into the same tiled format, such that it can be used in our framework. 2 Related Work Deep learning has been applied over the last decade to curated optical imagery with great success [30–33], including the recent development of large, self-supervised foundation models [34–37]. Such models have been extraordinarily successful in tasks such as semantic segmentation [38, 39], image classification [33], [38] [40] and object detection [38, 41]. EO data from optical sensors has similarly been the subject of success for deep learning practitioners. Early work focusing on small, fully supervised models, showed great promise in a huge range of tasks, including land cover classification [23], biomass measurement [24], road detection [25, 42] and flood mapping [43], although many models were limited in scope to a small geographic area [24, 44–46]. More recent work has focused 2 on the development of large foundation models, often self supervised [28, 47], which are readily adaptable to a range of downstream tasks and geographic areas [27]. Work applying these models to optical EO data has been enabled by the wealth of easily accessible open data. As well as being available as raw products from satellite data providers such as ESA [48], many datasets comprising optical satellite imagery in ML-ready formats exist [49–52], across a range of spatial resolutions [53, 54], and for multiple time-points [50, 55], although they may be limited in other scopes — for example, not having aligned task labels [55, 56] or being limited to a single region [50]. We provide data at a large scale (14.1% of the land surface of the Earth) with a diverse set of auxiliary labels. The application of deep learning to SAR data is less comprehensive. A body of work applying deep learning directly to SAR data exists [57–60] [61], but data limitations mean that geographic or temporal generalisability, often lacking in remote sensing models [62], has not clearly been shown. The development of foundation models for SAR may prove to be productive in obtaining geographic and temporal generalisability. To create foundation models, SAR is commonly applied alongside optical imagery in data fusion-based approaches [63, 64], but with little attribution regarding whether such approaches can work well without optical data. Some work exists exploiting schemes such as masked autoencoding [65, 66], contrastive learning [67], or knowledge distillation [68, 69] to develop foundation models for polarimetric SAR — and shows that strong geographic generalisabilty is obtainable when using SAR data at large scales [65, 69]. Many datasets providing ML-readable polarimetric SAR (polSAR) data exist [51, 55, 70–75], although most do not provide interferometric SAR (inSAR) data [74, 75]. The full diversity of inSAR datatypes, such as interferometry and coherence, have seen a number of applications in machine learning, and a rich tapestry of applications in other contexts. Interferometry, for example, is often used to track earthquakes [17], landslides [18] and glacial movement [19, 20], in a manner that is both more repeatable — being immune to adverse weather conditions — and more accurate — being able to track millimetre-scale height changes — than methods using optical data. Coherence has been used with success in urban damage assessment [15], flood detection [13, 14] and canopy height measurement [16]. A number of datasets exist making these datatypes available to deep learning users, many of which are focused on specific events, tasks or locations. Hephaestus [76], for example, contains 216K interferometric SAR (inSAR) patches localised to volcanoes annotated with various labels describing volcanic activity. ISSLIDE [77] contains inSAR data from the French alps describing slow landslides, and Pol-InSAR-Island [78] comprises inSAR data describing land cover on Baltrum, a Frisian island. UrbanSARFloods [79] contains Sentinel-1 interferometric coherence data from a diverse set of global locations, but is limited to specific events. S1SLC_CVDL [80] opts to provide complex-valued singlelook SAR data rather than processed interferometric data, from three manually selected Sentinel-1 scenes containing major population centres. We make inSAR and coherence data available at a multi-continental scale, alongside both polarimetric SAR and optical data, in M3LEO. 2.1 Comparison to Existing Datasets We provide a comparison between a number of popular large-scale Earth observation datasets, including M3LEO, in Table 1. We define tile to mean a fixed location or area on the surface of the Earth, and chip as the content of some data product over that tile. Unlike in many of these datasets, we provide acquisitions from different seasons within the same year for the same tile as channels, rather than separate chips. The described number of chips in Table 1 therefore appears relatively lower for M3LEO compared to, for example, SSL4EO-S12, for a fixed number of satellite acquisitions. We instead measure the number of timepoints in years, rounded up for part-years. Of the four datasets offering Sentinel-1 SAR data, only M3LEO offers data from multiple years, and only M3LEO offers inSAR data (although see Section 1 for a brief description of available task-specific or localised interferometric datasets). Regarding spatial coverage, of the datasets listed in Table 1 M3LEO is most similar in scale to SatlasPretrain and SSL4EO-L, with these three datasets being significantly larger than the remainder. Of these three datasets, only M3LEO provides SAR data of any modality. The temporal coverage of M3LEO sits between SatlasPretrain and SSL4EO-L, although the aggregate number of years does not give a full picture — in SatlasPretrain, some acquisitions are available for 3 a wider range of years for specific events, and in SSL4EO-L different data products are sometimes collected for non-overlapping years, meaning much of the dataset is not a parallel corpus. In M3LEO, the primary satellite data from Sentinel-1 and Sentinel-2 is available for 2018-2020 for all tiles. Several of these datasets contain auxiliary information in addition to satellite acquisitions. Land cover labels are common, included in SSL4EO-L, SEN12MS, BigEarthNet and SatlasPretrain [51, 56, 72, 73]. Additional auxiliary datasets are sometimes available — SSL4EO-L, for example, includes more detailed crop classification data. SatlasPretrain introduced a number of novel additional labelled datasets including building and road polygons. M3LEO currently includes 4 auxiliary datasets - Land cover, vegetation cover, aboveground biomass and Digital Elevation Models (DEMs). Many datasets are pre-sampled within their selected AOIs prior to distribution. Some datasets sample based on a manually specified distribution (for example, SSL4EO-S12 and SSL4EO-L sample locations based on Gaussian distributions centred on large cities), and some randomly. We include all available tiles within our AOIs — partly with a view to increasing data volume, but also to enable further research on sampling schemes. Selecting a good sampling strategy to diversify actively illuminated radar data with phase information is not straightforward, and it doubtful whether sampling schemes developed for optical EO imagery would transfer well to this data. The auxiliary data included in M3LEO, such as elevation and land use, may be useful for constructing such sampling schemes. Table 1: Existing Datasets. Summary of popular large-scale Earth observation pre-training datasets. Dataset SAR Years Num. Num. Tile Size Coverage Sampling Tiles Chips* km km2 SSL4EO-S12 [55] Y 1 251K 3M 2.64×2.64 1.75×106 Targeted SSL4EO-L [51] N 6 250K 5M 7.92×7.92 1.57×107 Targeted SEN12MS [70] Y 1 181K 542K 2.56×2.56 1.18×106 Random SeCo [71] N 2 200K 1M 2.65×2.65 1.40×106 Random BigEarthNet [72] N 1 590K 1.2M 1.20×1.20 8.50×105 Targeted SatlasPretrain [73] N 1** 856K N/A 5.12×5.12 2.13×107 None M3LEO Y 3 1.05M 17.2M 4.48×4.48 2.11×107 None * Heuristic only — some work, such as SSL4EO-S12, considers acquisitions at the same location from different seasons to be a seperate data chip. ** Contains additional historical images from 2016-2021 that are relevant to dynamic events such as floods. 3 Dataset & Framework 3.1 SAR Datasets The many benefits of SAR data are met with increased complexity compared to optical data. SAR sensors are active — that is, they emit microwave pulses (5.6 cm wavelength for Sentinel-1) and measure backscatter rather than imaging the Earth under passive illumination from the Sun. This enables day-night operation and the penetration of atmospheric obstructions such as cloud and dust. Unlike sunlight, the pulses emitted by SAR sensors are polarised, with the ability to emit pulses polarised either horizontally or vertically with respect to the Earth. SAR sensors are also able to measure the polarization of the backscatter, and capture both amplitude and phase. This signal, typically stored as a complex number, allows studying the geometry of surface-level objects, in addition to their reflectances. As an example, higher amplitudes are measured when surface features align with the polarisation of the emitted pulse. As with optical imagery, objects smaller than the measurement wavelength are invisible to the sensor. The use of SAR data is further complicated by the use of side-looking radar. SAR sensors operate with the emitter and receiver aimed laterally, rather than vertically, as for optical sensors. Since radar operates by measuring the time of arrival for a backscattered signal, aiming the sensor vertically would make it impossible to distinguish between targets at an equal distance to the left or right of the direction of travel. This is corrected by side-looking, with the sensor aimed laterally such that the entire field of view is to one side of the satellite. Although side-looking corrects directional ambiguity, it necessitates complex post-processing to correct the resulting geometrical distortions and 4 radiance redistribution [81–83], and results in the same terrain being imaged differently depending on the direction of satellite travel, as the terrain is illuminated from the opposite side. We provide data in both ascending (northwards) and descending (southwards) satellite directions in M3LEO. We provide three products derived from SAR data in this dataset — polarimetric amplitude, intereferograms, and coherence. We give a brief background on each of these datatypes below, although we omit most technical details regarding their construction as they are beyond the scope of this work. References are provided for users who are interested in further background on these datatypes. Amplitude Polarimetric amplitude measures the power of the backscattered signal received by the sensor. We provide amplitude data derived from ESA Sentinel-1 Level 1 Ground Range Detected SAR data (S1GRD), as available in Google Earth Engine (GEE)1. Phase information is not provided via GEE and it is therefore impossible to produce further SAR datatypes from data available on the GEE platform. We provide data measuring vertically polarised and horizontally polarised returns from a vertically polarised emission (referred to as VV and VH respectively). Data is provided for imagery from both ascending and descending trajectories. We refer users to [84] for a more detailed breakdown of the theory behind polarimetric SAR data. S1GRD data is of 10 m/pixel resolution and provided for 2018-2020 as four seasonal averages per-year. Interferometry Interformetry measures the phase difference between pairs of acquisitions over the same terrain. These phase differences provide data about small-scale displacements, which can be measured modulo the wavelength. Post-processed interferograms are unwrapped by computing accumulated modulo-wavelength displacements. As the scale at which these displacements can be measured depends on wavelength, rather than horizontal resolution, interferometric phase difference can be used to measure surface height changes with millimetre accuracy. We provide interferometric data computed using select pairs of Sentinel-1 acquisitions, from ASF ARIA Geocoded UNWrapped Inteferogram data (GUNW), as available in ASF vertex2[85]. We include all available acquisition pairs with time deltas of less than 72 days. We refer users to [86] for a detailed treatment of the processing steps required to construct interferograms in addition to the underlying physics. GUNW data is provided for 2020 at a resolution of approximately 90 m/pixel. Coherence The coherence of SAR imagery is calculated using the complex correlation between coincident pixels across two separate acquisitions. For a given complex valued pixel z(i) in SAR acquisitions at times 1 and 2, the coherence γ is defined as: γ = E[z(i) 1 (t)z(i) 2 (t)] q E[\|z(i) 1 (t)\|2]E[\|z(i) 2 (t)\|2] (1) To provide meaningful coherence values, expectations are computed within a small spatial window surrounding each pixel. The resulting resolution of coherence maps is therefore lower than that of the original acquisitions. Man-made structures typically exhibit high coherence, as they are stable across acquisitions. Forests and other vegetation larger than the wavelength of the instrument have lower coherence. Coherence is affected in all cases by additional decorrelation factors not relating directly to terrain, such as doppler centroid difference or thermal noise. [87] and [88] provide more detailed treatments of coherence estimation. We provide coherence estimates from the Global Seasonal Sentinel-1 Interferometric Coherence (GSSIC) dataset [89], using date-pairs with time deltas of 12, 24, 36 and 48 days, with one date-pair per tile per season. GSSIC data is of approximately 90 m/pixel resolution and from 2020. A decay model is included with the GSSIC coherence data. For both interferometry and coherence, the selection of acquisition pairs is critical. The acquisition pair selection process introduces a significant combinatorial challenge to managing SAR datasets — if every possible combination between all acquisitions were considered, the number of interferograms or coherence estimates would grow quadratically with the number of acquisitions. We pre-empt this issue by the provision of pre-selected date-pairs. 1developers.google.com/earth-engine/datasets/catalog/COPERNICUS_S1_GRD 2asf.alaska.edu/datasets/daac/aria-sentinel-1-geocoded-unwrapped-interferograms/ 5 Optical data We additionally provide data (S2SRM) sourced from the ESA Sentinel-2 mission3[48], as available in Google Earth Engine. Data is provided from the L2A product (surface reflectance). We do not include top-of-atmosphere L1C reflectances. We summarize this data as monthly means of cloud-free pixels. We include four monthly averages for 2018-2020 — March, June, September and December. We include all Sentinel-2 bands with a of 10 m/pixel (red, green, blue, NIR) or 20 m/pixel (vegetation red edge, SWIR 11/12). 3.2 Auxiliary Datasets ESA World Cover Land cover classification labels (semantic segmentation) were obtained from the ESA World Cover product (ESAWC) [90], as available in Google Earth Engine4. The resolution of ESAWC is 10 m/pixel, and it comprises 11 classes (See Appendix E). The ESAWC product has been independently validated as having an accuracy of approximately 75% [91]. Data is provided for 2020. ESA CCI Biomass A map of above ground biomass (AGB) in Mg ha−1, derived from the ESA Climate Change Initiative’s Biomass product5 [92] is provided for 2020 at a resolution of 90 m/pixel. The relative error of this data is 20% for areas with a measured biomass exceeding 50 Mg ha−1 [93] MODIS Vegetation Cover Tree cover, non-tree cover and non-vegetated (bare) percentage labels derived from the Terra MODIS Vegetation Continuous Fields product (MODISVEG)6[94], are provided at a resolution of 250 m/pixel. A limited amount of independent validation reports the RMSE of the MODISVEG data as approximately 10% [94]. Our dataset includes yearly maps for 2016-2020. GHS Built Surface Maps of built surface area (m2/pixel), derived from the Copernicus Global Human Settlement Built Surface (GHSBUILTS) product7[95] are provided at 100 m/pixel for 2020. The mean absolute error of this data has been evaluated using independent reference data has been estimated to be approximately 6%[96] (or 600 m2/pixel, at 100 m/pixel). SRTM Digital Elevation Maps Digital Elevation Maps, derived from the NASA Shuttle Radar Topopgraphic Mission (SRTM)8[97, 98] are provided at a resolution of 90 m/pixel. This data was measured in 2000, but we emphasise that terrain height changes relatively little at this resolution compared to other products such as ESAWC or GHSBUILTS. The RMSE of the SRTM data was originally reported as 16 m [99] although it has been measured as more accurate in some regions [100]. 3.3 Data coverage Our dataset covers six distinct geographic areas of interest (AOIs): the contiguous United States (CONUS), Europe, the Middle East, Pakistan and India (PAKIN), China, and South America. A visualisation is provided in Figure 1. We limit coverage to these regions for reasons relating to the acquisition parameters used by Sentinel-1. Firstly, Sentinel-1 operates with different acquisition modes depending on the region. We choose to only include areas where Sentinel-1 uses the Interferometric Wide (IW) swath acquisition mode. In polar regions, Sentinel-1 employs the Extra Wide (EW) swatch acquisition mode, which introduces systematic differences in how terrain is illuminated — such as the azimuth steering angle of the radar emitter being 0.8° for EW and 0.6° for IW acquisition. The polarisations used in polar regions are reversed (HH, HV vs. VV, VH) compared to other areas. We provide IW data with both VV and VH polarisations in the initial release of this dataset. Secondly, much of the terrestrial surface of the Earth is only covered by a single direction of satellite travel. This includes much of North America, Africa, continental Asia, Oceania and the Amazon rainforest. Unlike passively illuminated data such as Sentinel-2, SAR actively illuminates terrain with 3developers.google.com/earth-engine/datasets/catalog/sentinel-2 4developers.google.com/earth-engine/datasets/catalog/ESA\_WorldCover\_v200 5gee-community-catalog.org/projects/cci_agb/ 6developers.google.com/earth-engine/datasets/catalog/MODIS\_006\_MOD44B 7human-settlement.emergency.copernicus.eu/ghs_buS2023.php 8developers.google.com/earth-engine/datasets/catalog/CGIAR_SRTM90_V4 6 a radar emitter aimed laterally from one side of the satellite. Orbital direction therefore systematically determines whether terrain is illuminated from an easterly or westerly direction. By focusing on regions with consistent acquisition parameters, we aim to provide a dataset that is more uniform and suitable for training models without introducing additional complexities. While including other regions might reduce geographic bias, it is not straightforward to address the potential systematic biases introduced by varying acquisition modes and polarisations, or the systematic lack of directional coverage in some regions. These issues warrant a detailed treatment beyond the scope of this work, and we refer readers to the Copernicus Wiki for a full set of details on Sentinel-1 coverage9 (particularly Figures 20, 21). Processing additional coverage is technically straightforward using the provided tools, should it be required. We include data for Europe despite its absence of GUNW coverage due to interest from data providers operating in Europe. A total of 1,048,827 unique geographic tiles were generated, covering an area of 2.11 × 107 km2. A breakdown by AOI can be seen in Table 2. Tiling is uniform across all AOIs and datasets—the chips provided for each dataset cover exactly the same geographical areas. It is important to note that not all component datasets or specific parameterisations, such as date-pair ranges, were available for every geographic tile; however, the availability is still consistently high. See Appendix A for a full breakdown by dataset and AOI. We provide a reduced version of our dataset, M3LEO-miniset, spanning 5,000 tiles per AOI for rapid model iteration and use in tutorials. Figure 1: Data splits. Geographic bands for training, validation and test sets, at a ratio of 60:20:20. Table 2: Coverage statistics summarised for each AOI in M3LEO. AOI Total No. Tiles Area (km2) % Earth’s land surface CONUS 167403 3.360×106 2.3% Europe 200489 4.024×106 2.7% Middle East 163986 3.291×106 2.2% PAKIN 147791 2.966×106 2.0% China 285402 5.728×106 3.7% South America 83756 1.681×106 1.1% Total 1048827 21.05×106 14.1% Data splits We provide use geographic bands to define training, validation and test splits for use in the explorations foudn in this work, following [65]. Bands were split into training, validation and test sets at a ratio of 60:20:20, and can be seen visually in Figure 1. This method reduces distribution shifts and data leakage compared to single-band splits and fully randomized assignments, respectively. See Appendix A for details. Users should define train-test splits appropriate for their individual applications if their needs differ. 9sentiwiki.copernicus.eu/web/s1-mission 7 3.4 Framework Data downloading & processing For each AOI, we provide a .geojson file containing the geographical extent and unique ID for each tile. These definitions are applied to each dataset at the point of tiling such that any chip with a given identifier spans precisely the same geographic extent of a chip corresponding to a different dataset of the same identifier. Throughout this work, we use the term tile to refer to a fixed area of the Earth’s surface defined in the definition files, and the term chip to refer to the data from a single component dataset, such as S1GRD or ESAWC, within the extent of a given tile. To process datasets available via Google Earth Engine (GEE) (S2RGB, S1GRD, SRTM, ESAWC, MODISVEG), we introduce geetiles10. This tool extracts and tiles data from GEE as per definition files and configurations provided in the geetiles repository. The remaining datasets (GSSIC, GUNW, AGB, GHSBUILTS) were extracted using sartiles11, which contains specific code to download and tile each of GUNW and GSSIC, as well as the facility to tile general GeoTIFF files, such as those provided for AGB and GHSBUILTS, for integration with M3LEO. Both geetiles and sartiles were developed alongside M3LEO, and are provided such that users are able to seamlessly integrate any data available via Google Earth Engine (or as a GeoTIFF file) with our framework. Pipeline We accompany our dataset with a modular PyTorch Lightning [101] framework parameterised using Hydra [29]. We provide PyTorch Lightning datasets for each component of M3LEO. We also provide additional modules defining self-supervised approaches applied successfully to M3LEO in previous works (MAE [65], CLIP [67], DINO [68, 69]). Integrating custom models with M3LEO is straightforward, requiring the addition of a single file. geetiles (GEE datasets) Data Processing Tools ML-ready Datasets PyTorch Lightning Framework sartiles (other GeoTIFF datasets) Built-up Surface S1GRD MODISVEG S2RGB SRTM DEM ESAWC AGB GHSBUILTS GUNW GSSIC M3LEO (17.5 TB of coregistered data tiles spanning 6 AOIs) M3LEO-miniset (smaller dataset of uncompressed chips for rapid testing) M3LEO Code Base Lightning Datasets Lightning Models Figure 2: M3LEO dataset and framework. The M3LEO dataset consists of nine ML-ready component datasets and a PyTorch Lightning framework, parameterised by Hydra, for model training. 4 Analysis We provide five auxiliary datasets (ESAWC, AGB, MODISVEG, GHSBUILTS, SRTM) in addition to SAR and optical satellite data acquisitions. Although these data could be used as labelled tasks (see Appendix D, and also [65, 67–69] for analysis of this type on M3LEO), they could equally be considered as providing information on important surface properties that would be non-trivial to derive directly from satellite data. We compare the shift in the marginal distribution of these terrain properties y, p(y) in Figure 3 for four of these auxiliary datasets. We computed the normalised L1 distance between the discrete ESAWC distributions and the Wasserstein distance between the continuous GHSBUILTS, MODISVEG and SRTM distributions, across AOIs. The marginal distributions p(y) for each auxiliary dataset are shown in Appendix B. We also provide an early exploration of the ‘appearance shift’ of features between AOIs for S1GRD — the change in the distribution of embeddings x for a SAR tile with known terrain properties y, P(x\|y). 10github.com/rramosp/geetiles 11github.com/rramosp/sartiles 8 To generate embeddings, we trained a masked autoencoder on S1GRD polarimetry from all six AOIs, following previous work on M3LEO [65]. Training hyperparameters can be seen in Appendix C. We applied max-pooling along the sequence dimension at the output of the ViT encoder and further reduced the dimension to 2 using UMAP [102]. We computed the expected Wasserstein distance between the conditional distributions p(x\|y) with respect to the distribution p(y) on the test set, across AOIs and show the results in Figure 4. For ESAWC, we applied principal component analysis and conditioned on the first 3 principal components with 10 evenly spaced bins per dimension. On the remaining variables, we used 100 evenly spaced bins. We also display the sliced Wasserstein distance between the embedding distributions p(x) in the leftmost matrix of Figure 4. We plot these reduced embeddings, coloured according to each of the terrain properties, in Figure 5. A checkpoint for the MAE model is available in the data repository. Figure 3: Distribution shifts. Distribution shifts between terrain properties described by auxiliary datasets. Distribution shift for the ESAWC categorical data was quantified using L1 distance and continuous data using Wasserstein distance. Figure 4: Covariance and appearance shift. (Leftmost) The Sliced Wasserstein Distances (SWD) between reduced train and test set embeddings across AOIs, generated using a masked autoencoder. (Right four) The expected value of same metric conditioned on each of four terrain properties. The expectation was computed with respect to the distribution of the terrain property on the test set. We conditioned on the first three principal components of the ESAWC distribution. We observed that there was significant covariate shift in the distribution of the embeddings produced by the masked autoencoder, p(x), across AOIs (Figure 4, leftmost matrix). Given that there was also significant shift in terrain properties described by the auxiliary datasets between AOIs (Figure 3, this is not immediately surprising. Another factor that must be considered, however, is the shift in the embeddings produced by the masked autoencoder for tiles with similar terrain properties y, p(x\|y), across AOIs (appearance shift). It can be seen in the four right-hand matrices of Figure 4 that although there is some reduction in the most extreme cases, the expected value of sliced Wasserstein distance conditioned on similar terrain properties is usually not substantially lower than the unconditioned case. Explained from an Earth observation perspective - the masked autoencoder does not extract overly similar features for two tiles with, for example, very high vegetation when those tiles are taken from different geographic regions. This effect can be seen visually in Figure 5 - embeddings with 9 Figure 5: Embedding scatter plots. Scatter plots of 2D embeddings, reduced using UMAP, coloured according to different auxiliary datasets. high values for particular labels do not cluster obviously across different AOIs. In contrast, previous work applying DINO-based self-supervision to M3LEO [69] found that embeddings with similar labels clustered tightly in the embedding space across AOIs. Despite this observation, previous work applying MAE-based pretraining to M3LEO has shown strong generalisation to novel AOIs [65]. We did not explore the use of GSSIC coherence data here, as this is substantially technically challenging and requires detailed treatment. Given the lower resolution of this data, it may not be productive to use as a naive input to deep learning models. A small set of experiments evidencing this is provided in Appendix D. We point readers to previous work using coherence data from M3LEO productively in a self supervised setting - in contrastive learning [67] or in knowledge distillation-based approaches [69]. We note that the provision of GUNW interferometric coherence data without reference to specific events such as floods or fires is unusual compared to other datasets [76–78]. Although we aim to include event-based datasets in a future update, this large-scale dataset of interferograms is still highly desirable in self-supervised schemes. A number of unknowns remain regarding domain shift in M3LEO. Many auxiliary datasets, such as ESAWC, do not exist for 2018 and 2019 so it is difficult to provide a substantial exploration of temporal shifts, although we provide S1GRD and S2SRM for three years. It is unclear whether features learned from encoders trained on different polarizations or orbital directions are comparable, although D contains a limited set of experiments on the value-add of different polarisations. 5 Conclusions and Future Work In this work, we introduced M3LEO, a multi-continental Earth observation dataset including a comprehensive set of SAR data, alongside digital elevation models, RGB data and a suite of downstream tasks. To the best our knowledge, this is the largest ML-readable polSAR dataset by total number of tiles and geographic coverage, and the largest inSAR dataset by the same metrics. We additionally provide a modular PyTorch Lightning framework to enable the application of deep learning and encourage the uptake of these datatypes. We provide additional tools, geetiles and sartiles, to enable the integration of any data available in Google Earth Engine with our framework. We trained an MAE-based model on polSAR data and conducted a small exploration on the appearance shift of features corresponding to similar labels across AOIs. Despite the fact that this type of training has previously been shown to generalise well geographically [65], the shift in low-level features useful for the reconstruction pretext task was substantial between geographic regions. This is in contrast to previous work using M3LEO that found embeddings from DINO-based models with similar labels from different AOIs clustered tightly. 10 6 Acknowledgements This work has been enabled by Frontier Development Lab Europe (https://fdleurope.org) a public / private partnership between the European Space Agency (ESA), Trillium Technologies, the University of Oxford and leaders in commercial AI supported by Google Cloud and Nvidia, developing open science for all Humankind. L.M-F. was supported by the European Research Council (ERC) Synergy Grant “Understanding and Modelling the Earth System with Machine Learning (USMILE)” under the Horizon 2020 research and innovation programme (Grant agreement No. 855187). M. J. A. was supported by the UKRI Centre for Doctoral Training in Application of Artificial Intelligence to the study of Environmental Risks [EP/S022961/1]. We are also indebted to Nicolas Longépé, Carlos López-Martínez, Fabio A. González Osorio, Samuel Bancroft, Emma Hatton, Alison Lowndes, Alistair Francis, Ioanna Bouri and the rest of reviewers during the 2023 FDL-Europe sprint. References [1] Xikun Hu, Yifang Ban, and Andrea Nascetti. Sentinel-2 MSI Data for Active Fire Detection in Major Fire-Prone Biomes: A Multi-Criteria Approach. International Journal of Applied Earth Observation and Geoinformation, 101:102347, September 2021. ISSN 1569-8432. doi: 10.1016/j.jag.2021.102347. [2] M. C. Hansen, P. V. Potapov, R. Moore, M. Hancher, S. A. 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U-Net: Convolutional Networks for Biomedical Image Segmentation, May 2015. [104] Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization, January 2017. 18 Checklist 1. For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] See Appendix F. (c) Did you discuss any potential negative societal impacts of your work? [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] 3. If you ran experiments (e.g. for benchmarks)... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Our code is available at https://github.com/spaceml-org/M3LEO and data at https: //huggingface.co/M3LEO and https://huggingface.co/M3LEO-miniset (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We intend only for our results to demonstrate the utility of our framework and data rather than being a baseline for direct comparison. Additionally, computational cost was too high for enough repeat runs to produce meaningful error bars. (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix D.3 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes] See Section 3. (b) Did you mention the license of the assets? [Yes] See Appendix G. (c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See https://github.com/spaceml-org/M3LEO. (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] 19 A Dataset Summary A.1 Chip counts Table A.1: Summary of component datasets in M3LEO, including number of chips and dataset size for each region (per year). Totals at the bottom are adjusted for multi-year datasets. Input Datasets CONUS Europe China Chips Size/GB Chips Size/GB Chips Size/GB S1GRD (2018-2020) 167403 1003 200489 1228 285402 1740 GSSIC (2020) 167403 73 200489 104 285399 125 GUNW (2020) 554844 579 N/A* N/A* 1027451 854 S2SRM (2018-2020) 167406 1228 200489 1433 285402 1945 ESAWC (2020) 167406 32 200489 39 285402 55 AGB (2018-2020) 167406 8.5 200489 11 285402 14 MODISVEG (2020) 167406 151 200489 189 285402 220 GHSBUILTS (2020) 167406 0.7 200489 1.5 285402 2.4 SRTM (2000) 167406 126 200489 151 285402 215 Total 2,563,704 7,680.2 2,806,846 8,500.5 5,023,076 12568.4 Input Datasets Middle East PAKIN S. America Chips Size/GB Chips Size/GB Chips Size/GB S1GRD (2018-2020) 163986 983 147791 886 83756 502 GSSIC (2020) 158985 68 147791 69 83756 34 GUNW (2020) 608865 619 486914 309 226093 155 S2SRM (2018-2020) 163986 1126 147791 992 83756 529 ESAWC (2020) 163986 32 147791 29 83756 16 AGB (2018-2020) 163986 7.7 147791 7 83756 4.1 MODISVEG (2020) 163986 88 147791 113 83756 79 GHSBUILTS (2020) 163986 1.3 147791 1.2 83756 0.7 SRTM (2000) 163986 124 147791 112 83756 63 Total 2,899,668 7,282.4 2,555,988 6,288.2 1,398,677 3,453 * GUNW data unavailable for Europe. A.2 Banding As outlined in Section 3.3, we stratified our data into training, validation and test splits using geographic banding. 60 bands were constructed at a fixed orientation for each AOI. See Table A.2 for the angles used per-AOI. Specifically, we allocated three adjacent bands for the training set, followed by one adjacent band each for the validation and test sets, in sequence, until all bands were categorized. See Figure 1 for a visual representation. 20 Table A.2: Angles for the construction of training, validation and test splits by geographic banding. Angles are measured in radians, clockwise (axis pointed towards the Earth), with west corresponding to an angle of 0. AOI Band Angle CONUS 0.9 Europe 0.9 China 1.5 Middle East 1.5 PAKIN 1.5 South America 0.6 B Marginal Distributions The marginal distributions of ESAWC, MODISVEG, STRM mean elevation and GHSBUILTS can be seen in Figures B.1, B.2, B.3 and B.4 respectively. Figure B.1: Marginal distribution for ESAWC. Note log scale. 21 Figure B.2: Marginal distribution for MODISVEG. Note log scale. 22 Figure B.3: Marginal distribution for SRTM. Mean elevation per-chip. Note log scale. 23 Figure B.4: Marginal distribution for GHSBUILTS. Note log scale. 24 C MAE Hyperparameters Training hyperparameters for the MAE-based model can be seen in Table C.1. We followed [65], other than including additional AOIs in the pretraining set. A checkpoint for this model is available at huggingface.co/M3LEO. Table C.1: Hyperparameter details for MAE pretraining used in main text. MAE Pretraining Encoder (Params) ViT-B [33] (88.8M) Decoder (Params) Reconstruction [35] (5.5M) Loss Function MSE Input Image Size 448 × 448 Output Image Size 448 × 448 Patch Size 16 × 16 Masking Type Random Masking Ratio 0.75 Optimiser AdamW Learning Rate 1.00E-04 No. Epochs 75 Input Dataset S1GRD Channels Seasonal, VV, VH, VV-VH, Ascending AOIs CONUS, Europe, China, Middle East, PAKIN, S. America 25 D Supervised Experiments In addition to our explorations of distribution shift, we performed a small set of supervised experiments, reframing the auxiliary datasets as labelled tasks. S1GRD, GSSIC coherence and S2SRM were used as input datasets. For S1GRD, we trained models separately using the VV and VH bands only, and with the VV and VH bands stacked at the input to the model. For all S1GRD models, four seasonal summaries were used for each band, resulting in four input channels for the VV and VH models, and eight channels for the model using both VV and VH. For GSSIC coherence, the coherence band was used with a single date pair of delta 36 days taken for each season, resulting in four input channels. For the S2SRM models, we used the red, green and blue channels with one input channel from each of the months of March, June, September and December, totalling 12 input channels. We additionally trained models using both S2SRM RGB in combination with each of the other datasets separately, stacking the bands at the input to the model. All input data was taken from 2020. We upscaled low resolution input datasets to 448 × 448 px before input to the model. We excluded GUNW from use in our baseline experiments to avoid introducing the mixed availability of GUNW chips as a confounding factor. ESAWC, AGB and GHSBUILTS labelled datasets were used as targets. ESAWC was formulated as semantic segmentation spanning 11 land use classes, for which segmentation accuracy is reported as mean intersection-over-union (mIoU). ESAWC data was used at the original resolution of 448 × 448. Both AGB and GHSBUILTS are formulated as regression tasks (per-pixel). Results are reported using root mean square error (RMSE), in Mg ha−1 for AGB and in m2 built surface for GHSBUILTS. We resized labels for AGB and GHSBUILTS to a fixed size of 45×55, to account for minor differences in dimension from chip-to-chip. We note that this means our pixels may not span exactly 1 hectare and therefore that results for AGB measured in Mg ha−1 are a heuristic rather than an absolute measure of biomass. We excluded data from Europe in these experiments due to the absence of GUNW data in this region. This constraint was applied despite having not used GUNW in these baselines, as these experiments were completed prior to the exclusion of GUNW and repeating them was prohibitively expensive. All models were trained on the same random 10% subset of the data. The spatial coverage of this subset is still similar to other popular large-scale EO datasets. We used the entire test set to compute the final metrics. Models We used a UNet-style architecture for all baseline experiments, following [103] with two major changes — halving the number of channels for all layers and replacing the up-convolutions with bilinear upsampling. For the AGB and GHSBUILTS regression tasks, we applied average pooling to the single-channel 448×448 UNet output to achieve an output dimension of 45×55. We opted not to use data augmentation. Selecting augmentations for SAR data is challenging — common choices such as rotation or flipping may introduce invariance to information specific to different instrument polarisations or orbital direction, for example. For further details on training and hyperparameters, see Appendix D.3. D.1 Results Results for all tasks are reported in Table D.1. For the experiments using a single data source as input, S1GRD using both polarisations (VV+VH) achieved the best result for the ESAWC (MIoU: 0.4185) and AGB (RMSE: 27.467 Mg ha−1) tasks. S2RGB achieved the best result for GHSBUILTS (RMSE: 131.968 m2). GSSIC coherence produced the worst result in all three cases (ESAWC MIoU: 0.2906, AGB RMSE: 39.314 Mg ha−1, GHSBUILTS RMSE: 131.968 m2). The best results for experiments using multiple data sources as input were achieved by fusion of S1GRD and S2RGB in all cases (ESAWC MIoU: 0.4634, AGB RMSE: 25.137 Mg ha−1, GHSBUILTS RMSE: 124.852 m2), with significant improvements over either S1GRD or S2RGB individually. Fusion of S2RGB with GSSIC improved results marginally (ESAWC MIoU: 0.4198, AGB RMSE: 28.550 Mg ha−1, GHSBUILTS RMSE: 131.300 m2) compared to either individual data source individually in all cases. D.2 Discussion In all experiments, S1GRD combining VV and VH polarisations performed similarly to S2RGB, although the gap was small. The inclusion of both polarisations for S1GRD increased performance 26 Table D.1: Baseline Evaluation Results for ESAWC, AGB, and GHSBUILTS tasks using our UNet models, outlined in Section D. Input Dataset Bands ESAWC AGB GHSBUILTS MIoU RMSE (Mg ha−1) RMSE (m2 built) S1GRD VV 0.3976 28.573 152.259 S1GRD VH 0.3787 30.333 152.847 S1GRD VV+VH 0.4185 27.467 141.719 GSSIC Coherence 0.2906 39.314 196.242 S2RGB RGB 0.4094 28.689 131.968 S2RGB+S1GRD RGB+VV+VH 0.4634 25.137 124.852 S2RGB+GSSIC RGB+Coherence 0.4198 28.550 131.300 uniformly compared to either individually. Unlike optical data, reflected SAR pulses are polarised differently depending on the terrain, and each measured polarisation contains unique information about the geometry of the imaged area. In all cases, performance was significantly improved by using both S1GRD and S2RGB data in fusion, despite giving similar performances individually, confirming a common finding in previous work. Baseline experiments using GSSIC as the sole input data source performed significantly worse than for other input data types. This is likely explained in large part by the difference in resolution between GSSIC (90m, upsampled to 10m), S1GRD (10m) and S2RGB (10m). Performance improved slightly in all cases when GSSIC was included alongside in fusion with S2RGB. A nuanced approach is required for including coherence and interferometry, rather than naive direct input. Users may wish to use these data sources in self-supervised learning schemes — for example, pretraining models by constructing coherence or interferometry data from polarimetry pairs, or in a contrastive scheme alongside polarimetry data. Some work has been successful in using this type of pretraining [67]. Motivated by the success of polarimetry and coherence data in non-ML tasks [13–20], we suggest that these data types generated from date-paired SAR acquisitions are likely to show stronger performance on change detection-type tasks, which we did not demonstrate here. M3LEO may provide useful pretraining data for these tasks. D.3 Training Hyperparameters Hyperparameter details for supervised experiments can be seen in Table D.2. We did not perform significant hyperparameter tuning. Model selection was using best performance on the validation set. All models used approximately 7.9 × 106 trainable parameters, to the nearest 105. We show runtimes for each experiment on 2 NVIDIA V100 GPUs in Table D.3. Table D.2: Hyperparameter details for supervised experiments ESAWC AGB GHSBUILTS Task Type Semantic Segmentation Regression Regression Loss Function Cross Entropy RMSE RMSE Input Dimensions 448 × 448 448 × 448 448 × 448 Output Dimensions 448 × 448 45 × 55 45 × 55 Output Channels 11 1 1 Optimiser Adam [104] Adam [104] Adam [104] Learning Rate 1.00E-04 1.00E-04 1.00E-04 No. Epochs 75 75 75 Model Selection Best (val) Best (val) Best (val) 27 Table D.3: Runtimes for single-input and data fusion baseline experiments Input Dataset ESAWC AGB GHSBUILTS S1GRD (VV) 16h 23m 15h 38m 15h 45m S1GRD (VH) 16h 39m 15h 41m 15h 48m S1GRD (VV+VH) 16h 54m 17h 29m 23h 50m GSSIC 16h 29m 15h 33m 15h 41m S2 17h 51m 17h 59m 18h 2m S2+S1GRD (VV+VH) 22h 54m 24h 35m 23h 19m S2+GSSIC 19h 23m 68h 20m 19h 49m E ESA World Cover Land Cover Classes We outline the 11 classes that together comprise the ESAWC dataset in Table E.1, along with their pixel values in M3LEO. We opted to leave these pixel values as-found in the original ESAWC. Table E.1: ESAWC Classes along with pixel values for data as-found in M3LEO. Pixel Value Class 10 Tree Cover 20 Shrubland 30 Grassland 40 Cropland 50 Built-up 60 Bare/Sparse Vegetation 70 Snow and Ice 80 Permanent Water Bodies 90 Herbaceous Wetland 95 Mangroves 100 Moss and Lichen 28 F Limitations While we highlight that M3LEO comprises ML-ready data and and easy-to-use framework, we also call attention to a number of potential limitations regarding both the data and framework. F.1 Data Limitations Coverage While the M3LEO dataset is large, coverage is not global. We limited coverage to the area covered in the GUNW dataset, which is approximately equal to the regions in which Senintel-1 has dualpolarization, ascending-descending coverage. Generating further inteferometric data is substantially technically complex, computationally demanding and requires the use of SAR acquisitions with phase information (which are difficult to access compared to the amplitude data we provide). Were this data to be generated, cloud storage requirements for M3LEO would approach the petabyte scale, for which we are unable to provide a feasible long-term storage solution. Change Detection & Time Series Data We highlight the potential application of interferometric data to change detection tasks, but note that this data is not included in the initial relase of M3LEO. Two datasets that could be used for change detection tasks have been processed — namely, ESA CCI Burned Area MODIS12 and the Global Flood Database13 — but have not been tested extensively enough for inclusion here. We are unable to guarantee the release of these components simultaneously with the data advertised in the main text of this work, but aim to release them in the future. Multitemporal Data Data from S1GRD, S2SRM and AGB have been tiled for 2018-2020 additionally, but other datasets are provided for 2020 only. Many datasets, such as ESAWC, simply do not exist for 2018 or 2019. One satellite of the Sentinel-1 pair, Sentinel-1B, suffered a power unit failure in December 2021, so we cannot provide data with the same coverage as 2018-2020 from 2021 onwards. F.2 Framework Limitations Data Loading Data is currently loaded from the disk using xarray. We chose to use xarray as it easily accommodates handling the wealth of metadata associated with remote sensing imagery, but it is not performant for loading a large number of tiles quickly. For users who wish to access the same data many times — either over many epochs, or many experiments — we recommend caching the returned data arrays at the first encounter. We provide this facility using blosc2. The caching process can be computationally expensive for large datasets, but is relatively far cheaper than performing repeat runs using xarray. Slightly increased disk space requirements are incurred. We provide data for download as parquet files, but Apache Spark is not currently integrated with the framework and this data will need to be decompressed before use. Visualisation While we did not advertise data visualisation within the main body of this work, a small number of tools to visualise model outputs exist. We aim to include these in the initial code release but are unable to guarantee this. Tile Size We provide data chips at a fixed spatial size of 448 × 448 m. While we provide the facility to re-tile data straightforwardly at different sizes with geetiles and sartiles, this process is computationally expensive when tiling over large spatial areas. Benchmarking While we provided some baseline results (Appendix D) using our framework and data, we did not provide a benchmarking framework under which models could be compared. For example, we made no assertion as to whether models should be evaluated on chips with missing channels — we chose to fill any missing data with a dummy value of 0. We also did not include European data in our test set. Although comparisons could be made by copying our approach exactly, we encourage domain experts to evaluate models according to the needs of their particular application. 12developers.google.com/earth-engine/datasets/catalog/ESA_CCI_FireCCI_5_1 13developers.google.com/earth-engine/datasets/catalog/GLOBAL_FLOOD_DB_MODIS_EVENTS_V1 29 G Licenses The licenses under which each component of M3LEO was originally distributed are listed in Table G.1. We distribute our framework and data under the Creative Commons BY-SA 4.0 license.. Table G.1: Licenses for components of M3LEO Dataset License S1GRD Creative Commons BY-SA 3.0 IGO GSSIC Creative Commons BY 4.0 DEED GUNW Other (free use)1 S2RGB Creative Commons BY-SA 3.0 IGO ESAWC Creative Commons BY 4.0 DEED AGB Other (free use)2 MODISVEG No restrictions3 GHSBUILTS Other (free use)4 SRTM Other (free-use)15 1 https://www.jpl.nasa.gov/jpl-image-use-policy 2 https://artefacts.ceda.ac.uk/licences/specific_licences/esacci_biomass_terms_and_ conditions_v2.pdf 3 https://developers.google.com/earth-engine/datasets/catalog/MODIS_006_MOD44B# terms-of-use 4 https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32011D0833 5 https://developers.google.com/earth-engine/datasets/catalog/USGS_SRTMGL1_003# terms-of-use 30	2024	3324
4,487	Auditing Local Explanations is Hard Robi Bhattacharjee University of Tübingen and Tübingen AI Center robi.bhattacharjee@wsii.uni-tuebingen.de Ulrike von Luxburg University of Tübingen and Tübingen AI Center ulrike.luxburg@uni-tuebingen.de Abstract In sensitive contexts, providers of machine learning algorithms are increasingly required to give explanations for their algorithms’ decisions. However, explanation receivers might not trust the provider, who potentially could output misleading or manipulated explanations. In this work, we investigate an auditing framework in which a third-party auditor or a collective of users attempts to sanity-check explanations: they can query model decisions and the corresponding local explanations, pool all the information received, and then check for basic consistency properties. We prove upper and lower bounds on the amount of queries that are needed for an auditor to succeed within this framework. Our results show that successful auditing requires a potentially exorbitant number of queries – particularly in high dimensional cases. Our analysis also reveals that a key property is the “locality” of the provided explanations — a quantity that so far has not been paid much attention to in the explainability literature. Looking forward, our results suggest that for complex high-dimensional settings, merely providing a pointwise prediction and explanation could be insufficient, as there is no way for the users to verify that the provided explanations are not completely made-up. 1 Introduction Machine learning models are increasingly used to support decision making in sensitive contexts such as credit lending, hiring decisions, admittance to social benefits, crime prevention, and so on. In all these cases, it would be highly desirable for the customers/applicants/suspects to be able to judge whether the model’s predictions or decisions are “trustworthy”. New AI regulation such as the European Union’s AI Act can even legally require this. One approach that is often held up as a potential way to achieve transparency and trust is to provide local explanations, where every prediction/decision comes with a human-understandable explanation for this particular outcome (e.g., LIME (Ribeiro et al., 2016), SHAP (Lundberg and Lee, 2017), or Anchors (Ribeiro et al., 2018)). However, in many real-world scenarios, the explanation receivers may not necessarily trust the explanation providers (Bordt et al., 2022). Imagine a company that uses machine learning tools to assist in screening job applications. Because the company is well-advised to demonstrate fair and equitable hiring, it is plausible that it might bias its explanations towards depicting these properties. And this is easy to achieve: the company is under full control of the machine learning model and the setup of the explanation algorithm, and prior literature (Ghorbani et al., 2019; Dombrowski et al., 2019) has shown that current explainability tools can be manipulated to output desirable explanations. This motivates the question: what restrictions or procedures could be applied to prevent such explanation cheating, and more specifically, what are ways to verify that the provided explanations 38th Conference on Neural Information Processing Systems (NeurIPS 2024). (a) Insufficient data for auditing (b) Sufficient data for auditing Figure 1: Local explanations (see Section 2.2 for notation): In both panels, a set of training points x and their classifications f(x) (red/blue, decision boundary in green) are shown. For three training points (one centered at each ball), a local linear explanation (gx, Rx) is illustrated where gx is a local linear classifier (black decision boundary) and Rx is a local ball centered at x. Panel (a) depicts a regime where there is insufficient data for verifying how accurate the local explanations approximate the classifier f – none of the provided regions contain enough points to assess the accuracy of the linear explanations. Panel (b) depicts a regime with more training points allowing us to validate the accuracy of the linear explanations based on how closely they align with the points in their corresponding regions. are actually trustworthy? One approach is to require that the explanation providers completely publicize their models, thus allowing users or third-party regulators to verify that the provided explanations are faithful to the actual model being used. However, such a requirement would likely face stiff resistance in settings where machine learning models are valuable intellectual property. In this work, we investigate an alternative approach, where a third-party regulator or a collective of users attempt to verify the trustworthiness of local explanations, simply based on the predictions and explanations over a set of examples. The main idea is that by comparing the local explanations with the actual predictions across enough data one could, in principle, give an assessment on whether the provided explanations actually adhere to the explained model. The goal of our work is to precisely understand when this is possible. 1.1 Our contributions: data requirements for auditing. We begin by providing a general definition for local explainability that encompasses many popular explainability methods such as Anchors (Ribeiro et al., 2018), Smooth-grad (Smilkov et al., 2017), and LIME (Ribeiro et al., 2016). We define a local explanation for a classifier f at a point x as a pair (Rx, gx), where Rx is a local region surrounding x, and gx is a simple local classifier designed to approximate f over Rx. For example, on continuous data, Anchors always output (Rx, gx) where Rx is a hyper-rectangle around x and gx is a constant classifier; gradient-based explanations such as Smooth-grad or LIME implicitly approximate the decision function f by a linear function in a local region around x. Obviously, any human-accessible explanation that is being derived from such a local approximation can only be trustworthy if the local function gx indeed approximates the underlying function f on the local region Rx. Hence, a necessary condition for a local explanation to be trustworthy is that the function gx is close to f on the region Rx, and this should be the case for most data points x sampled from the underlying distribution. To measure how closely a set of local explanations adheres to the original classifier f, we propose an explainability loss function Lγ(E, f), which quantifies the frequency with which f differs by more 2 than γ from the local classifier gx over the local region Rx (see Sec. 2.2 for precise definitions). We then introduce a formalism for auditing local explanations where an auditor attempts to estimate the explainability loss Lγ(E, f). In our formalism, the auditor does so with access to the following objects: 1. A set of data points X = {x1, . . . , xn} drawn i.i.d from the underlying data distribution. 2. The outputs of a classifier on these points, f(X) = {f(x1), . . . , f(xn)}. 3. The provided local explanations for these points E(f, X) = {E(f, x1), . . . , E(f, xn)} Observe that in our formalism, the auditor has only restricted access to the machine learning model and the explanations: they can only interact with them through their evaluations at specific data-points. We have chosen this scenario because we believe it to be the most realistic one in many practical situations, where explanation providers try to disclose as little information on their underlying machine learning framework as possible. In our main result, Theorem 4.1, we provide a lower bound for the amount of data needed for an auditor to accurately estimate Lγ(E, f). A key quantity in our analysis is the locality of the provided explanations. We show that the smaller the provided local regions Rx are, the more difficult it becomes to audit the explainer. Intuitively, this holds because estimating the explainability loss relies on observing multiple points within these regions, as illustrated in Panel (b) of Figure 1. By contrast, if this fails to hold (Panel (a)), then there is no way to validate how accurate the local explanations are. We also complement our lower bound with an upper bound (Theorem 4.2) that demonstrates that reasonably large local regions enable auditing within our framework. Our results imply that the main obstacle to auditing local explanations in this framework is the locality of the provided explanations. As it turns out, this quantity is often prohibitively small in practice, making auditing practically impossible. In particular, for high-dimensional applications, the local regions Rx given by the explainer are often exponentially small in the data-dimension. Thus the explanations cannot be verified in cases where there does not exist any prior trust between the explanation provider and the explanation receivers. We stress that estimating the local loss Lγ(E, f) serves as a first baseline on the path towards establishing trustworthy explanations. It is very well possible that an explanation provider achieves a small local loss (meaning that the local classifiers closely match the global classifier f) but nevertheless provides explanations that are misleading in some other targeted manner. Thus, we view successful auditing in this setting as a necessary but not sufficient condition for trusting an explanation provider. Our results might have far-reaching practical consequences. In cases where explanations are considered important or might even be required by law, for example by the AI Act, it is a necessary requirement that explanations can be verified or audited (otherwise, they would be completely useless). Our results suggest that in the typical high dimensional setting of modern machine learning, auditing pointwise explanations is impossible if the auditor only has access to pointwise decisions and corresponding explanations. In particular, collectives of users, for example coordinated by non-governmental organizations (NGOs), are never in the position to audit explanations. The only way forward in auditing explanations would be to appoint a third-party auditor who has more power and more access to the machine learning model, be it access to the full specification of the model function and its parameters, or even to the training data. Such access could potentially break the fundamental issues posed by small local explainability regions in our restricted framework, and could potentially enable the third party auditor to act as a moderator to establish trust between explanation receivers and explanation providers. 1.2 Related Work Prior work (Yadav et al., 2022; Bhatt et al., 2020; Oala et al., 2020; Poland, 2022) on auditing machine learning models is often focused on applying explainability methods to audit the models, rather than the explanations themselves. However, there has also been recent work (Leavitt and Morcos, 2020; Zhou et al., 2021) arguing for more rigorous ways to evaluate the performance of various explanation methods. There are numerous approaches for doing so: including performance based on human-evaluation (Jesus et al., 2021; Poursabzi-Sangdeh et al., 2021), and robustness (Alvarez-Melis and Jaakkola, 2018). 3 There has also been a body of work that evaluates explanations based on the general notion of faithfulness between explanations and the explained predictor. Many approaches (Wolf et al., 2019; Poppi et al., 2021; Tomsett et al., 2020) examine neural-network specific measures, and typically rely on access to the neural network that would not be present in our setting. Others are often specialized to a specific explainability tool – with LIME (Visani et al., 2022; Botari et al., 2020) and Shap (Huang and Marques-Silva, 2023) being especially popular choices. By contrast, our work considers a general form of local explanation, and studies the problem of auditing such explanations in a restricted access setting, where the auditor only interacts with explanations through queries. To our knowledge, the only previous work in a similar setting is (Dasgupta et al., 2022), in which local explanations are similarly audited based on collecting them on a set of data sampled from a data distribution. They consider a quantity called the local sufficiency, which directly corresponds to our notion of local loss (Definition 2.3). However, their work is restricted to a discrete setting where local fidelity is evaluated based on instances that receive identical explanations. In particular, they attempt to verify that points receiving identical explanations also receive identical predictions. By contrast, our work lies within a continuous setting, where a local explanation is said to be faithful if it matches the underlying model over a local region. A central quantity to our analysis is the locality of an explanation, which is a measure of how large the local regions are. Prior work has rarely measured or considered this quantity, with a notable exception being Anchors method (Ribeiro et al., 2018) which utilizes it to assist in optimizing their constructed explanations. However, that work did not explore this quantity beyond treating it as a fixed parameter. Finally, we note that other recent work, such as (Bassan and Katz, 2023), provides avenues for providing explanations with certifiable correctness, meaning that they provide proof that their accurate reflect the underlying model. We view our work as complementary to such methods as our work demonstrates the necessity of such ideas by demonstrating difficulties with using generic local explanation methods. 2 Local Explanations 2.1 Preliminaries In this work, we restrict our focus to binary classification – we let µ denote a data distribution over Rd, and f : Rd →{±1} be a so-called black-box binary classifier that needs to be explained. We note that lower bounds shown for binary classification directly imply lower bounds in more complex settings such as multi-class classification or regression. For any measurable set, M ⊆Rd, we let µ(M) denote the probability mass µ assigns M. We will also let supp(µ) denote the support of µ, which is the set of all points x such that µ ({x′ : \|\|x −x′\|\| ≤r}) > 0 for all r > 0. We define a hyper-rectangle in Rd as a product of intervals, (a1, b1] × · · · × (ad, bd], and let Hd denote the set of all hyper-rectangles in Rd. We let Bd denote the set of all L2-balls in Rd, with the ball of radius r centered at point x being defined as B(x, r) = {x′ : \|\|x −x′\|\| ≤r}. We will utilize the following two simple hypothesis classes: Cd, which is the set of the two constant classifiers over Rd, and Ld, which is the set of all linear classifiers over Rd. These classes serve as important examples of simple and interpretable classifiers for constructing local explanations. 2.2 Defining local explanations and explainers One of the most basic and fundamental concepts in Explainable Machine Learning is the notion of a local explanation, which, broadly speaking, is an attempt to explain a complex function’s behavior at a specific point. In this section, we describe a general form that such explanations can take, and subsequently demonstrate that two widely used explainability methods, LIME and Anchors, adhere to it. We begin by defining a local explanation for a classifier at a given point. Definition 2.1. For x ∈Rd, and f : Rd →{±1}, a local explanation for f at x is a pair (Rx, gx) where Rx ⊆Rd is a region containing x, and gx : Rx →{±1} is a classifier. 4 Here, gx is typically a simple function intended to approximate the behavior of a complex function, f, over the region Rx. The idea is that the local nature of Rx simplifies the behavior of f enough to provide intuitive explanations of the classifier’s local behavior. Next, we define a local explainer as a map that outputs local explanations. Definition 2.2. E is a local explainer if for any f : Rd →{±1} and any x ∈Rd, E(f, x) is a local explanation for f at x. We denote this as E(f, x) = (Rx, gx). We categorize local explainers based on the types of explanations they output – if R denotes a set of regions in Rd, and G denotes a class of classifiers, Rd →{±1}, then we say E ∈E(R, G) if for all f, x, E(f, x) outputs (Rx, gx) with Rx ∈R and gx ∈G. Local explainers are typically constructed for a given classifier f over a given data distribution µ. In practice, different algorithms employ varying amounts of access to both f and µ – for example, SHAP crucially relies on data sampled from µ whereas gradient based methods often rely on knowing the actual parameters of the model, f. To address all of these situations, our work takes a black-box approach in which we make no assumptions about how a local explainer is constructed from f and µ. Instead we focus on understanding how to evaluate how effective a given explainer is with respect to a classifier f and a data distribution µ. 2.3 Examples of Explainers We now briefly discuss how various explainability tools in practice fit into our framework of local explanations. Anchors: The main idea of Anchors (Ribeiro et al., 2018) is to construct a region the input point in which the desired classifier to explain remains (mostly) constant. Over continuous data, it outputs a local explainer, E, such that E(x) = (Rx, gx), where gx is a constant classifier with gx(x′) = f(x) for all x′ ∈Rd, and Rx is a hyper-rectangle containing x. It follows say that the Anchors method outputs an explainer in the class, E(Hd, Cd). Gradient-Based Explanations: Many popular explainability tools (Smilkov et al., 2017; Agarwal et al., 2021; Ancona et al., 2018) explain a model’s local behavior by using its gradient. By definition, gradients have a natural interpretation as a locally linear model. Because of this, we argue that gradient-based explanations are implicitly giving local explanations of the form (Rx, gx), where Rx = B(x, r) is a small L2 ball centered at x, and gx is a linear classifier with coefficients based on the gradient. Therefore, while the radius r and the gradient gx being used will vary across explanation methods, the output can be nevertheless interpreted as an explainer in E (Bd, Ld), where Bd denotes the set of all L2-balls in Rd, and Ld denotes the set of all linear classifiers over Rd. LIME: At a high level, LIME (Ribeiro et al., 2016) also attempts to give local linear approximations to a complex model. However, unlike gradient-based methods, LIME includes an additional featurewise discretization step where points nearby the input point, x, are mapped into a binary representation in {0, 1}d based on how similar a point is to x. As a consequence, LIME can be construed as outputting local explanations of a similar form to those outputted by gradient-based methods. Finally, as an important limitation of our work, although many well-known local explanations fall within our definitions, this does not hold in all cases. Notably, Shapley-value (Lundberg and Lee, 2017) based techniques do not conform to the format given in Definition 2.1, as it is neither clear how to construct local regions that they correspond to, nor the precise local classifier being used. 2.4 A measure of how accurate an explainer is We now formalize what it means for a local classifier, gx, to “approximate" the behavior of f in Rx. Definition 2.3. For explainer E and point x, we let the local loss, L(E, f, x) be defined as the fraction of examples drawn from the region Rx such that gx and f have different outputs. More precisely, we set L(E, f, x) = Pr x′∼µ[gx(x′) ̸= f(x)\|x′ ∈Rx]. 5 µ is implicitly used to evaluate E, and is omitted from the notation for brevity. We emphasize that this definition is specific to classification, which is the setting of this work. A similar kind of loss can be constructed for regression tasks based on the mean-squared difference between gx and f. We contend that maintaining a low local loss across most data points is essential for any reasonable local explainer. Otherwise, the explanations provided by the tool can be made to support any sort of explanation as they no longer have any adherence to the original function f. To measure the overall performance of an explainer over an entire data distribution, it becomes necessary to aggregate L(E, f, x) over all x ∼µ. One plausible way to accomplish this would be to average L(E, f, x) over the entire distribution. However, this would leave us unable to distinguish between cases where E gives extremely poor explanations at a small fraction of points as opposed to giving mediocre explanations over a much larger fraction. To remedy this, we opt for a more precise approach in which a user first chooses a local error threshold, 0 < γ < 1, such that local explanations that incur an explainabiliy loss under γ are considered acceptable. They then measure the global loss for E by determining the fraction of examples, x, drawn from µ that incur explainability loss above γ. Definition 2.4. Let γ > 0 be a user-specified local error threshold. For local explainer E, we define the explainability loss Lγ(E, f) as the fraction of examples drawn from µ that incur a local loss larger than γ. That is, Lγ(E, f) = Pr x∼µ[L(E, f, x) ≥γ]. We posit that the quantity Lγ(E, f) serves as an overall measure of how faithfully explainer E adheres to classifier f, with lower values of Lγ(E, f) corresponding to greater degrees of faithfulness. 2.5 A measure of how large local regions are The outputted local region Rx plays a crucial role in defining the local loss. On one extreme, setting Rx to consist of a single point, {x}, can lead to a perfect loss of 0, as the explainer only needs to output a constant classifier that matches f at x. But these explanations would be obviously worthless as they provide no insight into f beyond its output f(x). On the other extreme, setting Rx = Rd would require the explainer to essentially replace f in its entirety with gx, which would defeat the purpose of explaining f (as we could simply use gx instead). Motivated by this observation, we define the local mass of an explainer at a point x as follows: Definition 2.5. The local mass of explainer E with respect to point x and function f, denoted Λ(E, f, x), is the probability mass of the local region outputted at x. That is, if E(f, x) = (Rx, gx), then Λ(E, f, x) = Pr x′∼µ[x′ ∈Rx]. Based on our discussion above, it is unclear what an ideal local mass is. Thus, we treat this quantity as a property of local explanations rather than a metric for evaluating their validity. As we will later see, this property is quite useful for characterizing how difficult it is to estimate the explainability loss of an explainer. We also give a global characterization of the local mass called locality. Definition 2.6. The locality of explainer E with respect to function f, denoted Λ(E, f), is the minimum local mass it incurs. That is, Λ(E, f) = infx∈supp(µ) Λ(E, f, x). 3 The Auditing Framework Recall that our goal is to determine how explanation receivers can verify provided explanations in situations where there isn’t mutual trust. To this end, we provide a framework for auditing local explanations, where an auditor attempts to perform this verification with as little access to the underlying model and explanations as possible. Our framework proceeds in with the following steps. 1. The auditor fixes a local error threshold γ. 2. A set of points X = {x1, . . . , xn} are sampled i.i.d from data distribution µ. 3. A black-box classifier f is applied to these points. We denote these values with f(X) = {f(x1), . . . , f(xn)}. 6 4. A local explainer E outputs explanations for f at each point. We denote these explanations with E(f, X) = {E(f, x1), . . . , E(f, xn)}. 5. The Auditor outputs an estimate A (X, f(X), E(f, X)) for the explainability loss. Observe that the auditor can only have access to the the model f and its corresponding explanations through the set of sampled points. Its only inputs are X, f(X), and E(f, X). In the context of the job application example discussed in Section 1, this would amount to auditing a company based on the decisions and explanations they provided over a set of applicants. In this framework, we can define the sample complexity of an auditor as the amount of data it needs to guarantee an accurate estimate for Lγ(E, f). More precisely, fix a data distribution, µ, a classifier, f, and an explainer E. Then we have the following: Definition 3.1. For tolerance parameters, ϵ1, ϵ2, δ > 0, and local error threshold, γ > 0, we say that an auditor, A, has sample complexity N(ϵ1, ϵ2, δ, γ) with respect to µ, E, f, if for any n ≥N(ϵ1, ϵ2, δ, γ), with probability at least 1 −δ over X = {x1, . . . , xn} ∼µn, A outputs an accurate estimate of the explainability loss, Lγ(E, f). That is, Lγ(1+ϵ1)(E, f) −ϵ2 ≤A (X, f(X), E(f, X)) ≤Lγ(1−ϵ1)(E, f) + ϵ2. Next, observe that our sample complexity is specific to the distribution, µ, the classifier, f, and the explainer, E. We made this choice to understand the challenges that different choices of µ, f, and E pose to an auditor. As we will later see, we will bound the auditing sample complexity using the locality (Definition 2.5), which is a quantity that depends on µ, f, and E. 4 How much data is needed to audit an explainer? 4.1 A lower bound on the sample complexity of auditing We now give a lower bound on the amount of data needed to successfully audit an explainer. That is, we show that for any auditor A and any data distribution µ we can find some explainer E and some classifier f such that A is highly likely to give an inaccurate estimate of the explainability loss. To state our theorem we use the following notation and assumptions. Recall that Hd denotes the set of hyper-rectangles in Rd, and that Cd denotes the set of the two constant binary classifiers over Rd. Additionally, we will include a couple of mild technical assumptions about the data distribution µ. We defer a detailed discussion of them to Appendices A.3 and A.1. We now state our lower bound. Theorem 4.1 (lower bound on the sample complexity of auditing). Let ϵ1, ϵ2 < 1 48 be tolerance parameters, and let γ < 1 3 be any local error threshold. Let µ be any non-degenerate distribution, and λ > 0 be any desired level of locality. Then for any auditor A there exists a classifier f : Rd →{±1} and an explainer E ∈E(Hd, Cd) such that the following conditions hold. 1. E has locality Λ(E, f) = λ. 2. There exist absolute constants c0, c1 > 0 such that if the auditor receives n ≤ c0 max(ϵ1, ϵ2)λ1−c1 max(ϵ1,ϵ2) many points, then with probability at least 1 3 over X = {x1, . . . , xn} ∼µn, A gives an inaccurate estimate of Lγ(E, f). That is, A (X, f(X), E(f, X)) /∈[Lγ(1+ϵ1)(E, f) −ϵ2, Lγ(1−ϵ1)(E, f) + ϵ2]. In summary, Theorem 4.1 says that auditing an explainer requires an amount of data that is inversely proportional to its locality. Notably, this result does not require the data-distribution to be adversarially chosen, and furthermore applies when the explainer E can be guaranteed to have a remarkably simple form being in E(Hd, Cd). Proof intuition of Theorem 4.1: The main intuition behind Theorem 4.1 is that estimating the local explainability loss, L(E, f, x), requires us to observe samples from the regions Rx. This would allow us to obtain an empirical estimate of L(E, f, x) by simply evaluating the fraction of points 7 from Rx that the local classifier, gx, misclassifies. This implies that the locality λ is a limiting factor as it controls how likely we are to observe data within a region Rx. However, this idea enough isn’t sufficient to obtain our lower bound. Although the quantity Ω 1 λ1−O(ϵ) does indeed serve as a lower bound on the amount of data needed to guarantee seeing a large number of points within a region, Rx, it is unclear what a sufficient number of observations within Rx is. Even if we don’t have enough points in any single region, Rx, to accurately estimate L(E, f, x), it is entirely plausible that aggregating loose estimates of L(E, f, x) over a sufficient number of points x might allow us to perform some type of estimation of Lγ(E, f). To circumvent this issue, the key technical challenge is constructing a distribution of functions f and fixing m = O 1 ϵ such that observing fewer than m points from a given region, Rx, actually provides zero information about which function was chosen. We include a full proof in Appendix A. 4.2 An upper bound on the sample complexity of auditing. We now show that if λ is reasonably large, then auditing the explainability loss Lγ(E, f) can be accomplished. As mentioned earlier, we stress that succeeding in our setting is not a sufficient condition for trusting an explainer – verifying that the local explanations gx match the overall function f is just one property that a good explainer would be expected to have. Thus the purpose of our upper bound in this section is to complement our lower bound, and further support that the locality parameter λ is the main factor controlling the sample complexity of an auditor. Our auditing algorithm proceeds by splitting the data into two parts, X1 and X2. The main idea is to audit the explanations given for points in X1 by utilizing the data from X2. If we have enough data, then it is highly likely for us to see enough points in each local region to do this. We defer full details for this procedure to Appendix B.1. We now give the an upper bound on its sample complexity. Theorem 4.2 (Upper Bound on Sample Complexity of Algorithm 1). There exists an auditor, A, for which the following holds. Let µ be a data distribution, f be a classifier, and E be an explainer. Suppose that E has locality λ with respect to µ and f. Let ϵ1, ϵ2, δ be tolerance parameters and let γ > 0 be a local error threshold. Then A has sample complexity at most N(ϵ1, ϵ2, δ, γ) = ˜O 1 ϵ2 2 + 1 λγ2ϵ2 1 . This bound shows that the locality is sufficient for bounding the sample complexity for auditing local explanations. We defer a full proof to Appendix B. Observe that the dependency on λ is O( 1 λ) which matches the dependency in our lower bound provided that ϵ1, ϵ2 →0. 5 The locality of practical explainability methods can be extremely small Theorems 4.1 and 4.2 demonstrate that the locality λ characterizes the amount of data needed for an Auditor to guarantee an accurate estimate of the explainability loss Lλ(E, f). It follows that if λ is extremely small, then auditing could require a prohibitive amount of data. This leads to the following question: how small is λ for practical explainability algorithms? To answer this, we will examine examine several commonly used algorithms that adhere to our framework. We begin with gradient-based methods, which can be construed as providing an explainer in the class E(Bd, Ld), where Bd denotes the set of L2 balls in Rd, and Ld denotes the set of linear classifiers. To understand the impact of dimension on the locality of such explainers, we begin with a simple theoretical example. Let µ be the data distribution over Rd that is a union of three concentric spheres. Specifically, x ∼µ is equally likely to be chosen at uniform from the sets S1 = {x : \|\|x\|\| = 1−α}, S2 = {x : \|\|x\|\| = 1}, and S3 = {x : \|\|x\|\| = 1 + β}, where α, β are small d-dependent constants (Defined in Appendix C). Let f : Rd →{±1} be any classifier such that f(x) = 1 if x ∈S1 ∪S3 and f(x) = −1 if x ∈S2. Observe that µ is a particularly simple data distribution over three spherical manifolds, and f is a simple classifier that distinguishes its two parts. We illustrate this distribution in panel (a) of Figure 2. Despite its simplicity, locally explaining f with linear classifiers faces fundamental challenges. We illustrate this in Figure 2. Choosing a large local neighborhood, as done at point A, leads to issues 8 Figure 2: An illustration of Theorem 5.1, with the concentric blue and red circles depicting the data distribution µ classified by f, and with local explanations being depicted at points A and B. Explanations are forced to either have large local loss (point A) or a low local mass (point B). posed by the curvature of the data distribution, meaning that it is impossible to create an accurate local linear classifier. On the other hand, choosing a neighborhood small enough for local linearity, as done in point B, leads to local regions that are exponentially small with respect to the data dimension. We formalize this in the following theorem. Theorem 5.1 (A high dimensional example). Let µ, f, be as described above, and let E be any explainer in E(Bd, Ld). Let x∗be any point chosen on the outer sphere, S3. Then E outputs an explanation at x∗that either has a large local loss, or that has a small local mass. That is, either L(E, f, x∗) ≥1 6, or Λ(E, f, x) ≤31−d. Theorem 5.1 demonstrates that if a locally linear explanation achieves even a remotely reasonable local loss, then it necessarily must have an extremely small local explanation. This suggests that, gradient based explanations will be exponentially local with respect to the data dimension, d. We believe that this is also exhibited in practice particularly over image data, where explanations are often verified based on perceptual validity, rather than relevance to practical training points beyond the point being explained. For example, the explanations given by SmoothGrad (Smilkov et al., 2017) are visualized as pixel by pixel saliency maps. These maps often directly correspond to the image being explained, and are clearly highly specific to the it (see e.g. Figure 3 of (Smilkov et al., 2017)). As a result, we would hardly expect the implied linear classifier to have much success over almost any other natural image. This in turn suggest that the locality would be extremely small. We also remark that a similar argument can be made for Lime, which also tends to validate its explanations over images perceptually (for example, see Figure 4 of Ribeiro et al. (2016)). Unlike the previous methods, Anchors (Ribeiro et al., 2018) explicitly seeks to maximize the local mass of its explanations. However, it abandons this approach for image classifiers, where it instead maximizes a modified form of locality based on super-imposing pixels from the desired image with other images. While this gives perceptually valid anchors, the types of other images that fall within the local region are completely unrealistic (as illustrated in Figure 3 of (Ribeiro et al., 2018)), and the true locality parameter is consequently extremely small. Thus, although Anchors can provide useful and auditable explanations in low-dimensional, tabular data setting, we believe that they too suffer from issues with locality for high-dimensional data. In particular, we note that it is possible to construct similar examples to Theorem 5.1 that are designed to force highly local Anchors-based explanations. 6 Conclusion Our results in Section 4 demonstrate that the locality of a local explainer characterizes how much data is needed to audit it; smaller local regions lead to larger amounts of data. Meanwhile, our 9 discussion in Section 5 shows that typical local explanations are extremely local in high-dimensional space. It follows that in many cases, auditing solely based on point-wise decisions and explanations is impossible. Thus, any entity without model access, such as a collective of users, are never in a position to guarantee trust for a machine learning model. We believe that the only way forward is through a more powerful third-party auditor that crucially as more access to the machine learning model, as this could potentially break the fundamental challenges posed by small explainability regions. We believe that investigating the precise types of access this would entail as an important direction for future work that might have broad practical consequences. Finally, although our definition of local explainers encompasses several widely used explanation methods, we do note that there are notable exceptions such as Shap (Lundberg and Lee, 2017), which does not fit into our paradigm. As a consequence, one important direction for future work is expanding our framework to encompass other local explanation methods and examine to what degree they can be audited. Acknowledgements This work has been supported by the German Research Foundation through the Cluster of Excellence “Machine Learning - New Perspectives for Science" (EXC 2064/1 number 390727645) and the Carl Zeiss Foundation through the CZS Center for AI and Law. References Agarwal, S., Jabbari, S., Agarwal, C., Upadhyay, S., Wu, S., and Lakkaraju, H. (2021). Towards the unification and robustness of perturbation and gradient based explanations. In International Conference on Machine Learning (ICML). Alvarez-Melis, D. and Jaakkola, T. S. (2018). On the robustness of interpretability methods. arxiv preprint 1806.08049. Ancona, M., Ceolini, E., Öztireli, C., and Gross, M. (2018). Towards better understanding of gradientbased attribution methods for deep neural networks. In International Conference on Learning Representations (ICLR). Bassan, S. and Katz, G. (2023). Towards formal XAI: formally approximate minimal explanations of neural networks. 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We believe this to be an extremely mild assumption considering that we typically operate in the regime where λ is exponentially small with the dimension, d, whereas the tolerance parameters are typically between 10−2 and 10−3. A.2 Main Proof Proof. Fix ϵ1, ϵ2, γ, λ, and µ, as given in the theorem statement. Our goal is to show the existence of classifier f and explainer E so that the auditor, A, is likely to incorrectly estimate the parameter, Lγ(E, f). To do so, our strategy will be instead to consider a distribution over choices of (E, f), and show that in expectation over this distribution, A estimates Lγ(E, f) poorly. To this end, we define the following quantities: 1. Let E be the explainer given in Section A.4. 2. Let f ∗be the random classifier defined in Section A.4. 3. Let n be any integer with n ≤ 1 2592λ(1−8max(ϵ1,ϵ2) . 4. Let X be a random variable for a set of points {x1, . . . , xn} drawn i.i.d from µ. 5. Let Y = f ∗(X) be a random variable for {f ∗(x1), f ∗(x2), . . . , f ∗(xn)}. Y has randomness over both f ∗and X∗. 6. Let ∆n = Rdn × {±1}n, and σ denote the measure over ∆n induced by (X, Y ). 7. By definition E’s output is independent of function, f ∗. Thus, we will abbreviate A’s output by writing A (X, f ∗(X), E(f ∗, X)) = A(X, Y, E). This emphasizes that both X and the output of E are independent of f ∗. 8. We let I∗denote the interval that the auditor seeks to output an estimate it. That is, I∗= Lγ(1 + ϵ1)(E, f ∗) −ϵ2, Lγ(1−ϵ1)(E, f ∗) + ϵ2 . Using this notation, we seek to lower bound that the auditor fails meaning we seek to lower bound, Pr f ∗,X [A(X, Y, E) /∈I∗] . To do so, let T1 denote the event T1 = 1 1 2 −ϵ2 < Lγ(1+ϵ1)(E, f ∗) ≤Lγ(1−ϵ1)(E, f ∗) < 1 2 + ϵ2 , and T2 denote the event T2 = 1 1 2 + 3ϵ2 < Lγ(1+ϵ1)(E, f ∗) ≤Lγ(1−ϵ1)(E, f ∗) < 1 2 + 5ϵ2 . The key observation is that any estimate, A(X, Y, E), can be inside at most one of the intervals, [ 1 2 −ϵ2, 1 2 + ϵ2] and [ 1 2 + 3ϵ2, 1 2 + 5ϵ2]. Using this, we can re-write our desired probability through the following integration. 12 Let (x, y) denote specific choices of (X, Y ). Note that in this context, x represents a set of points in (Rd)n, and y represents a set of labels in {±1}n. We then have the following: Pr f ∗,X[A(X, Y, E) /∈I∗] = Z ∆n Pr f ∗[A(x, y, E) /∈I∗\|X = x, Y = y] dσ(x, y) ≥ Z ∆n Pr f ∗[T1\|X = x, Y = y]1 (A(x, y, E) /∈I1) + Pr f ∗[T2\|X = x, Y = y]1 (A(x, y, E) /∈I1) dσ(x, y) ≥ Z ∆n min Pr f ∗[T1\|X = x, Y = y], Pr f ∗[T2\|X = x, Y = y] dσ(x, y), where the last equation holds because at least one of the events, A(x, y, E) /∈I1 and A(x, y, E) /∈I2, must hold. To bound this last quantity, we utilize Lemma A.9. Let S∗= (x, y) : Pr[T1\|X = x, Y = y], Pr[T0\|X = x, Y = y] ≥2 5 . By Lemma A.9, we have that σ(S∗) ≥5 6. It follows that Pr f ∗,X[A(X, Y, E) /∈I∗] ≥ Z ∆n min Pr f ∗[T1\|X = x, Y = y], Pr f ∗[T2\|X = x, Y = y] dσ(x, y) ≥ Z S∗min Pr f ∗[T1\|X = x, Y = y], Pr f ∗[T2\|X = x, Y = y] dσ(x, y) ≥ Z S∗ 2 5dσ(x, y) = 2 5σ(S∗) ≥1 3, which completes the proof as this implies with probability at least 1 3, the Auditors estimate is not sufficiently accurate. A.3 Non-degenerate Distributions Theorem 4.1 includes the assumption that µ is non-degenerate, which is defined as follows. Definition A.1. We say that data distribution µ over Rd is non-degenerate if for all x ∈Rd, there exists 1 ≤i ≤d such that µ ({x′ : x′ i = xi}) = 0. Being non-degenerate essentially means that at any point, x, the data distribution µ has a finite probability density with respect to some feature. This condition any distribution with a well-defined density over Rd (i.e. such as a Gaussian) and is also met for most practical data-sets in which any of the features is globally continuously distributed (i.e. mass in kg over a distribution of patients). We exclude data distributions with point masses because they can pose particularly simple cases in which there is a strict lower bound on how small the local region assigned to a given point can be. For example, in the extreme case where µ is concentrated on a single point, auditing any model or explanation over µ is trivial. We now show a useful property of non-degenerate distributions. Lemma A.2. Let µ be a non-degenerate distribution and R be a hyper-rectangle. Then R can be partitioned into two hyper-rectangles, R1, R2 such that µ(R1), µ(R2) ≥µ(R) 4 . Proof. Let R = (a1, b1] × (a2, b2] × · · · × (ad, bd]. First, suppose that there exists 1 ≤i ≤d such that for all r ∈(ai, bi], µ ({x : x = r} ∩R) ≤µ(R) 4 . 13 Let r∗= sup r : µ (R ∩{x : xi ≤r}) ≤µ(R) 4 . It follows that setting R1 = R ∩{x : xi ≤r∗} and R2 = R \ R1 will suffice as R1 will have probability mass at least µ(R) 4 and probability mass at most µ(R) 2 . Otherwise, suppose that no such i exists. Then, thus, there exists r1, r2, . . . , rd} such that µ(R ∩{x : xi = ri}) > 0. It follows that the point (r1, . . . , rd) violates Definition A.1, which is a contradiction. Thus some i exists, which allows us to apply the above argument, finishing the proof. A.4 Constructing f ∗and E We begin by partitioning the support of µ into hyper-rectangles such that each rectangle has probability mass in the interval [ α 4 , α]. We then further partition these rectangles into a large number of equal parts. Formally, we have the following: Lemma A.3. Let α > 0 be fixed, and K > 0 be any integer. Then for some integer L > 0, there exists a set of hyper-rectangles, {Rj i : 1 ≤i ≤L, 1 ≤j ≤K} such that the following hold: 1. R1 i , . . . RK i partition rectangle Ri. 2. For all 1 ≤i ≤L, α ≤µ(Ri) ≤4α. 3. For all 1 ≤i ≤L and 1 ≤j ≤K, µ(Ri) 4K ≤µ(Rj i) ≤µ(Ri) K . Proof. First construct R1, . . . , RL by using the following procedure: 1. Begin with the set A = {R∗} where R∗is a large rectangle containing the support of µ. 2. If A contains a rectangle, R, such that µ(R) > 4α, then apply Lemma A.2 to split R into two rectangles with mass at least µ(R) 4 and mass at most 3µ(R) 4 . 3. Repeat step 2 until no such rectangles, R, exist. This process clearly must terminate in a set of rectangles each of which has mass in the desired range, and also must terminate as a single rectangle can only be cut at most log 1 α log 3 4 times. Next, to construct R1 i , R2 i , RK i , we simply utilize an analogous procedure, this time starting with {Ri} and replacing α with µ(Ri) 4K . We now construct a fixed explainer, E. Definition A.4. Let E denote the explainer so that for all x ∈supp(µ), we have E(x) = (Rx, g+1) where Rx is the unique hyper-rectangle, Ri that contains x, and g+1 is the constant classifier that always outputs +1. We now construct a distribution over functions f, and let f ∗be a random function that follows this distribution. We have the following: Definition A.5. Let f ∗be a random classifier mapping Rd to {±1} constructed as follows. Let m be an integer and 0 ≤p1, . . . , p2m, q1, . . . , q2m ≤1 be real numbers that satisfy the conditions set forth in Lemma A.10. Then f ∗is constructed with the following steps: 1. Let P be a binary event that occurs with probability 1 2. 2. If P occurs, then set ri = pi for 1 ≤i ≤p2m. Otherwise set ri = qi. 3. If x /∈∪L i=1Ri, then f ∗(x) = +1. 4. For each rectangle Ri, randomly select 1 ≤k ≤2m at uniform. 14 5. For each sub-rectangle Rj i, with probability rk, set f(x) = −1 for all x ∈Rj i, and with probability 1 −rk, set f ∗(x) = +1 for all x ∈Rj i. Note that m is constructed based on ϵ1, ϵ2, and γ which we assume to be provided as in the statement of Theorem 4.1. A.5 Properties of f ∗and E We now prove several useful properties of this construction. To do so, we use the following notation: 1. We let f ∗denote the random variable representing the way f is generated. We use f ∗= f to denote the event that f ∗equals a specific function f : Rd →{±1}. 2. We let P denote the indicator variable for the binary event used in Section A.4 to construct f. 3. We let m denote the integer from Lemma A.10 that is used to construction f ∗. 4. We let X = (x1, . . . , xn) ∼µn denote a random variable of n i.i.d selected points from µ. We use x to denote a specific instance of X. 5. We let Y = (y1, . . . , yn) be a random variable over labels constructed by setting yi = f ∗(xi). We similarly use y to denote a specific instance of Y . 6. We let σ denote the measure over Rd × {±1} n associated with (X, Y ) . 7. We let ∆n denote the domain of the pair of random vectors (X, Y ) (as done in Section A.2) We begin with a bounds on the probability that we see any rectangle that has a large number of points selected from it. Lemma A.6. Let R1, . . . , RL be as defined in section A.4, and m as given. Let U denote the subset of ∆n such that U = {(x, y) : ∃1 ≤i ≤z, \|X ∩Ri\| ≥2m} . Then σ(U) ≤ 1 180. Proof. We bound the probability that a single rectangle, Ri, contains at least 2m points from X, and then apply a union bound over all L rectangles. By construction, µ(Ri) ≤4λ, which implies that for each point xj ∈X the probability that Xj falls within rectangle Ri is at most 4λ. Thus, for any set of 2m distinct points from X, the probability that they all fall within Ri is at most (4λ)2m. By taking a union bound over all n 2m subsets of 2m point from X, and substituting our assumed upper bound for n (point 3. of Section A.2), we have the following Pr[\|X ∩Ri\| ≥2m] ≤ n 2m (4λ)2m ≤ en 2m 2m (4λ)2m ≤ e 2m 1 2592 max(ϵ1, ϵ2)λ1−8 max(ϵ1,ϵ2) 2m (4λ)2m = (4λ) e 2m 41− 1 2m λ1− 1 2m 2592 max(ϵ1, ϵ2)λ1−8 max(ϵ1,ϵ2) !2m . 15 By definition (see Lemma A.10), m ≥ 1 16 max(ϵ1,ϵ2). Substituting this, and noting that λ1− 1 2m is increasing with respect to m (since λ < 1), we have Pr[\|X ∩Ri\| ≥2m] ≤(4λ) e 2m 41− 1 2m λ1− 1 2m 2592 max(ϵ1, ϵ2)λ1−8 max(ϵ1,ϵ2) !2m ≤(4λ) e8 max(ϵ1, ϵ2) 1 4λ1−8 max(ϵ1,ϵ2) 2592 max(ϵ1, ϵ2)λ1−8 max(ϵ1,ϵ2) 2m ≤(4λ) 96 2592 2m < λ 180. Finally, we apply a union bound over all rectangles. Observe that there are at most 1 λ such rectangles because by construction each rectangle has mass at most λ. Thus, our total probability is at most 1 λ λ 180 which is at most 1 180 as desired. Next, we leverage the properties from the construction of f to bound the conditional probability of P = 1 when (x, y) /∈U. Lemma A.7. Let (x, y) be in the support of σ so that (x, y) /∈U. Then Pr[P = 1\|(X, Y ) = (x, y)] = Pr[P = 0\|(X, Y ) = (x, y)] = 1 2. Proof. Our main idea will be to use Bayes-rule, and show that Pr[(X, Y ) = (x, y)\|P = 1] = Pr[(X, Y ) = (x, y)\|P = 0. This will suffice due to the fact that the prior distribution for P is uniform over {0, 1}. To do so, we first note that X is independent from P. For this reason, it suffices to show that Pr[Y = y\|P = 1, X = x] = Pr[Y = y\|P = 0, X = x]. To do so, we will express these probabilities in terms of the real numbers, p1, . . . , p2m and q1, . . . , q2m from which they were constructed (see Definition A.5). For each rectangle, Ri (see Lemma A.3), let Y ∩Ri denote the function values of all points in the set X ∩Ri. It follows from step 4 of Definition A.5 that the values in Y ∩Ri and Y ∩Rj are independent from each other. Thus, we can re-write our desired probability as Pr[Y = y\|P = 1, X = x] = L Y i=1 Pr (Y ∩Ri) = (y ∩Ri)\|P = 1, (X ∩Ri) = (x ∩Ri) . We now analyze the latter quantity for a rectangle, Ri. For convenience, let us relabel indices so that x ∩Ri = {x1, x2, . . . , xl} and y ∩Ri = {y1, . . . , yl} for some integer l ≥0. We also let X1, . . . , Xl and Y1, . . . , Yl denote the corresponding values for X ∩Ri and Y ∩Ri. We now further assume that that for all xa, xb ∈{x1, . . . , xl}, that xa and xb are contained within different sub-rectangles, Ra i , Rb i (see Definition A.5). If this isn’t the case, observe that we can simply remove the pair (xb, yb), as by the construction of f ∗, this will be forced to be identical to (xa, ya). By applying this assumption, we now have that for a given choice of the parameter rk (step 4 of Definition A.5), the values of y1, . . . , yl are mutually independent. Utilizing this, we have Pr (Y ∩Ri) = (y ∩Ri)\|P = 1, (X ∩Ri) = (x ∩Ri) = 1 2m 2m X j=1 lY k=1 yk 2 −ykpj + 1 2 = 1 2m 2m X j=1 F(pj), Where F is a polynomial of degree l. Here, the expression, yk 2 −ykpj + 1 2 simply evaluates to pj if yk = −1 (as pj is the probability of observing a −1) and 1 −pj otherwise. 16 Next, observe that the only difference when performing this computation for P = 0 is that we use the real numbers, q1, . . . q2m instead. Thus, we have, Pr (Y ∩Ri) = (y ∩Ri)\|P = 0, (X ∩Ri) = (x ∩Ri) = 1 2m 2m X j=1 lY k=1 yk 2 −ykqj + 1 2 = 1 2m 2m X j=1 F(qj), To show these two expression are equal, by assumption (x, y) /∈U which implies that l < 2m. Furthermore, by Lemma A.10, P2m k=1 pt k = P2m k=1 qt k,for all 0 ≤t ≤l. It follows that P2m k=1 F(pk) = P2m k=1 F(qk), which implies our desired result. Next, we bound the probability of events related to the value of Lγ, the parameter that the Auditor seeks to estimate. Lemma A.8. Let T1 denote the event that 1 2 −ϵ2 < Lγ(1+ϵ1) ≤Lγ(1−ϵ1) < 1 2 + ϵ2. Let T0 denote the event that 1 2 + 3ϵ2 < Lγ(1+ϵ1) < Lγ(1−ϵ1) ≤1 2 + 5ϵ2. Then taken over the randomness of the entire construction, Pr[T1, P = 1], Pr[T0, P = 0] ≥ 89 180, where P is the binary event defined above. Proof. By definition, Pr[P = 1] = Pr[P = 0] = 1 2. Thus, it suffices to show that Pr[T1\|P = 1], Pr[T0\|P = 0] ≥89 90. We begin with the case that P = 1 (the case for P = 0 will be similar). For each rectangle, Ri, let r(Ri) denote the choice of rk made for Ri in step 5 of Definition A.5. The crucial observation is that the value of r(Ri) nearly determines the local loss that E pays for points in Ri with respect to f ∗. In particular, if the number of sub-rectangles, K, is sufficiently large, then by the law of large numbers, we have that with high probability over the choice of f ∗, for all rectangles Ri and for all x ∈Ri, \|L(E, f ∗, x) −r(Ri)\| < 0.01γ(ϵ). (1) Let us fix K to be any number for which this holds, and assume that this value of K is set throughout our construction. Next, recall by Lemma A.10 that p1 < p2 < · · · < pm < γ(1 −2ϵ1) < γ(1 + 2ϵ1) < pm+1 < · · · < p2m. Recall that r(Ri) is chosen at uniform among {p1 . . . p2m} (step 5 of Definition A.5). It follows from Equation 1 that for any x ∈Ri, and for any α ∈{γ(1 −ϵ1), γ(1 + ϵ1)} that Pr f ∗[L(E, f ∗, x) ≥α for all x ∈Ri] = 1 2. Furthermore, because we are conditioning on P = 1, the value of f ∗within each rectangle, Ri, is independent. This implies that we can bound the behavior of Lα(E, f ∗) by expressing as a sum of independent variables. 17 Let α ∈{γ(1 −ϵ1), γ(1 + ϵ1)}, we have by Hoeffding’s inequality that Pr Lα(E, f ∗) ∈ 1 2 −ϵ2, 1 2 + ϵ2 = Pr " L X i=1 µ(Ri)1 (L(E, f ∗, x) ≥α for all x ∈Ri) ! ∈ 1 2 −ϵ2, 1 2 + ϵ2 # ≥1 −2 exp − 2ϵ2 2 PL i=1 µ(Ri)2 ! ≥1 −2 exp −2ϵ2 2 16λ ≥1 − 1 180 = 179 180; The penultimate inequality holds since µ(Ri) ≤4λ for each Ri, and because there are at most 1 λ such rectangles. The last inequality holds because λ < ϵ2 2 by the assumption in Section A.1. Thus by taking a union bound over both values of α, we have that Lα(E, f ∗) ∈ 1 2 −ϵ2, 1 2 + ϵ2 with probability at least 89 90. This completes our proof for the case P = 1. For P = 0, we can follow a nearly identical argument. The only difference is that the values of q (see Lemma A.10) are selected so that Pr f ∗[L(E, f ∗, x) ≥α] ≥1 2 + 4ϵ2. This results in the expected loss falling within a different interval, and an identical analysis using Hoeffding’s inequality gives the desired result. The main idea of proving Theorem 4.1 is to show that for many values of X, Y , the conditional probabilities of T1 and T0 occurring are both fairly large. This, in turn, will cause the Auditor to have difficulty as its estimate will necessarily fail for at least one of these events. To further assist with proving this, we have the following additional lemma. Lemma A.9. Let S∗denote the subset of Rd × {±1} n such that S∗= (x, y) : Pr[T1\|(X, Y ) = (x, y)], Pr[T0\|(X, Y ) = (x, y)] ≥2 5 . Then σ(S∗) ≥5 6. Proof. Let S′ 1 = {(x, y) : Pr[T1\|(X, Y ) = (x, y)] < 2 5}, and similarly S′ 2 = {(x, y) : Pr[T2\|(X, Y ) = (x, y)] < 2 5}. Then S∗= Rd × {±1} n \ (S′ 1 ∪S′ 2). Thus it suffices to upper bound the mass of S′ 1 and S′ 2. To do so, let U be the set defined in Lemma A.6. Then we have 89 180 ≤Pr[T1, P = 1] = Z (Rd×{±1})n Pr[T1, P = 1\|(X, Y ) = (x, y)]dσ ≤ Z S′ 1 Pr[T1, P = 1\|(X, Y ) = (x, y)]dσ + Z U\S′ 1 Pr[T1, P = 1\|(X, Y ) = (x, y)]dσ + Z ∆n\(S′ 1∪U) Pr[T1, P = 1\|(X, Y ) = (x, y)]dσ < 2 5σ(S′ 1) + (σ(U) −σ(U ∩S′ 1)) + 1 2 (σ(∆n \ U) −σ ((∆n \ U) ∩S′ 1)) ≤ 2 5 −1 2 σ(S′ 1) + 1 2σ(∆n \ U) + σ(U) ≤1 2 179 180 + 1 180 −σ(S′ 1) 10 18 Here we are simply leveraging the fact that Pr[P = 1\|X, Y = x, y] is precisely 1 2 when x, y are not in U, and consequently that the probability of T1 and P = 1 is at most 2 5, 1, and 1 2 when (x, y) is in the sets S′ 1, U \ S′ 1 and (∆n \ U) \ S′ 1 respectively. Finally, simplifying this inequality gives us σ(S′ 1) ≤ 1 12. A completely symmetric argument will similarly give us that σ(S′ 2) ≤ 1 12. Combining these with a union bound, it follows that σ(S∗) ≥5 6, as desired. A.6 Technical Lemmas Lemma A.10. For all 0 < γ, ϵ1, ϵ2 < 1 48, there exists m > 0, and real numbers 0 ≤ p1, p2, . . . , p2m, q1, . . . , q2m ≤1 such that the following four conditions hold: 1. For all 0 ≤t ≤2m −1, P2m i=1 pt i = P2m i=1 qt i. 2. p1 ≤p2 ≤· · · ≤pm < γ(1 −2ϵ1) < γ(1 + 2ϵ1) < pm+1 ≤. . . p2m. 3. q1 ≤q2 ≤· · · ≤qm−1 < qm = qm+1 = γ(1 + 2ϵ1) < qm+2 ≤. . . q2m. 4. 1 4ϵ2 ≥m ≥ 1 8 max(2ϵ1,ϵ2) + 1. Proof. Let l denote the largest integer that is strictly smaller than 1 8 max(2ϵ1,ϵ2), and let ϵ = 1 8l. It follows that ϵ ≥max(2ϵ1, ϵ2). Let Pl and Ql be as defined in Lemma A.11. Let m = 2l. Then it follows from the definitions of m, l that m = 2l ≤ 1 4 max(2ϵ1, max(ϵ2) ≤ 1 4ϵ2 , which proves the first part of property 4 in Lemma A.10. For the second part, by the definition of l, we have l ≥ 1 8 max(2ϵ1,ϵ2) −1. Since ϵ1, ϵ2 ≤ 1 48, it follows that l ≥2 which implies m = 2l ≥l + 2 ≥ 1 8 max(2ϵ1, ϵ2) + 1. Next, let p1, . . . , p4l denote the (sorted in increasing order) roots of the polynomial P ′ l (x) = Pl x −γ(1 + 2ϵ1) 2γϵ . Since the roots of Pl are explicitly given in Lemma A.11, it follows that the middle two roots of P ′ l (x) (which are the values of pm and pm+1) satisfy pm = γ(1 + 2ϵ1 −2ϵ), pm+1 = γ(1 + 2ϵ1 + 2ϵ). Because ϵ′ > ϵ, these values clearly satisfy the inequalities given by point 2 in the Lemma statement. Next, define q1, . . . , q4l as the (sorted in increasing order) roots of the polynomial, Q′ l(x) = Ql x −γ(1 + 2ϵ1) 2γϵ . Again using Lemma A.11, we see that qm = qm+1 = γ(1 + 2ϵ1), which satisfies point 3. To see that pi and qi are indeed in the desired range, we simply note that by substitution, both p1 and p2 must be larger than γ(1 + 2ϵ1) −4l(2γϵ). However, by definition, 4l(2γϵ) = γ. Thus, this quantity is larger than 0 which implies that p1 and q1 are both positive. Because γ < 1 10, a similar argument implies that p2m and q2m are at most 1. Finally, point 1 follows from the fact that p1, . . . , p2m and q1, . . . , q2m are the complete sets of roots of two polynomials that have matching coefficients for the first 2m coefficients. it follows by basic properties of Newton sums that P pt i = P qt i for 0 ≤i ≤2m −1, and this proves point 1. 19 Lemma A.11. For any l > 0, let Pl(x) = ((x + 1)(x + 3) . . . (x + 4l −1)) ((x −1)(x −3) . . . (x −4l + 1)) . Let Ql(x) = Pl(x)−Pl(0). Then Ql has 2l −1 distinct real roots over the interval (−4l, −1), 2l −1 distinct real roots over the interval (1, 4l), and a double root at x = 0. Proof. By symmetry, P ′ l (0) = Q′ l(0) = 0, and by definition Ql(0) = 0. It follows that x = 0 is a double root. Next, fix 1 ≤i ≤l −1. By definition, we have that Pl(4i −1) = Pl(4i + 1) = 0. We also have that Pl(4i) = 2(l+i) Y j=1 (2j −1) 2(l−i) Y j=1 (2j −1). Meanwhile, we also have that Pl(0) = Q2 j=1 l(2j −1) 2 . By directly comparing terms, it follows that Pl(i) is strictly larger than Pl(0). Thus, by the intermediate value theorem, Ql must have at least one root in both (4i −1, 4i) and (4i, 4i + 1). Using a similar argument, we can also show that Ql has at least one root in (4l −1, 4l). Since Pl is an even function, it follows that Ql is as well which means it symmetrically has roots in the intervals (−4i, −4i + 1) for 1 ≤i ≤l and (−4i −1, 4i) for 1 ≤i ≤l −1. Taken all together, we have constructed 2(l + l −1) = 4l −2 distinct intervals that each contain a root. Since Ql also has a double root at x = 0, it follows that this must account for all of its roots as deg(Ql) = deg(Pl) = 4l. B Proof of Theorem 4.2 B.1 Algorithm description The main idea of our auditor, simple_audit , is to essentially performs a brute-force auditing where we choose a set of points, X1, and attempt to assess the accuracy of their local explanations by using a a wealth of labeled data, X2, to validate it. Our algorithm uses the following steps (pseudocode given in Algorithm 1). 1. (lines 1 -3) We first partition X based on the tolerance parameters, ϵ1, ϵ2, δ. X1 will be the set of points that we validate, and X2 will be the set of points we use for validation. 2. (lines 8), For each point x in X1, we check whether there are enough points from X2 that fall within its local region, Rx, to accurate estimate its local loss. 3. (line 9-13) For each point satisfying the criteria in line 8, we evaluate its empirical local loss and then tally up how many points have a loss that is larger than γ. 4. (line 17) We output the proportion of points with loss larger than γ among all points whose loss we measured. At a high level, we expect this algorithm to succeed as long as we have enough data in each of the local regions induced from points in X1. B.2 Notation We use the following: 1. Let δ, ϵ1, ϵ2, γ be the tolerance parameters defined in Definition 3.1. 2. Let λ = Λ(E, f) denote the locality of E, f w.r.t. data distribution µ. 3. Let X1 be the set of points that are specifically being audited. 4. Let X2 be the set of points being used to audit. 5. Let \|X1\| = m. By definition, m > log 1 δ ϵ2 2 . 6. We set \|X2\| = n′ = n −m. By definition, n′ > stuff. 20 Algorithm 1 simple_audit(X, f(X), E(f, X), ϵ1, ϵ2, γ, δ) 1: m ←61 ϵ2 2 log 12 δ . 2: k ← 1 2γ2ϵ2 1 log 176 ϵ2δ . 3: X1 ←{x1, . . . , xm}, X2 ←X \ X1. 4: r′, b′ ←0. 5: for xi ∈X1 do 6: (Rxi, gxi) = E(xi, f) 7: Xi 1 = Rxi ∩X2. 8: if \|Xi 1\| ≥k then 9: ˆL(E, f, xi) ← 1 \|Xi 1\| P xj∈Xi 1 1 (gxi(xj) ̸= f(xj)) 10: if ˆL(E, f, xi) > γ then 11: r′ = r′ + 1. 12: else 13: b′ = b′ + 1. 14: end if 15: end if 16: end for 17: Return r′ r′+b′ . 7. For any x ∈Rd, we let E(x) = (Rx, gx) be the local explanation outputted for x by explainer E. We also define the following quantities related to estimating how frequently the local loss outputted by the explainer E is above the desired threshold, γ. 1. Let r∗= Prx∼µ[L(E, f, x) ≥γ(1 + ϵ1)]. 2. Let g∗= Prx∼µ[γ(1 −ϵ1) ≤L(E, f, x) ≤γ(1 + ϵ1)]. 3. Let b∗= Prx∼µ[L(E, f, x) ≤γ(1 −ϵ1)]. Here, r∗denotes the probability that a point has a large local error, b∗, the probability a point has a low local error, and g∗, the probability of an "in-between" case that is nearby the desired threshold, γ. By the definition of sample complexity (Definition 3.1), the goal of Algorithm 1 is to output an estimate that is inside the interval, [r∗−ϵ2, r∗+ g∗+ ϵ2]. Next, we define r, g, b as versions of these quantities that are based on the sample, X1. 1. Let r = Prx∼X1[L(E, f, x) ≥γ(1 + ϵ1)]. 2. Let g = Prx∼X1[γ(1 −ϵ1) ≤L(E, f, x) ≤γ(1 + ϵ1)]. 3. Let b = Prx∼X1[L(E, f, x) ≤γ(1 −ϵ1)]. Observe that while x is drawn at uniform from X1 in these quantities, we still use the true loss with respect toe µ, L(E, f, x), to determine whether it falls into r, g or b. Because of this, it becomes necessary to define two more fully empirical quantities that serve as estimates of r and b (we ignore g as it will merely contribute to a "margin" in our estimation terms). 1. Let r′ = Prx∼X1 " (Prx′∼X2[gx(x′) ̸= f(x′)\|x′ ∈Rx] > γ) and \|X2 ∩Rx\| ≥ log 176 ϵ2δ 2(γϵ1)2 # . 2. let b′ = Prx∼X1 " (Prx′∼X2[gx(x′) ̸= f(x′)\|x′ ∈Rx] ≤γ) and \|X2 ∩Rx\| ≥ log 176 ϵ2δ 2(γϵ1)2 # . The final estimate outputted by Algorithm 1 is precisely r′ r′+b′ . Thus, our proof strategy will be to show that for sufficiently large samples, r, g, b are relatively accurate estimates of r∗, g∗, b∗, and in turn r′ and b′ are relatively accurate estimates of r and b. Together, these will imply that our estimate is within the desired interval, [r∗, b∗]. 21 B.3 The main proof Proof. (Theorem 4.2) Let n ≥61 ϵ2 2 log 12 δ + log 176 ϵ2δ 2λγ2ϵ2 1 log 44 log 176 ϵ2δ ϵ2δγ2ϵ2 1 . By ignoring log factors, we see that n = ˜O 1 ϵ2 2 + 1 λγ2ϵ2 1 , thus satisfying the desired requirement in Theorem 4.2. Let X1 and X2 be as in Algorithm 1, and let m, n′ denote \|X1\| and \|X2 respectively. Directly from Algorithm 1, it follows that , m = 61 ϵ2 2 log 12 δ , n′ ≥ log 176 ϵ2δ 2λγ2ϵ2 1 log 44 log 176 ϵ2δ ϵ2δγ2ϵ2 1 . By letting ϵ = ϵ2 7 , and k = log 16 ϵδ 2(γϵ1)2 , we have that m ≥ 1 2ϵ2 log 12 δ , and that n′ ≥ log 176 ϵ2δ 2λγ2ϵ2 1 log 44 log 16 ϵδ ϵ2δγ2ϵ2 1 = log 16 ϵδ 2λγ2ϵ2 1 log 4 log 16 ϵδ ϵδγ2ϵ2 1 = k log 8k δϵ λ . Our bounds on m, k and n′ allow us to apply Lemmas B.1 and B.4 along with a union bound to get that the following equations hold simultaneously with probability at least 1 −δ over X ∼µn: \|r −r∗\|.\|g −g∗\|, \|b −b∗\| ≤ϵ, (2) r(1 −2ϵ) ≤r′ ≤r + g + bϵ (3) b(1 −2ϵ) ≤b′ ≤rϵ + g + b. (4) Recall that our goal is to show that r′ r′+b′ ∈[r∗−ϵ2, r∗+ g∗+ ϵ2] holds with probability at least 1 −δ. Thus, it suffices to show that this a simple algebraic consequence of equations 2, 3, and 4. To this end, we have r′ r′ + b′ (a) ≥ r(1 −2ϵ) r(1 −2ϵ) + rϵ + g + b ≥r(1 −2ϵ) r + b + g ≥ r r + b + g −2ϵ (b) ≥ r∗−ϵ r∗+ b∗+ g∗+ 3ϵ −2ϵ = r∗ 1 + 3ϵ − ϵ 1 + 3ϵ −2ϵ ≥r∗(1 −3ϵ) −ϵ −2ϵ ≥r∗−4ϵ −2ϵ (c) ≥r∗−ϵ2. Here step (a) follows from Equations 3 and 4, (b) from Equation 2, and (c) from the fact that ϵ = ϵ2 11. 22 For the other side of the inequality, we have r′ r′ + b′ (a) ≤ r + g + bϵ r + g + bϵ + b(1 −2ϵ) ≤ r + g + bϵ (r + g + b)(1 −ϵ) ≤ r + g (r + g + b)(1 −ϵ) + ϵ 1 −ϵ ≤ r + g r + g + b + 2ϵ + ϵ(1 + 2ϵ) (b) ≤ r∗+ g∗+ 2ϵ r∗+ g∗+ b∗−3ϵ + 3ϵ + 2ϵ2 = r∗+ g∗+ 2ϵ 1 −3ϵ + 3ϵ + 2ϵ2 (c) ≤(r∗+ g∗)(1 + 4ϵ) + 2ϵ(1 + 4ϵ) + 3ϵ + 2ϵ2 ≤r∗+ g∗+ 6ϵ + 8ϵ2 + 3ϵ + 2ϵ2 (d) ≤r∗+ g∗+ 7ϵ + 4ϵ (e) ≤r∗+ g∗+ ϵ2 Here step (a) follows from Equations 3 and 4, (b) from Equation 2, (c) from the fact that 1 1−3ϵ ≤1+4ϵ, (d) from ϵ = ϵ2 11 ≤1 8, 1 2, and (e) from the ϵ = ϵ2 11. B.4 Concentration lemmas In this section, we show several lemmas that allow us to bound the behavior of the random variables r, g, b, r′ and b′ (defined in section B.2). We also use m and n′ as they are defined in Section B.2 to be the sizes of \|X1\| and \|X2\| respectively. Finally, we also let ϵ = ϵ2 11. We begin by bounding the differences between r, g, b and r∗, g∗, b∗. Lemma B.1. Suppose that m ≥ 1 2ϵ2 log 12 δ . Then with probability at least 1 −δ 2 over X1 ∼µm, the \|r −r∗\|, \|g −g∗\|, \|b −b∗\| ≤ϵ. Proof. Observe that r is the average of m i.i.d binary variables each of which have expected value r∗. It follows by Hoeffding’s inequality that Pr[\|r −r∗\| > ϵ] ≤2 exp −2(ϵm)2 m ≤2 exp −2ϵ2 1 2ϵ2 log 12 δ = δ 6. By an identical argument, we see that the same holds for Pr[\|g −g∗\| > ϵ] and Pr[\|b −b∗\| > ϵ]. Applying a union bound over all three gives us the desired result. Next, we show that if n′ is sufficiently large, then it is highly likely that for any given point x, we observe a large number of points from X2 within the explanation region, Rx. Lemma B.2. Let x ∈supp(µ), and let k > 0 be an integer. Suppose that n′ ≥k log 8k δϵ λ . Then with probability at least 1 −δϵ 8 over X2 ∼µm, \|Rx ∩X2\| ≥k. 23 Proof. Partition X2 into k sets, X1 2, X2 2, . . . , Xk 2 each of which contain at least log 8k δϵ λ i.i.d points from µ. Because each point is drawn independently, we have that for any 1 ≤i ≤k, Pr[Xi 2 ∩Rx = ∅] = 1 −Pr x′∼µ[x′ ∈Rx] log 8k δϵ λ ≤(1 −λ) log 8k δϵ λ ≤exp −log 8k δϵ = δϵ 8k . Here we are using the definition of λ as a lower bound on the probability mass of Rx. Next, we show that if Rx has a sufficient number of points, then it is quite likely for the empirical estimate of the local loss at x to be accurate. Lemma B.3. Let x ∈supp(µ), and let k ≥ log 16 ϵδ 2(γϵ1)2 . Then conditioning on there being at least k elements from X2 in Rx, the empirical local loss at x differs from the true local loss by at most γϵ1 with probability at least 1 −δϵ 8 . That is, Pr X2∼µn′ "L(E, f, x) − 1 \|X2 ∩Rx\| X x′∈X2∩Rx 1 (gx(x′) ̸= f(x′)) > γϵ1 \|X2 ∩Rx\| ≥k # ≤δϵ 8 . Proof. The key idea of this lemma is that the distribution of k points drawn from µ conditioned on being in Rx is precisely the marginal distribution over which L(E, f, x) is defined. In particular, this means that the points in X2 ∩Rx can be construed as i.i.d drawn from the marginal distribution of µ over Rx. Given this observation, the rest of the proof is a straightforward application of Hoeffding’s inequality. Letting ˆL(E, f, x) = 1 \|X2∩Rx\| P x′∈X2∩Rx 1 (gx(x′) ̸= f(x′)) and K = \|X2 ∩Rx\|, we have Pr X2∼µn′ hL(E, f, x) −ˆL(E, f, x) > γϵ1 K ≥k i ≤2 exp −2(Kγϵ1)2 K ≤2 exp −log 16 δϵ = δϵ 8 , as desired. It follows by a union bound that the probability that least one of the sets in {Xi 2 ∩Rx : 1 ≤i ≤k} is empty is at most δϵ 8 . Thus with probability at least 1 −δϵ 8 , all the sets are non-empty which implies that \|Rx ∩X2\| ≥k, completing the proof. Finally, we use the previous two lemmas to show that r′ and b′ closely approximate r and b. Lemma B.4. Let k ≥ log 16 ϵδ 2(γϵ1)2 , and suppose that n′ ≥k log 8k δϵ λ . Then with probability at least 1 −δ 2 over X2 ∼µn′, the following equations holds: r(1 −2ϵ) ≤r′ ≤r + g + bϵ, b(1 −2ϵ) ≤b′ ≤rϵ + g + b. Proof. We begin by defining subsets of X1 that correspond to r, g, b, r′ and b′. We have 1. Let R = {x ∈X1 : L(E, f, x) ≥γ(1 + ϵ1)}. 2. Let G = {x ∈X1 : γ(1 −ϵ1) ≤L(E, f, x) ≤γ(1 + ϵ1)}. 24 3. Let B = {x ∈X1 : L(E, f, x) ≤γ(1 −ϵ1)]}. 4. Let R′ = x ∈X1 : Prx′∼X2[gx(x′) ̸= f(x′)\|x′ ∈Rx] > γ and \|X2 ∩Rx\| ≥ log 176 ϵ2δ 2(γϵ1)2 . 5. Let B′ = x ∈X1 : Prx′∼X2[gx(x′) ̸= f(x′)\|x′ ∈Rx] ≤γ and \|X2 ∩Rx\| ≥ log 176 ϵ2δ 2(γϵ1)2 . Observe that r, g, b, r′, and b′ are the probabilities that x ∼X1 is in the sets R, G, B, R′, and B′ respectively. Our strategy will be to use the previous lemmas to bound the sizes of the intersections, R′ ∩R, R′ ∩ B, B′ ∩R, B′ ∩B′. To this end, let x ∈R be an arbitrary point. By Lemma B.2, with probability at least 1 −δϵ 8 over X2 ∼µn′, x ∈R′ ∪B′. Furthermore, by Lemma B.3 (along with the definition of R), with probability at most δϵ 8 , x ∈B′. Applying linearity of expectation along with Markov’s inequality, we get the following two bounds: Pr X2 R ∩(X1 \ (R′ ∩B′) > \|R\|ϵ ≤EX2[\|R ∩(X1 \ (R′ ∩B′)\|] \|R\|ϵ ≤\|R\| δϵ 8 \|R\|ϵ = δ 8, Pr X2 R ∩B′ > \|R\|ϵ ≤EX2[\|R ∩B′\|] \|R\|ϵ ≤\|R\| δϵ 8 \|R\|ϵ = δ 8. Applying an analogous line of reasoning stating with x ∈B, we also have Pr X2 B ∩(X1 \ (R′ ∩B′) > \|B\|ϵ ≤δ 8, Pr X2 B ∩R′ > \|B\|ϵ ≤δ 8. Applying a union bound, none of these events occur with probability at least 1 −δ 2 over X2 ∼µn′. Thus, it suffices to show that they algebraically imply the desired inequalities. To this end, suppose none of them hold. Then we have, r′ = \|R′\| \|X1\| = \|R′ ∩B\| + \|R′ ∩G\| + \|R′ ∩R\| \|X1\| ≤\|B\|ϵ + \|G\| + \|R\| \|X1\| = bϵ + g + r, r′ = \|R′\| \|X1\| = \|R′ ∩B\| + \|R′ ∩G\| + \|R′ ∩R\| \|X1\| ≥0 + 0 + \|R\| −\|R ∩B′\| −\|R \ (B′ ∪R′)\| \|X1\| ≥\|R\| −\|R\|ϵ −\|R\|ϵ \|X1\| = r(1 −2ϵ). The upper and lower bounds on b′ are analogous. 25 C Proof of Theorem 5.1 C.1 Definitions and Notation Definition C.1. Let α = 1 3670016d4 and β = 1 3584d2 . Definition C.2. Let S1, S2, S3 be three (d −1)-spheres centered at the origin with radii (1 −α), 1, and 1 + β respectively for 0 < α, β. Definition C.3. Let µ denote the data distribution so that x ∼µ is selected by first selecting i ∈{1, 2, 3} at uniform, and then selecting x from Si at uniform. Definition C.4. Let f denote the classifier Rd →{±1} such that f(x) =    +1 \|\|x\|\|2 ≤1 −α 2 −1 1 −α 2 < \|\|x\|\|2 ≤1 + β 2 +1 \|\|x2\|\| > 1 + β 2 . Definition C.5. Let x∗be an arbitrary point chosen on S3, and let g be any linear classifier, and B(a, r) be any L2-ball that contains x∗. Lemma C.6. There exists x ∈S2 and 0 ≤θ1, θ2, θ3 ≤π such that S1 ∩B(a, r) = C(S1, x(1 −α), θ1), S2 ∩B(a, r) = C(S2, x, θ2), S3 ∩B(a, r) = C(S3, x(1 + β), θ3), where C(S, x, θ) denotes the spherical cap of angle θ centered at x on (d −1)-sphere S (see Definition C.18). C.2 Main Proof We begin by showing that the structure of the data distribution µ provides significant difficulty for linear classifiers. At a high level, the curvature of the spheres, S1, S2, S3, make separating them linearly only possible for small portions of the sphere. We formalize this with the following lemma. Lemma C.7. Let θ ≥π 4 . Let x be an arbitrary point on S2, and let T1(x, θ), T3(x, θ) denote the sets T1(x, θ) = C(S2, x, θ) ∪C(S1, x(1 −α), θ), T3(x, θ) = C(S2, x, θ) ∪C(S3, x(1 + β), θ). Let g : Rd →{±1} denote any linear classifier. Then g exhibits a loss of at least 1 3 over the conditional distribution of µ restricted to either T1 or T3. That is, Pr x′∼µ[g(x′) ̸= f(x′)\|x′ ∈T1(x, θ)], Pr x′∼µ[g(x′) ̸= f(x′)\|x′ ∈T3(x, θ)] ≥1 3. Next, we show that if the local explanation region B(a, r),contains a sufficiently large probability mass, then it also must include a region that takes the form given by T1 or T3 from Lemma C.7. Lemma C.8. Suppose that µ(B(a, r)) ≥31−d. Let T1 and T3 be as defined in Lemma C.7. Then there exist x ∈S2 and θ ≥π 4 such that at least one of the following hold: • T1(x, θ) ⊆B(a, r), and µ(T1(x,θ)) µ(B(a,r)) ≥1 2. • T3(x, θ) ⊆B(a, r), and µ(T3(x,θ)) µ(B(a,r)) ≥1 2. We are now prepared to prove Theorem 5.1. Proof. (Theorem 5.1) Suppose B(a, r) ≥31−d. Then by Lemma C.8, there exists θ ≥π 4 such that either T1(x, θ) or T3(x, θ) is a subset of B(a, r) and satisfies the conditions outlined above. Suppose that T1(x, θ) ⊆B(a, r) (the other case is analogous). 26 Let g be any linear classifier. Then it follows from Lemmas C.7 and C.8 that the loss g incurs over the conditional distribution of µ over B(a, r) can be bounded as follows: Pr z∼B(a,r)[g(z) ̸= f(z)] ≥Pr[z ∈T1(x, θ)] Pr[g(z) ̸= f(z)\|z ∈T1(x, θ)] ≥1 2 1 3 = 1 6, which concludes the proof. C.3 Proof of Lemma C.7 We will show that the claim holds for T3(x, θ) as the proof for T1(x, θ) is nearly identical (as α < β). Let w ∈Rd be a unit vector and b ∈R be a scalar such that g(z) = 1 ⟨w, z⟩≥b −1 ⟨w, z⟩< b . Our main strategy will be to find a large set of points within T3(x, θ) such that g(z) = g(z(1 + β)) for all z within this set. This will force g to misclassify either z or z(1 + β) which will lead to our desired error bound. To this end, define T ∗= n z ∈C(S2, x, θ) : g(z) = −1, g(z(1 + β)) = +1, \|⟨x, z⟩\| ≤cos π 8 o . Lemma C.9. µ(T ∗) µ(C(S2,x,θ)) ≤ 1 10. Proof. Let z be selected at uniform from C(S2, x, θ) \ C S2, x, π 8 ∪C S2, −x, π 8 . Note that z definitionally satisfies that \|⟨x, z⟩\| ≤cos π 8 . It suffices to upper bound the probability that g(z) ̸= g(z(1 + β)). Let Cϕ = {z : ⟨z, x⟩= cos ϕ}. Our main idea is to condition on z ∈Cϕ, and then integrate over all choices of ϕ. That is, if we let ϕ denote the random variable representing the angle between x and z, then Pr z [g(z) = −1, g(z(1 + β)) = +1] = Eϕ Pr z\|ϕ[g(z) = −1, g(z(1 + β)) = +1]. We will now bound the latter quantity. Fix any ϕ, and observe that the conditional distribution, z\|ϕ can be written as z = x cos ϕ + u sin ϕ where u is a random vector in Rd−1 that is uniformly distributed over the unit sphere, Sd−2 ⊆Rd−1. Rewriting the condition that g(z) ̸= g(z(1 + β)) in terms of u, observe that g(z) = −1, g(z(1 + β)) = +1 =⇒⟨w, z⟩≤b ≤⟨w, z(1 + β)⟩ =⇒ b 1 + β ≤⟨w, z⟩≤b =⇒ b 1 + β −⟨x cos ϕ, w⟩≤⟨w, u sin ϕ⟩≤b −⟨x cos ϕ, w⟩ =⇒⟨w, u⟩∈ s, s + β sin ϕ , where s is a constant that depends solely on b, w, x, and ϕ. Note that we are using the fact that \|b\| ≤(1 + β) as otherwise g would trivially output the same label over all z ∼µ). By applying Lemma C.17 along with the fact that (by definition of ϕ) β sin ϕ ≤ β sin π 8 ≤ 1 1370d2 , we have that Pr u u ∈ s, s + β sin ϕ ≤1 10, which implies the desired result. 27 Lemma C.10. µ(C(S2,x, π 8 )∪C(S2,−x, π 8 )) µ(C(S2,x,θ)) ≤ 7 30. Proof. By symmetry, µ C S2, x, π 8 = µ C S2, −x, π 8 so it suffices to bound one of them. Since θ ≥π 4 by assumption, applying Lemma C.20, we have µ C S2, x, π 8 ∪C S2, −x, π 8 µ (C(S2, x, θ)) ≤2µ C S2, x, π 8 µ (C(S2, x, θ)) ≤2µ C S2, x, π 8 µ C S2, x, π 4 ≤21 2 sin π 8 sin π 4 d−2 ≤7 30, as d ≥5 in the assumption of Theorem 5.1. We are now prepared to prove the main lemma. Proof. (Lemma C.7) Let A∗⊆C(S2, x, θ) be defined as the set of all points for which g classifies both the point and its image in (1 + β)S3 correctly. That is, A∗= {z ∈C(S2, x, θ) : g(z) = −1, g((1 + β)z) = +1}. By the previous two lemmas, we have µ(A∗) µ(C(S2, x, θ)) ≤µ(T ∗∪C(S2, x, π 8 ) ∪C(S2, −x, π 8 )) µ(C(S2, x, θ)) ≤1 10 + 7 30 = 1 3 Each z ∈A∗is a point for which both z and (1 + β)z are correctly classified, and each z ∈ C(S2, x, θ) \ A∗corresponds to either z being misclassified, or (1 + β)z being misclassified. It follows that the overall accuracy of g over T3(x, θ) is at most Pr z∼T3(x,θ)[g(z) = f(z)] ≤ Pr z∼C(S2,x,θ)[z ∈A∗] + 1 2 Pr z∼C(S2,x,θ)[z /∈A∗] ≤1 2 1 + Pr z∼C(S2,x,θ)[z ∈A∗] ≤2 3 Thus g must incur loss at least 1 3 over T3(x, θ), as desired. C.4 Proof of Lemma C.8 Throughout this section, we assume that µ(B(a, r)) ≥31−d. Lemma C.11. max(θ1, θ2, θ3) ≥π 3 . Proof. Assume towards a contradiction that this does not hold. Let x be as in Lemma C.6. Then by the Definition of µ (Definition C.3) and Lemma C.6, it follows that µ(B(a, r)) = µ(C(S1, x(1 −α), θ1)) + µ(C(S2, x, θ2)) + µ(C(S3, x(1 + β), θ3)) = 1 3 (Ψ(θ1) + Ψ(θ2) + Ψ(θ3)) < Ψ π 3 , where Ψ is as defined in Section C.6. However, Lemma C.19 implies that Ψ pi 3 ≤31−dΨ(π) = 31−d. This contradicts our assumption on µ(B(a, r)) and implies the desired result. 28 Lemma C.12. r ≥1 −α. Proof. Lemma C.11 implies that B(a, r) must intersect some sphere among S1, S2, S3 in a spherical cap of an angle at least π 3 . Basic geometry implies that r ≥min(rad(S1), rad(S2), rad(S3)) where rad(Si) denotes the radius of Si. The desired result follows from the fact that 1 −α −rad(S1) ≤ rad(S2), rad(S3). Lemma C.13. \|θ2 −max(θ1, θ3)\| ≤ 1 4d. Proof. We first compute θ1, θ2, θ3 in terms of a, r, α, and β. We begin with θ2, and note that the expressions for θ1 and θ3 can be similarly derived. To this end, we have S2 ∩B(a, r) = {x : \|\|x\|\| = 1, \|\|x −a\|\| ≤r} = x : \|\|x\|\| = 1, ⟨x, x⟩−2⟨x, a⟩+ ⟨a, a⟩≤r2 = x : \|\|x\|\| = 1, x \|\|x\|\|, a \|\|a\|\| ≥1 + a2 −r2 2a , where we use a to denote \|\|a\|\| in a slight abuse of notation. It follows from Lemma C.6 that cos θ2 = 1 + a2 −r2 2a . We can similarly show that cos θ1 = (1 −α)2 + a2 −r2 2(1 −α)a , cos θ3 = (1 + β)2 + a2 −r2 2(1 + β)a . Let h : R →R be the function defined as h(s) = s2+a2−r2 2sa . Thus, cos θ1 = h(1 −α), cos θ2 = h(1), cos θ3 = h(1 + β). Note that in cases where h is outside of the interval [−1, 1] (meaning θi would not be defined), we simply set θi equal to π and 0 respectively, as these quantities still accurately describe the intersection between B(a, r) and the corresponding sphere, Si. Case 1: 0 ≤a ≤β 2 By definition, B(a, r) contains x∗and therefore intersects S3. It follows from the triangle inequality that r ≥1 + β 2 . However, this implies that B(a, r) must contain the entirety of S2 and S1, which implies that θ1 = θ2 = max(θ1, θ3) = π, thus implying the lemma statement. Case 2: β 2 < a ≤1 −α If r > 1 + 2β, then B(a, r) will contain S1, S2 and S3, which implies θ1 = θ2 = θ3 = π (implying the lemma statement). Thus, assume r ≤1 + 2β. Differentiating h w.r.t. s gives h′(s) = 1 2a 1 + r2 −a2 s2 . 29 By Lemma C.12, r2 ≥a2, which implies that h′(s) is nonnegative for s ∈[1−α, 1+β]. Furthermore, we have that over the interval, [1 −α, 1 + β], h′(s) = 1 2a 1 + r2 −a2 s2 ≤1 β   1 + (1 + 2β)2 − β 2 2 (1 −α)2    = 1 β 1 + 1 + 4β + 3.75β2 (1 −α)2 ≤1 β 1 + 1 + 4(0.25) + 3.75(0.25)2 0.8752 ≤4 β . This is obtained by substituting appropriate upper and lower bounds for r, a, s, α, and β. Because h′(s) is nonnegative over the interval, we must have that h(1 −α) ≤h(1) ≤h(1 + β) which implies θ1 ≥θ2 ≥θ3 (as cos is a decreasing function). It follows from our upper bound on h′(s) that \| cos θ2 −cos (max(θ1, θ3)) \| = cos(θ2) −cos(θ1) = h(1) −h(1 −α) = Z 1 1−α h′(s)ds ≤ Z 1 1−α 4 β ds = 4α β . Applying Lemma C.14 implies that \|θ2 −max(θ1, θ3)\| ≤8 q α β = 1 4d, which implies the lemma statement. Case 3: a > 1 −α First suppose that \|a −r\| > 3. If r > a + 3, then the triangle inequality implies that S1, S2, S3 ⊆ B(a, r) which implies the desired result. On the other hand, if r < a −3, then we must have a > 3, and that B(a, r) is disjoint from S1, S2, S3 which again implies the desired result. Thus, we assume that \|a −r\| ≤3. We now use a similar strategy to the previous case, and bound the derivative, h′(s). By substituting that \|a −r\| ≤3, we have, for s ∈[1 −α, 1 + β], \|h′(s)\| = 1 2a 1 + r2 −a2 s2 = 1 2a 1 + (2a + 3)(3) s2 = 1 2a 1 + (2a + 3)(3) (1 −α)2 = 1 2a 1 + (2a + 3)(3) (0.875)2 ≤ 1 2a (1 + 4(2a + 3)) ≤ 1 2a (13 + 8a) ≤4 + 10 = 14. 30 Here we are exploiting the fact that 1 −α ≥ √ 3 2 , 0.65. It follows by the same argument given in Case 2 that \| cos θ2 −cos(max(θ1, θ3))\| ≤14β. Applying Lemma C.14 implies \|θ2 −max(θ1, θ3)\| ≤ 4√14β = 1 4d, as desired. Now we are ready to prove the lemma. Proof. (Lemma C.8) Let x be as in Lemma C.6, and let θ∗= max(θ1, θ2, θ3). Then by applying Lemma C.6 to the Definition of µ (Definition C.3) gives us µ (B(a, r)) = µ (C(S1, x(1 −α), θ1)) + µ (C(S2, x, θ2)) + µ (C(S3, x(1 + β), θ3)) = 1 3Ψ(θ1) + 1 3Ψ(θ2) + 1 3Ψ(θ3) ≤Ψ(θ∗). Here Ψ denotes the function defined in Section C.6. Next, let θ = min(max(θ1, θ3), θ2). Let T1(x, θ) and T3(x, θ) be as defined in Lemma C.7. Observe that if θ1 ≥θ3, then T1(x, θ) ⊆C(S1, x(1 −α), θ1) ∪C(S2, x, θ2) ⊆B(a, r), and otherwise, T3(x, θ) ⊆C(S3, x(1 + β), θ3) ∪C(S2, x, θ2) ⊆B(a, r). Thus, at least one of these sets is part of B(a, r). We now show that these sets have the desired mass. By the definition of θ∗, we have µ(T1(x, θ)) µ(B(a, r)) , µ(T3(x, θ)) µ(B(a, r)) ≥2µ(C(S2, x, θ)) 3µ(C(S2, x, θ∗)). Next, Lemma C.11 implies that θ∗≥π 3 , and Lemma C.13 implies that θ∗−θ ≤ 1 4d. It follows that θ ≥θ∗−1 4d ≥θ∗ 1 −1 4d . Substituting this, we find that 2µ(C(S2, x, θ)) 3µ(C(S2, x, θ∗)) = 2 3Ψ(θ) Ψ(θ∗) ≥2 3 Ψ θ∗1 −1 4d Ψ(θ∗) ≥2 3 1 −1 4d d−1 ≥2 3 1 e 1/4 ≥1 2, where the last steps follow from Lemmas C.19 and C.16. This completes the proof. C.5 Technical Lemmas Lemma C.14. Suppose ϕ1, ϕ2 ∈[0, π] such that \| cos(ϕ1) −cos(ϕ2)\| ≤c. Then \|ϕ1 −ϕ2\| ≤4√c. Proof. WLOG, suppose ϕ1 ≤ϕ2. Let x = ϕ2 −ϕ1. 31 Using the sum to product rules, it follows that α ≥\| cos ϕ1 −cos ϕ2\| = −2 sin ϕ1 −ϕ2 2 sin ϕ1 + ϕ2 2 ≥ 2 sin x 2 sin ϕ1 + ϕ2 2 . However, observe that π −ϕ1+ϕ2 2 ≥ϕ2 −ϕ1+ϕ2 2 = x 2 and that ϕ1+ϕ2 2 ≥0+0+x 2 = x 2. It follows that ϕ1+ϕ2 2 ∈[ x 2, π −x 2], which implies that sin ϕ1+ϕ2 2 ≥sin x 2. Substituting this, we have c ≥ 2 sin x 2 sin ϕ1 + ϕ2 2 ≥2 sin2 x 2 We now do casework based on x. First suppose that x ≥π 2 . Then c ≥2 sin2 π 4 = 1. By definition, x ≤π, so it follows that x ≤4√α, implying the desired result. Otherwise, if x ≤π 2 , then sin x 2 ≥x 4, as the function t 7→sin(t) −t 2 is nonnegative on the interval [0, π 2 ]. Substituting this, we see that c ≥x2 8 . Thus x ≤ √ 8c < 4√c, as desired. Lemma C.15. For 0 ≤c ≤1 and 0 ≤θ ≤π, sin(cθ) ≥c sin(θ). Proof. Let f(θ) = sin(cθ) −c sin(θ). Observe that f(0) = 0. Furthermore, for θ ∈[0, π], we have f ′(θ) = c cos(cθ) −c cos(θ) = c (cos(cθ) −cos(θ)) . Since cos is a decreasing function on the interval [0, π], it follows that cos(cθ) ≥cos(θ), which implies f ′(θ) ≥0. Thus f is non-decreasing on the interval, and the desired inequality holds. Lemma C.16. For all x > 1, 1 −1 x x−1 ≥1 e. Proof. Let f(x) = 1 −1 x x−1. It is well known that limx→∞f(x) = 1 e and limx→1+ f(x) = 1. Thus it suffices to show that f(x) is a non-increasing function. To do so, we will show that ln f(x) is non-increasing by taking its derivative. We have d (ln f(x)) dx = d dx (x −1) ln x −1 x = d dx ((x −1) ln(x −1) −(x −1) ln x) = ln(x −1) + x −1 x −1 − ln(x) + x −1 x = 1 x −(ln(x) −ln(x −1)) = 1 x − Z x x−1 1 t dt ≤1 x − Z x x−1 1 xdt = 1 x −1 x = 0. Lemma C.17. Let z be a point chosen at uniform over S2, and let w be a fixed unit vector. Then if t ≤ 1 1370d2 , then for any s ∈R, Pr z [⟨w, z⟩∈[s, s + t]] ≤1 10. 32 Proof. Let θ denote the random variable that represents the angle between w and z. Applying Lemma C.14, it follows that for some choice of s′ ∈R that Pr z [⟨w, z⟩∈[s, s + t] ≤Pr θ [θ ∈[s′, s′ + 4 √ t]. We now bound this quantity by utilizing the quantity Ψ (defined in Section C.6). We have, Pr θ [θ ∈[s′, s′ + 4 √ t] = R s′+4 √ t s′ sin(d−2) ϕdϕ R π 0 sin(d−2) ϕdϕ ≤ 2 R π 2 π 2 −2 √ t sin(d−2) ϕdϕ 2 R π 2 0 sin(d−2) ϕdϕ = 1 −Ψ π 2 −2 √ t Ψ π 2 . Here we have simply chosen the interval of length 4 √ t that maximizes the corresponding the integral. Next, we continue by applying Lemmas C.19 and C.16 to get Pr θ [θ ∈[s′, s′ + 4 √ t] ≤1 −Ψ π 2 −2 √ t Ψ π 2 ≤1 − 1 −4 √ t π d−1 ≤1 − 1 − 1 29d d−1 = 1 − 1 − 1 29d 29(d−1)!1/29 ≤1 − 1 − 1 29d 29d−1!1/29 ≤1 − 1 e 1/29 ≤1 10, as desired. C.6 Spherical Caps Definition C.18. Let S be a (d −1) sphere centered at the origin, let 0 ≤θ ≤π be an angle, and let x ∈S be a point. We let C(S, x, θ) denote the spherical cap with angle θ centered at x, and it consists of all points, x′ ∈Sd−1, such that ⟨x,x′⟩ \|\|x\|\|\|\|x′\|\| ≥cos ϕ. Here we take the convention of associating C(Si, xi, 0) with both the empty set and with {xi}. While these are distinct sets, they both have measure 0 under π. We also associate C(Si, xi, π) with the entirety of Si. We let Ψ(θ) denote the ratio of the (d−1)-surface area of the region, C(S, x, θ), to the (d−1)-surface area of the entire sphere. Thus, Ψ(θ) denotes the fraction of the sphere covered by a spherical cap of angle θ. By standard integration over spherical coordinates, we have Ψ(θ) = R θ 0 sin(d−2) ϕdϕ R π 0 sin(d−2) ϕdϕ . Next, we bound Ψ(θ) with the following inequality. 33 Lemma C.19. Let 0 ≤θ ≤π and let 0 ≤c ≤1. Then Ψ(cθ) Ψ(θ) ≥cd−1. Proof. By applying Lemma C.15 to the definition of Ψ, we have the following manipulations. Ψ(cϕ) Ψ(ϕ) = R cϕ 0 sind−2 θdθ R ϕ 0 sind−2 θdθ = R ϕ 0 sind−2(cu)(cdu) R ϕ 0 sind−2 θdθ ≥ R ϕ 0 (c sin(u))d−2 (cdu) R ϕ 0 sind−2 θdθ = cd−1. We similarly have an upper bound on this ratio. Lemma C.20. Let 0 ≤θ ≤π 2 and 0 ≤c ≤1. Then Ψ(cθ) Ψ(θ) ≤c sin cϕ sin ϕ d−2 . Proof. We similarly have, Ψ(cϕ) Ψ(ϕ) = R cϕ 0 sind−2 θdθ R ϕ 0 sind−2 θdθ = R ϕ 0 sind−2(cu)(cdu) R ϕ 0 sind−2 θdθ ≤ R ϕ 0 sin(u) sin cϕ sin ϕ d−2 (cdu) R ϕ 0 sind−2 θdθ = c sin cϕ sin ϕ d−2 . Here we are using the fact that t 7→sin ct sin t is a non-decreasing function for t ∈[0, π]. 34 NeurIPS Paper Checklist 1. 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4,488	SARAD: Spatial Association-Aware Anomaly Detection and Diagnosis for Multivariate Time Series Zhihao Dai Department of Computer Science University of Warwick Coventry, UK zhihao.dai@warwick.ac.uk Ligang He∗ Department of Computer Science University of Warwick Coventry, UK ligang.he@warwick.ac.uk Shuang-Hua Yang Department of Computer Science University of Reading Reading, UK shuang-hua.yang@reading.ac.uk Matthew Leeke School of Computer Science University of Birmingham Birmingham, UK m.leeke@bham.ac.uk Abstract Anomaly detection in time series data is fundamental to the design, deployment, and evaluation of industrial control systems. Temporal modeling has been the natural focus of anomaly detection approaches for time series data. However, the focus on temporal modeling can obscure or dilute the spatial information that can be used to capture complex interactions in multivariate time series. In this paper, we propose SARAD, an approach that leverages spatial information beyond data autoencoding errors to improve the detection and diagnosis of anomalies. SARAD trains a Transformer to learn the spatial associations, the pairwise inter-feature relationships which ubiquitously characterize such feedback-controlled systems. As new associations form and old ones dissolve, SARAD applies subseries division to capture their changes over time. Anomalies exhibit association descending patterns, a key phenomenon we exclusively observe and attribute to the disruptive nature of anomalies detaching anomalous features from others. To exploit the phenomenon and yet dismiss non-anomalous descent, SARAD performs anomaly detection via autoencoding in the association space. We present experimental results to demonstrate that SARAD achieves state-of-the-art performance, providing robust anomaly detection and a nuanced understanding of anomalous events. 1 Introduction Time series anomaly detection is critical for industrial automation (Rieth et al., 2018), intrusion detection (Mathur and Tippenhauer, 2016), and healthcare sensing (Goldberger et al., 2000). Anomaly detection in these contexts is typically treated as an unsupervised learning problem, owing to the novelty of anomalies and the scarcity of labeled anomalies. Temporal modeling is the mainstream basis of current time series anomaly detectors. By learning the dependencies between discrete time steps, temporal modeling can pinpoint the time spans of anomalies. During anomalies, unseen and peculiar temporal dependence patterns degrade data autoencoding (Wang et al., 2023b) or autoregression (Zhao et al., 2020) performance, thereby enabling detection. Alternatively, irregular temporal representations can be driven to breach learned enclosing ∗Corresponding Author 38th Conference on Neural Information Processing Systems (NeurIPS 2024). feature #12 feature #15 time range: [68964, 69639] (a) Raw time series before, during, and after an anomaly pi. #12 #15 features #12 #15 features range: [0.0032, 0.1190] (b) Pre-pi’s A. #12 #15 features #12 #15 features range: [0.0032, 0.1190] (c) In-pi’s A. #12 #15 features #12 #15 features range: [0.0032, 0.1190] (d) Post-pi’s A. #12 #15 features #12 #15 features range: [0.0000, 0.0398] (e) Pre-reduction. #12 #15 features #12 #15 features range: [0.0000, 0.0398] (f) Post-reduction. Figure 1: Spatial associations captured by Transformer on a service monitoring benchmark. 1a shows the raw time series right before, during, and after an anomaly (colored in red). Association mapping AL h by final L-th layer’s MHSA are averaged across heads to derive A before (1b), during (1c), and after (1d) the anomaly. Darker cells have larger values. Anomalous features (#12 and #15) are highlighted with red bounding boxes. The reduction-only changes from before or after the anomaly to during the anomaly are shown in 1e and 1f, i.e., ReLU(Apre −Ain) and ReLU(Apost −Ain). The anomaly leads to association reductions on anomalous features, prominently column-wise on A. hyperspheres (Shen et al., 2020), resulting in high anomaly scores that enable detection. Despite its temporal precision in anomaly detection, temporal modeling either assumes feature independence or combines variables of diverse physical nature, the former simplifying the modeling and the latter mitigating the multicollinearity issue. Such assumptions lead to either omission or dilution of spatial information crucial to anomaly detection. Specifically, it overlooks the long-time-range spatial associations, the relationships between various features which ubiquitously characterize normal behaviors of multivariate time series. Where anomaly detection pinpoints the temporal locations of an anomaly, anomaly diagnosis identifies the spatial locations, i.e., the anomalous feature set, of an anomaly. Temporal methods also restrict diagnostic capabilities, as the lack of spatial information mismatches autoencoding-based or autoregression-based anomaly criterion, which de facto measures temporal novelty, with its objective of capturing spatial novelty. Linear Linear Softmax Linear Linear Transpose Dot Product Dot Product Figure 2: MHSA. Furthermore, time series anomalies frequently dissolve spatial associations, motivating anomaly detection in the association space. Using an vanilla Transformer (Vaswani et al., 2017), we investigate the changes in spatial associations throughout anomalies. Applied on transposed time windows (the spatial dimension comes before the temporal), an encoder-only Transformer is trained to minimize reconstruction errors on unlabeled N-variate time series and, by doing so, learns to model the multivariate series spatially via the Multi-Head Self-Attention (MHSA) illustrated in Figure 2. MHSA at each stacked l-th layer computes an intermediate association mapping Al h ∈RN×N per h-th head, mapping back input X to produce attention scores. The last layer’s mapping AL h thus effectively captures the contributions of k-th feature to the reconstruction of j-th feature at each location (j, k), not least for its architectural proximity to the reconstructed output. Recent research (Liu et al., 2024) also highlights the important role MHSA plays in capturing the inter-feature associative relationships when applied on the multi-variate dimension. As new associations emerge and old ones dissolve over time, Figure 1 shows the association changes on a real-world benchmark. We observe that anomalies exhibit reductions for anomalous features, a phenomenon we herein coin as Spatial 2 Association Reduction (SAR). The rationale is that anomalies either originate from or result in dissolution of pre-existing associations, detaching anomalous features from their non-anomalous counterparts. Additionally, we make the observation that SAR is most prominent column-wise on AL h, since each j-th column characterizes the dropouts of j from associating with other, mostly non-anomalous, features. Due to lack of explicit spatial information, temporal modeling is inadequate for exploiting SAR. More examples are given in Appendix C. From a spatial modeling perspective, we propose SARAD to leverage spatial information and to exploit SAR for enabling robust time series anomaly detection and diagnosis. For quantifying anomalous spatial novelty in the data space, we train a Transformer on transposed time windows as an autoencoder. To capture the spatial association progression, the reduction-only changes of associations over time, the data reconstruction divides the input window by time into two halves to be processed in parallel. Consequently, the progression is the non-negative backward difference of the intermediate association mappings via MHSA. Subseries division circumvents memory storage of latest association mappings and enables time window shuffling during training, which reduces order bias, enhances generalization, and prevents catastrophic forgetting. For quantifying anomalous reduction novelty, we train a Multi-Layer Perceptron (MLP) as an antuencoder on progression in the association space. Whereas progression encompass all association reduction, autoencoding rules out those not caused by anomalies. The reconstruction errors via the data module measure data-only anomalous deviation from expected system behaviors and falter when such deviations are not prominent, e.g., at the start of an anomaly. The reconstruction errors via the progression module are sensitive to change in spatial associations, thus complementing the former. We develop a joint anomaly detection criterion that combines both. Experiments show SARAD delivers stateof-the-art detection and diagnosis performance with architectural elegance. Code is available at https://github.com/daidahao/SARAD/. We summarize our contributions as follows. • We reveal and extract spatial association descending patterns of time series anomalies with a bespoke Transformer and subseries division. The former learns the pairwise inter-feature associations via autoencoding in the data space and the latter enables shuffled autoencoding training and memory-less progression aggregation. • We propose progression autoencoding to quantify anomalous descent in the association space and a joint detection criterion in both data and association spaces, which complement each other. • Experimentally, SARAD performs state-of-the-art anomaly detection and diagnosis on multivariate time series and ablation studies support our design choices. 2 Related Work Influenced by the dominance of temporal modeling in time series forecasting (Wang et al., 2023a; Zhang et al., 2023; Wu et al., 2021), temporal modeling is also prevalent in time series anomaly detection. Recurrent neural networks such as LSTM (Hochreiter and Schmidhuber, 1997) have innate capabilities for handling sequential data. These approaches use hidden states for past input memorization, enabling detection (Li et al., 2019; Malhotra et al., 2015) and diagnosis (Qian et al., 2021). Transformer (Vaswani et al., 2017) network is widely adopted (Fan et al., 2023; Xu et al., 2022) approach that is commonly applied to model temporal associations between different time points using its attention mechanism. Linear regression (Zeng et al., 2023) and MLP (Wang et al., 2024; Audibert et al., 2020) directly model temporal dependencies. TranAD (Tuli et al., 2022) replaces the MLP in Audibert et al. (2020) with a Transformer, making the detection criterion more robust through its adversarial training paradigm. Temporal modeling, however, is restricted by the exceptionally small receptive field in time and adversely impacted by the timestamp misalignment across features. In the context of anomaly detection, temporal modeling helps capture anomalous temporal associations (Xu et al., 2022; Yang et al., 2023), but offers limited detection capabilities in absence of spatial information. In a diagnostic context, temporal detectors mismatch anomaly criterion of temporal novelty with spatial interpretation. Spatial associations characterize the multivariate time series commonly found in such supervisory systems for industrial control. The relationships range from strongly correlated, e.g., due to spatial proximity, to fully independent, e.g., due to mechanical disconnection. For forecasting, iTransformer (Liu et al., 2024) applies Transformer on the transposed time series to enable direct spatial 3 modeling. Crossformer (Zhang and Yan, 2023) screens the time series through custom Two-Stage Attention layers for more efficient spatial modeling. In terms of detection, GDN (Deng and Hooi, 2021) learns a directed graph of features for the prediction of last time points, whose errors serve as anomaly scores. GDN is partially limited by a mismatch between its singe-timestamp prediction target and the prevalent range-wise anomalies as well as unstable Top-K node selection during training. InterFusion (Li et al., 2021) learns compressed spatial and temporal dependencies, using a hierarchical Variational Auto-Encoder (Kingma and Welling, 2014) to reconstruct the series. Neither inspects temporal changes in associations throughout anomalies. On another front, Isolation Forest (IF) models build a binary decision tree ensemble by partitioning either the data space (Liu et al., 2008) or the deep embedding space (Xu et al., 2023) formed by randomized neural networks. They are constrained by the lack of temporal and spatial (in the former case) or spatial (in the latter case) information, and their anomaly scores are not reflective of the degrees of anomalies. We emphasize anomalous association descending patterns towards better time series detection and diagnosis. Different from previous work, we explicitly utilize the reduction in spatial associations over time during an anomaly, an insight we derived from the cyber-physical defense space. Dynamic watermarking (Satchidanandan and Kumar, 2017) and similar defense techniques (Dai et al., 2023) overlay actuation with private signals to reveal attacks resulting in correlational breakdowns. While their approaches are intrusive and actively alter system behaviors, our detector remains non-intrusive, passively monitors the spatial associations, and is applicable to any supervisory system. We refer to spatiality in this work as the multi-dimensional vector nature inherent to multivariate time series data. The terminology is also used in literature on time series related tasks (Gangopadhyay et al., 2021; Zheng et al., 2023). We note that spatiality may carry different meanings in other AI contexts, such as geographic positions or characteristics on Earth. We differentiate those meanings from our definition of spatiality, which traces its root to the spatial distribution of sensors and actuators in control systems where time series are routinely collected. 3 Method The problems of anomaly detection and diagnosis are specified as follows. Anomaly Detection Given a N-feature time series T = {x1, · · · , xN} where xn ∈RT is of the same length T, the objective is to predict the anomaly label yt ∈{0, 1} at each timestamp t. Anomaly Diagnosis Given the same time series T , the objective is to predict the diagnosis label gt ⊆[N], the set of anomalous features at each timestamp t. 3.1 Overview SARAD comprises two sequential modules; a Transformer for time series data reconstruction and a MLP network for spatial progression reconstruction. Table 1 decomposes the system framework of SARAD. The Transformer temporally divides by 2 and reconstructs the input time series to learn pairwise inter-feature associations and to enable order-free memory-efficient progression aggregation. The MLP reconstructs the aggregated progression to quantify anomalous association reduction while dismissing non-anomalous reduction. Towards robust anomaly detection, reconstruction errors from the two modules jointly serve as a criterion, sensitive to data-only anomalous deviation and anomalous association reduction. 3.2 Data Reconstruction In light of restricted capabilities of temporal detectors, here we adapt Transformer to spatially reconstruct the series data. The data module contains two components, Subseries Split & Merge and Subseries Reconstruction, shown in the first and second columns in Table 1. The former wraps around the second by temporally splitting a multivariate input series in half at its beginning and temporally merging at its end. Subseries division enables capturing of spatial progression within a single time window. Without the former, the model must store in memory the last association mappings at each step and keep to the time ordering during training, which is prone to overfitting and 4 Table 1: SARAD is a composition of two modules and three components: data reconstruction (subseries split and merge, and subseries reconstruction) and spatial progression reconstruction. Data Reconstruction Progression Recon. Subseries Split & Merge Subseries Recon. Architecture Encoder-only Transformer MLP Diagram Subseries Split: Subseries Merge: Time Feature Embedding MLP Linear Encoder MHSA ReLU Aggregate MLP Training Objective min \|\| ˆ X −X\|\|2 2 min \|\| ˆS −S\|\|2 2 Detection Criterion \|\| ˆ X −X\|\|2 2 \|\| ˆS −S\|\|2 2 Diagnosis Criterion \|\| ˆ X(j,·) −X(j,·)\|\|2 2 \|\| ˆ S(·,j) −S(·,j)\|\|2 2 catastrophic forgetting. The latter is an encoder-only Transformer composed of an embedding layer, a L-layer attention-based encoder, and finally a linear projection layer. Encoder-only Transformers are commonly found in Transformer-based detectors (Kang and Kang, 2024; Kim et al., 2023) due to its simplicity and the length uniformity of the target output, i.e., the reconstructed series. Ours exclusively models spatial associations, unconventional to the aforenamed detectors and most temporal forecasters (Nie et al., 2023; Zhou et al., 2022; Liu et al., 2022) and yet more aligned with recent spatial-aware forecasters (Liu et al., 2024; Zhang and Yan, 2023). At each encoding layer, MHSA computes pairwise association mappings, a representation of inter-feature dependencies which ubiquitously characterize the multivariate time series and are crucial to anomaly detection. Subseries Split and Merge We suppose the input series is a time window X ∈R2W ×N of length 2W, where 2 is for the convenience of a temporal split. Before reconstruction, X is split into two half multivariate subseries of equal temporal length: X = {X1 ∈RW ×N, X2 ∈RW ×N}. After subseries reconstruction, the two reconstructed subseries are concatenated to form the full reconstructed ˆX = { ˆX1 ∈RW ×N, ˆX2 ∈RW ×N}. Embedding To lead subseries reconstruction, each Xi is embedded as X 0 i = Ei + M, wherein Ei = Linear(XT i ) ∈RN×D and M = {mi ∈RD\|i ∈[N]} is a learnable feature-level embedding. Spatial-Aware Encoding A stack of L Transformer encoding layers is used to encode the series in the D-length attention space. Each layer is stacked with MHSA and MLP with residual connections: Zl i = LN(MHSA(X l−1 i ) + X l−1 i ), X l i = LN(MLP(Zl i) + Zl i) (1) where X l−1 i , Zl i, X l i ∈RN×D are l−1 layer’s output, l-th layer’s hidden state and output respectively and LN(·) is Layer Normalization (Ba et al., 2016). It is spatial-aware because MHSA explicitly learns a pairwise inter-feature association mapping to exchange information in between feature 5 representations. Notably within the MHSA with H heads as shown in Figure 2, each (j, k)-th element on the association mapping Ah ∈RN×N of its h-th head computes how much j-th feature’s attention scores should originate from the k-th feature’s key. From a broader perspective of data reconstruction, it is the quantification of the residual impact of k-th feature’s originals on the j-th feature’s reconstructions. We refactor MHSA implementation to enable parallel encoding of subseries. Linear Projection Output from the last encoding layer X L i is linearly projected and transposed to derive the reconstructed subseries ˆXi ∈RW ×N. 3.3 Spatial Progression Reconstruction To exploit SAR caused by anomalies, the module first extracts and aggregates the spatial progression, the non-negative backward difference of the association mappings via MHSA. In line with general anomaly detectors (Aggarwal, 2013), the module conducts autoencoding in the association space to quantify anomalous SAR and to dismiss non-anomalous SAR. Anomalous SAR occurs when, say, a compromised sensor’s readings are no longer correlating with its spatially adjacent or mechanically related counterparts. Association Progression We define association progression Sl h ∈RN×N at the h-th attention head in the l-th layer to be the non-negative backward difference in association mappings {Al 1,h, Al 2,h}: Sl h = ReLU(Al 1,h −Al 2,h) (2) where ReLU(·) passes through only non-negative values and outputs zeros otherwise. Progression Aggregation To center the detection on association dropouts, we aggregate the column sums of progression Sl h from all attention heads in the final L-th layer to form S ∈RH×N: S = { N X j=1 SL h,(j,k)\|h ∈[H], k ∈[N]} (3) We recall from the data module, each k-th column in Al i,h quantifies the impact of k-th feature on all features’ reconstruction. Taking the sum per each k-th column, we measure with S the dropout rates of k from participating all features’ reconstruction. As we have observed in Section 1, SAR at the column level is indicative of time series anomalies, more so than at the row level. The last layer’s progression is focused not least for its proximity to the final reconstructed output, whereafter no more information is exchanged between features. Liu et al. (2024) manifests that final layer’s mappings resemble closely with the inter-feature correlations of the target, in our case, the reconstructed. Autoencoding With S flattened as an one-dimensional vector, a 2-layer MLP is trained to output the reconstructed and reshaped ˆ S ∈RH×N, synchronously with the data module training. 3.4 Joint Training and Anomaly Detection Training Objective We train an end-to-end model with a joint minimization objective: LR = \|\| ˆ X −X\|\|2 2, LS = \|\| ˆ S −S\|\|2 2, L = LR + λLSLS (4) where LR is the data reconstruction loss, LS the progression reconstruction loss, and λLS a weight hyper-parameter. Gradients are stopped from flowing into S to prevent updates to the data module and collapses in association representation. We are training two anomaly detectors simultaneously, one working in the original data space, the other in the spatial progression space. Anomaly Detection Criterion For an input series X, the anomaly score s is a scalar defined to be: r = \|\| ˆ X −X\|\|2 2, p = \|\| ˆ S −S\|\|2 2, s = (r −µr)/σr + (p −µp)/σp (5) where r is the data reconstruction error, p the progression reconstruction error, µr, µp the means of r, p on the validation set, and σr, σp the standard deviation of r, p. The criterion takes into account the normalized errors in the data space and the progression space, each of which quantifies the anomalous magnitude in respective spaces and complment the other. 6 Table 2: Statistics of the main datasets. Training Test Anomalies Lengths Sampling Dataset Features Set Set Count Ratio min med max Period SMD 38 708,405 708,420 327 4.16% 2 11 3,161 1 min PSM 25 132,481 87,841 71 27.73% 1 5 8,861 1 min SWaT 51 496,800 449,919 34 12.02% 101 447 35,900 1 sec HAI 79 921,603 402,005 50 2.23% 17 162.5 422 1 sec Anomaly Diagnosis Criterion For an input series X and its j-th feature, its anomaly score sj is a scalar defined to be: sj = rj = \|\| ˆ X(j,·) −X(j,·)\|\|2 2, (6) where rj is feature j’s data reconstruction error. The criterion is sensitive to spatial novelty. 4 Experiments SARAD is compared against state-of-the-art detectors on real-world benchmarks for detection and diagnosis, the latter only when diagnostic labels are available. 4.1 Experimental Setup Datasets We evaluate on four real-world datasets collected under industrial control and service monitoring settings. These dataset are: 1) Server Machine Dataset (SMD) (Su et al., 2019b,a), 2) Pooled Server Metrics (PSM) dataset (Abdulaal et al., 2021a,b) 3) Secure Water Treatment (SWaT) dataset (Mathur and Tippenhauer, 2016; iTrust, 2023), and 4) Hardware-In-the-Loop-based Augmented ICS (HAI) dataset (Shin et al., 2021b,a). All training sets contain only unlabeled data and the test sets contain data with anomaly labels. Anomalies range from service outages to external cyber-physical attacks. We summarize the statistics of the datasets in Table 2. Descriptions of each dataset are detailed in Appendix E. Detection Metrics Real-world benchmarks are rife with range-wise anomalies spanning consecutive time points (Wagner et al., 2023). We use the range-based metrics proposed in (Paparrizos et al., 2022). Compared against their point-based counterparts, they provide robustness to labeling delay and scoring noises as well as performant detector separability and series consistency. We compute the threshold-independent AUC-ROC and AUC-PR scores to be rid of thresholding impact and fully parameter-free Volume Under the Surface (VUS) AUC-ROC and AUC-PR scores. Full details are discussed in Appendix I. Diagnosis Metrics Consistent with previous works (Tuli et al., 2022; Zhao et al., 2020), we use common metrics such as Hit Rate (HR) (Su et al., 2019b) and Normalized Discounted Cumulative Gain (NDCG) (Järvelin and Kekäläinen, 2002) where diagnosis labels are available. At the range level, we measure the Interpretation Score (IPS) initially proposed in Li et al. (2021) and here expanded to fit the P% parameterization. Full details are discussed in Appendix J. Baselines We compare SARAD against state-of-the-art anomaly detection baselines, including Isolation Forest-based IF (Liu et al., 2008), Deep IF (DIF) (Xu et al., 2023); MLP-based USAD (Audibert et al., 2020); graph-based GDN (Deng and Hooi, 2021); LSTM-based MAD-GAN (Li et al., 2019); CNN-based DiffAD (Xiao et al., 2023); and Transformer-based TranAD (Tuli et al., 2022), ATF-UAD (Fan et al., 2023), AT (Xu et al., 2022), DCdetector (Yang et al., 2023). Noticeably, GDN employs explicit spatial modeling in its graph construction although spatial associations are not directly involved in anoamly scoring. MAD-GAN emphasizes on anomaly detection within cyber-physical systems. All baselines are trained using official implementations where available and recommended hyperparameters from respectively papers are used. 7 Table 3: Anomaly detection performance. Threshold-independent AUC-ROC and AUC-PR metrics and fully parameter-free VUS-ROC and VUS-PR metrics are reported. All values are average percentages from five random seeds. The best values are in bold and the second best underlined. SMD PSM SWaT HAI Method AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR IF 53.81 7.27 53.56 7.25 58.08 41.51 57.99 41.48 86.11 66.52 84.39 63.57 72.90 10.03 71.65 9.81 DIF 60.27 10.30 59.84 10.23 52.00 36.61 51.88 36.55 89.38 73.19 87.88 70.54 82.10 35.86 81.13 34.16 TranAD 46.86 5.92 46.54 5.88 50.20 35.22 49.47 35.19 47.78 17.65 47.13 17.57 75.60 25.80 75.06 25.41 ATF-UAD 43.41 4.98 43.10 4.97 46.44 33.20 46.03 33.18 55.18 20.66 54.35 20.58 70.56 22.47 69.81 22.06 AT 50.01 5.42 49.97 5.36 37.66 26.49 36.82 26.47 46.77 12.95 46.45 12.79 47.41 5.85 47.24 5.85 DCdetector 49.47 4.51 49.10 4.50 45.94 24.76 46.01 24.82 50.80 14.47 50.76 14.37 N/A N/A N/A N/A USAD 50.20 6.93 50.01 6.91 42.45 33.23 42.30 33.20 80.36 60.06 78.53 57.33 72.59 23.27 71.84 22.73 GDN 66.37 9.40 66.07 9.34 63.51 40.66 63.13 40.53 79.30 28.12 78.79 28.17 84.79 35.82 84.03 35.05 MAD-GAN 64.35 9.77 64.16 9.74 57.50 40.08 57.37 40.03 86.51 61.95 86.10 62.03 84.92 49.06 84.09 48.14 DiffAD 58.71 7.22 58.40 7.19 51.60 32.10 51.02 32.01 27.02 9.22 26.45 9.21 86.96 21.95 86.25 21.74 Ours 79.97 15.09 79.67 15.02 61.87 41.06 61.77 41.01 88.29 72.90 87.52 70.68 96.87 67.78 96.17 64.70 4.2 Results Anomaly Detection Table 3 shows the anomaly detection performance in metrics defined in Section 4.1. It demonstrates that, despite its architectural elegance, SARAD either outperforms all baselines by significant margins on the threshold-independent VUS-ROC scores (SMD: +15.51% MAD-GAN, HAI: +9.92% DiffAD) or performs on par with current best detectors (PSM: −1.36% GDN, SWaT: −0.36% DIF). IF scrutinizes the distributional shifts of anomalies with random data partitions and delivers consistent performance across datasets. DIF extends IF into randomized deep representation spaces and archives decent improvements due to more flexible partitions and temporally local information extraction via dilated convolutions. Temporal modeling methods such as DiffAD, ATF-UAD, and AT rely solely or heavily on reconstruction errors and when the errors do not correspond the the underlying anomalies their performance plummet. Adversarial training in USAD and MAD-GAN amplifies reconstruction errors of anomalies to mitigate but not eliminate such issues and thus suffer less performance drops. In contrast, our SARAD additionally accounts for the SAR frequent with anomalies and independent of data distributional shifts, thus outperforming all. SARAD also overpasses GDN, which despite its explicit spatial modeling adopts prediction errors as its sole detection criterion, limiting its performance. SARAD’s top performance on SWaT and HAI underlines its ability to unravel complex spatial associations even in complex large-scale systems. Standard deviations of Table 3 are reported in Appendix K. Anomaly Diagnosis Table 4 shows the anomaly diagnosis performance in metrics defined in Section 4.1. DiffAD uses a subset of SMD features and thus is discarded from comparisons for fairness. SARAD outperforms baselines on the point-based HR@150% (SMD: +26.67% TranAD, SWaT: +5.10% USAD, HAI: +3.81% GDN) and NDCG@150% (SMD: +27.97% TranAD, SWaT: +4.76% GDN, HAI: +2.93% GDN). SARAD also outperforms on the range-based IPS@150% on most datasets (SMD: 43.19% TranAD, SWaT: 33.43% TranAD). Unlike SMD which performs forensic diagnosis to label anomalous features, SWaT and HAI label only the origins of cyberattacks as diagnosis labels. Consequentially, attack origins sometimes might not behave anomalously, e.g., attacks had failed, or the full set of anomalous features were not identified, thus diminishing the performance numbers on SWaT and HAI. SARAD generally outperforms detectors underpinned by temporal modeling due to its sensitivity to spatial associative changes. SARAD also outperforms spatial detectors such as GDN whose prediction errors limit its temporal scope to a single time point. Standard deviations of Table 4 are reported in Appendix L. Visualization Figure 3 visualizes a real-world anomaly example via SARAD. Our detector captures the significant SAR caused by the anomalous features. The loose reconstruction of the progression raises the progression-based score p and, in turn, the joint detection score s. Taking a broader view of the series in Fig. 3h, SAR significantly raises the scores at the start of the anomaly, even when the data-based errors r are small. SARAD exploits SAR to achieve more robust anomaly detection. 8 Table 4: Anomaly diagnosis performance. Point-based HR@P%, NDCG@P%, and range-based IPS@P% are reported. All values are average percentages from five random seeds. SMD SWaT HAI Method HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% P 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 TranAD 33.33 45.26 34.71 41.81 22.21 33.25 4.82 6.36 4.82 5.76 17.47 20.91 4.08 6.27 3.98 5.33 12.67 21.78 ATF-UAD 27.80 40.94 26.67 34.46 17.34 26.16 1.85 3.19 1.83 2.65 5.45 8.28 2.96 5.22 3.08 4.48 5.04 8.59 USAD 27.03 39.74 25.83 33.44 18.31 26.07 4.39 7.73 4.39 6.48 12.83 27.17 3.78 5.83 3.76 5.01 12.37 16.22 GDN 28.67 41.57 28.62 36.27 21.18 30.76 5.99 7.21 6.13 6.89 14.04 21.72 5.45 8.29 5.50 7.23 7.41 12.07 DiffAD N/A N/A N/A N/A N/A N/A 1.82 2.90 1.81 2.47 5.45 12.63 3.32 4.79 3.41 4.32 12.00 17.26 Ours 56.73 71.93 60.79 69.78 61.38 76.44 9.57 12.83 9.61 11.65 35.45 54.34 6.45 12.10 6.69 10.16 7.48 14.07 #1 #9 #10 #12 #13 #14 #15 time range: [15798, 15949] (a) features range: [0.0006, 0.2200] (b) features range: [0.0006, 0.2200] (c) heads range: [0.0000, 1.5716] (d) heads range: [0.0000, 1.5716] (e) (f) (g) (h) rj pj sj (i) Figure 3: Visualization of applying SARAD for detection on SMD. 3a shows the raw time series right before and during an anomaly pi (colored in red). An input time window for SARAD is bounded in the black box. 3b and 3c show the average association mapping ¯ AL via final L-th layer’s MHSA. 3d shows the aggregated progression S according to Eq. 3. 3e is its reconstruction. 3f, 3g, 3h show the scores p, r, and joint s according to Eq. 5 per feature. 3i shows the anomaly scores for 3a’s segment. Anomalous features (#1, #9, #10, #12, #13, #14 ,and #15) are highlighted with red bounding boxes. Complexity and Time Overheads SARAD incurs 32 mins for training and and 0.39 ms for inference per sample on HAI, the largest dataset, falling far below the data collection time and sampling frequency. Those numbers are comparable with baselines and detailed in Appendix N. 4.3 Ablation Studies Spatial Progression Reconstruction To evaluate the effectiveness of the progression module, we perform ablation studies on its submodules in Table 5. Standard deviations are reported in Appendix K. Removing the ReLU, i.e., to capture both association increases and reductions, in progression loses the focus on asscoation reduction and impairs the detection performance. Replacing the column sum operation in aggregation with the row sum which characterizes the disconnection of anomalous features from others and is shown to be less effectiveness than the column sum representing the drop out rates. Fully concatenating without sum operation dilutes the reduction patterns and significantly hurts the detection, at a cost of complexity. For the detection submodule, using the progression directly instead of the reconstruction errors registers reductions as anomalies directly and underperforms except on SMD due to its inability to rule out normal reduction patterns. Choice of Detection Criterion Table 6 compares the detection performance using Eq. 5 (Joint), using only data-based r (DR), and using only progression-based p (SPR). While the data reconstruction is a robust criterion of anomalousness, SARAD embeds the spatial information into the joint criterion and outperforms either single criterion overall. Standard deviations are reported in Appendix K. Additional ablation studies on the choice of diagnosis criterion are detailed in Appendix D. 9 Table 5: Anomaly detection performance under progression reconstruction changes. SMD PSM SWaT HAI Submodule Change AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR Ours 79.97 15.09 79.67 15.02 61.87 41.06 61.77 41.01 88.29 72.90 87.52 70.68 96.87 67.78 96.17 64.70 Prog. (Eq.2) no ReLU 75.43 12.38 75.16 12.35 60.35 40.32 60.21 40.24 87.48 67.88 86.75 66.04 95.59 64.81 94.93 61.87 Aggr. (Eq.3) row sum 79.47 15.87 79.17 15.82 63.36 42.41 63.20 42.33 88.19 70.12 87.40 68.12 96.56 67.26 95.80 64.21 Aggr. (Eq.3) no sum 57.31 6.09 56.98 6.08 47.50 33.92 47.34 33.86 86.10 69.69 85.26 67.71 91.63 57.49 90.70 55.01 Detection S directly 80.42 15.79 80.14 15.73 60.59 39.71 60.49 39.63 87.33 69.38 86.70 67.59 95.72 64.95 94.98 61.91 Table 6: Anomaly detection performance under different choices of detection criterion. SMD PSM SWaT HAI Method Criter. AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR Ours both 79.97 15.09 79.67 15.02 61.87 41.06 61.77 41.01 88.29 72.90 87.52 70.68 96.87 67.78 96.17 64.70 DR r only 73.21 13.53 72.72 13.44 61.53 39.99 61.45 39.93 86.76 68.66 86.15 67.03 96.28 66.38 95.56 63.33 SPR p only 79.71 15.44 79.39 15.36 62.41 41.47 62.20 41.36 65.87 20.10 65.31 19.53 95.54 61.73 94.77 59.07 5 Conclusion In this work, we propose SARAD for time series anomaly detection and diagnosis. The approach effectively exploits the spatial association descending patterns of anomalies. Data reconstruction with Transformer guides learning of spatial associations from data and captured as progression, while progression reconstruction quantifies the anomalous association descent and complements the insensitivity of the former to spatial disassociation during anomalies. SARAD experimentally demonstrates state-of-the-art detection and diagnosis performance and foreshadows the power of spatial modeling for related time series tasks. Acknowledgments and Disclosure of Funding Calculations were performed using the Sulis Tier 2 HPC platform hosted by the Scientific Computing Research Technology Platform at the University of Warwick. Sulis is funded by EPSRC Grant EP/T022108/1 and the HPC Midlands+ consortium. The authors acknowledge the use of the Batch Compute System in Department of Computer Science at the University of Warwick, and associated support services, in the completion of this work. References Ahmed Abdulaal, Zhuanghua Liu, and Tomer Lancewicki. 2021a. 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Curran Associates, Inc., 43322–43355. https://proceedings.neurips.cc/paper_files/paper/ 2023/file/86c17de05579cde52025f9984e6e2ebb-Paper-Conference.pdf 15 A Broader Impacts The broader impact of the work presented rests in the increasingly pervasive nature of industrial control systems, intrusion detection systems, and remote monitoring solutions in healthcare contexts, all of which commonly utilize some form of anomaly detection. Further to the time series analysis that is commonplace, the work presented in this paper demonstrates how Transformer can be used to learn the spatial associations that ubiquitously characterize these feedback-controlled systems, supplementing time series analysis to provide state-of-the-art performance. As such, the work presented has broad applicability, whilst explicitly targeting automated industrial control systems. B Limitations While the model size scales linearly with the number of features, the time complexity of SARAD is quadratic with respect to the features. SARAD could incur significant training and inference overheads when the supervisory system is extensively large. We caution that the overheads of the largest dataset in our experiments fall well below data collection overhead and sampling frequency (see Appendix N). To scale, we will explore hierarchical time series anomaly detection via clustering. Another limitation of this work is the scarcity of forensically labeled datasets like SMD for anomaly diagnosis, not least due to the intensive labor and domain knowledge implied. To address that gap, we will explore publicly available audit and operational time series data for sources. 16 feature #1 feature #9 feature #10 feature #12 feature #13 feature #14 feature #15 time range: [15303, 16941] (a) Raw time series before, during, and after an anomaly pi. #1 #9 #10 #12 #13 #14 #15 features #1 #9 #10 #12 #13 #14 #15 features range: [0.0035, 0.1441] (b) Pre-pi’s A. #1 #9 #10 #12 #13 #14 #15 features #1 #9 #10 #12 #13 #14 #15 features range: [0.0035, 0.1441] (c) In-pi’s A. #1 #9 #10 #12 #13 #14 #15 features #1 #9 #10 #12 #13 #14 #15 features range: [0.0035, 0.1441] (d) Post-pi’s A. #1 #9 #10 #12 #13 #14 #15 features #1 #9 #10 #12 #13 #14 #15 features range: [0.0000, 0.0367] (e) Pre-reduction. #1 #9 #10 #12 #13 #14 #15 features #1 #9 #10 #12 #13 #14 #15 features range: [0.0000, 0.0367] (f) Post-reduction. Figure 4: Spatial associations captured by Transformer on SMD (Su et al., 2019b). feature #12 feature #15 time range: [92799, 93210] (a) Raw time series before, during, and after an anomaly pi. #12 #15 features #12 #15 features range: [0.0014, 0.1178] (b) Pre-pi’s A. #12 #15 features #12 #15 features range: [0.0014, 0.1178] (c) In-pi’s A. #12 #15 features #12 #15 features range: [0.0014, 0.1178] (d) Post-pi’s A. #12 #15 features #12 #15 features range: [0.0000, 0.0415] (e) Pre-reduction. #12 #15 features #12 #15 features range: [0.0000, 0.0415] (f) Post-reduction. Figure 5: Spatial associations captured by Transformer on SMD (Su et al., 2019b). C Examples of Spatial Association Reduction As mentioned in Section 1, herein we provide more real-world examples of Spatial Association Reduction (SAR) exhibited by time series anomalies. Figures 4, 5, 6, 7 showcase the spatial associations captured within Transformer via MHSA. Subfigures in each aforementioned figure show, in that order, (a) the raw time series right before, during, and after an anomaly pi ∈P (colored in red), (b) association mapping AL h output by final L-th layer’s MHSA are averaged across heads to derive average association A before pi, (c) AL h during pi, (d) AL h after pi wherein brighter cells have smaller values and anomalous features are highlighted with red bounding boxes, (e) the reduction-only changes from before the anomaly to during the anomaly, i.e., ReLU(Apre −Ain), and finally (f) the reduction-only changes from after the anomaly to during the anomaly, i.e., ReLU(Apost −Ain). 17 feature #38 time range: [115748, 116933] (a) Raw time series before, during, and after an anomaly pi. #38 features #38 features range: [0.0044, 0.3045] (b) Pre-pi’s A. #38 features #38 features range: [0.0044, 0.3045] (c) In-pi’s A. #38 features #38 features range: [0.0044, 0.3045] (d) Post-pi’s A. #38 features #38 features range: [0.0000, 0.0136] (e) Pre-reduction. #38 features #38 features range: [0.0000, 0.0136] (f) Post-reduction. Figure 6: Spatial associations captured by Transformer on SWaT (Mathur and Tippenhauer, 2016). feature #4 time range: [371378, 371681] (a) Raw time series before, during, and after an anomaly pi. #4 features #4 features range: [0.0013, 0.3153] (b) Pre-pi’s A. #4 features #4 features range: [0.0013, 0.3153] (c) In-pi’s A. #4 features #4 features range: [0.0013, 0.3153] (d) Post-pi’s A. #4 features #4 features range: [0.0000, 0.0132] (e) Pre-reduction. #4 features #4 features range: [0.0000, 0.0132] (f) Post-reduction. Figure 7: Spatial associations captured by Transformer on SWaT (Mathur and Tippenhauer, 2016). 18 Table 7: Anomaly diagnosis performance under different choices of diagnosis criterion. SMD SWaT HAI Method Criter. HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% P 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 Ours rj only 56.73 71.93 60.79 69.78 61.38 76.44 9.57 12.83 9.61 11.65 35.45 54.34 6.45 12.10 6.69 10.16 7.48 14.07 SPR pj only 42.97 57.33 46.32 54.85 52.01 64.82 14.32 17.18 14.40 16.18 20.10 37.37 5.71 8.50 5.86 7.58 9.70 17.04 Joint both 48.91 61.49 53.12 60.59 56.56 70.06 2.60 3.50 2.66 3.20 16.16 20.40 10.28 15.42 10.74 13.92 12.59 16.52 D Choice of Diagnosis Criterion Concerning the rationality of data-only diagnosis criterion in Eq. 6, we consider an alternate joint diagnosis criterion in line with the detection criterion in Eq. 5. rj = \|\| ˆ X(j,·) −X(j,·)\|\|2 2, pj = \|\| ˆ S(·,j) −S(·,j)\|\|2 2, sj = (rj −µrj)/σrj + (pj −µpj)/σpj (7) where rj is feature j’s data reconstruction error, pj its progression reconstruction error, µrj, µpj the means of rj, pj on the validation set, and σrj, σpj the standard deviation of rj, pj. Table 7 compares the diagnosis performance using only rj (SARAD), using only pj (SPR), and using Eq. 7 (Joint). Unlike in anomaly detection, the great discrepancy between rj and pj more than often degrades the performance of the joint criterion. SARAD uses rj only which produces suboptimal and yet reliable performance in the longer anomalous horizons. Standard deviations are reported in Appendix L. 19 E Datasets We include four real-world datasets collected under industrial control and IT service monitoring settings for evaluations. Anomalies range from IT service outages to external cyber-physical attacks against control systems. E.1 Dataset Descriptions 1. Server Machine Dataset (SMD) (Su et al., 2019b,a) is a server metric dataset from a large-scale IT company. Engineers annotated anomalous events in the second half of the data with indicator-level attributions. 2. Pooled Server Metrics (PSM) dataset (Abdulaal et al., 2021a,b) captures key performance indicators of servers on an online shopping platform. Website engineers annotated anomalous events for data in the last eight weeks. 3. Secure Water Treatment (SWaT) dataset (Mathur and Tippenhauer, 2016; iTrust, 2023) contains sensor readings and actuator status on a minuscule real-world water treatment system during a six-day normal operational period. A knowledgeable attacker performed 36 cyber-physical attacks during a five-day attack period and they are labelled as anomalous accordingly. 4. HIL-based Augmented ICS (HAI) dataset (Shin et al., 2021b,a) records measurements and control actions within a Hardware-In-the-Loop (HIL) dual power (steam-turbine and hydropower) generation testbed during its two-week operation. Both single-point primitive and multi-point combined attacks are performed on the testbed to emulate a threat actor with cyber-physical capacities. We use the 21.03 version of HAI. All training sets contain only unlabeled data and the test sets contain data with anomaly labels. Statistics of the datasets are given in Table 2. E.2 Lengths of Anomalies We further characterize the detection datasets by the lengths of the anomalous events. Figure 8 shows the empirical cumulative distribution function of the anomalous lengths. SWaT has the longest median length of 447 among the four datasets considered, followed by HAI (162), SMD (11), and lastly PSM (5). The very short lengths on SMD and PSM benefit temporal detectors which tend to embed a single or few time points (see Appendix M), whereas SARAD adopts a half time window embedding strategy. The catch is that SARAD can learn spatial relationships with temporal aggregated information per feature, while temporal detectors could not, bringing about benefits of performing anomaly detection in the spatial association space. A more scalable approach for temporal aggregation is to be explored in the future, though variable window sizes or subseries splits might be implied. E.3 Lengths of Diagnosis Labels Figure 8 shows the empirical cumulative distribution function of the diagnosis label lengths. Whereas SMD forensically labels features which deviate from their normal behavioral patterns as anomalous, SWaT and HAI only label the points of attacks as anomalous as the their creators have advanced knowledge of such attacks. The latter labeling strategy results in incomplete sets of anomalous features and diminishes the diagnosis performance of all models, as evidence in Table 4 in Section 4.2. 20 101 102 103 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 4 25% 11 50% 72 75% min = 2 max = 3161 std = 238.42 (a) SMD. 100 101 102 103 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 2 25% 5 50% 88 75% min = 1 max = 8861 std = 1226.88 (b) PSM. 103 104 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 332 25% 447 50% 712 75% min = 101 max = 35900 std = 5982.59 (c) SWaT. 50 100 150 200 250 300 350 400 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 107 25% 162 50% 256 75% min = 17 max = 422 std = 95.13 (d) HAI. Figure 8: Empirical distribution function of the lengths of anomalies. 5 10 15 20 25 30 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 4 25% 8 50% 18 75% min = 1 max = 34 std = 7.88 (a) SMD. 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion min = 1 max = 3 std = 0.51 (b) SWaT. 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Length of Anomaly 0% 20% 40% 60% 80% 100% Proportion 2 50% 2 75% min = 1 max = 3 std = 0.72 (c) HAI. Figure 9: Empirical distribution function of the lengths of diagnosis labels. 21 F Implementation Details We implement SARAD in Python using pyTorch library (Paszke et al., 2019) and Hydra framework (Yadan, 2019). All experiments are run on a single NVIDIA A10 (24GB) GPU. Adam optimizer (Kingma and Ba, 2015) is used and learning rate is halved every epoch for 3 epochs to prevent over-fitting. The time window size is 2W = 100. The data reconstruction module has H = 8 attention heads per layer with attention length D = 512 and hidden length DF F = 2048. For hyperparameter tuning, training set is temporally partitioned into 80% for training and 20% for validation. On each dataset we first perform TPE sampling (Bergstra et al., 2011) for number of encoding layers L ∈{3, 5} and learning rate ∈[10−4, 10−2] to derive the best data reconstruction loss LR on the validation set. The progression module by default has hidden length of DP = 64. We then perform TPE sampling to search weight λLS ∈[10−2, 102] for the progression reconstruction loss LS on the validation set. G Open-accessed Code and Data During the review period, code is anonymized and openly available at https://github.com/ daidahao/SARAD/ with specific instructions and scripts to reproduce experimental results. All data used in our experiments can be openly accessed from public repositories or requested via original authors’ websites. Full links are provided in Appendix E. 22 10 20 50 100 200 400 60 70 80 90 VUS-ROC (%) SMD PSM SWaT HAI (a) Window size. 1 2 3 4 5 60 65 70 75 80 85 90 95 VUS-ROC (%) SMD PSM SWaT HAI (b) Training epochs. 64 128 256 512 1024 2048 60 70 80 90 VUS-ROC (%) SMD PSM SWaT HAI (c) Attention length D. 8 16 32 64 128 256 60 65 70 75 80 85 90 95 VUS-ROC (%) SMD PSM SWaT HAI (d) Progression hidden length DP . Figure 10: Hyperparameter Sensitivity of detection performance in VUS-ROC scores. H Hyperparameter Sensitivity We examine the hyperparameter sensitivity of SARAD’s detection performance. Concretely, we consider the effects of the sliding window size (default is 100), the number of training epochs (3), the attention length of Transformer encoding layers D (512), and the hidden length of the progression reconstruction module DP (64). Figure 10 and 11 present the results. On the window size, a sliding window too small confines temporal modeling to small temporal receptive fields and contains anomaly detection in the association space. However, a sliding window too large incurs higher computational costs, although unlike temporal modeling the costs here are linear. On datasets with shorter anomalous lengths such as SMD and PSM, the anomalous patterns are diluted even further, resulting in performance degradation. On the number of training epochs, fewer epochs lead to model underfitting, and yet overfitting is largely prevented with more epochs due to the aggressive learning rate halving per epoch. On the attention length D, a larger Transformer is prone to overfitting with visible performance drops on SMD and PSM as D passes the default 512, both of whose monitored systems are smaller in scale. On the hidden length DP , a more complex progression anomaly detector does not adversely impact the performance, suggesting that the association space is less prone to detection overfitting than the data space. 23 10 20 50 100 200 400 10 20 30 40 50 60 70 VUS-PR (%) SMD PSM SWaT HAI (a) Window size. 1 2 3 4 5 10 20 30 40 50 60 70 VUS-PR (%) SMD PSM SWaT HAI (b) Training epochs. 64 128 256 512 1024 2048 10 20 30 40 50 60 70 VUS-PR (%) SMD PSM SWaT HAI (c) Attention length D. 8 16 32 64 128 256 20 30 40 50 60 70 VUS-PR (%) SMD PSM SWaT HAI (d) Progression hidden length DP . Figure 11: Hyperparameter Sensitivity of detection performance in VUS-PR scores. 24 I Detection Metrics Conventional metrics such as precision, recall, F1 and the threshold-independent Area Under the Curve (AUC) scores are commonly point-based, i.e., predicted labels are scored individually by time points (Zhou et al., 2023; Zhang et al., 2022; Xu et al., 2018). Real-world benchmarks are rife with range-wise anomalies spanning consecutive time points (Wagner et al., 2023). Point-based metrics are generally ill-suited for evaluating detection performance due to the continuous-discrete conversion and the series-label misalignment (labeling anomalies precisely is hard) (Garg et al., 2022; Tatbul et al., 2018). Here, we use the range-based metrics proposed in (Paparrizos et al., 2022). Compared against their point-based counterparts, they provide robustness to labeling delay and scoring noises as well as performant detector separability and series consistency. Given a set of anomalous ranges P = {pi = (si, ei)} wherein each anomaly pi starts at timestamp si and ends at ei, we enclose each range with uniform l/2-length preceding and succeeding buffers. Given the anomaly label yt ∈{0, 1} at each timestamp t, we derive a new soft label ˜yt ∈[0, 1] as per the minimum temporal distance of t to any anomaly pi ∈P: ˜yt =    p 1 −\|si −t\|/l, ∃pi ∈P, t ∈[si −l/2, si) p 1 −\|t −ei\|/l, ∃pi ∈P, t ∈(ei, ei + l/2] yt, otherwise (8) where l is the buffer length, normally set to the median segment length in P. Within the buffers, ˜yt monotonically increases from √ 2/2 to 1 as the distance decreases. With the new soft label series ˜Y = {˜yt}, we define the True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) accordingly: TP = ˜YT · ˆY, FP = (1 −˜Y)T · ˆY, TN = (1 −˜Y)T · (1 −ˆY), FN = ˜YT · (1 −ˆY) (9) We then compute the threshold-independent AUC for the Receiver Operating Characteristic (AUCROC), i.e., TP rate vs FP rate, and the Precision-Recall (AUC-PR) curves respectively to be rid of thresholding impact. Fully parameter-free Volume Under the Surface (VUS) scores for AUC-ROC (VUS-ROC) and AUC-PR (VUS-PR) are also computed under different buffer lengths ˆl ∈[0, 2l]. 25 J Diagnosis Metrics In line with previous works (Tuli et al., 2022; Zhao et al., 2020), we use common metrics such as Hit Rate (HR) (Su et al., 2019b) and Normalized Discounted Cumulative Gain (NDCG) (Järvelin and Kekäläinen, 2002) to measure performance where diagnosis labels are available. Given a set of anomalous features Gi ⊆[N] at an anomalous timestamp t ∈pi ∈P as a diagnosis ground-truth and the set of top k-ranked features Γt@P% according to Eq. 5 where k = ⌈\|Gi\| × P%⌉, say k = 5 when \|Gi\| = 3 and P = 150, the HR at P% (P ≥100) features is the overlap ratio between the two: HRt@P% = \|Gi ∩Γt@P%\| \|Gi\| (10) In information retrieval, DCG measures the cumulative utility of retrieved documents by their ranking order up to a certain position. NDCG normalizes the DCG by the maximum possible DCG. They are parameterized by P% to determine the location in our evaluation and calculated as follows. DCGt@P% = k X j=1 rj log2(j + 1), IDCGt = \|Gi\| X j=1 1 log2(j + 1), NDCGt@P% = DCGt@P% IDCGt (11) where rj ∈{0, 1} is the relevance value of the j-th element and, in this case, the membership of Γt@P%’s j-th feature in Gi. NDCG has a value strictly between 0 and 1. At the range level, we measure the Interpretation Score (IPS) initially proposed in Li et al. (2021) and here expanded to fit the P% parameterization. For each anomalous range pi, the IPS score is: IPSi@P% = \|Gi ∩Ωi@P%\| \|Gi\| (12) where Ωi@P% is the top k-ranked features according to maxt∈pi sj,t, the j-th feature’s maximum anomaly score during pi and k = ⌈\|Gi\| × P%⌉. It is the HR equivalence at the range level as per the highest anomalous scores per feature. 26 Table 8: Standard deviations of anomaly detection performance in Table 3. Standard deviations of threshold-independent AUC-ROC and AUC-PR metrics and fully parameter-free VUS-ROC and VUS-PR metrics are reported. All values are percentages. SMD PSM SWaT HAI Method AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR IF 0.45 0.11 0.40 0.11 1.15 1.11 1.13 1.10 1.42 2.49 1.56 2.58 1.30 0.94 1.25 0.79 DIF 0.81 0.16 0.76 0.15 0.94 0.64 1.00 0.64 0.26 0.73 0.33 0.90 1.65 3.31 1.64 3.18 TranAD 0.21 0.11 0.23 0.10 0.66 0.29 0.62 0.28 12.15 5.30 11.92 5.30 8.89 9.83 8.83 9.61 ATF-UAD 2.23 0.12 2.19 0.12 4.19 1.75 4.33 1.74 12.06 6.39 11.92 6.27 10.83 12.11 10.77 11.86 AT 0.39 0.50 0.39 0.48 3.05 0.87 3.22 0.86 3.51 1.43 3.61 1.43 5.50 1.96 5.24 1.90 DCdetector 0.39 0.12 0.39 0.12 1.72 0.13 1.57 0.13 0.06 0.23 0.06 0.24 N/A N/A N/A N/A USAD 1.43 0.48 1.44 0.48 0.40 0.21 0.40 0.21 6.41 7.42 6.43 6.77 7.41 8.02 7.17 7.79 GDN 0.81 1.56 0.78 1.55 2.58 1.26 2.39 1.20 1.35 3.21 1.17 3.14 1.63 4.62 1.66 4.46 MAD-GAN 2.00 0.50 1.99 0.50 4.65 4.08 4.71 4.09 4.15 4.09 4.17 4.13 2.05 1.76 1.99 1.72 DiffAD 0.40 0.17 0.31 0.18 1.15 0.42 0.89 0.40 0.26 0.09 0.30 0.12 0.60 0.47 0.56 0.47 Ours 0.98 0.75 1.01 0.76 1.07 0.69 1.09 0.69 0.70 1.34 0.60 1.05 0.47 1.08 0.56 1.11 Table 9: Standard deviations of anomaly detection performance in Table 5. Standard deviations of threshold-independent AUC-ROC and AUC-PR metrics and fully parameter-free VUS-ROC and VUS-PR metrics are reported. All values are percentages. SMD PSM SWaT HAI Submodule Change AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR Ours 0.98 0.75 1.01 0.76 1.07 0.69 1.09 0.69 0.70 1.34 0.60 1.05 0.47 1.08 0.56 1.11 Prog. (Eq.2) no ReLU 1.00 0.45 1.00 0.45 0.76 0.61 0.73 0.61 0.52 5.17 0.51 4.87 0.19 0.33 0.17 0.33 Aggr. (Eq.3) row sum 0.87 0.61 0.88 0.62 1.30 1.30 1.29 1.28 1.39 3.62 1.40 3.61 0.22 0.67 0.21 0.60 Aggr. (Eq.3) no sum 6.34 1.21 6.37 1.20 0.71 0.21 0.69 0.21 0.48 0.98 0.48 1.00 0.51 0.42 0.56 0.39 Detection S directly 0.70 0.51 0.69 0.50 0.85 0.42 0.85 0.42 1.17 1.20 1.12 1.12 1.74 4.39 1.81 4.18 K Standard Deviations of Detection Performance Table 8 reports the standard deviations of anomaly detetion performance as reported in Table 3 in Section 4.2. Table 9 reports the standard deviations of anomaly diagnosis performance as reported in Table 5 in Section 4.3. Table 10 reports the standard deviations of anomaly detection performance as reported in Table 6 in Section 4.3. Table 10: Standard deviations of anomaly detection performance in Table 3. Standard deviations of threshold-independent AUC-ROC and AUC-PR metrics and fully parameter-free VUS-ROC and VUS-PR metrics are reported. All values are percentages. SMD PSM SWaT HAI Method Criter. AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR AROC APR VROC VPR Ours both 0.98 0.75 1.01 0.76 1.07 0.69 1.09 0.69 0.70 1.34 0.60 1.05 0.47 1.08 0.56 1.11 DR r only 0.82 0.19 0.56 0.15 1.76 1.00 1.76 1.01 1.07 1.52 1.11 1.72 0.44 0.86 0.46 0.81 SPR p only 0.99 0.83 1.02 0.85 0.39 0.18 0.42 0.20 1.61 0.67 1.64 0.63 1.10 3.30 1.21 3.32 27 Table 11: Standard deviations of anomaly diagnosis performance in Table 4. Standard deviations of point-based HR@P%, NDCG@P%, and range-based IPS@P% are reported. All values are percentages. SMD SWaT HAI Method HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% P 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 TranAD 0.08 0.29 0.22 0.20 0.25 0.69 1.10 2.50 1.13 1.99 4.25 7.37 0.84 2.16 0.54 1.36 1.47 3.07 ATF-UAD 1.62 1.68 2.60 2.39 2.67 2.73 1.19 2.45 1.17 1.92 4.42 5.81 2.37 4.39 2.68 3.93 5.06 6.52 USAD 0.37 0.81 0.90 1.06 0.50 0.77 1.73 1.54 1.75 1.59 5.77 3.06 0.52 0.27 0.72 0.56 2.04 1.98 GDN 1.41 1.27 1.63 1.57 1.08 1.20 1.36 1.33 1.36 1.32 2.51 3.15 2.08 2.59 2.38 2.70 2.37 6.06 DiffAD N/A N/A N/A N/A N/A N/A 0.09 0.16 0.09 0.12 3.18 3.11 0.37 0.52 0.35 0.44 0.67 1.45 Ours 0.46 0.46 0.56 0.58 1.61 0.96 0.98 1.68 1.03 1.45 4.20 2.52 0.64 0.96 0.74 0.88 1.71 1.57 Table 12: Standard deviations of anomaly diagnosis performance in Table 7. Standard deviations of point-based HR@P%, NDCG@P%, and range-based IPS@P% are reported. All values are percentages. SMD SWaT HAI Method HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% HR@P% ND@P% IPS@P% P 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 100 150 Ours 0.46 0.46 0.56 0.58 1.61 0.96 0.98 1.68 1.03 1.45 4.20 2.52 0.64 0.96 0.74 0.88 1.71 1.57 SPR 1.38 1.58 1.28 1.36 2.05 2.55 21.26 24.13 21.31 23.10 8.55 10.41 0.95 1.35 0.99 1.20 2.94 4.36 Joint 1.63 1.27 1.60 1.39 1.73 2.46 0.20 0.92 0.16 0.62 2.83 5.36 0.63 1.20 0.65 0.94 1.72 1.94 L Standard Deviations of Diagnosis Diagnosis Table 11 reports the standard deviations of anomaly diagnosis performance as reported in Table 4 in Section 4.2. Table 12 reports the standard deviations of anomaly diagnosis performance as reported in Table 7 in Appendix D. 28 Table 13: Anomaly detection point-based performance. Threshold-dependent Precision, Recall, F1 scores and threshold-independent AUC-ROC and AUC-PR scores are reported. All values are average percentages from five random seeds. The best values are in bold and the second best underlined. SMD PSM SWaT HAI Method P R F1 AROC APR P R F1 AROC APR P R F1 AROC APR P R F1 AROC APR IF 15.22 18.53 16.61 66.26 12.67 39.12 76.75 51.50 71.00 45.61 99.54 59.06 74.13 84.25 72.86 12.87 22.00 16.12 70.68 8.20 DIF 29.62 21.67 24.87 69.20 16.47 34.17 93.14 49.98 65.00 40.68 96.79 61.24 75.01 87.37 75.63 62.67 39.75 48.59 79.71 30.62 TranAD 18.54 12.79 14.12 52.34 8.44 34.04 96.53 50.33 62.09 45.74 26.45 74.01 36.62 57.92 16.60 65.27 30.33 39.05 73.22 31.03 ATF-UAD 6.07 24.06 9.49 52.23 4.71 30.50 91.67 45.56 57.70 38.35 30.09 73.37 40.54 62.47 17.89 73.25 25.37 37.18 68.62 26.67 AT 4.16 100.00 7.98 49.97 4.55 27.73 100.00 43.42 45.79 26.21 12.02 100.00 21.46 42.43 10.23 6.86 8.88 7.60 35.82 4.40 DCdetector 4.20 100.00 8.05 49.12 4.19 25.01 100.00 40.01 49.47 24.82 11.82 100.00 21.14 49.93 11.82 N/A N/A N/A N/A N/A USAD 14.15 22.20 17.12 63.14 10.08 30.04 97.68 45.95 57.96 42.64 95.99 62.00 75.30 83.26 72.59 69.20 28.80 39.54 71.35 29.18 GDN 16.47 26.84 17.87 65.30 9.62 39.08 84.15 53.25 69.09 42.18 38.04 71.12 49.53 76.89 24.10 68.95 45.61 52.64 82.39 39.43 MAD-GAN 15.31 21.39 17.63 63.31 10.53 30.80 92.42 46.08 63.07 43.75 84.83 69.53 76.42 86.63 63.69 79.97 49.00 60.76 81.07 49.02 DiffAD 10.51 19.89 13.75 60.34 7.67 27.76 100.00 43.45 55.09 33.04 12.02 100.00 21.46 18.71 7.27 46.76 16.53 24.37 80.96 20.74 Ours 17.06 41.26 24.10 79.82 15.10 41.55 58.60 48.60 65.42 43.28 96.20 66.92 78.92 86.91 74.81 65.42 62.63 63.99 93.40 49.22 Table 14: Standard deviations of anomaly detection point-based performance in Table 14. Standard deviations of threshold-dependent Precision, Recall, F1 scores and threshold-independent AUC-ROC and AUC-PR scores are reported. All values are percentages. SMD PSM SWaT HAI Method P R F1 AROC APR P R F1 AROC APR P R F1 AROC APR P R F1 AROC APR IF 1.16 2.02 0.47 0.72 1.50 3.06 9.03 1.54 1.12 1.70 0.22 0.11 0.03 0.91 0.86 1.43 2.09 0.59 0.67 0.90 DIF 4.06 0.98 0.98 0.50 1.05 1.59 1.06 1.65 1.52 2.38 1.87 0.73 0.60 0.34 0.56 1.91 3.22 2.62 1.15 1.94 TranAD 5.98 3.31 0.46 0.28 0.50 0.33 0.62 0.30 0.62 0.74 12.41 16.00 12.29 19.93 6.80 14.08 11.13 6.63 7.58 8.08 ATF-UAD 1.00 7.60 1.05 1.21 0.25 2.70 9.45 2.52 4.88 3.10 12.68 9.36 12.34 15.50 6.36 9.99 10.18 12.94 8.39 11.35 AT 0.00 0.00 0.00 0.44 0.43 0.00 0.00 0.00 1.95 1.01 0.00 0.00 0.00 5.39 1.35 3.33 3.13 3.18 5.81 1.59 DCdetector 0.00 0.00 0.00 0.78 0.13 0.00 0.00 0.00 0.32 0.15 0.00 0.00 0.00 0.04 0.21 N/A N/A N/A N/A N/A USAD 2.10 1.91 1.17 2.68 1.20 0.44 0.43 0.50 0.61 0.27 3.30 1.41 0.70 2.64 1.25 13.69 6.17 4.12 4.73 5.44 GDN 7.60 12.09 3.46 0.58 2.19 2.50 11.27 4.26 2.87 2.06 1.75 1.49 1.19 1.50 1.56 14.18 11.91 5.65 1.60 4.23 MAD-GAN 2.46 2.16 1.08 1.98 0.52 2.31 5.94 2.08 4.54 5.54 0.91 0.93 0.60 2.44 2.22 0.86 0.90 0.87 1.42 1.27 DiffAD 0.38 0.61 0.28 0.34 0.39 0.00 0.01 0.00 0.74 0.45 0.00 0.00 0.00 0.20 0.03 5.14 0.87 1.24 0.77 0.47 Ours 0.76 4.22 1.20 0.56 0.51 1.65 0.65 1.06 0.79 1.09 1.44 1.27 0.93 0.48 1.11 0.38 0.38 0.23 0.60 0.62 M Point-based Evaluations In addition to the model evaluations using range-based metrics such as VUS-ROC and VUS-PR in Sections 4.2 and 4.3, we conduct point-based evaluations herein using point-based metrics exclusively. Table 13 reports report the Precision, Recall, and F1 scores under the threshold where the method of interest achieves the best F1 score. To mitigate the impact of thresholding protocol, Table 13 also reports the threshold-independent AUC-ROC and AUC-PR scores. Table 14 reports the standard deviations. SARAD achieves state-of-the-art performance on most datasets except PSM. The enlarged temporal receptive field of SARAD (half input window) contributes to its underperformance by point-based metrics, especially on datasets where anomalous ranges are short (see Table 2), when compared against most others’ receptive filed of single or few time points (typical 1D Conv kernel size is 3). We again underline that most real-world anomalies are continuous and point-based metrics are mismatched for such anomalies (see Appendix I for discussion). 29 Table 15: Model complexity and overheads. The total training time (in minutes), the inference time per sample (in milliseconds), and the total number of parameters (where applicable) are reported. The best values are in bold and the second best underlined. SMD PSM SWaT HAI T = 708K T = 132K T = 497K T = 922K N = 38 N = 25 N = 51 N = 79 T = 1 min T = 1 min T = 1 s T = 1 s Train. Infer. Param. Train. Infer. Param. Train. Infer. Param. Train. Infer. Param. Method (mins) (ms) (mins) (ms) (mins) (ms) (mins) (ms) IF 0.01 0.01 N/A 0.00 0.01 N/A 0.01 0.01 N/A 0.03 0.01 N/A DIF 10.13 1.66 874K 1.91 1.52 853K 7.67 1.68 895K 15.47 1.87 940K TranAD 6.80 0.04 127K 0.96 0.03 57K 6.06 0.05 226K 17.15 0.07 531K ATF-UAD 14.20 0.06 414K 1.90 0.05 408K 12.77 0.06 421K 16.42 0.06 436K AT 0.74 0.00 867K 9.25 0.00 4.80M 36.53 0.00 910K 60.57 0.00 4.91M DCdetector 102.16 0.01 867K 25.72 0.03 895K 214.56 0.02 910K Out of Memory USAD 4.34 0.01 803K 0.86 0.01 441K 3.74 0.01 1.26M 9.34 0.01 2.54M GDN 36.04 0.44 3K 9.64 0.34 3K 19.65 0.46 4K 65.69 0.68 6K MAD-GAN 42.56 0.40 268K 45.06 0.46 261K 1416.26 0.40 274K 165.01 0.47 289K DiffAD 11.93 1.49 38.85M 4.94 7.91 38.85M 15.32 1.69 38.85M 42.87 2.80 38.85M Ours 1.47 0.11 9.57M 2.32 0.12 15.85M 10.87 0.16 9.59M 31.97 0.39 15.94M N Model Complexity and Overheads We study and compare the complexity and time overheads of all baselines and SARAD. Concretely, we evaluate the model complexity by the number of parameters being used and the total training time as well as the inference time per sample. All experiments on time overheads are performed on a compute node with AMD EPYC 7443 (48 cores, 96 threads) CPU, NVIDIA A10 (24GB) GPU, and 512 GB RAM. Table 15 reports the total training time (in minutes), the inference time per sampling point, and the number of network parameters used (where applicable) of all baselines on the main datasets. The size of the training set T, the number of features N, and the sampling period T are also reported per dataset for easy reference. SARAD, while not the fastest nor the lightest model, incurs moderate time overheads and model complexity. We note that even SWaT, the smallest dataset in terms of actual clock time, spans approximately 6 days for collection. This is far exceeding most detectors’ training time besides MAD-GAN and levigates concerns for training overheads for them. All detectors also incur an inference per sample time several magnitudes below the smallest sampling frequency of 1 second, guaranteeing real-time deployment of all detectors once trained. 30 O Baselines We trained all baselines using official implementations where available and recommended hyperparameters from respectively papers are used. Some baselines, such as DiffAD and AT, have dataset-specific hyperparameters and here we adopted them as well. The open-accessed URLs of the baselines used are listed as followed. • IF (ICDM’08) (Liu et al., 2008): https://github.com/xuhongzuo/deep-iforest. • DIF (TKDE’23) (Xu et al., 2023): https://github.com/xuhongzuo/deep-iforest. • TranAD (VLDB’22) (Tuli et al., 2022): https://github.com/imperial-qore/ TranAD. • ATF-UAD (NN’23) (Fan et al., 2023): https://github.com/wzhSteve/ATF-UAD. • AT (ICLR’22) (Xu et al., 2022): https://github.com/thuml/Anomaly-Transformer. • DCdetector (KDD’23) (Yang et al., 2023): https://github.com/DAMO-DI-ML/ KDD2023-DCdetector. • USAD (KDD’20) (Audibert et al., 2020): https://github.com/manigalati/usad. • GDN (AAAI’21) (Deng and Hooi, 2021): https://github.com/d-ailin/GDN. • MAD-GAN (ICANN’19) (Li et al., 2019): https://github.com/LiDan456/MAD-GANs. Official implementation was migrated to pyTorch for uniform environmental set-up. See our codebase for details. • DiffAD (KDD’23) (Xiao et al., 2023): https://github.com/ChunjingXiao/DiffAD. 31 10 2 10 1 100 101 Data Reconstruction Error (r) 10 3 10 2 10 1 Progression Recon. Error (p) Anomalous Normal (a) SMD. 10 2 10 1 100 Data Reconstruction Error (r) 10 3 10 2 Progression Recon. Error (p) Anomalous Normal (b) PSM. 10 2 10 1 Data Reconstruction Error (r) 10 3 10 2 10 1 100 Progression Recon. Error (p) Anomalous Normal (c) SWaT. 10 3 10 2 10 1 100 101 102 Data Reconstruction Error (r) 10 3 10 2 10 1 Progression Recon. Error (p) Anomalous Normal (d) HAI. Figure 12: Joint detection criterion s of data reconstruction error r and progression reconstruction error p. P Joint Detection Criterion Figure 12 visualizes the two components of the joint detection criterion s in Eq. 5, i.e., the data reconstruction error r and the progression reconstruction error p. Balanced resampling is applied here. Recall that s is the sum of normalized r and p. Most anomalous samples (input series) either has high r or high p, and oftentimes both. The former measures the magnitude of anomalousness in the data space, the latter in the spatial association space. The basis underpins the formalization of the joint detection criterion. 32 10 6 10 5 10 4 10 3 10 2 10 1 100 101 102 Anomaly Score 0.0 0.2 0.4 0.6 Density Normal Anomaly (a) SMD. 10 4 10 3 10 2 10 1 100 101 102 Anomaly Score 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Density Normal Anomaly (b) PSM. 10 5 10 3 10 1 101 103 Anomaly Score 0.0 0.5 1.0 1.5 2.0 2.5 Density Normal Anomaly (c) SWaT. 10 4 10 2 100 102 104 Anomaly Score 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Density Normal Anomaly (d) HAI. Figure 13: Distributions of anomalies scores. Q Distributions of Anomaly Scores Figure 13 the distributions of the joint anomaly scores. On SMD and PSM, the scores of the normal and anomalous samples (input series) overlap more heavily than on SWaT and HAI. The observations here highlight the difficulty of anomaly detection in the service monitoring space, more so than in industrial control where measurements and actuation result from well-defined control logics. The hardness is evident is Table 3 of detection performance and 4 of diagnosis performance. 33 Table 16: Anomaly detection performance under different thresholds. TPRs under thresholds set by pre-defined FPRs ∈{1%, 5%, 10%} are reported. Higher TPRs are better. Similarly, FPRs under thresholds set by pre-defined TPRs ∈{90%, 95%, 99%} are reported. Lower FPRs are better. All values are average percentages from five random seeds. The best values are in bold and the second best underlined. SMD PSM SWaT HAI TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) Method 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% IF 0.29 3.02 7.22 78.49 86.46 93.63 0.50 3.50 11.09 80.94 89.64 94.26 32.88 56.45 65.77 48.48 56.06 64.54 4.15 18.14 28.67 57.31 61.96 68.40 DIF 1.56 8.43 15.17 79.37 87.22 92.33 0.23 1.77 4.63 78.51 83.80 91.86 48.07 63.25 69.63 41.18 50.41 59.62 31.32 41.90 49.40 48.56 54.21 58.81 TranAD 0.86 4.25 8.64 87.86 91.53 95.21 1.15 2.58 7.62 75.94 87.46 96.24 0.12 0.57 1.61 75.78 81.11 85.72 21.39 31.48 37.79 53.60 58.50 66.99 ATF-UAD 0.04 1.04 4.42 90.65 93.75 96.61 0.43 2.05 5.67 82.90 89.02 95.95 0.23 1.02 2.16 69.66 75.93 81.20 16.93 24.21 31.12 60.42 65.08 69.25 AT 1.47 3.78 3.94 99.77 99.77 99.77 0.56 2.73 5.43 99.77 99.77 99.77 1.43 4.07 7.68 98.46 98.46 98.46 4.37 11.61 17.51 95.60 96.91 99.11 DCdetector 0.32 3.40 8.10 97.47 98.43 99.07 0.25 1.88 4.32 99.85 99.85 99.85 0.91 2.19 2.19 98.48 98.48 98.48 N/A N/A N/A N/A N/A N/A USAD 0.38 3.48 7.55 84.13 89.58 94.09 0.55 1.85 5.13 87.22 90.07 92.05 33.13 45.56 52.37 60.13 70.99 74.63 17.82 25.89 33.17 57.63 61.71 70.61 GDN 1.75 12.03 22.41 70.54 77.88 85.00 0.50 3.12 11.77 68.10 74.60 84.05 1.34 1.91 2.57 41.72 50.87 59.36 35.91 48.13 55.29 44.44 50.43 55.77 MAD-GAN 2.66 15.92 24.81 78.00 84.93 90.21 3.28 8.89 15.61 79.78 85.25 93.30 22.43 60.97 67.95 46.78 59.60 70.56 42.69 52.38 58.18 47.93 54.24 59.33 DiffAD 0.55 7.74 16.10 81.87 88.12 93.06 0.58 3.83 8.22 86.78 93.18 98.33 0.51 3.01 5.88 98.46 98.46 98.46 9.31 27.85 48.94 29.57 34.39 37.69 Ours 1.44 16.42 36.41 48.19 57.97 68.38 0.71 5.85 9.60 82.33 90.10 96.65 53.87 60.11 68.83 42.85 50.24 59.05 58.26 83.03 88.82 11.13 19.76 27.01 Table 17: Standard deviations of anomaly detection performance under different thresholds in Table 16. Standard deviations of TPRs under thresholds set by pre-defined FPRs ∈{1%, 5%, 10%} are reported. Similarly, standard deviations of FPRs under thresholds set by pre-defined TPRs ∈{90%, 95%, 99%} are reported. All values are percentages. SMD PSM SWaT HAI TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) TPR↑(@FPR) FPR↓(@TPR) Method 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99% IF 0.08 0.52 0.46 1.34 1.82 0.91 0.09 0.84 1.00 3.07 1.03 2.23 4.29 4.61 2.39 2.68 2.88 2.74 0.51 1.04 1.11 2.00 1.46 2.98 DIF 0.48 0.90 1.44 1.27 1.45 1.48 0.16 0.66 0.38 2.11 1.73 1.91 4.64 2.05 2.18 1.48 0.78 0.84 2.91 3.61 3.00 2.37 2.74 2.17 TranAD 0.08 0.40 0.35 0.74 0.23 0.23 0.22 0.30 0.23 0.38 1.15 0.29 0.06 0.15 0.09 9.61 8.01 5.79 10.67 13.44 14.65 8.42 7.38 11.20 ATF-UAD 0.04 0.91 1.66 2.12 1.87 1.70 0.30 0.97 2.09 7.55 7.46 2.80 0.17 0.25 0.21 6.07 5.91 3.89 10.48 13.55 16.69 6.00 4.69 4.35 AT 0.06 0.51 0.67 0.00 0.00 0.00 0.05 0.53 0.91 0.00 0.00 0.00 0.62 0.81 1.31 0.00 0.00 0.00 2.24 4.63 5.58 3.32 2.06 0.00 DCdetector 0.04 0.61 1.39 5.15 3.00 1.57 0.03 0.42 1.04 0.00 0.00 0.00 0.02 0.60 0.60 0.00 0.00 0.00 N/A N/A N/A N/A N/A N/A USAD 0.22 0.74 1.26 3.34 3.14 3.31 0.12 0.00 0.01 0.28 0.44 0.34 9.68 11.59 8.33 15.06 14.38 15.27 7.29 10.77 11.55 6.59 5.71 5.00 GDN 1.51 5.42 4.85 1.77 2.13 2.75 0.42 2.05 2.87 8.93 8.37 5.87 1.25 1.57 1.72 3.10 6.71 7.48 12.76 7.33 5.96 3.83 4.28 3.06 MAD-GAN 1.23 1.54 1.30 4.22 4.57 3.55 1.74 2.58 3.12 6.60 6.30 2.91 19.27 5.15 6.39 15.81 11.37 11.48 1.92 2.97 2.88 4.87 4.66 4.20 DiffAD 0.11 0.75 0.52 1.18 0.74 0.23 0.09 0.28 0.26 0.98 0.48 0.29 0.07 0.32 0.34 0.01 0.00 0.00 0.48 2.26 0.99 1.98 2.66 2.54 Ours 0.18 1.76 4.81 0.48 0.87 1.35 0.35 0.80 0.97 1.42 1.25 0.73 3.57 3.14 0.45 0.70 1.08 1.69 1.51 2.45 2.73 2.80 2.36 2.12 R Detection Performance Under Thresholds Effectiveness of each anomaly detector is influenced by the selection of thresholds. Previously, we reported the range-based and point-based threshold-independent performance metrics. Here, we study the influence of threshold selection on the detection performance. Table 16 reports the range-based True Positive Rates (TPRs) under thresholds set by pre-defined False Positive Rates (FPRs)as well as the FPRs under different thresholds set by pre-defined TPRs. Table 17 reports the standard deviations. 34 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The main claims clearly reflect the paper’s contributions and scope of time series anomaly detection in the abstract and Section 1. 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4,489	Leveraging Visual Tokens for Extended Text Contexts in Multi-Modal Learning Alex Jinpeng Wang1, Linjie Li2, Yiqi Lin1, Min Li3, Lijuan Wang2, and Mike Zheng Shou1B 1Show Lab, National University of Singapore 2Microsoft 3Central South University Abstract Training models with longer in-context lengths is a significant challenge for multimodal machine learning due to substantial GPU memory and computational costs. This exploratory study does not present state-of-the-art models; rather, it introduces an innovative method designed to increase in-context text length in multi-modality large language models (MLLMs) efficiently. We present Visualized In-Context Text Processing (VisInContext), which processes long in-context text using visual tokens. This technique significantly reduces GPU memory usage and floating point operations (FLOPs) for both training and inferenceing stage. For instance, our method expands the pre-training in-context text length from 256 to 2048 tokens with nearly same FLOPs for a 56 billion parameter MOE model. Experimental results demonstrate that model trained with VisInContext delivers superior performance on common downstream benchmarks for in-context few-shot evaluation. Additionally, VisInContext is complementary to existing methods for increasing in-context text length and enhances document understanding capabilities, showing great potential in document QA tasks and sequential document retrieval. The code is available at https://github.com/showlab/VisInContext. 1 Introduction Large Language Models (LLMs), such as OPT, Mistral, and LLaMA-2 [4, 5, 6], have significantly advanced the field of Natural Language Processing (NLP). These advancements are partly due to the increased capability of LLMs to process long contexts, from 512 tokens [7] up to 16K tokens [6]. Building on these developments, recent multi-modal learning research [1, 8, 9, 10] has shifted focus from simple image-text pairs, like those in CC3M [11] and LAION-400M [12], to more complex and lengthy interleaved document datasets. Examples include web corpora like MMC4 [13] and the OBELICS [14] dataset, as well as PDF corpora like DocVQA [15]. However, training models on these complex datasets presents significant challenges due to the increased GPU memory and computational demands of extended contexts. For instance, while processing just 5M data items from MMC4 and 10M from the OBELICS dataset, OpenFlamingo9B [9] resorted to sub-sampling text and processing only 256 tokens at a time, yet it still requires 32 80GB A100 GPUs for over three days. This highlights the need for more computation-efficient methods to handle long context lengths effectively. In the domain of LLMs, two popular methods to extend context length are the use of memorizing banks [16] and novel self-attention mechanisms [17, 18]. These methods have inspired advancements in the multi-modality domain as well. For example, the Large World Model [19] introduces Ring Attention [18], and MA-LMM [20] employs memory banks to process long video understanding tasks. While these techniques have shown promise, our approach aims to increase in-context text length B: Corresponding Author. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). (a) GPU memory consumption. (b) The flops comparison. Figure 1: VisInContext significantly increases the in-context text length from 256 to 2048 during pretraining on NVIDIA H100 GPU. For our method, we incorporate VisInContext after 128 text tokens. We implement PyTorch Flamingo [1] models with different in-context length during pre-training. The language model is a 56B MOE [2] model loaded with 4-bit quantization and the batch size on each GPU is 32 with FP16. We train the model with DeepSpeed [3] Zero-2. by leveraging the strengths of visual encoders in MLLMs. We first observe that existing MLLMs usually exploit a much lighter visual encoders, compared to its text decoders. For instance, Flamingo-9B consists of a 304.4M ViT-L/16 [21] as image encoder, and a 7.1B Chinchilla [1] model as the text decoder. Additionally, previous works [22, 23] have demonstrated that visual encoders trained on paired image-text data also exhibit emergent OCR capabilities. Motivated by these observations, we propose Visualized In-Context Text Processing (VisInContext), a method that uses visual tokens to process extended textual contexts, which is complementary of existing methods in extending context length. Specifically, we convert long textual content into images and use the visual encoders to extract textual representations. In this way, we can efficiently and effectively enable models with much longer text contexts, as shown in Figure 1. With VisInContext, we show that the in-context text length can be increased by 7 times over the competing baseline. Additionally, we observe almost the same overall computation FLOPs even as in-context length extends significantly. Our extensive experiments will also show that VisInContext renders superior model performance on conventional in-context few-shot evaluations and document understanding, with much lower computational cost. Contributions. In summary, our contributions are as follows: i. We introduce Visualized InContext Text Processing (VisInContext), a novel method that increases in-context text length using visual tokens. VisInContext directly compresses text context at input-level, which is complementary to existing techniques with improved self-attention or memory banks. ii. We demonstrate that VisInContext is effective for both training and inference stage with much lower computational cost. iii. With extended text context brought by VisInContext, our model improves the average in-context few-shot performance from 55.8% to 57.8% over the competing baseline. iv. As a byproduct, our method also shows great potential in document understanding on popular document QA tasks and our newly proposed sequential document retrieval task. 2 Method The goal of VisInContext is to process in-context text using visual tokens so that the model can handle long text context more efficiently. We primarily base our study on Flamingo-based architecture [1, 9, 14], as it has shown success in improving a model’s ability to learn from long multimodal context that contains arbitrarily interleaved text and images. 2.1 Terminology Before diving into model details, we define the following terms: 2 Figure 2: VisInContext Pipeline. The VisInContext pipeline builds upon the Flamingo model for in-context few-shot modeling (represented in gray). VisInContext processes interleaved image-text data by rendering portions of the in-context text into images. This approach maintains the Text Token Length of the model while allowing for a significantly extended In-context Text Length. In-context Text Length: The actual length of text tokens observed by the model within a document. Text Token Length: The length of the text sequence input directly to the LLM, corresponding to the token count of this sequence. With VisInContext, the In-context Text Length is greater than the text token length, as part of the text is represented using visual tokens. 2.2 Overall Architecture The implementation and architecture of VisInContext are shown in Figure 2. It is based on a dual-stream encoder model that integrates both visual and textual data. To effectively handle long interleaved data, we use a pre-sampling strategy as in Flamingo-style works [1, 9, 14]. We sample m images I1, I2, . . . , Im ∈I with corresponding texts T1, T2, . . . , Tm ∈T. Tokens are concatenated in the form < visual1 >< text1 > . . . < visualm >< textm >, where < visual > is a single placeholder token. A random 256-token sequence is then sampled. However, since the overall length of a web document is generally much longer than 256 tokens (In-context Text Length ≥Text Token Length), this sampling approach can lead to the omission of a lot of related text context. To address this issue, we convert these omitted text context into visual signals by rendering them into images. We first concatenate all omitted text segments and divide them into K parts to render text images, named T ′ 1, T ′ 2, . . . , T ′ m ∈T ′. Both the original images and the text-rendered images are then processed through a shared frozen vision encoder. Then, we employ two learnable resamplers to extract a fixed number of tokens from both the raw and text-rendered image features, respectively. To facilitate the model to learn from rendered text images, we introduce two novel model designs, Token Masking mechanism and Text-Centric Contrastive Learning (TCCL). Token Masking allows the model to only read from text image tokens by masking the raw image tokens with masking ratio 1, which ensures that the model won’t simply be ignoring the text images during training, hence can learn the association between the rendered text images {T ′ i } and the text tokens {Ti}. TCCL aligns the visual text representation from the resampler with the embeddings extracted from text tokenizers in LLM, which reduces the gap between our visual text tokens and the text tokens the LLM is trained to perceive. With these designs, VisInContext not only reduces computational demands—as evidenced by a reduction in flops and inference time—but also improves the OCR ability, as we will show in our experiments. 2.3 Text Rendering This module converts textual data into a visually rich RGB format, specifically rendering the text into an image size of ph × npw, where n is the number of patches. We employ the HERSHEY font at a size of 10px. On average, one 16x16 patch accommodates approximately 1.5 OPT text tokens. A 224x224 text image contains about 294 text tokens. Consequently, a visual encoder operating on this rendered text image requires only 1/3 of tokens to encode an equivalent amount of text, compared 3 to the text tokenizer in language models. The vision encoder is quite lightweight ViT-L (340M) compared to language model MOE (56B), which makes the processing of rendered text images significantly more efficient than directly inputting the text into a language model. 2.4 Token Masking In our initial experiments, we find that combining tokens from raw images and text images directly led to the network disregarding the text-image input. To address this issue, we introduce a Token Masking strategy to force the model to learn text semantics from visual inputs. During pretraining, the raw image and text image are first encoded into the same number of tokens after resampler, and then we mask the raw image tokens with a pre-defined probability. When masking out the raw image tokens, the model can focus on learning the association between rendered text images and the complementary text tokens. At inference time, we add the text-image tokens and image tokens together, to allow the model effectively leverage information from both sources. 2.5 Text-Centric Contrastive Loss (TCCL) Motivation. Given that the vision encoder, typically a frozen Vision Transformer (ViT) [24], never observes rendered text images during pretraining, it may struggle to derive text semantics from pixels. To mitigate this issue, we introduce a new training objective, Text-Centric Contrastive Loss (TCCL). This objective aims to guide the resampler on rendered text images to interpret visual representations of text with a proficiency comparable to traditional text tokenizers, so that the textual semantics can be effective extracted from the rendered text images. Mechanism. TCCL utilizes raw text token embeddings from the text tokenizer as soft supervision signals to supervise the resampler to learn text-centric representation. To reduce the global semantic gap between text image embeddings and text token embeddings, we first aggregate these embeddings with average pooling and then align them with TCCL. Intuitively, TCCL is designed to turn the joint of the vision encoder and resampler into a “visual" text tokenizer, as it promotes the text image embeddings to share a similar global semantic as the text token embeddings. The core of TCCL is formulated as a contrastive loss: Lij = −log exp(sim(fvi, ftj)/τ) PN k=1 exp(sim(fvi, ftk)/τ) ! (1) Where Lij denotes the contrastive loss for comparing the ith text image against the jth text, fvi and ftj represent the feature embeddings of the ith text image and jth text, respectively. τ is a parameter that control the sharpness of the output distribution. Note that fvi and fti are different features extracted from the same text, as the ith text image is a direct rendering of the ith text. 3 Experiment 3.1 Experimental Setup Pretraining. We validate VisInContext with Open-Flamingo [9] and CosMo [25]. To enhance computational efficiency, all models utilize float16 precision. For the 56B MOE [2] model, we employ DeepSpeed’s [3] Zero-2 stage with CPU offloading and further optimize the model by quantizing it to 4-bit precision 1. We also use Flash Attention [17] to further improve memory efficiency. For all other experiments, we train the model using DeepSpeed Zero-2 without CPU off-loading. The Open-Flamingo 9B baseline is based on Mistral7B [5]. Our pretraining dataset includes a 180M subset of DataComp1B [26], MMC4 [13], the OBELICS [14] dataset, and OCR Rendered Text [27]. (More details are provided in the Appendix B.1) For each input document or image-text pair, we render a text sequence into an image with a fixed size of 16x8192 (512 patches) by default, with ph = pw = 16. 1The implementation is from https://github.com/TimDettmers/bitsandbytes. 4 Method Text ICL Tokens↑ Shots VQA Caption Classi. Mean okvqa textvqa vizwiz vqav2 coco flickr HM OpenFlamingo MOE [9]† Raw Text 256 0 40.2 21.3 23.3 47.8 82.3 59.4 60.4 47.8 4 42.5 22.2 32.2 49.8 90.5 63.5 63.8 52.1 32 46.8 23.2 40.5 49.9 98.2 66.2 66.0 55.8 +VisInContext +Rendered Image 2048 0 39.5 26.4 26.3 48.5 84.4 60.5 62.2 49.7 4 44.3 28.9 32.0 50.3 94.2 65.3 65.5 54.4 32 46.3 31.2 41.2 51.0 101.3 68.4 65.2 57.8 Table 1: Increasing in-context text length with VisInContext significantly improves performance on multi-modality downstream tasks. The model is pre-trained with a 56B MOE model. ICL stands for in-context text length. HM is short for hatefulmemes. With VisInContext, we increase the ICL from 256 to 2048, leading to clear improvements over the baseline. † indicates our implementation. Method Text Source Text Tokens T-Shots VQA Caption Mean okvqa textvqa vizwiz vqav2 coco flickr OpenFlamingo9B Baseline [9] †Raw Text 10 0 18.1 14.8 21.5 26.5 40.1 32.1 25.5 62 4 23.8 18.1 23.7 40.5 57.5 35.3 33.2(7.7↑) 426 32 25.2 16.4 25.5 34.6 66.1 38.5 34.4(8.9↑) +VisInContext Rendered Image 10 0 16.2 16.8 15.4 30.6 42.3 33.5 25.8 10 4 17.2 21.8 19.7 35.2 52.4 35.2 30.3(4.5↑) 10 32 21.3 22.6 21.5 38.8 60.3 37.0 33.6(7.8↑) Table 2: VisInContext effectively incorporates in-context text with visual tokens, demonstrating significant performance improvements with consistent token usage. Here, T-shots refer to text-only in-context examples. Tokens indicate the length of the input to the LLM. Text source describes the preprocessing method for in-context examples. † denotes our implementation on 180M pretraining data. Downstream Evaluation. Our objective is to demonstrate that in-context length can be extended using visual tokens, thereby enhancing the understanding of complex multimodal documents. Consequently, we focus primarily on tasks related to long-context understanding. To evaluate the long-context understanding ability, we adopt the few-shot evaluation setting in Flamingo [1]. We report answer accuracy on the OK-VQA [28], TextVQA [29], VizWiz [30], and VQAV2 [31]. Additionally, we assess performance on captioning tasks using COCO [32] and Flickr30K [33]. Moreover, we also propose a setting named text-only in context few-shots to explore text-only in-context evaluation. For this setting, we use in-context sampling without visual input to generate long-context inputs and the visual input is not observed by the model. In order to illustrate the impact of having long in-context text, we evaluate the model for document understanding on DocVQA [15] and OCR VQA [34]. Lastly, we introduce a new task, sequential multimodal document retrieval. This dataset is based on the existing interleaved OBELICS [14] dataset. Further details are provided in the Sec. D of the appendix. 3.2 In-context Few-shot Evaluation Impact of Extended In-Context Text Length. Interleaved document datasets typically contain long texts. For instance, the OBELICS [14] dataset has an average token length of 815 tokens per document. Due to GPU memory constraints, Flamingo-like models [14, 9] only sub-sample 256 tokens during pretraining, which leads to a significant loss of context information. We compare the baseline model pre-trained with 256 tokens, against our method with an increasing In-context Text Length to 2048 tokens. Table 1 shows a clear advantage of VisInContext. For example, on TextVQA, accuracy improves from 23.2% to 31.2% with 32-shot. Similarly, the average model performance across all datasets show an increase from 55.8% to 57.8%. These findings demonstrate that VisInContext effectively increases the In-context Text Length to improve multi-modality understanding. Few-shot Evaluation with Text-only In-context Examples. As downstream tasks often differ in format from pretraining data, several works [1, 9, 14] have tested the few-shot abilities of models 5 Method Text Source DocVQA OCR VQA val test Open-Flamingo-9B Baseline [9] Raw Text 45.3 48.2 51.5 +VisInContext Rendered Image 48.5(3.2↑) 52.2(4.0↑) 58.4(6.9↑) Table 3: VisInContext clearly boosting the baseline on document understanding tasks. Figure 3: VisInContext significantly improves the OCR ability of LLM. We present the Rendered Text [27] images and the corresponding next-word prediction accuracy on the validation set. Using the same pre-training steps, VisInContext achieves significantly better results in predicting words in visual images, even when the fonts are difficult to recognize. using in-context examples. For instance, in the VQA dataset, a few question-and-answer pairs are provided as in-context examples with visual signals. However, for zero-shot evaluation, two question-and-answer pairs are added as in-context examples without visual signals in [1, 9, 14]. Follow the zero-shot setting, we examine the effect of having text-only in-context examples and extend it to multi-shot setting, by leaving out the corresponding images (See Appendix .E for more details). We compare model performance of the baseline Open-Flamingo 9B and our method under the same setting, where the differences lie in how these text-only in-context examples are processed. Specifically, Open-Flamingo directly takes in them as text tokens, while VisInContext takes in the corresponding rendered text images. Figure 4: VisInContext extends the incontext text length of MOE based MLLM from 1k to 9k at inference stage. Table 2 summarizes the results across four VQA benchmarks and two captioning benchmarks. Notably, compared to the text-only 0-shot setting, our VisInContext with 32shot significantly is improved on all VQA and captioning benchmarks considered. Though the 32-shot performance of VisInContext is slightly lower than the competing baseline, we cut down the input tokens to the LLM from 426 to only 10 Text Token Length, which lead to significant reduction in the inference cost. These outcomes highlight two key points: i. VisInContext can effectively understand text rendered in images. ii. Text rendered as images can be comparably effective as raw text, when used as text-only in-context examples. Comparison on Inference Cost. We then analyze the inference cost of VisInContext and compare to the baseline. Both models are based on a 56B MOE LLM with a batch size of one to explore the maximum manageable In-context Text Length. The results, shown in Figure 4, demonstrate that the In-context Text Length can be extended up to 9192 tokens for the 56B MOE model on 80GB H100 GPUs with our method at inference stage. This result 6 Figure 5: Sequential multi-modal retrieval example. The input sequence is I1, T1, R1, I2, T2, R2 that from interleaved document in OBELICS [14] dataset. Visual Input Text Input Surrounding Text Input Seq-I Seq-T Raw Image Raw Text 16.3 64.8 Raw Image Raw Text Raw Text 18.9 67.5 Raw Image Raw Text Rendered Text Image 22.7 66.5 Table 4: The model pretrain with VisInContext significantly improves sequence understanding ability. We report the sequence retrieval result on OBELICS-Hybrid6. highlights the efficiency and advantages of VisInContext, also show its potential in understanding very long document. 3.3 Document understanding In this section, we evaluate the model on document understanding tasks. Unlike common visionlanguage tasks that usually short-form pairs, this task requires comprehension of long and complex document data. We evaluate our model on DocVQA and OCRVQA. All document images are of size 384 × 384. Following Pix2Struct [35], we finetune the model on DocVQA train data and report performance on the average normalized Levenshtein similarity (ANLS) metric. Results in Table 3 show that our method significantly outperforms the baseline. For instance, we achieve a 6.9% improvement on OCRVQA. To further analyze why our method enhances document understanding, we present the validation accuracy of the LLM on the Rendered Text [27] dataset during pretraining in Figure 3. We observe a substantial improvement in next word prediction accuracy, with top-1 accuracy increasing from 67.37% to 85.25% (a 16% improvement) and top-5 accuracy rising from 80.76% to 93.38%. These findings indicate that the LLM can effectively understand text embedded in visual signals with VisInContext. 3.4 Sequential Multi-modal Retrieval In order to further analyze the benefit of having long text context in multimodal modeling, we propose a new task – Sequential Multimodal Retrieval (SMR), based on document data from interleaved OBELICS [14] dataset. The document is composed of interleaved data, consisting of images and texts arranged in a meaningful sequence. We show one sample in Figure 5 and define the input and output of this task as below: Input: Given a pair of content items, an image and a corresponding text (I1, T1, R1, I2, T2, R2), from a document D. I is Image, T is the matched text and R is the surrounding text. Output: The task is to retrieve the next image I2 and the next text T2 in the sequence. Named as Seq-I and Seq-T, correspondingly. We sample the first 1K documents that contain data like I1, T1, R1, I2, T2, R2 from OBELICS [14] and named it as OBELICS-Hybrid6, which have at least three frames and three texts. (See Sec. E in appendix for more details.) This task encourages the model to leverage the contextual and semantic relationship in interleaved sequences to effectively predict and retrieve the subsequent pair. To enable our model with retrieval, we follow CosMo [25] to add a simple contrastive head between visual embedding and language embedding from the middle layers. Recall that visual embeddings are either from raw images or rendered images or the addition of the two in our method. Table 4 7 Method Pretrain Text Source Task DocVQA-val FuYu9B [8]† Raw-Text 42.3 + VisInContext +Rendered Image 44.5 (2.2↑) Table 5: Pretraining with VisInContext helps on long-context understanding task for FuYu model. † means our implementation on 180M data. reports the results from our model with several input variants. We observe taking surrounding text input as rendered text image performs much better on the Sequence to image retrieval, while on par on Sequence to text retrieval, when compared with taking surrounding text input as raw text. These results further support the designs of VisInContext in the context of document understanding. 3.5 Extension to MLLM with Linear Embedding Beyond utilizing the visual encoder, some works [36, 8] also employ linear embedding to extract visual features directly from raw images. To show the generality of our method, we also explore FuYu [8] model as a baseline and integrate VisInContext into the model. (See Sec. A in the appendix for more details.) As indicated in Table 5, our method is successful in improving the performances on DocVQA dataset that require long-context understanding. Text Image Token Masking TCCL Ok-VQA TextVqa VizWiz VqaV2 11.5 15.3 8.7 24.2 ✓ 11.3 15.0 9.4 30.1 ✓ ✓ 17.8 18.3 15.3 33.5 ✓ ✓ 13.5 15.3 10.3 30.9 ✓ ✓ ✓ 17.2 21.8 19.7 35.2 Table 6: Ablation study of the component in our pipeline for text-only 4-shot example. Font Size 4 6 8 10 12 TextVQA 15.4 17.2 18.5 21.8 20.3 DocVQA 39.8 42.5 45.6 44.3 36.2 Table 7: Font size ablation. We report the result on DocVQA val dataset. Dataset 2 4 8 16 32 TextVQA 21.8 20.5 21.3 18.5 15.3 DocVQA 44.3 43.2 39.4 40.5 36.6 Table 8: Font interval thresh ablation. Larger thresh leads to few texts in general. 3.6 Ablation Study Ablations on Model Design. We conduct ablation studies on the following modeling components to demonstrate their effectiveness: Text Image, TCCL, and Token Masking. Results are detailed in Table 6, which reveal two findings: 1. Token Masking is crucial for the model to learn from rendered text images. Without Token Masking, the model can only perform comparably to the baseline. Forcing the model to learn text semantics from rendered text images via token masking significantly improves model performance. 2. Utilizing TCCL with Token Masking yields better performance than using Token Masking alone. Ablations on Font Size and Interval Threshold. As shown in Table 7, optimal performance varies with changes in font size. We found that adjusting the font size impacts performance similarly to altering the patch size—both methods effectively increase the contextual information within each patch. We prefer modifying the font size over the patch size because it allows for more intuitive adjustments. Our findings indicate that the model does not need a highly detailed understanding of each word to perform effectively. Another important factor is the font interval threshold. As shown in Table 8, we observed that a too-large interval leads to inferior results. This is intuitive because a larger threshold results in fewer texts in the rendered text image. 8 4 Related Work Multimodal Language Models. Current mainstream Multimodal Large Language Models (MLLMs) [37, 38, 22, 39, 40, 41] leverage the capabilities of Large Language Models (LLMs) [42, 6] due to their strong reasoning abilities, as demonstrated by recent advancements. These models typically adopt one of two primary designs for integrating visual information. The first approach involves the effective adaptation of visual representations, which are acquired via a separate visual encoder, into the text-based LLM framework like CLIP, GIT, and BLIP2 [22, 43, 37]. The representative method in this category incorporates visual representations into the language model using cross-attention, as seen in the Flamingo series models [1, 9, 14]. Along this line, recently some works like LLaVA [40], EMU2 [44], InternVL [45], DeepSeeker [10], and QWen [41] lead to superior results on multi-modality tasks with supervised finetuning on high-quality data. The second approach uses visual embeddings directly as input "tokens" for the LLMs, bypassing the traditional use of a separate visual encoder. This method processes visual patches with a linear layer and uses the resulting embeddings as direct inputs to the LLM, as implemented in models like ViLT [36] and FuYu [8]. This strategy omits the need for an additional visual encoder and simplifies the architecture. In this work, we adopt the Flamingo [1] architecture as our main baseline for the following reasons: First, the Flamingo model emphasizes in-context few-shot learning ability and designs comprehensive few-shot evaluation strategies. Second, our focus is on extending the in-context text length during pre-training rather than on supervised fine-tuning. Enhancing Text Understanding through Visual Inputs. Traditional text tokenization processes raw text efficiently, but it faces challenges such as vulnerability to spelling errors and limited crosslingual transferability [46, 47]. These issues have prompted the exploration of tokenizer-free models, which aim to improve robustness and facilitate better cross-language applicability. For instance, a single spelling error can lead to entirely different tokens using traditional tokenization methods, impacting model performance. Recent developments have seen innovative approaches like the Pixel model [46], which proposes processing text as an image using both an image encoder and an image decoder. This approach has sparked a series of studies that process not only textual data but also images, charts, and tables through a unified visual input system [35, 46, 48, 47]. These models are trained on a diverse array of visual data, such as webpage screenshots and user interface images, sourced extensively from the internet. They are specifically designed to handle visually-situated text in an end-to-end manner, offering the potential to support a wide range of applications. Long Context Modeling. The challenge of incorporating more tokens into LLMs is an active area of research [49, 50]. Common approaches involve novel self-attention mechanisms [18, 51, 52] , compressed token [53, 54, 55] or memory banks [16]. Some works [56] exploit tensor parallelism or sequence parallelism to reduce memory costs. There also have some works focus on position embedding [57, 58]. In multi-modality research, closed-source models like Gemini [59] and GPT4V [60] support long context inference up to millions of tokens. Open-source models such as MA-LMM for Long-Term Video Understanding [20] can process up to one hour of video using a long memory bank. The most relevant work Large World Model [19] extends token length using Ring Attention. In contrast to these methods, our method utilizes off-the-shelf LLMs and compresses text tokens into visual tokens for efficient processing. Our method is complementary to these existing techniques and can be integrated with them to achieve lower computational cost and longer context length. 5 Conclusion and Limitations This paper centers on multi-modality learning and addresses the in-context length limitations presented by heavy computational cost of LLMs in MLLMs. Our contribution is a novel and efficient method named VisInContext, which enables the model to perceive long text context as rendered text images. Comprehensive experiments show that VisInContext is effective on conventional in-context few-shot evaluations and document understanding, while being much more efficient. 9 One limitation of our method is, currently our method requires processing a fixed size image even for brief texts. 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Chartqa: A benchmark for question answering about charts with visual and logical reasoning. arXiv preprint arXiv:2203.10244, 2022. 13 A Extending VisInContext to MLLM Using Only Linear Embedding In Multimodal Large Language Model (MLLM), besides using visual encoders, there are several works [36, 8] that utilize only linear embeddings to encode visual information. To demonstrate the generality of our method, we first extend it to the FuYu [8] model in this section. We evaluated our methodology using the FuYu [8] architecture, a prominent model that leverages visual information through simple linear embeddings. This approach utilizes a large language model framework without the need for a visual encoder, instead employing linear embedding to process visual information. A.1 Methodology Our system architecture, based on FuYu [8], incorporates a single decoder where both visual and textual inputs are converted into token embeddings and processed by the same Transformer structure. Inputs are divided into two segments: the first is a sequence of image patches forming a screenshot and image patches from text image, and the second a sequence of textual tokens that contextualize the screenshot. The configuration of this uni-modal approach is depicted in Figure 6. Figure 6: The main pipeline is based on Fuyu [8]. What’s different is we introduce an additional text image. During pre-training, the rendered text image and original image is also alternatively. The DCSE is preserved. We show one image-text pair here for simplicity. Input Format. The model inputs are formatted such that a text sequence of m tokens is divided into screenshot segment (ms) tokens and text segment (mt) tokens, each comprising 256 tokens. The screenshot dimensions are defined as ph × pw pixels. Special tokens are integrated to guide the model’s understanding of segment differentiation. The rendering strategies are consistent with those employed in our visual encoder-based methods. Architecture. The architecture follows the FuYu model [8], a widely used framework among visual encoder-free multi-modal language models (LMs), utilizing a 9 billion parameter model. Image patches are transformed into embeddings through a linear projection, while textual inputs utilize corresponding word embeddings. These embeddings are subsequently processed together in the Transformer blocks. To keep contrastive loss, we compute the average token metric after 3rd layer. A.2 Evaluation Settings The primary objective of this experiment is to assess whether autoregressive screenshot language models (LMs) can accurately interpret text within screenshots using only linear embedding for the rendered text image. In this setup, the screenshot LM processes 256 text tokens derived from the screenshot context along with an additional 25 text tokens. A.3 Training Settings Our training setup follows the Flamingo model, using image-text data sourced from DataComp [26]. For interleaved data, since FuYu does not support interleaved input, we sample one frame and a fixed length of text during pretraining. The model is trained with DeepSpeed Zero 2 optimization [3] and uses fp16 data type. We initialize the Language Model from Persimmon-8B weight. B Pretraining & Downstream Task Evaluation details B.1 Pretraining Data Details The associated data statistics for pretraining, presented comprehensively in Table 9, mainly include Datacomp [26] subset, MMC4 [13], Obelics [14] and Rendered Text [27]. 14 Data Type Dataset Sample Image-Text DataComp1B [26] 108M Interleaved Image-Text MMC4 [13] 30M Obelics [14] 30M Synthetic Data Rendered Text [27] 12M Total 180M Table 9: Statistics of the Pre-training Dataset: Subsets are random sampling. The comparison between our method and Open-flamingo baseline, utilizing equivalent-scale pretraining data, consistently demonstrates the superior performance of our approach across diverse tasks. VisInContext 3B VisInContext 9B VisInContext 57B Model Language Model Backbone OPT-IML-1.8B [61] Mistral-7B [5] MOE 56B [2] Vision Model Backbone openai/clip -vit-large -patch14 openai/clip -vit-large -patch14 laion/CLIP-ViT -H-14-laion2B -s32B-b79K Cross-Layer Interval 2 4 4 Training Text Sequence Length 128 128 128 ICL Text Length 2048 2048 2048 Effective Batch Size 3072 1536 768 Max Training Steps 200K 200K 500K Weight Decay 0.1 0.1 0.1 Optimizer adamw(0.9, 0.999) adamw(0.9, 0.999) adamw(0.9, 0.999) Gradient Clipping 1.0 1.0 1.0 Learning Rate Initial Max 5e-5 3e-5 3e-5 Decay Schedule Cosine Cosine Cosine Linear warmup Steps 5000 5000 5000 Table 10: The hyperparameters used in pre-training for three distinct VisInContext variations. The learning rate and batch size is smaller for sine the GPU memory limitation is 32GB. B.2 Hyperparameter Configuration In this subsection, we outline the essential training details required for reproducibility. Our experiments included three different model sizes, with larger models requiring smaller batch sizes due to GPU memory limitations. We employed DeepSpeed ZeRO-2 optimization with fp16 precision and adjusted gradient accumulation steps to match the data type count. Comprehensive results are presented in Table 10. The text input sequence length is 128 as default. Since we adopt VisInContext to increase the in-context length, the ICL length can be increased to 2048, around 15 text images. The τ for TCCL is 0.07. B.3 Parameter Details The Flamingo [1] baseline include Resampler, Language Model, Visual Encoder, Cross-attention. Both the Language Model and Visual Encoder are frozen during pretraining. We mainly train the Gated Cross Attention layer and Resampler layer. 15 Model Language Vision Gated Cross Attention Resampler Flamingo-9B 7.1B 428M 0.8B 194M Flamingo-9B Baseline † 7B 307M 0.35B 194M MOE Baseline† 56B 307M 0.5B 194M Table 11: Parameter counts for each component in MLLM. † means our implementation. Method Text Source Parameter Text Tokens↓ T-Shots VQA Mean ok-vqa textvqa vizwiz vqav2 OPT1.3B [4] Rendered Image 1.3B 10 0 11.2 15.8 5.4 33.6 16.5 10 4 17.2 21.8 7.8 33.2 20.0 (3.5↑) 10 32 21.3 22.6 11.5 35.8 22.8(6.3↑) MPT Rendered Image 7B 10 0 28.5 23.2 24.4 37.7 28.5 10 4 30.1 23.2 28.4 40.3 30.5(2.0↑) 10 32 32.5 25.4 30.3 41.8 32.5(4.0↑) Table 12: VisInContext performs well over different Language Models. B.4 Extension to Other Language Models Our research extends beyond the Mistral model, incorporating other language models such as OPT and MoE. The comparative results are summarized in Table 12. We noted marked improvements in performance across all models with the use of more in-context examples. This indicates that VisInContext’s effectiveness is not highly dependent on the specific language model used, showcasing the broad applicability and robustness of our methodology. C Document Understanding Example In this experiment, we present examples to demonstrate our method. As shown in Figure 7, we provide samples from the validation sets of DocumentVQA [15] and ChartQA [62]. Using VisInContext, we observe that the method answers questions more accurately, even when the font is unclear. For instance, consider the first pdf image have low resolution. D Sequential Multi-modal Retrieval Details Data Collection We retrieve 1,000 samples from the OBELICS dataset, each sample consisting of six segments in the fixed order: I1, T1, T2, I2, T3, T4. Each image has one matching text segment and additional surrounding text. We use relative positioning to indicate which text is matched and which is surrounding. Retrieval Details To perform the retrieval task, we incorporate contrastive loss during pretraining, following the approach of CosMo [25]. We add a contrastive head for the uni-modality text and vision embeddings. Using the mean of the text and image features as a query, we retrieve the next image or text segment. This task tests the model’s ability to handle long in-context text. We compute the dot product similarity and rank the scores to determine the final result. When processing surrounding text as "Raw Text," we concatenate T1 and the surrounding text R1 directly to the LLM to obtain the text embedding. We then use the mean of this text embedding and the image embedding of I1 to retrieve the next image or text. For our method, we render the surrounding text R1 into an image and use the sum of two resampler outputs as the image embedding. We compute the mean of this image embedding and the T1 embedding and use this mean vector to retrieve the next image or text segment. 16 Figure 7: The document understanding example of our method. E In-Context Few-Shot Example In this work, we primarily follow the methods from the Flamingo series [1, 14, 9], as these provide comprehensive support for in-context pretraining. For zero-shot evaluation, the input sequence is formatted as follows: <Visual><Question><Answer> For few-shot evaluation, such as a two-shot evaluation, the input includes two in-context examples. The sequence then becomes: <Visual1><Question1><Answer1><Visual2><Question2><Answer2> <Visual><Question><Answer> For the text-only input sequence in a text-to-image few-shot setting, the format is: <Question1><Answer1><Question2><Answer2> <Visual><Question><Answer> Note that we remove all visual tokens to create a longer input sequence, which can then be rendered into a text image for text-to-image few-shot evaluation. F Activating the Visual Encoder One approach involves activating the visual encoder during pretraining, allowing the model to independently learn visual OCR information within the vision encoder. As shown in Table 13, this method significantly enhances performance in document understanding tasks. However, it also introduces considerable instability during pretraining and requires extended 17 Method DocVQA OCR VQA Classification val test Hatefulmems Frozen 48.5 52.2 58.4 61.3 Learnable 50.3(1.9↑) 54.0(1.8↑) 59.0(0.6↑) 59.4(1.9↓) Table 13: The impact of opening visual encoder during pre-training. iterations (from 200k to 500k) for convergence. Additionally, it decreases performance in classification tasks. Therefore, we use the frozen visual encoder by default, as the token resampler alone suffices to develop document understanding capabilities. G Boarder Impact This work introduces VisInContext, a method to enhance token efficiency in multi-modality large language models (MLLMs) by using visual tokens to process extended textual contexts. Positive Impacts: VisInContext can democratize access to advanced NLP technologies by reducing computational resources required for long text sequences. This improvement promotes sustainable AI practices by lowering energy consumption and allows researchers with limited resources to utilize powerful MLLMs for applications in education, healthcare, and content generation. Negative Impacts: Potential negative impacts include the misuse of efficient text processing for spreading misinformation or creating deepfake content. Additionally, reliance on visual tokens may introduce biases if training data is not diverse. Mitigation Strategies: To mitigate these risks, we recommend implementing content moderation, developing ethical AI usage guidelines, and ensuring diverse and balanced training datasets. Continuous monitoring and auditing of AI systems using VisInContext can also help address unintended consequences. In summary, VisInContext offers significant advancements in token efficiency and computational sustainability, but it is essential to consider and address its broader societal impacts responsibly. 18 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: Reflected in Section1 contribution. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,490	Reshuffling Resampling Splits Can Improve Generalization of Hyperparameter Optimization Thomas Nagler∗ Lennart Schneider∗ Bernd Bischl Matthias Feurer t.nagler@lmu.de Department of Statistics, LMU Munich Munich Center for Machine Learning (MCML) Abstract Hyperparameter optimization is crucial for obtaining peak performance of machine learning models. The standard protocol evaluates various hyperparameter configurations using a resampling estimate of the generalization error to guide optimization and select a final hyperparameter configuration. Without much evidence, paired resampling splits, i.e., either a fixed train-validation split or a fixed crossvalidation scheme, are often recommended. We show that, surprisingly, reshuffling the splits for every configuration often improves the final model’s generalization performance on unseen data. Our theoretical analysis explains how reshuffling affects the asymptotic behavior of the validation loss surface and provides a bound on the expected regret in the limiting regime. This bound connects the potential benefits of reshuffling to the signal and noise characteristics of the underlying optimization problem. We confirm our theoretical results in a controlled simulation study and demonstrate the practical usefulness of reshuffling in a large-scale, realistic hyperparameter optimization experiment. While reshuffling leads to test performances that are competitive with using fixed splits, it drastically improves results for a single train-validation holdout protocol and can often make holdout become competitive with standard CV while being computationally cheaper. 1 Introduction Hyperparameters have been shown to strongly influence the performance of machine learning models (van Rijn & Hutter, 2018; Probst et al., 2019). The primary goal of hyperparameter optimization (HPO; also called tuning) is the identification and selection of a hyperparameter configuration (HPC) that minimizes the estimated generalization error (Feurer & Hutter, 2019; Bischl et al., 2023). Typically, this task is challenged by the absence of a closed-form mathematical description of the objective function, the unavailability of an analytic gradient, and the large cost to evaluate HPCs, categorizing HPO as a noisy, black-box optimization problem. An HPC is evaluated via resampling, such as a holdout split or M-fold cross-validation (CV), during tuning. These resampling splits are usually constructed in a fixed and instantiated manner, i.e., the same training and validation splits are used for the internal evaluation of all configurations. On the one hand, this is an intuitive approach, as it should facilitate a fair comparison between HPCs and reduce the variance in the comparison.1 On the other hand, such a fixing of train and validation splits might steer the optimization, especially after a substantial budget of evaluations, towards favoring HPCs ∗Equal contribution. 1This approach likely originates from the concept of paired statistical tests and the resulting variance reduction, but in our literature search we did not find any references discussing this in the context of HPO. For example, when comparing the performance of two classifiers on one dataset, paired tests are commonly 38th Conference on Neural Information Processing Systems (NeurIPS 2024). which are specifically tailored to the chosen splits. Such and related effects, where we "overoptimize" the validation performance without effective reward in improved generalization performance have been sometimes dubbed "overtuning" or "oversearching". For a more detailed discussion of this topic, including related work, see Section 5 and Appendix B. The practice of reshuffling resampling splits during HPO is generally neither discussed in the scientific literature nor HPO software tools.2 To the best of our knowledge, only Lévesque (2018) investigated reshuffling train-validation splits for every new HPC. For both holdout and M-fold CV using reshuffled resampling splits resulted in, on average, slightly lower generalization error when used in combination with Bayesian optimization (BO, Garnett, 2023) or CMA-ES (Hansen & Ostermeier, 2001) as HPO algorithms. Additionally, reshuffling was used by a solution to the NeurIPS 2006 performance prediction challenge to estimate the final generalization performance (Guyon et al., 2006). Recently, in the context of evolutionary optimization, reshuffling was applied after every generation (Larcher & Barbosa, 2022). In this paper, we systematically examine the effect of reshuffling on HPO performance. Our contributions can be summarized as follows: 1. We show theoretically that reshuffling resampling splits during HPO can result in finding a configuration with better overall generalization performance, especially when the loss surface is rather flat and its estimate is noisy (Section 2). 2. We confirm these theoretical insights through controlled simulation studies (Section 3). 3. We demonstrate in realistic HPO benchmark experiments that reshuffling splits can lead to a real-world improvement of HPO (Section 4). Especially in the case of reshuffled holdout, we find that the final generalization performance is often on par with 5-fold CV under a wide range of settings. We discuss results, limitations, and avenues for future research in Section 5. 2 Theoretical Analysis 2.1 Problem Statement and Setup Machine learning (ML) aims to fit a model to data, so that it generalizes well to new observations of the same distribution. Let D = {Zi}n i=1 be the observed dataset consisting of i.i.d. random variables from a distribution P, i.e., in the supervised setting Zi = (Xi, Yi).3,4 Formally, an inducer g configured by an HPC λ ∈Λ maps a dataset D to a model from our hypothesis space h = gλ(D) ∈H. During HPO, we want to find a HPC that minimizes the expected generalization error, i.e., find λ∗= arg min λ∈Λ µ(λ), where µ(λ) = E[ℓ(Z, gλ(D))], where ℓ(Z, h) is the loss of model h on a fresh observation Z. In practice, there is usually a limited computational budget for each HPO run, so we assume that there is only a finite number of distinct HPCs Λ = {λ1, . . . , λJ} to be evaluated, which also simplifies the subsequent analysis. Naturally, we cannot optimize the generalization error directly, but only an estimate of it. To do so, a resampling is constructed. For every HPC λj, draw M random sets I1,j, . . . , IM,j ⊂{1, . . . , n} of validation indices with nvalid = ⌈αn⌉instances each. The random index draws are assumed to be independent of the observed data. The data is then split accordingly into pairs Vm,j = {Zi}i∈Im,j, Tm,j = {Zi}i/∈Im,j of disjoint validation and training sets. Define the validation loss on the m-th fold L(Vm,j, gλj(Tm,j)) = 1 nvalid X i∈Im,j ℓ(Zi, gλj(Tm,j)), employed that implicitly assume that differences between the performance of classifiers on a given CV fold are comparable (Dietterich, 1998; Nadeau & Bengio, 1999, 2003; Demšar, 2006). 2In Appendix B, we present an overview of how resampling is addressed in tutorials and examples of standard HPO libraries and software. We conclude that usually fixed splits are used or recommended. 3Throughout, we use bold letters to indicate (fixed and random) vectors. 4We provide a notation table for symbols used in the main paper in Table 2 in the appendix. 2 and the M-fold validation loss as bµ(λj) = 1 M M X m=1 L(Vm,j, gλj(Tm,j)). Since µ is unknown, we minimize bλ = arg minλ∈Λ bµ(λ), hoping that µ(bλ) will also be small. Typically, the same splits are used for every HPC, so Im,j = Im for all j = 1, . . . , J and m = 1, . . . , M. In the following, we investigate how reshuffling train-validation splits (i.e., Im,j ̸= Im,j′ for j ̸= j′) affects the HPO problem. 2.2 How Reshuffling Affects the Loss Surface We first investigate how different validation and reshuffling strategies affect the empirical loss surface bµ. In particular, we derive the limiting distribution of the sequence √n(bµ(λj) −µ(λj))J j=1. This limiting regime will not only reveal the effect of reshuffling on the loss surface, but also give us a tractable setting to study HPO performance. Theorem 2.1. Under regularity conditions stated in Appendix C.1, it holds √n (bµ(λj) −µ(λj))J j=1 →N(0, Σ) in distribution, where Σi,j = τi,j,MK(λi, λj), τi,j,M = lim n→∞ 1 nM 2α2 n X s=1 M X m=1 M X m′=1 Pr(s ∈Im,i ∩Im′,j), and K(λi, λj) = lim n→∞Cov[¯ℓn(Z′, λi), ¯ℓn(Z′, λj)], ¯ℓn(z, λ) = E[ℓ(z, gλ(T ))] −E[ℓ(Z, gλ(T ))], where the expectation is taken over a training set T of size n and two fresh samples Z, Z′ from the same distribution. The regularity conditions are rather mild and discussed further in Appendix C.1. The kernel K reflects the (co-)variability of the losses caused by validation samples. The contribution of training samples only has a higher-order effect. The validation scheme enters the distribution through the quantities τi,j,M. In what follows, we compute explicit expressions for some popular examples. The following list provides formal definitions for the index sets Im,j. (i) (holdout) Let M = 1 and I1,j = I1 for all j = 1, . . . , J, and some size-⌈αn⌉index set I1. (ii) (reshuffled holdout) Let M = 1 and I1,1, . . . , I1,J be independently drawn from the uniform distribution over all size-⌈αn⌉subsets from {1, . . . , n}. (iii) (M-fold CV) Let α = 1/M and I1, . . . , IM be a disjoint partition of {1, . . . , n}, and Im,j = Im for all j = 1, . . . , J. (iv) (reshuffled M-fold CV) Let α = 1/M and (I1,j, . . . , IM,j), j = 1, . . . , J, be independently drawn from the uniform distribution over disjoint partitions of {1, . . . , n}. (v) (M-fold holdout) Let Im, m = 1, . . . , M, be independently drawn from the uniform distribution over size-⌈αn⌉subsets of {1, . . . , n} and set Im,j = Im for all m = 1, . . . , M, j = 1, . . . , J. (vi) (reshuffled M-fold holdout) Let Im,j, m = 1, . . . , M, j = 1, . . . , J, be independently drawn from the uniform distribution over size-⌈αn⌉subsets of {1, . . . , n}. The value of τi,j,M for each example is computed explicitly in Appendix E. In all these examples, we in fact have τi,j,M = σ2, i = j τ 2σ2, i ̸= j. , (1) for some method-dependent parameters σ, τ shown in Table 1. The parameter σ2 captures any increase in variance caused by omitting an observation from the validation sets. The parameter τ quantifies a potential decrease in correlation in the loss surface due to reshuffling. More precisely, 3 Table 1: Exemplary parametrizations in Equation (1) for resamplings; see Appendix E for details. Method σ2 τ 2 holdout (HO) 1/α 1 reshuffled HO 1/α α M-fold CV 1 1 reshuffled M-fold CV 1 1 M-fold HO (subsampling / Monte Carlo CV) 1 + (1 −α)/Mα 1 reshuffled M-fold HO 1 + (1 −α)/Mα 1/(1 + (1 −α)/Mα) the observed losses bµ(λi), bµ(λj) at distinct HPCs λi ̸= λj become less correlated when τ is small. Generally, an increase in variance leads to worse generalization performance. The effect of a correlation decrease is less obvious and is studied in detail in the following section. We make the following observations about the differences between methods in Table 1: • M-fold CV incurs no increase in variance (σ2 = 1) and — because every HPC uses the same folds — no decrease in correlation. Interestingly, the correlation does not even decrease when reshuffling the folds. In any case, all samples are used exactly once as validation and training instance. At least asymptotically, this leads to the same behavior, and reshuffling should have almost no effect on M-fold CV. • The two (1-fold) holdout methods bear the same 1/α increase in variance. This is caused by only using a fraction α of the data as validation samples. Reshuffled holdout also decreases the correlation parameter τ 2. In fact, if HPCs λi ̸= λj are evaluated on largely distinct samples, the validation losses bµ(λi) and bµ(λj) become almost independent. • M-fold holdout also increases the variance, because some samples may still be omitted from validation sets. This increase is much smaller for large M. Accordingly, the correlation is also decreased by less in the reshuffled variant. 2.3 How Reshuffling Affects HPO Performance In practice, we are mainly interested in the performance of a model trained with the optimal HPC bλ. To simplify the analysis, we explore this in the large-sample regime derived in the previous section. Assume bµ(λj) = µ(λj) + ϵ(λj) (2) where ϵ(λ) is a zero-mean Gaussian process with covariance kernel Cov(ϵ(λ), ϵ(λ′)) = K(λ, λ) if λ = λ′, τ 2K(λ, λ′) else. (3) Let Λ ⊆{λ ∈Rd : ∥λ∥≤1} with \|Λ\| = J < ∞be the set of hyperparameters. Theorem 2.2 ahead gives a bound on the expected regret E[µ(bλ)−µ(λ∗)]. It depends on several quantities characterizing the difficulty of the HPO problem. The constant κ = sup ∥λ∥,∥λ′∥≤1 \|K(λ, λ) −K(λ, λ′)\| K(λ, λ)∥λ −λ′∥2 . can be interpreted as a measure of correlation of the process ϵ. In particular, Corr(ϵ(λ), ϵ(λ′)) ≥ 1 −κ∥λ −λ′∥2. The constant is small when ϵ is strongly correlated, and large otherwise. Further, define η as the minimal number such that any η-ball contained in {∥λ∥≤1} contains at least one element of Λ. It measures how densely the set of candidate HPCs Λ covers set of all possible HPCs. If Λ is a deterministic uniform grid, we have about η ≈J−1/d. Similarly, Lemma D.1 in the Appendix shows that η ≲J−1/2d when randomly sampling HPCs. Finally, the constant m = sup λ∈Λ \|µ(λ) −µ(λ∗)\| ∥λ −λ∗∥2 , 4 −2 −1 0 1 2 3 0.00 0.25 0.50 0.75 1.00 λ Loss Surface Type True Empirical (τ = 1) Empirical (τ = 0.3) (a) High signal-to-noise ratio −2 0 2 0.00 0.25 0.50 0.75 1.00 λ Loss Surface Type True Empirical (τ = 1) Empirical (τ = 0.3) (b) Low signal-to-noise ratio Figure 1: Example of reshuffled empirical loss yielding a worse (left) and better (right) minimizer. measures the local curvature at the minimum of the loss surface µ. Finding an HPC λ close to the theoretical optimum λ∗is easier when the minimum is more pronounced (large m). On the other hand, the regret µ(λ) −µ(λ∗) is also punishing mistakes more quickly. Defining log(x)+ = max{0, log(x)}, we can now state our main result. Theorem 2.2. Let bµ follow the Gaussian process model (2). Suppose κ < ∞, 0 < σ2 ≤Var[ϵ(λ)] ≤ σ2 < ∞for all λ ∈Λ, and m > 0. Then E[µ(bλ) −µ(λ∗)] ≤σ √ d[8 + B(τ) −A(τ)]. where B(τ) = 48 hp 1 −τ 2p log J + τ p 1 + log(3κ)+ i , A(τ) = p 1 −τ 2(σ/σ) s log σ 2mη2 + . The numeric constants result from several simplifications in a worst-case analysis, which lowers their practical relevance. A qualitative analysis of the bound is still insightful. The bound is increasing in σ and d, indicating that the HPO problem is harder when there is a lot of noise or there are many parameters to tune. The terms B(τ) and A(τ) have conceptual interpretations: • The term B(τ) quantifies how likely it is to pick a bad bλ because of bad luck: a λ far away from λ∗had such a small ϵ(λ) that it outweighs the increase in µ. Such events are more likely when the process ϵ is weakly correlated. Accordingly, B(τ) is decreasing in τ and increasing in κ. • The term A(τ) quantifies how likely it is to pick a good bλ by luck: a λ close to λ∗had such a small ϵ(λ) that it overshoots all the other fluctuations. Also such events are more likely when the process ϵ is weakly correlated. Accordingly, the term A(τ) is decreasing in τ. The B, as stated, is unbounded, but a closer inspection of the proof shows that it is upper bounded by √log J. This bound is attained only in the unrealistic scenario when the validation losses are essentially uncorrelated across all HPCs. The term A is bounded from below by zero, which is also the worst case because the term enters our regret bound with a negative sign. Both A and B are decreasing in the reshuffling parameter τ. There are two regimes. If σ/2mη2 ≤e, then A(τ) = 0 and reshuffling cannot lead to an improvement of the bound. The term σ/mη2 can be interpreted as noise-to-signal ratio (relative to the grid density). If the signal is much stronger than the noise, the HPO problem is so easy that reshuffling will not help. This situation is illustrated in Figure 1a. If on the other hand σ/mη2 > e, the terms A(τ) and B(τ) enter the bound with opposing signs. This creates tension: reshuffling between HPCs increases B(τ), which is countered by a decrease in A(τ). So which scenarios favor reshuffling? When the process ϵ is strongly correlated, κ is small and reshuffling (decreasing τ) incurs a high cost in B(τ). This is intuitive: When there is strong 5 correlation, the validation loss surface bµ is essentially just a vertical shift of µ. Finding the optimal λ is then almost as easy as if we would know µ, and decorrelating the surface through reshuffling would make it unnecessarily hard. When ϵ is less correlated (κ large) however, reshuffling does not hurt the term B(τ) as much, but we can reap all the benefits of increasing A(τ). Here, the effect of reshuffling can be interpreted as hedging against the catastrophic case where all bµ(λ) close to the optimal λ∗are simultaneously dominated by a region of bad hyperparameters. This is illustrated in Figure 1b. 3 Simulation Study To test our theoretical understanding of the potential benefits of reshuffling resampling splits during HPO, we conduct a simulation study. This study helps us explore the effects of reshuffling in a controlled setting. 3.1 Design We construct a univariate quadratic loss surface function µ : Λ ⊂R 7→R, λ →m(λ −0.5)2/2 which we want to minimize. The global minimum is given at µ(0.5) = 0. Combined with a kernel for the noise process ϵ as in Equation (3), this allows us to simulate an objective as observed during HPO by sampling bµ(λ) = µ(λ) + ϵ(λ). We use a squared exponential kernel K(λ, λ′) = σ2 K exp (−κ(λ −λ′)2/2) that is plugged into the covariance kernel of the noise process ϵ in Equation (3). The parameters m and κ in our simulation setup correspond exactly to the curvature and correlation constants from the previous sections. Recall that Theorem 2.2 states that the effect of reshuffling strongly depends on the curvature m of the loss surface µ (a larger m implies a stronger curvature) and the constant κ as a measure of correlation of the noise ϵ (a larger κ implies weaker correlation). Combined with the possibility to vary τ in the covariance kernel of ϵ, we can systematically investigate how curvature of the loss surface, correlation of the noise and the extent of reshuffling affect optimization performance. In each simulation run, we simulate the observed objective ˆµ(λ), identify the minimizer ˆλ = arg minλ∈Λ ˆµ(λ), and calculate its true risk, µ(ˆλ). We repeat this process 10000 times for various combinations of τ, m, and κ. 3.2 Results Figure 2 visualizes the true risk of the configuration ˆλ that minimizes the observed objective. We observe that for a loss surface with low curvature (i.e., m ≤2), reshuffling is beneficial (lower values of τ resulting in a better true risk of the configuration that optimizes the observed objective) as long as the noise process is not too correlated (i.e., κ ≥1). As soon as the noise process is more strongly correlated, even flat valleys of the true risk µ remain clearly visible in the observed risk bµ, and reshuffling starts to hurt the optimization performance. Moving to scenarios of high curvature, the general relationship of m and κ remains the same, but reshuffling starts to hurt optimization performance already with weaker correlation in the noise. In summary, the simulations show that in cases of low curvature of the loss surface, reshuffling (reducing τ) tends to improve the true risk of the optimized configuration, especially when the loss surface is flat (small m) and the noise is not strongly correlated (i.e., κ is large). This exactly confirms our theoretical predictions from the previous section. 4 Benchmark Experiments In this section, we present benchmark experiments of real-world HPO problems where we investigate the effect of reshuffling resampling splits during HPO. First, we discuss the experimental setup. Second, we present results for HPO using random search (Bergstra & Bengio, 2012). Third, we also show the effect of reshuffling when applied in BO using HEBO (Cowen-Rivers et al., 2022) and SMAC3 (Lindauer et al., 2022). Recall that our theoretical insight suggests that 1) reshuffling might be beneficial during HPO and 2) holdout should be affected the most by reshuffling and other resamplings should only be affected to a lesser extent. 6 m: 20 κ: 0.04 m: 20 κ: 1 m: 20 κ: 4 m: 20 κ: 100 m: 10 κ: 0.04 m: 10 κ: 1 m: 10 κ: 4 m: 10 κ: 100 m: 2 κ: 0.04 m: 2 κ: 1 m: 2 κ: 4 m: 2 κ: 100 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.070 0.075 0.080 0.085 0.090 0.175 0.200 0.225 0.250 0.275 0.21 0.24 0.27 0.30 0.08 0.10 0.12 0.18 0.19 0.20 0.100 0.125 0.150 0.175 0.200 0.07 0.08 0.09 0.10 0.11 0.12 0.08 0.12 0.16 0.05 0.10 0.15 0.20 0.02 0.04 0.06 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 τ μ(λ^) Figure 2: Mean true risk (lower is better) of the configuration minimizing the observed objective systematically varied with respect to curvature m, correlation strength κ of the noise (a larger κ implying weaker correlation), and extent of reshuffling τ (lower τ increasing reshuffling). A τ of 1 indicates no reshuffling. Error bars represent standard errors. 4.1 Experimental Setup As benchmark tasks, we use a set of standard HPO problems defined on small- to medium-sized tabular datasets for binary classification. We suspect the effect of the resampling variant used and whether the resampling is reshuffled to be larger for smaller datasets, where the variance of the validation loss estimator is naturally higher. Furthermore, from a practical perspective, this also ensures computational feasibility given the large number of HPO runs in our experiments. We systematically vary the learning algorithm, optimized performance metric, resampling method, whether the resampling is reshuffled, and the size of the dataset used for training and validation during HPO. Below, we outline the general experimental design and refer to Appendix F for details. We used a subset of the datasets defined by the AutoML benchmark (Gijsbers et al., 2024), treating these as data generating processes (DGPs; Hothorn et al., 2005). We only considered datasets with less than 100 features to reduce the required computation time and required the number of observations to be between 10000 and 1000000; for further details see Appendix F.1. Our aim was to robustly measure the generalization performance when varying the size n, which, as defined in Section 2 denotes the size of the combined data for model selection, so one training and validation set combined. First, we sampled 5000 data points per dataset for robust assessment of the generalization error; these points are not used during HPO in any way. Then, from the remaining points we sampled tasks with n ∈{500, 1000, 5000}. We selected CatBoost (Prokhorenkova et al., 2018) and XGBoost (Chen & Guestrin, 2016) for their state-of-the-art performance on tabular data (Grinsztajn et al., 2022; Borisov et al., 2022; McElfresh et al., 2023; Kohli et al., 2024). Additionally, we included an Elastic Net (Zou & Hastie, 2005) to represent a linear baseline with a smaller search space and a funnel-shaped MLP (Zimmer et al., 2021) as a cost-effective neural network baseline. We provide details regarding training pipelines and search spaces in Appendix F.2. We conduct a random search with 500 HPC evaluations for every resampling strategy we described in Table 1, for both fixed and reshuffled splits. We always use 80/20 train-validation splits for holdout 7 500 1000 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 −0.730 −0.725 −0.720 −0.715 −0.70 −0.69 −0.68 −0.67 −0.66 −0.68 −0.67 −0.66 −0.65 −0.64 −0.63 No. HPC Evaluations Mean Test Performance Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 3: Average test performance (negative ROC AUC) of the incumbent for XGBoost on dataset albert for increasing n (train-validation sizes, columns). Shaded areas represent standard errors. and 5-fold CVs, so that training set size (and negative estimation bias) are the same. Anytime test performance of an HPO run is assessed by re-training the current incumbent (i.e. the best HPC until the current HPO iteration based on validation performance) on all available train and validation data and evaluating its performance on the outer test set. Note we do this for scientific evaluation in this experiment; obviously, this is not possible in practice. Using random search allows us to record various metrics and afterwards simulate optimizing for different ones, specifically, we recorded accuracy, area under the ROC curve (ROC AUC) and logloss. We also investigated the effect of reshuffling on two state-of-the-art BO variants (Eggensperger et al., 2021; Turner et al., 2021), namely HEBO (Cowen-Rivers et al., 2022) and SMAC3 (Lindauer et al., 2022). The experimental design was the same as for random search, except for the budget, which we reduced from 500 HPCs to 250 HPCs, and only optimized ROC AUC. 4.2 Experimental Results In the following, we focus on the results obtained using ROC AUC. We present aggregated results over different tasks, learning algorithms and replications to get a general understanding of the effects. Unaggregated results and results involving accuracy and logloss can be found in Appendix G. Results of Reshuffling Different Resamplings For each resampling (holdout, 5-fold holdout, 5-fold CV, and 5x 5-fold CV), we empirically analyze the effect of reshuffling train and validation splits during HPO. In Figure 3 we exemplarily show how test performance develops over the course of an HPO run on a single task for different resamplings (with and without reshuffling). Naturally, test performance does not necessarily increase in a monotonic fashion, and especially holdout without reshuffling tends to be unstable. Its reshuffled version results in substantially better test performance. Next, we look at the relative improvement (compared to standard 5-fold CV, which we consider our baseline) with respect to test ROC AUC performance of the incumbent over time in Figure 4, i.e., the difference in test performance of the incumbent between standard 5-fold CV and a different resampling protocol; hence a positive difference tells us how much better in test error we are, if we would have chosen the other protocol instead 5-fold CV. We observe that reshuffling generally results in equal or better performance compared to the same resampling protocol without reshuffling. For 5-fold holdout and especially 5-fold CV and 5x 5-fold CV, reshuffling has a smaller effect on relative test performance improvement, as expected. Holdout is affected the most by reshuffling and results in substantially better relative test performance compared to standard holdout. We also observe that an HPO protocol based on reshuffled holdout results in similar final test performance as standard 5-fold CV while overall being substantially cheaper due to requiring less model fits per HPC evaluation. In Appendix G.2, we further provide an ablation study on the number of folds when using M-fold holdout, where we observed that – in line with our theory – the more folds are used, the less reshuffling affects M-fold holdout. 8 500 1000 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 −0.50 −0.25 0.00 0.25 −1.2 −0.8 −0.4 0.0 0.4 −1.0 −0.5 0.0 0.5 No. HPC Evaluations Mean Test Improvement Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 4: Average improvement (compared to standard 5-fold CV) with respect to test performance (ROC AUC) of the incumbent over different tasks, learning algorithms and replications separately for increasing n (train-validation sizes, columns). Shaded areas represent standard errors. However, this general trend can vary for certain combinations of classifier and performance metric, see Appendix G. Especially for logloss, we observed that reshuffling rarely is beneficial; see the discussion in Section 5. Finally, the different resamplings generally behave as expected. The more we are willing to invest compute resources into a more intensive resampling like 5-fold CV or 5x 5-fold CV, the better the generalization performance of the final incumbent. Results for BO and Reshuffling Figure 5 shows that, generally HEBO and SMAC3 outperform random search with respect to generalization performance (i.e., comparing HEBO and SMAC3 to random search under standard holdout, or comparing under reshuffled holdout). More interestingly, HEBO, SMAC3 and random search all strongly benefit from reshuffling. Moreover, the performance gap between HEBO and random search but also SMAC3 and random search narrows when the resampling is reshuffled, which is an interesting finding of its own: As soon as we are concerned with generalization performance of HPO and not only investigate validation performance during optimization, the choice of optimizer might have less impact on final generalization performance compared to other choices such as whether the resampling is reshuffled during HPO or not. We present results for BO and reshuffling for different resamplings in Appendix G. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 No. HPC Evaluations Mean Test Improvement Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 5: Average improvement (compared to random search on standard holdout) with respect to test performance (ROC AUC) of the incumbent over tasks, learning algorithms and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 5 Discussion In the previous sections, we have shown theoretically and empirically that reshuffling can enhance generalization performance of HPO. The main purpose of this article is to draw attention to this 9 surprising fact about a technique that is simple but rarely discussed. Our work goes beyond a preliminary experimental study on reshuffling (Lévesque, 2018), in that we also study the effect of reshuffling on random search, multiple metrics and learning algorithms, and most importantly, for the first time, we provide a theoretical analysis that explains why reshuffling can be beneficial. Limitations To unveil the mechanisms underlying the reshuffling procedures, our theoretical analysis relies on an asymptotic approximation of the empirical loss surface. This allows us to operate on Gaussian loss surfaces, which exhibit convenient concentration and anti-concentration properties required in our proof. The latter are lacking for general distributions, which explains our asymptotic approach. The analysis was further facilitated by a loss stability assumption regarding the learning algorithms that is generally rather mild; see the discussion in Bayle et al. (2020). However, it typically fails for highly sensitive losses, which has practical consequences. In fact, Figure 9 in Appendix G shows that reshuffling usually hurts generalization for the logloss and small sample sizes. It is still an open question whether this problem can be fixed by less naive implementations of the technique. Another limitation is our focus on generalization after search through a fixed, finite set of candidates. This largely ignores the dynamic nature of many HPO algorithms, which would greatly complicate our analysis. Finally, our experiments are limited in that we restricted ourselves to tabular data and binary classification and we avoided extremely small or large datasets. Relation to Overfitting The fact that generalization performance can decrease during HPO (or computational model selection in general) is sometimes known as oversearching, overtuning, or overfitting to the validation set (Quinlan & Cameron-Jones, 1995; Escalante et al., 2009; Koch et al., 2010; Igel, 2012; Bischl et al., 2023), but has arguably not been studied very thoroughly. Given recent theoretical (Feldman et al., 2019) and empirical (Purucker & Beel, 2023) findings, we expect less overtuning on multi-class datasets, making it interesting to see how reshuffling would affect the generalization performance. Several works suggest strategies to counteract this effect. First, LOOCVCV proposes a conservative choice of incumbents (Ng, 1997) at the cost of leave-one-out analysis or an additional hyperparameter. Second, it is possible to use an extra selection set (Igel, 2012; Lévesque, 2018; Mohr et al., 2018) at the cost of reduced training data, which was found to lead to reduced overall performance (Lévesque, 2018). Third, by using early stopping one can stop hyperparameter optimization before the generalization performance degrades again. This was so far demonstrated to be able to save compute budget at only marginally reduced performance, but also requires either a sensitivity hyperparameter or correct estimation of the variance of the generalization estimate and was only developed for cross-validation so far (Makarova et al., 2022). Reshuffling itself is orthogonal to these proposals and a combination with the above-mentioned methods might result in further improvements. Outlook Generally, the related literature detects overfitting to the validation set either visually (Ng, 1997) or by measuring it (Koch et al., 2010; Igel, 2012; Fabris & Freitas, 2019). Developing a unified formal definition of the above-mentioned terms and thoroughly analyzing the effect of decreased generalization performance after many HPO iterations and how it relates to our measurements of the validation performance is an important direction for future work. We further found, both theoretically and experimentally, that investing more resources when evaluating each HPC can result in better final HPO performance. To reduce the computational burden on HPO again, we suggest further investigating the use of adaptive CV techniques, as proposed by AutoWEKA (Thornton et al., 2013) or under the name Lazy Paired Hyperparameter Tuning (Zheng & Bilenko, 2013). Designing more advanced HPO algorithms exploiting the reshuffling effect should be a promising avenue for further research. Acknowledgments and Disclosure of Funding We thank Martin Binder and Florian Karl for helpful discussions. Lennart Schneider is supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics - Data - Applications (ADACenter) within the framework of BAYERN DIGITAL II (20-3410-2-9-8). 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Xi Random vector, describing the features Yi Random variable, describing the target Zi = (Xi, Yi) Data point D = {Zi}n i=1 Dataset consisting of iid random variables n Number of observations g Inducer/ML algorithm h Model, created by the inducer via h = gλ(D) λ Hyperparameter configuration Λ Finite set of all hyperparameter configurations J \|Λ\|, i.e., the number of hyperparameter configurations gλj Hyperparameterized inducer µ(λ) Expected loss of a hyperparameterized inducer on the distribution of a dataset ℓ(Z, h) Loss of a model h on a fresh observation Z M Number of folds in M-fold cross-validation α Percentage of samples to be used for validation I1,j, . . . , IM,j ⊂{1, . . . , n} M sets of validation indices, to be used for evaluating λj Vm,j Validation data for fold m and configuration λj Tm,j Training data for fold m and configuration λj L(Vm,j, gλj (Tm,j)) Validation loss for fold m and configuration λj bµ(λj) M-fold validation loss σ2 Increase in variance of validation loss caused by resampling τ 2 Decrease in correlation among validation losses caused by reshuffling τi,j,M Resampling-related component of validation loss covariance K(·, ·) Kernel capturing the covariance of the pointwise losses between two HPCs ϵ(λj) Zero-mean Gaussian process, see Equation (2) d Number of hyperparameters κ Curvature constant of covariance kernel η Density of hyperparameter set Λ m Local curvature at the minimum of the loss surface µ σ Lower bound on the noise level B(τ) Part of the regret bound penalizing reshuffling A(τ) Part of the regret bound rewarding reshuffling B Extended Related Work Due to the black box nature of the HPO problem (Feurer & Hutter, 2019; Bischl et al., 2023), gradient free, zeroth-order optimization algorithms such as BO (Garnett, 2023), Evolutionary Strategies (Loshchilov & Hutter, 2016) or a simple random search (Bergstra & Bengio, 2012) have become standard optimization algorithms to tackle vanilla HPO problems. In the last decade, most research on HPO has been concerned with constructing new algorithms that excel at finding configurations with a low estimated generalization error. Examples include BO variants such as as HEBO (Cowen-Rivers et al., 2022) or SMAC3 (Lindauer et al., 2022). Another direction of HPO research has been concerned with speeding up the HPO process to allow more efficient spending of compute resources. Multifidelity HPO, for example, turns the black box optimization problem into a gray box one by making use of lower fidelity approximations to the target function, i.e., using fewer numbers of epochs or subsets of the data for cheap low-fidelity evaluations that approximate the costly high-fidelity evaluation. Examples include bandit-based budget allocation algorithms such as Successive Halving (Jamieson & Talwalkar, 2016), Hyperband (Li et al., 2018) and their extensions that use non-random search mechanisms (Falkner et al., 2018; Awad et al., 2021; Mallik et al., 2023) or algorithms making use of multi-fidelity information in the context of BO (Swersky et al., 2014; Klein et al., 2017; Wu et al., 2020; Kadra et al., 2023). Several works address the problem of speeding up cross-validation techniques and use techniques that could be described as grey box optimization techniques. Besides the ones mentioned in the main paper (Thornton et al., 2013; Zheng & Bilenko, 2013), it is possible to employ racing techniques for model selection in machine learning as demonstrated by Lang et al. (2015), and there has been a recent interest in methods that adapt the cost of running full cross-validation procedures (Bergman et al., 2024; Buczak et al., 2024). When addressing the problem of HPO, we must acknowledge an inherent mismatch between the explicit objective we optimize – namely, the estimated generalization performance of a model – and the actual implicit optimization goal, which is to identify a configuration that yields the best 16 generalization performance on new, unseen data. Typically, evaluations and comparisons of different HPO algorithms focus exclusively on the final best validation performance (i.e., the objective that is directly optimized), even though an unbiased estimate of performance on an external unseen test set might be available. While this approach is logical for assessing the efficacy of an optimization algorithm based on the metric it seeks to improve, relying solely on finding an optimal validation configuration is beneficial only if there is reason to assume a strong correlation between the optimized validation performance and true generalization ability on new, unseen test data. This discrepancy can be found deeply within the HPO community, where the evaluation of HPO algorithms on standard benchmark libraries is usually done solely with respect to the validation performance (Eggensperger et al., 2021; Pineda Arango et al., 2021; Salinas et al., 2022; Pfisterer et al., 2022).5 This relationship between validation performance (i.e., the estimated generalization error derived from resampling) and true generalization performance (e.g., assessed through an outer holdout test set or additional resampling) of an optimal validation configuration found during HPO remains a largely unexplored area of research. In general, little research has focused on the selection of resampling types, let alone the automated selection of resampling types (Guyon et al., 2010; Feurer et al., 2022). While we usually expect that a more intensive resampling will reduce the variance of the estimated generalization error and thereby improve the (rank) correlation between optimized validation and unbiased outer test performance within HPCs, this benefit is naturally offset by a higher computational expense. Overall, there is little research on which resampling method to use in practice for model selection, and we only know of a study for support vector machines (Wainer & Cawley, 2017), a simulation study for clinical prediction models (Dunias et al., 2024), a study on feature selection (Molinaro et al., 2005) and a study on fast CV (Bergman et al., 2024). In addition, ML-Plan (Mohr et al., 2018) proposed a two-stage procedure. In a first stage (search), the tool uses planning on hierarchical task networks to find promising machine learning pipelines on 70% of the training data. In a second step (selection), it uses 100% of the training data and retrains the most promising candidates from the search step. Finally, it uses a combination of the internal generalization error estimation that was used during search and the 0.75 percentile of the generalization error estimation from the selection step to make a more unbiased selection of the final model. The paper found that this improves performance over using only regular cross-validation for search and selection. The general consensus, that is in agreement with our findings, is that CV or repeated CV generally leads to better generalization performance. In addition, while there are theoretical works that compare the accuracy of estimating the generalization error of holdout and CV (Blum et al., 1999), our goals is to correctly identify a single solution, which generalizes well, see the excellent survey by Arlot & Celisse (2010) for a discussion on this topic. Bouthillier et al. (2021) studied the sources of variance in machine learning experiments, and find that the split into training and test data has the largest impact. Consequently, they suggest to reshuffle the data prior to splitting it into the training, which is then used for HPO, and the test set. We followed their suggestion when designing our experiments and draw a new test sample for every replication, see Section 4.1 and Appendix F. This dependence on the exact split was further already discussed in the context of how much the outcome of a statistical test on results of machine learning experiments depended on the exact train-test split (Bouckaert, 2004). Finally, the first warning against comparing too many hypothesis using cross-validation was raised by Schaffer (1993), and in addition to the works discussed in Section 5 in the main paper, also picked up by Rao et al. (2008); Cawley & Talbot (2010). Moreover, the problem of finding a correct "upper objective" in a bilevel optimization problem has been noted (Guyon et al., 2010, 2015, 2019). Also, in the related field of algorithm configuration the problem has been identified (Eggensperger et al., 2019). B.1 Current Treatment of Resamplings in HPO Libraries and Software In Table 3, we provide a brief summary of how resampling is handled in popular HPO libraries and software.6 For each library, we checked whether the core functionality, examples, or tutorials mention 5We admit that these benchmark libraries implement efficient benchmarking methods such as surrogate (Eggensperger et al., 2018; Pfisterer et al., 2022) or tabular benchmarks (Pineda Arango et al., 2021). It would be possible to adapt them to return the test performance, however, changes in the HPO evaluation protocol, such as the one we propose, would not be feasible. 6This summary is not exhaustive but reflects the general consensus observed in widely-used software. 17 the possibility of reshuffling the resampling during HPO or if the resampling is considered fixed. If reshuffling is used in an example, mentioned, or if core functionality uses it, we mark it with a ✓. If it is unclear or inconsistent across examples and core functionality, we mark it with a ?. Otherwise, we use a ✗. Our conclusion is that the concept of reshuffling resampling generally receives little attention. Table 3: Exemplary Treatment of Resamplings in HPO Libraries and Software Software Reshuffled? Reference(s) sklearn ✗ GridSearchCV1/ RandomizedSearchCV2 HEBO ✗ sklearn_tuner3 optuna ? Inconsistency between examples4,5,6 bayesian-optimization ✗ sklearn Example7,8 ax ✗ CNN Example9 spearmint ✗ No official HPO Examples scikit-optimize ✗ BO for GBT Example7,10 SMAC3 ✗ SVM Example7,11 dragonfly ✗ Tree Based Ensemble Example12 aws sagemaker ✗ Blog Post13 raytune ? Inconsistency between examples14,15 hyperopt(-sklearn) ? Cost Function Logic16 ✗: no reshuffling, ?: both reshuffling and no reshuffling or unclear, ✓: reshuffling 1 https://github.com/scikit-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3/sklearn/m odel_selection/_search.py#L1263 2 https://github.com/scikit-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3/sklearn/m odel_selection/_search.py#L1644 3 https://github.com/huawei-noah/HEBO/blob/b60f41aa862b4c5148e31ab4981890da6d41f2b1/HEBO/hebo/sklearn_t uner.py#L73 4 https://github.com/optuna/optuna-integration/blob/15e6b0ec6d9a0d7f572ad387be8478c56257bef7/optuna_in tegration/sklearn/sklearn.py#L223 here sklearn’s cross_validate is used which by default does not reshuffle the resampling https://github.com/scikit-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3/sklearn/m odel_selection/_validation.py#L186 5 https://github.com/optuna/optuna-examples/blob/dd56b9692e6d1f4fa839332edbcdd93fd48c16d8/pytorch/py torch_simple.py#L79 here, data loaders for train and valid are instantiated within the objective of the trial but the data within the loaders is fixed 6 https://github.com/optuna/optuna-examples/blob/dd56b9692e6d1f4fa839332edbcdd93fd48c16d8/xgboost/xgbo ost_simple.py#L22 here, the train validation split is performed within the objective of the trial and no seed is set which results in reshuffling https://github.com/scikit-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3/s klearn/model_selection/_split.py#L2597 7 functionality relies on sklearn’s cross_val_score which by default does not reshuffle the resampling https://github.com/sciki t-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3/sklearn/model_selection/_validati on.py#L631 8 https://github.com/bayesian-optimization/BayesianOptimization/blob/c7e5c3926944fc6011ae7ace29f7b5ed0f 9c983b/examples/sklearn_example.py#L32 9 https://github.com/facebook/Ax/blob/ac44a6661f535dd3046954f8fd8701327f4a53e2/tutorials/tune_cnn_serv ice.ipynb#L39 and https://github.com/facebook/Ax/blob/ac44a6661f535dd3046954f8fd8701327f4a53e2/ax/util s/tutorials/cnn_utils.py#L154 10 https://github.com/scikit-optimize/scikit-optimize/blob/a2369ddbc332d16d8ff173b12404b03fea472492/ex amples/hyperparameter-optimization.py#L82C21-L82C36 11 https://github.com/automl/SMAC3/blob/9aaa8e94a5b3a9657737a87b903ee96c683cc42c/examples/1_basics/2_sv m_cv.py#L63 12 https://github.com/dragonfly/dragonfly/blob/3eef7d30bcc2e56f2221a624bd8ec7f933f81e40/examples/tree_r eg/skltree.py#L111 13 https://aws.amazon.com/blogs/architecture/field-notes-build-a-cross-validation-machine-learning-mod el-pipeline-at-scale-with-amazon-sagemaker/ 14 https://github.com/ray-project/ray/blob/3f5aa5c4642eeb12447d9de5dce22085512312f3/doc/source/tune/exa mples/tune-pytorch-cifar.ipynb#L120 here, data loaders for train and valid are instantiated within the objective but the data within the loaders are fixed 15 https://github.com/ray-project/ray/blob/3f5aa5c4642eeb12447d9de5dce22085512312f3/doc/source/tune/exa mples/tune-xgboost.ipynb#L335 here, the train validation split is performed within the objective and no seed is set which results in reshuffling https://github.com/scikit-learn/scikit-learn/blob/8721245511de2f225ff5f9aa5f5fadce663cd4a3 /sklearn/model_selection/_split.py#L2597 16 https://github.com/hyperopt/hyperopt-sklearn/blob/4bc286479677a0bfd2178dac4546ea268b3f3b77/hpsklearn /estimator/_cost_fn.py#L144 dependence on random seed which by default is not set and there is no discussion of reshuffling and behavior is somewhat unclear 18 C Proofs of the Main Results C.1 Proof of Theorem 2.1 We impose stability assumptions on the learning algorithm similar to Bayle et al. (2020); Austern & Zhou (2020). Let Z, Z1, . . . , Zn, Z′ 1, be iid random variables. Define T = {Zi}n i=1, and T ′ as T but with Zn replaced by the independent copy Z′ n. Define eℓn(z, λ) = ℓ(z, gλ(T )) −E[ℓ(Z, gλ(T )) \| T ], assume that each gλ(T ) is invariant to the ordering in T , ℓis bounded, and max λ∈Λ E{[eℓ(Z, gλ(T )) −eℓ(Z, gλ(T ′))]2} = o(1/n). (4) This loss stability assumption is rather mild, see Bayle et al. (2020) for an extensive discussion. Further, define the risk R(g) = E[ℓ(Z, g)] and assume that for every λ ∈Λ, there is a prediction rule g∗ λ such that max λ∈Λ E [\|R(gλ(T )) −R(g∗ λ)\|] = o(1/√n). (5) This assumption requires gλ(T ) to converge to some fixed prediction rule sufficiently fast and serves as a reasonable working condition for our purposes. It is satisfied, for example, when ℓis the square loss and gλ is an empirical risk minimizer over a hypothesis class Gλ with finite VC-dimension. For further examples, see, e.g., Bousquet & Zhivotovskiy (2021), van Erven et al. (2015), and references therein. The assumption could be relaxed, but this would lead to a more complicated limiting distribution but with the same essential interpretation. Theorem C.1. Under assumptions (4) and (5), it holds √n (bµ(λj) −µ(λj))J j=1 →d N(0, Σ), where Σj,j′ = τi,j,M lim n→∞Cov[¯ℓn(Z, λj), ¯ℓn(Z, λj′)], τj,j′,M = lim n→∞ 1 nM 2α2 n X i=1 M X m=1 M X m′=1 Pr(i ∈Im,j ∩Im′,j′). Proof. Define eµ(λj) = 1 M M X m=1 E[L(Vm,j, gλj(Tm,j)) \| Tm,j]. By the triangle inequality (first and second step), Jensen’s inequality (third step), and (5) (last step), E[\|eµ(λj) −µ(λj)\|] ≤ max 1≤m≤M E E[L(Vm,j, gλj(Tm,j)) \| Tm,j] −E[L(Vm,j, gλj(Tm,j))] ≤ max 1≤m≤M E hE[L(Vm,j, gλj(Tm,j)) \| Tm,j] −E[L(Vm,j, g∗ λj)] i + max 1≤m≤M E hE[L(Vm,j, gλj(Tm,j))] −E[L(Vm,j, g∗ λj)] i ≤2 max 1≤m≤M E hE[L(Vm,j, gλj(Tm,j)) \| Tm,j] −E[L(Vm,j, g∗ λj)] i = 2 max 1≤m≤M E hR(gλj(Tm,j)) −R(g∗ λj) i = o(1/√n). Next, assumption (4) together with Theorem 2 and Proposition 3 of Bayle et al. (2020) yield √n (bµ(λj) −eµ(λj)) −1 M M X m=1 1 α√n X i∈Im,j ¯ℓn(Zi, λj) →p 0. 19 Now rewrite 1 Mα√n M X m=1 X i∈Im,j ¯ℓn(Zi, λj) = 1 Mα√n n X i=1 M X m=1 1(i ∈Im,j)¯ℓn(Zi, λj) \| {z } :=ξ(j) i,n . The sequence (ξi,n)n i=1 = (ξ(j) i,n, . . . , ξ(j) i,n)n i=1 is a triangular array of independent, centered, and bounded random vectors. Because 1(Zi ∈Vm,j) and Zi are independent, it holds Cov(ξ(j) i,n, ξ(j′) i,n ) = M X m=1 M X m′=1 E[1(i ∈Im,j ∩Im′,j′)]E[¯ℓn(Zi, λj)¯ℓn(Zi, λj′)], so lim n→∞Cov " 1 Mα√n n X i=1 ξ(j) i,n, 1 Mα√n n X i=1 ξ(j′) i,n # = lim n→∞ 1 nM 2α2 n X i=1 Cov h ξ(j) i,n, ξ(j′) i,n i = Σj,j′. Now the result follows from Lindeberg’s central limit theorem for triangular arrays (e.g., van der Vaart, 2000, Proposition 2.27). C.2 Proof of Theorem 2.2 We want to bound the probability that µ(ˆλ) −µ(λ∗) is large. For some δ > 0, define the set of ‘good’ hyperparameters Λδ = {λj : µ(λj) −µ(λ∗) ≤δ}. Now Pr µ(bλ) −µ(λ∗) > δ = Pr bλ /∈Λδ = Pr min λ/∈Λδ bµ(λ) < min λ∈Λδ bµ(λ) ≤Pr min λ/∈Λδ bµ(λ) < min λ∈Λδ/2 bµ(λ) = Pr min λ/∈Λδ µ(λ) + ϵ(λ) < min λ∈Λδ/2 µ(λ) + ϵ(λ) ≤Pr δ + min λ/∈Λδ ϵ(λ) < δ/2 + min λ∈Λδ/2 ϵ(λ) = Pr min λ/∈Λδ ϵ(λ) −min λ∈Λδ/2 ϵ(λ) < −δ/2 = Pr max λ/∈Λδ ϵ(λ) −max λ∈Λδ/2 ϵ(λ) > δ/2 . (ϵ d= −ϵ) There is a tension between the two maxima. The more λ’s there are in Λδ/2 and the less they are correlated, the more likely it is to find one ϵ(λ) that is large. This makes the probability small. However, the less ϵ is correlated, the larger is maxλ/∈Λδ ϵ(λ), making the probability large. To formalize this, use the Gaussian concentration inequality (Talagrand, 2005, Lemma 2.1.3): Pr max λ/∈Λδ ϵ(λ) −max λ∈Λδ/2 ϵ(λ) > δ/2 ≤Pr 2 max λ∈Λ ϵ(λ) −E max λ∈Λ ϵ(λ) > δ/2 −E max λ∈Λδ/2 ϵ(λ) + E max λ/∈Λδ ϵ(λ) ≤2 exp ( − δ/2 −E maxλ∈Λδ/2 ϵ(λ) + E [maxλ/∈Λδ ϵ(λ)] 2 8σ2 ) , provided δ/2−E maxλ∈Λδ/2 ϵ(λ) +E [maxλ/∈Λδ ϵ(λ)] ≥0. We bound the two maxima separately. 20 Lower Bound for Maximum over the Good Set Recall the definition of m right before Theorem 2.2 and observe Λδ/2 = {λ: µ(λ) −µ(λ∗) ≤δ/2} ⊃{λ: m∥λ −λ∗∥2 ≤δ/2} = {λ: ∥λ −λ∗∥≤(δ/2m)1/2} = B(λ∗, (δ/2m)1/2). Pack the ball B(λ∗, (δ/2m)1/2) with smaller balls with radius η. We can always construct such a packing with at least (δ/2mη2)d/2 elements. By assumption, each small ball contains at least one element of Λ. Pick one element from each small ball and collect them into the set Λ′ δ/2. By construction, \|Λ′ δ/2\| ≥(δ/2mη2)d/2 and min λ̸=λ′∈Λ′ δ/2\| ∥λ −λ′∥≥η. Sudakov’s minoration principle (e.g., Wainwright, 2019, Theorem 5.30) gives E max λ∈Λδ/2 ϵ(λ) ≥1 2 q log \|Λ′ δ/2\| min {λ̸=λ′}∩Λ′ δ/2 p Var [ϵ(λ) −ϵ(λ′)] ≥1 2 q log \|Λ′ δ/2\| min ∥λ−λ′∥≥η p Var [ϵ(λ) −ϵ(λ′)]. In general, Var [ϵ(λ) −ϵ(λ′)] = K(λ, λ) + K(λ′, λ′) −2τ 2K(λ, λ′) = (1 −τ 2)[K(λ, λ) + K(λ′, λ′)] + τ 2[K(λ, λ) −K(λ, λ′)] + τ 2[K(λ′, λ′) −K(λ, λ′)] ≥2σ2(1 −τ 2). Hence, we have min ∥λ−λ′∥≥η Var [ϵ(λ) −ϵ(λ′)] ≥2σ2(1 −τ 2), which implies E max λ∈Λδ/2 ϵ(λ) ≥1 2σ √ d p 1 −τ 2p log(δ/2mη2) =: σ √ dA(τ, δ)/2. Upper Bound for Maximum over the Bad Set Dudley’s entropy bound (e.g., Giné & Nickl, 2016, Theorem 2.3.6) gives E max λ/∈Λδ ϵ(λ) ≤12 Z ∞ 0 p log N(s)ds, where N(s) is the minimum number of points λ1, . . . , λN(s) such that sup λ∈Λ min 1≤k≤N(s) p Var [ϵ(λ) −ϵ(λk)] ≤s. Note that sup λ,λ′∈Λ p Var [ϵ(λ) −ϵ(λ′)] ≤2σ, so N(s) = 1 for all s ≥2σ. For s2 ≤4σ2(1 −τ 2), we can use the trivial bound N(s) ≤J. For s2 > 4σ2(1 −τ 2), cover Λ with ℓ2-balls of size (s/2στκ). We can do this with less than N(s) ≤(6σκ/s)d ∨1 such balls. Let λ1, . . . , λN be the centers of these balls. In general, it holds Var [ϵ(λ) −ϵ(λ′)] = K(λ, λ) + K(λ′, λ′) −2τ 2K(λ, λ′) = (1 −τ 2)[K(λ, λ) + K(λ′, λ′)] + τ 2[K(λ, λ) −K(λ, λ′)] + τ 2[K(λ′, λ′) −K(λ, λ′)] ≤2(1 −τ 2)σ2 + 2τ 2σ2κ2∥λ −λ′∥2. 21 For s2 > 4σ2(1 −τ 2), we thus have sup λ∈Λ min 1≤k≤N(s) Var [ϵ(λ) −ϵ(λk)] ≤ sup ∥λ−λ′∥2≤(s/2τσκ)2 Var [ϵ(λ) −ϵ(λ′)] ≤2(1 −τ 2)σ2 + 2τ 2σ2κ2(s/2τσκ)2 ≤s2, as desired. Now decompose the integral Z ∞ 0 p log N(s)ds = Z 2σ √ 1−τ 2 0 p log N(s)ds + Z 2σ 2σ √ 1−τ 2 p log N(s)ds ≤2σ √ d p 1 −τ 2p log J + Z 2σ 2σ √ 1−τ 2 p log N(s)ds. For the second term, compute Z 2σ σ √ 1−τ 2 p log N(s)ds ≤ √ d Z 2σ 2σ √ 1−τ 2 p log(6σκ/s)+ ds = σ √ d Z 2 2 √ 1−τ 2 p log(6κ/s)+ ds ≤σ √ d Z 2 0 log(6κ/s)+ ds 1/2 2(1 − p 1 −τ 2) 1/2 = σ √ d p 2 + 2 log(3κ)+ 2(1 − p 1 −τ 2) 1/2 = 2σ √ d p 1 + log(3κ)+ τ (1 + √ 1 −τ 2)1/2 ≤2σ √ dτ p 1 + log(3κ)+. We have shown that E max λ/∈Λδ ϵ(λ) ≤24σ √ d hp 1 −τ 2p log J + τ p 1 + log(3κ)+ i =: σ √ dB(τ)/4. Integrating Probabilities Summarizing the two previous steps, we have Pr µ(bλ) −µ(λ∗) > δ ≤2 exp      − δ −σ √ d[B(τ) −A(τ, δ)] 2 36σ2      , provided t ≥σ √ d[B(τ) −A(τ, δ)]. Now for any s ≥0 and t ≥2es2mη2, it holds A(τ, s) ≥(σ/σ) p 1 −τ 2s =: A(τ)s. In particular, if t ≥2es2mη2 + σ √ d[B(τ) −A(τ)s] =: C, we have Pr µ(bλ) −µ(λ∗) > δ ≤4 exp      − δ −σ √ d[B(τ) −A(τ)s] 2 36σ2      . 22 Integrating the probability gives E[µ(bλ) −µ(λ∗)] = Z ∞ 0 Pr µ(bλ) −µ(λ∗) > δ dδ = Z C 0 Pr µ(bλ) −µ(λ∗) > δ dδ + Z ∞ C Pr µ(bλ) −µ(λ∗) > δ dδ ≤C + Z ∞ C exp      − δ −σ √ d[B(τ) −A(τ)s] 2 36σ2      dδ ≤C + √ 36σ = 2es2mη2 + σ √ d[B(τ) −A(τ)s] + 6σ. Simplifying The bound can be optimized with respect to s, but the solution involves the Lambert W-function, which has no analytical expression. Instead choose s for simplicity as s = s log σ 2mη2 + . which gives E[µ(bλ) −µ(λ∗)] ≤σ √ d " 8 + B(τ) −A(τ) s log σ 2mη2 # . D Additional Results on the Density of Random HPC Grids Lemma D.1. Suppose that the J elements in Λ are drawn independently from a continuous density p with c := min∥λ∥≤1 p(λ) > 0. Then with probability at least 1 −δ, η ≲ p log(1/δ)/J 1/d , and with probability 1, η ≲ p log(J)/J 1/d , for all J sufficiently large. Proof. We want to bound the probability that there is a λ such that \|B(λ, η) ∩Λ\| = 0. In what follows λ is silently understood to have norm bounded by 1. Let eλ1, . . . , eλN the centers of η/2-balls covering {∥λ∥≤1}, for which we may assume N ≤(6/η)d. For eλk the closest center to λ, it holds ∥λ′ −λ∥≤∥λ′ −eλk∥+ ∥eλk −λ∥≤∥λ′ −eλk∥+ η/2, so ∥λ′ −eλk∥≤η/2 implies ∥λ′ −λ∥≤η. We thus have Pr(∃λ: \|B(λ, η) ∩Λ\| = 0) = Pr inf λ J X i=1 1{∥λi −λ∥≤η} ≤0 ! ≤Pr min 1≤k≤N J X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! . 23 Further Pr min 1≤k≤N J X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! = Pr max 1≤k≤N J X i=1 −1{∥λi −eλk∥≤η/2} ≥0 ! ≤Pr max 1≤k≤N J X i=1 E h 1{∥λi −eλk∥≤η/2} i −1{∥λi −eλk∥≤η/2} ≥J inf λ E [1{∥λi −λ∥≤η/2}] ! . It holds E [1{∥λi −λ∥≤η/2}] = Pr (∥λi −λ∥≤η/2) = Z ∥λ′−λ∥≤η/2 p(λ′)dλ′ ≥c vol(B(0, η/2)) = cvd(η/2)d, where vd = vol(B(0, 1)). Now the union bound and Hoeffding’s inequality give Pr min 1≤k≤N J X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! ≤N exp −Jc2v2 d(η/2)2d 2 ≤(6/η)d exp −Jc2v2 d(η/2)2d 2 . Choosing η = 2 q 2 log(3d√ Jcvd/δ)/ √ Jcvd 1/d gives Pr(∃λ: \|B(λ, η) ∩Λ\| = 0) ≤δ/ q 2 log(3d√ Jcvd), which is bounded by δ when √ J ≥e1/2/3dcvd. Further, setting η = 2( p 6 log(J)/ √ Jcvd)1/d gives Pr min 1≤k≤N J X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! ≲J−5/2, so that ∞ X J=1 Pr min 1≤j≤J min 1≤k≤N j X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! ≤ ∞ X J=1 J Pr min 1≤k≤N J X i=1 1{∥λi −eλk∥≤η/2} ≤0 ! ≲ ∞ X J=1 1 J3/2 < ∞. Now the Borel-Cantelli lemma (e.g., Kallenberg, 1997, Theorem 4.18) implies that, with probability 1, \|B(λ, η) ∩Λ\| ≥1, for all J sufficiently large. 24 E Selected Validation Schemes E.1 Definition of Index Sets Recall: (i) (holdout) Let M = 1 and I1,j = I1 for all j = 1, . . . , J, and some size-⌈αn⌉index set I1. (ii) (reshuffled holdout) Let M = 1 and I1,1, . . . , I1,J be independently drawn from the uniform distribution over all size-⌈αn⌉subsets from {1, . . . , n}. (iii) (M-fold CV) Let α = 1/M and I1, . . . , IM be a disjoint partition of {1, . . . , n}, and Im,j = Im for all j = 1, . . . , J. (iv) (reshuffled M-fold CV) Let α = 1/M and (I1,j, . . . , IM,j), j = 1, . . . , J, be independently drawn from the uniform distribution over disjoint partitions of {1, . . . , n}. (v) (M-fold holdout) Let Im, m = 1, . . . , M, be independently drawn from the uniform distribution over size-⌈αn⌉subsets of {1, . . . , n} and set Im,j = Im for all m = 1, . . . , M, j = 1, . . . , J. (vi) (reshuffled M-fold holdout) Let Im,j, m = 1, . . . , M, j = 1, . . . , J, be independently drawn from the uniform distribution over size-⌈αn⌉subsets of {1, . . . , n}. E.2 Derivation of Reshuffling Parameters in Limiting Distribution Recall τi,j,M = 1 nM 2α2 n X s=1 M X m=1 M X m′=1 Pr(s ∈Im,i ∩Im′,j). For all schemes in the proposition, the probabilities are independent of the index s, so the average over s = 1, . . . , n can be omitted. We now verify the constants σ, τ from Table 1. (i) It holds Pr(s ∈I1,i ∩I1,j) = Pr(s ∈I1) = α. Hence, τi,j,1 = 1/α = 1/α × 1 = σ2 × τ 2. (ii) (reshuffled holdout) This is a special case of part (vi) with M = 1. (iii) (M-fold CV) It holds Pr(s ∈Im,i ∩Im′,j) = Pr(s ∈Im ∩Im′) = 1/M, m = m′, 0, m ̸= m′. Only M probabilities in the double sum are non-zero, whence τi,j,M = 1 M 2α2 × M/M = 1/α2M 2 = 1 × 1 = σ2 × τ 2, where we used α = 1/M. (iv) (reshuffled M-fold CV) It holds Pr(s ∈Im,i ∩Im′,j) =        1/M, m = m′, i = j 0, m ̸= m′, i = j 1/M 2, m = m′, i ̸= j 1/M 2, m ̸= m′, i ̸= j. For i = j, only M probabilities in the double sum are non-zero. Also using α = 1/M, we get τi,j,M = 1 M 2α2 × M × 1/M = 1 = σ2. For i ̸= j, τi,j,M = 1 M 2α2 × M 2 × 1/M 2 = 1 × 1 = σ2 × τ 2. 25 (v) (M-fold holdout) It holds Pr(s ∈Im,i ∩Im′,j) = Pr(s ∈Im ∩Im′) = α, m = m′, α2, else. This gives τi,j,M = 1 M 2α2 × [M × α + (M −1)M × α2] = [1/αM + (M −1)/M] × 1 = σ2 × τ 2. for all i, j. (vi) (reshuffled M-fold holdout) It holds Pr(s ∈Im,i ∩Im′,j) = α, m = m′, i = j α2, else. For i = j, this gives τi,j,M = 1 M 2α2 × [M × α + (M −1)M × α2] = 1/αM + (M −1)/M. For i ̸= j, τi,j,M = 1 M 2α2 × (M 2 × α2) = 1. This implies that (1) holds with σ2 = 1/Mα + (M −1)/M, τ 2 = 1/(1/Mα + (M −1)/M). Remark E.1. Although not technically covered by Theorem 2.1, performing independent bootstraps for each λj correspond to reshuffled n-fold holdout with α = 1/n. Accordingly, σ ≈ √ 2 and τ ≈ p 1/2. F Details Regarding Benchmark Experiments F.1 Datasets We list all datasets used in the benchmark experiments in Table 4. Table 4: List of datasets used in benchmark experiments. All information can be found on OpenML (Vanschoren et al., 2014). OpenML Dataset ID Dataset Name Size (n × p) 23517 numerai28.6 96320 × 21 1169 airlines 539383 × 7 41147 albert 425240 × 78 4135 Amazon_employee_access 32769 × 9 1461 bank-marketing 45211 × 16 1590 adult 48842 × 14 41150 MiniBooNE 130064 × 50 41162 kick 72983 × 32 42733 Click_prediction_small 39948 × 11 42742 porto-seguro 595212 × 57 Note that datasets serve as data generating processes (DGPs; Hothorn et al., 2005). As we are mostly concerned with the actual generalization performance of the final best HPC found during HPO based on validation performance we rely on a comparably large held out test set that is not used during HPO. We therefore use 5000 data points sampled from a DGP as an outer test set. To further be able to measure the generalization performance robustly for varying data sizes available during HPO, we construct concrete tasks based on the DGPs by sampling subsets of (train_valid; n) size 500, 1000 and 5000 from the DGPs. This results in 30 tasks in total (10 DGPS × 3 train_valid sizes). For more details and the concrete implementation of this procedure, see Appendix F.3. We also collected another 5000 data points as an external validation set, but did not use it. Therefore, we had to tighten the restriction to 10000 data points mentioned in the main paper to 15000 data points as the lower bound on data points. To allow for stronger variation over different replications, we decided to use 20000 as the final lower bound. 26 F.2 Learning Algorithms Here we briefly present training pipeline details and search spaces of the learning algorithms used in our benchmark experiments. The funnel-shaped MLP is based on sklearn’s MLP Classifier and is constructed in the following way: The hidden layer size for each layer is determined by num_layers and max_units. We start with max_units and half the number of units for every subsequent layer to create a funnel. max_batch_size is the largest power of 2 that is smaller than the number of training samples available. We use ReLU as activation function and train the network optimizing logloss as a loss function via SGD using a constant learning rate and Nesterov momentum for 100 epochs. Table 5 lists the search space (inspired from Zimmer et al. (2021)) used during HPO. The Elastic Net is based on sklearn’s Logistic Regression Classifier. We train it for a maximum of 1000 iterations using the "saga" solver. Table 6 lists the search space used during HPO. The XGBoost and CatBoost search spaces are listed in Table 7 and Table 8, both inspired from their search spaces used in McElfresh et al. (2023). For both the Elastic Net and Funnel MLP, missing values are imputed in the preprocessing pipeline (mean imputation for numerical features and adding a new level for categorical features). Categorical features are target encoded in a cross-validated manner using a 5-fold CV. Features are then scaled to zero mean and unit variance via a standard scaler. For XGBoost, we impute missing values for categorical features (adding a new level) and target encode them in a cross-validated manner using a 5-fold CV. For CatBoost, no preprocessing is performed. XGBoost and CatBoost models are trained for 2000 iterations and stop early if the validation loss (using the default internal loss function used during training, i.e., logloss) does not improve over a horizon of 20 iterations. For retraining the best configuration on the whole train and validation data, the number of boosting iterations is set to the number of iterations used to find the best validation performance prior to the stopping mechanism taking action.7 F.3 Exact Implementation In the following, we outline the exact implementation of performing one HPO run for a given learning algorithm on a concrete task (dataset × train_valid size) and a given resampling. We release all code to replicate benchmark results and reproduce our analyses via https://github.com/slds-l mu/paper_2024_reshuffling. For a given replication (in total 10): 1. We sample (without replacement) train_valid size (500, 1000 or 5000 points) and test size (always 5000) points from the DGP (i.e. a concrete dataset in Table 4). These are shared for every learning algorithm (i.e. all learning algorithms are evaluated on the same data). 2. A given HPC is evaluated in the following way: • The resampling operates on the train validation8 set of size train_valid. • The learning algorithm is configured by the HPC. • The learning algorithm is trained on training splits and evaluated on validation splits according to the resampling strategy. In case reshuffling is turned on, the training and validation splits are recreated for every HPO. We compute the Accuracy, ROC AUC and logloss when using a random search and compute ROC AUC when using HEBO or SMAC3 and average performance over all folds for resamplings involving multiple folds. • For each HPC we then always re-train the model on all train_valid data being available and evaluate the model on the held-out test set to compute an outer estimate of generalization performance for each HPC (regardless of whether it is the incumbent for a given iteration or not). 7For CV and repeated holdout we take the average number of boosting iterations over the models trained on the different folds. 8With train validation we refer to all data being available during HPO which is then further split by a resampling into train and validation sets. 27 Table 5: Search Space for Funnel-Shaped MLP Classifier. Parameter Type Range Log num_layers Int. 1 to 5 No max_units Int. 64, 128, 256, 512 No learning_rate Num. 1 × 10−4 to 1 × 10−1 Yes batch_size Int. 16, 32, ..., max_batch_size No momentum Num. 0.1 to 0.99 No alpha Num. 1 × 10−6 to 1 × 10−1 Yes Table 6: Search Space for Elastic Net Classifier. Parameter Type Range Log C Num. 1 × 10−6 to 1 × 104 Yes l1_ratio Num. 0.0 to 1.0 No Table 7: Search Space for XGBoost Classifier. Parameter Type Range Log max_depth Int. 2 to 12 Yes alpha Num. 1 × 10−8 to 1.0 Yes lambda Num. 1 × 10−8 to 1.0 Yes eta Num. 0.01 to 0.3 Yes Table 8: Search Space for CatBoost Classifier. Parameter Type Range Log learning_rate Num. 0.01 to 0.3 Yes depth Int. 2 to 12 Yes l2_leaf_reg Num. 0.5 to 30 Yes 3. We evaluate 500 HPCs when using random search and 250 HPC when using HEBO or SMAC3 (SMAC4HPO facade). As resamplings, we use holdout with a 80/20 train-validation split and 5 folds for CV, so that the holdout strategy is just one fold of the CV and the fraction of data points being used for training and respectively validation are the same across different resampling strategies. 5-fold holdout simply repeats the holdout procedure five times and 5x 5-fold CV repeats the 5-fold CV five times. Each of the four resamplings can be reshuffled or not (standard). As mentioned above, the test set is only varied for each of the 10 replica (repetitions with different seeds), but consistent for different tasks (i.e. the different learning algorithms are evaluated on the same test set, similarly, also the different dataset subsets all share the same test set). This allows for fair comparisons of different resamplings on a concrete problem (i.e. a given dataset, train_valid size and learning algorithm). Additionally, for the random search, the 500 HPCs evaluated for a given learning algorithm are also fixed over different dataset and train_valid size combinations. This is done to allow for an isolation of the effect, the concrete resampling (and whether it is reshuffled or not) has on generalization performance, reducing noise arising due to different HPCs. Learning algorithms themselves are not explicitly seeded to allow for variation during model training over different replications. Resamplings and partitioning of data are always performed in a stratified manner with respect to the target variable. For the random search, we only ran (standard and reshuffled) holdout and (standard and reshuffled) 5x 5-fold CV experiments (because we can simulate 5-fold CV and 5-fold holdout experiments based 28 on the results obtained from the 5x 5-fold CV (by only considering the first repeat or the first fold for each of the five repeats).9 For running HEBO or SMAC3, each resampling (standard and reshuffled for holdout, 5-fold holdout, 5-fold CV, 5x 5-fold CV) has to be actually run due to the adaptive nature of BO. For the random search experiments, this results in 10 (DGPs) × 3 (train_valid sizes) × 4 (learning algorithms) × 2 (holdout or 5x 5-fold CV) × 2 (standard or reshuffled) × 10 (replications) = 4800 HPO runs,10 each involving the evaluation of 500 HPCs and each evaluation of an HPC involving either 2 (for holdout; due to retraining on train validation data) or 26 (for 5x 5-fold CV; due to retraining on train validation data) model fits. In summary, the random search experiments involve the evaluation of 2.4 Million HPCs with in total 33.6 Million model fits. Similarly, for the HEBO and SMAC3 experiments, this each results in 10 (DGPs) × 3 (train_valid sizes) × 4 (learning algorithms) × 4 (holdout, 5-fold CV, 5x 5-fold CV or 5-fold holdout) × 2 (standard or reshuffled) × 10 (replications) = 9600 HPO runs11, each involving the evaluation of 250 HPCs and each evaluation of an HPC involving either 2 (for holdout; due to retraining on train validation data), 6 (for 5-fold CV or 5-fold holdout; due to retraining on train validation data) or 26 (for 5x 5-fold CV; due to retraining on train validation data) model fits. In summary, the HEBO and SMAC3 experiments each involve the evaluation of 2.4 Million HPCs with in total 24 Million model fits. F.4 Compute Resources We estimate our total compute time for the random search, HEBO and SMAC3 experiments to be roughly 11.86 CPU years. Benchmark experiments were run on an internal HPC cluster equipped with a mix of Intel Xeon E5-2670, Intel Xeon E5-2683 and Intel Xeon Gold 6330 instances. Jobs were scheduled to use a single CPU core and were allowed to use up to 16GB RAM. Total emissions are estimated to be an equivalent of roughly 6508.67 kg CO2. G Additional Benchmark Results Visualizations G.1 Main Experiments In this section, we provide additional visualizations of the results of our benchmark experiments. Figure 6 illustrates the trade-off between the final number of model fits required by different resamplings and the final average normalized test performance (AUC ROC) after running random search for a budget of 500 hyperparameter configurations. We can see that the reshuffled holdout on average comes close to the final test performance of the overall more expensive 5-fold CV. Below, we give an overview of the different types of additional analyses and visualizations we provide. Normalized metrics, i.e., normalized validation or test performance refer to the measure being scaled to [0, 1] based on the empirical observed minimum and maximum values obtained on the raw results level (ADTM; see Wistuba et al., 2018). More concretely, for each scenario consisting of a learning algorithm that is run on a given task (dataset × train_valid size) given a certain performance metric, the performance values (validation or test) for all resamplings and optimizers are normalized on the replication level to [0, 1] by subtracting the empirical best value and dividing by the range of performance values. Therefore a normalized performance value of 0 is best and 1 is worst. Note that we additionally provide further aggregated results on the learning algorithm level and raw results of validation and test performance via https://github.com/slds-lmu/paper_2024_reshuffl ing. • Random search – Normalized validation performance in Figure 7. 9We even could have simulated the vanilla holdout from the 5x 5-fold CV experiments by choosing an arbitrary fold and repeat but choose not to do so, to have some sanity checks regarding our implementation by being able to compare the "true" holdout with a the simulated holdout. 10Note that we do not have to take the 3 different metrics into account because random search allows us to simulate runs for different metric post hoc. 11Note that HEBO and SMAC3 were only run for ROC AUC as the performance metric. 29 500 1000 5000 300 1000 3000 10000 300 1000 3000 10000 300 1000 3000 10000 0.20 0.25 0.30 0.35 0.40 0.25 0.30 0.35 0.40 0.45 0.30 0.35 0.40 0.45 No. Final Model Fits Mean Normalized Test Performance Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 6: Trade-off between the final number of model fits required by different resamplings and the final average normalized test performance (AUC ROC) after running random search for a budget of 500 hyperparameter configurations. Averaged over different tasks, learning algorithms and replications separately for increasing n (train-validation sizes, columns). Shaded areas represent standard errors. – Normalized test performance in Figure 8. – Improvement in test performance over 5-fold CV in Figure 9. – Rank w.r.t. test performance in Figure 10. • HEBO and SMAC3 vs. random search holdout – Normalized validation performance in Figure 11. – Normalized test performance in Figure 12. – Improvement in test performance over standard holdout in Figure 13. – Rank w.r.t. test performance in Figure 14. • HEBO and SMAC3 vs. random search 5-fold holdout – Normalized validation performance in Figure 15. – Normalized test performance in Figure 16. – Improvement in test performance over standard 5-fold holdout in Figure 17. – Rank w.r.t. test performance in Figure 18. • HEBO and SMAC3 vs. random search 5-fold CV – Normalized validation performance in Figure 19. – Normalized test performance in Figure 20. – Improvement in test performance over 5-fold CV in Figure 21. – Rank w.r.t. test performance in Figure 22. • HEBO and SMAC3 vs. random search 5x 5-fold CV – Normalized validation performance in Figure 23. – Normalized test performance in Figure 24. – Improvement in test performance over 5x 5-fold CV in Figure 25. – Rank w.r.t. test performance in Figure 26. 30 Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 No. HPC Evaluations Mean Normalized Validation Performance Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 7: Random search. Average normalized performance over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.25 0.30 0.35 0.40 0.45 0.50 0.1 0.2 0.3 0.4 0.5 No. HPC Evaluations Mean Normalized Test Performance Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 8: Random search. Average normalized test performance over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 31 Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 −0.4 −0.3 −0.2 −0.1 0.0 0.1 −0.50 −0.25 0.00 0.25 −0.3 −0.2 −0.1 0.0 0.1 −0.4 −0.2 0.0 −1.2 −0.8 −0.4 0.0 0.4 −1.5 −1.0 −0.5 0.0 −0.75 −0.50 −0.25 0.00 0.25 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 No. HPC Evaluations Mean Test Improvement Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 9: Random search. Average improvement (compared to standard 5-fold CV) with respect to test performance of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 4.0 4.5 5.0 5.5 3.5 4.0 4.5 5.0 5.5 6.0 4 5 4.0 4.5 5.0 5.5 3.5 4.0 4.5 5.0 5.5 6.0 4 5 6 4.0 4.5 5.0 4.0 4.5 5.0 5.5 4 5 6 No. HPC Evaluations Mean Rank (Test Performance) Reshuffling FALSE TRUE Resampling Holdout 5−fold CV 5−fold Holdout 5x 5−fold CV Figure 10: Random search. Average ranks (lower is better) with respect to test performance over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 32 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 No. HPC Evaluations Mean Normalized Validation Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 11: HEBO and SMAC3 vs. random search for holdout. Average normalized validation performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.30 0.35 0.40 0.45 0.50 No. HPC Evaluations Mean Normalized Test Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 12: HEBO and SMAC3 vs. random search for holdout. Average normalized test performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 −0.5 0.0 0.5 1.0 No. HPC Evaluations Mean Test Improvement Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 13: HEBO and SMAC3 vs. random search for holdout. Average improvement (compared to standard holdout) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 33 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 3.25 3.50 3.75 4.00 3.00 3.25 3.50 3.75 4.00 3.00 3.25 3.50 3.75 4.00 No. HPC Evaluations Mean Rank (Test Performance) Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 14: HEBO and SMAC3 vs. random search for holdout. Average ranks (lower is better) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.2 0.4 0.6 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 No. HPC Evaluations Mean Normalized Validation Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 15: HEBO and SMAC3 vs. random search for 5-fold holdout. Average normalized validation performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.25 0.35 0.45 0.55 0.3 0.4 0.5 0.25 0.30 0.35 0.40 0.45 0.50 No. HPC Evaluations Mean Normalized Test Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 16: HEBO and SMAC3 vs. random search for 5-fold holdout. Average normalized test performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 34 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 No. HPC Evaluations Mean Test Improvement Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 17: HEBO and SMAC3 vs. random search for 5-fold holdout. Average improvement (compared to standard 5-fold holdout) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 3.00 3.25 3.50 3.75 4.00 3.25 3.50 3.75 3.25 3.50 3.75 No. HPC Evaluations Mean Rank (Test Performance) Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 18: HEBO and SMAC3 vs. random search for 5-fold holdout. Average ranks (lower is better) with respect to test performance (ROC AUC) of the incumbent tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.2 0.4 0.6 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 No. HPC Evaluations Mean Normalized Validation Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 19: HEBO and SMAC3 vs. random search for 5-fold CV. Average normalized validation performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 35 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.30 0.35 0.40 0.45 0.50 No. HPC Evaluations Mean Normalized Test Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 20: HEBO and SMAC3 vs. random search for 5-fold CV. Average normalized test performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 No. HPC Evaluations Mean Test Improvement Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 21: HEBO and SMAC3 vs. random search for 5-fold CV. Average improvement (compared to standard 5-fold CV) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 3.25 3.50 3.75 3.2 3.4 3.6 3.8 3.3 3.5 3.7 No. HPC Evaluations Mean Rank (Test Performance) Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 22: HEBO and SMAC3 vs. random search for 5-fold CV. Average ranks (lower is better) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 36 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.8 No. HPC Evaluations Mean Normalized Validation Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 23: HEBO and SMAC3 vs. random search for 5x 5-fold CV. Average normalized validation performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.3 0.4 0.5 No. HPC Evaluations Mean Normalized Test Performance Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 24: HEBO and SMAC3 vs. random search for 5x 5-fold CV. Average normalized test performance (ROC AUC) over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 −0.4 0.0 0.4 0.8 1.2 No. HPC Evaluations Mean Test Improvement Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 25: HEBO and SMAC3 vs. random search for 5x 5-fold CV. Average improvement (compared to standard 5x 5-fold CV) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 37 500 1000 5000 1 50 100 150 200 250 1 50 100 150 200 250 1 50 100 150 200 250 3.00 3.25 3.50 3.75 3.25 3.50 3.75 4.00 3.2 3.4 3.6 3.8 No. HPC Evaluations Mean Rank (Test Performance) Optimizer Random Search HEBO SMAC3 Reshuffling FALSE TRUE Figure 26: HEBO and SMAC3 vs. random search for 5x 5-fold CV. Average ranks (lower is better) with respect to test performance (ROC AUC) of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 38 G.2 Ablation on M-fold holdout Based on the 5x 5-fold CV results we further simulated different M-fold holdout resamplings (standard and reshuffled) by taking M repeats from the first fold of the 5x 5-fold CV. This allows us to get an understanding of the effect more folds have on M-fold holdout, especially in the context of reshuffling. Regarding normalized validation performance we observe that more folds generally result in a less optimistically biased validation performance (see Figure 27). Looking at normalized test performance (Figure 28) we observe the general trend that more folds result in better test performance – which is expected. Reshuffling generally results in better test performance compared to the standard resampling (with the exception of logloss where especially in the case of a single holdout, reshuffling can hurt generalization performance). This effect is smaller, the more folds are used, which is in line with our theoretical results presented in Table 1. Looking at improvement compared to standard 5-fold holdout with respect to test performance and ranks with respect to test performance, we observe that often reshuffled 2-fold holdout results that are highly competitive with standard 3, 4 or 5-fold holdout. Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 0.25 0.50 0.75 0.0 0.2 0.4 0.6 0.8 0.25 0.50 0.75 No. HPC Evaluations Mean Normalized Validation Performance Holdout 1−fold 2−fold 3−fold 4−fold 5−fold Reshuffling FALSE TRUE Figure 27: Random search. Average normalized validation performance over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 39 Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 0.25 0.30 0.35 0.40 0.45 0.50 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.25 0.30 0.35 0.40 0.45 0.30 0.35 0.40 0.45 0.50 0.1 0.2 0.3 0.4 0.5 No. HPC Evaluations Mean Normalized Test Performance Holdout 1−fold 2−fold 3−fold 4−fold 5−fold Reshuffling FALSE TRUE Figure 28: Random search. Average normalized test performance over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 −0.4 −0.3 −0.2 −0.1 0.0 0.1 −0.6 −0.4 −0.2 0.0 0.2 −0.3 −0.2 −0.1 0.0 0.1 −0.4 −0.3 −0.2 −0.1 0.0 0.1 −1.0 −0.5 0.0 −1.5 −1.0 −0.5 0.0 −0.50 −0.25 0.00 0.25 0.50 −1.0 −0.5 0.0 −2.0 −1.5 −1.0 −0.5 0.0 No. HPC Evaluations Mean Test Improvement Holdout 1−fold 2−fold 3−fold 4−fold 5−fold Reshuffling FALSE TRUE Figure 29: Random search. Average improvement (compared to standard 5-fold holdout) with respect to test performance of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 40 Logloss, 500 Logloss, 1000 Logloss, 5000 ROC AUC, 500 ROC AUC, 1000 ROC AUC, 5000 Accuracy, 500 Accuracy, 1000 Accuracy, 5000 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 1 100 200 300 400 500 5.0 5.5 6.0 6.5 5.0 5.5 6.0 6.5 7.0 4.5 5.0 5.5 6.0 6.5 5.0 5.5 6.0 6.5 4.5 5.0 5.5 6.0 6.5 7.0 5 6 7 4.8 5.2 5.6 6.0 5.0 5.5 6.0 6.5 5 6 7 No. HPC Evaluations Mean Rank (Test Performance) Holdout 1−fold 2−fold 3−fold 4−fold 5−fold Reshuffling FALSE TRUE Figure 30: Random search. Average ranks (lower is better) with respect to test performance of the incumbent over tasks, learners and replications for different n (train-validation sizes, columns). Shaded areas represent standard errors. 41 NeurIPS Paper Checklist 1. 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4,491	Probabilistic Emulation of a Global Climate Model with Spherical DYffusion Salva R¨uhling Cachay UC San Diego Brian Henn Allen Institute for AI Oliver Watt-Meyer Allen Institute for AI Christopher S. Bretherton Allen Institute for AI Rose Yu UC San Diego Abstract Data-driven deep learning models are transforming global weather forecasting. It is an open question if this success can extend to climate modeling, where the complexity of the data and long inference rollouts pose significant challenges. Here, we present the first conditional generative model that produces accurate and physically consistent global climate ensemble simulations by emulating a coarse version of the United States’ primary operational global forecast model, FV3GFS. Our model integrates the dynamics-informed diffusion framework (DYffusion) with the Spherical Fourier Neural Operator (SFNO) architecture, enabling stable 100-year simulations at 6-hourly timesteps while maintaining low computational overhead compared to single-step deterministic baselines. The model achieves near gold-standard performance for climate model emulation, outperforming existing approaches and demonstrating promising ensemble skill. This work represents a significant advance towards efficient, data-driven climate simulations that can enhance our understanding of the climate system and inform adaptation strategies.1 1 Introduction Climate models are foundational tools used to understand how the Earth system evolves over long time periods and how it may change as a response to possible greenhouse gas emission scenarios. Such climate simulations are currently very expensive to generate due to the computational complexity of the underlying physics-based climate models, which must be run on supercomputers. As a result, scientists and policymakers are limited to exploring only a small subset of possibilities for different mitigation and adaptation strategies [48]. Figure 1: Weather performance (xaxis) is not a strong indicator of climate performance (y-axis). Each dot corresponds to a distinct sample or checkpoint epoch. Training relatively cheap-to-run data-driven surrogates to emulate global climate models could provide a compelling alternative [15]. Although recent deep learning models are on the verge of transforming the conceptually similar field of medium-range weather forecasting [5, 38, 11, 51], these advances do not directly transfer to long-term climate projections [37]. Indeed, most such models only report forecasts up to two weeks into the future and may diverge or become physically inconsistent over longer simulations. In contrast, climate projections demand accurate and stable simulations of the global Earth system spanning decades or centuries, requiring reliable reproduction of long-term statistics. 1Code is available at https://github.com/Rose-STL-Lab/spherical-dyffusion 38th Conference on Neural Information Processing Systems (NeurIPS 2024). In Figure 1 we quantitatively show this divergence between the medium-range weather forecasting skill of ML models (measured as the average RMSE on 5-day forecasts) and their performance on longer climate time scales (measured as the RMSE of the 10-year time-mean). We have verified that this finding holds regardless of the analyzed variable and the proxy used for weather performance, which we discuss in more detail in Appendix E.3. Heuristically, optimizing weather skill ensures that a climate model takes a locally accurate path around the climate ’attractor’, but it does not guarantee that small but systematic errors may not build up to distort that simulated attractor to have biased long-term climate statistics. While this is a little-discussed observation in the ML community, the climate modeling community has documented it for physics-based models [17, 54]. A recent breakthrough is a deterministic surrogate called ACE (Ai2 Climate Emulator) [67], which remains remarkably stable and physically consistent over 10-year simulations at 6-hourly time steps, forced by time-varying specified sea-surface temperature and sea-ice. Its success can be attributed to careful data processing, problem design, and the Spherical Fourier Neural Operator (SFNO) [8] architecture. ACE is trained to emulate the United States’ primary operational global forecast model, the physics-based FV3GFS [73], which is operationally used at the US National Weather Service and US National Centers for Environmental Prediction. ACE produces encouragingly small ten-year mean climate biases (i.e. biased long-term averages), but they are still significantly larger than the theoretical minimum imposed by internal variability of the reference physics-based model. ACE’s deterministic nature restricts its ability to model the full distribution of climate states or to facilitate ensemble simulations, which involve drawing multiple samples from the same model. These capabilities are crucial for climate modeling, as they enable better uncertainty quantification, more robust and physically consistent predictions, and a deeper understanding of potential future climate scenarios and associated risks [32]. While it is possible to ensemble a deterministic model by perturbing its inputs, this approach often leads to under-dispersed (i.e. overly confident) ensembles compared to generative or physics-based approaches [57]. Even then, the problem remains that due to optimizing them on MSE-based loss functions, the deterministic predictions may degrade to a mean prediction for longer forecast time scales and underestimate unlikely events [9]. A generative modeling approach, particularly the use of diffusion models [59, 25], appears to be a promising solution to these challenges. However, standard diffusion models are computationally intensive to train and sample from. This complexity poses significant problems for climate modeling because: 1) atmospheric data is extremely high-dimensional, making the use of video diffusion models [63, 27, 69, 58, 26, 23] prohibitive, even more so as this class of models still struggle with videos longer than a few seconds; and 2) the sampling speed of standard diffusion models is particularly problematic for long, sequential inference rollouts. For instance, generating a single 10year-long simulation, as in our experiments, with a standard autoregressive diffusion model [35, 51] that uses N diffusion steps would require 14600×N neural network forward passes. If a second-order solver for sampling is used [31, 51], this number doubles. Even with N as small as 30, this results in half a million forward passes to generate a single sample trajectory, severely limiting the potential of data-driven models to serve as fast surrogates for expensive physics-based models. As a solution to this computational problem, we build upon the dynamics-informed diffusion model framework, DYffusion, from R¨uhling Cachay et al. [57], which caps the computational overhead at inference time (as measured by the number of neural net forward passes) to less than 3× as much as for a deterministic next-step forecasting model such as SFNO or ACE. Unfortunately, the original DYffusion method relies on an UNet-based architecture designed for Euclidean data rather than physical fields on a sphere. As we show in Figure 1, this mismatch of inductive biases becomes more problematic at the long climate time scales that we focus on in this paper. We address these limitations by carefully integrating the DYffusion framework with the SFNO architecture from Bonev et al. [8], and the data and evaluation procedure from Watt-Meyer et al. [67]. To achieve this integration, we extend SFNO with time conditioning and inference stochasticity modules. Our proposed framework, Spherical DYffusion, achieves strong results: On average, across all 34 predicted fields, our model reduces climate biases to within 50% of the reference model, which is more than 2× and 4× lower than the best baselines. For critical fields, such as the derived total water path quantity, our method achieves results within 20% of the reference model, representing a 5× improvement over the next best baseline (see Fig. 2). Additionally, our method proves effective for ensemble climate simulations, reproducing climate variability consistent with the reference model and further reducing climate biases towards the theoretical minimum through ensemble-averaging. 2 Figure 2: RMSE of 10-year time-means for a subset of important fields. The leftmost bar in the first two subplots shows the reference noise floor, determined by comparing ten independent 10-year reference FV3GFS simulations with the validation simulation. The scores computed using the mean over these ten simulations (a proxy for an ”ensemble prediction”) are shown in light shade. The subsequent bars show the corresponding scores for our method and the deep-learning baselines, using a 25-member ensemble for the probabilistic methods (all except ACE, which only reports scores for its single deterministic prediction). Scores computed using the ensemble-mean prediction are shown in light shade. The dark shaded bar on top indicates the performance drop when using a single member’s prediction only, with error bars representing the standard deviation over the 25 different member choices. The rightmost subplot displays the average time-mean RMSE of the ML-based emulators relative to the reference across all 34 variables. On average, our method’s time-mean RMSEs are 50% higher than the noise floor, which is less than half the average RMSE of the next best method, ACE. When using the 25-member ensemble mean prediction, this reduces to 29.28%. Our generative model is a leap forward toward purely ML-based large ensemble climate projections that are both efficient and accurate. Our main contributions are: 1. We present the first conditional generative model for probabilistic emulation of a realistic climate model, with minimal computational overhead over deterministic baselines. 2. We carefully integrate two distinct frameworks, ACE and DYffusion, including additional modifications to the SFNO architecture such as time-conditioning modules. 3. We show that our integrated method performs considerably better than relevant baselines in terms of reduced climate biases, ensemble-based climate modeling, and consistent variability of the climate predictions. 4. We show that short-term weather performance does not necessarily translate to accurate reproduction of long-term climate statistics. 2 Related Work ML for weather and climate modeling. There are fundamental differences in weather and climate modeling. Climate refers to the average weather over long periods of time2. While weather forecasting focuses on short time scales in the order of days or weeks, climate modeling simulates longer periods of decades to centuries. Weather forecasting is primarily an initial-value problem, for which it is important to analyze short-term time-specific predictions. Climate modeling is primarily a boundarycondition (or forcing-driven) problem [65], characterized by long-term averages and distributions. Deep learning-based models have emerged as a much more computationally efficient alternative to traditional physics-based numerical weather prediction (NWP) models, showing impressive skill for deterministic medium-range weather forecasting [49, 33, 5, 8, 10, 47, 38]. This success has been more recently extended to ensemble-based probabilistic weather forecasting [34, 51]. An alternative approach is hybrid modeling, where a physics-based component is complemented by ML-based parameterizations or corrections [52, 71, 56, 1, 36, 70, 34]. At longer lead times, when weather becomes chaotic and less predictable, the ensemble mean prediction of a physics-based or probabilistic ML-based ensemble improves deterministic metrics such as root mean squared error (RMSE) over non-ensembled methods [51, 34, 53]. However, advances in weather forecasting hardly transfer to long-term climate projections. Fully datadriven models fail to maintain stability beyond two-week-ahead forecasts, as errors accumulate over their autoregressive rollouts. Weyn et al. [68] and Bonev et al. [8] showed stable forecasts for horizons of up to six weeks and one year, respectively. Only recently, Watt-Meyer et al. [67] notably achieved stable and accurate 10-year simulations, followed by another deterministic SFNO-based climate emulator showing promising results using four prognostic variables [22]. Easier, but less flexible and 2For example, see https://oceanservice.noaa.gov/facts/weather climate 3 informative, alternatives to full-scale temporal modeling of atmospheric dynamics, include emulation of annual means given an emission scenario [66, 30, 46, 42], temporal super-resolution of monthly means [4], or debiasing climate model output [3, 7, 45]. Diffusion models. Diffusion models [25, 59–61] have demonstrated significant success in generating data such as natural images and videos. While traditionally formulated for finite-dimensional spaces, these models have been extended to function spaces [40]. Their direct applications to autoregressive forecasting [35, 51] and downscaling [64, 43, 20] of physical data have shown promising results. However, these approaches inherit the computational complexity associated with training and sampling from standard diffusion models. This is particularly prohibitive for autoregressive predictions on climate time scales, as the total number of neural network forward passes increases proportionally with the number of sampling steps, typically ranging from 20 to 1000. Consequently, recent research that leverages insights from diffusion models to balance predictive performance and sampling speed appears more promising for assessing their viability in climate simulations [57, 41]. While diffusion models traditionally rely on U-Net architectures [55, 13], vision transformers have shown promising results in image synthesis [50, 29, 24]. Our work explores a different, neural operator-based, architecture for Earth data. 3 Background We first define the problem and then introduce the key components in our framework, namely DYffusion and SFNO. We abbreviate a time series of tensors y0, . . . , yt with y0:t. 3.1 Problem Setting Our goal is to learn the probability distribution P(x1:H \| x0, f 0:H) over a horizon of H time steps, conditional on initial conditions x0 and a scenario of forcing variables f 0:H (i.e. time-varying boundary conditions). In our paper, these forcings correspond to prescribed sea surface temperatures and incoming solar radiation (see Section 5.1), leaving it to future work to force based on greenhouse gas emission scenarios explicitly. Each xt ∈RD represents the state of the atmosphere at a given timestep, t, consisting of two- and three-dimensional surface and atmospheric variables across a latitude-longitude grid. These variables, which serve as both input and output, are referred to as prognostic variables. We assume a constant time interval between successive time steps t and t + 1. To make training feasible, it is necessary to train on a much shorter horizon h, i.e. learn the distribution P(xt+1:t+h \| xt, f t:t+h), and apply the model autoregressively. This process begins with P(x1:h \| x0, f 0:h) and continues until reaching time step H at inference time. 3.2 Diffusion Models and DYffusion Diffusion models can be seen as a general paradigm to learn the target distribution p(s(0)), by iterating over N diffusion steps of a forward or reverse process. We denote the states of each diffusion step with s(n), using a superscript n to clearly distinguish them from the physical time steps of the data xt. Standard diffusion models [59, 25, 31], initialize the reverse process from a simple isotropic Gaussian distribution s(N) ∼N(0, I) so that as n →0 the intermediate states s(n) are gradually denoised towards a real data sample s(0). In Cold Diffusion [2], this paradigm is extended to more general data corruption processes such as blurring. R¨uhling Cachay et al. [57] propose DYffusion, by adapting cold diffusion models to forecasting problems. The key idea is to make the forward and reverse processes dynamics-informed by directly coupling them to the physical time steps of the data. That is, the reverse process is initialized with s(N) = x0 and iteratively evolves jointly with the dynamics of the data x1, . . . , xh−1 to reach the data at some target time step, s(0) = xh. In DYffusion, the forward and reverse processes are informed by temporal dynamics in the data and do not rely on data corruption. Their only source of stochasticity comes from using a stochastic neural network as an operator for the forward process and is implemented by using Monte Carlo (MC) dropout [19]. This forward process essentially corresponds to a temporal interpolator network, while the reverse process is represented by a multi-step forecasting network. Thus, compared to standard diffusion models, DYffusion requires training one more neural network, which they propose doing in separate stages, beginning with the interpolator model. Due to its dynamics-informed 4 Figure 3: The diagram shows how our proposed approach functions at inference time. Given an initial condition xt and forcings f t:t+h, our method uses the DYffusion framework, integrated with two SFNO backbone networks, to generate predictions for the next h time steps based on an alternation of direct multi-step forecasts and temporal interpolations. To simplify the visualization, we exclude the facts that the interpolator network, SFNOϕ, is conditioned on xt and f t in addition to an estimate of xt+h. We also exclude the time-conditioning of both networks. To forecast more time steps beyond t + h, our method is applied autoregressively. Figure 4: Diagram of one of the blocks of the modified SFNO architecture for our proposed method. The full architecture consists of a sequence of 8 such blocks. Our newly introduced time-conditioning modules correspond to the Time Embedding, followed by the MLP on the right, and the scale-shift operation. Our method relies on dropout, which is part of the two-layer MLP on the top. SFNO-based baselines use the same architecture and hyperparameters without the time embedding module. nature, DYffusion was shown to be faster at sampling time and more memory-efficient than standard diffusion models, while matching or outperforming their accuracy. 3.3 Spherical Fourier Neural Operator (SFNO) The SFNO architecture [8] extends the FNO framework from Li et al. [39] to spherical data and symmetries such as the Earth. FNOs efficiently model long-range interactions in the Fourier space, but because the underlying Fast Fourier Transform is defined on a Euclidean domain, this can lead to modeling artifacts. SFNOs overcome this issue by using the spherical harmonic transform (SHT) [14], a generalization of the Fourier transform, instead. The SFNO model achieves higher long-term stability of autoregressive rollouts than the FNO model, showing stable forecasts of Earth’s atmospheric dynamics for up to 1-year-long rollouts at six-hourly time steps. The ACE model from Watt-Meyer et al. [67] is based on the SFNO architecture, modifying some of the hyperparameters and the grid used for the first and last SHT of the SFNO. We use the SFNO configuration from ACE in our experiments. 4 Spherical DYffusion SFNO and ACE are deterministic models that cannot be readily used for uncertainty quantification or ensemble-based climate modeling. DYffusion introduces an efficient diffusion-based approach specifically for forecasting problems but only for Euclidean data. Thus, we propose Spherical DYffusion, a deep generative model for data-driven probabilistic climate simulations that carefully integrates SFNO and DYffusion into an unified framework. DYffusion requires two neural networks that are used for temporal interpolation and direct multi-step forecasts. In the original framework, these are UNet-like networks. For our approach, we propose to replace them with modified versions of the SFNO architecture, which we denote by SFNOϕ and SFNOθ, respectively. Training. We follow the original training procedure from DYffusion, complementing it with the use of the input-only forcing variables. That is, for a specified training horizon h, these networks are 5 trained in two stages such that for sequences of prognostic data xt:t+h and forcings f t:t+h SFNOϕ (xt, xt+h, f t, i \|ξ) ≈xt+i SFNOθ(SFNOϕ (xt, xt+h, f t, j \|ξ) , f t+j, j) ≈xt+h, where i ∈{1, . . . , h −1} and we use j ∈{0, 1, . . . , h −1}, defining SFNOϕ (xt, ·, ·, 0 \|ξ) = xt. In our experiments, we use h = 6. Here, ξ refers to the random variable representing the interpolator network’s inference stochasticity. We discuss its implementation further below. The forecaster network, SFNOθ, is deterministic. The full training scheme is defined in Algorithm 1. Inference. At inference time, we follow the DYffusion sampling scheme based on cold sampling [2]. Essentially, we start with the initial conditions x0 to generate a first forecast of time step h through a forward pass of the forecaster network, i.e. ˆxh = SFNOθ(x0, f 0, 0). Given this prediction, we can now use the interpolator network to interpolate ˆx1 = SFNOϕ (x0, ˆxh, f 0, 1 \|ξ). In practice, cold sampling applies a correction term to this estimate. The prior forecast of xh can now be refined with ˆxh = SFNOθ(ˆx1, f 1, 1). The alternation between forecasting and interpolation continues until SFNOϕ predicts ˆxh−1 and the forecaster network performs a last refinement forecast of time step h, conditioned on the time j = h −1 and interpolated sample ˆxh−1. After this final forecast of xh, the process is repeated autoregressively, starting with xh as the new initial condition. This slightly simplified sampling process is illustrated in Figure 3 and fully described in Algorithm 2. Repeating this sampling process multiple times using the same initial conditions will lead to an ensemble of samples, thanks to the interpolator network being stochastic. SFNO time-conditioning. To use SFNO as described above, it is necessary to implement timeconditioning modules that allow the interpolator and forecaster networks to be conditioned on the time i and j, respectively, given that the original SFNO architecture does not support this. We follow the same approach taken by standard diffusion models [13], which consists of transforming the time condition into a vector of sine/cosine Fourier features at 32 frequencies with base period 16, then pass them through a 2-layer MLP to obtain 128-dimensional time encodings that are mapped by a linear layer into the learnable scale and offset parameters. We scale and shift the neural representations of every SFNO block directly following the normalization layer and preceding the application of the SFNO spectral filter, as shown in Figure 4. SFNO inference stochasticity. A stochastic interpolator network, made explicitly through the random variable ξ above, was shown to be a key design choice in the original DYffusion framework. However, to the best of our knowledge, the SFNO model has been only used for deterministic modeling. We overcome this issue through MC dropout [19], i.e. enabling dropout modules [62] at inference time. Following the original SFNO implementation (of training-time-only dropout), we propose to use a dropout module inside the MLP of each SFNO block. In addition, we enable stochastic depth [28]–also known as drop path–at inference time at a rate of 0.1. Stochastic depth randomly skips a whole SFNO block. When this happens the whole block reduces to the identity function, since only the residual connection is enabled. To the best of our knowledge, this has not been explored before as a source of inference stochasticity. 5 Experiments 5.1 Dataset and Experimental Setup To compare our proposed method against ACE [67], we use the same dataset, training and evaluation setup. The dataset consists of 11 distinct 10-year-long simulations from the state-of-the-art global atmospheric model FV3GFS [73], saved every 6 hours. The forcings consist of annually repeating climatological sea surface temperature (1982-2012 average) and incoming solar radiation. Greenhouse gas and aerosol concentrations are kept fixed. The data was regridded conservatively from the cubedsphere geometry of FV3GFS to a 1◦Gaussian grid, and filtered with a spherical harmonic transform round-trip to remove artifacts in the high latitudes. We train on 100 years of simulated data from FV3GFS, and evaluate the models on how well they can emulate a distinct 10-year-long validation simulation (i.e. H = 14600 = 10 × 365 × 4). The 11 simulations form an initial-condition ensemble, where each simulation is independent of the other–after some discarded spinup time–due to the chaoticity of the atmosphere [32]. For more details, see Appendix B. 6 5.2 Baselines We compare with the following baselines for climate projection. • ACE [67] applied the SFNO architecture to the FV3GFS dataset described above. • ACE-STO: We re-train ACE but use MC dropout, in the same way how it is applied in SFNOϕ for our method, to generate stochastic predictions. • DYffusion [57]: We train DYffusion using the original UNet-based architecture as its interpolator and forecaster neural networks. • Reference [73]: physics-based FV3GFS climate model simulations. We use the ten training simulations to create a 10-member reference ensemble that we use to more robustly estimate the ‘noise floor’ introduced in [67] and to compare the variability of the reference ensemble with sample simulations from our method. Note that this reference is not appropriate for weather forecasts given that it is initialized from different initial conditions. It is worth noting that ACE also compared their results against a physics-based baseline called C48, which corresponds to running FV3GFS at half the original spatial resolution. This makes C48 around 8× less computationally costly to run compared to the reference simulations but was shown to underperform ACE, which our method is shown to outperform in the experiments below. For ACE, we directly use the pre-trained model from the original paper. ACE was trained on a next-step forecasting objective based on a MSE loss. For ACE-STO, we re-train ACE from scratch with the only difference being that we use a dropout rate of 10% for the MLP in the SFNO architecture. We use the same dropout rate for the interpolator model, SFNOϕ, in our method. For both DYffusion and our approach, we choose h = 6. That is, these models are trained to forecast up to 36 hours into the future. We use the same training and sampling procedures for both, the only difference being the underlying neural architectures. Table 1: Computational complexity of the different deep learning methods in terms of: 1) the number of neural function evaluations (NFEs) needed to predict h time steps. and 2) Total inference runtime (simulating 10 years), including the time needed to compute metrics (in hours:minutes). N refers to the number of diffusion steps which usually ranges between 20 to 1000. Method NFE Runtime ACE / SFNO h 01:08 Standard diffusion Nh N/A Ours 3(h −1) 02:56 Physics-based FV3GFS N/A 78:04 FV3GFS (2× coarser) N/A 45:38 Runtime analysis. In Table 1, we report the computational complexity in terms of the number of neural function evaluations (NFEs) needed to predict h time steps, and the wall clock runtime for simulating one complete validation trajectory of 10 years. For our method, NFEs is not 3h because in the first and last iteration we do not need to actually run line 8 and lines 7 & 8 in Algorithm 2, respectively. Our runtime analysis confirms that the computational overhead at inference time for using our method, is less than 3× as much as for a deterministic next-step forecasting model such as SFNO or ACE. This enables our method to provide significant 25× speed-ups and associated energy savings over using the emulated physics-based model, FV3GFS. All models were trained on A6000 GPUs using distributed training on 2 up to 8 GPUs, ensuring that the effective batch size remains the same (see Figure 8). For a fair inference runtime comparison measuring the wall clock time needed to simulate 10 years (i.e. one full validation rollout), we run all deep-learning baselines on one A100 GPU. We also include the runtime for the physics-based FV3GFS climate model which was run on 96 cores (24 cores for the 2× coarser version) of AMD EPYC 7H12 processors. The deep learning methods are not only much faster, but also much more energy-efficient than FV3GFS. For illustrative purposes, we also report the complexity of a standard autoregressive diffusion model [25, 35, 51] approach in terms of the number of neural function evaluations (NFEs) needed to predict h time steps, totaling to Nh where N is the number of sampling steps required to reverse the diffusion process. N usually ranges from between 20 to 1000. This makes the use of such an approach less attractive for climate emulation since the resulting inference runtime would not offer as significant speed-ups over the physics-based reference model. 7 Figure 5: Global maps of the 10-year time-mean biases of a single sample from the reference noise floor simulation, our model, and the ACE baseline for the total water path field. Each subplot reports the global mean RMSE and bias of the respective bias map. Our model reproduces biases of similar location and magnitude to the reference noise floor, suggesting they are mainly due to internal climate variability rather than model bias, while the baseline exhibits larger climate biases. 5.3 Climate Biases Metrics. The most crucial quality of an ML-based climate model is its ability to reproduce the climatology of the emulated reference system, i.e. the long-term average (“time-mean”) of weather states. The time-mean of the validation simulation is defined as 1 H PH t=1 xt. The time-mean for each model is defined as 1 H PH t=1 ˆxt, where ˆxt is the model’s prediction for time step t. These two quantities are then compared against each other using the bias, i.e. prediction - target, and root mean squared error (RMSE) as key metrics of interest for analyzing climate biases. For the probabilistic methods, i.e. ours, DYffusion, and ACE-STO, we generate simulation ensembles by sampling from the model multiple times using the same initial conditions. Unless specified otherwise, all ensemble results are based on E = 25 ensemble members. We evaluate the ensemble performance using two metrics: the RMSE of the ensemble-mean prediction ( 1 EH PE e=1 PH t=1 ˆxt,e) and the RMSE of member-wise time-means ( 1 H PH t=1 ˆxt,e), where e indexes individual ensemble members. For the latter, standard deviations are computed over the member-wise errors. The corresponding “optimal noise floor” for the ML-based emulators is estimated by comparing the validation simulation with the 10-member reference ensemble. All metrics, which are fully defined in Appendix D, are weighted by grid cell area. It is important to acknowledge the potential for improving the estimate of the “noise floor” based on statistical significance testing and improved metrics [21]. Quantitative analysis. Our method and all baselines consistently produce stable long-term climate simulations without diverging. In Figure 2, we compare the RMSE of the time-means of the reference, our method, and all baselines. Our method significantly reduces climate biases compared to baseline methods across most fields, with errors often closer to the reference simulation’s noise floor than to the next best baseline. The performance of ACE is notably degraded when made stochastic through MC dropout. Similarly, a direct application of DYffusion fails to accurately reproduce long-term climate statistics. Both these baselines are unable to outperform or even match the scores of the deterministic ACE baseline. Only our proposed careful integration of these two paradigms leads to a skillful climate model emulator: On average, our method’s time-mean RMSEs are only 49.36% higher than the noise floor, which is less than half the average RMSE (110.47%) of the next best method, ACE. Ensemble averaging significantly enhances our method’s performance, reducing climate biases by 29.28% on average across all variables. As shown by the light shading in Fig. 2, the ensemble-mean predictions consistently achieve lower time-mean RMSEs compared to single-member predictions (dark shading). This ensemble-based improvement distinguishes our approach from ACE-STO and DYffusion, where ensemble averaging proves less effective, and from ACE, where initial-condition perturbations would be required for ensembling. Additional results for more fields are available in Figure 9 of the Appendix. Our comprehensive evaluation in Table 4 includes ensemble metrics such as the Continuous Ranked Probability Score (CRPS) and spread-skill ratio. The results demonstrate that our method outperforms alternatives in emulating the 10-year time-mean climatology of the reference model for most variables and metrics. However, some challenges remain, particularly in matching the reference ensemble’s performance for stratospheric (level 0) variables and in achieving better ensemble scores. 8 Qualitative analysis. In Figure 5 we show the corresponding global maps of the time-mean biases for the total water path (TWP) field. Our model reproduces small biases of remarkably similar location and magnitude to the ”perfect-model” reference simulation, with spatial pattern RMSEs of approximately 1% of the global-time-mean TWP. The perfect-model bias is due to unforced random decadal variability in the mean climate of the reference model - each 10-year period has randomly different weather, leading to a slight difference in 10-year time-mean averages across this weather. The reference bias is due to comparing one such decade simulated with the reference model with other simulated decades; its spatial pattern depends strongly on which decade is used for computing the reference model climatology. That our model (trained on 100 years of output) reproduces this pattern suggests that it emulates the long-term (e.g. century-long) time-mean statistics of the reference model even more accurately than a 10-year-mean RMSE can reliably resolve. On the other hand, the baseline ACE model exhibits somewhat larger climate biases, indicative of an actual, albeit small model deficiency that is already evident with a single 10-year estimate of climatology. In Appendix E.4, we visualize two sample 10-year trajectories simulated by Spherical DYffusion as well as the corresponding validation simulation from FV3GFS. Supplementary videos demonstrate the full temporal evolution of key derived variables: near-surface wind speed3 and total water path4. The emulated fields demonstrate high realism, closely mimicking the patterns and variability observed in actual climate model outputs. This showcases Spherical DYffusion’s capability to generate plausible and physically consistent climate scenarios over decadal timescales. Table 2: Global area-weighted mean of the spread of an ensemble of 10-yr time-mean’s for surface pressure, total water path, air temperature, zonal wind, and meridional wind (the last three at the near-surface level). The climate variability of our method is consistent with the reference model. Model ps TWP T7 u7 v7 Reference 19.96 0.199 0.090 0.142 0.110 Ours 23.52 0.214 0.094 0.167 0.121 DYffusion 24.75 0.223 0.082 0.169 0.127 ACE-STO 30.32 0.256 0.135 0.192 0.131 Climate variability. Above, we have verified that sampling 10-yearlong trajectories from our model produces encouragingly low ensemble mean and member-wise time-mean biases. An important feature of climate is its natural variability on time scales of years, decades, or even centuries even when external forcings (e.g. sunlight or greenhouse gas concentrations) remain unchanged. For instance, multi-decadal periods of relative drought have stressed many past human civilizations. The present simulations are more constrained than natural climate variability because they employ a repeating cycle of sea-surface temperature and thus do not allow for feedbacks between the atmosphere, ocean, vegetation, and cryosphere. Nevertheless, an important quality of an ML emulator of the global atmosphere suitable for climate studies is that it simulates a similar level of low-frequency climate variability as the reference model. Here, we verify that our time-mean ensemble passes this challenging test, measured using the intra-ensemble variability of time-mean averages of a few important climate statistics simulated by 25-member ensembles of the emulators vs. the ten reference simulations. We measure this variability by computing the area-weighted average of the standard deviation of time-means across the ensemble dimension. In Table 2 we show that the resulting global mean variability of the ensemble of time-means of our method is within 10-20% of those of the reference simulations for all tabulated variables (and other predicted fields). DYffusion achieves similarly accurate ensemble variability, while ACE-STO In Appendix E.1.2 we show that the corresponding global maps of the time-mean variability reveal similar spatial patterns. That is, our method generates ensemble climate simulations with decadal variability consistent with the underlying climate model. 100-year-long simulation. We evaluate the long-term stability of Spherical DYffusion through a 100-year simulation, a critical timescale for many climate modeling applications. Figure 6 demonstrates the model’s robustness through time series of key global mean variables from a single (random) simulation, which completed in approximately 26 hours of wall-clock time. The model generates physically consistent temporal patterns in response to annually repeating forcings. Notably, Spherical 3https://youtu.be/7lHra7gBiBo 4https://youtu.be/Hac xGsJ1qY 9 Figure 6: Comparison of 100-year global mean simulations between Spherical DYffusion and ACE. From top to bottom: near-surface air temperature (T7), total water path (TWP), and surface pressure (ps). Both models are driven by identical annually repeating forcings. Spherical DYffusion demonstrates more stable trajectories, particularly evident in the surface pressure predictions, while maintaining physically realistic variability patterns. The consistent behavior across all variables indicates the model’s robustness for long-term climate simulations. DYffusion exhibits improved variability patterns compared to the baseline ACE model, which suffers from unrealistic annual fluctuations (e.g. see surface pressure). 6 Conclusion We introduce Spherical DYffusion, a novel approach that combines efficient diffusion modeling with a spherical-aware neural architecture to probabilistically emulate complex global climate dynamics across decadal to centennial timescales. Our model achieves lower climate biases than relevant deterministic and probabilistic baselines, getting significantly closer to the optimal performance provided by the emulated climate model. For climate model emulation problems, our approach presents a unique solution for balancing generative modeling, computational efficiency, and low climate biases. This opens up the ability to perform fully data-driven ensemble climate simulations. Limitations. To achieve real-world impact, the dataset will need to be expanded so that ML emulators can be evaluated (and trained) on climate change scenarios/simulations. This will require using time-varying climate change forcings such as greenhouse gas and aerosol concentrations. Although our use of the state-of-the-art FV3GFS atmospheric model enables generation of such training data, any emulator will inherently reflect biases present in the base model. Additionally, we only considered emulating the atmosphere, but to achieve a full Earth System Model (ESM) we also need to emulate (or couple to a physics-based model of) other components such as ocean, land, sea-ice, etc. It is important to stress that while our method is more than 25× faster than the reference physics-based climate model, it is still slower than deterministic emulators such as ACE. Though our method characterizes model uncertainty through its generative design, extending it to incorporate initial condition uncertainty—a key component of traditional ensemble physics-based models—could further enhance its capabilities. The method also needs extension to handle output-only variables like precipitation, either through dedicated prediction heads or modifications to the DYffusion framework. 10 Acknowledgements This work was supported in part by the U.S. Army Research Office under Army-ECASE award W911NF-23-1-0231, the U.S. Department Of Energy, Office of Science, IARPA HAYSTAC Program, CDC-RFA-FT-23-0069, DARPA AIE FoundSci, DARPA YFA, NSF Grants #2205093, #2100237,#2146343, and #2134274. S.R.C. acknowledges generous support from a summer internship and subsequent collaboration with the Allen Institute for AI (Ai2), which is primarily funded by the estate of Paul G. Allen. 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Bulletin of the American Meteorological Society, 100(7):1225 – 1243, 2019. doi:https://doi.org/10.1175/BAMS-D-17-0246.1. 2, 6, 7, 18 16 Appendix A Broader Impact The goal of this work is to advance the application of machine learning to climate modeling, specifically for generating fast and cheap ML-based climate simulations. This could significantly democratize climate modeling, improve scientific understanding of the earth system, and enhance decisionand policy-making in a changing climate. However, to realize this goal, the reliability and limitations of such ML models will need to be much better understood. B Dataset In the subsections below, we elaborate on the dataset and variables that we use, including background information on FV3GFS and how it was configured in order to generate the training and validation data. Any listed data preprocessing steps below are also described in appendix A from [67]. The final training and validation data can be downloaded from Google Cloud Storage following the instructions of the ACE paper at https://zenodo.org/records/10791087. The data are licensed under Creative Commons Attribution 4.0 International. B.1 Input, output and forcing variables Table 3: Input and output variables used in this work. The table was adapted based on Table 1 of [67]. The k subscript refers to a vertical layer index and ranges from 0 to 7 starting at the top of the atmosphere and increasing towards the surface. The two prognostic surface variables, Ts and ps, do not have this additional vertical dimension. Each of their snapshots is a 2D latitude-longitude matrix. The Time column indicates whether a variable represents the value at a particular time step (“Snapshot”), the average across the 6-hour time step (“Mean”), or a quantity that does not depend on time (“Invariant”). “TOA” denotes “Top Of Atmosphere”, the climate model’s upper boundary. Prognostic variables (input and output) Symbol Description Units Time Is 3D? Tk Air temperature K Snapshot Yes qT k Specific total water (vapor + condensates) kg/kg Snapshot Yes uk Wind speed in eastward direction m/s Snapshot Yes vk Wind speed in northward direction m/s Snapshot Yes Ts Skin temperature of land or sea-ice K Snapshot No ps Atmospheric pressure at surface Pa Snapshot No Forcing variables (input-only) Symbol Description Units Time DSWRFT OA Downward shortwave radiative flux at TOA W/m2 Mean Ts Skin temperature of open ocean K Snapshot Additional input-only variables Symbol Description Units Time zs Surface height of topography m Invariant fl Land grid cell fraction − Invariant fo Ocean grid cell fraction − Snapshot fsi Sea-ice grid cell fraction − Snapshot Derived, evaluation-only, variables Symbol Description Units Time Is 3D? WSk Wind speed m/s Snapshot Yes TWP Total water path mm Snapshot No 17 The complete list of input, output, and forcing variables used in this work is given in Table 3. The only difference to the work from [67] is that we do not consider diagnostic (output only) variables. The forcings consist of annually repeating climatological sea surface temperature (1982-2012 average), Ts, and incoming solar radiation, DSWRFT OA. Prescribed sea surface temperatures are simply “overwritten” on the skin temperature predictions of the ML models over all open ocean locations (when rolling out the ML-based simulation). The other forcing or input-only variables are added as an additional channel dimension. Derived variables are computed from the (predicted) prognostic variables as described below. Derived variables. For evaluation, we also consider the derived variable called total water path which is computed as TWP = 1 g P k qT k dpk, i.e. as a function of surface pressure and the profile of specific total water. Its units are mm (or kg/m2, assuming that water has a density of 1000 kg/m3). The derived wind speed variable for level k is computed based on the simulated meridional and zonal wind variables as WSk = p u2 k + v2 k. Its units are m/s. B.2 Background on FV3GFS Our dataset and physics-based baselines (including our “noise-floor” reference baseline) are based on simulations from a comprehensive global atmospheric model called Finite-Volume on a CubedSphere Global Forecasting System (FV3GFS) [73]. It was developed by the National Oceanic and Atmospheric Administration (NOAA) Geophysical Fluid Dynamics Laboratory (GFDL)5. A very similar model version is operationally used by the US National Centers for Environmental Prediction (NCEP) and the US weather forecasting service6. Its scalability to horizontal grid spacings as fine as 3 km [12] makes it an excellent candidate for generating training data for future ML-based climate model emulators, including out-of-distribution climate change simulations that may be necessary to train ML emulators on so that they can generalize. B.3 FV3GFS configuration for data generation In the following we summarize the reference data for this study, as also discussed in Section 2.1 of [67]. The training and validation data is generated by running an ensemble of 11 10-year (after discarding a 3-month spinup period) FV3GFS simulations on a C96 cubed-sphere grid (approximately 100 km horizontal grid spacing) with 63 vertical levels. The simulations are an initial-condition ensemble. That is, they are identical except for using different initial atmospheric states. Initialcondition ensembles are a popular tool in climate modeling [32]. Discarding a 3-month spinup period ensures that each simulation is independent of each other due to the chaoticity of the atmosphere7. Each simulation is forced by repeating annual cycles of sea-surface temperature and insolation. The temperature, humidity, two wind components at each grid point, and selected vertical fluxes at the surface and top of the atmosphere in each grid column are saved every six hours. For ML training, the temperature, humidity, and two wind components are averaged along FV3GFS’s 63 levels to 8 vertical layers, and the data are interpolated to a latitude-longitude grid of 180 × 360 dimensions. C Implementation details All methods and baselines are conditioned on the forcings, f t, by simple concatenation of the forcings with the remaining input variables across the channel dimension. We use PyTorch Lightning [16] and Weights & Biases [6] as part of our software stack. C.1 Training and inference pseudocode In Algorithms 1 and 2 we provide the procedures used to train and sample from our proposed method, respectively. 5https://www.gfdl.noaa.gov/fv3/ 6https://www.weather.gov/news/fv3 7E.g. see Kay et al. [32], who note that: “After initial condition memory is lost, which occurs within weeks in the atmosphere, each ensemble member evolves chaotically, affected by atmospheric circulation fluctuations characteristic of a random, stochastic process (e.g., Lorenz 1963; Deser et al. 2012b)”. 18 Algorithm 1 Spherical DYffusion, Training Input: networks SFNOϕ, SFNOθ, norm ∥·∥, horizon h = 6 Stage 1: Train interpolator network, SFNOϕ 1. Sample i ∼Uniform ({1, . . . , h −1}) 2. Sample xt, xt+i, xt+h ∼RD, and corresponding forcing f t 3. Sample network stochasticity (dropout), ξ 4. Optimize minϕ ∥SFNOϕ (xt, xt+h, f t, i \|ξ) −xt+i∥2 Stage 2: Train forecaster network, SFNOθ 1. Freeze SFNOϕ and enable its inference stochasticity ξ 2. Sample j ∼Uniform({0, . . . , h −1}) and xt, xt+h ∼RD 3. Retrieve corresponding forcings f t, f t+j 4. ˆxt+j ←SFNOϕ (xt, xt+h, f t, j \|ξ) # with ˆxt+j := xt for j = 0 5. Optimize minθ SFNOθ ˆxt+j, f t+j, j −xt+h 2 Algorithm 2 Spherical DYffusion, Inference 1: Input: Initial conditions ˆx0 := x0, training and inference horizon h and H = 14600, forcings f 0:H 2: # Autoregressive loop: 3: for t = 0, h, 2 · h, . . . , (⌈H/h⌉−1) · h do 4: # Sampling loop for time steps t + 1, . . . , t + h: 5: for j = 0, 1, . . . , h −1 do 6: ˆxt+h ←SFNOθ ˆxt+j, f t+j, j # (Refine) forecast 7: ˜xt+j+1 ←SFNOϕ (ˆxt, ˆxt+h, f t, j + 1 \|ξ) # Interpolate 8: ˆxt+j+1 = ˜xt+j+1 + ˆxt+j −SFNOϕ (ˆxt, ˆxt+h, f t, j \|ξ′) # Cold sampling 9: end for 10: end for 11: Return: ˆx1:H C.2 Discussion on the training horizon The training horizon, h, is a critical hyperparameter for both DYffusion and our proposed method. Throughout this study, we use h = 6 (corresponding to 36 hours) for both approaches. While we initially explored other horizons, we chose h = 6 as it strikes an optimal balance: A smaller horizon (e.g., h = 3) reduces the number of sampling steps since the reverse sampling process directly corresponds to physical time steps, potentially degrading performance. Conversely, a larger horizon makes the forecasting task more challenging, as predicting xt+h from xt becomes increasingly difficult for the forecasting model. Our choice is further supported by the DYffusion paper, which successfully used h = 7 for sea surface temperature forecasting. While we believe that values close to h = 6 would likely perform similarly well, comprehensive ablation studies would require re-training two neural networks sequentially, making such experiments computationally expensive to run. C.3 Hyperparameters Architectural hyperparameters. To fairly compare against the deterministic SFNO model from [67], we use exactly the same hyperparameters for training the interpolator and forecasting networks for our method, as described in Table 78. For the stochastic version of ACE, ACE-STO, we re-train ACE from scratch with the only difference being that we use a dropout rate of 10% for the MLP in the SFNO architecture. We train the stochastic interpolator model, SFNOϕ, in our method using the same dropout rate. Both of these stochastic models are run using MC dropout 8Names correspond to the definition of the SphericalFourierNeuralOperatorNet class found at: https://github.com/ai2cm/modulus/blob/94f62e1ce2083640829ec12d80b00619c40a47f8/ modulus/models/sfno/sfnonet.py#L292. Unless specified otherwise, defaults are used. 19 Figure 7: Table is directly taken from [67], and reports the SFNO hyperparameters used for ACE as well as the interpolator and forecasting networks of our method. Name Value embed_dim 256 filter_type linear num_layers 8 operator_type dhconv scale_factor 1 spectral_layers 3 Figure 8: Optimization hyperparameters. The effective batch size is calculated as data loader batch size × number of GPUs × number of gradient accumulation steps, and is ensured to be the same for all our trained models regardless of the number of GPUs used. Name Value Optimizer AdamW Initial learning rate 4 × 10−4 Weight decay 5 × 10−3 Learning rate schedule Cosine annealing Number of epochs 60 Effective batch size 72 Exponential moving average decay rate 0.9999 Gradient clipping 0.5 (i.e. enabling the dropout layers at inference time). For our interpolator network, we also use a 10% rate for stochastic depth [28], which is also enabled at inference time. This choice was informed by preliminary experiments focused on training a good interpolator network. There, we found the addition of stochastic depth to slightly improve the interpolator’s validation CRPS scores (for the interpolated timesteps 1 to 5) and significantly improve the calibration of the interpolation ensemble based on the spread-skill ratio (averaged across variables from around 0.26 to 0.35). We found worse results when using stochastic depth for ACE-STO at inference time. Optimization hyperparameters. We train the interpolator networks for DYffusion and our method on the same relative L2 loss function used for the baseline from [67], and the corresponding forecaster networks on the L1 loss. The models that we train on our own, i.e. the interpolation and forecasting networks of DYffusion and our method are trained with mixed precision. Inference is always run at full precision. For the non-interpolation networks, we perform early stopping based on the best CRPS averaged over a 500-step (125 days) rollout. More optimization-related hyperparameters are discussed in Table 8. D Metrics Unless specified otherwise, all ensemble results are based on E = 25 ensemble members. All metrics are area-weighted according to the size of the grid cell, as described below. D.1 Preliminaries Let X ∈RE×I×J denote an ensemble of predictions, and Y ∈RI×J the corresponding targets, where E is the number of ensemble members, I is the number of latitudes, and J the number of longitudes in the grid. In the context of this paper, Y usually corresponds to the validation, reference 10-year time-mean and X corresponds to an ensemble of 10-year time-means simulated by the reference climate model (excluding the validation time-mean), our proposed method, or any of the baselines. Let w(i) denote the normalized latitude-dependent area weights at latitude i, such that 1 I PI i w(i) = 1, which ensure that spatial means are not biased towards the polar regions (see e.g. Rasp et al. [53]). 20 D.2 Member-wise Metrics We report the average, member-wise area-weighted bias, Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), which are defined as follows Bias = 1 EIJ E X e=1 X i,j w(i)(Xe,i,j −Yi,j) (1) MAE = 1 EIJ E X e=1 w(i)\|Xe,i,j −Yi,j\| (2) RMSE = 1 E E X e=1 s 1 IJ X i,j w(i)(Xe,i,j −Yi,j)2 (3) For the bias, closer to zero is better, for the MAE and RMSE lower is better. D.3 Ensemble Metrics Ensemble-mean RMSE. For a skillful ensemble, the magnitude of the average, member-wise RMSE (see above) can be reduced by computing the RMSE on the ensemble-mean prediction, defined as ¯Xi,j = 1 E PE e=1 xe,i,j, instead. RMSEens = s 1 IJ X i,j w(i)( ¯Xi,j −Yi,j)2 (4) Spread-Skill Ratio (SSR). Following Fortin et al. [18], the spread-skill ratio is defined as the ratio between the ensemble spread and the ensemble-mean RMSE. The ensemble spread is defined as the the square root of the ensemble variance Spread = s 1 IJ X i,j w(i)vare(Xe,i,j), (5) where vare computes the variance of the ensemble.Then, we can compute the spread-skill ratio simply as SSR = r E + 1 E Spread RMSEens , (6) where q E+1 E is a correction factor which is especially important to include for small ensemble sizes. Note that this factor is omitted in e.g. WeatherBench-2 [53]. The SSR serves as a simple measure of the reliability of the ensemble, where values smaller than 1 indicate underdispersion (i.e. the model is overconfident in its predictions), and larger values overdispersion. That is, closer to 1 is better. Continuous Ranked Probability Score (CRPS). Following Zamo and Naveau [72], we use the unbiased version of the CRPS [44], which is a proper scoring rule: CRPS = 1 IJ X i,j w(i)  1 E E X e=1 \|Xe,i,j −Yi,j\| − 1 2E(E −1) E X e=1 E X f=1 \|Xe,i,j −Xf,i,j\|   (7) where the first term represents the skill, and the second term represents the spread. The biased CRPS averages over the spread with the factor 1 2E2 , which is biased–especially for small ensemble sizes–compared to using the unbiased version with the factor 1 2E(E−1). Note that common Python packages such as xskillscore and properscoring use the biased version. Lower is better. Note on deterministic models. For deterministic models like ACE without initial condition ensembling, the ensemble size is trivially E = 1, causing ¯Xi,j to be identical to X. This results in RMSEens reducing to standard RMSE, MAE equaling CRPS, and a zero spread-skill ratio. To accurately 21 Figure 9: Same as Figure 2 but showing more fields for the RMSE of 10-year time-mean’s. Bars (left to right) show 1) the noise floor calculated from the pairwise differences of ten independent 10-year reference model simulations with respect to the validation simulation. In light shade we report the score computed using the mean over the ten reference simulations as ”prediction”, 2) the memberwise scores of an 25-member ensemble of our method in dark shade, and the corresponding ensemble mean score in light shade, 3) the score of the deterministic ACE baseline, 4) the member-wise scores of an 25-member ensemble of the DYffusion baseline in dark shade, and the corresponding ensemble mean score in light shade. The standard deviation error bar is computed over the set of pairwise (member-wise) time-mean RMSEs for the reference (Ours and DYffusion). ACE does not have a standard deviation since it is a deterministic model. Turning ACE stochastic through MC dropout (ACE-STO) degrades its performance. Our method significantly reduces climate biases over the baseline methods and can be effectively ensembled to reduce its climate biases further, approaching the theoretical lower limit imposed by the noise floor of the reference simulation. reflect that ensemble metrics are not meaningful for single-member deterministic predictions, we denote these metrics as −for ACE in Table 4. While incorporating initial condition ensembling would enhance ACE’s performance on these metrics beyond the naive deterministic baseline, such techniques are orthogonal to the model-based ensembling approaches explored in this work. We leave this extension to future research, noting that initial condition ensembling could potentially improve results for all models in our comparison, including the inherently stochastic ones. E Additional results and figures E.1 Climate Biases We quantitatively analyze the 10-year time mean biases of our model and the baselines in terms of the global mean RMSE in Figure 9. The time-mean prediction is the average over the 14,600 predicted snapshots during the 10 years. Our method significantly reduces climate biases over the baseline methods across most fields. Notably, the errors of our method are often closer to the noise floor of the reference simulation than to the next best baseline. We also show that our method can be effectively ensembled to further reduce climate biases, its ensemble-mean reliably improving time-mean scores across all fields. Interestingly, the stochastic version of ACE, ACE-STO, significantly underperforms the deterministic version. Similarly, the direct application DYffusion fails to match the deterministic ACE baseline, even after ensemble averaging. This shows that MC dropout and DYffusion alone are not the reason for the encouraging performance of our method, but rather the holistic integration of all components, including MC dropout in our SFNO-based interpolator network. In Table 4, we report a comprehensive evaluation of the (ensemble) of 10-year time-means of each method for a subset of ten representative variables. We report the mean bias error, mean absolute error (MAE), root mean square error (RMSE), ensemble-mean RMSE, spread-skill ratio, and Continuous Ranked Probability Score (CRPS), which are rigorously defined in Appendix D. 22 Table 4: Comprehensive evaluation of simulated 10-year time-means. Bias, RMSE, and MAE represent average member-wise scores. For Bias (Spread-skill ratio; SSR) closer to 0 (1) is better. For the other metrics, lower is better, with relative changes from the reference shown in parentheses. See Appendix D for mathematical formulations and Table 3 for variable descriptions and units. Variable Metric Reference Ours ACE ACE-STO DYffusion TWP Bias 0.004 -0.043 0.021 0.017 0.686 RMSE 0.336 0.404 (+20%) 0.653 (+94%) 1.372 (+308%) 0.965 (+187%) RMSEens 0.249 0.327 (+31%) 1.206 (+385%) 0.934 (+276%) SSR 1.017 0.760 0.574 0.273 MAE 0.245 0.303 (+24%) 0.459 (+88%) 0.957 (+291%) 0.768 (+214%) CRPS 0.125 0.178 (+43%) 0.639 (+413%) 0.644 (+417%) ps Bias 0.036 4.820 45.47 34.82 -126.4 RMSE 39.37 48.79 (+24%) 103.5 (+163%) 131.1 (+233%) 151.5 (+285%) RMSEens 31.50 39.59 (+26%) 120.1 (+281%) 149.0 (+373%) SSR 0.847 0.766 0.470 0.190 MAE 26.26 35.60 (+36%) 71.69 (+173%) 93.14 (+255%) 134.8 (+413%) CRPS 14.44 21.91 (+52%) 66.48 (+360%) 121.6 (+742%) T7 Bias 0.011 -0.049 0.121 0.369 0.311 RMSE 0.172 0.290 (+69%) 0.349 (+103%) 0.831 (+383%) 0.692 (+302%) RMSEens 0.124 0.267 (+114%) 0.734 (+490%) 0.684 (+450%) SSR 1.065 0.474 0.634 0.158 MAE 0.108 0.187 (+73%) 0.224 (+108%) 0.510 (+373%) 0.408 (+278%) CRPS 0.054 0.132 (+147%) 0.343 (+540%) 0.360 (+573%) T5 Bias 0.005 -0.068 0.079 0.173 0.377 RMSE 0.171 0.244 (+42%) 0.333 (+94%) 0.610 (+256%) 0.540 (+215%) RMSEens 0.132 0.211 (+60%) 0.525 (+299%) 0.527 (+301%) SSR 0.933 0.619 0.657 0.228 MAE 0.117 0.171 (+46%) 0.243 (+108%) 0.451 (+286%) 0.388 (+232%) CRPS 0.060 0.110 (+84%) 0.299 (+402%) 0.330 (+455%) T0 Bias 0.000 0.034 0.162 -0.127 0.517 RMSE 0.124 0.220 (+78%) 0.444 (+259%) 0.767 (+520%) 0.592 (+379%) RMSEens 0.074 0.202 (+174%) 0.674 (+815%) 0.585 (+695%) SSR 1.533 0.517 0.599 0.161 MAE 0.084 0.150 (+78%) 0.316 (+277%) 0.550 (+555%) 0.526 (+527%) CRPS 0.034 0.102 (+200%) 0.348 (+921%) 0.481 (+1312%) u7 Bias 0.012 0.038 -0.170 -0.023 0.077 RMSE 0.240 0.307 (+28%) 0.456 (+90%) 0.935 (+289%) 0.462 (+92%) RMSEens 0.178 0.239 (+35%) 0.874 (+391%) 0.427 (+140%) SSR 1.012 0.846 0.412 0.438 MAE 0.173 0.226 (+31%) 0.343 (+98%) 0.693 (+300%) 0.339 (+96%) CRPS 0.087 0.129 (+48%) 0.519 (+494%) 0.249 (+185%) v7 Bias 0.005 0.015 0.009 0.044 -0.067 RMSE 0.196 0.224 (+14.3%) 0.299 (+53%) 0.592 (+202%) 0.320 (+64%) RMSEens 0.152 0.178 (+17.0%) 0.548 (+260%) 0.292 (+92%) SSR 0.910 0.802 0.439 0.471 MAE 0.138 0.164 (+18.6%) 0.224 (+62%) 0.440 (+218%) 0.247 (+79%) CRPS 0.072 0.094 (+30%) 0.325 (+351%) 0.179 (+148%) WS7 Bias 0.003 -0.053 -0.017 -0.080 -0.001 RMSE 0.243 0.303 (+24%) 0.437 (+79%) 0.886 (+264%) 0.450 (+85%) RMSEens 0.183 0.238 (+30%) 0.823 (+349%) 0.415 (+126%) SSR 0.976 0.830 0.430 0.445 MAE 0.175 0.224 (+28%) 0.331 (+89%) 0.659 (+277%) 0.334 (+91%) CRPS 0.089 0.128 (+44%) 0.488 (+449%) 0.244 (+175%) WS5 Bias 0.022 0.058 -0.104 -0.036 -0.081 RMSE 0.324 0.398 (+23%) 0.626 (+93%) 1.128 (+248%) 0.591 (+82%) RMSEens 0.240 0.311 (+30%) 1.030 (+329%) 0.543 (+126%) SSR 1.013 0.837 0.475 0.452 MAE 0.248 0.311 (+25%) 0.492 (+98%) 0.878 (+254%) 0.456 (+84%) CRPS 0.124 0.176 (+42%) 0.636 (+412%) 0.329 (+165%) WS0 Bias 0.151 -0.022 -0.167 0.854 1.642 RMSE 0.450 0.944 (+110%) 2.163 (+381%) 2.661 (+491%) 3.158 (+602%) RMSEens 0.307 0.887 (+189%) 2.349 (+664%) 3.142 (+922%) SSR 1.203 0.397 0.573 0.110 MAE 0.336 0.752 (+124%) 1.626 (+384%) 2.035 (+506%) 2.044 (+509%) CRPS 0.152 0.589 (+287%) 1.354 (+789%) 1.874 (+1131%) 23 E.1.1 Zonal time-means In this section, we analyze the absolute magnitudes of the simulated time-means by examining their zonal averages (aggregated over the longitude dimension). We also visualize the standard deviation of the respective ensembles of time- and zonal-means for the reference and stochastic methods. We visualize these in Figures 10 and 11. For several fields, including surface pressure, total water path (not shown), and near-surface temperature (top left subplot in Fig. 10), differences between the simulations are not visually noticeable, except for polar biases in baseline methods. However, discrepancies become pronounced in higher-altitude and wind fields, where our method generally achieves the closest agreement with the reference model. Although near-surface fields are the most relevant for society and decision-making, the clear biases of the baseline method at high-altitude levels might contribute to long-term biases, especially in longer simulations, due to the interactions of atmospheric dynamics across all levels. This observation may partly explain why our method achieves the lowest time-mean biases and RMSEs, as discussed in Appendix E.1. Figure 10: Zonal means of the simulated 10-year time-mean climatologies for a representative subset of four temperature fields. Level 7 represents near-surface conditions, while Level 0 corresponds to the highest altitude. Our method generally provides the closest emulation to the reference data. The most notable biases in the emulations occur at Levels 2 and 0, indicating greater discrepancies at higher altitudes. Emulation challenges are also significant near the poles, including at near-surface levels, particularly for DYffusion. Figure 11: Zonal-means of the simulated 10-year time-mean climatologies for a representative subset of four northward (meridional) wind fields. Our method generally provides the closest emulation to the reference data, except for the level-0 polar latitudes. 24 E.1.2 Climate variability In Fig. 12 we show the global maps corresponding to the global means of Table 2. Our method shows a consistent ensemble variability in terms of the simulated climate that also largely reflects the spatial patterns and magnitudes of the reference ensemble. Figure 12: Global maps of the standard deviation of the 10-year time-mean of the reference ensemble and a 25-member ensemble of our method. The climate variability of our method is consistent with the reference model, and largely follows similar spatial patterns with adequate magnitudes. The global mean standard deviation is reported in Table 2. E.2 Weather forecasting While we focus on climate time scales in this work, climate is formed by the statistics of weather, so it is important to verify that our method also generates reasonable forecasts of the weather simulated by the reference model. In Figure 13, we analyze the medium-range forecasting skill of our method and the baselines for lead times up to two weeks. Interestingly, ACE and DYffusion show persistent biases for the surface pressure field that are clearly visible from the first few days of forecasts already but do not seem to reflect on the RMSEs at weather time scales. Such persistent biases, however, may be magnified over longer simulations and could explain why the baselines have problems reproducing accurate long-term climate statistics. In terms of RMSE, the deterministic model ACE generally has a slight edge over our method and DYffusion, especially on lead times of less than a week. After that, 25 Figure 13: Comparison of medium-range weather forecasting skill between Spherical DYffusion (25-member ensemble and single forecast), DYffusion (25-member ensemble and single forecast), and ACE (single, deterministic forecast). Our method generates competitive probabilistic ensemble weather forecasts, a necessary but not sufficient prerequisite for achieving good climate simulations. Figure 14: We visualize the performances of DYffusion and Spherical DYffusion (in different marker colors) at multiple checkpoint epochs and for multiple generated samples. We plot the 10-year time-mean RMSE (“climate skill”) of three example fields versus the time-step-wise near-surface temperature RMSE averaged out over the first 20 forecasts (5 days; “weather skill”). The 5-day weather forecast performance shows no correlation with the long-term climate biases (indeed, there seems to exist an inverse correlation). This has important implications for practitioners, implying that optimizing for short-term forecasts alone – as is current practice for most ML-based weather forecasting models – may be suboptimal for attaining accurate climate simulations. We have verified that the behavior shown above holds for fields other than near-surface temperature too (not shown). the ensembles of our method and DYffusion perform best in terms of ensemble-mean RMSE. As expected, the ensemble mean significantly reduces the RMSE compared to using a single sample from our method or DYffusion, especially at longer lead times. The ensemble metrics, CRPS, and spread / RMSE ratio show that our method’s and DYffusion’s ensemble perform quite similarly, even though they are based on completely different ML architectures. Both ensembles tend to be underdispersed (Spread / RMSE < 1) on short time scales but quickly converge to a well-dispersed ensemble at longer lead times which persists for the whole 10-year climate simulations (not shown). E.3 Weather vs. climate performance In Figure 14, we illustrate that weather performance does not correlate with the climate biases of the same model. We plot the average RMSE over the first 5 days of simulation (here, using the nearsurface temperature field) against the 10-year time-mean RMSE of various fields, and do not observe any correlation between the two metrics. We have verified that this observation holds independently of the analyzed field. This is a little-discussed observation that has important implications for ML 26 practitioners since it implies that optimizing for short-term forecasts alone – as is current practice for most ML-based weather forecasting models – may be suboptimal for attaining accurate climate simulations. Heuristically, optimizing weather skill ensures that a climate model takes a locally accurate path around the climate ’attractor’, but it does not guarantee that small but systematic errors may not build up to distort that simulated attractor to have biased time-mean statistics. This observation has been documented for the case of physics-based climate models [17, 54]. E.4 Qualitative samples Figures 15, 16, and 17 compare near-surface air temperature, near-surface wind speed. and total water path between the FV3GFS validation simulation, two randomly selected 10-year trajectories generated by Spherical DYffusion, and the trajectory predicted by ACE. For both variables, we show the final ten snapshots of each simulation. The complete temporal evolutions of these simulations for near-surface wind speed and total water path can be viewed at https://youtu.be/7lHra7gBiBo and https://youtu.be/Hac xGsJ1qY, respectively. The emulated fields demonstrate high realism, closely mimicking the patterns and variability observed in actual climate model outputs. This showcases Spherical DYffusion’s capability to generate plausible and physically consistent climate scenarios over decadal timescales. Figure 15: We visualize the final 10 predictions from two random 10-year trajectory samples (i.e. the end of the ninth year) generated by Spherical DYffusion (middle rows) and ACE (bottom row). Here, we show the near-surface air temperature variable, Tk for level k = 7. It is important to note that at these extended time scales, simulated trajectories are expected to diverge significantly from one another for any given time step. 27 Figure 16: We visualize the final 10 predictions from two random 10-year trajectory samples generated by Spherical DYffusion (middle rows) and ACE (bottom row). Here, we show the derived nearsurface wind speed variable, WSk for level k = 7. It is important to note that at these extended time scales, simulated trajectories are expected to diverge significantly from one another for any given time step. A video visualizing the full 10-year simulations is accessible at https://youtu.be/7lHra7gBiBo. Figure 17: Same as Figure 16 but for the derived total water path variable, TWP. A video visualizing the full 10-year simulations is accessible at https://youtu.be/Hac xGsJ1qY. 28 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: The paper’s main contributions are enumerated at the end of the introduction. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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4,492	Looks Too Good To Be True: An Information-Theoretic Analysis of Hallucinations in Generative Restoration Models Regev Cohen Idan Kligvasser Ehud Rivlin Daniel Freedman Verily AI (Google Life Sciences), Israel regevcohen@google.com Abstract The pursuit of high perceptual quality in image restoration has driven the development of revolutionary generative models, capable of producing results often visually indistinguishable from real data. However, as their perceptual quality continues to improve, these models also exhibit a growing tendency to generate hallucinations – realistic-looking details that do not exist in the ground truth images. Hallucinations in these models create uncertainty about their reliability, raising major concerns about their practical application. This paper investigates this phenomenon through the lens of information theory, revealing a fundamental tradeoff between uncertainty and perception. We rigorously analyze the relationship between these two factors, proving that the global minimal uncertainty in generative models grows in tandem with perception. In particular, we deﬁne the inherent uncertainty of the restoration problem and show that attaining perfect perceptual quality entails at least twice this uncertainty. Additionally, we establish a relation between distortion, uncertainty and perception, through which we prove the aforementioned uncertainly-perception tradeoff induces the well-known perception-distortion tradeoff. We demonstrate our theoretical ﬁndings through experiments with super-resolution and inpainting algorithms. This work uncovers fundamental limitations of generative models in achieving both high perceptual quality and reliable predictions for image restoration. Thus, we aim to raise awareness among practitioners about this inherent tradeoff, empowering them to make informed decisions and potentially prioritize safety over perceptual performance. Perception Uinherent 2Uinherent Uncertainty Lower Bound Impossible Region ( Better) ( Better) Figure 1: Illustration of Theorem 3. In restoration tasks, the minimal attainable uncertainty is lower bounded by a function that begins at the inherent uncertainty UInherent of the problem (Deﬁnition 2) and graudally increases up to twice this value as the recovery approaches perfect perceptual quality. 38th Conference on Neural Information Processing Systems (NeurIPS 2024). Better Perception Higher Uncertainty (Hallucinations) Figure 2: Image inpainting results. Algorithms are ordered from low to high perception (left to right). Note the corresponding increased hallucinations and distortion. See Section 5 for details. 1 Introduction Restoration tasks and inverse problems impact many scientiﬁc and engineering disciplines, as well as healthcare, education, communication and art. Generative artiﬁcial intelligence [80, 38, 10] has transformed the ﬁeld of inverse problems due to its unprecedented ability to infer missing information and restore corrupted data. In the realm of image restoration, the quest for high perceptual quality has led to a new generation of generative models, capable of producing outputs of remarkable realism, virtually indistinguishable from true images. While powerful, growing empirical evidence indicates that generative models are susceptible to hallucinations [30], characterized by the generation of seemingly authentic content that deviates from the original input data, hindering applications where faithfulness is crucial. The root cause of hallucination lies in the ill-posed nature of restoration problems, where multiple possible solutions can explain the observed measurements, leading to uncertainty in the estimation process. Concerns surrounding hallucinations have prompted the development of uncertainty quantiﬁcation methods, designed to evaluate the reliability of generated outputs. These approaches offer crucial insights into the model’s conﬁdence in its predictions, empowering users to assess potential deviations from the original data and make informed decisions. Despite this progress, the relationship between achieving high perceptual quality and the extent of uncertainty remains an understudied area. This paper establishes the theoretical relationship between uncertainty and perception, demonstrating through rigorous analysis that the global minimal uncertainty in generative models increases with the level of desired perceptual quality (see illustration in Figure 1). Leveraging information theory, we quantify uncertainty using the entropy of the recovery error [19], while we measure perceptual quality via conditional divergence between the distributions of the true and recovered images [58]. Our main contribution are as follows: 1. We introduce a deﬁnition for the inherent uncertainty UInherent of an inverse problem, and formulate the uncertainty-perception (UP) function, seeking the minimal attainable uncertainty for a given perceptual index. We prove the UP function is globally lower-bounded by UInherent (Theorem 1). 2. We prove a fundamental trade-off between uncertainty and perception under any underlying data distribution, restoration problem or model (Theorem 1). Speciﬁcally, the entropy power of the recovery error exhibits a lower bound inversely related to the Rényi divergence between the true and recovered image distributions (Theorem 3). This shows that perfect perceptual quality requires at least twice the inherent uncertainty UInherent. 3. We establish a relationship between uncertainty and mean squared error (MSE) distortion, demonstrating that the uncertainty-perception trade-off induces the well-known distortion-perception trade-off [14] (Theorem 4). 4. We empirically validate all theoretical ﬁndings through experiments on image super-resolution and inpainting (Section 5), covering a broad spectrum of recovery algorithms, diverse metrics and data distributions. Our experimental results for image inpainting are illustrated in Figure 2. 2 We aim to provide practitioners with a deeper understanding of the tradeoff between uncertainty and perceptual quality, allowing them to strategically navigate this balance and prioritize safety when deploying generative models in real-world, sensitive applications. 2 Related Work Recent work in image restoration has made signiﬁcant strides in both perceptual quality assessment and uncertainty quantiﬁcation, largely independently. Below, we outline the main trends in research on these topics, laying the foundation for our framework. Perception Quantiﬁcation Perceptual quality in restoration tasks encompasses how humans perceive the output, considering visual ﬁdelity, similarity to the original, and absence of artifacts. While traditional metrics like PSNR and SSIM [82] capture basic similarity, they miss ﬁner details and higher-level structures. Learned metrics like LPIPS [87], VGG-loss [72], and DISTS [22] offer improvements but still operate on pixel or patch level, potentially overlooking holistic aspects. Recently, researchers have leveraged image-level embeddings from large vision models like DINO [17] and CLIP [62] to capture high-level similarity. Further advancements include HyperIQA [74] that leverages self-adaptive hyper networks to blindly assess image quality in the wild, while LIQE [88] and QAlign [84] utilize large language models to capture high-level semantic similarity and alignment between the restored and original images. Here, we follow previous works [58, 14, 31] and adopt a mathematical notion of perceptual quality deﬁned as the divergence between probability densities. Uncertainty Quantiﬁcation Uncertainty quantiﬁcation techniques can be broadly categorized into two main paradigms: Bayesian estimation and frequentist approaches. The Bayesian paradigm deﬁnes uncertainty by assuming a distribution over the model parameters and/or activation functions [1]. The most prevalent approach is Bayesian neural networks [52, 78, 34], which are stochastic models trained using Bayesian inference. To improve efﬁciency, approximation methods have been developed, including Monte Carlo dropout [24, 25], stochastic gradient Markov chain Monte Carlo [67, 18], Laplacian approximations [63] and variational inference [16, 51, 60]. Alternative Bayesian techniques encompass deep Gaussian processes [20], deep ensembles [7, 33], and deep Bayesian active learning [26]. In contrast to Bayesian methods, frequentist approaches operate assume ﬁxed model parameters with no underlying distribution. Examples of such distribution-free techniques are model ensembles [44, 59], bootstrap [36, 2], interval regression [59, 37, 83] and quantile regression [27, 64]. An emerging approach in recent years is conformal prediction [3, 70], which leverages a labeled calibration dataset to convert point estimates into prediction regions. Conformal methods require no retraining, computationally efﬁcient, and provide coverage guarantees in ﬁnite samples [49]. These works include conformalized quantile regression [64, 69, 6], conformal risk control [5, 8, 4], and semantic uncertainty intervals for generative adversarial networks [68]. The authors of [42] introduce the notion of conformal prediction masks, interpretable image masks with rigorous statistical guarantees for image restoration, highlighting regions of high uncertainty in the recovered images. Please see [75] for an extensive survey of distribution-free conformal prediction methods. A recent approach [11] introduces a principal uncertainty quantiﬁcation method for image restoration that considers spatial relationships within the image to derive uncertainty intervals that are guaranteed to include the true unseen image with a user-deﬁned conﬁdence probabilities. While the above studies offer a variety of approaches for quantifying uncertainty, a rigours analysis of the relationship between uncertainty and perception remains underexplored in the context of image restoration. The Distortion-Perception Tradeoff The most relevant studies to our research are the work on the distortion-uncertainty tradeoff [14] and its follow-ups [23, 15, 13]. A key ﬁnding in [14] establishes a convex tradeoff between perceptual quality and distortion in image restoration, applicable to any distortion measure and distribution. Moreover, perfect perceptual quality comes at the expense of no more than 3dB in PSNR. The work in [23] extends this, providing closed-form expressions for the tradeoff when MSE distortion and Wasserstein-2 distance are considered as distortion and perception measures respectively. In [58], it is shown that the Lipschitz constant of any deterministic estimator grows to inﬁnity as it approaches perfect perception. This work uniquely emphasizes uncertainty in image restoration, distinguishing it from distortion. While distortion measures how close a restored image is to the original, uncertainty quantiﬁes the conﬁdence in the restoration itself. This distinction is crucial for decision-making, as high uncertainty can hinder informed choices, complementing existing research on perceptual quality and robustness. 3 3 Problem Formulation We adopt a Bayesian perspective to address inverse problems, wherein we seek to recover a random vector X ∈Rd from its observations, represented by another random vector Y = M(X) ∈Rd′. Here M : Rd →Rd′ is a non-invertible degradation function, implying X cannot be perfectly recovered from Y . Formally: Deﬁnition 1. A degradation function M said to be invariable if, the conditional probability pX\|Y (·\|y) is a Dirac delta function for almost every y in the support of the distribution pY of Y . The restoration process involves constructing a estimator ˆX ∈Rd to estimate X from Y , inducing conditional probability p ˆ X\|Y . The estimation process forms a Markov chain X →Y →ˆX, implying that X and ˆX are statistically independent given Y . In this paper, we analyze estimators ˆX with respect to two performance criteria: perception and uncertainty. To assess perceptual quality, we follow a theoretical approach, similar to previous works [85, 14], and measure perception using conditional divergence1 between X and ˆX deﬁned as Dv(X, ˆX Y ) ≜Ey∼pY h Dv pX\|Y =y, p ˆ X\|Y =y i , (1) where Dv stands for general divergence function. When an estimator attains a low value of the metric above, we say it exhibits high perceptual quality. When it comes to uncertainty, there are diverse practical methods to quantify it [28, 1]. However, for our analysis, we aim to identify a fundamental understanding of uncertainty. Therefore, we adopt the concept of entropy power from information theory, which assesses the statistical spread of a random variable. For the deﬁnition of entropy power and other relevant background, we refer the reader to Appendix B. Utilizing entropy power, we formally deﬁne the inherent uncertainty intrinsic to the restoration problem as follows Deﬁnition 2. The inherent uncertainty in estimating X from Y is deﬁned as: UInherent ≜N(X\|Y ) = 1 2πee 2 d h(X\|Y ), where h(X\|Y ) denotes the entropy of X given Y . The inherent uncertainty quantiﬁes the information irrevocably lost during observation, acting as a fundamental limit on the recovery of X from Y , regardless of the estimation method. Notably, when the degradation process is invertible, this inherent uncertainty becomes zero UInherent = 0, reﬂecting the possibility of perfect recovery of X with complete conﬁdence. We now turn our attention to the main focus of this paper, the uncertainty-perception (UP) function: U(P) ≜min p ˆ X\|Y n N( ˆX −X\|Y ) : Dv(X, ˆX Y ) ≤P o . (2) In essence, U(P) represents the minimum uncertainty achievable by an estimator with perception quality of at least P, given the side information within the observation Y . In contrast to the perception-distortion function [14], the above objective prioritizes the information content of error signals over their mere energy, and its minimization promotes concentrated errors for robust and reliable predictions. The following example offers intuition into the typical behavior of this function. Example 1. Consider Y = X + W where X ∼N(0, 1) and W ∼N(0, σ2) are independent. Let the perception measure be the symmetric Kullback–Leibler (KL) divergence DSKL and assume stochastic estimators of the form ˆX = E [X\|Y ] + Z where Z ∼N(0, σ2 z) is independent of Y . As derived in Appendix C, the UP function admits a closed form expression in this case, given by U(P) = N(X\|Y ) h 1 + P + 1 − p (P + 1)2 −1 2 i , where N(X\|Y ) = σ2/(1 + σ2). The above result, illustrated in Appendix C, demonstrates the minimal attainable uncertainty increases as the perception quality improves. Moreover, The above example suggests a structure for uncertainty-perception function U(P), which fundamentally relies on the inherent uncertainty 1See Appendix A for a brief explanation of how conditional divergence relates to human perception. 4 N(X\|Y ). Remarkably, the following section shows that this dependency generalizes beyond the speciﬁc example presented here, where its particular form is determined by the underlying distributions, along with the speciﬁc perception measure employed. Remark One may consider the following alternative formulation ˜U(P) ≜min p ˆ X\|Y n N( ˆX −X) : Dv(X, ˆX Y ) ≤P o . (3) The alternative objective quantiﬁes uncertainty as the entropy power of the error, independent of the side information Y . While potentially insightful, this approach may overestimate uncertainty since N( ˆX −X\|Y ) ≤N( ˆX −X) where equality holds if and only if the error E = ˆX −X is independent of Y . Although further investigation is warranted, we hypothesize that the behavior of function (3) mirrors that of the UP function (2), which we examine in detail in the following section. 4 The Uncertainty-Perception Tradeoff Thus far, we have formulated the uncertainty-perception function and elucidated its underlying rationale. We now proceed to derive its key properties, including a detailed analysis for the case where Rényi divergence serves as the measure of perceptual quality. Subsequently, we establish a direct link between the UP function and the well-known distortion-perception tradeoff. Finally, we demonstrate our theoretical ﬁndings through experiments on image super-resolution. 4.1 The Uncertainty-Perception Plane The following theorem establishes general properties of the uncertainty-perception function, U(P), irrespective of the speciﬁc distributions and divergence measures chosen. Theorem 1. The uncertainty-perception function U(P) displays the following properties 1. Quasi-linearity (monotonically non-increasing and continuous): min U(P1), U(P2) ≤U λP1 + (1 −λ)P2 ≤max U(P1), U(P2) , ∀λ ∈[0, 1] 2. Boundlessness: N(X\|Y ) ≤U(P) ≤2N(XG\|Y ), where XG is a zero-mean Gaussian random variable with covariance identical to X. The inherent uncertainty is upper bounded by N(XG\|Y ), which depends on the deviation of X from Gaussianity. The theorem establishes a fundamental tradeoff between perceptual quality and uncertainty in image restoration, regardless of the speciﬁc divergence measure, data distributions, or restoration model employed. This tradeoff is fundamentally linked to the inherent uncertainty N(X\|Y ) arising from the information loss during the observation process. Notably, the upper bound can be expressed as N(XG\|Y ) = N(X\|Y )e 2 d DKL(X,XG\|Y ). (4) This shows that as X approaches Gaussianity, N(X\|Y ) approaches N(XG\|Y ). However, concurrently, it implies in general higher values of N(X\|Y ) due to Lemma 1 of Appendix B. This ﬁnding yields a surprising insight: for multivariate Gaussian distributions, perfect perceptual quality comes at the expense of exactly twice the inherent uncertainty of the problem. Next, we show that for a ﬁxed perceptual index P, the optimal algorithms lie on the boundary of the constraint set. This facilitates the optimization, as it restricts the search space to the boundary points. Theorem 2. Assume Dv(X, ˆX Y ) is convex in its second argument. Then, for any P ≥0, the minimum is attained on the boundary where Dv(X, ˆX Y ) = P. Note that the assumption of the convexity of Dv in its second argument is not a restrictive condition. In fact, most widely-used divergence functions, notably all f-divergences (such as KL divergence, total variation distance, Hellinger distance, and Chi-square divergence), exhibit this property. 5 While the above theorems describe important characteristics of the uncertainty-perception function, additional assumptions are needed to gain deeper insights. Therefore, we now focus on Rényi divergence as our perception measure. Rényi divergence is a versatile family of divergence functions parameterized by an order 0 ≤r, encompassing the well-known KL divergence as a special case when r = 1. This divergence plays a critical role in in analyzing Bayesian estimators and numerous information theory calculations [79]. Importantly, it is also closely related to other distance metrics used in probability and statistics, such as the Wasserstein and Hellinger distances. Focusing on the case where r = 1/2, we arrive at: U(P) = min p ˆ X\|Y n N( ˆX −X\|Y ) : D1/2(X, ˆX Y ) ≤P o . (5) While we set r = 1/2 to facilitate our derivations, it is important to note that all orders r ∈(0, 1) are equivalent (see Appendix B). Consequently, given this equivalence and the close relationship between Rényi divergence and other metrics, analyzing the speciﬁc formulation provided by (5) may yield valuable insights applicable to a wide range of divergence measures. The following theorem provides lower and upper bounds for the UP function. Theorem 3. The uncertainty-perception function is conﬁned to the following region η(P) · N(X\|Y ) ≤U(P) ≤η(P) · N(XG\|Y ) where 1 ≤η(P) ≤2 is a convex function w.r.t the perception index and is given by η(P) = 2e 2P d − q (2e 2P d −1)2 −1 . Noteworthy, Theorem 3 holds true regardless of the underlying distributions of X and Y , thereby providing a universal characterization of the UP function in terms of perception. Furthermore, as depicted in Figure 3, Theorem 3 gives rise to the uncertainty-perception plane, which divides the space into three distinct regions: 1. Impossible region, where no estimator can reach. 2. Optimal region, encompassing all estimators that are optimal according to (5). 3. Suboptimal region of estimators which exhibit overly high uncertainty. The existence of an impossible region highlights the uncertainty-perception tradeoff, proving no estimator can achieve both high perception and low uncertainty simultaneously. This ﬁnding underscores the importance of practitioners being aware of this tradeoff, enabling them to make informed decisions when prioritizing between perceptual quality and uncertainty in their applications. The uncertainty-perception plane could serve as a valuable framework for evaluating estimator performance in this context. While not a comprehensive metric, it may offer insights into areas where improvements can be made, guiding practitioners towards estimators that strike a more desirable balance between perception and uncertainty. For certain estimators residing in the suboptimal region, it may be possible to achieve lower uncertainty without sacriﬁcing perceptual quality. Thus, we believe that our proposed uncertainty-perception plane can serve as a valuable starting point for further research and practical applications, ultimately leading to the development of safer and reliable image restoration algorithms. Next, we analyze how the dimensionality of the underlying data affects the uncertainty-perception tradeoff. To achieve this, we extend the function η(P) to include a dimension parameter d, denoted as η(P; d). As shown in Fig. 4, η(P; d) exhibits a rapid incline as perception improves and it attain higher values in higher dimensions. This observation suggests that in high-dimensional settings, the uncertainty-perception tradeoff becomes more severe, implying that any marginal improvement in perception for an algorithm is accompanied by a dramatic increase in uncertainty. Finally, we conjecture that the general form of the tradeoff, given by the inequality in Theorem 3, holds for different divergence measures, with the speciﬁc form of η(P) capturing the nuances of each chosen measure. For instance, considering the Hellinger distance as our perception measure, we obtain the same inequality as in Theorem 3 but with η(P) deﬁned for 0 ≤P ≤1 as2 ηHellinger(P) = 2 (1 −P)4/d − s 2 (1 −P)4/d −1 2 −1. (6) 2The case of P = 1 is obtained by taking the limit lim P →1 η(P) = 1. 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 D1/2(X, X\|Y) N(X\|Y) 2 N(X\|Y) 2 N(XG\|Y) N(X X\|Y) Suboptimal Optimal Impossible Figure 3: The uncertainty-perception plane (Theorem 3). The impossible region demonstrates the inherent tradeoff between perception and uncertainty, while other regions may guide practitioners toward estimators that better balance the two factors, highlighting potential areas for improvement. 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Perceptual P 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 (P; d) d=64 d=128 d=256 d=512 d=1024 d=2048 Figure 4: Impact of dimensionality, as revealed in Theorem 3, demonstrates that the uncertaintyperception tradeoff intensiﬁes in higher dimensions. This implies that even minor improvements in perceptual quality for an algorithm may come at the cost of a signiﬁcant increase in uncertainty. 4.2 Revisiting the Distortion-Perception Tradeoff Having established the uncertainty-perception tradeoff and its characteristics, we now broaden our analysis to estimation distortion, particularly the mean squared-error. A well-known result in estimation theory states that for any random variable X and for any estimator ˆX based upon side information Y , the following holds true [19]: E h \|\| ˆX −X\|\|2i ≥ 1 2πee2h(X\|Y ). (7) This inequality, related to the uncertainty principle, serves as a fundamental limit to the minimal MSE achieved by any estimator. However, it does not consider the estimation uncertainty of ˆX as the right hand side is independent of ˆX. Thus, we extend the above in the following theorem. Theorem 4. For any random variable X, observation Y and unbiased estimator ˆX, it holds that 1 dE h \|\| ˆX −X\|\|2i ≥N ˆX −X Y . Notice that for any estimator ˆX we have N( ˆX −X\|Y ) ≥N(X\|Y ), implying 1 dE[∥ˆX −X∥2] ≥N(X\|Y ) = 1 2πee 2 d h(X\|Y ). (8) 7 The above result aligns with equation (7), demonstrating that Theorem 4 serves as a generalization of inequality (7), incorporating the uncertainty associated with the estimation. Furthermore, by viewing the estimator ˆX as a function of perception index P, we arrive at the next corollary. Corollary 1. Deﬁne the following distortion-perception function D(P) ≜min p ˆ X\|Y n1 dE h \|\| ˆX −X\|\|2i : Dv(X, ˆX Y ) ≤P o . Then, for any perceptual index P, we have D(P) ≥U(P). As uncertainty increases with improving perception, the corollary implies that distortion also increases. Thus, when utilizing MSE as a measure of distortion, the uncertainty-perception tradeoff induces a distortion-perception tradeoff [14], offering a novel interpretation of this well-known phenomenon. 5 Experiments Setup. Our theoretical framework is grounded in empirical observations, leading us to validate our ﬁndings through experiments on common benchmark tasks: image super-resolution and inpainting. We analyze performance through the lens of uncertainty, alongside established measures of perceptual quality and distortion. To assess perceptual quality, we employ state-of-the-art metrics including HyperIQA [74], LIQE [88] and Q-ALIGN [84]. Distortion is evaluated using traditional measures: MSE, peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM) [82]. Accurately estimating entropy in high-dimensional spaces presents signiﬁcant challenges [46]; hence, we utilize an upper bound for uncertainty, N( ˆXG −XG\|Y ), as detailed in Appendix F. This practical alternative simpliﬁes computation to calculating the geometric mean of the singular values of the error covariance. For super-resolution, we utilize the BSD100 benchmark dataset [55], aiming to predict a highresolution image from its low-resolution counterpart obtained via 4× bicubic downsampling. Our evaluation spans a diverse range of recovery algorithms, including EDSR [50], ESRGAN [81], SinGAN [71], SANGAN [39], DIP [77], SRResNet/SRGAN variants [47], EnhanceNet [66], and Latent Diffusion Models (LDMs) with parameter β ∈[0, 1] [65], where β = 0 recovers DDIM [32] and β = 1 recovers DDPM [73]. In the context of image inpainting, we leverage the SeeTrue dataset [86], an image-text alignment benchmark known for its diverse collection of real and synthetic text-image pairs. Here, we focus our analysis on diffusion models due to their state-of-the-art performance and growing popularity in the ﬁeld. Results. Figure 5 presents our super-resolution analysis. As observed in the top row, across various perceptual measures, an unattainable blank region exists in the lower right corner, indicating that no model simultaneously achieves both low uncertainty and high perceptual quality. Furthermore, an anti-correlation emerges near this region, where modest improvements in perceptual quality translate to dramatic increases in uncertainty. This observation suggests the existence of a tradeoff between uncertainty and perception. Additionally, the bottom row showcases a strong relationship between uncertainty and distortion across diverse measures, demonstrating that any increase in uncertainty leads to a signiﬁcant rise in distortion.3 Figure 6 displays similar trends for image inpainting, consistent with our super-resolution analysis and reinforcing the validity of our ﬁndings across diverse restoration tasks and data distributions. This is further visualized in Figure 2, which presents outputs from selected algorithms ordered by perceptual quality. The results clearly demonstrate an increase in hallucination (uncertainty) and distortion with increasing perceptual quality. Finally, Appendix H presents additional results obtained via direct estimation of statistics in high dimensions, further supporting our theoretical analysis. 6 Conclusion This study established the uncertainty-perception tradeoff in generative restoration, demonstrating that high perceptual quality leads to increased hallucination (uncertainty), particularly in high dimensions. We characterized this tradeoff and its fundamental relation to the inherent uncertainty of the problem, 3Note that MSE is a measure of distortion, whereas PSNR and SSIM are measures of inverse distortion; this accounts for the negative slope in the ﬁrst two ﬁgures, and the positive slope in the third. 8 1.0 1.5 2.0 2.5 3.0 3.5 LIQE 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 QALIGN 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN 3 4 5 6 HYPERIQA 1e 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 PSNR 1e1 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 SSIM 1e 1 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Uncertainty 1e 3 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 MSE 1e 2 Bicubic DDIM DIP EDSR ESRGAN EnhanceNet DDPM LDM0.2 LDM0.5 LDM0.7 SANGAN SRGAN-VGG22 SRGAN-VGG54 SRResNet-MSE SRResNet-VGG22 SinGAN Figure 5: Evaluation of SR algorithms. Top: Uncertainty-perception plane showing the tradeoff between perceptual quality and uncertainty (y-axis) for various perceptual measures. Bottom: Uncertainty-distortion plane showing the relationship between uncertainty and various distortion measures. Axis placement differs in the two rows to highlight the distinct roles of uncertainty. introducing the uncertainty-perception plane which may guide practitioners in understanding estimator performance. By extending our analysis to MSE distortion, we showed that the distortion-perception tradeoff emerges as a direct consequence of the uncertainty-perception tradeoff. Experimental results conﬁrmed our theoretical ﬁndings, highlighting the importance of this tradeoff in image restoration. 7 Limitations Our analysis is grounded in the theoretical framework of entropy as a measure of uncertainty. Information theory offers a powerful framework for quantifying uncertainty and dependencies in data, handling multivariate and heterogeneous data types, and capturing complex patterns. However, its wider adoption has been limited by the challenge of estimating information-theoretic measures in high dimensions. The curse of dimensionality makes accurate density estimation infeasible [12, 48], leading many to rely on simpler second-order statistics. The development of practical tools for estimating statistics in high-dimensional data remains an active area of research [76]. While initial approaches assumed exponential family distributions (e.g., Gaussian) for tractable calculations [57], their performance degrades for long-tailed distributions. Non-parametric methods like binning strategies, including KDE and kNN estimators [61, 40, 29], offer more ﬂexibility but are data-dependent and sensitive to parameter choices. Alternative approaches involve ensemble estimation [43] or von Mises Expansions [35], the distributional analog of the Taylor expansion. Rotation-Based Iterative Gaussianization [46] presents a promising direction by transforming data into a multivariate Gaussian domain, simplifying density estimation. However, its application to images has been limited to small patches due to the computational challenges of learning rotations based on principal or independent component analysis. A recent extension addresses this by utilizing convolutional rotations, enabling efﬁcient processing of entire images [45]. While accurately estimating high-dimensional entropy remains an active research area, Section 5 utilizes a tractable upper bound. This alternative calls for further investigation into its potential for quantifying uncertainty and analyzing algorithm performance. Moreover, incorporating this bound into the design of new algorithms could enable explicit control over the uncertainty-perception trade-off, potentially leading to more reliable solutions. 9 4.09 4.10 4.11 4.12 4.13 4.14 4.15 LIQE 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM 3.81 3.82 3.83 3.84 3.85 QALIGN 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM 5.88 5.90 5.92 5.94 5.96 5.98 HYPERIQA 1e 1 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 PSNR 1e1 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 8.300 8.325 8.350 8.375 8.400 8.425 8.450 8.475 SSIM 1e 1 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 Uncertainty 1e 3 1.20 1.25 1.30 1.35 1.40 MSE 1e 2 DDIM LDM0.1 LDM0.2 LDM0.3 LDM0.4 LDM0.5 LDM0.6 LDM0.7 LDM0.8 LDM0.9 DDPM Figure 6: Evaluation of LDMs on image inpainting, highlighting the trade-off between uncertainty and perceptual quality (top) and the uncertainty-distortion relationship (bottom). No model achieves both low uncertainty and high perceptual quality, with higher uncertainty generally leading to increased distortion. Differing axis placements emphasize the distinct roles of uncertainty. Lastly, we focused our empirical validation on image super-resolution and inpainting, two benchmark problems in image restoration. Our analysis, however, applies to any restoration task with noninvertible degradation. Hence, expanding the experiments to additional image-to-image tasks and domains such as audio, video, and text may reveal broader implications and applications of our work. 8 Broader Impact Our work revealing a fundamental tradeoff between uncertainty and perception in image restoration carries signiﬁcant societal impact. Developers across various ﬁelds, including healthcare and autonomous systems, often integrate cutting-edge models into their applications, prioritizing stateof-the-art performance and perceptual quality. However, our work aims to highlight a crucial factor often overlooked: the inherent tradeoff between uncertainty and perception. By raising awareness of this tradeoff, we empower developers to make informed decisions that prioritize safety and reliability over purely perceptual enhancements. For instance, in healthcare, potential restoration algorithms can be evaluated by plotting them on the uncertainty-perception plane, facilitating the identiﬁcation of methods that strike the optimal balance for speciﬁc clinical needs. Furthermore, by understanding this inherent trade-off, practitioners can consider trading performance for better safety and resilience against potential misuse and misinterpretations. While primarily theoretical, our analysis yields a practical measure of uncertainty (or entropy), used in our experiments to visually and quantitatively illustrate our ﬁndings. 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From a Bayesian perspective, the optimal decision rule maximizing psuccess yields ([56] Section 2): psuccess = 1 2 + 1 2DTV(pX,Y , p ˆ X,Y ) where DTV(pX,Y , p ˆ X,Y ) is the total-variation (TV) distance. When D(pX,Y , p ˆ X,Y ) = 0, the two pairs are indistinguishable (psuccess = 0.5), implying perfect perception quality. We generalize this beyond the total-variation (TV) distance to any conditional divergence, recognizing that the divergence that best relates to human perception remains an open question. B Information-Theory Preliminaries To make the paper self-contained, we brieﬂy overview the essential deﬁnitions and results in information-theory. Let X, Y and Z be continuous random variables with probability density functions pX(x), pY (y) and pZ(z) respectively. The space of probability density functions is denoted by Ω. We assume the quantities described below, which involve integrals, are well-deﬁned and ﬁnite. Deﬁnition 3 (Entropy). The differential entropy of X, whose support is a set Sx, is deﬁned by h(X) ≜− Z SX pX(x) log pX(x)dx. Deﬁnition 4 (Rényi Entropy). The Rényi entropy of order r ≥0 of X is deﬁned by hr(X) ≜ 1 1 −r log Z pr X(x)dx. The above quantity generalizes various notions of entropy, including Hartley entropy, collision entropy, and min-entropy. In particular, for r = 1 we have h1(X) ≜lim r→1 hr(X) = h(X). Deﬁnition 5 (Entropy Power). Let be h(X) be the differential entropy of X ∈Rd. Then, the entropy Power of X is given by N(X) ≜ 1 2πee 2 d h(X). Deﬁnition 6 (Divergence). A statistical divergence is any function Dv : Ω×Ω→R+ which satisﬁes the following conditions for all p, q ∈Ω: 1. Dv(p, q) ≥0. 2. Dv(p, q) = 0 iff p = q almost everywhere. Table 1: Formulas for Multivariate Gaussian Distribution Distribution Quantity Closed-Form Expression X ∼N(µx, Σx) h(X) 1 2 ln{(2πe)d \|Σx\|}. X ∼N(µx, Σx) N(X) \|Σx\|1/n . X ∼N(µx, Σx) h 1 2 (X) 1 2 ln{(8π)d \|Σx\|}. X ∼N(µx, Σx), Y ∼N(µy, Σy) D1/2(X, Y ) 1 4(µx −µy)T Σx+Σy 2 −1 (µx −µy) + ln Σx+Σy 2 √ \|Σx\|\|Σy\| . 16 Deﬁnition 7 (Rényi Divergence). The Rényi divergence of order r ≥0 between pX and pY is Dr(X, Y ) ≜ 1 r −1 log Z pr X(x)p1−r Y (x)dx. The above establishes a spectrum of divergence measures, generalising the Kullback–Leibler divergence as D1(X, Y ) = DKL(X, Y ). Furthermore, it is important to note that all orders r ∈(0, 1) are equivalent [79], since r t 1 −t 1 −rDt(·, ·) ≤Dr(·, ·) ≤Dt(·, ·), ∀0 < r ≤t < 1. (9) Deﬁnition 8 (Conditioning). Consider the joint probability pXY and the conditional probabilities pX\|Y (x\|y) and pZ\|Y (z\|y). The conditional differential entropy of X ∈Rd given Y is deﬁned as h(X\|Y ) ≜− Z SXY pXY (x, y) log pX\|Y (x\|y)dxdy = Ey∼pY [h(X\|Y = y)] where SXY is the support set of pXY . Then, the conditional entropy power of X given Y is N(X\|Y ) = 1 2πee 2 d h(X\|Y ). Similarly, the conditional divergence between X and Z given Y is deﬁned as Dv(X, Z Y ) ≜Ey∼pY [Dv(X\|Y = y, Z\|Y = y)] . For example, the conditional Rényi divergence is given by Dr(X,Z Y ) ≜ Z 1 r −1 log Z pr X\|Y (x\|y)p1−r Z\|Y (x\|y)dx pY dy. Table 1 summarizes closed-form expressions for several quantities relevant to the multivariate Gaussian distribution. Below we present two fundamental results that form the basis of our analysis. Lemma 1 (Maximum Entropy Principle [19]). Let X ∈Rd be a continuous random variable with zero mean and covariance Σx. Deﬁne XG ∼N(0, Σx) to be a Gaussian random variable, independent of X, with the identical covariance matrix ΣxG = Σx. Then, h(X) ≤h(XG), N(X) ≤N(XG) = \|Σx\|1/d . Lemma 2 (Entropy Power Inequality [53]). Let X and Y be independent continuous random variables. Then, the following inequality holds N(X) + N(Y ) ≤N(X + Y ), where equality holds iff X and Y are multivariate Gaussian random variables with proportional covariance matrices. Equivalently, let Xg and Yg be deﬁned as independent, isotropic multivariate Gaussian random variables satisfying h(Xg) = h(X) and h(Yg) = h(Y ). Then, h(X) + h(Y ) = h(Xg) + h(Yg) = h(Xg + Yg) ≤h(X + Y ). C Derivation of Example 1 Since ˆX = E [X\|Y ] + Z, then ˆX\|Y ∼N(E [X\|Y ] , σ2 z). Moreover, X\|Y ∼N(E [X\|Y ] , σ2 q) where σ2 q = σ2 1+σ2 . Thus, the conditional error entropy is given by N( ˆX −X\|Y ) = σ2 q + σ2 z and the symmetric KL divergence is DSKL(X, ˆX Y ) = σ2 q+σ2 z 2σzσq −1, leading the following problem U(P) = min σz n σ2 q + σ2 z : σ2 q + σ2 z 2σzσq −1 ≤P o . (10) 17 0 1 2 3 4 5 DSKL(X, X\|Y) N(X\|Y) 1.5 N(X\|Y) 2 N(X\|Y) N(X X\|Y) Figure 7: The Uncertainty-Perception function for Example 1. As perception quality improves, the minimal achievable uncertainty increases, suggesting a tradeoff governed by the inherent uncertainty. Therefore, we seek the minimal value of σz that satisﬁes the constraint. Note that the minimal value is attained at the boundary of the constraint set, where the inequality becomes an equality σ2 q + σ2 z 2σzσq −1 = P ⇒σ2 z −2σq(P + 1)σz + σ2 q = 0. (11) The solution to the aforementioned quadratic problem is σ∗ z = σq P + 1 − p (P + 1)2 −1 . Substituting the later into the objective function, we obtain U(P) = σ2 q h 1 + P + 1 − p (P + 1)2 −1 2 i . (12) Finally, the entropy power of an univariate Gaussian distribution equals its variance σ2 q = N(X\|Y ). Figure 7 visualizes the resulting uncertainty-perception tradeoff. D Proof of Theorem 1 First, the constraint C(P) ≜{ ˆX : Dv(X, ˆX Y ) ≤P} deﬁnes a compact set which is continuous in P. Hence, by the Maximum Theorem [19], U(P) is continuous. In addition, U(P) is the minimal error entropy power obtained over a constraint set whose size does not decrease with P, thus, U(P) is non-increasing in P. Any continuous non-increasing function is quasi-linear. For the lower bound consider the case where P = ∞, leading to the following unconstrained problem U(∞) ≜min p ˆ X\|Y N( ˆX −X\|Y ). (13) For any P ≥0 it holds that U(∞) ≤U(P), and by Lemma 2 we have N(X\|Y ) + min p ˆ X\|Y N( ˆX\|Y ) ≤U(∞). (14) Since minp ˆ X\|Y N( ˆX\|Y ) ≥0 we obtain ∀P ≥0 : N(X\|Y ) ≤U(P). (15) Next, we have U(P) ≤U(0) = N( ˆX0 −X\|Y ) where p ˆ X0\|Y = pX\|Y . Deﬁne V ≜ˆX0 −X, then Σv\|y = Σˆx\|y + Σx\|y = 2Σx\|y where we use that X and ˆX are independent given Y . Thus, U(0) = N(V \|Y ) ≤N(VG\|Y ) = Σv\|y 1/d = 2Σx\|y 1/d = 2 Σx\|y 1/d = 2N(XG\|Y ), (16) where the ﬁrst inequality is due to Lemma 1. Finally, for any P ≥0 it holds that U(P) ≤U(0) which implies U(0) ≤2N(XG\|Y ), completing the proof. 18 E Proof of Theorem 2 Assuming Dv(X, ˆX Y ) is convex in its second argument, the constraint represent a compact, convex set. Moreover, h( ˆX −X\|Y ) is strictly-concave w.r.t p ˆ X\|Y as a composition of a linear function (convolution) with a strictly-concave function (entropy). Therefore, we minimize a log-concave function over a convex domain and thus the global minimum is attained on the set boundary where Dv(X, ˆX Y ) = P. F Proof of Theorem 3 We begin with applying Lemma 1 and Lemma 2 to bound the objective function as follows N( ˆXg\|Y ) + N(Xg\|Y ) = N( ˆXg −Xg\|Y ) ≤N( ˆX −X\|Y ) ≤N( ˆXG −XG\|Y ). (17) Note that the bounds are tight as the upper bound is attained when ˆX\|Y and X\|Y are multivariate Gaussian random variables, while the lower bound is attained if we further assume they are isotropic. Thus, we can bound the uncertainty-perception function as follows Ug(P) ≤U(P) ≤UG(P) (18) where we deﬁne Ug(P) ≜min p ˆ Xg\|Y n N( ˆXg\|Y ) + N(Xg\|Y ) : D1/2(Xg, ˆXg Y ) ≤P o , UG(P) ≜min p ˆ XG\|Y n N( ˆXG −XG\|Y ) : D1/2(XG, ˆXG Y ) ≤P o . (19) The above quantities can be expressed in closed form. We start with minimization problem of the upper bound which can be written as UG(P) = min p ˆ XG\|Y n 1 2πee 2 d E[h( ˆ XG−XG\|Y =y)] : E h D1/2(XG, ˆXG Y = y) i ≤P o , (20) where the expectation is over y ∼Y . Substituting the expressions for h(XG −XG\|Y = y) and D1/2(XG, ˆXG Y = y), we get UG(P) = min {Σˆx\|y} ( 1 2πee 2 d E " 1 2 log n (2πe)d\|Σˆx\|y+Σx\|y\| o# : E  log Σˆx\|y + Σx\|y /2 qΣˆx\|y Σx\|y  ≤P ) . (21) Notice the optimization is with respect to the covariance matrices {Σˆx\|y}. Simplifying the above, we can equivalently solve the following minimization min {Σˆx\|y} E log Σˆx\|y + Σx\|y s.t. E  log Σˆx\|y + Σx\|y /2 qΣˆx\|y Σx\|y  ≤P. (22) The solution of a constrained optimization problem can be found by minimization the Lagrangian L {Σˆx\|y}, λ ≜E log Σˆx\|y + Σx\|y + λ  E  log Σˆx\|y + Σx\|y /2 qΣˆx\|y Σx\|y  −P  . (23) Since expectation is a linear operation and using that P = E [P], we rewrite the above as L {Σˆx\|y}, λ = E  log Σˆx\|y + Σx\|y + λ  log Σˆx\|y + Σx\|y /2 qΣˆx\|y Σx\|y −P    . (24) The expression within the expectation can be written as log Σˆx\|y + Σx\|y + λ log Σˆx\|y + Σx\|y /2 −1 2 log Σˆx\|y −1 2 log Σx\|y −P . (25) 19 Next, according to KKT conditions the solutions should satisfy ∂L ∂Σˆx\|y = 0. Using the linearity of the expectation and differentiating (25) w.r.t Σˆx\|y we obtain Σˆx\|y + Σx\|y −1 + λ Σˆx\|y + Σx\|y −1 −1 2Σ−1 ˆx\|y = 0 (26) Multiplying both sides by Σˆx\|y + Σx\|y , we have I + λI −λ 2 I −λ 2 Σx\|yΣ−1 ˆx\|y = 0 ⇒(1 + λ 2 )I = λ 2 Σx\|yΣ−1 ˆx\|y ⇒(λ + 2)Σˆx\|y = λΣx\|y ⇒Σˆx\|y = λ λ + 2Σx\|y. (27) Deﬁne γ = λ λ+2, so Σˆx\|y = γΣx\|y. Substituting the latter into the constraint we get log γΣx\|y + Σx\|y /2 −1 2 log γΣx\|y −1 2 log Σx\|y = P ⇒n log 1 + γ 2 −n 2 log γ = P ⇒(1 + γ)2 4γ = e 2 d P ⇒γ2 + 2γ + 1 = 4γe 2 d P ⇒γ(P) = 2e 2 d P −1 − q (2e 2 d P −1)2 −1. (28) Thus, we obtain that UG(P) = η(P) · N(XG\|Y ) (29) where η(P) = γ(P) + 1 = 2e 2 d P − q (2e 2 d P −1)2 −1. (30) Notice that η(0) = 2, while limP →∞η(P) = 1, so 1 ≤η(P) ≤2. Following similar steps where we replace Σˆx\|y and Σx\|y with N( ˆX\|Y ) and N(X\|Y ) respectively, we derive Ug(P) = η(P) · N(X\|Y ). (31) G Proof of Theorem 4 Deﬁne E ≜ˆX −X. Then, 1 dE h \|\| ˆX −X\|\|2i = (a) E 1 dE h \|\| ˆX −X\|\|2Y i = E 1 dE \|\|E\|\|2Y = E 1 dE ET E Y = E 1 dTr E EET Y = E 1 dTr Σε\|y ≥ (b) E hΣε\|y 1/di = E hΣˆx\|y + Σx\|y 1/di ≥ (c) E 1 2πee 2 d h( ˆ X−X\|Y =y) ≥ (d) 1 2πee 2 d E[h( ˆ X−X\|Y =y)] = 1 2πee 2 d h( ˆ X−X\|Y ) = N ˆX −X Y , where (a) is by the law of total expectation, (b) is due to the inequality of arithmetic and geometric means, (c) follows Lemma 1, and (d) is according to Jensen’s inequality. 20 H Results via Direct Estimation Estimating high-dimensional statistics is prone to errors [46]. we used practical measures for perceptual quality and a tractable upper bound for uncertainty. Here, we supplement those results with direct computations of entropy and divergence in a high-dimensional setting. Following prior work [14, 23], we treat images as stationary sources and extract 9 × 9 patches. To estimate Rényi divergence for perceptual quality assessment, we ﬁrst model the probability density functions using kernel density estimation. Subsequently, we compute the divergence through empirical expectations. Uncertainty is estimated using the Kozachenko-Leonenko estimator, which calculates the patch sample differential entropy based on nearest neighbor distances [41, 21, 9, 54]. Results, shown in Figure 8, strongly align with the trends observed in Figure 5. 0.5 0.6 0.7 0.8 0.9 Perception 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Uncertainty 1e 2 DDIM LDM0.2 LDM0.5 LDM0.7 DDPM SinGAN EnhanceNet ESRGAN SANGAN DIP Bicubic EDSR SRResNet-MSE SRGAN-VGG22 SRGAN-VGG54 SRResNet-VGG22 0.4 0.6 0.8 1.0 1.2 Uncertainty 1e 2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Distortion 1e 2 DDIM LDM0.2 LDM0.5 LDM0.7 DDPM SinGAN EnhanceNet ESRGAN SANGAN DIP Bicubic EDSR SRResNet-MSE SRGAN-VGG22 SRGAN-VGG54 SRResNet-VGG22 Figure 8: Evaluation of SR algorithms via direct estimation of high-dimensional statistics. Left: Uncertainty-perception plane demonstrating the tradeoff between perceptual quality and uncertainty. Right: Uncertainty-distortion plane illustrating the relation between uncertainty and distortion. Results are consistent with the ﬁnding in Figure 5. 21 NeurIPS Paper Checklist 1. 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4,493	The Sample-Communication Complexity Trade-off in Federated Q-Learning Sudeep Salgia Carnegie Mellon University ssalgia@andrew.cmu.edu Yuejie Chi Carnegie Mellon University yuejiechi@cmu.edu Abstract We consider the problem of Federated Q-learning, where M agents aim to collaboratively learn the optimal Q-function of an unknown infinite horizon Markov Decision Process with finite state and action spaces. We investigate the trade-off between sample and communication complexity for the widely used class of intermittent communication algorithms. We first establish the converse result, where we show that any Federated Q-learning that offers a linear speedup with respect to number of agents in sample complexity needs to incur a communication cost of at least Ω( 1 1−γ ), where γ is the discount factor. We also propose a new Federated Q-learning algorithm, called Fed-DVR-Q, which is the first Federated Q-learning algorithm to simultaneously achieve order-optimal sample and communication complexities. Thus, together these results provide a complete characterization of the sample-communication complexity trade-off in Federated Q-learning. 1 Introduction Reinforcement Learning (RL) [Sutton and Barton, 2018] refers to an online sequential decision making paradigm where the learning agent aims to learn an optimal policy, i.e., a policy that maximizes the long-term reward, through repeated interactions with an unknown environment. RL finds applications across a diverse array of fields including, but not limited to, autonomous driving, games, recommendation systems, robotics and Internet of Things (IoT) [Kober et al., 2013, Yurtsever et al., 2020, Silver et al., 2016, Lim et al., 2020]. The primary hurdle in RL applications is often the high-dimensional nature of the decision space that necessitates the learning agent to have to access to an enormous amount of data in order to have any hope of learning the optimal policy. Moreover, the sequential collection of such an enormous amount of data through a single agent is extremely time-consuming and often infeasible in practice. Consequently, practical implementations of RL involve deploying multiple agents to collect data in parallel. This decentralized approach to data collection has fueled the design and development of distributed or federated RL algorithms that can collaboratively learn the optimal policy without actually transferring the collected data to a centralized server. Such a federated approach to RL, which does not require the transfer of local data, is gaining interest due to lower bandwidth requirements and lower security and privacy risks. In this work, we focus on federated variants of Q-learning algorithms where the agents collaborate to directly learn the optimal Q-function without forming an estimate of the underlying unknown environment. A particularly important aspect of designing Federated RL algorithms, including Federated Q-learning algorithms, is to address the natural tension between sample and communication complexity. At one end of the spectrum lies the naïve approach of running a centralized algorithm with optimal sample complexity after transferring and combining all the collected data at a central facility/server. Such an approach trivially achieves the optimal sample complexity while suffering from a very high and infeasible communication complexity. On the other hand, several recently proposed 38th Conference on Neural Information Processing Systems (NeurIPS 2024). algorithms [Khodadadian et al., 2022, Woo et al., 2023] operate in more practical regimes, offering significantly lower communication complexities as compared to the naïve approach at the cost of sub-optimal sample complexities. These results suggest the existence of underlying trade-off between sample and communication complexities of Federated RL algorithms. The primary goal of this work is to better understand this trade-off in context of Federated Q-learning by investigating these following fundamental questions: • Fundamental limit of communication: What is the minimum amount of communication required by a federated Q-learning algorithm to achieve any statistical benefit of collaboration? • Optimal algorithm design: How does one design a federated Q-Learning algorithm that simultaneously offers optimal order sample and communication complexity guarantees i.e., operates on the optimal frontier of sample-communication complexity trade-off? 1.1 Main Results We consider a setup where M distributed agents collaborate to learn the optimal Q-function of an infinite horizon Markov Decision Process which is defined over a finite state space S and a finite action set A, and has a discount factor of γ ∈(0, 1). We consider a commonly considered setup in federated learning called the intermittent communication setting, where the clients intermittently share information among themselves with the help of a central server. In this work, we provide a complete characterization of the trade-off between sample and communication complexity under the aforementioned setting by providing answers to both the questions. The main result of this work is twofold and is summarized below. • Fundamental bounds on communication complexity of Federated Q-learning: We establish lower bounds on the communication complexity of Federated Q-learning, both in terms of number of communication rounds and the overall number of bits that need to be transmitted in order to achieve any speed up in convergence with respect to the number of agents. Specifically, we show that in order for an intermittent communication algorithm to obtain any benefit of collaboration, i.e., any order of speed up w.r.t. the number of agents, the number of communication rounds must be least Ω( 1 (1−γ) log2 N ) and the number of bits sent by each agent to the server must be least Ω( \|S\|\|A\| (1−γ) log2 N ), where N denotes the number of samples taken by the algorithm for each state-action pair. • Achieving the optimal sample-communication complexity trade-off: We propose a new Federated Q-Learning algorithm called Federated Doubly Variance Reduced Q Learning, Fed-DVR-Q for short, that simultaneously achieves optimal order of sample complexity and the minimal order of communication as dictated by the lower bound. We show that Fed-DVR-Q learns an ε-optimal Q-function in the ℓ∞sense with ˜O \|S\|\|A\| Mε2(1−γ)3 i.i.d. samples from the generative model at each agent while incurring a total communication cost of ˜O \|S\|\|A\| (1−γ) bits per agent across ˜O 1 (1−γ) rounds of communication. Thus, Fed-DVR-Q not only improves upon both the sample and communication complexities of existing algorithms, but also is the first algorithm to achieve both order-optimal sample and communication complexities (See Table 1 for a comparison). 1.2 Related Work Single agent Q-Learning. Q-Learning has been extensively studied in the single-agent setting in terms of both its asymptotic convergence [Jaakkola et al., 1993, Tsitsiklis, 1994, Szepesvári, 1997, Borkar and Meyn, 2000] and its finite-time sample complexity in both synchronous [Even-Dar and Mansour, 2004, Beck and Srikant, 2012, Wainwright, 2019a, Chen et al., 2020, Li et al., 2023] and asynchronous settings [Chen et al., 2021b, Li et al., 2023, 2021, Qu and Wierman, 2020]. Distributed RL. There has also been a considerable effort towards developing distributed and federated RL algorithms. The distributed variants of the classical TD learning algorithm have been investigated in a series of studies [Chen et al., 2021c, Doan et al., 2019, 2021, Sun et al., 2020, Wai, 2020, Wang et al., 2020, Zeng et al., 2021b]. The impact of environmental heterogeneity in federated TD learning was studied in Wang et al. [2023]. A distributed version of actor-critic 2 Algorithm/Reference Number of Agents Sample Complexity Communication Complexity Q-learning [Li et al., 2023] 1 \|S\|\|A\| (1 −γ)4ε2 N/A Variance Reduced Q-learning [Wainwright, 2019b] 1 \|S\|\|A\| (1 −γ)3ε2 N/A Fed-SynQ [Woo et al., 2023] M \|S\|\|A\| M(1 −γ)5ε2 M 1 −γ Fed-DVR-Q (This work) M \|S\|\|A\| M(1 −γ)3ε2 1 1 −γ Lower bound ([Azar et al., 2013], This work) M \|S\|\|A\| M(1 −γ)3ε2 1 1 −γ Table 1: Comparison of sample and communication complexity of various single-agent and Federated Q-learning algorithms for learning an ε-optimal Q-function under the synchronous setting. We hide logarithmic factors and burn-in costs for all results for simplicity of presentation. In the above table, S and A represent state and action spaces respectively and γ denotes the discount factor. We report the communication complexity only in terms of number of rounds as other algorithms assume transmission of real numbers and hence do not report bit level costs. For the lower bound, Azar et al. [2013] and this work establish the bound for sample and communication complexity respectively. algorithms was studied by Shen et al. [2023] where the authors established convergence of their algorithm and demonstrated a linear speed up in the number of agents in their sample complexity bound. Chen et al. [2022] proposed a new distributed actor-critic algorithm which improved the dependence of sample complexity on the error ε and incurs a communication cost of ˜O(ε−1). Chen et al. [2021a] have proposed a communication efficient distributed policy gradient algorithm and have analyzed its convergence and established a communication complexity of O(1/(Mε)). Xie and Song [2023] adopts a distributed policy optimization perspective, which is different from the Q-learning paradigm considered in this work. Moreover, the algorithm in Xie and Song [2023] obtains a linear communication cost, which is worse than that obtained in our work. Similarly, Zhang et al. [2024] focuses on on-policy learning and incurs a communication cost that depends polynomially on the required error ε. Several other studies [Yang et al., 2023, Zeng et al., 2021a, Lan et al., 2024] have also developed and analyzed other distributed/federated variants of the classical natural policy gradient method [Kakade, 2001]. Assran et al. [2019], Espeholt et al. [2018], Mnih et al. [2016] have developed distributed algorithms to train deep RL networks more efficiently. Distributed Q-learning. Federated Q-learning has been explored relatively recently. Khodadadian et al. [2022] proposed and analyzed a federated Q-learning algorithm in the asynchronous setting with a sample complexity of ˜O \|S\|2 Mµ5 min(1−γ)9ε2 , where µmin is the minimum entry of the stationary state-action occupancy distribution of the sample trajectories over all agents. Jin et al. [2022] study the impact of environmental heterogeneity across clients in Federated Q-learning. They propose an algorithm where the local environments are different at each client but each client knows their local environment. Under this setting, they propose an algorithm that achieves a sample and communication complexity of O( 1 (1−γ)3ε) and O( 1 (1−γ)3ε) rounds respectively. Woo et al. [2023] proposed new algorithms with improved analysis for Federated Q-learning under both synchronous and asynchronous settings. Their proposed algorithm achieves a sample complexity and communication complexity of ˜O( \|S\|\|A\| M(1−γ)5ε2 ) and ˜O( M\|S\|\|A\| 1−γ ) real numbers respectively under the synchronous setting and that of ˜O( 1 Mµavg(1−γ)5ε2 ) and ˜O M\|S\|\|A\| 1−γ real numbers respectively under the asynchronous setting. Here, µavg denotes the minimum entry of the average stationary state-action occupancy distribution of all agents. In a follow up work, Woo et al. [2024] propose a Federated Qlearning for offline RL in finite horizon setting and establish a sample and communication complexity of ˜O( H7\|S\|Cavg Mε2 ) and ˜O(H), where H denotes the length of the horizon and Cavg denotes the average single-policy concentrability coefficient of all agents. 3 Accuracy-Communication Trade-off in Federated Learning. The trade-off between communication complexity and accuracy (equivalently, sample complexity) has been studied in various federated and distributed learning problems, including stochastic approximation algorithms for convex optimization. Duchi et al. [2014], Braverman et al. [2016] establish the celebrated inverse linear relationship between the error and the communication cost the problem of distributed mean estimation. Similar trade-off for distributed stochastic optimization, multi-armed bandits and linear bandits has been studied and established across numerous studies [Woodworth et al., 2018, 2021, Tsitsiklis and Luo, 1987, Shi and Shen, 2021, Salgia and Zhao, 2023]. 2 Problem Formulation and Preliminaries In this section, we provide a brief background of Markov Decision Processes, outline the performance measures for Federated Q-learning algorithms and describe the class of intermittent communication algorithms considered in this work. 2.1 Markov Decision Processes In this work, we focus on an infinite-horizon Markov Decision Process (MDP), denoted by M, over a state space S and an action space A and with a discount factor γ ∈(0, 1). Both the state and action spaces are assumed to be finite sets. In an MDP, the state s evolves dynamically under the influence of actions based on a probability transition kernel, P : (S × A) × S →[0, 1]. The entry P(s′\|s, a) denotes the probability of moving to state s′ when an action a is taken in the state s. An MDP is also associated with a deterministic reward function r : S × A →[0, 1], where r(s, a) denotes the immediate reward obtained for taking the action a in the state s. Thus, the transition kernel P along with the reward function r completely characterize an MDP. In this work, we consider the synchronous setting, where each agent has access to an independent generative model or simulator from which they can draw independent samples from the unknown underlying distribution P(·\|s, a) for each state-action pair (s, a) [Kearns and Singh, 1998]. A policy π : S →∆(A) is a rule for selecting actions across different states, where ∆(A) denotes the simplex over A and π(a\|s) denotes the probability of choosing action a in a state s. Each policy π is associated with a state value function and a state-action value function, or the Q-function, denoted by V π and Qπ respectively. V π and Qπ measure the expected discounted cumulative reward attained by π starting from a particular state s and state-action pair (s, a) respectively. Mathematically, V π and Qπ are given as V π(s) := E " ∞ X t=0 γtr(st, at) s0 = s # ; Qπ(s, a) := E " ∞ X t=0 γtr(st, at) s0 = s, a0 = a # , (1) where at ∼π(·\|st) and st+1 ∼P( · \|st, at) for all t ≥0. The expectation is taken w.r.t. the randomness in the trajectory {st, at}∞ t=1. Since the rewards lie in [0, 1], it follows immediately that both the value function and Q-function lie in the range [0, 1 1−γ ]. An optimal policy π⋆is a policy that maximizes the value function uniformly over all the states and it has been shown that such an optimal policy π⋆always exists [Puterman, 2014]. The optimal value and Q-functions are those corresponding to that of an optimal policy π⋆are denoted as V ⋆:= V π⋆ and Q⋆:= Qπ⋆respectively. The optimal Q-function, Q⋆, is also the unique fixed point of the Bellman operator T : S × A →S × A, given by (T Q)(s, a) = r(s, a) + γ · Es′∼P (·\|s,a) max a′∈A Q(s′, a′) . (2) Q-learning [Watkins and Dayan, 1992] aims to learn the optimal policy by first learning Q⋆as the solution to the fixed point equation T Q = Q and then obtain a deterministic optimal policy via the maximization π⋆(s) = arg maxa Q⋆(s, a). Let Z ∈S\|S\|\|A\| be a random vector whose (s, a)th coordinate is drawn from the distribution P(·\|s, a), independently of all other coordinates. We define the random operator TZ : (S × A) →(S × A) as (TZQ)(s, a) = r(s, a) + γV (Z(s, a)), (3) 4 where V (s′) = maxa′∈A Q(s′, a′). The operator TZ can be interpreted as the sample Bellman Operator, where we have the relation T Q = EZ[TZQ] for all Q-functions. Lastly, the federated learning setup considered in this work consists of M agents, where all the agents face a common, unknown MDP, i.e., the transition kernel and the reward functions are the same across agents, which is popularly known as the homogeneous setting. For a given value of ε ∈(0, 1 1−γ ), the objective of agents is to collaboratively learn an ε-optimal estimate (in the ℓ∞sense) of the optimal Q-function of the unknown MDP. 2.2 Performance Measures We measure the performance of a Federated Q-learning algorithm A using two metrics — sample complexity and communication complexity. For a given MDP M, let bQM(A , N, M) denote the estimate of Q⋆ M, the optimal Q-function of the MDP M, returned by an algorithm A , when given access to N i.i.d. samples from the generative model for each (s, a) pair at all the M agents. The minimax error rate of the algorithm A , denoted by ER(A ; N, M), is defined as ER(A ; N, M) := sup M=(P,r) E h ∥bQM(A , N, M) −Q⋆ M∥∞ i , (4) where the expectation is taken over the samples and any randomness in the algorithm. Given a value of ε > 0, the sample complexity of A , denoted by SC(A ; ε, M) is given as SC(A ; ε, M) := \|S\|\|A\| · min{N ∈N : ER(A ; N, M) ≤ε}. (5) Similarly, we can also define a high-probability version for any δ ∈(0, 1) as follows: SC(A ; ε, M, δ) := \|S\|\|A\| · min{N ∈N : Pr(sup M ∥bQM(A , N, M) −Q⋆ M∥∞≤ε) ≥1 −δ}. We measure the communication complexity of any federated learning algorithm both in terms of frequency of information exchange and total number of bits uploaded by the agents. For each agent m, let Cm round(A ; N) and Cm bit(A ; N) respectively denote the number of times agent m sends a message to the server and the total number of bits uploaded by agent m to the server when an algorithm A is run with N i.i.d. samples from the generative model for each (s, a) pair at each agent. The communication complexity of A , when measured in terms of frequency of communication and total number of bits exchanged, is given by CCround(A ; N) := 1 M M X m=1 Cm round(A ; N); CCbit(A ; N) := 1 M M X m=1 Cm bit(A ; N), (6) respectively. Similarly, for a given value of ε ∈(0, 1 1−γ ), we can also define CCround(A ; ε) and CCbit(A ; ε) based on when A is run to guarantee a minimax error of at most ε. 2.3 Intermittent Communication Algorithms Algorithm 1: A generic algorithm A 1: Input : T, R, {ηt}T t=1, C = {tr}R r=1, B 2: Set Qm 0 ←0 for all agents m 3: for t = 1, 2, . . . , T do 4: for m = 1, 2, . . . , M do 5: Compute Qm t−1 2 according to Eqn. 7 6: Compute Qm t according to Eqn. 8 7: end for 8: end for 9: return QT In this work, we consider a popular class of federated learning algorithms referred to as algorithms with intermittent communication. The intermittent communication setting provides a natural framework to extend single agent Qlearning algorithms to the distributed setting. As the name suggests, under this setting, the agents intermittently communicate with each other, sharing their updated beliefs about Q⋆. Between two communication rounds, each agent updates their belief about Q⋆using stochastic fixed point iteration based on the locally available data, similar to a single agent setup. Such intermittent communication algorithms have been extensively studied and used to establish lower bounds on communication complexity of distributed stochastic convex optimization [Woodworth et al., 2018, 2021]. 5 A generic Federated Q-learning algorithm with intermittent communication is outlined in Algorithm 1. It is characterized by the following five parameters: (i) total number of updates T; (ii) the number of communication rounds R; (iii) a step size schedule {ηt}T t=1; (iv) a communication schedule {tr}R r=1; (v) batch size B. During the tth iteration, each agent m computes {bTZb(Qm t−1)}B b=1, a minibatch of sample Bellman operators at the current estimate Qm t−1, using B samples from the generative model for each (s, a) pair, and obtains an intermediate local estimate using the Q-learning update as follows: Qm t−1 2 = (1 −ηt)Qm t−1 + ηt B B X b=1 TZb(Qm t−1). (7) Here ηt ∈(0, 1] is the step-size chosen corresponding to the tth time step. The intermediate estimates are averaged based on a communication schedule C = {tr}R r=1 consisting of R rounds, i.e., Qm t = ( 1 M PM j=1 Qj t−1 2 if t ∈C, Qm t−1 2 otherwise. (8) In the above equation, the averaging step can also be replaced with any distributed mean estimation routine that includes compression to control the bit level costs. Without loss of generality, we assume that Qm 0 = 0 for all agents m and tR = T, i.e., the last iterates are always averaged. It is straightforward to note that the number of samples taken by an intermittent communication algorithm is BT, i.e, N = BT and the communication complexity CCround = R. 3 Lower Bound In this section, we investigate the first of the two questions regarding the lower bound on communication complexity. The following theorem establishes a lower bound on the communication complexity of a Federated Q-learning algorithm with intermittent communication. Theorem 1. Assume that γ ∈[5/6, 1) and the state and action spaces satisfy \|S\| ≥4 and \|A\| ≥ 2. Let A be a Federated Q-learning algorithm with intermittent communication that is run for T ≥max{16, 1 1−γ } steps with a step size schedule of either ηt := 1 1+cη(1−γ)t or ηt := η for all 1 ≤t ≤T. If R = CCround(A ; N) ≤ c0 (1 −γ) log2 N ; or CCbit(A ; N) ≤ c1\|S\|\|A\| (1 −γ) log2 N for some universal constants c0, c1 > 0 then, for all choices of communication schedule, batch size B, cη > 0 and η ∈(0, 1), the minimax error of A satisfies ER(A ; N, M) ≥ Cγ log3 N √ N , for all M ≥2 and N = BT. Here Cγ > 0 is a constant that depends only on γ. The above theorem states that in order for an intermittent communication algorithm to obtain any benefit of collaboration, i.e., for the error rate ER(A ; N, M) to decrease w.r.t. number of agents, the number of communication rounds must be least Ω( 1 (1−γ) log2 N ). This implies that any Federated Q-learning algorithm that offers order optimal sample complexity, and thereby also a linear speed up with respect to the number of agents, must have at least Ω( 1 (1−γ) log2 N ) rounds of communication and transmit Ω( \|S\|\|A\| (1−γ) log2 N ) bits of information per agent. This characterizes the converse relation for the sample-communication tradeoff in Federated Q-learning. We would like to point out that our lower bound extends to the asynchronous setting as the assumption of i.i.d. noise corresponding to a generative model is a special case of Markovian noise observed in asynchronous setting. The lower bound on the communication complexity of Federated Q-learning is a consequence of the bias-variance trade-off that governs the convergence of the algorithm. While a careful choice of step-sizes alone is sufficient to balance this trade-off in the centralized setting, the choice of communication schedule also plays an important role in balancing this trade-off in the federated setting. The local steps between two communication rounds induce a positive estimation bias that 6 depends on the standard deviation of the iterates and is a well-documented issue of “over-estimation” in Q-learning [Hasselt, 2010]. Since such a bias is driven by local updates, it does not reflect any benefit of collaboration. During a communication round, the averaging of iterates across agents allows the algorithm an opportunity to counter this bias by reducing the effective variance of the updates through averaging. In our analysis, we show that if the communication is infrequent, the local bias becomes the dominant term and averaging of iterates is insufficient to counter the impact of the positive bias induced by the local steps. As a result, we do not observe any statistical gains when the communication is infrequent. The analysis is inspired the analysis of Q-learning by Li et al. [2023] and is based on analyzing the convergence of an intermittent communication algorithm on a specifically chosen “hard” instance of MDP. Please refer to Appendix B for a detailed proof. Remark 1 (Communication complexity of policy evaluation). Several recent studies [Liu and Olshevsky, 2023, Tian et al., 2024] established that a single round of communication is sufficient to achieve linear speedup of TD learning for policy evaluation, which do not contradict with our results focusing on Q-learning for policy learning. The latter is more involved due to the nonlinearity of the Bellman optimality operator. Specifically, if the operator whose fixed point is to be found is linear in the decision variable (e.g., the value function in TD learning) then the fixed point update only induces a variance term corresponding to the noise. However, if the operator is non-linear, then in addition to the variance term, we also obtain a bias term in the fixed point update. While the variance term can be controlled with one-shot averaging, more frequent communication is necessary to ensure that the bias term is small enough. Remark 2 (Extension to asynchronous Q-learning). We would like to point out that our lower bound extends to the asynchronous setting [Li et al., 2023] as the assumption of i.i.d. noise corresponding to a generative model is a special case of Markovian noise observed in the asynchronous setting. 4 The Fed-DVR-Q algorithm Having characterized the lower bound on the communication complexity of Federated Q-learning, we explore our second question of interest — designing a federated Q-learning algorithm that achieves this lower bound while simultaneously offering an optimal order of sample complexity. We propose a new Federated Q-learning algorithm, Fed-DVR-Q, that achieves not only a communication complexity of CCround = ˜O( 1 1−γ ) and CCbit = ˜O( \|S\|\|A\| 1−γ ) but also the optimal order of sample complexity (upto logarithmic factors), thereby providing a tight characterization of the achievability frontier that matches with the converse result derived in the previous section. 4.1 Algorithm Description Algorithm 2: Fed-DVR-Q 1: Input : Error bound ε > 0, failure probability δ > 0 2: k ←1, Q(0) ←0 3: // Set parameters as described in Sec. 4.1.3 4: for k = 1, 2, . . . , K do 5: Q(k) ← REFINEESTIMATE(Q(k−1), B, I, Lk, Dk, J) 6: k ←k + 1 7: end for 8: return Q(K) Fed-DVR-Q proceeds in epochs. During an epoch k ≥1, the agents collaboratively update Q(k−1), the estimate of Q⋆obtained at the end of previous epoch, to a new estimate Q(k), with the aid of the sub-routine called REFINEESTIMATE. The sub-routine REFINEESTIMATE is designed to ensure that the suboptimality gap, ∥Q(k) −Q⋆∥∞, reduces by a factor of 2 at the end of every epoch. Thus, at the end of K = O(log(1/ε)) epochs, Fed-DVR-Q obtains a ε-optimal estimate of Q⋆, which is then set to be the output of the algorithm. Please refer to Alg. 2 for a pseudocode. 4.1.1 The REFINEESTIMATE sub-routine REFINEESTIMATE, starting from Q, an initial estimate of Q⋆, uses variance reduced Q-learning updates to obtain an improved estimate of Q⋆. It is characterized by four parameters — the initial estimate Q, the number of local iterations I, the recentering sample size L and the batch size B, 7 which can be appropriately tuned to control the quality of the returned estimate. Additionally, it also takes input two parameters D and J required by the compressor. The first step in REFINEESTIMATE is to collaboratively approximate T Q for the variance reduced updates. To this effect, each agent m builds an approximation of T Q as follows: eT (m) L (Q) := 1 ⌈L/M⌉ ⌈L/M⌉ X l=1 TZ(m) l (Q), (9) where {Z(m) 1 , Z(m) 2 , . . . , Z(m) ⌈L/M⌉} are ⌈L/M⌉i.i.d. samples with Z(m) 1 ∼Z. Each agent sends C eT (m) L (Q) −Q; D, J , a compressed version of the difference eT (m) L (Q)−Q, to the server, which collects all the estimates from the agents and constructs the estimate eTL(Q) = Q + 1 M M X m=1 C eT (m) L (Q) −Q; D, J (10) and sends it back to the agents for the variance reduced updates. We defer the description of the compression routine to the end of this section. Equipped with the estimate eTL(Q), REFINEESTIMATE constructs a sequence {Qi}I i=1 using the following iterative update scheme initialized with Q0 = Q. During the ith iteration, each agent m carries out the following update: Qm i−1 2 = (1 −η)Qi−1 + η h bT (m) i Qi−1 −bT (m) i Q + eTL(Q) i . (11) In the above equation, η ∈(0, 1) is the step size and bT (m) i Q := 1 B P z∈Z(m) i TzQ, where Z(m) i is the minibatch of B i.i.d. random variables drawn according to Z, independently at each agent m for all iterations i. Each agent then sends a compressed version of the update, i.e., C Qm i−1 2 −Qi−1; D, J , to the server, which uses them to compute the next iterate Qi = Qi−1 + 1 M M X m=1 C Qm i−1 2 −Qi−1; D, J , (12) and broadcast it to the clients. After I such updates, the obtained iterate QI is returned by the routine. A pseudocode of the REFINEESTIMATE routine is given in Algorithm 3 in Appendix A. 4.1.2 The Compression Operator The compressor, C (·; D, J), used in the proposed algorithm Fed-DVR-Q is based on the popular stochastic quantization scheme. In addition to the input vector Q to be quantized, the quantizer C takes two input parameters D and J. D corresponds to an upper bound on ℓ∞norm of Q, i.e., ∥Q∥∞≤D. J corresponds to the resolution of the compressor, i.e., number of bits used by the compressor to represent each coordinate of the output vector. The compressor first splits the interval [0, D] into 2J −1 intervals of equal length where 0 = d1 < d2, · · · < d2J = D correspond to end points of the intervals. Each coordinate of Q is then separately quantized as follows. The value of the nth coordinate, C (Q)[n], is set to be djn−1 with probability djn−Q[n] djn−djn−1 and to djn with the remaining probability, where jn := min{j : dj < Q[i] ≤dj+1}. It is straightforward to note that each coordinate of C (Q) can be represented using J bits. 4.1.3 Setting the parameters The desired convergence of the iterates {Q(k)} is obtained by carefully choosing the parameters of the sub-routine REFINEESTIMATE and the compression operator C . For all epochs k ≥1, we set the number of iterations I and the batch size B of REFINEESTIMATE and the number of bits J of the compressor C to be ⌈ 2 η(1−γ)⌉, ⌈2 M ( 12γ (1−γ))2 log( 8KI\|S\|\|A\| δ )⌉and ⌈log2( 70 η(1−γ) q 2 M log( 8KI\|S\|\|A\| δ ))⌉respectively. The total number of epochs is set to K = ⌈1 2 log2( 1 1−γ )⌉+ ⌈1 2 log2( 1 (1−γ)ε2 )⌉. The recentering sample sizes Lk and bounds Dk are set to be the following functions of epoch index k: Lk := 19600 (1 −γ)2 log 8KI\|S\|\|A\| δ · 4k if k ≤K0 4k−K0 if k > K0 ; Dk := 16 · 2−k 1 −γ , (13) 8 where K0 = ⌈1 2 log2( 1 1−γ )⌉. The piecewise definition of Lk is crucial to obtain the optimal dependence with respect to 1 1−γ , similar to the two-step procedure outlined in Wainwright [2019b]. 4.2 Performance Guarantees The following theorem characterizes the sample and communication complexity of Fed-DVR-Q. Theorem 2. Consider any δ ∈(0, 1) and ε ∈(0, 1]. Under the federated learning setup described in Section 2.1, the sample and communication complexities of the Fed-DVR-Q algorithm, when run with the choice of parameters described in Sec. 4.1.3 and a learning rate η ∈(0, 1), satisfy the following relations for some universal constant C1 > 0: SC(Fed-DVR-Q; ε, M, δ) ≤ C1 ηM(1 −γ)3ε2 log2 1 (1 −γ)ε log 8KI\|S\|\|A\| δ , CCround(Fed-DVR-Q; ε, δ) ≤ 16 η(1 −γ) log2 1 (1 −γ)ε , CCbit(Fed-DVR-Q; ε, δ) ≤32\|S\|A\| η(1 −γ) log2 1 (1 −γ)ε log2 70 η(1 −γ) s 2 M log 8KI\|S\|\|A\| δ ! . A proof of Theorem 2 can be found in Appendix C. A few implications of the theorem are in order. Optimal Sample-Communication complexity trade-off. As shown by the above theorem, FedDVR-Q offers a linear speed up in the sample complexity with respect to the number of agents while simultaneously achieving the same order of communication complexity as dictated by the lower bound derived in Theorem 1, both in terms of frequency and bit level complexity. Moreover, Fed-DVR-Q is the first Federated Q-Learning algorithm that achieves a sample complexity with optimal dependence on all the salient parameters, i.e., \|S\|, \|A\| and 1 1−γ , in addition to linear speedup w.r.t. to number of agents and thereby bridges the existing gap between upper and lower bounds on sample complexity for Federated Q-learning. Thus, Theorem 1 and 2 together provide a characterization of optimal operating point of the sample-communication complexity trade-off in Federated Q-learning. Role of Minibatching. The commonly adopted approach in intermittent communication algorithm is to use a local update scheme that takes multiple small (i.e., B = O(1)), noisy updates between communication rounds, as evident from the algorithm design in Khodadadian et al. [2022], Woo et al. [2023] and even numerous FL algorithms for stochastic optimization McMahan et al. [2017], Haddadpour et al. [2019], Khaled et al. [2020]. In Fed-DVR-Q, we replace the local update scheme of taking multiple small, noisy updates by a single, large update with smaller variance, obtained by averaging the noisy updates over a minibatch of samples. The use of updates with smaller variance in variance reduced Q-learning yields the algorithm its name. While both the approaches result in similar sample complexity guarantees, the local update scheme requires more frequent averaging across clients to ensure that the bias of the estimate, also commonly referred to as “client drift”, is not too large. On the other hand, the minibatching approach does not encounter the problem of bias accumulation from local updates and hence can afford more infrequent averaging allowing Fed-DVR-Q to achieve optimal order of communication complexity. Compression. Fed-DVR-Q is the first algorithm in Federated Q-Learning to analyze and establish communication complexity at the bit level. All existing studies on Federated RL focus only on the frequency of communication and assume transmission of real numbers with infinite bit precision. On the other hand, the our analysis provides a more holistic view point of communication complexity and provides bounds at the bit level, which is of great practical significance. While some recent other studies [Wang et al., 2023] also consider quantization in Federated RL, their objective is to understand the impact of message size on convergence with no constraint on the frequency of communication, unlike the holistic viewpoint adopted in this work. 9 5 Conclusion and Future Directions We presented a complete characterization of the sample-communication trade-off for Federated Q-learning algorithms with intermittent communication. We showed that no Federated Q-learning algorithm with intermittent communication can achieve a linear speedup with respect to the number of agents if its number of communication rounds are sublinear in 1 1−γ . We also proposed a new Federated Q-learning algorithm called Fed-DVR-Q that uses variance reduction along with minibatching to achieve optimal-order sample and communication complexities. In particular, we showed that FedDVR-Q has a sample complexity of ˜O( \|S\|\|A\| M(1−γ)3ε2 ), which is order-optimal in all salient problem parameters, and a communication complexity of ˜O( 1 1−γ ) rounds and ˜O( \|S\|\|A\| 1−γ ) bits. The results in this work raise several interesting questions that are worth exploring. While we focus on the tabular setting in this work, it is of great interest to investigate to the trade-off in other settings where we use function approximation to model the Q⋆and V ⋆functions. 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Algorithm 3: REFINEESTIMATE(Q, B, I, L, D, J) 1: Input: Initial estimate Q, batch size B, Number of iterations I, recentering sample size L, quantization bound D, message size J 2: // Build an approximation for T Q which is to be used for variance reduced updates 3: for m = 1, 2, . . . , M do 4: Draw ⌈L/M⌉i.i.d. samples from the generative model for each (s, a) pair and evaluate eT (m) L (Q) according to Eqn. (9) 5: Send C (eT (m) L (Q) −Q; D, J) to the server 6: Receive 1 M PM m=1 C (eT (m) L (Q) −Q; D, J) from the server and compute eTL(Q) according to Eqn. (10) 7: end for 8: Q0 ←Q 9: // Variance reduced updates with minibatching 10: for i = 1, 2, . . . , I do 11: for m = 1, 2, . . . , M do 12: Draw B i.i.d. samples from the from the generative model for each (s, a) pair 13: Compute Qm i−1 2 according to Eqn. (11) 14: Send C (Qm i−1 2 −Qi−1; D, J) to the server 15: Receive 1 M PM m=1 C (Qm i −Qi−1; D, J) from the server and compute Qi according to Eqn. (12) 16: end for 17: end for 18: return QI B Proof of Theorem 1 In this section, we prove the main result of the paper, the lower bound on the communication complexity of federated Q-learning algorithms. At a high level, the proof consists of the following three steps. Introducing the “hard” MDP instance. The proof builds upon analyzing the behavior of a generic algorithm A outlined in Algorithm 1 over a particular instance of MDP. The particular choice of MDP is inspired by, and borrowed from, other lower bound proofs in the single-agent setting [Li et al., 2023] and helps highlight core issues that lie at the heart of the sample-communication complexity trade-off. Following Li et al. [2023], the construction is first over a small state-action space that allows us to focus on a simpler problem before generalizing it to larger state-action spaces. Establishing the performance of intermittent communication algorithms. In the second step, we analyze the error of the iterates generated by an intermittent communication algorithm A . The analysis is inspired by the single-agent analysis in Li et al. [2023], which highlights the underlying bias-variance trade-off. Through careful analysis of the algorithm dynamics in the federated setting, we uncover the impact of communication on the bias-variance trade-off and the resulting error of the iterates to obtain the lower bound on the communication complexity. Generalization to larger MDPs. As the last step, we generalize our construction of the “hard” instance to more general state-action space and extend our insights to obtain the statement of the theorem. 14 B.1 Introducing the “hard” instance We first introduce an MDP instance Mh that we will use throughout the proof to establish the result. Note that this MDP is identical to the one considered in Li et al. [2023] to establish the lower bounds on the performance of single-agent Q-learning algorithm. It consists of four states S = {0, 1, 2, 3}. Let As denote the action set associated with the state s. The probability transition kernel and the reward function of Mh is given as follows: A0 = {1} P(0\|0, 1) = 1 r(0, 1) = 0, (14a) A1 = {1, 2} P(1\|1, 1) = p P(0\|1, 1) = 1 −p r(1, 1) = 1, (14b) P(1\|1, 2) = p P(0\|1, 2) = 1 −p r(1, 2) = 1, (14c) A2 = {1} P(2\|2, 1) = p P(0\|2, 1) = 1 −p r(2, 1) = 1, (14d) A3 = {1} P(3\|3, 1) = 1 r(3, 1) = 1, (14e) where the parameter p = 4γ −1 3γ . We have the following results about the optimal Q and V functions of this hard MDP instance. Lemma 1 ([Li et al., 2023, Lemma 3]). Consider the MDP Mh constructed in Eqn. (14). We have, V ⋆(0) = Q⋆(0, 1) = 0 V ⋆(1) = Q⋆(1, 1) = Q⋆(1, 2) = V ⋆(2) = Q⋆(2, 1) = 1 1 −γp = 3 4(1 −γ) V ⋆(3) = Q⋆(3, 1) = 1 1 −γ . Throughout the next section of the proof, we focus on this MDP with four states and two actions. In Appendix B.4, we generalize the proof to larger state-action spaces. B.2 Notation and preliminary results For convenience, we first define some notation that will be used throughout the proof. Useful relations of the learning rates. We consider two kinds of step size sequences that are commonly used in Q-learning — the constant step size schedule, i.e., ηt = η for all t ∈{1, 2, . . . , T} and the rescaled linear step size schedule, i.e., ηt = 1 1+cη(1−γ)t, where cη > 0 is a universal constant that is independent of the problem parameters. We define the following quantities: η(t) k = ηk tY i=k+1 (1 −ηi(1 −γp)) for all 0 ≤k ≤t, (15) where we take η0 = 1 and use the convention throughout the proof that if a product operation does not have a valid index, we take the value of that product to be 1. For any integer 0 ≤τ < t, we have the following relation, which will be proved at the end of this subsection for completeness: tY k=τ+1 (1 −ηk(1 −γp)) + (1 −γp) t X k=τ+1 η(t) k = 1. (16) Similarly, we also define, eη(t) k = ηk tY i=k+1 (1 −ηi) for all 0 ≤k ≤t, (17) which satisfies the relation tY k=τ+1 (1 −ηk) + t X k=τ+1 eη(t) k = 1. (18) 15 for any integer 0 ≤τ < t. The claim follows immediately by plugging p = 0 in (16). Note that for constant step size, the sequence eη(t) k is clearly increasing. For the rescaled linear step size, we have, eη(t) k−1 eη(t) k = ηk ηk−1(1 −ηk) = 1 −(1 −cη(1 −γ))ηk 1 −cη(1 −γ)ηk ≤1 (19) whenever cη ≤ 1 1−γ . Thus, eη(t) k is an increasing sequence as long as cη ≤ 1 1−γ . Similarly, η(t) k is also clearly increasing for the constant step size schedule. For the rescaled linear step size schedule, we have, η(t) k−1 η(t) k = ηk ηk−1(1 −ηk(1 −γp)) ≤ ηk ηk−1(1 −ηk) ≤1, whenever cη ≤ 1 1−γ . The last bound follows from Eqn. (19). Proof of (16). We can show the claim using backward induction. For the base case, note that, (1 −γp)η(t) t + (1 −γp)η(t) t−1 = (1 −γp)ηt + (1 −γp)ηt−1(1 −(1 −γp)ηt) = 1 −(1 −ηt(1 −γp))(1 −ηt−1(1 −γp)) = 1 − tY k=t−1 (1 −ηk(1 −γp)), as required. Assume (16) is true for some τ. We have, (1 −γp) t X k=τ η(t) k = (1 −γp)ηt τ + (1 −γp) t X k=τ+1 η(t) k = (1 −γp)ητ tY k=τ+1 (1 −ηk(1 −γp)) + 1 − tY k=τ+1 (1 −ηk(1 −γp)) = 1 − tY k=τ (1 −ηk(1 −γp)), thus completing the induction step. Sample transition matrix. Recall Z ∈S\|S\|\|A\| is a random vector whose (s, a)-th coordinate is drawn from the distribution P(·\|s, a). We use bP m t to denote the sample transition at time t and agent m obtained by averaging B i.i.d. samples from the generative model. Specifically let {Zm t,b}B b=1 denote a collection of B i.i.d. copies of Z collected at time t at agent m. Then, for all s, a, s′, bP m t (s′\|s, a) = 1 B B X b=1 P m t,b(s′\|s, a), (20) where P m t,b(s′\|s, a) = 1{Zm t,b(s, a) = s′} for s′ ∈S. Preliminary relations of the iterates. We state some preliminary relations regarding the evolution of the Q-function and the value function across different agents that will be helpful for the analysis later. We begin with the state 0, where we have Qm t (0, 1) = V m t (0) = 0 for all agents m ∈[M] and t ∈[T]. This follows almost immediately from the fact that state 0 is an absorbing state with zero reward. Note that Qm 0 (0, 1) = V m 0 (0) = 0 holds for all clients m ∈[M]. Assuming that Qm t−1(0, 1) = V m t−1(0) = 0 for all clients for some time instant t −1, by induction, we have, Qm t−1/2(0, 1) = (1 −ηt)Qm t−1(0, 1) + ηt(γV m t−1(0)) = 0. Consequently, Qm t (0, 1) = 0 and V m t (0) = 0, for all agents m, irrespective of whether there is averaging. 16 For state 3, the iterates satisfy the following relation: Qm t−1/2(3, 1) = (1 −ηt)Qm t−1(3, 1) + ηt(1 + γV m t−1(3)) = (1 −ηt)Qm t−1(3, 1) + ηt(1 + γQm t−1(3, 1)) = (1 −ηt(1 −γ))Qm t−1(3, 1) + ηt, where the second step follows by noting V m t (3) = Qm t (3, 1). Once again, one can note that averaging step does not affect the update rule implying that the following holds for all m ∈[M] and t ∈[T]: V m t (3) = Qm t (3, 1) = t X k=1 ηk tY i=k+1 (1 −ηi(1 −γ)) ! = 1 1 −γ " 1 − tY i=1 (1 −ηi(1 −γ)) # , (21) where the last step follows from Eqn. (16) with p = 1. Similarly, for state 1 and 2, we have, Qm t−1/2(1, 1) = (1 −ηt)Qm t−1(1, 1) + ηt(1 + γ bP m t (1\|1, 1)V m t−1(1)), (22) Qm t−1/2(1, 2) = (1 −ηt)Qm t−1(1, 2) + ηt(1 + γ bP m t (1\|1, 2)V m t−1(1)), (23) Qm t−1/2(2, 1) = (1 −ηt)Qm t−1(2, 1) + ηt(1 + γ bP m t (2\|2, 1)V m t−1(2)). (24) Since the averaging makes a difference in the update rule, we further analyze the update as required in later proofs. B.3 Main analysis We first focus on establishing a bound on the number of communication rounds, i.e., CCround(A ) (where we drop the dependency with other parameters for notational simplicity), and then use this lower bound to establish the bound on the bit level communication complexity CCbit(A ). To establish the lower bound on CCround(A ) for any intermittent communication algorithm A , we analyze the convergence behavior of A on the MDP Mh. We assume that the averaging step in line 6 of Algorithm 1 is carried out exactly. Since the use of compression only makes the problem harder, it is sufficient for us to consider the case where there is no loss of information in the averaging step for establishing a lower bound. Lastly, throughout the proof, without loss of generality we assume that log N ≤ 1 1 −γ , (25) otherwise, the lower bound in Theorem 1 reduces to the trivial lower bound. We divide the proof into following three parts based on the choice of learning rates and batch sizes: 1. Small learning rates: For constant learning rates, 0 ≤η < 1 (1−γ)T and for rescaled linear learning rates, the constant cη satisfies cη ≥log T. 2. Large learning rates with small ηT /(BM): For constant learning rates, η ≥ 1 (1−γ)T and for rescaled linear learning rates, the constant cη satisfies 0 ≤cη ≤log T ≤ 1 1−γ (c.f. (25)). Additionally, the ratio ηT BM satisfies ηT BM ≤1−γ 100 . 3. Large learning rates with large ηT /(BM): We have the same condition on the learning rates as above. However, in this case the ratio ηT BM satisfies ηT BM > 1−γ 100 . We consider each of the cases separately in the following three subsections. B.3.1 Small learning rates In this subsection, we prove the lower bound for small learning rates, which follow from similar arguments in Li et al. [2023]. 17 For this case, we focus on the dynamics of state 2. We claim that the same relation established in Li et al. [2023] continues to hold, which will be established momentarily: E[V m T (2)] =  1 M M X j=1 E[V j T (2)]  = T X k=1 η(t) k = 1 −η(T ) 0 1 −γp . (26) Consequently, for all m ∈[M], we have V ⋆(2) −E[V m T (2)] = η(T ) 0 1 −γp. (27) To obtain lower bound on V ⋆(2) −E[V m T (2)], we need to obtain a lower bound on η(T ) 0 , which from [Li et al., 2023, Eqn. (120)] we have log(η(T ) 0 ) ≥−1.5 T X t=1 η(1 −γp) ≥−2 T X t=1 1 t log T ≥−2 =⇒ η(T ) 0 ≥e−2 when T ≥16 for both choices of learning rates. On plugging this bound in (27), we obtain, E[∥Qm T −Q⋆∥∞] ≥E[\|Q⋆(2) −Qm T (2)\|] ≥V ⋆(2) −E[V m T (2)] ≥ 3 4e2(1 −γ) √ N (28) holds for all m ∈[M], N ≥1 and M ≥2. Thus, it can be noted that the error rate ER(A ; N, M) is bounded away from a constant value irrespective of the number of agents and the number of communication rounds. Thus, even with CCround = Ω(T), we will not observe any collaborative gain if the step size is too small. Proof of (26). Recall that from (24), we have, Qm t−1/2(2, 1) = (1 −ηt)V m t−1(2) + ηt(1 + γ bP m t (2\|2, 1)V m t−1(2)). Here, Qm t−1(2, 1) = V m t−1(2) as the second state has only a single action. • When t is not an averaging instant, we have, V m t (2) = Qm t (2, 1) = (1 −ηt)V m t−1(2) + ηt(1 + γ bP m t (2\|2, 1)V m t−1(2)). (29) On taking expectation on both sides of the equation, we obtain, E[V m t (2)] = (1 −ηt)E[V m t−1(2)] + ηt(1 + γE[ bP m t (2\|2, 1)V m t−1(2)]) = (1 −ηt)E[V m t−1(2)] + ηt 1 + γE[ bP m t (2\|2, 1)]E[V m t−1(2)] = (1 −ηt)E[V m t−1(2)] + ηt 1 + γpE[V m t−1(2)] = (1 −ηt(1 −γp))E[V m t−1(2)] + ηt. (30) In the second step, we used the fact that bP m t (2\|2, 1) is independent of V m t−1(2). • Similarly, if t is an averaging instant, we have, V m t (2) = Qm t (2, 1) = 1 M M X j=1 Qj t−1/2(2, 1) = (1 −ηt) 1 M M X j=1 V j t−1(2) + 1 M M X j=1 ηt(1 + γ bP j t (2\|2, 1)V j t−1(2)). (31) Once again, upon taking expectation we obtain, E[V m t (2)] = (1 −ηt) 1 M M X j=1 E[V j t−1(2)] + 1 M M X j=1 ηt(1 + γE[ bP j t (2\|2, 1)V j t−1(2)]) 18 = (1 −ηt) 1 M M X j=1 E[V j t−1(2)] + 1 M M X j=1 ηt(1 + γpE[V j t−1(2)]) = (1 −ηt(1 −γp))  1 M M X j=1 E[V j t−1(2)]  + ηt. (32) Eqns. (30) and (32) together imply that for all t ∈[T], 1 M M X m=1 E[V m t (2)] ! = (1 −ηt(1 −γp)) 1 M M X m=1 E[V m t−1(2)] ! + ηt. (33) On unrolling the above recursion with V m 0 = 0 for all m ∈[M], we obtain the desired claim (26). B.3.2 Large learning rates with small ηT BM In this subsection, we prove the lower bound for case of large learning rates when the ratio ηT BM is small. For the analysis in this part, we focus on the dynamics of state 1. Unless otherwise specified, throughout the section we implicitly assume that the state is 1. We further define a key parameter that will play a key role in the analysis: τ := min{k ∈N : ∀t ≥k, ηt ≤ηk ≤3ηt}. (34) It can be noted that for constant step size sequence τ = 1 and for rescaled linear stepsize τ = T/3. Step 1: introducing an auxiliary sequence. We define an auxiliary sequence bQm t (a) for a ∈ {1, 2} and all t = 1, 2, . . . , T to aid our analysis, where we drop the dependency with state s = 1 for simplicity. The evolution of the sequence bQm t is defined in Algorithm 4, where bV m t = maxa∈{1,2} bQm t (a). In other words, the iterates { bQm t } evolve exactly as the iterates of Algorithm 1 except for the fact that sequence { bQm t } is initialized at the optimal Q-function of the MDP. We would like to point out that we assume that the underlying stochasticity is also identical in the sense that the evolution of both Qm t and bQm t is governed by the same bP m t matrices. The following lemma controls the error between the iterates Qm t and bQm t , allowing us to focus only on bQm t . Algorithm 4: Evolution of bQ 1: Input : T, R, {ηt}T t=1, C = {tr}R r=1, B 2: Set bQm 0 (a) ←Q⋆(1, a) for a ∈{1, 2} and all agents m // Different initialization 3: for t = 1, 2, . . . , T do 4: for m = 1, 2, . . . , M do 5: Compute bQm t−1 2 according to Eqn. (7) 6: Compute bQm t according to Eqn. (8) 7: end for 8: end for Lemma 2. The following relation holds for all agents m ∈[M], all t ∈[T] and a ∈{1, 2}: Qm t (1, a) −bQm t (a) ≥− 1 1 −γ tY i=1 (1 −ηi(1 −γ)). By Lemma 2, bounding the error of the sequence bQm t allows us to obtain a bound on the error of Qm t . To that effect, we define the following terms for any t ≤T and all m ∈[M]: ∆m t (a) := bQm t (a) −Q⋆(1, a); a = 1, 2; 19 ∆m t,max = max a∈{1,2} ∆m t (a). In addition, we use ∆t = 1 M PM m=1 ∆m t to denote the error of the averaged iterate1, and similarly, ∆t,max := max a∈{1,2} ∆t(a). (35) We first derive a basic recursion regarding ∆m t (a). From the iterative update rule in Algorithm 4, we have, ∆m t (a) = (1 −ηt)∆m t−1(a) + ηt(1 + γ bP m t (1\|1, a)bV m t−1 −Q⋆(1, a)) = (1 −ηt)∆m t−1(a) + ηtγ( bP m t (1\|1, a)bV m t−1 −pV ⋆(1)) = (1 −ηt)∆m t−1(a) + ηtγ(p(bV m t−1 −V ⋆(1)) + ( bP m t (1\|1, a) −p)bVt−1) = (1 −ηt)∆m t−1(a) + ηtγ(p∆m t−1,max + ( bP m t (1\|1, a) −p)bV m t−1). Here in the last line, we used the following relation: ∆m t,max = max a∈{1,2}( bQm t (a) −Q⋆(1, a)) = max a∈{1,2} bQm t (a) −V ⋆(1) = bV m t−1 −V ⋆(1), as Q⋆(1, 1) = Q⋆(1, 2) = V ⋆(1). Recursively unrolling the above expression, and using the expression (17), we obtain the following relation: for any t′ < t when there is no averaging during the interval (t′, t) ∆m t (a) = tY k=t′+1 (1 −ηk) ! ∆m t′ (a) + t X k=t′+1 eη(t) k γ(p∆m k−1,max + ( bP m k (1\|1, a) −p)bV m k−1). (36) For any t′, t with t′ < t, we define the notation φt′,t := tY k=t′+1 (1 −ηk), (37) ξm t′,t(a) := t X k=t′+1 eη(t) k γ( bP m k (1\|1, a) −p)bV m k−1, a = 1, 2; (38) ξm t′,t,max := max a∈{1,2} ξm t′,t(a). (39) Note that by definition, E[ξm t′,t(a)] = 0 for a ∈{1, 2} and all m, t′ and t. Plugging them into the previous expression leads to the simplified expression ∆m t (a) = φt′,t∆m t′ (a) + " t X k=t′+1 eη(t) k γp∆m k−1,max # + ξm t′,t(a). We specifically choose t′ and t to be the consecutive averaging instants to analyze the behaviour of ∆m t across two averaging instants. Consequently, we can rewrite the above equation as ∆m t (a) = φt′,t∆t′(a) + " t X k=t′+1 eη(t) k γp∆m k−1,max # + ξm t′,t(a). (40) Furthermore, after averaging, we obtain, ∆t(a) = φt′,t∆t′(a) + 1 M M X m=1 " t X k=t′+1 eη(t) k γp∆m k−1,max # + 1 M M X m=1 ξm t′,t(a). (41) 1We use this different notation in appendix as opposed to the half-time indices used in the main text to improve readability of the proof. 20 Step 2: deriving a recursive bound for E[∆t,max]. Bounding (40), we obtain, ∆m t,max ≥φt′,t∆t′,max + " t X k=t′+1 eη(t) k γp∆m k−1,max # + ξm t′,t,max −φt′,t\|∆t′(1) −∆t′(2)\|, (42a) ∆m t,max ≤φt′,t∆t′,max + " t X k=t′+1 eη(t) k γp∆m k−1,max # + ξm t′,t,max, (42b) where in the first step we used the fact that max{a1 + b1, a2 + b2} ≥min{a1, a2} + max{b1, b2} = max{a1, a2} + max{b1, b2} −\|a1 −a2\|. (43) On taking expectation, we obtain, E[∆m t,max] ≥φt′,tE[∆t′,max] + " t X k=t′+1 eη(t) k γpE[∆m k−1,max] # + E[ξm t′,t,max] −φt′,tE[\|∆t′(1) −∆t′(2)\|], (44a) E[∆m t,max] ≤φt′,tE[∆t′,max] + " t X k=t′+1 eη(t) k γpE[∆m k−1,max] # + E[ξm t′,t,max]. (44b) Similarly, using (41) and (43) we can write, ∆t,max ≥φt′,t∆t′,max + 1 M M X m=1 " t X k=t′+1 eη(t) k γp∆m k−1,max # −φt′,t\|∆t′(1) −∆t′(2)\| + max ( 1 M M X m=1 ξm t′,t(1), 1 M M X m=1 ξm t′,t(2) ) (45a) =⇒E[∆t,max] ≥φt′,tE[∆t′,max] + 1 M M X m=1 " t X k=t′+1 eη(t) k γpE[∆m k−1,max] # −φt′,tE[\|∆t′(1) −∆t′(2)\|] + E " max ( 1 M M X m=1 ξm t′,t(1), 1 M M X m=1 ξm t′,t(2) )# . (45b) On combining (44b) and (45b), we obtain, E[∆t,max] ≥1 M M X m=1 E[∆m t,max] −E[ξm t′,t,max] −φt′,tE[\|∆t′(1) −∆t′(2)\|] + E " max ( 1 M M X m=1 ξm t′,t(1), 1 M M X m=1 ξm t′,t(2) )# . (46) In order to simplify (46), we make use of the following lemmas. Lemma 3. Let t′ < t be two consecutive averaging instants. Then for all m ∈[M], E[∆m t,max] −E[ξm t′,t,max] ≥ tY k=t′+1 (1 −ηk(1 −γp)) ! E[∆t′,max] + E[ξm t′,t,max] " t X k=t′+1 η(t) k −1 # + −φt′,tE[\|∆t′(1) −∆t′(2)\|], where [x]+ = max{x, 0}. Lemma 4. For all consecutive averaging instants t′, t satisfying t −max{t′, τ} ≥1/ητ and all m ∈[M], we have, E[ξm t′,t,max] ≥ 1 240 log 180B ηT (1−γ) · ν ν + 1, 21 E " max ( 1 M M X m=1 ξm t′,t(1), 1 M M X m=1 ξm t′,t(2) )# ≥ 1 240 log 180BM ηT (1−γ) · ν ν + √ M , where ν := r 20ηT B(1 −γ). Lemma 5. For all t ∈{tr}R r=1, we have E[\|∆t(1) −∆t(2)\|] ≤ s 8ηT 3BM(1 −γ). Thus, on combining the results from Lemmas 3, 4, and 5 and plugging them into (46), we obtain the following relation for t, t′ ≥τ: E[∆t,max] ≥ tY k=t′+1 (1 −ηk(1 −γp)) ! E[∆t′,max] + E[ξm t′,t,max] " t X k=t′+1 η(t) k −1 # + −2φt′,tE[\|∆t′(1) −∆t′(2)\|] + E " max ( 1 M M X m=1 ξm t′,t(1), 1 M M X m=1 ξm t′,t(2) )# ≥(1 −ητ(1 −γp))t−t′E[∆t′,max] +   1 −(1 −ητ(1 −γp))t−t′ 5760 log 180B ηT (1−γ) (1 −γp)  · ν ν + 1 · 1 t −t′ ≥8 ητ −2(1 −ηT )t−t′ s 8ηT 3BM(1 −γ) + 1 240 log 180BM ηT (1−γ) · ν ν + √ M · 1 t −t′ ≥8 ητ , (47) where we used the relation φt′,t ≤(1 −ηT )t−t′, as well as the value of ν as defined in Lemma 4 along with the fact t X k=t′+1 η(t) k −1 ≥1 −(1 −ητ(1 −γp))t−t′ 24(1 −γp) (48) for all t, t′ ≥τ such that t −t′ ≥8/ητ. Proof of (48). We have, t X k=t′+1 η(t) k −1 = t X k=t′+1 ηk tY i=k+1 (1 −ηi(1 −γp)) ! −1 ≥ t X k=t′+1 ηt tY i=k+1 (1 −ητ(1 −γp)) ! −1 ≥ηt t X k=t′+1 (1 −ητ(1 −γp))t−k −1 ≥ηt · 1 −(1 −ητ(1 −γp))t−t′ ητ(1 −γp) ! −1 ≥1 −(1 −ητ(1 −γp))t−t′ 3(1 −γp) −1. (49) To show (48), it is sufficient to show that 1 −(1 −ητ(1 −γp))t−t′ 3(1 −γp) ≥8 7 for t −t′ ≥8/ητ. Thus, for t −t′ ≥8/ητ we have, 1 −(1 −ητ(1 −γp))t−t′ 3(1 −γp) ≥1 −exp(−ητ(1 −γp) · (t −t′)) 3(1 −γp) 22 ≥1 −exp(−8(1 −γp)) 3(1 −γp) . (50) Since γ ≥5/6, 1 −γp ≤2/9. For x ≤2/9, the function f(x) = 1−e−8x 3x ≥8/7, proving the claim. Step 3: lower bounding E[∆T,max]. We are now interested in evaluating E[∆T,max] based on the recursion (47). To this effect, we introduce some notation to simplify the presentation. Let Rτ := min{r : tr ≥τ}. (51) For r = Rτ, . . . , R, we define the following terms: xr := E[∆tr,max], αr := (1 −ητ(1 −γp))tr−tr−1, βr := (1 −ηT )tr−tr−1, Ir := {r ≥r′ > Rτ : tr′ −tr′−1 ≥8/ητ}, C1 := 1 5760 log 180B ηT (1−γ) (1 −γp) · ν ν + 1, C2 := s 32ηT 3BM(1 −γ), C3 := 1 240 log 180BM ηT (1−γ) · ν ν + √ M . With these notations in place, the recursion in (47) can be rewritten as xr ≥αrxr−1 −βrC2 + C31{r ∈Ir} + (1 −αr)C11{r ∈Ir}, (52) for all r ≥Rτ. We claim that xr satisfies the following relation for all r ≥Rτ + 1 (whose proof is deferred to the end of this step): xr ≥ r Y i=Rτ +1 αi ! xRτ − r X k=Rτ +1 βk r Y i=k+1 αi ! C2 + r X k=Rτ +1 r Y i=k+1 αi ! 1{k ∈Ik}C3 + C1  Y i/∈Ir αi   1 − Y i∈Ir αi ! , (53) where we recall that if there is no valid index for a product, its value is taken to be 1. Invoking (53) for r = R and using the relation xRτ −1 ≥0, we obtain, xR ≥− R X k=Rτ βk R Y i=k+1 αi ! C2 + R X k=Rτ R Y i=k+1 αi ! C31{k ∈Ik} + C1  Y i/∈IR αi   1 − Y i∈IR αi ! ≥−RC2 + C1  Y i/∈IR αi   1 − Y i∈IR αi ! ≥−R · s 32ηT 3BM(1 −γ) +  Y i/∈IR αi   1 − Y i∈IR αi ! · 1 5760 log 180B ηT (1−γ) (1 −γp) · ν ν + 1, (54) where we used the fact βk QR i=k+1 αi ≤1 and that C3 ≥0. Consider the expression Y i/∈IR αi = Y i/∈IR (1 −ητ(1 −γp))ti−ti−1 ≥1 −ητ(1 −γp) · X i/∈IR (ti −ti−1) \| {z } =:T1 . (55) 23 Consequently, 1 − Y i∈IR αi ! = 1 −(1 −ητ(1 −γp))T −τ−T1 ≥1 −exp (−ητ(1 −γp) (T −τ −T1)) . (56) Note that T1 satisfies the following bound T1 := X i/∈IR (ti −ti−1) ≤(R −\|IR\|) · 8 ητ ≤8R ητ . (57) We split the remainder of the analysis based on the step size schedule. • For the constant step size schedule, i.e., ηt = η ≥ 1 (1−γ)T , we have, Rτ = 0, with τ = 0 and t0 = 0 (as all agents start at the same point). If R ≤ 1 96000(1−γ) log( 180B η(1−γ)), then, (55), (56) and (57) yield the following relations: T1 ≤8R η ≤ T 12000 log(180N), Y i/∈IR αi ≥1 −η(1 −γp) · T1 ≥1 −32R(1 −γ) 3 ≥1 − 1 9000 log(180N), 1 − Y i∈IR αi ! ≥1 −exp (−η(1 −γp) (T −T1)) ≥1 −exp −4 3 1 − 1 9000 log(180N) . On plugging the above relations into (54), we obtain xR ≥ √ 40 96000 log 180B η(1−γ) (1 −γ) · ν ν + 1 − ν 5 √ M (58) where recall that ν := r 20η 3B(1 −γ). Consider the function f(x) = x x+1 − x 5 √ M . We claim that for x ∈[0, √ M] and all M ≥2, f(x) ≥7 20 min{x, 1}. (59) The proof of the above claim is deferred to the end of the section. In light of the above claim, we have, xR ≥ √ 40 96000 log 180B η(1−γ) (1 −γ) · 7 20 · min ( 1, s 20η 3B(1 −γ) ) ≥ √ 40 96000 log (180N) · 7 20 · min ( 1 1 −γ , s 20 3(1 −γ)4N ) , (60) where we used the fact that M ≥2, √x log(1/x) is an increasing function and the relation ν M = 20η 3BM(1 −γ) ≤1 15 ≤1. • Next, we consider the rescaled linear step size schedule, where τ = T/3 (cf. (34)). To begin, we assume tRτ ≤max{ 3T 4 , T − 1 6ητ (1−γp)}. It is straightforward to note that max 3T 4 , T − 1 6ητ(1 −γp) = ( 3T 4 if cη ≥3 T − 1 6ητ (1−γp) if cη < 3. If R ≤ 1 384000(1−γ) log 180B ηT (1−γ) ·(5+cη) then, (55), (56) and (57) yield the following relations: T1 ≤8R ητ , Y i/∈IR αi ≥1 −ητ(1 −γp) · T1 ≥1 −32R(1 −γ) 3 ≥1 − 1 36000. 24 For cη ≥3, we have, 1 − Y i∈IR αi ! ≥1 −exp (−ητ(1 −γp) (T −tRτ −T1)) ≥1 −exp − (1 −γ)T (3 + cη(1 −γ)T) + 32R(1 −γ) 3 ≥ 1 2(3 + cη), where we used T ≥ 1 1−γ in the second step. Similarly, for cη < 3, we have, 1 − Y i∈IR αi ! ≥1 −exp (−ητ(1 −γp) (T −tRτ −T1)) ≥1 −exp −1 6 + 32R(1 −γ) 3 ≥1 10. On plugging the above relations into (54), we obtain xR ≥ 18 √ 1.6 384000 log 180B ηT (1−γ) (1 −γ)(5 + cη) · ν ν + 1 − ν 18 √ M ≥ 18 √ 1.6 384000 log 180B ηT (1−γ) (1 −γ)(5 + cη) · 7 20 · min ( 1, s 20ηT 3B(1 −γ) ) ≥ 18 √ 1.6 384000 log 180B ηT (1−γ) (5 + cη) · 7 20 · min ( 1 1 −γ , s 20ηT 3B(1 −γ)3 ) ≥ 18 √ 1.6 384000 log (180N(1 + log N)) (5 + log N) · 7 20 · min ( 1 1 −γ , s 20 3B(1 + log N)(1 −γ)4N ) , (61) where we again used the facts that M ≥2, cη ≤log N, √x log(1/x) is an increasing function and the relation ν M = 20ηT 3BM(1 −γ) ≤1. • Last but not least, let us consider the rescaled linear step size schedule case when tRτ > max{ 3T 4 , T − 1 6ητ (1−γp)}. The condition implies that the time between the communication rounds Rτ −1 and Rτ is at least T0 := max{ 5T 12 , 2T 3 − 1 6ητ (1−γp)}. Thus, (47) yields that E[∆tRτ ] ≥   1 −(1 −ητ(1 −γp))T0 5760 log 180 BηT (1−γ) (1 −γp)  · ν ν + 1 −2(1 −ηT )T0 s 8ηT 3BM(1 −γ). (62) Using the above relation along with (53), we can conclude that xR ≥(1 −ητ(1 −γp))T −tRτ   1 −(1 −ητ(1 −γp))T0 5760 log 180 BηT (1−γ) (1 −γp)  · ν ν + 1 −2(1 −ηT )T0 · (1 −ητ(1 −γp))T −tRτ s 8ηT 3BM(1 −γ) −RC2. (63) 25 In the above relation, we used the trivial bounds C1, C3 ≥0 and a crude bound on the term corresponding to C2, similar to (54). Let us first consider the case of cη ≥3. We have, 1 −(1 −ητ(1 −γp))T0 ≥1 −exp (−ητ(1 −γp)5T/12) ≥1 −exp − 5(1 −γ)T 3(3 + cη(1 −γ)T) ≥ 1 3 + cη , (1 −ητ(1 −γp))T −tRτ ≥1 −ητ(1 −γp)T 4 ≥1 − (1 −γ)T (3 + cη(1 −γ)T) ≥1 −1 cη ≥2 3. Similarly, for cη < 3, we have, 1 −(1 −ητ(1 −γp))T0 ≥1 −exp −ητ(1 −γp)2T 3 + 1 6 ≥1 −exp − 8(1 −γ)T 3(3 + cη(1 −γ)T) + 1 6 ≥1 −e−5/18, (1 −ητ(1 −γp))T −tRτ ≥1 −ητ(1 −γp) 6ητ(1 −γp) ≥5 6. The above relations implies that (1 −ητ(1 −γp))T −tRτ (1 −(1 −ητ(1 −γp))T0) ≥c for some constant c, which only depends on cη. On plugging this into (63), we obtain a relation that is identical to that in (54) up to leading constants. Thus, by using a similar sequence of argument as used to obtain (61), we arrive at the same conclusion as for the case of tRτ ≤max{ 3T 4 , T − 1 6ητ (1−γp)}. Step 4: finishing up the proof. Thus, (60), (61) along with the above conclusion together imply that there exists a numerical constant c0 > 0 such that E[\|bV m T (1) −V ⋆(1)\|] ≥E[∆T,max] ≥ c0 log3 N · min ( 1 1 −γ , s 1 (1 −γ)4N ) . (64) The above equation along with Lemma 2 implies E[\|V m T −V ⋆(1)\|] ≥ c0 log3 N · min ( 1 1 −γ , s 1 (1 −γ)4N ) − 1 1 −γ T Y i=1 (1 −ηi(1 −γ)). (65) On the other hand, from (21) we know that E[\|V m T (3) −V ⋆(3)\|] ≥ 1 1 −γ T Y i=1 (1 −ηi(1 −γ)). (66) Hence, E[∥Qm T −Q⋆∥∞] ≥E [max {\|V m T (3) −V ⋆(3)\|, \|V m T (1) −V ⋆(1)\|}] ≥max {E [\|V m T (3) −V ⋆(3)\|] , E [\|V m T (1) −V ⋆(1)\|]} ≥max ( 1 1 −γ T Y i=1 (1 −ηi(1 −γ)), min ( 1 1 −γ , s 1 (1 −γ)4N ) − 1 1 −γ T Y i=1 (1 −ηi(1 −γ)) ) ≥1 2 min ( 1 1 −γ , s 1 (1 −γ)4N ) , (67) where the third step follows from (65) and (66) and the fourth step uses max{a, b} ≥(a + b)/2. Thus, from (28) and (67) we can conclude that whenever CCround = O 1 (1−γ) log2 N , ER(A ; N, M) = Ω 1 log3 N √ N for all values of M ≥2. In other words, for any algorithm to achieve any collaborative gain, its communication complexity should satisfy CCround = Ω 1 (1−γ) log2 N , as required. 26 Proof of (53). We now return to establish (53) using induction. For the base case, (52) yields xRτ +1 ≥αRτ +1xRτ −βRτ +1C2 + C31{Rτ + 1 ∈IRτ +1} + (1 −αRτ +1)C11{Rτ + 1 ∈IRτ +1}. (68) Note that this is identical to the expression in (53) for r = Rτ + 1 as   Y i/∈IRτ +1 αi    1 − Y i∈IRτ +1 αi  = (1 −αRτ +1)1{Rτ + 1 ∈IRτ +1} based on the adopted convention for products with no valid indices. For the induction step, assume (53) holds for some r ≥Rτ + 1. On combining (52) and (53), we obtain, xr+1 ≥αr+1xr −βr+1C2 + C31{(r + 1) ∈Ir+1} + (1 −αr+1)C11{r + 1 ∈Ir+1} ≥αr+1 r Y i=Rτ +1 αi ! xRτ −αr+1 r X k=Rτ +1 βk r Y i=k+1 αi ! C2 + αr+1 r X k=Rτ +1 r Y i=k+1 αi ! C31{k ∈Ik} + αr+1C1  Y i/∈Ir αi   1 − Y i∈Ir αi ! −βr+1C2 + C31{(r + 1) ∈Ir+1} + (1 −αr+1)C11{(r + 1) ∈Ir+1} ≥ r+1 Y i=Rτ +1 αi ! xRτ − r+1 X k=Rτ +1 βk r+1 Y i=k+1 αi ! C2 + r+1 X k=Rτ +1 r+1 Y i=k+1 αi ! C31{k ∈Ik} + αr+1C1  Y i/∈Ir αi   1 − Y i∈Ir αi ! + (1 −αr+1)C11{(r + 1) ∈Ir+1}. (69) If (r + 1) /∈ Ir+1, then 1 −Q i∈Ir αi = 1 −Q i∈Ir+1 αi and αr+1 Q i/∈Ir αi = Q i/∈Ir+1 αi . Consequently, αr+1C1  Y i/∈Ir αi   1 − Y i∈Ir αi ! + (1 −αr+1)C11{(r + 1) ∈Ir+1} = C1  Y i/∈Ir+1 αi    1 − Y i∈Ir+1 αi  . (70) On the other hand, if (r + 1) ∈Ir+1, then Q i/∈Ir αi = Q i/∈Ir+1 αi . Consequently, we have, αr+1C1  Y i/∈Ir αi   1 − Y i∈Ir αi ! + (1 −αr+1)C11{(r + 1) ∈Ir+1} = αr+1C1  Y i/∈Ir+1 αi   1 − Y i∈Ir αi ! + (1 −αr+1)C1 ≥C1  Y i/∈Ir+1 αi   " αr+1 1 − Y i∈Ir αi ! + (1 −αr+1) # ≥C1  Y i/∈Ir+1 αi    1 − Y i∈Ir+1 αi  . (71) Combining (69), (70) and (71) proves the claim. Proof of (59). To establish this result, we separately consider the cases x ≤1 and x ≥1. • When x ≤1, we have f(x) = x x + 1 − 1 5 √ M ≥x · 1 2 − x 5 √ M ≥7x 20 , (72) where in the last step, we used the relation M ≥2. 27 • Let us now consider the case x ≥1. The second derivative of f is given by f ′′(x) = − 1 2(x+1)3 . Clearly, for all x ≥1, f ′′ < 0 implying that f is a concave function. It is well-known that a continuous, bounded, concave function achieves its minimum values over a compact interval at the end points of the interval (Bauer’s minimum principle). For all M ≥2, we have, f(1) = 1 2 − 1 5 √ M ≥7 20; f( √ M) = √ M √ M + 1 −1 5 ≥7 20. Consequently, we can conclude that for all x ∈[1, √ M], f(x) ≥7 20. (73) Combining (72) and (73) proves the claim. B.3.3 Large learning rates with large ηT BM In order to bound the error in this scenario, note that ηT BM controls the variance of the stochastic updates in the fixed point iteration. Thus, when ηT BM is large, the variance of the iterates is large, resulting in a large error. To demonstrate this effect, we focus on the dynamics of state 2. This part of the proof is similar to the large learning rate case of Li et al. [2023]. For all t ∈[T], define: V t(2) := 1 M M X m=1 V m t (2). (74) Thus, from (33), we know that E[V t(2)] obeys the following recursion: E[V t(2)] = (1 −ηt(1 −γp))E[V t−1(2)] + ηt. Upon unrolling the recursion, we obtain, E[V T (2)] = T Y k=t+1 (1 −ηk(1 −γp)) ! E[V t(2)] + T X k=t+1 η(T ) k . Thus, the above relation along with (16) and the value of V ⋆(2) yields us, V ⋆(2) −E[V T (2)] = T Y k=t+1 (1 −ηk(1 −γp)) 1 1 −γp −E[V t(2)] . (75) Similar to Li et al. [2023], we define τ ′ := min 0 ≤t′ ≤T −2 E[(V t)2] ≥ 1 4(1 −γ)2 for all t′ + 1 ≤t ≤T . If such a τ ′ does not exist, it implies that either E[(V T )2] < 1 4(1−γ)2 or E[(V T −1)2] < 1 4(1−γ)2 . If the former is true, then, V ⋆(2) −E[V T (2)] = 3 4(1 −γ) − q E[(V T )2] > 1 4(1 −γ). (76) Similarly, if E[(V T −1)2] < 1 4(1−γ)2 , it implies E[V T −1] < 1 2(1−γ). Using (33), we have, E[V T (2)] = (1 −ηT (1 −γp))E[V T −1(2)] + ηT ≤E[V T −1(2)] + 1 < 1 2(1 −γ) + 1 6(1 −γ) = 2 3(1 −γ). Consequently, V ⋆(2) −E[V T (2)] > 3 4(1 −γ) − 2 3(1 −γ) > 1 12(1 −γ). (77) For the case when τ ′ exists, we divide the proof into two cases. 28 • We first consider the case when the learning rates satisfy: T Y k=τ ′+1 (1 −ηk(1 −γp)) ≥1 2. (78) The analysis for this case is identical to that considered in Li et al. [2023]. We explicitly write the steps for completeness. Specifically, V ⋆(2) −E[V T (2)] = T Y k=τ ′+1 (1 −ηk(1 −γp)) ! 1 1 −γp −E[V τ ′(2)] ≥1 2 · 3 4(1 −γ) − q E[(V τ ′(2))2] ≥1 2 · 3 4(1 −γ) − 1 2(1 −γ) ≥ 1 8(1 −γ), (79) where the first line follows from (75), the second line from the condition on step sizes and the third line from the definition of τ ′. • We now consider the other case where, 0 ≤ T Y k=τ ′+1 (1 −ηk(1 −γp)) < 1 2. (80) Using [Li et al., 2023, Eqn.(134)], for any t′ < t and all agents m, we have the relation V m t (2) = 1 1 −γp − tY k=t′+1 (1 −ηk(1 −γp)) 1 1 −γp −V m t′ (2) + X k=t′+1 η(t) k γ( ˆP m k (2\|2) −p)V m k−1(2). The above equation is directly obtained by unrolling the recursion in (24) along with noting that Qt(2, 1) = Vt(2) for all t. Consequently, we have, V T (2) = 1 1 −γp − T Y k=t′+1 (1 −ηk(1 −γp)) 1 1 −γp −V t′(2) + 1 M M X m=1 T X k=t′+1 η(T ) k γ( ˆP m k (2\|2) −p)V m k−1(2). (81) Let {Ft}T t=0 be a filtration such that Ft is the σ-algebra corresponding to {{ ˆP m s (2\|2)}M m=1}t s=1. It is straightforward to note that n 1 M PM m=1 η(T ) k γ( ˆP m k (2\|2) −p)V m k−1(2) o k is a martingale sequence adapted to the filtration Fk. Thus, using the result from [Li et al., 2023, Eqn.(139)], we can conclude that Var(V T (2)) ≥E " T X k=τ ′+2 Var 1 M M X m=1 η(T ) k γ( ˆP m k (2\|2) −p)V m k−1(2) Fk−1 !# . (82) We have, Var 1 M M X m=1 η(T ) k γ( ˆP m k (2\|2) −p)V m k−1(2) Fk−1 ! = 1 M 2 M X m=1 Var η(T ) k γ( ˆP m k (2\|2) −p)V m k−1(2) Fk−1 = (η(T ) k )2 BM γ2p(1 −p) 1 M M X m=1 (V m k−1(2))2 ! 29 ≥(1 −γ)(4γ −1) 9BM · (η(T ) k )2 · (V k−1(2))2, (83) where the first line follows from that fact that variance of sum of i.i.d. random variables is the sum of their variances, the second line from variance of Binomial random variable and the third line from Jensen’s inequality. Thus, (82) and (83) together yield, Var(V T (2)) ≥(1 −γ)(4γ −1) 9BM · T X k=τ ′+2 (η(T ) k )2 · E[(V k−1(2))2] ≥(1 −γ)(4γ −1) 9BM · 1 4(1 −γ)2 · T X k=max{τ,τ ′}+2 (η(T ) k )2, (84) where the second line follows from the definition of τ ′. We focus on bounding the third term in the above relation. We have, T X k=max{τ ′,τ}+2 η(T ) k 2 ≥ T X k=max{τ ′,τ}+2 ηk T Y i=k+1 (1 −ηi(1 −γp) !2 ≥ T X k=max{τ ′,τ}+2 ηT tY i=k+1 (1 −ητ(1 −γp)) !2 = η2 T T X k=max{τ ′,τ}+2 (1 −ητ(1 −γp))2(t−k) ≥η2 T · 1 −(1 −ητ(1 −γp))2(T −max{τ ′,τ}−1) ητ(1 −γp)(2 −ητ(1 −γp)) ≥ηT · 1 4(1 −γ) · c′, (85) where the second line follows from monotonicity of ηt and the numerical constant c′ in the fifth step is given by the following claim whose proof is deferred to the end of the section: 1 −(1 −ητ(1 −γp))2(T −max{τ ′,τ}−1) ≥ ( 1 −e−8/9 for constant step sizes, 1 −exp − 8 3 max{1,cη} for linearly rescaled step sizes . (86) Thus, (84) and (85) together imply Var(V T (2)) ≥ (4γ −1) 36BM(1 −γ) · T X k=τ ′+2 (η(T ) k )2 ≥c′(4γ −1) 144(1 −γ) · ηT BM(1 −γ) ≥c′(4γ −1) 144(1 −γ) · 1 100, (87) where the last inequality follows from the bound on ηT BM . Thus, for all N ≥1, we have, E[(V ⋆(2) −V T (2))2] = E[(V ⋆(2) −E[V T (2)])2] + Var(V T (2)) ≥ c′′ (1 −γ)N , for some numerical constant c′′. Similar to the small learning rate case, the error rate is bounded away from a constant value irrespective of the number of agents and the number of communication rounds. Thus, even with CCround = Ω(T), we will not observe any collaborative gain in this scenario. Proof of (86). To establish the claim, we consider two cases: 30 • τ ′ ≥τ: Under this case, we have, (1 −ητ(1 −γp))2(T −max{τ ′,τ}−1) = (1 −ητ(1 −γp))2(T −τ ′−1) ≤(1 −ητ(1 −γp))T −τ ′ ≤ T Y k=τ ′+1 (1 −ηk(1 −γp)) ≤1 2, (88) where the last inequality follows from (80). • τ ≥τ ′: For this case, we have (1 −ητ(1 −γp))2(T −max{τ ′,τ}−1) = (1 −ητ(1 −γp))2(T −τ−1) ≤(1 −ητ(1 −γp))T −τ ≤exp −2Tητ(1 −γp) 3 . (89) For the constant stepsize schedule, we have, exp −2Tητ(1 −γp) 3 ≤exp −2T 3 · 1 (1 −γ)T · 4(1 −γ) 3 = exp −8 9 (90) For linearly rescaled stepsize schedule, we have, exp −2Tητ(1 −γp) 3 ≤exp −2T 3 · 1 1 + cη(1 −γ)T/3 · 4(1 −γ) 3 = exp − 8 3 max{1, cη} (91) On combining (88), (89), (90) and (91), we arrive at the claim. B.4 Generalizing to larger state action spaces We now elaborate on how we can extend the result to general state-action spaces along with the obtaining the lower bound on the bit level communication complexity. For the general case, we instead consider the following MDP. For the first four states {0, 1, 2, 3}, the probability transition kernel and reward function are given as follows. A0 = {1} P(0\|0, 1) = 1 r(0, 1) = 0, (92a) A1 = {1, 2, . . . , \|A\|} P(1\|1, a) = p P(0\|1, a) = 1 −p r(1, a) = 1, ∀a ∈A (92b) A2 = {1} P(2\|2, 1) = p P(0\|2, 1) = 1 −p r(2, 1) = 1, (92c) A3 = {1} P(3\|3, 1) = 1 r(3, 1) = 1, (92d) where the parameter p = 4γ −1 3γ . The overall MDP is obtained by creating \|S\|/4 copies of the above MDP for all sets of the form {4r, 4r+1, 4r+2, 4r+3} for r ≤\|S\|/4−1. It is straightforward to note that the lower bound on the number of communication rounds immediately transfers to the general case as well. Moreover, note that the bound on CCround implies the bound CCbit = Ω 1 (1−γ) log2 N as every communication entails sending Ω(1) bits. To obtain the general lower bound on bit level communication complexity, note that we can carry out the analysis in the previous section for all \|A\|/2 pairs of actions in state 1 corresponding to the set of states {0, 1, 2, 3}. Moreover, the algorithm A , needs to ensure that the error is low across all the \|A\|/2 pairs. Since the errors are independent across all these pairs, each of them require Ω 1 (1−γ) log2 N bits of information to be transmitted during the learning horizon leading to a lower bound of Ω \|A\| (1−γ) log2 N . Note that since we require a low ℓ∞error, A needs to ensure that the error is low across all the pairs, resulting in a communication cost linearly growing with \|A\|. Upon repeating the argument across all \|S\|/4 copies of the MDP, we arrive at the lower bound of CCbit = Ω \|S\|\|A\| (1−γ) log2 N . 31 B.5 Proofs of auxiliary lemmas B.5.1 Proof of Lemma 2 Note that a similar relationship is also derived in Li et al. [2023], but needing to take care of the averaging over multiple agents, we present the entire arguments for completeness. We prove the claim using an induction over t. It is straightforward to note that the claim is true for t = 0 and all agents m ∈{1, 2, . . . , M}. For the inductive step, we assume that the claim holds for t −1 for all clients. Using the induction hypothesis, we have the following relation between V m t−1(1) and bV m t−1: V m t−1(1) = max a∈{1,2} Qm t−1(1, a) ≥max a∈{1,2} bQm t−1(a) − 1 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)) = bV m t−1 − 1 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)). (93) For t /∈{tr}R r=1 and a ∈{1, 2}, we have, Qm t (1, a) −bQm t (a) = Qm t−1/2(1, a) −bQm t−1/2(a) = (1 −ηt)Qm t−1(1, a) + ηt(1 + γ bP m t (1\|1, a)V m t−1(1)) − h (1 −ηt) bQm t−1(a) + ηt(1 + γ bP m t (1\|1, a)bV m t−1) i = (1 −ηt)(Qm t−1(1\|1, a) −bQm t−1(a)) + ηtγ bP m t (1\|1, a)(V m t−1(1) −bV m t−1) ≥−(1 −ηt) 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)) −bP m t (1\|1, a) · ηtγ 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)) ≥−(1 −ηt) 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)) −ηtγ 1 −γ t−1 Y i=1 (1 −ηi(1 −γ)) ≥− 1 1 −γ tY i=1 (1 −ηi(1 −γ)). (94) For t ∈{tr}R r=1 and a ∈{1, 2}, we have, Qm t (1, a) −bQm t (a) = 1 M M X m=1 Qm t−1/2(1, a) −1 M M X m=1 bQm t−1/2(a) = 1 M M X m=1 h (1 −ηt)Qm t−1(1, a) + ηt(1 + γ bP m t (1\|1, a)V m t−1(1)) i −1 M M X m=1 h (1 −ηt) bQm t−1(a) + ηt(1 + γ bP m t (1\|1, a)bV m t−1) i = 1 M M X m=1 h (1 −ηt)(Qm t−1(1, a) −bQm t−1(a)) + ηtγ bP m t (1\|1, a)(V m t−1(1) −bV m t−1) i ≥− 1 1 −γ tY i=1 (1 −ηi(1 −γ)), (95) where the last step follows using the same set of arguments as used in (94). The inductive step follows from (94) and (95). B.5.2 Proof of Lemma 3 In order to bound the term E[∆m t,max] −E[ξm t′,t,max], we make use of the relation in (44a), which we recall E[∆m t,max] ≥φt′,tE[∆t′,max] + " t X k=t′+1 eη(t) k γpE[∆m k−1,max] # + E[ξm t′,t,max] −φt′,tE[\|∆t′(1) −∆t′(2)\|]. 32 • To aid the analysis, we consider the following recursive relation for any fixed agent m: yt = (1 −ηt)yt−1 + ηt(γpyt−1 + E[ξm t′,t,max]). (96) Upon unrolling the recursion, we obtain, yt = tY k=t′+1 (1 −ηk) ! yt′ + t X k=t′+1 ηk tY i=k+1 (1 −ηi) ! γpyk−1 + t X k=t′+1 ηk tY i=k+1 (1 −ηi) ! E[ξm t′,t,max] = φt′,tyt′ + t X k=t′+1 eη(t) k γpyk−1 + t X k=t′+1 eη(t) k E[ξm t′,t,max]. (97) Initializing yt′ = E[∆t′,max] in (97) and plugging this into (44a), we have E[∆m t,max] ≥yt −φt′,tE[\|∆t′(1) −∆t′(2)\|], where we used Pt k=t′+1 eη(t) k ≤1 (cf. (18)). We now further simply the expression of yt. By rewriting (96) as yt = (1 −ηt(1 −γp))yt−1 + ηtE[ξm t′,t,max], it is straight forward to note that yt is given as yt = tY k=t′+1 (1 −ηk(1 −γp)) ! yt′ + E[ξm t′,t,max] " t X k=t′+1 η(t) k # . (98) Consequently, we have, E[∆m t,max] −E[ξm t′,t,max] ≥ tY k=t′+1 (1 −ηk(1 −γp)) ! E[∆t′,max] + E[ξm t′,t,max] " t X k=t′+1 η(t) k −1 # −φt′,tE[\|∆t′(1) −∆t′(2)\|]. (99) • We can consider a slightly different recursive sequence defined as wt = (1 −ηt)wt−1 + ηt(γpwt−1). (100) Using a similar sequence of arguments as outlined in (96)-(98), we can conclude that if wt′ = E[∆t′,max], then E[∆m t,max] ≥wt + E[ξm t′,t,max] −φt′,tE[\|∆t′(1) −∆t′(2)\|] and consequently, E[∆m t,max] ≥ tY k=t′+1 (1 −ηk(1 −γp)) ! E[∆t′,max] + E[ξm t′,t,max] −φt′,tE[\|∆t′(1) −∆t′(2)\|]. (101) On combining (99) and (101), we arrive at the claim. B.5.3 Proof of Lemma 4 We begin with bounding the first term E[ξm t′,t,max]; the second bound follows in an almost identical derivation. 33 Step 1: applying Freedman’s inequality. Using the relation max{a, b} = a+b+\|a−b\| 2 , we can rewrite E[ξm t′,t,max] as E[ξm t′,t,max] = E ξm t′,t(1) + ξm t′,t(2) 2 + ξm t′,t(1) −ξm t′,t(2) 2 = 1 2E ξm t′,t(1) −ξm t′,t(2) 2 = 1 2E " t X k=t′+1 eη(t) k γ( bP m k (1\|1, 1) −bP m k (1\|1, 2))bV m k−1 \| {z } =:ζm t′,t # , (102) where we used the definition in (38) and the fact that E[ξm t′,t(1)] = E[ξm t′,t(2)] = 0. Decompose ζm t′,t as ζm t′,t = t X k=t′+1 B X b=1 eη(t) k γ B (P m k,b(1\|1, 1) −P m k,b(1\|1, 2))bV m k−1 =: L X l=1 zl, (103) where for all 1 ≤l ≤L zl := γ B (P m k(l),b(l)(1\|1, 1) −P m k(l),b(l)(1\|1, 2))bV m k(l)−1 with k(l) := ⌊l/B⌋+ t′ + 1; b(l) = ((l −1) mod B) + 1; L = (t −t′)B. Let {Fl}L l=1 be a filtration such that Fl is the σ-algebra corresponding to {P m k(j),b(j)(1\|1, 1), P m k(j),b(j)(1\|1, 2)}l j=1. It is straightforward to note that {zl}L l=1 is a martingale sequence adapted to the filtration {F}L l=1. We will use the Freedman’s inequality [Freedman, 1975, Li et al., 2023] to obtain a high probability bound on \|ζm t′,t\|. • To that effect, note that sup l \|zl\| ≤sup l eη(t) k(l) · γ B · (P m k(l),b(l)(1\|1, 1) −P m k(l),b(l)(1\|1, 2)) · bV m k(l)−1 ≤eη(t) k(l) · γ B(1 −γ) ≤ ηt B(1 −γ), (104) where the second step follows from the bounds \|(P m k(l),b(l)(1\|1, 1) −P m k(l),b(l)(1\|1, 2))\| ≤1 and bV m k(l)−1 ≤ 1 1−γ and the third step uses cη ≤ 1 1−γ and the fact that eη(T ) k is increasing in k in this regime. (cf. (19)). • Similarly, Var(zl\|Fl−1) ≤ eη(t) k(l) 2 γ2 B2 · bV m k(l)−1 2 · Var(P m k(l),b(l)(1\|1, 1) −P m k(l),b(l)(1\|1, 2)) ≤ eη(t) k(l) 2 γ2 B2(1 −γ)2 · 2p(1 −p) ≤ 2 eη(t) k(l) 2 3B2(1 −γ). (105) Using the above bounds (104) and (105) along with Freedman’s inequality yield that Pr  \|ζm t′,t\| ≥ v u u t 8 log(2/δ) 3B2(1 −γ) L X l=1 eη(t) k(l) 2 + 4ηt log(2/δ) 3B(1 −γ)  ≤δ. (106) 34 Setting δ0 = (1−γ)2 2 · E[\|ζm t′,t\|2], with probability at least 1 −δ0, it holds \|ζm t′,t\| ≥ v u u t8 log(2/δ0) 3B(1 −γ) t X k=t′+1 eη(t) k 2 + 4ηt log(2/δ0) 3B(1 −γ) =: D. (107) Consequently, plugging this back to (102), we obtain E[ξm t′,t,max] = 1 2E[\|ζm t′,t\|] ≥1 2E[\|ζm t′,t\|1{\|ζm t′,t\| ≤D}] ≥ 1 2DE[\|ζm t′,t\|21{\|ζm t′,t\| ≤D}] ≥ 1 2D E[\|ζm t′,t\|2] −E[\|ζm t′,t\|21{\|ζm t′,t\| > D}] ≥ 1 2D E[\|ζm t′,t\|2] − Pr(\|ζm t′,t\| > D) (1 −γ)2 ≥ 1 4D · E[\|ζm t′,t\|2]. (108) Here, the penultimate step used the fact that \|ζm t′,t\| ≤ t X k=t′+1 eη(t) k (1 −γ) ≤ 1 (1 −γ), and the last step used the definition of δ0. Thus, it is sufficient to obtain a lower bound on E[\|ζm t′,t\|2] in order obtain a lower bound for E[ξm t′,t,max]. Step 2: lower bounding E[\|ζm t′,t\|2]. To proceed, we introduce the following lemma pertaining to lower bounding bV m t that will be useful later. Lemma 6. For all time instants t ∈[T] and all agent m ∈[M]: E bV m t 2 ≥ 1 2(1 −γ)2 . We have, E[\|ζm t′,t\|2] = E " L X l=1 Var (zl\|Fl−1) # = E " L X l=1 E z2 l \|Fl−1 # ≥ L X l=1 eη(t) k(l) 2 γ2 B2 · 2p(1 −p) · E ˆV m k(l)−1 2 ≥ L X l=1 eη(t) k(l) 2 γ2 B2 · 2p(1 −p) · 1 2(1 −γ)2 ≥ 2 9B(1 −γ) · t X k=max{t′,τ}+1 eη(t) k 2 , (109) where the third line follows from Lemma 6 and the fourth line uses γ ≥5/6. Step 3: finishing up. We finish up the proof by bounding Pt k=max{t′,τ}+1 eη(t) k 2 for t − max{t′, τ} ≥1/ητ. We have t X k=max{t′,τ}+1 eη(t) k 2 ≥ t X k=max{t′,τ}+1 ηk tY i=k+1 (1 −ηi) !2 35 (i) ≥ t X k=max{t′,τ}+1 ηt tY i=k+1 (1 −ητ) !2 = η2 t t X k=max{t′,τ}+1 (1 −ητ)2(t−k) ≥η2 t · 1 −(1 −ητ)2(t−max{t′,τ}) ητ(2 −ητ) ≥ηt · 1 −exp(−2) 6 ≥ηt 10 ≥ηT 10 , (110) where (i) follows from the monotonicity of ηk. Plugging (110) into the expressions of D (cf. (107)) we have D = v u u t8 log(2/δ0) 3B(1 −γ) t X k=t′+1 eη(t) k 2 + 4ηt log(2/δ0) 3B(1 −γ) ≤9 2E[\|ζm t′,t\|2] · r 8 log(2/δ0) 3 1 B(1 −γ) t X k=t′+1 eη(t) k 2 !−1/2 + 60 · E[\|ζm t′,t\|2] · log(2/δ0) ≤3E[\|ζm t′,t\|2] · log(2/δ0) "s 60B(1 −γ) ηt + 20 # ≤60E[\|ζm t′,t\|2] · log(2/δ0) "s 3B(1 −γ) 20ηT + 1 # , where the second line follows from (109) and (110), and the third line follows from (110). On combining the above bound with (108), we obtain, E[ξm t′,t,max] ≥ 1 240 log(2/δ0) · ν ν + 1, (111) where ν := r 20ηT 3B(1 −γ). Note that we have, δ0 = (1 −γ)2 2 · E[\|ζm t′,t\|2] ≥(1 −γ) 9B · t X k=t′+1 eη(t) k 2 ≥ηT (1 −γ) 90B . Combining the above bound with (111) yields us the required bound. Step 4: repeating the argument for the second claim. We note that second claim in the theorem, i.e., the lower bound on E h max n 1 M PM m=1 ξm t′,t(1), 1 M PM m=1 ξm t′,t(2) oi follows through an identical series of arguments where the bounds in Eqns. (104) and (105) contain an additional factor of M in the denominator (effectively replacing B with BM), which is carried through in all the following steps. B.5.4 Proof of Lemma 5 Using Eqns. (41) and (38), we can write ∆t(1) −∆t(2) = tY k=t′+1 (1 −ηk) ! (∆t′(1) −∆t′(2)) + 1 M M X m=1 t X k=t′+1 ηk tY i=k+1 (1 −ηi) ! γ( bP m k (1\|1, 1) −bP m k (1\|1, 2))bV m k−1. 36 Upon unrolling the recursion, we obtain, ∆t(1) −∆t(2) = t X k=1 M X m=1 ηk tY i=k+1 (1 −ηi) ! γ M ( bP m k (1\|1, 1) −bP m k (1\|1, 2))bV m k−1. If we define a filtration Fk as the σ-algebra corresponding to { bP 1 l (1\|1, 1), bP 1 l (1\|1, 2), . . . , bP M l (1\|1, 1), bP M l (1\|1, 2)}k l=1, then it is straightforward to note that {∆t(1) −∆t(2)}t is a martingale sequence adapted to the filtration {Ft}t. Using Jensen’s inequality, we know that if {Zt}t is a martingale adapted to a filtration {Gt}t, then for a convex function f such that f(Zt) is integrable for all t, {f(Zt)}t is a sub-martingale adapted to {Gt}t. Since f(x) = \|x\| is a convex function, {\|∆t(1) −∆t(2)\|}t is a submartingale adapted to the filtration {Ft}t. As a result, sup 1≤t≤T E[\|∆t(1) −∆t(2)\|] ≤E[\|∆T (1) −∆T (2)\|] ≤ E[(∆T (1) −∆T (2))2] 1/2 . (112) We use the following observation about a martingale sequence {Xi}t i=1 adapted to a filtration {Gi}t i=1 to evaluate the above expression. We have, E   t X i=1 Xi !2 = E  E   t X i=1 Xi !2 Gt−1     = E  E  X2 t + 2Xt t−1 X i=1 Xi ! + t−1 X i=1 Xi !2 Gt−1     = E X2 t + E   t−1 X i=1 Xi !2  = t X i=1 E X2 i , (113) where the third step uses the facts that Pt−1 i=1 Xi is Gt−1 measure and E[Xt\|Gt−1] = 0 and fourth step is obtained by recursively applying second and third steps. Using the relation in Eqn. (113) in Eqn. (112), we obtain, sup 1≤t≤T E[\|∆t(1) −∆t(2)\|] ≤ E[(∆T (1) −∆T (2))2] 1/2 ≤   T X k=1 E   M X m=1 eη(T ) k · γ M · ( bP m k (1\|1, 1) −bP m k (1\|1, 2)) ˆV m k−1 !2    1/2 ≤ T X k=1 eη(T ) k 2 · 2γ2p(1 −p) BM 2 · M X m=1 E ˆV m k−1 2!1/2 ≤ T X k=1 eη(T ) k 2 · 2γ2p(1 −p) BM(1 −γ)2 !1/2 . (114) Let us focus on the term involving the step sizes. We separately consider the scenario for constant step sizes and linearly rescaled step sizes. For constant step sizes, we have, T X k=1 eη(T ) k 2 = T X k=1 ηk T Y i=k+1 (1 −ηi) !2 = T X k=1 η2(1 −η)2(T −k) ≤ η2 1 −(1 −η)2 ≤η. (115) Similarly, for linearly rescaled step sizes, we have, T X k=1 eη(T ) k 2 = τ X k=1 eη(T ) k 2 + T X k=τ+1 ηk T Y i=k+1 (1 −ηi) !2 37 ≤ τ X k=1 eη(T ) τ 2 + T X k=τ+1 η2 k(1 −ηT )2(T −k) ≤η2 τ(1 −ηT )2(T −τ) · τ + η2 τ · 1 ηT (2 −ηT ) ≤3ηT · ηT · T · exp −4TηT 3 + 3ηT ≤9 4eηT + 3ηT ≤4ηT , (116) where the second step uses cη ≤log N ≤ 1 1−γ and the fact that eη(T ) k is increasing in k in this regime. (See Eqn. (19)) and fifth step uses xe−4x/3 ≤3/4e. On plugging results from Eqns. (115) and (116) into Eqn. (114) along with the value of p, we obtain, sup 1≤t≤T E[\|∆t(1) −∆t(2)\|] ≤ s 8ηT 3BM(1 −γ), (117) as required. B.5.5 Proof of Lemma 6 For the proof, we fix an agent m. In order to obtain the required lower bound on bV m t , we define an auxiliary sequence Q m t that evolves as described in Algorithm 5. Essentially, Q m t evolves in a manner almost identical to bQm t except for the fact that there is only one action and hence there is no maximization step in the update rule. Algorithm 5: Evolution of Q 1: r ←1, Q m 0 = Q⋆(1, 1) for all m ∈{1, 2, . . . , M} 2: for t = 1, 2, . . . , T do 3: for m = 1, 2, . . . , M do 4: Q m t−1/2 ←(1 −ηt)Q m t−1(a) + ηt(1 + bP m t (1\|1, 1)Q m t−1) 5: Compute Q m t according to Eqn. (8) 6: end for 7: end for It is straightforward to note that bQm t (1) ≥Q m t , which can be shown using induction. From the initialization, it follows that bQm 0 (1) ≥Q m 0 . Assuming the relation holds for t −1, we have, bQm t−1/2(1) = (1 −ηt) bQm t−1(1) + ηt(1 + γ bP m t (1\|1, 1)bV m t−1) ≥(1 −ηt) bQm t−1(1) + ηt(1 + γ bP m t (1\|1, 1) bQm t−1(1)) ≥(1 −ηt)Q m t−1 + ηt(1 + γ bP m t (1\|1, 1)Q m t−1) = Q m t−1/2. Since bQm t and Q m t follow the same averaging schedule, it immediately follows from the above relation that bQm t (1) ≥Q m t . Since bV m t ≥bQm t (1) ≥Q m t , we will use the sequence Q m t to establish the required lower bound on bV m t . We claim that for all time instants t and all agents m, E[Q m t ] = 1 1 −γp. (118) Assuming (118) holds, we have E[(bV m t )2] ≥ E[bV m t ] 2 ≥ E[Q m t ] 2 ≥ 1 1 −γp 2 ≥ 1 2(1 −γ)2 , 38 as required. In the above expression, the first inequality follows from Jensen’s inequality, the second from the relation bV m t ≥Q m t ≥0 and the third from (118). We now move now to prove the claim (118) using induction. For the base case, E[Q m 0 ] = 1 1−γp holds by choice of initialization. Assume that E[Q m t−1] = 1 1−γp holds for some t −1 for all m. • If t is not an averaging instant, then for any client m, Q m t = (1 −ηt)Q m t−1 + ηt(1 + γ bP m t (1\|1, 1)Q m t−1) =⇒E[Q m t ] = (1 −ηt)E[Q m t−1] + ηt(1 + γE[ bP m t (1\|1, 1)Q m t−1]) = (1 −ηt)E[Q m t−1] + ηt(1 + γpE[Q m t−1]) = (1 −ηt) 1 −γp + ηt 1 + γp 1 −γp = 1 1 −γp. (119) The third line follows from the independence of bP m t (1\|1, 1) and Q m t−1 and the fourth line uses the inductive hypothesis. • If t is an averaging instant, then for all clients m, Q m t = (1 −ηt) M M X j=1 Q j t−1 + ηt 1 M M X j=1 (1 + γ bP j t (1\|1, 1)Q j t−1) =⇒E[Q m t ] = (1 −ηt) M M X j=1 E[Q j t−1] + ηt 1 M M X j=1 (1 + γE[ bP j t (1\|1, 1)Q j t−1]) = (1 −ηt) M M X j=1 1 1 −γp + ηt 1 M M X j=1 1 + γp 1 −γp = 1 1 −γp, (120) where we again make use of independence and the inductive hypothesis. Thus, (119) and (120) taken together complete the inductive step. C Analysis of Fed-DVR-Q In this section, we prove Theorem 2 that outlines the performance guarantees of Fed-DVR-Q. There are two main parts of the proof. The first part deals with establishing that for the given choice of parameters described in Section 4.1.3, the output of the algorithm is an ε-optimal estimate of Q⋆with probability 1 −δ. The second part deals with deriving the bounds on the sample and communication complexity based on the choice of prescribed parameters. We begin with the second part, which is easier of the two. C.1 Establishing the sample and communication complexity bounds Establishing the communication complexity. We begin with bounding CCround. From the description of Fed-DVR-Q, it is straightforward to note that each epoch, i.e., each call to the REFINEESTIMATE routine, involves I + 1 rounds of communication, one for estimating T Q and the remaining ones during the iterative updates of the Q-function. Since there are a total of K epochs, CCround(Fed-DVR-Q; ε, M, δ) ≤(I + 1)K ≤ 16 η(1 −γ) log2 1 (1 −γ)ε , where the second bound follows from the prescribed choice of parameters in Sec. 4.1.3. Similarly, since the quantization step is designed to compress each coordinate into J bits, each message transmitted by an agent has a size of no more than J · \|S\|\|A\| bits. Consequently, CCbit(Fed-DVR-Q; ε, M, δ) ≤J · \|S\|\|A\| · CCround(Fed-DVR-Q; ε, M, δ) ≤32\|S\|A\| η(1 −γ) log2 1 (1 −γ)ε log2 70 η(1 −γ) s 4 M log 8KI\|S\|\|A\| δ ! , where once again in the second step we plugged in the choice of J from Sec. 4.1.3. 39 Establishing the sample complexity. In order to establish the bound on the sample complexity, note that during epoch k, each agent takes a total of ⌈Lk/M⌉+ I · B samples, where the first term corresponds to approximating eTL(Q(k−1)) and the second term corresponds to the samples taken during the iterative update scheme. Thus, the total sample complexity is obtained by summing up over all the K epochs. We have, SC(Fed-DVR-Q; ε, M, δ) ≤ K X k=1 Lk M + I · B ≤I · B · K + 1 M K X k=1 Lk + K. To continue, notice that 1 M K X k=1 Lk ≤ 39200 M(1 −γ)2 log 8KI\|S\|\|A\| δ K0 X k=1 4k + K X k=K0+1 4k−K0 ! ≤ 39200 3M(1 −γ)2 log 8KI\|S\|\|A\| δ 4K0 + 4K−K0 ≤ 156800 3M(1 −γ)2 log 8KI\|S\|\|A\| δ 1 1 −γ + 1 (1 −γ)ε2 , where the first line follows from the choice of Lk in Sec. 4.1.3 and the last line follows from K0 = ⌈1 2 log2( 1 1−γ )⌉. Plugging this relation and the choices of I and B (cf. Sec. 4.1.3) into the previous bound yields SC(Fed-DVR-Q; ε, M, δ) ≤ 4608 ηM(1 −γ)3 log2 1 (1 −γ)ε log 8KI\|S\|\|A\| δ + K + 156800 3M(1 −γ)2 log 8KI\|S\|\|A\| δ 1 1 −γ + 1 (1 −γ)ε2 ≤ 313600 ηM(1 −γ)3ε2 log2 1 (1 −γ)ε log 8KI\|S\|\|A\| δ + K. Plugging in the choice of K finishes the proof. C.2 Establishing the error guarantees In this section, we show that the Q-function estimate returned by the Fed-DVR-Q algorithm is ε-optimal with probability at least 1 −δ. We claim that the estimates of the Q-function generated by the algorithm across different epochs satisfy the following relation for all k ≤K with probability 1 −δ: ∥Q(k) −Q⋆∥∞≤2−k 1 −γ . (121) The required bound on ∥Q(K) −Q⋆∥∞immediately follows by plugging in the value of K. Thus, for the remainder of the section, we focus on establishing the above claim. Step 1: fixed-point contraction of REFINEESTIMATE. Firstly, note that the variance-reduced update scheme carried out during the REFINEESTIMATE routine resembles that of the classic Qlearning scheme, i.e., fixed-point iteration, with a different operator defined as follows: H(Q) := T (Q) −T (Q) + eTL(Q), for some fixed Q. (122) Thus, the update scheme at step i ≥1 in (11) can then be written as Qm i−1 2 = (1 −η)Qi−1 + η b H(m) i (Qi−1), (123) where b H(m) i (Q) := bT (m) i (Q) −bT (m) i (Q) + eTL(Q) is a stochastic, unbiased estimate of the operator H, similar to bT (m) i (Q). Let Q⋆ H denote the fixed point of H. Then the update scheme in (123) drives the sequence {Qm i }i≥0 to Q⋆ H; further, as long as ∥Q⋆−Q⋆ H∥∞is small, the required error ∥Qi −Q⋆∥∞can also be controlled. The following lemmas formalize these ideas and pave the path to establish the claim in (121). The proofs are deferred to Appendix C.3. 40 Lemma 7. Let δ ∈(0, 1). Consider the REFINEESTIMATE routine described in Algorithm 3 and let Q⋆ H denote the fixed point of the operator H defined in (122) for some fixed Q. Then the iterates generated by REFINEESTIMATE QI satisfy ∥QI −Q⋆ H∥∞≤1 6 ∥Q −Q⋆∥∞+ ∥Q⋆−Q⋆ H∥∞ + D 70 with probability 1 − δ 2K . Lemma 8. Consider the REFINEESTIMATE routine described in Alg. 3 and let Q⋆ H denote the fixed point of the operator H defined in Eqn. (122) for a fixed Q. The following relation holds with probability 1 − δ 2K : ∥Q⋆ H −Q⋆∥∞≤∥Q −Q⋆∥∞· s 16κ′ L(1 −γ)2 + s 64κ′ L(1 −γ)3 + 2κ′√ 2 3L(1 −γ)2 + D 70, whenever L ≥32κ′, where κ′ = log 12K\|S\|\|A\| δ . Step 2: establishing the linear contraction. We now leverage the above lemmas to establish the desired contraction in (121). Instantiating the operator (122) at each k-th epoch by setting Q := Q(k−1) and L := Lk, we define Hk(Q) := T (Q) −T (Q(k−1)) + eTLk(Q(k−1)), (124) whose fixed point is denoted as Q⋆ Hk. Using the results from Lemmas 7 and 8 with D := Dk and H = Hk, we obtain ∥Q(k) −Q⋆∥∞≤∥Q(k) −Q⋆ Hk∥∞+ ∥Q⋆ H −Q⋆ Hk∥∞ ≤1 6 ∥Q(k−1) −Q⋆∥∞+ ∥Q⋆−Q⋆ Hk∥∞ + Dk 70 + ∥Q⋆ Hk −Q⋆∥∞ = 1 6 ∥Q(k−1) −Q⋆∥∞+ 7∥Q⋆−Q⋆ Hk∥∞ + Dk 70 ≤∥Q(k−1) −Q⋆∥∞ 1 6 + 7 6 s 16κ′ Lk(1 −γ)2 ! + 7 6 s 64κ′ Lk(1 −γ)3 + 2 √ 2κ′ 3Lk(1 −γ)2 ! + 13Dk 420 ≤∥Q(k−1) −Q⋆∥∞ 1 6 + 7 6 s 16κ′ Lk(1 −γ)2 ! + 7 6 s 100κ′ Lk(1 −γ)3 + 13Dk 420 , (125) holds with probability 1 −δ K . Here, we invoke Lemma 7 in the second step and Lemma 8 in the fourth step corresponding to the REFINEESTIMATE routine during the k-th epoch. In the last step, we used the fact that Lk(1−γ)2 κ′ ≥1. We now use induction along with the recursive relation in (125) to establish the required claim (121). Let us first consider the case 0 ≤k ≤K0. The base case, ∥Q(0) −Q⋆∥∞≤ 1 1−γ , holds by definition. Let us assume the relation holds for k −1. Then, from (125) and choice of Lk (Sec. 4.1.3), we have ∥Q(k) −Q⋆∥∞≤∥Q(k−1) −Q⋆∥∞ 1 6 + 7 6 s 16κ′ Lk(1 −γ)2 ! + 7 6 s 100κ′ Lk(1 −γ)3 + 13Dk 420 ≤2−(k−1) 1 −γ 1 6 + 2−k · 7 6 r 8 19600 ! + 2−k · 7 6 s 50 19600(1 −γ) + 104 420 · 2−(k−1) 1 −γ ≤2−(k−1) 1 −γ 1 6 + 7 6 r 91 39200 + 1 4 ! ≤2−k 1 −γ . (126) 41 Now we move to the second case, for k > K0. From (125) and choice of Lk (Sec. 4.1.3), we have ∥Q(k) −Q⋆∥∞≤∥Q(k−1) −Q⋆∥∞ 1 6 + 7 6 s 16κ′ Lk(1 −γ)2 ! + 7 6 s 100κ′ Lk(1 −γ)3 + 13Dk 420 ≤2−(k−1) 1 −γ 1 6 + 2−(k−K0) · 7 6 r 8 19600 ! + 2−(k−K0) · 7 6 s 50 19600(1 −γ) + 104 420 · 2−(k−1) 1 −γ ≤2−(k−1) 1 −γ 1 6 + 7 6 r 1 196 + 1 4 ! ≤2−k 1 −γ . (127) By a union bound argument, we can conclude that the relation ∥Q(k) −Q⋆∥∞≤2−k 1−γ holds for all k ≤K with probability at least 1 −δ. Step 3: confirm the compressor bound. The only thing left to verify is that the inputs to the compressor are always bounded by Dk during the k-th epoch, for all 1 ≤k ≤K. The following lemma provides a bound on the input to the compressor during any run of the REFINEESTIMATE routine. Lemma 9. Consider the REFINEESTIMATE routine described in Algorithm 3 with some for some fixed Q. For all i ≤I and all agents m, the following bound holds with probability 1 − δ 2K : ∥Qm i−1 2 −Qi−1∥∞≤η∥Q −Q⋆ H∥∞ 7 6 · (1 + γ) + 2γ + ηD(1 + γ) 70 . For the k-th epoch, it follows that η∥Q(k−1) −Q⋆ Hk∥∞ 7 6 · (1 + γ) + 2γ + ηDk(1 + γ) 70 ≤13 3 ∥Q(k−1) −Q⋆∥∞+ ∥Q⋆−Q⋆ Hk∥∞ + Dk(1 + γ) 70 ≤13 3 · 15 14 · ∥Q(k−1) −Q⋆∥∞+ 2Dk 70 ≤ 195 42 + 16 70 · 2−(k−1) 1 −γ ≤8 · 2−(k−1) 1 −γ := Dk. In the third step, we used the same sequence of arguments as used in (126) and (127) and, in the fourth step, we used the bound on ∥Q(k−1) −Q⋆∥∞from (121) and the prescribed value of Dk. C.3 Proof of auxiliary lemmas C.3.1 Proof of Lemma 7 Let us begin with analyzing the evolution of the sequence {Qi}I i=1 during a run of the REFINEESTIMATE routine. The sequence {Qi}I i=1 satisfies the following recursion: Qi = Qi−1 + 1 M M X m=1 C Qm i−1 2 −Qi−1; D, J = Qi−1 + 1 M M X m=1 Qm i−1 2 −Qi−1 + ζm i 42 = 1 M M X m=1 Qm i−1 2 + ζm i = (1 −η)Qi−1 + η M M X m=1 b H(m) i (Qi−1) + 1 M M X m=1 ζm i \| {z } =:ζi . (128) In the above expression, ζm i denotes the quantization noise introduced at agent m in the i-th update. Subtracting Q⋆ H from both sides of (128), we obtain Qi −Q⋆ H = (1 −η)(Qi−1 −Q⋆ H) + η M M X m=1 b H(m) i (Qi−1) −Q⋆ H + ζi = (1 −η)(Qi−1 −Q⋆ H) + η M M X m=1 b H(m) i (Qi−1) −b H(m) i (Q⋆ H) + η M M X m=1 b H(m) i (Q⋆ H) −H(Q⋆ H) + ζi. (129) Consequently, ∥Qi −Q⋆ H∥∞≤(1 −η)∥Qi−1 −Q⋆ H∥∞+ η M M X m=1 b H(m) i (Qi−1) −b H(m) i (Q⋆ H) ∞ + η M M X m=1 b H(m) i (Q⋆ H) −H(Q⋆ H) ∞ + ∥ζi∥∞, (130) which we shall proceed to bound each term separately. • Regarding the second term, it follows that b H(m) i (Q) −b H(m) i (Q⋆ H) ∞= bT (m) i (Q) −bT (m) i (Q⋆ H) ∞≤γ ∥Q −Q⋆ H∥∞, (131) which holds for all Q since bT (m) i is a γ-contractive operator. • Regarding the third term, notice that 1 M M X m=1 b H(m) i (Q⋆ H) −H(Q⋆ H) = 1 MB M X m=1 X z∈Z(m) i Tz(Q⋆ H) −Tz(Q) −T (Q⋆ H) + T (Q) . Note that Tz(Q⋆ H) −Tz(Q) −T (Q⋆ H) + T (Q) is a zero-mean random vector satisfying ∥Tz(Q⋆ H) −Tz(Q) −T (Q⋆ H) + T (Q)∥∞≤2γ∥Q −Q⋆ H∥∞. (132) Thus, each of its coordinate is a (2γ∥Q −Q⋆ H∥∞)2-sub-Gaussian vector. Applying the tail bounds for a maximum of sub-Gaussian random variables [Vershynin, 2018], we obtain that 1 M M X m=1 b H(m) i (Q⋆ H) −H(Q⋆ H) ∞ ≤2γ∥Q −Q⋆ H∥∞· s 2 MB log 8KI\|S\|\|A\| δ (133) holds with probability at least 1 − δ 4KI . • Turning to the last term, by the construction of the compression routine described in Section 4.1.2, it is straightforward to note that ζm i is a zero-mean random vector whose coordinates are independent, D2 · 4−J-sub-Gaussian random variables. Thus, ζi is also a zero-mean random vector whose coordinates are independent, D2 M·4J -sub-Gaussian random variables. Hence, we can similarly conclude that ∥ζi∥∞≤D · 2−J · s 2 M log 8KI\|S\|\|A\| δ (134) holds with probability at least 1 − δ 4KI . 43 Combining the above bounds into (130), and introducing the short-hand notation κ := log 8KI\|S\|\|A\| δ , we obtain with probability at least 1 − δ 2KI , ∥Qi −Q⋆ H∥∞≤(1 −η(1 −γ))∥Qi−1 −Q⋆ H∥∞+ 2ηγ∥Q −Q⋆ H∥∞· r 2κ MB + D · 2−J · r 2κ M . Unrolling the above recursion over i = 1, . . . , I yields the following relation, which holds with probability at least 1 − δ 2K : ∥QI −Q⋆ H∥∞≤(1 −η(1 −γ))I ∥Q0 −Q⋆ H∥∞+ r 2κ M 2ηγ √ B ∥Q −Q⋆ H∥∞+ D · 2−J · I X i=1 (1 −η(1 −γ))I−i ≤(1 −η(1 −γ))I ∥Q −Q⋆ H∥∞+ 1 η(1 −γ) r 2κ M 2ηγ √ B ∥Q −Q⋆ H∥∞+ D · 2−J ≤∥Q −Q⋆ H∥∞ (1 −η(1 −γ))I + 2γ (1 −γ) r 2κ MB ! + D · 2−J η(1 −γ) · r 2κ M (135) ≤∥Q −Q⋆ H∥∞ 6 + D 70 ≤1 6 ∥Q −Q⋆∥∞+ ∥Q⋆−Q⋆ H∥∞ + D 70. (136) Here, the fourth step is obtained by plugging in the prescribed values of B, I and J in Sec. 4.1.3. C.3.2 Proof of Lemma 8 Intuitively, the error ∥Q⋆ H −Q⋆∥∞depends on the error term eTL(Q) −T (Q). If the latter is small, then H(Q) is close to T (Q) and consequently so are Q⋆ H and Q⋆. Thus, we begin with bounding the term eTL(Q) −T (Q). We have, eTL(Q) −T (Q) = Q + 1 M M X m=1 C eT (m) L (Q) −Q −T (Q) = 1 M M X m=1 eT (m) L (Q) + ˜ζ(m) L −T (Q) = 1 M M X m=1 eT (m) L (Q) −eT (m) L (Q⋆) −T (Q) + T (Q⋆) + 1 M M X m=1 ˜ζ(m) L + 1 M M X m=1 eT (m) L (Q⋆) −T (Q⋆) , (137) where once again ˜ζ(m) L := eT (m) L (Q) −Q −C eT (m) L (Q) −Q denotes the quantization error at agent m. Similar to the arguments of (133) and (134), we can conclude that each of the following relations hold with probability at least 1 − δ 6K : 1 M M X m=1 eT (m) L (Q) −eT (m) L (Q⋆) −T (Q) + T (Q⋆) ∞ ≤2γ∥Q −Q⋆∥∞· s 2 L log 12K\|S\|\|A\| δ , (138) 1 M M X m=1 ˜ζ(m) L ∞ ≤D · 2−J · s 2 M log 12K\|S\|\|A\| δ . (139) For the third term, we can rewrite it as 1 M M X m=1 eT (m) L (Q⋆) −T (Q⋆) = 1 M⌈L/M⌉ M X m=1 ⌈L/M⌉ X l=1 TZ(m) l (Q⋆) −T (Q⋆) . 44 We will use Bernstein inequality element wise to bound the above term. Let σ⋆∈R\|S\|×\|A\| be such that [σ⋆(s, a)]2 = Var(TZ(Q⋆)(s, a)), i.e., (s, a)-th element of σ denotes the standard deviation of the random variable TZ(Q⋆)(s, a). Since ∥TZ(Q⋆) −T (Q⋆)∥∞≤ 1 1−γ a.s., Bernstein inequality gives us that 1 M M X m=1 eT (m) L (Q⋆)(s, a) −T (Q⋆)(s, a) ≤σ⋆(s, a) s 2 L log 6K\|S\|\|A\| δ + 2 3L(1 −γ) log 6K\|S\|\|A\| δ . (140) holds simultaneously for all (s, a) ∈S × A with probability at least 1 − δ 6K . On combining (137), (138), (139) and (140), we obtain that eTL(Q)(s, a) −T (Q)(s, a) = ∥Q −Q⋆∥∞· r 8κ′ L + σ⋆(s, a) r 2κ′ L + 2κ′ 3L(1 −γ) + D · 2−J · r 2κ′ M , (141) holds simultaneously for all (s, a) ∈S × A with probability at least 1 − δ 2K , where κ′ = log 12K\|S\|\|A\| δ . We use this bound in (141) to obtain a bound on ∥Q⋆ H −Q⋆∥∞using the following lemma. Lemma 10 (Wainwright [2019b]). Let π⋆and π⋆ H respectively denote the optimal policies w.r.t. Q⋆ and Q⋆ H. Then, ∥Q⋆ H −Q⋆∥∞≤max n (I −γP π⋆)−1 eTL(Q) −T (Q) , (I −γP π⋆ H)−1 eTL(Q) −T (Q) o . Here, for any deterministic policy π, P π ∈ R\|S\|\|A\|×\|S\|\|A\| is given by (P πQ)(s, a) = P s′∈S P(s′\|s, a)Q(s′, π(s′)). Furthermore, it was shown in Wainwright [2019b, Proof of Lemma 4] that if the error \|eTL(Q)(s, a) − T (Q)(s, a)\| satisfies eTL(Q)(s, a) −T (Q)(s, a) ≤z0∥Q −Q⋆∥∞+ z1σ⋆(s, a) + z2 (142) for some z0, z1, z2 ≥0 with z1 < 1, then the bound in Lemma 10 can be simplified to ∥Q⋆ H −Q⋆∥∞≤ 1 1 −z1 z0 1 −γ ∥Q −Q⋆∥∞+ z1 (1 −γ)3/2 + z2 1 −γ . (143) On comparing, (141) with (142), we obtain z0 ≡ r 8κ′ L ; z1 ≡ r 2κ′ L ; z2 ≡ 2κ′ 3L(1 −γ) + D · 2−J · r 2κ′ M . Moreover, the condition L ≥32κ′ implies that z1 < 1 and 1 1−z1 ≤ √ 2. Thus, on plugging in the above values in (143), we can conclude that ∥Q⋆ H −Q⋆∥∞≤∥Q −Q⋆∥∞· s 16κ′ L(1 −γ)2 + s 64κ′ L(1 −γ)3 + 2κ′√ 2 3L(1 −γ)2 + D · 2−J (1 −γ) · r 4κ′ M ≤∥Q −Q⋆∥∞· s 8κ′ L(1 −γ)2 + s 32κ′ L(1 −γ)3 + 2 √ 2κ′ 3L(1 −γ)2 + D 40, (144) where once again we use the value of J in the last step. C.3.3 Proof of Lemma 9 From the iterative update rule in (123), for any agent m we have, Qm i−1 2 −Qi−1 = η( b H(m) i−1(Qi−1) −Qi−1) 45 = η( b H(m) i−1(Qi−1) −b H(m) i−1(Q⋆ H) + b H(m) i−1(Q⋆ H) −H(Q⋆ H) + Q⋆ H −Qi−1). Thus, ∥Qm i−1 2 −Qi−1∥∞≤η ∥b H(m) i−1(Qi−1) −b H(m) i−1(Q⋆ H)∥∞+ ∥b H(m) i−1(Q⋆ H) −H(Q⋆ H)∥∞+ ∥Q⋆ H −Qi−1∥∞ ≤η γ∥Qi−1 −Q⋆ H∥∞+ 2γ∥Q −Q⋆ H∥∞+ ∥Q⋆ H −Qi−1∥∞ = η (1 + γ)∥Qi−1 −Q⋆ H∥∞+ 2γ∥Q −Q⋆ H∥∞ ≤η∥Q −Q⋆ H∥∞ 7 6 · (1 + γ) + 2γ + ηD(1 + γ) 70 , holds with probability 1 − δ 2KI . Here, the second inequality follows from (131) and (132), The last step in the above relation follows from (135) evaluated at a general value of i and the prescribed value of J. By a union bound argument, the above relation holds for all i with probability at least 1 − δ 2K . D Numerical Experiments In this section, we corroborate our theoretical results through simulations. For the simulations, we consider an MDP with 3 states and two actions, i.e., S = {0, 1, 2} and A = {0, 1}. The discount parameter is set to γ = 0.9. The reward and transition kernel of the MDP is based on the hard instance constructed in Appendix B. Specifically, the reward and transition kernel of state 0 is given by the expression in Eqn. (14a). Similarly, the reward and transition kernel corresponding to state 1 and 2 are identical and given by Eqns. (14b) and (14c) with p = 0.8. 106 107 10−3 10−2 10−1 100 Error vs Samples Taken Fed-DVR-Q Fed-Syn-Q (a) Sample Complexity 0.0 0.2 0.4 0.6 0.8 1.0 ×107 0 5000 10000 15000 20000 25000 30000 Communication Cost vs Samples Taken Fed-DVR-Q Fed-Syn-Q (b) Communication Complexity Figure 1: Comparison between sample and communication complexities of Fed-DVR-Q and the algorithm Fed-SynQ from Woo et al. [2023]. We perform three empirical studies. In the first study, we compare the proposed algorithm Fed-DVRQ to the Fed-SynQ algorithm proposed in Woo et al. [2023]. We consider a Federated Q-learning setting with 5 agents. The parameters for both the algorithms were set to the suggested values in the respective papers. Both the algorithms were run with 107 samples at each agent. For the communication cost of Fed-SynQ we assume that each real number is expressed using 32 bits. In Fig 1a, we plot the error rate of the algorithm as a function of the number of samples used. In Fig. 1b we plot the corresponding communication complexities. As evident from Fig 1a, Fed-DVR-Q achieves a smaller error than Fed-SynQ under the same sample budget. Similarly, as suggested by Fig. 1b, Fed-DVR-Q also requires much less communication (measured in terms of the number of bits transmitted) than Fed-SynQ, demonstrating the effectiveness of the proposed approach and corroborating our theoretical results. In the second study, we examine the effect of the number of agents on the sample and communication complexity of Fed-DVR-Q. We vary the number of agents from 5 to 25 in multiples of 5 and record the sample and communication complexity to achieve an error rate of ε = 0.03. The sample 46 5 10 15 20 25 0.2 0.4 0.6 0.8 1.0 ×107 Sample Complexity vs Number of Agents Fed-DVR-Q (a) Sample Complexity 5 10 15 20 25 10200 10400 10600 10800 11000 11200 Communication Complexity vs Number of Agents Fed-DVR-Q (b) Communication Complexity Figure 2: Dependence of sample and communication complexities of Fed-DVR-Q on the number of agents. 3 4 5 6 7 8 9 10 3000 4000 5000 6000 7000 8000 Communication Complexity vs Eﬀective horizon Fed-DVR-Q Figure 3: Communication complexity of Fed-DVR-Q as a function of effective horizon, i.e., 1 1−γ . and communication complexities as a function of number of agents are plotted in Figs. 2a and 2b respectively. The sample complexity decreases as 1/M while the communication complexity is independent of the number of agents. This corroborates the linear speedup phenomenon suggested by our theoretical results and the independence between communication complexity and the number of agents. In the last study, we compare the communication complexity of Fed-DVR-Q as function of the discount parameter γ. We consider the same setup as in the first study and vary the values of γ from 0.7 to 0.9 in steps of 0.05. We run the algorithm to achieve an accuracy of ε = 0.1 with parameter choices prescribed in Sec. 4.1.3. We plot the communication cost of Fed-DVR-Q against the effective horizon, i.e., 1 1−γ in Fig. 3. As evident from the figure, the communication scales linearly with the effective horizon, which matches the theoretical claim in Theorem 2. 47 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? 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