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2024/Towards Lossless Dataset Distillation via Difficulty-Aligned Trajectory Matching/9f241194-6197-49e1-ae17-9a1c98b06f82_origin.pdf

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2024/Towards Lossless Dataset Distillation via Difficulty-Aligned Trajectory Matching/full.md

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+# TOWARDS LOSSLESS DATASET DISTILLATION VIA DIFFICULTY-ALIGNED TRAJECTORY MATCHING
+
+Ziyao Guo $^{1,3,4}$  Kai Wang $^{1\dagger}$  George Cazenavette $^{2}$  Hui Li $^{4}$  Kaipeng Zhang $^{3\dagger}$  Yang You $^{1\dagger}$
+
+$^{1}$ National University of Singapore  $^{2}$ Massachusetts Institute of Technology  
+$^{3}$ Shanghai Artificial Intelligence Laboratory  $^{4}$ Xidian University
+
+gzyaftermath@outlook.com, {kai.wang, youy}@comp.nus.edu.sg, gcaz@mit.edu
+
+# ABSTRACT
+
+The ultimate goal of Dataset Distillation is to synthesize a small synthetic dataset such that a model trained on this synthetic set will perform equally well as a model trained on the full, real dataset. Until now, no method of Dataset Distillation has reached this completely lossless goal, in part because they only remain effective when the total number of synthetic samples is extremely small. Since only so much information can be contained in such a small number of samples, it seems that to achieve truly lossless dataset distillation, we must develop a distillation method that remains effective as the size of the synthetic dataset grows. In this work, we present such an algorithm and elucidate why existing methods fail to generate larger, high-quality synthetic sets. Current state-of-the-art methods rely on trajectory-matching, or optimizing the synthetic data to induce similar long-term training dynamics as the real data. We empirically find that the training stage of the trajectories we choose to match (i.e., early or late) greatly affects the effectiveness of the distilled dataset. Specifically, early trajectories (where the teacher network learns easy patterns) work well for a low-cardinality synthetic set since there are fewer examples wherein to distribute the necessary information. Conversely, late trajectories (where the teacher network learns hard patterns) provide better signals for larger synthetic sets since there are now enough samples to represent the necessary complex patterns. Based on our findings, we propose to align the difficulty of the generated patterns with the size of the synthetic dataset. In doing so, we successfully scale trajectory matching-based methods to larger synthetic datasets, achieving lossless dataset distillation for the very first time. Code and distilled datasets are available at https://github.com/NUS-HPC-AI-Lab/DATM.
+
+# 1 INTRODUCTION
+
+Dataset distillation (DD) aims at distilling a large dataset into a small synthetic one, such that models trained on the distilled dataset will have similar performance as those trained on the original dataset. In recent years, several algorithms have been proposed for this important topic, such as gradient matching (Zhao et al., 2020; Kim et al., 2022; Zhang et al., 2023; Liu et al., 2023b), kernel inducing points (Nguyen et al., 2020; 2021), distribution matching (Wang et al., 2022; Zhao & Bilen, 2023; Zhao et al., 2023), and trajectory matching (Cazenavette et al., 2022; Cui et al., 2023; Du et al., 2023). So far, dataset distillation has achieved great success in the regime of extremely small synthetic sets. For example, MTT (Cazenavette et al., 2022) achieves  $71.6\%$  test accuracy on CIFAR-10 using only  $1\%$  of the original data size. This impressive performance led to its application in a variety of downstream tasks such as continual learning (Masarczyk & Tautkute, 2020; Rosasco et al., 2021), privacy protection (Zhou et al., 2020; Sucholutsky & Schonlau, 2021a; Dong et al., 2022; Chen et al., 2022; Xiong et al., 2023), and neural architecture search (Such et al., 2020; Wang et al., 2021).
+
+However, although previous DD methods have achieved great success with very few IPC (images-per-class), there still remains a significant gap between the performance of their distilled datasets and the full, real counterparts. To minimize this gap, one would intuitively think to increase the size of the
+
+![](images/fcd77e280f07cc385b482fae171a288b8b6bee8fd579748cfd2aa55fee1f9481.jpg)  
+Figure 1: (a) Illustration of the objective of dataset distillation. (b) The optimization in dataset distillation can be viewed as the process of generating informative patterns on the synthetic dataset. (c) We align the difficulty of the synthetic patterns with the size of the distilled dataset, to enable our method to perform well in both small and large IPC regimes. (d) Comparison of the performance of multiple dataset distillation methods on CIFAR-10 with different IPC. As IPC increases, the performance of previous methods becomes worse than random selection.
+
+![](images/8d5022f78f6b7c05aebe6e2efbb19fb2aa43a612fbe5c109453b8c442f7b56a9.jpg)
+
+![](images/e8b2b81a6f385a262b72942c35753319cc0968bed840111cbed4139876a59896.jpg)
+
+synthetic dataset. Unfortunately, as IPC increases, previous distillation methods mysteriously become less effective, even performing worse than random selection (Cui et al., 2022; Zhou et al., 2023). In this paper, we offer an answer as to why previous dataset distillation methods become ineffective as IPC increases and, in doing so, become the first to circumvent this issue, allowing us to achieve lossless dataset distillation.
+
+We start our work by observing the patterns learned by the synthetic data, taking trajectories matching (TM) based distillation methods (Cazenavette et al., 2022; Du et al., 2023) as an example. Generally, the process of dataset distillation can be viewed as the embedding of informative patterns into a set of synthetic samples. For TM-based distillation methods, the synthetic data learns patterns by matching the training trajectories of surrogate models optimized over the synthetic dataset and the real one. According to (Arpit et al., 2017), deep neural networks (DNNs) typically learn to recognize easy patterns early in training and hard patterns later on. As a result, we note that the properties of the data generated by TM-based methods vary widely depending on from which teacher training stage we sample our trajectories from (early or late). Specifically, matching early or late trajectories causes the synthetic data to learn easy or hard patterns respectively.
+
+We then empirically show that the effect of learning easy and hard patterns varies with the size of the synthetic dataset (i.e., IPC). In low-IPC settings, easy patterns prove the most beneficial since they explain a larger portion of the real data distribution than an equivalent number of hard samples. However, with a sufficiently large synthetic set, learning hard samples becomes optimal since their union covers both the easy and "long-tail" hard samples of the real data. In fact, learning easy patterns in the high-IPC setting performs worse than random selection since the synthetic images collapse towards the mean patterns of the distribution and can no longer capture the long-tail parts. Previous distillation methods default toward distilling easy patterns, leading to their ineffectiveness in high-IPC cases.
+
+The above findings motivate us to manage to align the difficulty of the learned patterns with the size of the distilled dataset, in order to keep our method effective in both low and high IPC cases. Our experiments show that, for TM-based methods, we can control the difficulty of the generated patterns by only matching the trajectories of a specified training phase. By doing so, our method is able to work well in both low and high IPC settings. Furthermore, we propose to learn easy and hard patterns sequentially, making the optimization stable enough for learning soft labels during the distillation, bringing further significant improvement. Our method achieves state-of-the-art performance in both low and high IPC cases. Notably, we distill CIFAR-10 and CIFAR-100 to 1/5 and Tiny ImageNet to 1/10 of their original sizes without any performance loss on ConvNet, offering the first lossless method of dataset distillation.
+
+# 2 PRELIMINARY
+
+For a given large, real dataset  $\mathcal{D}_{\mathrm{real}}$ , dataset distillation aims to synthesize a smaller dataset  $\mathcal{D}_{\mathrm{syn}}$  such that models trained on  $\mathcal{D}_{\mathrm{syn}}$  will have similar test performance as models trained on  $\mathcal{D}_{\mathrm{real}}$ .
+
+![](images/0cfca549942c83df58ec46951346998dc2d81029f1a94f8fd8993835d00a8fc3.jpg)  
+(a)
+
+![](images/1293fd14068c272a8f6bea93a28a2251c9f205046a239ef09ad494f81ee58d03.jpg)  
+(b)
+
+![](images/e86ceefb4ff8ff27f51e7ecfc57de0140298d8d324a095a9c36128cd4696c092.jpg)  
+(c)  
+Figure 2: We train expert models on CIFAR-10 for 40 epochs. Then the distillation is performed under different IPC settings by matching either early trajectories  $\{\theta_t^* | 0 \leq t \leq 20\}$ , late trajectories  $\{\theta_t^* | 20 \leq t \leq 40\}$ , or all trajectories  $\{\theta_t^* | 0 \leq t \leq 40\}$ . As IPC increases, matching late trajectories becomes beneficial while matching early trajectories tends to be harmful.
+
+![](images/4b0f734a97588c68263e5c221ff34cb460c2b63972ce835cc0680f1800ec8678.jpg)  
+(d)
+
+For trajectory matching (TM) based methods, the distillation is performed by matching the training trajectories of the surrogate models optimized over  $\mathcal{D}_{\mathrm{real}}$  and  $\mathcal{D}_{\mathrm{syn}}$ . Specifically, let  $\tau^{*}$  denote the expert training trajectories, which is the time sequence of parameters  $\{\theta_t^*\}_{0}^n$  obtained during the training of a network on the real dataset  $\mathcal{D}_{\mathrm{real}}$ . Similarly,  $\hat{\theta}_t$  denotes the parameters of the network trained on the synthetic dataset  $\mathcal{D}_{\mathrm{syn}}$  at training step  $t$ .
+
+In each iteration of the distillation,  $\theta_t^*$  and  $\theta_{t + M}^*$  are randomly sampled from a set of expert trajectories  $\{\tau^{*}\}$  as the start parameters and target parameters used for the matching, where  $M$  is a preset hyperparameter. Then TM-based distillation methods optimize the synthetic dataset  $\mathcal{D}_{\mathrm{syn}}$  by minimizing the following loss:
+
+$$
+\mathcal {L} = \frac {\left\| \hat {\theta} _ {t + N} - \theta_ {t + M} ^ {*} \right\| _ {2} ^ {2}}{\left\| \theta_ {t} ^ {*} - \theta_ {t + M} ^ {*} \right\| _ {2} ^ {2}}, \tag {1}
+$$
+
+where  $N$  is a preset hyper-parameter and  $\hat{\theta}_{t + N}$  is obtained in the inner optimization with cross-entropy (CE) loss  $\ell$  and the trainable learning rate  $\alpha$ :
+
+$$
+\hat {\theta} _ {t + i + 1} = \hat {\theta} _ {t + i} - \alpha \nabla \ell \left(\hat {\theta} _ {t + i}, \mathcal {D} _ {\text {s y n}}\right), \text {w h e r e} \hat {\theta} _ {t} := \theta_ {t} ^ {*}. \tag {2}
+$$
+
+# 3 METHOD
+
+In this section, we first analyze the influence of matching trajectories from different training stages. Then, we introduce our method and its carefully designed modules.
+
+# 3.1 EXPLORATION
+
+TM-based methods generate patterns on the synthetic data by matching training trajectories. According to Arpit et al. (2017), DNNs tend to learn easy patterns early in training, then the harder ones later on. Motivated by this, we start our work by exploring the effect of matching trajectories from different training phases. Specifically, we train expert models for 40 epochs and roughly divide their training trajectories into two parts: the early trajectories  $\{\theta_t^* | 0 \leq t \leq 20\}$  and the latter ones  $\{\theta_t^* | 20 \leq t \leq 40\}$ . Then we perform the distillation by matching these two sets of trajectories under various IPC settings. Experimental results are reported in Figure 2. Our observations and relevant analyses are presented as follows.
+
+Observation 1. As shown in Figure 2, matching early trajectories works better with small synthetic datasets, but matching late trajectories performs better as the size of the synthetic set grows larger.
+
+Analysis 1. Since DNNs learn to recognize easy patterns early in training and hard patterns later on, we infer that matching early trajectories yields distilled data with easy patterns while matching late trajectories produces hard ones. Combined with the empirical results from Figure 2, we can conclude that distilled data with easy patterns perform well for small synthetic sets while data with hard features work better with larger sets. Perhaps unsurprisingly, this highly coincides with a common observation in the area of dataset pruning: preserving easy samples works better when very
+
+few samples are kept, while keeping hard samples works better when the pruned dataset is larger (Sorscher et al., 2022).
+
+Observation 2. As can be observed in Figure 2 (a), matching late trajectories leads to poor performance in the low IPC setting. When IPC is high, matching early trajectories will consistently undermine the performance of the synthetic dataset as the distillation goes on, as can be observed in Figure 2 (d). Also, as reflected in Figure 2, simply choosing to match all trajectories is not a good strategy.
+
+Analysis 2. In low IPC settings, due to distilled data's limited capacity, it is challenging to learn data that models the outliers (hard samples) without neglecting the more plentiful easy samples; since easy samples make up most of the real data distribution, modeling these samples is more efficient performance-wise when IPC is low. Therefore, matching early trajectories (which will generate easy patterns) performs better than matching later ones (for low IPC). Conversely, in high IPC settings, distilling data that models only the easy samples is no longer necessary, and will even perform worse than a random subset of real samples. Thus, we must now consider the less-common hard samples by matching late trajectories (Figure 4). Since previous distillation methods focus on extremely small IPC cases, they tend to be biased towards generating easy patterns, leading to their ineffective in large IPC cases.
+
+Based on the above analyses, to keep dataset distillation effective in both low and high IPC cases, we must calibrate the difficulty of the generated patterns (i.e., avoid generating patterns that are too easy or too difficult). To this end, we propose our method: Difficulty-Aligned Trajectory Matching, or DATM.
+
+# 3.2 DIFFICULTY-ALIGNED TRAJECTORY MATCHING
+
+Since patterns learned by matching earlier trajectories are easier than the later ones, we can control the difficulty of the generated patterns by restricting the trajectory-matching range. Specifically, let  $\tau^{*} = \{\theta_{t}^{*}|0\leq t\leq n\}$  denote an expert trajectory. To control the matching range flexibly, we set a lower bound  $T^{-}$  and an upper bound  $T^{+}$  on the sample range of  $t$ , such that only parameters within  $\{\theta_t^* |T^- \leq t\leq T^+\}$  can be sampled for the matching. Then the trajectory segment used for the matching can be formulated as:
+
+$$
+\tau^ {*} = \left\{\underbrace {\theta_ {0} ^ {*}, \theta_ {1} ^ {*}} _ {\text {t o o e a s y}}, \dots , \underbrace {\theta_ {T -} ^ {*} , \cdots , \theta_ {T +} ^ {*}} _ {\text {m a t c h i n g r a n g e}}, \underbrace {\cdots , \theta_ {n} ^ {*}} _ {\text {t o o h a r d}} \right\}. \tag {3}
+$$
+
+To further enrich the information contained in the synthetic dataset, an intuitive choice is using soft labels (Hinton et al., 2015). Recently, Cui et al. (2023) show that using soft labels to guide the distillation can bring non-trivial improvement for the performance. However, their soft labels are not optimized during the distillation, leading to poor consistency between synthetic data and soft labels. To enable learning labels, we find the following challenges need to be solved:
+
+Mislabeling. We use logits  $L_{i} = f_{\theta^{*}}(x_{i})$  to initialize soft labels, which are generated by the pretrained model  $f_{\theta^{*}}$  sampled from expert trajectories. However, labels initialized in this way might be incorrect (i.e., target class doesn't have the highest logit score). To avoid mislabeling, we sift through  $\mathcal{D}_{\mathrm{real}}$  to find samples that can be correctly classified by model  $f_{\theta^{*}}$  and use them to construct the subset  $\mathcal{D}_{\mathrm{sub}}$ . Then we randomly select samples from  $\mathcal{D}_{\mathrm{sub}}$  to initialize  $\mathcal{D}_{\mathrm{syn}} = \{(x_{i},\hat{y}_{i} = \mathrm{softmax}(L_{i}))\}$ , such that we can avoid the distillation being misguided by the wrong label.
+
+Instability. During the experiments, we found that optimizing soft labels will increase the instability of the distillation when the IPC is low. In low IPC settings, the distillation loss tends to be higher and less stable overall since the smaller synthetic set struggles to induce a proper training trajectory. This issue becomes fatal when labels are optimized during the distillation, as the labels are too fragile to take the wrong guidance brought by the mismatch, leading to increased instability. To alleviate this, we propose to generate only easy patterns in the early distillation phase. After enough easy patterns are embedded into the synthetic data for surrogate models to learn them well, we then gradually generate harder ones. By applying this sequential generation (SG) strategy, the surrogate model can match the expert trajectories better. Accordingly, the distillation becomes more stable.
+
+In practice, to generate only easy patterns at the early distillation stage, we set a floating upper bound  $T$  on the sample range of  $t$ , which is set to be relatively small in the beginning and will be gradually increased as the distillation progresses until it reaches its upper bound  $T^{+}$ . Overall, the
+
+<table><tr><td>Dataset</td><td colspan="5">CIFAR-10</td><td colspan="4">CIFAR-100</td><td colspan="3">Tiny ImageNet</td></tr><tr><td>IPC</td><td>1</td><td>10</td><td>50</td><td>500</td><td>1000</td><td>1</td><td>10</td><td>50</td><td>100</td><td>1</td><td>10</td><td>50</td></tr><tr><td>Ratio</td><td>0.02</td><td>0.2</td><td>1</td><td>10</td><td>20</td><td>0.2</td><td>2</td><td>10</td><td>20</td><td>0.2</td><td>2</td><td>10</td></tr><tr><td>Random</td><td>15.4±0.3</td><td>31.0±0.5</td><td>50.6±0.3</td><td>73.2±0.3</td><td>78.4±0.2</td><td>4.2±0.3</td><td>14.6±0.5</td><td>33.4±0.4</td><td>42.8±0.3</td><td>1.4±0.1</td><td>5.0±0.2</td><td>15.0±0.4</td></tr><tr><td>DC</td><td>28.3±0.5</td><td>44.9±0.5</td><td>53.9±0.5</td><td>72.1±0.4</td><td>76.6±0.3</td><td>12.8±0.3</td><td>25.2±0.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DM</td><td>26.0±0.8</td><td>48.9±0.6</td><td>63.0±0.4</td><td>75.1±0.3</td><td>78.8±0.1</td><td>11.4±0.3</td><td>29.7±0.3</td><td>43.6±0.4</td><td>-</td><td>3.9±0.2</td><td>12.9±0.4</td><td>24.1±0.3</td></tr><tr><td>DSA</td><td>28.8±0.7</td><td>52.1±0.5</td><td>60.6±0.5</td><td>73.6±0.3</td><td>78.7±0.3</td><td>13.9±0.3</td><td>32.3±0.3</td><td>42.8±0.4</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CAFE</td><td>30.3±1.1</td><td>46.3±0.6</td><td>55.5±0.6</td><td>-</td><td>-</td><td>12.9±0.3</td><td>27.8±0.3</td><td>37.9±0.3</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>KIP1</td><td>49.9±0.2</td><td>62.7±0.3</td><td>68.6±0.2</td><td>-</td><td>-</td><td>15.7±0.2</td><td>28.3±0.1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FRePo1</td><td>46.8±0.7</td><td>65.5±0.4</td><td>71.7±0.2</td><td>-</td><td>-</td><td>28.7±0.1</td><td>42.5±0.2</td><td>44.3±0.2</td><td>-</td><td>15.4±0.3</td><td>25.4±0.2</td><td>-</td></tr><tr><td>RCIG1</td><td>53.9±1.0</td><td>69.1±0.4</td><td>73.5±0.3</td><td>-</td><td>-</td><td>39.3±0.4</td><td>44.1±0.4</td><td>46.7±0.3</td><td>-</td><td>25.6±0.3</td><td>29.4±0.2</td><td>-</td></tr><tr><td>MTT2</td><td>46.2±0.8</td><td>65.4±0.7</td><td>71.6±0.2</td><td>X</td><td>X</td><td>24.3±0.3</td><td>39.7±0.4</td><td>47.7±0.2</td><td>49.2±0.4</td><td>8.8±0.3</td><td>23.2±0.2</td><td>28.0±0.3</td></tr><tr><td>TESLA2</td><td>48.5±0.8</td><td>66.4±0.8</td><td>72.6±0.7</td><td>X</td><td>X</td><td>24.8±0.4</td><td>41.7±0.3</td><td>47.9±0.3</td><td>49.2±0.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FTD2,3</td><td>46.0±0.4</td><td>65.3±0.4</td><td>73.2±0.2</td><td>X</td><td>X</td><td>24.4±0.4</td><td>42.5±0.2</td><td>48.5±0.3</td><td>49.7±0.4</td><td>10.5±0.2</td><td>23.4±0.3</td><td>28.2±0.4</td></tr><tr><td>DATM (Ours)</td><td>46.9±0.5</td><td>66.8±0.2</td><td>76.1±0.3</td><td>83.5±0.2</td><td>85.5±0.4</td><td>27.9±0.2</td><td>47.2±0.4</td><td>55.0±0.2</td><td>57.5±0.2</td><td>17.1±0.3</td><td>31.1±0.3</td><td>39.7±0.3</td></tr><tr><td>Full Dataset</td><td colspan="5">84.8±0.1</td><td colspan="4">56.2±0.3</td><td colspan="3">37.6±0.4</td></tr></table>
+
+Table 1: Comparison with previous dataset distillation methods on CIFAR-10, CIFAR-100 and Tiny ImageNet. ConvNet is used for the distillation and evaluation. Hilighted results indicate we achieve lossless distillation. Our method consistently out-performs prior works and is the only to achieve lossless distillation.  
+$^{1}$  Kernel-based methods use a much larger neural network; we underline their results when they perform best.  
+${}^{2}$  Previous TM-based methods perform worse than random initialization in higher IPC cases, indicated by  $\mathbb{N}$  .  
+For a fair comparison, we reproduce FTD without using EMA (exponential moving average).
+
+process of sampling the start parameters  $\theta_t^*$  can be formulated as:
+
+$$
+\theta_ {t} ^ {*} \sim \mathcal {U} \left(\left\{\theta_ {T ^ {-}} ^ {*}, \dots , \theta_ {T} ^ {*} \right\}\right), \text {w h e r e} T \rightarrow T ^ {+}. \tag {4}
+$$
+
+In each iteration, after deciding the value of  $t$ , we then sample  $\theta_t^*$  and  $\theta_{t + M}^*$  from expert trajectories as the start parameters and the target parameters for the matching. Then  $\hat{\theta}_{t + N}$  can be obtained by Eq. 2. Subsequently, after calculating the matching loss using Eq. 1, we perform backpropagation to calculate the gradients and then use them to update the synthetic data  $x_{i}$  and  $L_{i}$ , where  $(x_{i},\hat{y}_{i} = \mathrm{softmax}(L_{i}))\in \mathcal{D}_{\mathrm{syn}}$ . See Algorithm 1 for the pseudocode of our method.
+
+# 4 EXPERIMENTS
+
+# 4.1 SETUP
+
+We compare our method with several representative distillation methods including DC (Zhao et al., 2020), DM (Zhao & Bilen, 2023), DSA (Zhao & Bilen, 2021), CAFE (Wang et al., 2022), KIP (Nguyen et al., 2020), FRePo (Zhou et al., 2022), RCIG (Loo et al., 2023), MTT (Cazenavette et al., 2022), TESLA (Cui et al., 2023), and FTD (Du et al., 2023). The evaluations are performed on several popular datasets including CIFAR-10, CIFAR-100 (Krizhevsky et al., 2009), and Tiny ImageNet (Le & Yang, 2015). We generate expert trajectories in the same way as FTD without modifying the involved hyperparameters. We also use the same suite of differentiable augmentations (Zhao & Bilen, 2021) in the distillation and evaluation stage, which is generally utilized in previous works (Zhao & Bilen, 2021; Wang et al., 2022; Cazenavette et al., 2022; Du et al., 2023).
+
+Consistent with previous works, we use networks with instance normalization by default, while networks with batch normalization are indicated with "-BN" (e.g., ConvNet-BN). Without particular specification, we perform distillation using a 3-layer ConvNet for CIFAR-10 and CIFAR-100, while we move up to a depth-4 ConvNet for Tiny ImageNet. We also use LeNet (LeCun et al., 1998), AlexNet (Krizhevsky et al., 2012), VGG11 (Simonyan & Zisserman, 2015), and ResNet18 (He et al., 2016) for cross-architecture experiments. More details can be found in Section A.8.
+
+# 4.2 MAIN RESULTS
+
+CIFAR-10/100 and Tiny ImageNet. As the results reported in Table 1, our method outperforms other methods with the same network architecture in all settings but CIFAR-10 with  $\mathrm{IPC} = 1$ . As can be observed, the improvements brought by previous distillation methods are quickly saturated as the distillation ratio approaches  $20\%$ . Especially in CIFAR-10, almost all previous methods have similar or even worse performance than random selection when the ratio is greater than  $10\%$ . Benefiting from our difficulty alignment strategy, our method remains effective in high IPC cases. Notably, we
+
+![](images/422b4a1aafb184cd5ee2b6efe612de6e8c3cd68239b48aec1323c8dec7308180.jpg)  
+(a)
+
+![](images/9f83cc1230d02d5430b1381f9cca9a97fcdd6a66c5a9b6ddc3798444f1d455f0.jpg)
+
+![](images/cf8dc45e2b0b66f94b2bba3cd1ac4b6da0cb1e4e262ff71552abf65795a5a082.jpg)  
+(b)  
+Figure 3: (a): (CIFAR-100, IPC=10) Synthetic datasets are initialized by randomly sampling data from the original dataset (random selection) or a subset of data that can be correctly classified (ours). Our strategy makes the optimization converge faster. (b): (CIFAR-10, IPC=50) Ablation on learning soft labels, where soft labels are initialized with expert models trained after different epochs. Learning labels relieves us from carefully selecting the labeling expert. (c): (CIFAR-10) The optimization with higher IPC converges in fewer iterations.
+
+![](images/08b8e27c6de6f98749788e03bde3a1ecc0b537374063e20480ca7cd9336fcbb1.jpg)  
+(c)
+
+<table><tr><td>Method</td><td>ConvNet</td><td>ResNet18</td><td>VGG</td><td>AlexNet</td></tr><tr><td>Random</td><td>33.46</td><td>31.95</td><td>32.18</td><td>26.65</td></tr><tr><td>MTT</td><td>45.68</td><td>42.56</td><td>41.22</td><td>40.29</td></tr><tr><td>FTD</td><td>48.90</td><td>46.65</td><td>43.24</td><td>42.20</td></tr><tr><td>DATM</td><td>55.03</td><td>51.71</td><td>45.38</td><td>45.74</td></tr></table>
+
+(a) CIFAR-100, IPC=50  
+
+<table><tr><td>Soft Label</td><td>Difficulty Alignment</td><td>Acc</td></tr><tr><td></td><td></td><td>48.50</td></tr><tr><td></td><td>✓</td><td>50.79</td></tr><tr><td>✓</td><td></td><td>52.96</td></tr><tr><td>✓</td><td>✓</td><td>55.03</td></tr></table>
+
+(b) CIFAR-100, IPC=50  
+
+<table><tr><td>Label Learning</td><td>Sequential Gen.</td><td>Acc</td></tr><tr><td></td><td></td><td>72.8</td></tr><tr><td>✓</td><td></td><td>75.0</td></tr><tr><td></td><td>✓</td><td>75.6</td></tr><tr><td>✓</td><td>✓</td><td>76.1</td></tr></table>
+
+successfully distill CIFAR-10 and CIFAR-100 to 1/5, and Tiny ImageNet to 1/10 their original size without causing any performance drop.
+
+Cross-architecture generalization. Here we evaluate the generalizability of our distilled datasets in various IPC settings. As the results reported in Table 2 (Left), our distilled dataset performs best on unseen networks when IPC is small, reflecting the good generalizability of the data and labels distilled by our method. Furthermore, we evaluate the generalizability in high IPC settings and compare the performance with two representative coreset selection methods including Glister (Killamsetty et al., 2021) and Forgetting (Toneva et al., 2018). As shown in Table 4, although coreset selection methods are applied case by case, they are not universally beneficial for all networks. Notably, although our synthetic dataset is distilled with ConvNet, it generalizes well on all networks, bringing non-trivial improvement. Furthermore, on CIFAR-100, the improvement of AlexNet is even higher than that of ConvNet. This reflects the overfitting problem of synthetic datasets to distillation networks is somewhat alleviated in higher IPC situations.
+
+# 4.3 ABLATION
+
+Ablation on components of our method. We perform ablation studies by adding the components of our methods one by one to measure their effect. As the results reported in Table 2 (b,c), all the components of our method bring non-trivial improvement. Especially, when soft labels are utilized, the distillation becomes unstable if our proposed sequential generation strategy is not applied, leading to poor performance and sometimes program crashes.
+
+Soft label. In our method, we use logits generated by the pre-trained model to initialize soft labels, since having an appropriate distribution before the softmax is critical for the optimization of the soft label (Section A.2). However, using logits will introduce additional information to the distilled dataset (Hinton et al.,
+
+(c) CIFAR-10, IPC=50  
+Table 2: (a): Cross-Architecture evaluation. Our distilled dataset performs well across various unseen networks. (b): Ablation studies on the components of our method; all bring non-trivial improvement. (c): Ablation on learning soft labels and our sequential generation (SG) strategy.  
+
+<table><tr><td>Dataset</td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">Tiny ImageNet</td></tr><tr><td>IPC</td><td>1</td><td>10</td><td>50</td><td>1</td><td>10</td><td>50</td><td>1</td><td>10</td><td>50</td></tr><tr><td>FTD</td><td>46.0</td><td>65.3</td><td>73.2</td><td>24.4</td><td>42.5</td><td>48.5</td><td>10.5</td><td>23.4</td><td>28.2</td></tr><tr><td>FTD+ASL</td><td>45.5</td><td>66.1</td><td>72.8</td><td>24.2</td><td>44.5</td><td>51.2</td><td>12.4</td><td>26.7</td><td>30.9</td></tr><tr><td>DATM</td><td>46.9</td><td>66.8</td><td>76.1</td><td>27.9</td><td>47.2</td><td>53.0</td><td>17.1</td><td>29.0</td><td>33.7</td></tr></table>
+
+Table 3: We assign soft labels (ASL) for datasets distilled by FTD. For fairness, our difficult alignment strategy is not utilized here. Results in red indicate the case when ASL is harmful.
+
+<table><tr><td>Dataset</td><td>Ratio</td><td>Method</td><td>ConvNet</td><td>ConvNet-BN</td><td>ResNet18</td><td>ResNet18-BN</td><td>VGG11</td><td>AlexNet</td><td>LeNet</td><td>MLP</td><td>Avg.</td></tr><tr><td rowspan="5">CIFAR-10</td><td rowspan="5">20%</td><td>Random</td><td>78.38</td><td>80.25</td><td>84.58</td><td>87.21</td><td>80.81</td><td>80.75</td><td>61.85</td><td>50.98</td><td>75.60</td></tr><tr><td>Glister</td><td>62.46</td><td>70.52</td><td>81.10</td><td>74.59</td><td>78.07</td><td>70.55</td><td>56.56</td><td>40.59</td><td>66.81</td></tr><tr><td>Forgetting</td><td>76.27</td><td>80.06</td><td>85.67</td><td>87.18</td><td>82.04</td><td>81.35</td><td>64.59</td><td>52.21</td><td>76.17</td></tr><tr><td>DATM</td><td>85.50</td><td>85.23</td><td>87.22</td><td>88.13</td><td>84.65</td><td>85.14</td><td>66.70</td><td>52.40</td><td>79.37</td></tr><tr><td>↑</td><td>+7.12</td><td>+4.98</td><td>+2.64</td><td>+0.92</td><td>+3.84</td><td>+4.39</td><td>+4.85</td><td>+1.42</td><td>+3.77</td></tr><tr><td rowspan="5">CIFAR-100</td><td rowspan="5">20%</td><td>Random</td><td>42.80</td><td>46.38</td><td>47.48</td><td>55.62</td><td>42.69</td><td>38.05</td><td>25.91</td><td>20.66</td><td>39.95</td></tr><tr><td>Glister</td><td>35.45</td><td>37.13</td><td>42.49</td><td>46.14</td><td>43.06</td><td>28.58</td><td>23.33</td><td>17.08</td><td>34.16</td></tr><tr><td>Forgetting</td><td>45.52</td><td>49.99</td><td>51.44</td><td>54.65</td><td>43.28</td><td>43.47</td><td>27.22</td><td>22.90</td><td>42.30</td></tr><tr><td>DATM</td><td>57.50</td><td>57.75</td><td>57.98</td><td>63.34</td><td>55.10</td><td>55.69</td><td>33.57</td><td>26.39</td><td>50.92</td></tr><tr><td>↑</td><td>+14.70</td><td>+11.37</td><td>+10.50</td><td>+7.72</td><td>+12.41</td><td>+17.64</td><td>+7.66</td><td>+5.73</td><td>+10.97</td></tr><tr><td rowspan="5">TI</td><td rowspan="5">10%</td><td>Random</td><td>15.00</td><td>24.21</td><td>17.73</td><td>28.07</td><td>22.51</td><td>14.03</td><td>9.25</td><td>5.85</td><td>17.08</td></tr><tr><td>Glister</td><td>17.32</td><td>19.77</td><td>18.84</td><td>23.12</td><td>19.10</td><td>11.68</td><td>8.84</td><td>3.86</td><td>15.32</td></tr><tr><td>Forgetting</td><td>20.04</td><td>23.83</td><td>19.38</td><td>28.88</td><td>23.77</td><td>12.13</td><td>12.06</td><td>5.54</td><td>18.20</td></tr><tr><td>DATM</td><td>39.68</td><td>40.32</td><td>36.12</td><td>43.14</td><td>38.35</td><td>35.10</td><td>12.41</td><td>9.02</td><td>31.76</td></tr><tr><td>↑</td><td>+24.68</td><td>+16.11</td><td>+18.39</td><td>+15.07</td><td>+15.84</td><td>+21.07</td><td>+3.16</td><td>+3.17</td><td>+14.68</td></tr></table>
+
+Table 4: We evaluate our distilled lossless datasets on unseen networks and compare them with two coreset selection methods. Results worse than random selection are indicated with red color.  $\uparrow$  denotes the performance improvement brought by our method compared with random selection. TI denotes Tiny ImageNet.
+
+2015). To see if this information can be directly integrated into the distilled datasets, we assign soft labels for datasets distilled by FTD (Du et al., 2023).
+
+As shown in Table 3, directly assigning soft labels for the distilled datasets will even hurt its performance when the number of categories in the classification problem is small. For CIFAR-100 and Tiny ImageNet, although assigning soft labels slightly improves the performance of FTD, there is still a huge gap between its performance and ours. This is because the soft labels synthesized by our method are optimized constantly during the distillation, leading to better consistency between the synthetic data and their labels.
+
+Furthermore, the information contained in logits varies with the capacity of the teacher model (Zong et al., 2023; Cui et al., 2023). In the experiments reported in Figure 3 (b), we use models trained after different epochs to initialize the soft labels. In the beginning, this difference has non-trivial influences on the performance of the synthetic datasets. However, the performance gaps soon disappear as the distillation goes on if labels are optimized during the distillation. This indicates learning labels relieves us from carefully selecting models to initialize soft labels. Moreover, as can be observed in Figure 3 (b), when soft labels are not optimized, the distillation becomes less stable, leading to the poor performance of the distilled dataset. Because using unoptimized soft labels will enlarge the discrepancy between the training trajectories over the synthetic dataset and the original one, considering the experts are trained with one-hot labels. More analyses are attached in Section A.2.2.
+
+Synthetic data initialization. To avoid mislabeling, we initialize the synthetic dataset by randomly sampling data from a subset of the original dataset, which only contains samples that can be correctly classified by the model used for initializing soft labels. The process of constructing the subset can be viewed as a simple coreset selection. Here we perform an ablation study to see its effect. Specifically, synthetic datasets are either initialized by randomly sampling data from the original dataset (random selection) or a subset of data that can be correctly classified by a pre-trained ConvNet (ours).
+
+As shown in Fig 3 (a), our initialization strategy can significantly speed up the convergence of the optimization. This is because data selected by our strategy are relatively easier for DNNs to learn. Thus, models trained on these easier samples will perform better when only limited training data are provided (Sorscher et al., 2022). Although this gap is gradually bridged as the distillation goes on, our initialization strategy can be utilized as a distillation speed-up technique.
+
+# 5 EXTENSION
+
+# 5.1 VISUALIZATION
+
+For a better understanding of easy patterns and hard patterns, we visualize the distilled images and discuss their properties. In Figure 4, we visualize the images synthesized by matching early trajectories and late trajectories under the same IPC setting, where easy patterns and hard ones are learned respectively. In Figure 5, we visualize the images distilled under different IPC settings.
+
+![](images/f6e042323f60bcf13809248bfa82791f284f289cba8b9edaa463569e32f18f20.jpg)  
+Figure 4: We perform the distillation on CIFAR-10 with IPC=50 by matching either early trajectories  $\{\theta_t|0\leq t\leq 10\}$  or late trajectories  $\{\theta_t|30\leq t\leq 40\}$ . All synthetic images are optimized 1000 times. Matching earlier trajectories will blur the details of the target object and change the color more drastically.
+
+![](images/502afab62b95cb344d0059be07d8e02a2ab9e4c2d56bd9e29c6964f47ad215a5.jpg)
+
+![](images/edcd11513b28d0d5772988b464a5d6528448308366deaaa02223e109e5c1324c.jpg)
+
+<table><tr><td>E/D</td><td>Random</td><td>ConvNet</td><td>ResNet18</td></tr><tr><td>ConvNet</td><td>31.00</td><td>68.28</td><td>44.54</td></tr><tr><td>ResNet18</td><td>26.70</td><td>48.66</td><td>45.28</td></tr><tr><td>VGG11</td><td>31.63</td><td>45.93</td><td>39.68</td></tr><tr><td>MLP</td><td>26.86</td><td>33.39</td><td>29.17</td></tr></table>
+
+$\mathrm{IPC} = 10$  
+
+<table><tr><td>E\D</td><td>Random</td><td>ConvNet</td><td>ResNet18</td></tr><tr><td>ConvNet</td><td>55.55</td><td>76.08</td><td>59.08</td></tr><tr><td>ResNet18</td><td>54.96</td><td>66.27</td><td>61.18</td></tr><tr><td>VGG11</td><td>48.72</td><td>59.43</td><td>50.72</td></tr><tr><td>MLP</td><td>36.66</td><td>33.29</td><td>34.41</td></tr></table>
+
+IPC=50  
+
+<table><tr><td>E\D</td><td>Random</td><td>ConvNet</td><td>ResNet18</td></tr><tr><td>ConvNet</td><td>78.38</td><td>85.50</td><td>84.64</td></tr><tr><td>ResNet18</td><td>84.58</td><td>87.22</td><td>87.70</td></tr><tr><td>VGG11</td><td>80.81</td><td>84.65</td><td>84.85</td></tr><tr><td>MLP</td><td>50.98</td><td>52.40</td><td>54.01</td></tr></table>
+
+$\mathrm{IPC} = 1000$
+
+Table 5: We use ConvNet and ResNet18 to perform the distillation (D) on CIFAR-10 with various IPC settings. Then evaluations (E) are performed using networks with various architectures. As IPC increases, datasets distilled using ResNet18 perform relatively better.
+
+As can be observed in Figure 4, compared with hard patterns, the learned easy patterns drive the synthetic images to move farther from their initialization and tend to blend the target object into the background. Although this process seems to make the image more informative, it blurs the texture and fine geometric details of the target object, making it harder for networks to learn to identify non-typical samples. This helps explain why generating easy patterns turned out to be harmful in high IPC cases. However, generating easy patterns performs well in the regime of low IPC, where the optimal solution is to model the most dense areas of the target category's distribution given the limited data budget. For example, as shown in Figure 5, the synthetic images collapse to almost only contain color and vague shape information when IPC is extremely low, which helps networks learn to identify easy samples from this basic property.
+
+Furthermore, we find matching late trajectories yields distilled images that contain more fine details. For example, as can be observed in Figure 4, matching late trajectories transforms the simple background in the dog images into a more informative one and gives more texture details to the dog and horse. This transformation helps networks learn to identify outlier (hard) samples; hence, matching late trajectories is a better choice in high IPC cases.
+
+# 5.2 DISTILLATION COST
+
+In this work, we scale dataset distillation to high IPC cases. Surprisingly, we find the distillation cost does not increase linearly with IPC, since the optimization converges faster in large IPC cases. This is because we match only late trajectories in high IPC cases, where the learned hard patterns only make a few changes on the images, as we have analyzed in Section 5.1 and can be observed in Figure 5. In practice, as reflected in Figure 3 (c), although the distillation with  $\mathrm{IPC} = 1000$  needs to optimize  $20\mathrm{x}$  more data than the case with  $\mathrm{IPC} = 50$ , the former one's cost is only 1.05 times higher.
+
+# 5.3 DISTILLATION BACKBONE NETWORKS
+
+So far, almost all representative distillation methods choose to use ConvNet to perform the distillation (Zhao & Bilen, 2021; Cazenavette et al., 2022; Loo et al., 2023). Using other networks as the distillation backbone will result in non-trivial performance degradation (Liu et al., 2023b). What makes ConvNet so effective for distillation remains an open question.
+
+Here we offer an answer from our perspective: part of the specialness of ConvNet comes from its low capacity. In general, networks with more capacity can learn more complex patterns. Accordingly, when used as distilling networks, their generated patterns are relatively harder for DNNs to learn.
+
+![](images/b04653312d3a49560d8c06b0d3f7c6d751c451e96f42615cec986cc430c5f8b7.jpg)  
+Figure 5: Visualization of the synthetic datasets distilled with different IPC settings. As IPC increases, synthetic images move less far from their initialization.
+
+![](images/83b2e879ee4a6b3792faa722c400376da239a0197c60c36f3962aa824d50a4e3.jpg)
+
+As we have analyzed in section 3.1, in small IPC cases (where previous distillation methods focus their attention), most improvement comes from the easy patterns generated on the synthetic data. Thus networks with more capacity such as ResNet will perform worse than ConvNet when IPC is low because their generated patterns are harder for DNNs to learn. However, in high IPC cases, where hard patterns play an important role, using stronger networks should perform relatively better. To verify this, we use ResNet18 and ConvNet to perform distillation on CIFAR-10 with different IPC settings. As shown in Table 5, when IPC is low, ConvNet performs much better than ResNet18 as the distillation network. However, when IPC reaches 1000, ResNet18 has a comparable or even better performance compared with ConvNet.
+
+# 6 RELATED WORK
+
+Dataset distillation introduced by Wang et al. (2018) is naturally a bi-level optimization problem, which aims at distilling a large dataset into a small one without causing performance drops. The following works can be divided into two types according to their mechanism:
+
+Kernel-based distillation methods use kernel ridge-regression with NTK (Lee et al., 2019) to obtain a closed-form solution for the inner optimization (Nguyen et al., 2020). By doing so, dataset distillation can be formulated as a single-level optimization problem. The following works have significantly reduced the training cost (Zhou et al., 2022) and improved the performance (Loo et al., 2022; 2023). However, since the heavy resource consumption of inversing matrix operation, it is hard to scale kernel-based methods to larger IPC.
+
+Matching-based methods minimize defined metrics of surrogate models learned from the synthetic dataset and the original one. According to the definition of the metric, they can be divided into four categories: based on matching gradients (Zhao et al., 2020; Kim et al., 2022; Zhang et al., 2023), features (Wang et al., 2022), distribution (Zhao & Bilen, 2023; Zhao et al., 2023), and training trajectories (Cazenavette et al., 2022; Cui et al., 2023; Du et al., 2023). So far, trajectory matching-based methods have shown impressive performance on every benchmark with low IPC (Cui et al., 2022; Yu et al., 2023). In this work, we further explore and show its great power in higher IPC cases.
+
+# 7 CONCLUSION AND DISCUSSION
+
+In this work, we find the difficulty of patterns generated by dataset distillation algorithms should be aligned with the size of the synthetic dataset, which is the key to keeping them effective in both low- and high-IPC cases. Building upon this insight, our method excels not only in low IPC cases but also maintains its efficacy in high IPC scenarios, achieving lossless dataset distillation for the first time.
+
+However, our distilled data are only lossless for the distillation backbone network: when evaluating them with other networks, the performance drops still exist. We think this is because models with different capacities need varying amounts of training data. How to overcome this issue is still a challenging problem. Moreover, it is hard to scale TM-based methods to large datasets due to its high distillation cost. How to improve its efficiency would be the goal of our future work.
+
+Acknowledgements. This work is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-PhD-2021-08-008). This work is also supported in part by the National Key R&D Program of China (NO.2022ZD0160100 and NO.2022ZD0160101). Part of Hui Li's work is supported by the National Natural Science Foundation of China (61932015), Shaanxi Innovation Team project (2018TD-007), Higher Education Discipline Innovation 111 project (B16037). Yang You's research group is being sponsored by NUS startup grant (Presidential Young Professorship), Singapore MOE Tier-1 grant, ByteDance grant, ARCTIC grant, SMI grant (WBS number: A-8001104-00-00), Alibaba grant, and Google grant for TPU usage. We thank Bo Zhao for valuable discussions and feedback.
+
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+
+# A APPENDIX
+
+# A.1 SOFT LABEL DISTRIBUTION
+
+To observe the changes in soft labels' distribution during the distillation, we record the standard deviation (std) of soft labels (after softmax) for each synthetic image, and report the average value of their std in Figure 6. As can be observed, for all datasets, their labels' std tends to increase as the distillation goes on. However, this increase does not arise due to the diversity between the values of soft labels becoming larger, but the values of non-target categories' labels are suppressed. Since the value of the target category is much higher than others, to facilitate observation, we only report the values of non-target categories in Figure 7. As can be observed, after the distillation, the values of non-target categories' labels are suppressed more drastically when IPC is smaller. This is because, in low IPC cases, basic patterns of target category are embedded into the synthetic data since only early expert trajectories are used for the matching. Accordingly, the model becomes more confident that the generated sample belongs to the target category.
+
+# A.2 SOFT LABEL INITIALIZATION
+
+We have tried to initialize soft labels with the original one-hot labels and directly optimize their values during the distillation, but the distillation soon crashed. We also have tried to add a softmax layer on it. However, the distillation is still not stable. After initializing labels with class probabilities calculated using softmax and logits output by a pre-trained model, finally, soft labels can be optimized stably during the distillation.
+
+For labels with a given distribution, their values before softmax can be different. In this case, due to the utilization of softmax, when performing backpropagation, the gradients of pre-softmax logits will also be different even if their values after softmax are the same. Through experiments, we find using an appropriate distribution before softmax to initialize soft labels is crucial to maintaining the stability of the distillation. We have also tried to modify the distribution of logits without changing their values after softmax. However, we see this operation will greatly increase the instability of distillation. Moreover, we have tried to scale the values of logits during the initialization, which also leads to the degradation of performance.
+
+# A.2.1 STABABILITY
+
+In the manuscript, we propose to generate easy and hard patterns sequentially to make the distillation stable enough for learning soft labels. The insight here is enabling the surrogate model to learn more easy patterns through the finite training steps and limited samples in inner optimization, such that the model can match the expert trajectories better. We find that simply increasing the update times in inner optimization can also stabilize the distillation. Because the surrogate model can learn easy patterns better through a longer learning process. However, increasing the update times in inner optimization will increase the memory requirement and the training cost. Thus we use the method introduced in the manuscript by default.
+
+# A.2.2 SOFT LABEL OPTIMIZATION
+
+We can choose to only replace original one-hot labels with soft labels but don't optimize them during the distillation. However, this will make the surrogate model harder to match the expert training trajectories, because the expert trajectories are trained with one-hot labels. As reflected in Figure 8 (Left), when soft labels are not optimized, the matching loss is higher than using one-hot labels. Although using unoptimized soft labels still performs better than one-hot labels because of the additional information contained in the soft labels, its performance is undermined by the under-matching.
+
+The under-matching issue can be alleviated by optimizing soft labels during the distillation. As can be observed in Figure 8 (Left), when soft labels are optimized during the distillation, the matching loss becomes lower than using one-hot labels. Accordingly, the performance is improved, as can be observed in Figure 8 (Right).
+
+![](images/44389614985c7782ede51da810ded676c7bc2c95466a8f2ffc63d94c711b3fdb.jpg)  
+Figure 6: Visualization of the changing of soft labels in various datasets where IPC=50. The average standard deviation of soft labels tends to increase as the distillation goes on.
+
+![](images/52e6e9a53dd0c271767e4382c9f322f222467e61ce1dc333e4b0df47dd94a956.jpg)
+
+![](images/ccbfb7ce0c8d68840eaba8c32911cac7d7d3e79bf4f739a40a11b533012c39a4.jpg)
+
+![](images/321f4cfb67352ca92d52d473e2ab017c2eb4722aa110a99279dc8bfd9f873896.jpg)  
+Figure 7: Visualization of the distribution of non-target soft labels of a synthetic image initialized with the same soft labels and image but distilled with different IPC settings. The values of non-target categories are suppressed more drastically when IPC is smaller.
+
+![](images/233f0df0d1cb489193cc1e27e3fe922e896aa53d81672967f5cb0965cae70ee2.jpg)
+
+![](images/8841cd9c3aed2bee32a270c4372fc822098e90d1964b2924ecab22bfb9a6fb43.jpg)
+
+# A.3 ABLATION ON SYNTHETIC STEPS
+
+We find increasing the synthetic steps  $N$  (Algorithm 1) will bring more performance gain when soft labels are utilized, as reflected in Figure 9 (Left). This is because the optimization of the surrogate model in the inner optimization affects how well it can match the expert trajectories. When soft labels are not utilized, the information contained in the synthetic dataset is relatively limited, thus the surrogate barely benefits from a longer learning process. Moreover, we find increasing the synthetic steps  $N$  can also bring improvement in high IPC cases. As shown in Figure 9 (middle), setting  $N = 80$  performs best in the case with IPC=1000, where the batch size is set to 1000. In this case, in every iteration, the parameters of the surrogate model are optimized over 80K images (10K unduplicated images) contained in the synthetic dataset, while the target parameters are obtained by the optimization over 100k images (50k unduplicated images) contained in the original datasets. In this case, although the length of the training trajectory over the synthetic dataset  $\mathcal{D}_{\mathrm{syn}}$  and the original dataset  $\mathcal{D}_{\mathrm{real}}$  are similar, matching trajectories still can improve the training performance of the synthetic datasets. Based on this observation, we conjecture that the key to keeping TM-based methods effective is to ensure the number of unduplicated images contained in  $\mathcal{D}_{\mathrm{syn}}$  is smaller than that of  $\mathcal{D}_{\mathrm{real}}$ , rather than use the short trajectory trained on  $\mathcal{D}_{\mathrm{syn}}$  to match the longer one optimized over  $\mathcal{D}_{\mathrm{real}}$ .
+
+# A.4 PREVIOUS TM-BASED METHODS IN LARGE IPC SETTINGS
+
+We have tried to use previous TM-based methods to perform the distillation in larger IPC settings. The distillation logs of FTD (Du et al., 2023) are reported in Figure 9. As can be observed, FTD will undermine the training performance of the datasets in larger IPC cases. We have tried to tune its hyper-parameters including the learning rate, batch size, synthetic steps, and the upper bound of the sample range, but the effort can only slow down the rate of performance degradation it arose. Without aligning the difficulty of the generated patterns with the size of the synthetic datasets, previous TM-based methods can not keep being effective in high IPC settings.
+
+![](images/d39e5e1a61e4b346de26d3061f4bbe28fff6f4c9c8692b0b610e9940026b04c3.jpg)  
+Figure 8: Left: Logs of the matching loss (smoothed with EMA), where the labels of the synthetic dataset are either one-hot labels, unoptimized soft labels, or soft labels that are optimized during the distillation. Right: Logs of performance of the distilled datasets. Learning soft labels during the distillation enables surrogate models to match expert trajectories better. Accordingly, its synthetic datasets have a better performance.
+
+![](images/9f4e03686e63ffcf69a96faab6eade70484ff685160d304879b236ea3bb952c5.jpg)
+
+![](images/3252e0d54d795121b77b6221e25742df57c1b4e8975fe053265b87cf73892284.jpg)
+
+![](images/aa79bdd547656188f1ff489441e9b5078a6af44bacd7f9d947faa0fc4373058c.jpg)  
+Figure 9: Left: (CIFAR-100,  $\mathrm{IPC} = 10$ ) Ablation on synthetic step  $N$ , distillation with soft labels benefits more from a larger  $N$ . Middle: (CIFAR-10) Distillation in larger IPC settings can still benefit from a higher larger  $N$ . Right: Distillation log of FTD on CIFAR-10 with larger IPC. The performance of the synthetic datasets keeps being degraded as the distillation goes on.
+
+![](images/5578d037f88c9c162efba2f10bfdaf908f7fa7b32d2151faf72efbd141e660ed.jpg)
+
+# A.5 MORE DETAILED ANALYSIS ON MATCHING LOSS
+
+Here we provide more results and additional analysis about the matching loss over expert trajectories. As can be observed in Figure 10, initially, the matching loss over the former part of the trajectories is always lower than the one over the later part. This indicates earlier trajectories are relatively easier to match compared with the later ones. In other words, the patterns that surrogate models need to learn to match the early trajectories are relatively easier.
+
+Moreover, it is interesting to observe that matching later trajectories can also reduce the matching loss over early trajectories in high IPC cases. We hypothesize that this is because, in the late training phases, DNNs just prefer to learn hard patterns to help identify the outliers, while a few easy patterns are also learned in late training phases. From this perspective, TM-based methods might not be the most efficient way to distill datasets with large IPC, since matching later trajectories still will generate a few easy patterns.
+
+We can also find that when IPC is small, matching late trajectories will raise the matching loss over the early trajectories. This indicates generating hard patterns is harmful for the model to learn basic (easy) patterns to obtain the basic capacity to perform the classification when data is limited. This coincide with the observation in the dataset pruning area: preserving hard samples perform worse when only limited samples are kept (Sorscher et al., 2022).
+
+# A.6 GUIDANCE FOR ALIGNING DIFFICULTY
+
+Our difficulty alignment aims at letting the models trained on the synthetic dataset learn as many hard patterns as possible, without compromising their capacity to classify easy patterns. For TM-based methods, this can be quantified by the matching loss over a distillation-uninvolved expert trajectory, as we have analyzed in section A.5. Specifically, we want to add patterns that can help to reduce the matching loss over the latter part of the expert trajectory without increasing the matching
+
+Algorithm 1: Pipeline of our method  
+Input:  $\{\tau^{*}\}$  : set of expert parameter trajectories.  $N$  : update times of the surrogate network in each inner optimization.  $M$  : update times between the start and target expert parameters.  $T^{-}, T, T^{+}$  : lower, current upper, final upper bound of the sample range of  $t$ .  $\mathcal{D}_{\mathrm{real}}$  : original dataset.  $I$  : interval for expanding the sampling range. Sample a model  $f_{\theta^{*}}$  from  $\{\tau^{*}\}$ . Construct  $\mathcal{D}_{\mathrm{sub}} = \{(x_i, \text{softmax}(L_i)) | (x_i, y_i) \in \mathcal{D}_{\mathrm{real}} \text{ and } \arg \max(L_i) == y_i\}$ , where  $L_i = f_{\theta^{*}}(x_i)$ . Randomly sample data from  $\mathcal{D}_{\mathrm{sub}}$  to initialize synthetic dataset  $\mathcal{D}_{\mathrm{syn}}$ . For iteration  $\leftarrow 0$  to max_iteration do
+- Randomly sample an expert training trajectory  $\tau^{*} \in \{\tau^{*}\}$  with  $\tau^{*} = \{\theta_i^*\}_0^n$ 
+- Select random start timestamp  $t$ , where  $T^{-} \leq t \leq T$ 
+- Sample  $\theta_t^*, \theta_{t + M}^*$  from  $\tau^{*}$ , initialize  $\hat{\theta}_t = \theta_t^*$ 
+- for  $i \leftarrow 0$  to N-1 do
+- \(\begin{array}{l}
+b_{t + i} \sim \mathcal{D}_{\mathrm{syn}} \\
+\hat{\theta}_{t + i + 1} = \hat{\theta}_{t + i} - \alpha \nabla \ell(\hat{\theta}_{t + i}, b_{t + i})
+\end{array}\)
+- update surrogate model with CE loss
+- Compute matching loss between  $\hat{\theta}_{t + N}$  and  $\theta_{t + M}^*$  with Eq. 1
+- Update  $(x_i, \text{softmax}(L_i)) \in \{b\}_t^{t + N - 1}$  and  $\alpha$  with respect to the matching loss
+- if (iteration %  $I == 0$ ) and  $(T < T^{+})$  then
+-  $T = T + 1$
+
+loss over the former part. In practice, we realize it by only matching a certain part of the expert trajectories during the distillation.
+
+Here we introduce how to tune the values of  $T^{-}$ ,  $T$ , and  $T^{+}$  (Algorithm 1). For an initialized synthetic dataset  $\mathcal{D}_{\mathrm{syn}}$ , we first set  $T^{-} = 0$  and  $T^{+} = T^{-} + 10$  to perform the distillation for 50 iterations, where the matching loss over a distillation-uninvolved expert trajectory is recorded. Then we simultaneously increase the values of  $T^{-}$  and  $T^{+}$  until the distillation will not increase the matching loss over the latter part of the expert trajectory. Subsequently, we increase the value of  $T^{+}$  until the distillation will increase the matching loss over the former part of the expert trajectory. After deciding the values of  $T^{-}$  and  $T^{+}$ , we let them respectively be the lower- and upper-bound of the sample range and perform the distillation. During the distillation, we record the value of  $t$  if the matching loss is larger than 1, which denotes the surrogate model can not match the expert trajectory. Then  $T$  is set as the minimum recorded value, to avoid matching too hard trajectories in the beginning.
+
+# A.7 MORE RELATED WORK
+
+Two early works (Bohdal et al., 2020; Sucholutsky & Schonlau, 2021b) in the dataset distillation area also focus on optimizing the labels of the datasets. Specifically, based on the dataset distillation algorithm proposed by Wang et al. (2018), Sucholutsky & Schonlau (2021b) propose to optimize labels during the distillation, while Bohdal et al. (2020) choose to only distill soft labels without optimizing the training data. Different from them, we use the pre-trained model to initialize soft labels, which contain more information. Furthermore, our method is based on matching training trajectories (Cazenavette et al., 2022) rather than the method proposed by Wang et al. (2018).
+
+Recently, several methods are proposed to improve the performance, efficiency and suitability of dataset distillation. For example, Liu et al. (2022) proposed to use hallucinations to enlarge the synthetic datasets in the deployment stage. Subsequently, Wang et al. (2023) achieved this goal by distilling the target dataset into a generative model. Moreover, (Zhang et al., 2023; Liu et al., 2023b) were proposed to accelerate the distillation and Liu et al. (2023a) proposed a method that allows adjusting the size of the distilled dataset during the deployment stage. Recently, Chen et al. (2024) proposed to improve the quality of the synthetic dataset with a carefully designed distillation schedule. Moreover, dataset distillation has also been successfully applied in condensing graph data (Jin et al., 2022; Yang et al., 2023; Zhang et al., 2024a;b) and multi-modality data (Wu et al., 2023).
+
+![](images/878fe379101c84b811dfa8e1071a1a7d5b02932a0d77edf795401ab6c012a4e7.jpg)  
+Figure 10: More detailed results of experiments reported in Figure 2. We train the expert models on CIFAR-10 for 40 epochs. Then we distill datasets with two strategies: (1) matching the early part of expert training trajectories, where  $\theta_t^* \in \{\theta_0^* \dots \theta_{20}^*\}$ . (2) matching the latter part of expert training trajectories, where  $\theta_t^* \in \{\theta_{20}^* \dots \theta_{40}^*\}$ . The first row shows the matching loss over a distillation-uninvolved expert trajectory, where the distillation is performed with strategy 1, and the second row shows the loss of strategy 2. In the first two rows, lines with darker color indicate the matching loss over more later part of the trajectories.  $t$  denotes the timestamp of the start parameters used for matching (Algorithm 1). The last row shows the performance of the datasets distilled by different strategies with various IPC settings.
+
+# A.8 SETTINGS
+
+Distillation. Consistent with previous works (Cazenavette et al., 2022; Du et al., 2023), we perform the distillation for 10000 iterations to make sure the optimization is fully converged. We use ZCA whitening in all the involved experiments by default.
+
+Evaluation. We keep our evaluation process consistent with previous works (Cazenavette et al., 2022; Du et al., 2023). Specifically, we train a randomly initialized network on the distilled dataset and then evaluate its performance on the entire validation set of the original dataset. Following previous works (Cazenavette et al., 2022; Du et al., 2023), the evaluation networks are trained for 1000 epochs to make sure the optimization is fully converged. All the results are the average over five trials. For fairness, experimental results of previous distillation methods in low IPC settings are obtained from (Du et al., 2023), while their results in high IPC cases come from (Cui et al., 2022).
+
+Since the exponential moving average (EMA) used in FTD (Du et al., 2023) is a plug-and-play technique that hasn't been utilized by previous matching-based methods, for a fair comparison, we reproduce FTD with the official released code without using EMA. Accordingly, we do not use EMA in our method.
+
+Network. We use various networks to evaluate the generalizability of our distilled datasets. Specifically, to scale ResNet, LeNet, and AlexNet to Tiny-ImageNet, we increase the stride of their first convolution layer from 1 to 2. For VGG, we increase the stride of its last max pooling layer from 1 to 2. The MLP utilized in our evaluation has one hidden layer with 128 units.
+
+Hyper-parameters. We report the hyper-parameters of our method in Table 6. Additionally, for all the experiments with optimizing soft labels, we set its momentum to 0.9. We find learning labels with a low momentum will somewhat increase the instability of the distillation. We conjecture this is because the optimized soft labels are easy to overfit the expert trajectories considering we only match one trajectory in each iteration.
+
+Compute resources. Our experiments are run on 4 NVIDIA A100 GPUs, each with 80 GB of memory. The heavy reliance on GPU memory can be alleviated by TESLA (Cui et al., 2023) or simply reducing the synthetic steps  $N$ , which will not cause too much performance degradation. For example, reducing the synthetic steps  $N$  from 80 to 40 saves about half the GPU memory, while it only makes the performance drop by around  $0.8\%$  for CIFAR-10 with  $\mathrm{IPC} = 1000$ ,  $0.7\%$  for CIFAR-100 with  $\mathrm{IPC} = 100$ , and  $0.4\%$  for TinyImageNet with  $\mathrm{IPC} = 50$ .
+
+<table><tr><td>Dataset</td><td>IPC</td><td>N</td><td>M</td><td>T-</td><td>T</td><td>T+</td><td>Interval</td><td>Synthetic Batch Size</td><td>Learning Rate (Label)</td><td>Learning Rate (Pixels)</td></tr><tr><td rowspan="5">CIFAR-10</td><td>1</td><td>80</td><td>2</td><td>0</td><td>4</td><td>4</td><td>-</td><td>10</td><td>5</td><td>100</td></tr><tr><td>10</td><td>80</td><td>2</td><td>0</td><td>10</td><td>20</td><td>100</td><td>100</td><td>2</td><td>100</td></tr><tr><td>50</td><td>80</td><td>2</td><td>0</td><td>20</td><td>40</td><td>100</td><td>500</td><td>2</td><td>1000</td></tr><tr><td>500</td><td>80</td><td>2</td><td>40</td><td>60</td><td>60</td><td>-</td><td>1000</td><td>10</td><td>50</td></tr><tr><td>1000</td><td>80</td><td>2</td><td>40</td><td>60</td><td>60</td><td>-</td><td>1000</td><td>10</td><td>50</td></tr><tr><td rowspan="4">CIFAR-100</td><td>1</td><td>40</td><td>3</td><td>0</td><td>10</td><td>20</td><td>100</td><td>100</td><td>10</td><td>1000</td></tr><tr><td>10</td><td>80</td><td>2</td><td>0</td><td>30</td><td>50</td><td>100</td><td>1000</td><td>10</td><td>1000</td></tr><tr><td>50</td><td>80</td><td>2</td><td>20</td><td>70</td><td>70</td><td>-</td><td>1000</td><td>10</td><td>1000</td></tr><tr><td>100</td><td>80</td><td>2</td><td>30</td><td>70</td><td>70</td><td>-</td><td>1000</td><td>10</td><td>50</td></tr><tr><td rowspan="3">TI</td><td>1</td><td>60</td><td>2</td><td>0</td><td>15</td><td>20</td><td>400</td><td>200</td><td>10</td><td>10000</td></tr><tr><td>10</td><td>60</td><td>2</td><td>10</td><td>50</td><td>50</td><td>-</td><td>250</td><td>10</td><td>100</td></tr><tr><td>50</td><td>80</td><td>2</td><td>40</td><td>70</td><td>70</td><td>-</td><td>250</td><td>10</td><td>100</td></tr></table>
+
+Table 6: Hyper-parameters for different datasets.
+
+![](images/73c9d1d2cff64aac3343b110837a1865238e917577b6e0925b73ae4ddf6a6d47.jpg)  
+Figure 11: (Tiny ImageNet,  $\mathrm{IPC} = 1$ ) Visualization of distilled images (1/2).
+
+![](images/29cae6b56167765c7cb7c32abd15c74efaa31b368833f1682a8f8cee2a990d19.jpg)  
+Figure 12: (Tiny ImageNet,  $\mathrm{IPC} = 1$ ) Visualization of distilled images (2/2).
+
+![](images/dec701e4ebd2c2fd767cf2106c01932436933ae9d6235578f074138d5f012747.jpg)  
+Figure 13: (Tiny ImageNet, IPC=50) Visualization of distilled images (1/2).
+
+![](images/f85f4a60941373fc060def3c36c8a971375a0b8a5db323dbf3dc0a5f9da534d4.jpg)  
+Figure 14: (Tiny ImageNet, IPC=50) Visualization of distilled images (2/2).

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2024/Towards Non-Asymptotic Convergence for Diffusion-Based Generative Models/b4e2a016-0bfa-45bf-93f6-59d4737b9813_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "TOWARDS NON-ASYMPTOTIC CONVERGENCE FOR DIFFUSION-BASED GENERATIVE MODELS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gen Li*",
+        "bbox": [
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+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuting Wei",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuxin Chen†",
+        "bbox": [
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+            167,
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+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuejie Chi\\*",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to  $\\ell_2$ -accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to  $1 / T$  (with  $T$  the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to  $1 / \\sqrt{T}$ , matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how  $\\ell_2$  score estimation errors affect the quality of the data generation process. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
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+            520
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diffusion models have emerged as a cornerstone in contemporary generative modeling, a task that learns to generate new data instances (e.g., images, text, audio) that look similar in distribution to the training data (Ho et al., 2020; Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Dhariwal & Nichol, 2021; Jolicoeur-Martineau et al., 2021; Chen et al., 2021; Kong et al., 2021; Austin et al., 2021). Originally proposed by Sohl-Dickstein et al. (2015) and later popularized by Song & Ermon (2019); Ho et al. (2020), the mainstream diffusion generative models — e.g., denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020) and denoising diffusion implicit models (DDIMs) (Song et al., 2020a) — have underpinned major successes in content generators like DALL-E 2 (Ramesh et al., 2022), Stable Diffusion (Rombach et al., 2022) and Imagen (Saharia et al., 2022), claiming state-of-the-art performance in the now broad field of generative artificial intelligence (AI). See Yang et al. (2022); Croitoru et al. (2023) for overviews of recent development.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In a nutshell, a diffusion generative model is based upon two stochastic processes in  $\\mathbb{R}^d$ :",
+        "bbox": [
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+            753,
+            710
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1) a forward process",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nX _ {0} \\rightarrow X _ {1} \\rightarrow \\dots \\rightarrow X _ {T} \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            444,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "that starts from a sample drawn from the target data distribution (e.g., of natural images) and gradually diffuses it into a noise-like distribution (e.g., standard Gaussians);",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2) a reverse process",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T} \\rightarrow Y _ {T - 1} \\rightarrow \\dots \\rightarrow Y _ {0} \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            439,
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+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Department of Statistics, The Chinese University of Hong Kong, Hong Kong.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "†Department of Statistics and Data Science, Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA.",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{\\ddagger}$ Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA.",
+        "bbox": [
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+    {
+        "type": "page_footnote",
+        "text": "$^{\\S}$ Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA.",
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "that starts from pure noise (e.g., standard Gaussians) and successively converts it into new samples sharing similar distributions as the target data distribution.",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Transforming data into noise in the forward process is straightforward, often hand-crafted by increasingly injecting more noise into the data at hand. What is challenging is the construction of the reverse process: how to generate the desired information out of pure noise? To do so, a diffusion model learns to build a reverse process (2) that imitates the dynamics of the forward process (1) in a time-reverse fashion; more precisely, the design goal is to ascertain distributional proximity<sup>1</sup>",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {t} \\stackrel {\\mathrm {d}} {\\approx} X _ {t}, \\quad t = T, \\dots , 1 \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            220,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "through proper learning based on how the training data propagate in the forward process. Encouragingly, there often exist feasible strategies to achieve this goal as long as faithful estimates about the (Stein) score functions — the gradients of the log marginal density of the forward process — are available, an intriguing fact that can be illuminated by the existence and construction of reverse-time stochastic differential equations (SDEs) (Anderson, 1982; Haussmann & Pardoux, 1986) (see Section 2.2 for more precise discussions). Viewed in this light, a diverse array of diffusion models are frequently referred to as score-based generative modeling (SGM). The popularity of SGM was initially motivated by, and has since further inspired, numerous recent studies on the problem of learning score functions, a subroutine that also goes by the name of score matching (e.g., Hyvarinen (2005; 2007); Vincent (2011); Song et al. (2020b); Koehler et al. (2023)).",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Nonetheless, despite the mind-blowing empirical advances, a mathematical theory for diffusion generative models is still in its infancy. Given the complexity of developing a full-fledged end-to-end theory, a divide-and-conquer approach has been advertised, decoupling the score learning phase (i.e., how to estimate score functions from training data) and the generative sampling phase (i.e., how to generate new data given the score estimates). In particular, the past two years have witnessed growing interest and remarkable progress from the theoretical community towards understanding the sampling phase (Block et al., 2020; De Bortoli et al., 2021; Liu et al., 2022; De Bortoli, 2022; Lee et al., 2023; Pidstrigach, 2022; Chen et al., 2022b;a; Tang, 2023; Chen et al., 2023c; Tang & Zhao, 2024; Li et al., 2024a). For instance, polynomial-time convergence guarantees have been established for stochastic samplers (e.g., Chen et al. (2022b;a); Benton et al. (2023a)) and deterministic samplers (e.g., Chen et al. (2023c); Benton et al. (2023b)), both of which accommodated a fairly general family of data distributions.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper. The present paper contributes to this growing list of theoretical endeavors by developing a new suite of non-asymptotic theory for several score-based generative modeling algorithms. We concentrate on two types of samplers (Song et al., 2021b) in discrete time: (i) a deterministic sampler based on a sort of ordinary differential equations (ODEs) called probability flow ODEs (which is closely related to the DDIM); and (ii) a DDPM-type stochastic sampler motivated by reverse-time SDEs. We impose only minimal assumptions on the target data distribution (e.g., no smoothness condition is needed), and would like to quantify the impact of  $\\ell_2$  score estimation errors. In comparisons to past works, our main contributions are three-fold.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Non-asymptotic convergence guarantees. For a popular deterministic sampler, we demonstrate that the number of steps needed to yield  $\\varepsilon$ -accuracy — meaning that the total variation (TV) distance between the distribution of  $X_{1}$  and that of  $Y_{1}$  is no larger than  $\\varepsilon$  — is proportional to  $1 / \\varepsilon$  (in addition to other polynomial dimension dependency). This improves upon prior convergence guarantees (Chen et al., 2023c) and does not exhibit exponential dependency on the smoothness assumption as in Chen et al. (2023c); Benton et al. (2023b). For another DDPM-type stochastic sampler, we establish an iteration complexity proportional to  $1 / \\varepsilon^{2}$ , matching existing theory Chen et al. (2022b;a); Benton et al. (2023a) in terms of the  $\\varepsilon$ -dependency.",
+        "bbox": [
+            169,
+            695,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Score estimation errors for the deterministic sampler. In our convergence guarantees for the deterministic sampler, the TV distance between  $X_{1}$  and  $Y_{1}$  are shown to be proportional to the  $\\ell_{2}$  score estimation error as well as the associated Jacobian errors. As far as we know, this is the first result for this deterministic sampler that accounts for score estimation errors in discrete time. In comparison, other theoretical results that accommodate score errors for the probability flow ODE approach",
+        "bbox": [
+            169,
+            813,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_footnote",
+        "text": "Two random vectors  $X$  and  $Y$  are said to obey  $X \\stackrel{\\mathrm{d}}{=} Y$  (resp.  $X \\stackrel{\\mathrm{d}}{\\approx} Y$ ) if they are equivalent (resp. close) in distribution.",
+        "bbox": [
+            169,
+            893,
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+            922
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "either study certain stochastic variations of this deterministic sampler (Chen et al., 2023b) or fall short of accommodating discretization errors (Benton et al., 2023b).",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An elementary non-asymptotic analysis framework. From the technical viewpoint, the analysis framework laid out in this paper is fully non-asymptotic in nature. In contrast to prior analyses that take a detour to study the continuum limits and then control the discretization error, our approach tackles the discrete-time processes directly using elementary analysis strategies. No knowledge of SDEs or ODEs is required for establishing our theory, thereby resulting in a more versatile framework and sometimes lowering the technical barrier towards understanding diffusion models.",
+        "bbox": [
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+            138,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notation. For any two functions  $f(d, T)$  and  $g(d, T)$ , we adopt the notation  $f(d, T) \\lesssim g(d, T)$  or  $f(d, T) = O(g(d, T))$  (resp.  $f(d, T) \\gtrsim g(d, T)$ ) to mean that there exists some universal constant  $C_1 > 0$  such that  $f(d, T) \\leq C_1 g(d, T)$  (resp.  $f(d, T) \\geq C_1 g(d, T)$ ) for all  $d$  and  $T$ ; moreover, the notation  $f(d, T) \\asymp g(d, T)$  indicates that  $f(d, T) \\lesssim g(d, T)$  and  $f(d, T) \\gtrsim g(d, T)$  hold at once. The notation  $\\widetilde{O}(\\cdot)$  is defined similar to  $O(\\cdot)$  except that it hides the logarithmic dependency. Additionally, the notation  $f(d, T) = o(g(d, T))$  means that  $f(d, T) / g(d, T) \\to 0$  as  $d, T$  tend to infinity. For any two probability measures  $P$  and  $Q$ , the total variation (TV) distance between them is defined to be  $\\mathsf{TV}(P, Q) := \\frac{1}{2} \\int |\\mathrm{d}P - \\mathrm{d}Q|$ . Throughout the paper,  $p_X(\\cdot)$  (resp.  $p_{X|Y}(\\cdot|\\cdot))$  denotes the probability density function of  $X$  (resp.  $X$  given  $Y$ ). For any matrix  $A$ , we denote by  $\\|A\\|$  (resp.  $\\|A\\|_{\\mathbb{F}}$ ) the spectral norm (resp. Frobenius norm) of  $A$ . Also, for any vector-valued function  $f$ , we let  $J_f$  or  $\\frac{\\partial f}{\\partial x}$  represent the Jacobian matrix of  $f$ .",
+        "bbox": [
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+            237,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            416,
+            339,
+            431
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce the basics of diffusion generative models. The ultimate goal of a generative model can be concisely stated: given data samples drawn from an unknown distribution of interest  $p_{\\mathrm{data}}$  in  $\\mathbb{R}^d$ , we wish to generate new samples whose distributions closely resemble  $p_{\\mathrm{data}}$ .",
+        "bbox": [
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+            446,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 DIFFUSION GENERATIVE MODELS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            506,
+            447,
+            518
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Towards achieving the above goal, a diffusion generative model typically encompasses two Markov processes: a forward process and a reverse process, as described below.",
+        "bbox": [
+            169,
+            532,
+            823,
+            561
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The forward process. In the forward chain, one progressively injects noise into the data samples to diffuse and obscure the data. The distributions of the injected noise are often hand-picked, with the standard Gaussian distribution receiving widespread adoption. Specifically, the forward Markov process produces a sequence of  $d$ -dimensional random vectors  $X_{1} \\to X_{2} \\to \\dots \\to X_{T}$  as follows:",
+        "bbox": [
+            169,
+            574,
+            823,
+            631
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nX _ {0} \\sim p _ {\\text {d a t a}}, \\tag {4a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            633,
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+            650
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nX _ {t} = \\sqrt {1 - \\beta_ {t}} X _ {t - 1} + \\sqrt {\\beta_ {t}} W _ {t}, \\quad 1 \\leq t \\leq T, \\tag {4b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            654,
+            823,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\{W_t\\}_{1 \\leq t \\leq T}$  indicates a sequence of independent noise vectors drawn from  $W_t \\stackrel{\\text{i.i.d.}}{\\sim} \\mathcal{N}(0, I_d)$ . The hyper-parameters  $\\{\\beta_t \\in (0, 1)\\}$  represent prescribed learning rate schedules that control the variance of the noise injected in each step. If we define",
+        "bbox": [
+            169,
+            679,
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+            724
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\alpha_ {t} := 1 - \\beta_ {t}, \\quad \\overline {{\\alpha}} _ {t} := \\prod_ {k = 1} ^ {t} \\alpha_ {k}, \\quad 1 \\leq t \\leq T, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            729,
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+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "then it can be straightforwardly verified that for every  $1 \\leq t \\leq T$ ,",
+        "bbox": [
+            169,
+            772,
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+            787
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nX _ {t} = \\sqrt {\\bar {\\alpha} _ {t}} X _ {0} + \\sqrt {1 - \\bar {\\alpha} _ {t}} \\bar {W} _ {t} \\quad \\text {f o r s o m e} \\bar {W} _ {t} \\sim \\mathcal {N} (0, I _ {d}). \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            790,
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+            806
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Clearly, if the covariance of  $X_0$  is also equal to  $I_d$ , then the covariance of  $X_t$  is preserved throughout the forward process; for this reason, this forward process (4) is sometimes referred to as variance-preserving (Song et al., 2021b). Throughout this paper, we employ the notation",
+        "bbox": [
+            169,
+            810,
+            823,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nq _ {t} := \\operatorname {l a w} \\left(X _ {t}\\right) \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            447,
+            856,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "to denote the distribution of  $X_{t}$ . As long as  $\\overline{\\alpha}_{T}$  is vanishingly small, one has the following property for a general family of data distributions:",
+        "bbox": [
+            169,
+            876,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nq _ {T} \\approx \\mathcal {N} (0, I _ {d}). \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            907,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+            948,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The reverse process. The reverse chain  $Y_{T} \\to Y_{T-1} \\to \\ldots \\to Y_{1}$  is designed to (approximately) revert the forward process, allowing one to transform pure noise into new samples with matching distributions as the original data. To be more precise, by initializing it as",
+        "bbox": [
+            169,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T} \\sim \\mathcal {N} (0, I _ {d}), \\tag {9a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            148,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "we seek to design a reverse-time Markov process with nearly identical marginals as the forward process, namely,",
+        "bbox": [
+            169,
+            169,
+            823,
+            198
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\text {g o a l}\\right) \\quad Y _ {t} \\stackrel {\\mathrm {d}} {\\approx} X _ {t}, \\quad t = T, T - 1, \\dots , 1. \\tag {9b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            198,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Throughout the paper, we often employ the following notation to indicate the distribution of  $Y_{t}$ :",
+        "bbox": [
+            171,
+            219,
+            802,
+            234
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np _ {t} := \\operatorname {l a w} \\left(Y _ {t}\\right). \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            238,
+            825,
+            255
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 DETERMINISTIC VS. STOCHASTIC SAMPLERS: A CONTINUOUS-TIME INTERPRETATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            268,
+            807,
+            282
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Evidently, the most crucial step of the diffusion model lies in effective design of the reverse process. Two mainstream approaches stand out:",
+        "bbox": [
+            169,
+            295,
+            823,
+            325
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- Deterministic samplers. Starting from  $Y_{T} \\sim \\mathcal{N}(0, I_{d})$ , this approach selects a set of functions  $\\{\\Phi_t(\\cdot)\\}_{1 \\leq t \\leq T}$  and computes:",
+        "bbox": [
+            215,
+            334,
+            823,
+            364
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {t - 1} = \\Phi_ {t} (Y _ {t}), \\quad t = T, \\dots , 1. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            367,
+            823,
+            385
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Clearly, the sampling process is fully deterministic except for the initialization  $Y_{T}$ .",
+        "bbox": [
+            227,
+            387,
+            772,
+            402
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- Stochastic samplers. Initialized again at  $Y_{T} \\sim \\mathcal{N}(0, I_{d})$ , this approach computes another collection of functions  $\\{\\Psi_{t}(\\cdot, \\cdot)\\}_{1 \\leq t \\leq T}$  and performs the updates:",
+        "bbox": [
+            215,
+            405,
+            823,
+            436
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {t - 1} = \\Psi_ {t} \\left(Y _ {t}, Z _ {t}\\right), \\quad t = T, \\dots , 1, \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            439,
+            823,
+            455
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the  $Z_{t}$  's are independent noise vectors obeying  $Z_{t}\\stackrel {\\mathrm{i.i.d.}}{\\sim}\\mathcal{N}(0,I_{d})$",
+        "bbox": [
+            228,
+            463,
+            702,
+            482
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to elucidate the feasibility of the above two approaches, we find it helpful to look at the continuum limit through the lens of SDEs and ODEs. It is worth emphasizing, however, that the development of our main theory does not rely on any knowledge of SDEs and ODEs.",
+        "bbox": [
+            169,
+            489,
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+            532
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- The forward process. A continuous-time analog of the forward process can be modeled as",
+        "bbox": [
+            215,
+            542,
+            821,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {d} X _ {t} = f \\left(X _ {t}, t\\right) \\mathrm {d} t + g (t) \\mathrm {d} W _ {t} \\quad (0 \\leq t \\leq T), \\quad X _ {0} \\sim p _ {\\text {d a t a}} \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            561,
+            823,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for some functions  $f(\\cdot, \\cdot)$  and  $g(\\cdot)$  (denoting respectively the drift and diffusion coefficient), where  $W_{t}$  denotes a  $d$ -dimensional standard Brownian motion. As a special example, the continuum limit of (4) takes the following form² (Song et al., 2021b)",
+        "bbox": [
+            227,
+            580,
+            823,
+            625
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {d} X _ {t} = - \\frac {1}{2} \\beta (t) X _ {t} \\mathrm {d} t + \\sqrt {\\beta (t)} \\mathrm {d} W _ {t} \\quad (0 \\leq t \\leq T), \\quad X _ {0} \\sim p _ {\\text {d a t a}} \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            627,
+            823,
+            655
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for some function  $\\beta(t)$ . As before, we denote by  $q_{t}$  the distribution of  $X_{t}$  in (13).",
+        "bbox": [
+            227,
+            659,
+            763,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- The reverse process. As it turns out, the following two reverse processes are both capable of reconstructing the distribution of the forward process, motivating the design of two distinctive samplers. Here and throughout, we use  $\\nabla \\log q_t(X)$  to abbreviate  $\\nabla_X \\log q_t(X)$  for notational simplicity.",
+        "bbox": [
+            215,
+            676,
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+            734
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- One feasible approach is to the so-called probability flow ODE (Song et al., 2021b)",
+        "bbox": [
+            243,
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+            810,
+            752
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {d} Y _ {t} ^ {\\text {o d e}} = \\left(- f \\left(Y _ {t} ^ {\\text {o d e}}, T - t\\right) + \\frac {1}{2} g (T - t) ^ {2} \\nabla \\log q _ {T - t} \\left(Y _ {t} ^ {\\text {o d e}}\\right)\\right) \\mathrm {d} t \\quad (0 \\leq t \\leq T), \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            755,
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+            797
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with  $Y_0^{\\mathrm{ode}}\\sim q_T$  , which exhibits matching distributions as follows:",
+        "bbox": [
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+            803,
+            700,
+            819
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T - t} ^ {\\mathsf {o d e}} \\stackrel {{\\mathrm {d}}} {{=}} X _ {t}, \\qquad 0 \\leq t \\leq T.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            823,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The deterministic nature of this approach often enables faster sampling. It has been shown that this family of deterministic samplers is closely related to the DDIM sampler (Karras et al., 2022; Song et al., 2021b).",
+        "bbox": [
+            259,
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+            32,
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+            47
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "2To see its connection with (4), it suffices to derive from (4) that  $X_{t} - X_{t - \\mathrm{dt}} = \\sqrt{1 - \\beta_{t}} X_{t - \\mathrm{dt}} - X_{t - \\mathrm{dt}} + \\sqrt{\\beta_{t}} W_{t}\\approx -\\frac{1}{2}\\beta_{t}X_{t - \\mathrm{dt}} + \\sqrt{\\beta_{t}} W_{t}$ .",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
+            493,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- In view of the classical results Anderson (1982); Haussmann & Pardoux (1986), one can also construct a \"reverse-time\" SDE",
+        "bbox": [
+            245,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {d} Y _ {t} ^ {\\mathrm {s d e}} = \\left(- f \\left(Y _ {t} ^ {\\mathrm {s d e}}, T - t\\right) + g (T - t) ^ {2} \\nabla \\log q _ {T - t} \\left(Y _ {t} ^ {\\mathrm {s d e}}\\right)\\right) \\mathrm {d} t + g (T - t) \\mathrm {d} Z _ {t} ^ {\\mathrm {s d e}} \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            135,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "for  $0 \\leq t \\leq T$ , with  $Y_0^{\\mathrm{sde}} \\sim q_T$  and  $Z_t^{\\mathrm{sde}}$  being a standard Brownian motion. Strikingly, this process also satisfies",
+        "bbox": [
+            261,
+            181,
+            823,
+            212
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T - t} ^ {\\mathsf {s d e}} \\stackrel {{\\mathrm {d}}} {{=}} X _ {t}, \\qquad 0 \\leq t \\leq T.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The popular DDPM sampler (Ho et al., 2020) falls under this category.",
+        "bbox": [
+            261,
+            243,
+            728,
+            258
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, in addition to the functions  $f$  and  $g$  that define the forward process, construction of both (15) and (16) relies only upon the knowledge of the gradient of the log density  $\\nabla \\log q_t(\\cdot)$  of the intermediate steps of the forward diffusion process — often referred to as the (Stein) score function. Consequently, a key enabler of the above paradigms lies in reliable learning of the score function, and hence the name score-based generative modeling.",
+        "bbox": [
+            169,
+            268,
+            826,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 ALGORITHMS AND MAIN RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            359,
+            491,
+            375
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze a couple of diffusion generative models, including both deterministic and stochastic samplers. While the proofs for our main theory are all postponed to the appendix, it is worth emphasizing upfront that our analysis framework directly tackles the discrete-time processes without resorting to any toolbox of SDEs and ODEs tailored to the continuous-time limits. This elementary approach might potentially be versatile for analyzing a broad class of variations of these samplers. For instance, prior ODE-based theory (e.g., Chen et al. (2023b;c)) encountered certain technical challenges when analyzing the deterministic sampler directly, and our elementary approach is able to shed new light on the convergence of this important sampler.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 ASSUMPTIONS AND LEARNING RATES",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Before proceeding, we impose some assumptions on the score estimates and the target data distributions, and specify the hypter-parameters  $\\{\\alpha_t\\}$ , which shall be adopted throughout all cases.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Score estimates. Given that the score functions are an essential component in score-based generative modeling, we assume access to faithful estimates of the score functions  $\\nabla \\log q_t(\\cdot)$  across all intermediate steps  $t$ , thus disentangling the score learning phase and the data generation phase. Towards this end, let us first formally introduce the true score function as follows.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Score function). The score function, denoted by  $s_t^\\star : \\mathbb{R}^d \\to \\mathbb{R}^d$ , is defined as",
+        "bbox": [
+            169,
+            646,
+            767,
+            662
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {t} ^ {\\star} (X) := \\nabla \\log q _ {t} (X), \\quad 1 \\leq t \\leq T. \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As has been pointed out by previous works concerning score matching (e.g., Hyvarinen (2005); Vincent (2011); Chen et al. (2022b)), the score function  $s_t^\\star$  admits an alternative form as follows (owing to properties of Gaussian distributions):",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {t} ^ {\\star} := \\arg \\min  _ {s: \\mathbb {R} ^ {d} \\rightarrow \\mathbb {R} ^ {d}} \\underset {W \\sim \\mathcal {N} (0, I _ {d}), X _ {0} \\sim p _ {\\mathrm {d a t a}}} {\\mathbb {E}} \\left[\\left\\| s \\left(\\sqrt {\\bar {\\alpha} _ {t}} X _ {0} + \\sqrt {1 - \\bar {\\alpha} _ {t}} W\\right) + \\frac {1}{\\sqrt {1 - \\bar {\\alpha} _ {t}}} W \\right\\| _ {2} ^ {2} \\right], \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            744,
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+            785
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which takes the form of the minimum mean square error estimator for  $-\\frac{1}{\\sqrt{1 - \\overline{\\alpha}_t}} W$  given  $\\sqrt{\\overline{\\alpha}_t} X_0 + \\sqrt{1 - \\overline{\\alpha}_t} W$  and is often more amenable to training.",
+        "bbox": [
+            169,
+            792,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With Definition 1 in place, we can readily introduce the following assumptions that capture the quality of the score estimate  $\\{s_t\\}_{1\\leq t\\leq T}$  we have available.",
+        "bbox": [
+            169,
+            832,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. Suppose that the score function estimate  $\\{s_t\\}_{1\\leq t\\leq T}$  obeys",
+        "bbox": [
+            171,
+            864,
+            665,
+            880
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {1}{T} \\sum_ {t = 1} ^ {T} \\mathbb {E} _ {X \\sim q _ {t}} \\left[ \\left\\| s _ {t} (X) - s _ {t} ^ {\\star} (X) \\right\\| _ {2} ^ {2} \\right] \\leq \\varepsilon_ {\\text {s c o r e}} ^ {2}. \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            886,
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+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2. For each  $1 \\leq t \\leq T$ , assume that  $s_t(\\cdot)$  is continuously differentiable, and denote by  $J_{s_t^*} = \\frac{\\partial s_t^*}{\\partial x}$  and  $J_{s_t} = \\frac{\\partial s_t}{\\partial x}$  the Jacobian matrices of  $s_t^*(\\cdot)$  and  $s_t(\\cdot)$ , respectively. Assume that the score function estimate  $\\{s_t\\}_{1 \\leq t \\leq T}$  obeys",
+        "bbox": [
+            169,
+            103,
+            823,
+            154
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {1}{T} \\sum_ {t = 1} ^ {T} \\underset {X \\sim q _ {t}} {\\mathbb {E}} \\left[ \\left\\| J _ {s _ {t}} (X) - J _ {s _ {t} ^ {*}} (X) \\right\\| \\right] \\leq \\varepsilon_ {\\text {J a c o b i}}. \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            157,
+            823,
+            199
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In a nutshell, Assumption 1 reflects the  $\\ell_2$  score estimation error, whereas Assumption 2 concerns the estimation error in terms of the corresponding Jacobian matrix. Both assumptions consider the average estimation errors over all  $T$  steps. Our theory for the deterministic sampler relies on both Assumptions 1 and 2, while the theory for the stochastic sampler requires only Assumption 1. We shall discuss in Section 3.2 the insufficiency of Assumption 1 alone for the deterministic sampler.",
+        "bbox": [
+            169,
+            210,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Target data distributions. Our goal is to uncover the effectiveness of diffusion models in generating a broad family of data distributions. Throughout this paper, the only assumptions we need to impose on the target data distribution  $p_{\\mathrm{data}}$  are the following:",
+        "bbox": [
+            169,
+            295,
+            823,
+            339
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "-  $X_0$  is an absolutely continuous random vector, and",
+        "bbox": [
+            215,
+            348,
+            568,
+            363
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {P} \\left(\\left\\| X _ {0} \\right\\| _ {2} \\leq T ^ {c _ {R}} \\mid X _ {0} \\sim p _ {\\mathrm {d a t a}}\\right) = 1 \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            367,
+            823,
+            386
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for some arbitrarily large constant  $c_{R} > 0$ .",
+        "bbox": [
+            228,
+            388,
+            511,
+            405
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This assumption allows the radius of the support of  $p_{\\mathrm{data}}$  to be exceedingly large (given that the exponent  $c_R$  can be arbitrarily large).",
+        "bbox": [
+            169,
+            414,
+            823,
+            444
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Learning rate schedule. Let us also take a moment to specify the learning rates to be used for our theory and analyses. For some large enough numerical constants  $c_{0}, c_{1} > 0$ , we set",
+        "bbox": [
+            169,
+            458,
+            823,
+            487
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\beta_ {1} = 1 - \\alpha_ {1} = \\frac {1}{T ^ {c _ {0}}}; \\tag {22a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            491,
+            823,
+            518
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\beta_ {t} = 1 - \\alpha_ {t} = \\frac {c _ {1} \\log T}{T} \\min  \\left\\{\\beta_ {1} \\left(1 + \\frac {c _ {1} \\log T}{T}\\right) ^ {t}, 1 \\right\\}. \\tag {22b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            521,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. As we shall see in our analysis, the discretization error depends crucially upon the quantity  $\\frac{1 - \\alpha_t}{1 - \\overline{\\alpha}_t}$ , whereas the initialization error relies heavily upon  $\\overline{\\alpha}_{1}$  and  $\\overline{\\alpha}_T$ . Based on these observations, our learning rate schedule (22) is designed to make  $\\frac{1 - \\alpha_t}{1 - \\overline{\\alpha}_t}$  as small as possible, while making sure  $\\overline{\\alpha}_{1}$  (resp.  $\\overline{\\alpha}_T$ ) is close to 1 (resp. 0). These properties will be shown in (43). Moreover, it is worth noting that our analysis can be easily extended to accommodate a much broader class of learning rates, although the resulting convergence rates might vary.",
+        "bbox": [
+            169,
+            566,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 AN ODE-BASED DETERMINISTIC SAMPLER",
+        "text_level": 1,
+        "bbox": [
+            171,
+            672,
+            516,
+            686
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We begin by analyzing a deterministic sampler: a discrete-time version of the probability flow ODE.",
+        "bbox": [
+            169,
+            696,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Armed with the score estimates  $\\{s_t\\}_{1\\leq t\\leq T}$ , a discrete-time version of the probability flow ODE approach (cf. (15)) adopts the following update rule:",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T} \\sim \\mathcal {N} (0, I _ {d}), \\quad Y _ {t - 1} = \\Phi_ {t} \\left(Y _ {t}\\right) \\quad \\text {f o r} t = T, \\dots , 1, \\tag {23a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            752,
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+            770
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\Phi_t(\\cdot)$  is taken to be",
+        "bbox": [
+            171,
+            773,
+            346,
+            789
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Phi_ {t} (x) := \\frac {1}{\\sqrt {\\alpha_ {t}}} \\left(x + \\frac {1 - \\alpha_ {t}}{2} s _ {t} (x)\\right). \\tag {23b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            795,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This approach, based on the probability flow ODE (15), often achieves faster sampling compared to the stochastic counterpart (Song et al., 2021b). Despite the empirical advances, however, the theoretical understanding of this type of deterministic samplers remained far from mature.",
+        "bbox": [
+            169,
+            832,
+            823,
+            875
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first derive non-asymptotic convergence guarantees — measured by the total variation distance between the forward and the reverse processes — for the above deterministic sampler (23). The proof of this result can be found in Li et al. (2023, Section 5.2).",
+        "bbox": [
+            169,
+            881,
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+            925
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Suppose that (21) holds true. Assume that the score estimates  $s_t(\\cdot)$  ( $1 \\leq t \\leq T$ ) satisfy Assumptions 1 and 2. Then the sampling process (23) with the learning rate schedule (22) satisfies",
+        "bbox": [
+            169,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {T V} \\left(q _ {1}, p _ {1}\\right) \\leq C _ {1} \\frac {d ^ {2} \\log^ {4} T}{T} + C _ {1} \\frac {d ^ {6} \\log^ {6} T}{T ^ {2}} + C _ {1} \\sqrt {d \\log^ {3} T} \\varepsilon_ {\\text {s c o r e}} + C _ {1} d (\\log T) \\varepsilon_ {\\text {J a c o b i}} \\tag {24}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            199,
+            138,
+            825,
+            172
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "for some universal constants  $C_1 > 0$ , where  $p_1$  (resp.  $q_1$ ) represents the distribution of  $Y_1$  (resp.  $X_1$ ).",
+        "bbox": [
+            168,
+            176,
+            823,
+            194
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implications. Let us highlight the main implications of Theorem 1. Before proceeding, note that our theory is concerned with convergence to  $q_{1}$ . Given that  $X_{1} \\sim q_{1}$  and  $X_{0} \\sim q_{0}$  are very close due to the choice of  $\\alpha_{1}$ , focusing on the convergence w.r.t.  $q_{1}$  instead of  $q_{0}$  remains practically relevant.",
+        "bbox": [
+            169,
+            203,
+            823,
+            247
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(a) Iteration complexity. Consider first the scenario that has access to perfect score estimates (i.e.,  $\\varepsilon_{\\mathrm{score}} = 0$ ). In order to achieve  $\\varepsilon$ -accuracy (in the sense that  $\\mathsf{TV}(q_1,p_1)\\leq \\varepsilon$ ), the number of steps  $T$  only needs to exceed",
+        "bbox": [
+            169,
+            253,
+            825,
+            294
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\widetilde {O} \\left(d ^ {2} / \\varepsilon + d ^ {3} / \\sqrt {\\varepsilon}\\right). \\tag {25}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            313
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(b) Stability. Turning to the more general case with imperfect score estimates (i.e.,  $\\varepsilon_{\\mathrm{score}} > 0$ ), the deterministic sampler (23) yields a distribution whose distance to the target distribution (measured again by the TV distance) scales proportionally with  $\\varepsilon_{\\mathrm{score}}$  and  $\\varepsilon_{\\mathrm{Jacobi}}$ . It is noteworthy that in addition to the score estimation errors, we are in need of an assumption on the stability of the associated Jacobian matrices, which plays a pivotal in ensuring that the reverse-time deterministic process does not deviate considerably from the desired process.",
+            "(c) Insufficiency of the score estimation error assumption alone. The careful reader might wonder why we are in need of additional assumptions beyond the  $\\ell_2$  score error stated in Assumption 1. To answer this question, we find it helpful to look at a simple example below."
+        ],
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "- Example. Consider the case where  $X_0 \\sim \\mathcal{N}(0,1)$ , and hence  $X_1 \\sim \\mathcal{N}(0,1)$ . Suppose that the reverse process for time  $t = 2$  can lead to the desired distribution if exact score function is employed, namely,",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {1} ^ {\\star} := \\frac {1}{\\sqrt {\\alpha_ {2}}} \\left(Y _ {2} - \\frac {1 - \\alpha_ {2}}{2} s _ {2} ^ {\\star} (Y _ {2})\\right) \\sim \\mathcal {N} (0, 1).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, suppose that the score estimate  $s_2(\\cdot)$  we have available obeys",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {2} \\left(y _ {2}\\right) = s _ {2} ^ {\\star} \\left(y _ {2}\\right) + \\frac {2 \\sqrt {\\alpha_ {2}}}{1 - \\alpha_ {2}} \\left\\{y _ {1} ^ {\\star} - L \\left\\lfloor \\frac {y _ {1} ^ {\\star}}{L} \\right\\rfloor \\right\\} \\quad \\text {w i t h} y _ {1} ^ {\\star} := \\frac {1}{\\sqrt {\\alpha_ {2}}} \\left(y _ {2} - \\frac {1 - \\alpha_ {2}}{2} s _ {2} ^ {\\star} \\left(y _ {2}\\right)\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "for  $L > 0$ , where  $\\lfloor z \\rfloor$  is the greatest integer not exceeding  $z$ . It follows that",
+        "bbox": [
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+            723,
+            638
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {1} = Y _ {1} ^ {\\star} + \\frac {1 - \\alpha_ {2}}{2 \\sqrt {\\alpha_ {2}}} \\left[ s _ {2} ^ {\\star} (Y _ {2}) - s _ {2} (Y _ {2}) \\right] = L \\left\\lfloor \\frac {Y _ {1} ^ {\\star}}{L} \\right\\rfloor .\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            679
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Clearly, the score error  $\\mathbb{E}_{X_2\\sim \\mathcal{N}(0,1)}\\bigl [|s_2(X_2) - s_2^\\star (X_2)|^2\\bigr ]$  can be made arbitrarily small by taking  $L\\to 0$ . However, the discrete nature of  $Y_{1}$  forces  $\\mathsf{T}\\mathsf{V}(Y_1,X_1) = 1$ .",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This example demonstrates that, for the deterministic sampler, the TV distance between  $Y_{1}$  and  $X_{1}$  might not improve as the score error decreases. If we wish to relax Assumption 2, one potential way is to resort to other metrics (e.g., Wasserstein distance) instead of TV distance between  $Y_{1}$  and  $X_{1}$ .",
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+    },
+    {
+        "type": "text",
+        "text": "(d) Relaxing the boundedness assumption on  $X_0$ . As it turns out, the assumption (21) can also be relaxed. Supposing that  $\\mathbb{P}\\big(\\|X_0\\|_2 \\leq B \\mid X_0 \\sim p_{\\mathrm{data}}\\big) = 1$  for some quantity  $B > 0$  (which is allowed to grow faster than a polynomial in  $T$ ), we can readily extend our analysis to obtain",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {T V} \\left(q _ {1}, p _ {1}\\right) \\leq C _ {1} \\frac {d ^ {2} \\log^ {4} T \\log^ {2} B}{T} + C _ {1} \\frac {d ^ {6} \\log^ {6} T \\log^ {3} B}{T ^ {2}} + C _ {1} \\sqrt {d \\log^ {3} T \\log B} \\varepsilon_ {\\text {s c o r e}} + C _ {1} d (\\log T) \\varepsilon_ {\\text {J a c o b i}}.\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Importantly, the convergence rate depends only logarithmically in  $B$ .",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with past works. To the best of our knowledge, the only non-asymptotic analysis for the discretized probability flow ODE approach in prior literature was derived by a very recent work Chen et al. (2023c), which established the first non-asymptotic convergence guarantees that exhibit",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "polynomial dependency in both  $d$  and  $1 / \\varepsilon$  (see, e.g., Chen et al. (2023c, Theorem 4.1)). However, it fell short of providing concrete polynomial dependency in  $d$  and  $1 / \\varepsilon$ , suffered from exponential dependency in the Lipschitz constant of the score function, and relied on exact score estimates. In contrast, our result in Theorem 1 uncovers a concrete  $d^2 /\\varepsilon$  scaling (ignoring lower-order and logarithmic terms) without imposing any smoothness assumption on the target data distribution, and makes explicit the effect of score estimation errors, both which were previously unavailable for such discrete-time deterministic samplers. Another recent work Benton et al. (2023b) studied the convergence of the probability flow ODE approach without accounting for the discretization error; the result therein also exhibited exponential dependency on the Lipschitz constant. Finally, while we were wrapping up the current paper, we became aware of the independent work Chen et al. (2023b) establishing improved polynomial dependency for two variants of the probability flow ODE. By inserting an additional stochastic corrector step — based on overdamped (resp. underdamped) Langevin diffusion — in each iteration of the probability flow ODE (so strictly speaking, these variations are no longer deterministic samplers), Chen et al. (2023b) showed that  $\\widetilde{O} (L^{3}d / \\varepsilon^{2})$  (resp.  $\\widetilde{O} (L^2\\sqrt{d} /\\varepsilon)$ ) steps are sufficient, where  $L$  denotes the Lipschitz constant of the score function. In comparison, our result demonstrates for the first time that the plain probability flow ODE already achieves the  $1 / \\varepsilon$  scaling without requiring either a corrector step; one limitation of our result, however, is the sub-optimal  $d$ -dependency compared to the variants studied in Chen et al. (2023b).",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.3 A DDPM-TYPE STOCHASTIC SAMPLER",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Armed with the score estimates  $\\{s_t\\}$ , we can readily introduce the following stochastic sampler that operates in discrete time, motivated by the reverse-time SDE (16):",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nY _ {T} \\sim \\mathcal {N} \\left(0, I _ {d}\\right), \\quad Y _ {t - 1} = \\Psi_ {t} \\left(Y _ {t}, Z _ {t}\\right) \\quad \\text {f o r} t = T, \\dots , 1 \\tag {26a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where  $Z_{t}\\stackrel {\\mathrm{i.i.d.}}{\\sim}\\mathcal{N}(0,I_{d})$  , and",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Psi_ {t} (y, z) = \\frac {1}{\\sqrt {\\alpha_ {t}}} \\left(y + \\left(1 - \\alpha_ {t}\\right) s _ {t} (y)\\right) + \\sigma_ {t} z \\quad \\text {w i t h} \\sigma_ {t} ^ {2} = \\frac {1}{\\alpha_ {t}} - 1. \\tag {26b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The key difference between this sampler and the deterministic sampler (23) is that: (i) there exists an additional pre-factor of  $1/2$  on  $s_t$  in the deterministic sampler; and (ii) the stochastic sampler injects additional noise  $Z_t$  in each step.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In contrast to deterministic samplers, the stochastic samplers have received more theoretical attention, with the state-of-the-art results established by Chen et al. (2022b;a) as well as a very recent paper Benton et al. (2023a). The elementary approach developed in the current paper is also applicable towards understanding this type of samplers, leading to the following non-asymptotic theory.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Suppose (21) holds true. Equipped with the estimates in Assumption 1 and the learning rate schedule (22), the stochastic sampler (26) achieves, for some universal constants  $C_1 > 0$ ,",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {T V} \\left(q _ {1}, p _ {1}\\right) \\leq \\sqrt {\\frac {1}{2} \\mathrm {K L} \\left(q _ {1} \\| p _ {1}\\right)} \\leq C _ {1} \\frac {d ^ {2} \\log^ {3} T}{\\sqrt {T}} + C _ {1} \\sqrt {d} \\varepsilon_ {\\text {s c o r e}} \\log^ {2} T. \\tag {27}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 establishes non-asymptotic convergence guarantees for the stochastic sampler (26). As asserted by the theorem, if we have access to perfect score estimates, then the number of steps needed to attain  $\\varepsilon$ -accuracy (measured by the TV distance between  $p_1$  and  $q_1$ ) is proportional to  $1 / \\varepsilon^2$ , matching the state-of-the-art  $\\varepsilon$ -dependency derived in Chen et al. (2022a), albeit exhibiting a worse dimensional dependency. In addition, in the presence of score estimation error, the sampler achieves a TV distance proportional to  $\\varepsilon_{\\mathrm{score}}$ , again consistent with prior results. Our analysis follows a completely different path compared with the SDE-based approach in Chen et al. (2022a), thus offering complementary interpretations for this important sampler.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 OTHER RELATED WORKS",
+        "text_level": 1,
+        "bbox": [
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+            837,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theory for SGMs. Early theoretical efforts in understanding the convergence of score-based stochastic samplers suffered from being either not quantitative (De Bortoli et al., 2021; Liu et al., 2022; Pidstrigach, 2022), or the curse of dimensionality (e.g., exponential dependencies in the convergence guarantees) (Block et al., 2020; De Bortoli, 2022). Lee et al. (2022) provided the first",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "polynomial convergence guarantees with  $L_{2}$ -accurate score estimates, for any smooth distribution satisfying the log-Sobolev inequality. Chen et al. (2022b); Lee et al. (2023); Chen et al. (2022a) subsequently lifted such a stringent data distribution assumption. More concretely, Chen et al. (2022b) accommodated a broad family of data distributions under the premise that the score functions over the entire trajectory of the forward process are Lipschitz; Lee et al. (2023) only required certain smoothness assumptions but came with worse dependence on the problem parameters; and more recent results in Chen et al. (2022a) applied to literally any data distribution with bounded second-order moment. In addition, Wibisono & Yang (2022) also established a convergence theory for score-based generative models, assuming that the error of the score estimator has a bounded moment generating function and that the data distribution satisfies the log-Sobolev inequality. Turning attention to samplers based on the probability flow ODE, Chen et al. (2023c) derived the first non-asymptotic bounds for this type of samplers. Improved convergence guarantees have recently been provided by a concurrent work Chen et al. (2023b), with the assistance of additional corrector steps inerspersed in each iteration of the probability flow ODE. It is worth noting that the corrector steps proposed therein are based on Langevin-type diffusion and inject additive noise, and hence the resulting sampling processes are not deterministic. Additionally, theoretical justifications for DDPM in the context of image in-painting have been developed by Rout et al. (2023). Moreover, convergence results based on the Wasserstein distance have recently been derived as well (Tang, 2023; Benton et al., 2023b), although these results typically exhibit exponential dependency on the Lipschitz constants of the score functions. Theoretical guarantees are also extended to accommodate popular methods like consistency models and diffusion guidance (Li et al., 2024b; Wu et al., 2024).",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Score matching. Hyvarinen (2005) showed that the score function can be estimated via integration by parts, a result that was further extended in Hyvarinen (2007). Song et al. (2020b) proposed sliced score matching to tame the computational complexity in high dimension. The consistency of the score matching estimator was studied in Hyvarinen (2005), with asymptotic normality established in Forbes & Lauritzen (2015). Optimizing the score matching loss has been shown to be intimately connected to minimizing upper bounds on the Kullback-Leibler divergence (Song et al., 2021a) and Wasserstein distance (Kwon et al., 2022) between the generated distribution and the target data distribution. From a non-asymptotic perspective, Koehler et al. (2023) studied the statistical efficiency of score matching by connecting it with the isoperimetric properties of the distribution.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Other theory for diffusion models. Oko et al. (2023) studied the approximation and generalization capabilities of diffusion modeling for distribution estimation. Assuming that the data are supported on a low-dimensional linear subspace, Chen et al. (2023a) developed a sample complexity bound for diffusion models. Moreover, Ghimire et al. (2023) adopted a geometric perspective and showed that the forward and backward processes of diffusion models are essentially Wasserstein gradient flows. Recently, the idea of stochastic localization, which is closely related to diffusion models, is adopted to sample from posterior distributions (Montanari & Wu, 2023; El Alaoui et al., 2022), which has been implemented using approximate message passing (Donoho et al., 2009; Li & Wei, 2022).",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have developed a new suite of non-asymptotic theory for establishing the convergence and faithfulness of diffusion generative modeling, assuming access to reliable estimates of the (Stein) score functions. Our analysis framework seeks to track the dynamics of the reverse process directly using elementary tools, which eliminates the need to look at the continuous-time limit and invoke the SDE and ODE toolboxes. Only the very minimal assumptions on the target data distribution are imposed. The analysis framework laid out in the current paper might shed light on how to analyze other variants of score-based generative models as well. Moving forward, there are plenty of questions that require in-depth theoretical understanding. For instance, the dimension dependency in our convergence results remains sub-optimal; can we further refine our theory in order to reveal tight dependency in this regard? Can we establish sharp convergence results in terms of the Wasserstein distance, which could sometimes be \"closer\" to how humans differentiate pictures and might potentially help relax Assumption 2 in the case of deterministic samplers? It would also be of paramount interest to establish end-to-end performance guarantees that take into account both the score learning phase and the sampling phase.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "ref_text",
+        "text": "Y. Wei is supported in part by the NSF grants DMS-2147546/2015447, CAREER award DMS-2143215, CCF-2106778, and the Google Research Scholar Award. Y. Chen is supported in part by the Alfred P. Sloan Research Fellowship, the Google Research Scholar Award, the AFOSR grant FA9550-22-1-0198, the ONR grant N00014-22-1-2354, and the NSF grants CCF-2221009 and CCF-1907661. Y. Chi is supported in part by the grants ONR N00014-19-1-2404, NSF CCF-2106778, DMS-2134080 and ECCS-2126634.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
+        "bbox": [
+            173,
+            238,
+            287,
+            253
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
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+            "Joe Benton, George Deligiannidis, and Arnaud Doucet. Error bounds for flow matching methods. arXiv preprint arXiv:2305.16860, 2023b.",
+            "Adam Block, Youssef Mroueh, and Alexander Rakhlin. Generative modeling with denoising autoencoders and Langevin sampling. arXiv preprint arXiv:2002.00107, 2020.",
+            "Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. arXiv preprint arXiv:2211.01916, 2022a.",
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+            "Sitan Chen, Giannis Daras, and Alexandros G Dimakis. Restoration-degradation beyond linear diffusions: A non-asymptotic analysis for DDIM-type samplers. arXiv preprint arXiv:2303.03384, 2023c.",
+            "Florinel-Alin Croitoru, Vlad Hondru, Radu Tudor Ionescu, and Mubarak Shah. Diffusion models in vision: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023.",
+            "Valentin De Bortoli. Convergence of denoising diffusion models under the manifold hypothesis. arXiv preprint arXiv:2208.05314, 2022.",
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+        ],
+        "bbox": [
+            171,
+            261,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
+            490,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
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+        ],
+        "page_idx": 10
+    },
+    {
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+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+        ],
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+    },
+    {
+        "type": "list",
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+        ],
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+            171,
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2024/Towards Non-Asymptotic Convergence for Diffusion-Based Generative Models/b4e2a016-0bfa-45bf-93f6-59d4737b9813_origin.pdf

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2024/Towards Non-Asymptotic Convergence for Diffusion-Based Generative Models/full.md

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+# TOWARDS NON-ASYMPTOTIC CONVERGENCE FOR DIFFUSION-BASED GENERATIVE MODELS
+
+Gen Li*
+
+Yuting Wei
+
+Yuxin Chen†
+
+Yuejie Chi\*
+
+# ABSTRACT
+
+Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to  $\ell_2$ -accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to  $1 / T$  (with  $T$  the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to  $1 / \sqrt{T}$ , matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how  $\ell_2$  score estimation errors affect the quality of the data generation process. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs.
+
+# 1 INTRODUCTION
+
+Diffusion models have emerged as a cornerstone in contemporary generative modeling, a task that learns to generate new data instances (e.g., images, text, audio) that look similar in distribution to the training data (Ho et al., 2020; Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Dhariwal & Nichol, 2021; Jolicoeur-Martineau et al., 2021; Chen et al., 2021; Kong et al., 2021; Austin et al., 2021). Originally proposed by Sohl-Dickstein et al. (2015) and later popularized by Song & Ermon (2019); Ho et al. (2020), the mainstream diffusion generative models — e.g., denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020) and denoising diffusion implicit models (DDIMs) (Song et al., 2020a) — have underpinned major successes in content generators like DALL-E 2 (Ramesh et al., 2022), Stable Diffusion (Rombach et al., 2022) and Imagen (Saharia et al., 2022), claiming state-of-the-art performance in the now broad field of generative artificial intelligence (AI). See Yang et al. (2022); Croitoru et al. (2023) for overviews of recent development.
+
+In a nutshell, a diffusion generative model is based upon two stochastic processes in  $\mathbb{R}^d$ :
+
+1) a forward process
+
+$$
+X _ {0} \rightarrow X _ {1} \rightarrow \dots \rightarrow X _ {T} \tag {1}
+$$
+
+that starts from a sample drawn from the target data distribution (e.g., of natural images) and gradually diffuses it into a noise-like distribution (e.g., standard Gaussians);
+
+2) a reverse process
+
+$$
+Y _ {T} \rightarrow Y _ {T - 1} \rightarrow \dots \rightarrow Y _ {0} \tag {2}
+$$
+
+that starts from pure noise (e.g., standard Gaussians) and successively converts it into new samples sharing similar distributions as the target data distribution.
+
+Transforming data into noise in the forward process is straightforward, often hand-crafted by increasingly injecting more noise into the data at hand. What is challenging is the construction of the reverse process: how to generate the desired information out of pure noise? To do so, a diffusion model learns to build a reverse process (2) that imitates the dynamics of the forward process (1) in a time-reverse fashion; more precisely, the design goal is to ascertain distributional proximity<sup>1</sup>
+
+$$
+Y _ {t} \stackrel {\mathrm {d}} {\approx} X _ {t}, \quad t = T, \dots , 1 \tag {3}
+$$
+
+through proper learning based on how the training data propagate in the forward process. Encouragingly, there often exist feasible strategies to achieve this goal as long as faithful estimates about the (Stein) score functions — the gradients of the log marginal density of the forward process — are available, an intriguing fact that can be illuminated by the existence and construction of reverse-time stochastic differential equations (SDEs) (Anderson, 1982; Haussmann & Pardoux, 1986) (see Section 2.2 for more precise discussions). Viewed in this light, a diverse array of diffusion models are frequently referred to as score-based generative modeling (SGM). The popularity of SGM was initially motivated by, and has since further inspired, numerous recent studies on the problem of learning score functions, a subroutine that also goes by the name of score matching (e.g., Hyvarinen (2005; 2007); Vincent (2011); Song et al. (2020b); Koehler et al. (2023)).
+
+Nonetheless, despite the mind-blowing empirical advances, a mathematical theory for diffusion generative models is still in its infancy. Given the complexity of developing a full-fledged end-to-end theory, a divide-and-conquer approach has been advertised, decoupling the score learning phase (i.e., how to estimate score functions from training data) and the generative sampling phase (i.e., how to generate new data given the score estimates). In particular, the past two years have witnessed growing interest and remarkable progress from the theoretical community towards understanding the sampling phase (Block et al., 2020; De Bortoli et al., 2021; Liu et al., 2022; De Bortoli, 2022; Lee et al., 2023; Pidstrigach, 2022; Chen et al., 2022b;a; Tang, 2023; Chen et al., 2023c; Tang & Zhao, 2024; Li et al., 2024a). For instance, polynomial-time convergence guarantees have been established for stochastic samplers (e.g., Chen et al. (2022b;a); Benton et al. (2023a)) and deterministic samplers (e.g., Chen et al. (2023c); Benton et al. (2023b)), both of which accommodated a fairly general family of data distributions.
+
+This paper. The present paper contributes to this growing list of theoretical endeavors by developing a new suite of non-asymptotic theory for several score-based generative modeling algorithms. We concentrate on two types of samplers (Song et al., 2021b) in discrete time: (i) a deterministic sampler based on a sort of ordinary differential equations (ODEs) called probability flow ODEs (which is closely related to the DDIM); and (ii) a DDPM-type stochastic sampler motivated by reverse-time SDEs. We impose only minimal assumptions on the target data distribution (e.g., no smoothness condition is needed), and would like to quantify the impact of  $\ell_2$  score estimation errors. In comparisons to past works, our main contributions are three-fold.
+
+Non-asymptotic convergence guarantees. For a popular deterministic sampler, we demonstrate that the number of steps needed to yield  $\varepsilon$ -accuracy — meaning that the total variation (TV) distance between the distribution of  $X_{1}$  and that of  $Y_{1}$  is no larger than  $\varepsilon$  — is proportional to  $1 / \varepsilon$  (in addition to other polynomial dimension dependency). This improves upon prior convergence guarantees (Chen et al., 2023c) and does not exhibit exponential dependency on the smoothness assumption as in Chen et al. (2023c); Benton et al. (2023b). For another DDPM-type stochastic sampler, we establish an iteration complexity proportional to  $1 / \varepsilon^{2}$ , matching existing theory Chen et al. (2022b;a); Benton et al. (2023a) in terms of the  $\varepsilon$ -dependency.
+
+Score estimation errors for the deterministic sampler. In our convergence guarantees for the deterministic sampler, the TV distance between  $X_{1}$  and  $Y_{1}$  are shown to be proportional to the  $\ell_{2}$  score estimation error as well as the associated Jacobian errors. As far as we know, this is the first result for this deterministic sampler that accounts for score estimation errors in discrete time. In comparison, other theoretical results that accommodate score errors for the probability flow ODE approach
+
+either study certain stochastic variations of this deterministic sampler (Chen et al., 2023b) or fall short of accommodating discretization errors (Benton et al., 2023b).
+
+An elementary non-asymptotic analysis framework. From the technical viewpoint, the analysis framework laid out in this paper is fully non-asymptotic in nature. In contrast to prior analyses that take a detour to study the continuum limits and then control the discretization error, our approach tackles the discrete-time processes directly using elementary analysis strategies. No knowledge of SDEs or ODEs is required for establishing our theory, thereby resulting in a more versatile framework and sometimes lowering the technical barrier towards understanding diffusion models.
+
+Notation. For any two functions  $f(d, T)$  and  $g(d, T)$ , we adopt the notation  $f(d, T) \lesssim g(d, T)$  or  $f(d, T) = O(g(d, T))$  (resp.  $f(d, T) \gtrsim g(d, T)$ ) to mean that there exists some universal constant  $C_1 > 0$  such that  $f(d, T) \leq C_1 g(d, T)$  (resp.  $f(d, T) \geq C_1 g(d, T)$ ) for all  $d$  and  $T$ ; moreover, the notation  $f(d, T) \asymp g(d, T)$  indicates that  $f(d, T) \lesssim g(d, T)$  and  $f(d, T) \gtrsim g(d, T)$  hold at once. The notation  $\widetilde{O}(\cdot)$  is defined similar to  $O(\cdot)$  except that it hides the logarithmic dependency. Additionally, the notation  $f(d, T) = o(g(d, T))$  means that  $f(d, T) / g(d, T) \to 0$  as  $d, T$  tend to infinity. For any two probability measures  $P$  and  $Q$ , the total variation (TV) distance between them is defined to be  $\mathsf{TV}(P, Q) := \frac{1}{2} \int |\mathrm{d}P - \mathrm{d}Q|$ . Throughout the paper,  $p_X(\cdot)$  (resp.  $p_{X|Y}(\cdot|\cdot))$  denotes the probability density function of  $X$  (resp.  $X$  given  $Y$ ). For any matrix  $A$ , we denote by  $\|A\|$  (resp.  $\|A\|_{\mathbb{F}}$ ) the spectral norm (resp. Frobenius norm) of  $A$ . Also, for any vector-valued function  $f$ , we let  $J_f$  or  $\frac{\partial f}{\partial x}$  represent the Jacobian matrix of  $f$ .
+
+# 2 PRELIMINARIES
+
+In this section, we introduce the basics of diffusion generative models. The ultimate goal of a generative model can be concisely stated: given data samples drawn from an unknown distribution of interest  $p_{\mathrm{data}}$  in  $\mathbb{R}^d$ , we wish to generate new samples whose distributions closely resemble  $p_{\mathrm{data}}$ .
+
+# 2.1 DIFFUSION GENERATIVE MODELS
+
+Towards achieving the above goal, a diffusion generative model typically encompasses two Markov processes: a forward process and a reverse process, as described below.
+
+The forward process. In the forward chain, one progressively injects noise into the data samples to diffuse and obscure the data. The distributions of the injected noise are often hand-picked, with the standard Gaussian distribution receiving widespread adoption. Specifically, the forward Markov process produces a sequence of  $d$ -dimensional random vectors  $X_{1} \to X_{2} \to \dots \to X_{T}$  as follows:
+
+$$
+X _ {0} \sim p _ {\text {d a t a}}, \tag {4a}
+$$
+
+$$
+X _ {t} = \sqrt {1 - \beta_ {t}} X _ {t - 1} + \sqrt {\beta_ {t}} W _ {t}, \quad 1 \leq t \leq T, \tag {4b}
+$$
+
+where  $\{W_t\}_{1 \leq t \leq T}$  indicates a sequence of independent noise vectors drawn from  $W_t \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0, I_d)$ . The hyper-parameters  $\{\beta_t \in (0, 1)\}$  represent prescribed learning rate schedules that control the variance of the noise injected in each step. If we define
+
+$$
+\alpha_ {t} := 1 - \beta_ {t}, \quad \overline {{\alpha}} _ {t} := \prod_ {k = 1} ^ {t} \alpha_ {k}, \quad 1 \leq t \leq T, \tag {5}
+$$
+
+then it can be straightforwardly verified that for every  $1 \leq t \leq T$ ,
+
+$$
+X _ {t} = \sqrt {\bar {\alpha} _ {t}} X _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} \bar {W} _ {t} \quad \text {f o r s o m e} \bar {W} _ {t} \sim \mathcal {N} (0, I _ {d}). \tag {6}
+$$
+
+Clearly, if the covariance of  $X_0$  is also equal to  $I_d$ , then the covariance of  $X_t$  is preserved throughout the forward process; for this reason, this forward process (4) is sometimes referred to as variance-preserving (Song et al., 2021b). Throughout this paper, we employ the notation
+
+$$
+q _ {t} := \operatorname {l a w} \left(X _ {t}\right) \tag {7}
+$$
+
+to denote the distribution of  $X_{t}$ . As long as  $\overline{\alpha}_{T}$  is vanishingly small, one has the following property for a general family of data distributions:
+
+$$
+q _ {T} \approx \mathcal {N} (0, I _ {d}). \tag {8}
+$$
+
+The reverse process. The reverse chain  $Y_{T} \to Y_{T-1} \to \ldots \to Y_{1}$  is designed to (approximately) revert the forward process, allowing one to transform pure noise into new samples with matching distributions as the original data. To be more precise, by initializing it as
+
+$$
+Y _ {T} \sim \mathcal {N} (0, I _ {d}), \tag {9a}
+$$
+
+we seek to design a reverse-time Markov process with nearly identical marginals as the forward process, namely,
+
+$$
+\left(\text {g o a l}\right) \quad Y _ {t} \stackrel {\mathrm {d}} {\approx} X _ {t}, \quad t = T, T - 1, \dots , 1. \tag {9b}
+$$
+
+Throughout the paper, we often employ the following notation to indicate the distribution of  $Y_{t}$ :
+
+$$
+p _ {t} := \operatorname {l a w} \left(Y _ {t}\right). \tag {10}
+$$
+
+# 2.2 DETERMINISTIC VS. STOCHASTIC SAMPLERS: A CONTINUOUS-TIME INTERPRETATION
+
+Evidently, the most crucial step of the diffusion model lies in effective design of the reverse process. Two mainstream approaches stand out:
+
+- Deterministic samplers. Starting from  $Y_{T} \sim \mathcal{N}(0, I_{d})$ , this approach selects a set of functions  $\{\Phi_t(\cdot)\}_{1 \leq t \leq T}$  and computes:
+
+$$
+Y _ {t - 1} = \Phi_ {t} (Y _ {t}), \quad t = T, \dots , 1. \tag {11}
+$$
+
+Clearly, the sampling process is fully deterministic except for the initialization  $Y_{T}$ .
+
+- Stochastic samplers. Initialized again at  $Y_{T} \sim \mathcal{N}(0, I_{d})$ , this approach computes another collection of functions  $\{\Psi_{t}(\cdot, \cdot)\}_{1 \leq t \leq T}$  and performs the updates:
+
+$$
+Y _ {t - 1} = \Psi_ {t} \left(Y _ {t}, Z _ {t}\right), \quad t = T, \dots , 1, \tag {12}
+$$
+
+where the  $Z_{t}$  's are independent noise vectors obeying  $Z_{t}\stackrel {\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,I_{d})$
+
+In order to elucidate the feasibility of the above two approaches, we find it helpful to look at the continuum limit through the lens of SDEs and ODEs. It is worth emphasizing, however, that the development of our main theory does not rely on any knowledge of SDEs and ODEs.
+
+- The forward process. A continuous-time analog of the forward process can be modeled as
+
+$$
+\mathrm {d} X _ {t} = f \left(X _ {t}, t\right) \mathrm {d} t + g (t) \mathrm {d} W _ {t} \quad (0 \leq t \leq T), \quad X _ {0} \sim p _ {\text {d a t a}} \tag {13}
+$$
+
+for some functions  $f(\cdot, \cdot)$  and  $g(\cdot)$  (denoting respectively the drift and diffusion coefficient), where  $W_{t}$  denotes a  $d$ -dimensional standard Brownian motion. As a special example, the continuum limit of (4) takes the following form² (Song et al., 2021b)
+
+$$
+\mathrm {d} X _ {t} = - \frac {1}{2} \beta (t) X _ {t} \mathrm {d} t + \sqrt {\beta (t)} \mathrm {d} W _ {t} \quad (0 \leq t \leq T), \quad X _ {0} \sim p _ {\text {d a t a}} \tag {14}
+$$
+
+for some function  $\beta(t)$ . As before, we denote by  $q_{t}$  the distribution of  $X_{t}$  in (13).
+
+- The reverse process. As it turns out, the following two reverse processes are both capable of reconstructing the distribution of the forward process, motivating the design of two distinctive samplers. Here and throughout, we use  $\nabla \log q_t(X)$  to abbreviate  $\nabla_X \log q_t(X)$  for notational simplicity.
+
+- One feasible approach is to the so-called probability flow ODE (Song et al., 2021b)
+
+$$
+\mathrm {d} Y _ {t} ^ {\text {o d e}} = \left(- f \left(Y _ {t} ^ {\text {o d e}}, T - t\right) + \frac {1}{2} g (T - t) ^ {2} \nabla \log q _ {T - t} \left(Y _ {t} ^ {\text {o d e}}\right)\right) \mathrm {d} t \quad (0 \leq t \leq T), \tag {15}
+$$
+
+with  $Y_0^{\mathrm{ode}}\sim q_T$  , which exhibits matching distributions as follows:
+
+$$
+Y _ {T - t} ^ {\mathsf {o d e}} \stackrel {{\mathrm {d}}} {{=}} X _ {t}, \qquad 0 \leq t \leq T.
+$$
+
+The deterministic nature of this approach often enables faster sampling. It has been shown that this family of deterministic samplers is closely related to the DDIM sampler (Karras et al., 2022; Song et al., 2021b).
+
+- In view of the classical results Anderson (1982); Haussmann & Pardoux (1986), one can also construct a "reverse-time" SDE
+
+$$
+\mathrm {d} Y _ {t} ^ {\mathrm {s d e}} = \left(- f \left(Y _ {t} ^ {\mathrm {s d e}}, T - t\right) + g (T - t) ^ {2} \nabla \log q _ {T - t} \left(Y _ {t} ^ {\mathrm {s d e}}\right)\right) \mathrm {d} t + g (T - t) \mathrm {d} Z _ {t} ^ {\mathrm {s d e}} \tag {16}
+$$
+
+for  $0 \leq t \leq T$ , with  $Y_0^{\mathrm{sde}} \sim q_T$  and  $Z_t^{\mathrm{sde}}$  being a standard Brownian motion. Strikingly, this process also satisfies
+
+$$
+Y _ {T - t} ^ {\mathsf {s d e}} \stackrel {{\mathrm {d}}} {{=}} X _ {t}, \qquad 0 \leq t \leq T.
+$$
+
+The popular DDPM sampler (Ho et al., 2020) falls under this category.
+
+Interestingly, in addition to the functions  $f$  and  $g$  that define the forward process, construction of both (15) and (16) relies only upon the knowledge of the gradient of the log density  $\nabla \log q_t(\cdot)$  of the intermediate steps of the forward diffusion process — often referred to as the (Stein) score function. Consequently, a key enabler of the above paradigms lies in reliable learning of the score function, and hence the name score-based generative modeling.
+
+# 3 ALGORITHMS AND MAIN RESULTS
+
+In this section, we analyze a couple of diffusion generative models, including both deterministic and stochastic samplers. While the proofs for our main theory are all postponed to the appendix, it is worth emphasizing upfront that our analysis framework directly tackles the discrete-time processes without resorting to any toolbox of SDEs and ODEs tailored to the continuous-time limits. This elementary approach might potentially be versatile for analyzing a broad class of variations of these samplers. For instance, prior ODE-based theory (e.g., Chen et al. (2023b;c)) encountered certain technical challenges when analyzing the deterministic sampler directly, and our elementary approach is able to shed new light on the convergence of this important sampler.
+
+# 3.1 ASSUMPTIONS AND LEARNING RATES
+
+Before proceeding, we impose some assumptions on the score estimates and the target data distributions, and specify the hypter-parameters  $\{\alpha_t\}$ , which shall be adopted throughout all cases.
+
+Score estimates. Given that the score functions are an essential component in score-based generative modeling, we assume access to faithful estimates of the score functions  $\nabla \log q_t(\cdot)$  across all intermediate steps  $t$ , thus disentangling the score learning phase and the data generation phase. Towards this end, let us first formally introduce the true score function as follows.
+
+Definition 1 (Score function). The score function, denoted by  $s_t^\star : \mathbb{R}^d \to \mathbb{R}^d$ , is defined as
+
+$$
+s _ {t} ^ {\star} (X) := \nabla \log q _ {t} (X), \quad 1 \leq t \leq T. \tag {17}
+$$
+
+As has been pointed out by previous works concerning score matching (e.g., Hyvarinen (2005); Vincent (2011); Chen et al. (2022b)), the score function  $s_t^\star$  admits an alternative form as follows (owing to properties of Gaussian distributions):
+
+$$
+s _ {t} ^ {\star} := \arg \min  _ {s: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d}} \underset {W \sim \mathcal {N} (0, I _ {d}), X _ {0} \sim p _ {\mathrm {d a t a}}} {\mathbb {E}} \left[\left\| s \left(\sqrt {\bar {\alpha} _ {t}} X _ {0} + \sqrt {1 - \bar {\alpha} _ {t}} W\right) + \frac {1}{\sqrt {1 - \bar {\alpha} _ {t}}} W \right\| _ {2} ^ {2} \right], \tag {18}
+$$
+
+which takes the form of the minimum mean square error estimator for  $-\frac{1}{\sqrt{1 - \overline{\alpha}_t}} W$  given  $\sqrt{\overline{\alpha}_t} X_0 + \sqrt{1 - \overline{\alpha}_t} W$  and is often more amenable to training.
+
+With Definition 1 in place, we can readily introduce the following assumptions that capture the quality of the score estimate  $\{s_t\}_{1\leq t\leq T}$  we have available.
+
+Assumption 1. Suppose that the score function estimate  $\{s_t\}_{1\leq t\leq T}$  obeys
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} _ {X \sim q _ {t}} \left[ \left\| s _ {t} (X) - s _ {t} ^ {\star} (X) \right\| _ {2} ^ {2} \right] \leq \varepsilon_ {\text {s c o r e}} ^ {2}. \tag {19}
+$$
+
+Assumption 2. For each  $1 \leq t \leq T$ , assume that  $s_t(\cdot)$  is continuously differentiable, and denote by  $J_{s_t^*} = \frac{\partial s_t^*}{\partial x}$  and  $J_{s_t} = \frac{\partial s_t}{\partial x}$  the Jacobian matrices of  $s_t^*(\cdot)$  and  $s_t(\cdot)$ , respectively. Assume that the score function estimate  $\{s_t\}_{1 \leq t \leq T}$  obeys
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \underset {X \sim q _ {t}} {\mathbb {E}} \left[ \left\| J _ {s _ {t}} (X) - J _ {s _ {t} ^ {*}} (X) \right\| \right] \leq \varepsilon_ {\text {J a c o b i}}. \tag {20}
+$$
+
+In a nutshell, Assumption 1 reflects the  $\ell_2$  score estimation error, whereas Assumption 2 concerns the estimation error in terms of the corresponding Jacobian matrix. Both assumptions consider the average estimation errors over all  $T$  steps. Our theory for the deterministic sampler relies on both Assumptions 1 and 2, while the theory for the stochastic sampler requires only Assumption 1. We shall discuss in Section 3.2 the insufficiency of Assumption 1 alone for the deterministic sampler.
+
+Target data distributions. Our goal is to uncover the effectiveness of diffusion models in generating a broad family of data distributions. Throughout this paper, the only assumptions we need to impose on the target data distribution  $p_{\mathrm{data}}$  are the following:
+
+-  $X_0$  is an absolutely continuous random vector, and
+
+$$
+\mathbb {P} \left(\left\| X _ {0} \right\| _ {2} \leq T ^ {c _ {R}} \mid X _ {0} \sim p _ {\mathrm {d a t a}}\right) = 1 \tag {21}
+$$
+
+for some arbitrarily large constant  $c_{R} > 0$ .
+
+This assumption allows the radius of the support of  $p_{\mathrm{data}}$  to be exceedingly large (given that the exponent  $c_R$  can be arbitrarily large).
+
+Learning rate schedule. Let us also take a moment to specify the learning rates to be used for our theory and analyses. For some large enough numerical constants  $c_{0}, c_{1} > 0$ , we set
+
+$$
+\beta_ {1} = 1 - \alpha_ {1} = \frac {1}{T ^ {c _ {0}}}; \tag {22a}
+$$
+
+$$
+\beta_ {t} = 1 - \alpha_ {t} = \frac {c _ {1} \log T}{T} \min  \left\{\beta_ {1} \left(1 + \frac {c _ {1} \log T}{T}\right) ^ {t}, 1 \right\}. \tag {22b}
+$$
+
+Remark 1. As we shall see in our analysis, the discretization error depends crucially upon the quantity  $\frac{1 - \alpha_t}{1 - \overline{\alpha}_t}$ , whereas the initialization error relies heavily upon  $\overline{\alpha}_{1}$  and  $\overline{\alpha}_T$ . Based on these observations, our learning rate schedule (22) is designed to make  $\frac{1 - \alpha_t}{1 - \overline{\alpha}_t}$  as small as possible, while making sure  $\overline{\alpha}_{1}$  (resp.  $\overline{\alpha}_T$ ) is close to 1 (resp. 0). These properties will be shown in (43). Moreover, it is worth noting that our analysis can be easily extended to accommodate a much broader class of learning rates, although the resulting convergence rates might vary.
+
+# 3.2 AN ODE-BASED DETERMINISTIC SAMPLER
+
+We begin by analyzing a deterministic sampler: a discrete-time version of the probability flow ODE.
+
+Armed with the score estimates  $\{s_t\}_{1\leq t\leq T}$ , a discrete-time version of the probability flow ODE approach (cf. (15)) adopts the following update rule:
+
+$$
+Y _ {T} \sim \mathcal {N} (0, I _ {d}), \quad Y _ {t - 1} = \Phi_ {t} \left(Y _ {t}\right) \quad \text {f o r} t = T, \dots , 1, \tag {23a}
+$$
+
+where  $\Phi_t(\cdot)$  is taken to be
+
+$$
+\Phi_ {t} (x) := \frac {1}{\sqrt {\alpha_ {t}}} \left(x + \frac {1 - \alpha_ {t}}{2} s _ {t} (x)\right). \tag {23b}
+$$
+
+This approach, based on the probability flow ODE (15), often achieves faster sampling compared to the stochastic counterpart (Song et al., 2021b). Despite the empirical advances, however, the theoretical understanding of this type of deterministic samplers remained far from mature.
+
+We first derive non-asymptotic convergence guarantees — measured by the total variation distance between the forward and the reverse processes — for the above deterministic sampler (23). The proof of this result can be found in Li et al. (2023, Section 5.2).
+
+Theorem 1. Suppose that (21) holds true. Assume that the score estimates  $s_t(\cdot)$  ( $1 \leq t \leq T$ ) satisfy Assumptions 1 and 2. Then the sampling process (23) with the learning rate schedule (22) satisfies
+
+$$
+\mathrm {T V} \left(q _ {1}, p _ {1}\right) \leq C _ {1} \frac {d ^ {2} \log^ {4} T}{T} + C _ {1} \frac {d ^ {6} \log^ {6} T}{T ^ {2}} + C _ {1} \sqrt {d \log^ {3} T} \varepsilon_ {\text {s c o r e}} + C _ {1} d (\log T) \varepsilon_ {\text {J a c o b i}} \tag {24}
+$$
+
+for some universal constants  $C_1 > 0$ , where  $p_1$  (resp.  $q_1$ ) represents the distribution of  $Y_1$  (resp.  $X_1$ ).
+
+Implications. Let us highlight the main implications of Theorem 1. Before proceeding, note that our theory is concerned with convergence to  $q_{1}$ . Given that  $X_{1} \sim q_{1}$  and  $X_{0} \sim q_{0}$  are very close due to the choice of  $\alpha_{1}$ , focusing on the convergence w.r.t.  $q_{1}$  instead of  $q_{0}$  remains practically relevant.
+
+(a) Iteration complexity. Consider first the scenario that has access to perfect score estimates (i.e.,  $\varepsilon_{\mathrm{score}} = 0$ ). In order to achieve  $\varepsilon$ -accuracy (in the sense that  $\mathsf{TV}(q_1,p_1)\leq \varepsilon$ ), the number of steps  $T$  only needs to exceed
+
+$$
+\widetilde {O} \left(d ^ {2} / \varepsilon + d ^ {3} / \sqrt {\varepsilon}\right). \tag {25}
+$$
+
+(b) Stability. Turning to the more general case with imperfect score estimates (i.e.,  $\varepsilon_{\mathrm{score}} > 0$ ), the deterministic sampler (23) yields a distribution whose distance to the target distribution (measured again by the TV distance) scales proportionally with  $\varepsilon_{\mathrm{score}}$  and  $\varepsilon_{\mathrm{Jacobi}}$ . It is noteworthy that in addition to the score estimation errors, we are in need of an assumption on the stability of the associated Jacobian matrices, which plays a pivotal in ensuring that the reverse-time deterministic process does not deviate considerably from the desired process.  
+(c) Insufficiency of the score estimation error assumption alone. The careful reader might wonder why we are in need of additional assumptions beyond the  $\ell_2$  score error stated in Assumption 1. To answer this question, we find it helpful to look at a simple example below.
+
+- Example. Consider the case where  $X_0 \sim \mathcal{N}(0,1)$ , and hence  $X_1 \sim \mathcal{N}(0,1)$ . Suppose that the reverse process for time  $t = 2$  can lead to the desired distribution if exact score function is employed, namely,
+
+$$
+Y _ {1} ^ {\star} := \frac {1}{\sqrt {\alpha_ {2}}} \left(Y _ {2} - \frac {1 - \alpha_ {2}}{2} s _ {2} ^ {\star} (Y _ {2})\right) \sim \mathcal {N} (0, 1).
+$$
+
+Now, suppose that the score estimate  $s_2(\cdot)$  we have available obeys
+
+$$
+s _ {2} \left(y _ {2}\right) = s _ {2} ^ {\star} \left(y _ {2}\right) + \frac {2 \sqrt {\alpha_ {2}}}{1 - \alpha_ {2}} \left\{y _ {1} ^ {\star} - L \left\lfloor \frac {y _ {1} ^ {\star}}{L} \right\rfloor \right\} \quad \text {w i t h} y _ {1} ^ {\star} := \frac {1}{\sqrt {\alpha_ {2}}} \left(y _ {2} - \frac {1 - \alpha_ {2}}{2} s _ {2} ^ {\star} \left(y _ {2}\right)\right)
+$$
+
+for  $L > 0$ , where  $\lfloor z \rfloor$  is the greatest integer not exceeding  $z$ . It follows that
+
+$$
+Y _ {1} = Y _ {1} ^ {\star} + \frac {1 - \alpha_ {2}}{2 \sqrt {\alpha_ {2}}} \left[ s _ {2} ^ {\star} (Y _ {2}) - s _ {2} (Y _ {2}) \right] = L \left\lfloor \frac {Y _ {1} ^ {\star}}{L} \right\rfloor .
+$$
+
+Clearly, the score error  $\mathbb{E}_{X_2\sim \mathcal{N}(0,1)}\bigl [|s_2(X_2) - s_2^\star (X_2)|^2\bigr ]$  can be made arbitrarily small by taking  $L\to 0$ . However, the discrete nature of  $Y_{1}$  forces  $\mathsf{T}\mathsf{V}(Y_1,X_1) = 1$ .
+
+This example demonstrates that, for the deterministic sampler, the TV distance between  $Y_{1}$  and  $X_{1}$  might not improve as the score error decreases. If we wish to relax Assumption 2, one potential way is to resort to other metrics (e.g., Wasserstein distance) instead of TV distance between  $Y_{1}$  and  $X_{1}$ .
+
+(d) Relaxing the boundedness assumption on  $X_0$ . As it turns out, the assumption (21) can also be relaxed. Supposing that  $\mathbb{P}\big(\|X_0\|_2 \leq B \mid X_0 \sim p_{\mathrm{data}}\big) = 1$  for some quantity  $B > 0$  (which is allowed to grow faster than a polynomial in  $T$ ), we can readily extend our analysis to obtain
+
+$$
+\mathrm {T V} \left(q _ {1}, p _ {1}\right) \leq C _ {1} \frac {d ^ {2} \log^ {4} T \log^ {2} B}{T} + C _ {1} \frac {d ^ {6} \log^ {6} T \log^ {3} B}{T ^ {2}} + C _ {1} \sqrt {d \log^ {3} T \log B} \varepsilon_ {\text {s c o r e}} + C _ {1} d (\log T) \varepsilon_ {\text {J a c o b i}}.
+$$
+
+Importantly, the convergence rate depends only logarithmically in  $B$ .
+
+Comparisons with past works. To the best of our knowledge, the only non-asymptotic analysis for the discretized probability flow ODE approach in prior literature was derived by a very recent work Chen et al. (2023c), which established the first non-asymptotic convergence guarantees that exhibit
+
+polynomial dependency in both  $d$  and  $1 / \varepsilon$  (see, e.g., Chen et al. (2023c, Theorem 4.1)). However, it fell short of providing concrete polynomial dependency in  $d$  and  $1 / \varepsilon$ , suffered from exponential dependency in the Lipschitz constant of the score function, and relied on exact score estimates. In contrast, our result in Theorem 1 uncovers a concrete  $d^2 /\varepsilon$  scaling (ignoring lower-order and logarithmic terms) without imposing any smoothness assumption on the target data distribution, and makes explicit the effect of score estimation errors, both which were previously unavailable for such discrete-time deterministic samplers. Another recent work Benton et al. (2023b) studied the convergence of the probability flow ODE approach without accounting for the discretization error; the result therein also exhibited exponential dependency on the Lipschitz constant. Finally, while we were wrapping up the current paper, we became aware of the independent work Chen et al. (2023b) establishing improved polynomial dependency for two variants of the probability flow ODE. By inserting an additional stochastic corrector step — based on overdamped (resp. underdamped) Langevin diffusion — in each iteration of the probability flow ODE (so strictly speaking, these variations are no longer deterministic samplers), Chen et al. (2023b) showed that  $\widetilde{O} (L^{3}d / \varepsilon^{2})$  (resp.  $\widetilde{O} (L^2\sqrt{d} /\varepsilon)$ ) steps are sufficient, where  $L$  denotes the Lipschitz constant of the score function. In comparison, our result demonstrates for the first time that the plain probability flow ODE already achieves the  $1 / \varepsilon$  scaling without requiring either a corrector step; one limitation of our result, however, is the sub-optimal  $d$ -dependency compared to the variants studied in Chen et al. (2023b).
+
+# 3.3 A DDPM-TYPE STOCHASTIC SAMPLER
+
+Armed with the score estimates  $\{s_t\}$ , we can readily introduce the following stochastic sampler that operates in discrete time, motivated by the reverse-time SDE (16):
+
+$$
+Y _ {T} \sim \mathcal {N} \left(0, I _ {d}\right), \quad Y _ {t - 1} = \Psi_ {t} \left(Y _ {t}, Z _ {t}\right) \quad \text {f o r} t = T, \dots , 1 \tag {26a}
+$$
+
+where  $Z_{t}\stackrel {\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,I_{d})$  , and
+
+$$
+\Psi_ {t} (y, z) = \frac {1}{\sqrt {\alpha_ {t}}} \left(y + \left(1 - \alpha_ {t}\right) s _ {t} (y)\right) + \sigma_ {t} z \quad \text {w i t h} \sigma_ {t} ^ {2} = \frac {1}{\alpha_ {t}} - 1. \tag {26b}
+$$
+
+The key difference between this sampler and the deterministic sampler (23) is that: (i) there exists an additional pre-factor of  $1/2$  on  $s_t$  in the deterministic sampler; and (ii) the stochastic sampler injects additional noise  $Z_t$  in each step.
+
+In contrast to deterministic samplers, the stochastic samplers have received more theoretical attention, with the state-of-the-art results established by Chen et al. (2022b;a) as well as a very recent paper Benton et al. (2023a). The elementary approach developed in the current paper is also applicable towards understanding this type of samplers, leading to the following non-asymptotic theory.
+
+Theorem 2. Suppose (21) holds true. Equipped with the estimates in Assumption 1 and the learning rate schedule (22), the stochastic sampler (26) achieves, for some universal constants  $C_1 > 0$ ,
+
+$$
+\mathrm {T V} \left(q _ {1}, p _ {1}\right) \leq \sqrt {\frac {1}{2} \mathrm {K L} \left(q _ {1} \| p _ {1}\right)} \leq C _ {1} \frac {d ^ {2} \log^ {3} T}{\sqrt {T}} + C _ {1} \sqrt {d} \varepsilon_ {\text {s c o r e}} \log^ {2} T. \tag {27}
+$$
+
+Theorem 2 establishes non-asymptotic convergence guarantees for the stochastic sampler (26). As asserted by the theorem, if we have access to perfect score estimates, then the number of steps needed to attain  $\varepsilon$ -accuracy (measured by the TV distance between  $p_1$  and  $q_1$ ) is proportional to  $1 / \varepsilon^2$ , matching the state-of-the-art  $\varepsilon$ -dependency derived in Chen et al. (2022a), albeit exhibiting a worse dimensional dependency. In addition, in the presence of score estimation error, the sampler achieves a TV distance proportional to  $\varepsilon_{\mathrm{score}}$ , again consistent with prior results. Our analysis follows a completely different path compared with the SDE-based approach in Chen et al. (2022a), thus offering complementary interpretations for this important sampler.
+
+# 4 OTHER RELATED WORKS
+
+Theory for SGMs. Early theoretical efforts in understanding the convergence of score-based stochastic samplers suffered from being either not quantitative (De Bortoli et al., 2021; Liu et al., 2022; Pidstrigach, 2022), or the curse of dimensionality (e.g., exponential dependencies in the convergence guarantees) (Block et al., 2020; De Bortoli, 2022). Lee et al. (2022) provided the first
+
+polynomial convergence guarantees with  $L_{2}$ -accurate score estimates, for any smooth distribution satisfying the log-Sobolev inequality. Chen et al. (2022b); Lee et al. (2023); Chen et al. (2022a) subsequently lifted such a stringent data distribution assumption. More concretely, Chen et al. (2022b) accommodated a broad family of data distributions under the premise that the score functions over the entire trajectory of the forward process are Lipschitz; Lee et al. (2023) only required certain smoothness assumptions but came with worse dependence on the problem parameters; and more recent results in Chen et al. (2022a) applied to literally any data distribution with bounded second-order moment. In addition, Wibisono & Yang (2022) also established a convergence theory for score-based generative models, assuming that the error of the score estimator has a bounded moment generating function and that the data distribution satisfies the log-Sobolev inequality. Turning attention to samplers based on the probability flow ODE, Chen et al. (2023c) derived the first non-asymptotic bounds for this type of samplers. Improved convergence guarantees have recently been provided by a concurrent work Chen et al. (2023b), with the assistance of additional corrector steps inerspersed in each iteration of the probability flow ODE. It is worth noting that the corrector steps proposed therein are based on Langevin-type diffusion and inject additive noise, and hence the resulting sampling processes are not deterministic. Additionally, theoretical justifications for DDPM in the context of image in-painting have been developed by Rout et al. (2023). Moreover, convergence results based on the Wasserstein distance have recently been derived as well (Tang, 2023; Benton et al., 2023b), although these results typically exhibit exponential dependency on the Lipschitz constants of the score functions. Theoretical guarantees are also extended to accommodate popular methods like consistency models and diffusion guidance (Li et al., 2024b; Wu et al., 2024).
+
+Score matching. Hyvarinen (2005) showed that the score function can be estimated via integration by parts, a result that was further extended in Hyvarinen (2007). Song et al. (2020b) proposed sliced score matching to tame the computational complexity in high dimension. The consistency of the score matching estimator was studied in Hyvarinen (2005), with asymptotic normality established in Forbes & Lauritzen (2015). Optimizing the score matching loss has been shown to be intimately connected to minimizing upper bounds on the Kullback-Leibler divergence (Song et al., 2021a) and Wasserstein distance (Kwon et al., 2022) between the generated distribution and the target data distribution. From a non-asymptotic perspective, Koehler et al. (2023) studied the statistical efficiency of score matching by connecting it with the isoperimetric properties of the distribution.
+
+Other theory for diffusion models. Oko et al. (2023) studied the approximation and generalization capabilities of diffusion modeling for distribution estimation. Assuming that the data are supported on a low-dimensional linear subspace, Chen et al. (2023a) developed a sample complexity bound for diffusion models. Moreover, Ghimire et al. (2023) adopted a geometric perspective and showed that the forward and backward processes of diffusion models are essentially Wasserstein gradient flows. Recently, the idea of stochastic localization, which is closely related to diffusion models, is adopted to sample from posterior distributions (Montanari & Wu, 2023; El Alaoui et al., 2022), which has been implemented using approximate message passing (Donoho et al., 2009; Li & Wei, 2022).
+
+# 5 DISCUSSION
+
+In this paper, we have developed a new suite of non-asymptotic theory for establishing the convergence and faithfulness of diffusion generative modeling, assuming access to reliable estimates of the (Stein) score functions. Our analysis framework seeks to track the dynamics of the reverse process directly using elementary tools, which eliminates the need to look at the continuous-time limit and invoke the SDE and ODE toolboxes. Only the very minimal assumptions on the target data distribution are imposed. The analysis framework laid out in the current paper might shed light on how to analyze other variants of score-based generative models as well. Moving forward, there are plenty of questions that require in-depth theoretical understanding. For instance, the dimension dependency in our convergence results remains sub-optimal; can we further refine our theory in order to reveal tight dependency in this regard? Can we establish sharp convergence results in terms of the Wasserstein distance, which could sometimes be "closer" to how humans differentiate pictures and might potentially help relax Assumption 2 in the case of deterministic samplers? It would also be of paramount interest to establish end-to-end performance guarantees that take into account both the score learning phase and the sampling phase.
+
+# ACKNOWLEDGEMENTS
+
+Y. Wei is supported in part by the NSF grants DMS-2147546/2015447, CAREER award DMS-2143215, CCF-2106778, and the Google Research Scholar Award. Y. Chen is supported in part by the Alfred P. Sloan Research Fellowship, the Google Research Scholar Award, the AFOSR grant FA9550-22-1-0198, the ONR grant N00014-22-1-2354, and the NSF grants CCF-2221009 and CCF-1907661. Y. Chi is supported in part by the grants ONR N00014-19-1-2404, NSF CCF-2106778, DMS-2134080 and ECCS-2126634.
+
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+# TOWARDS OPTIMAL FEATURE-SHAPING METHODS FOR OUT-OF-DISTRIBUTION DETECTION
+
+Qinyu Zhao $^{1}$ , Ming Xu $^{1}$ , Kartik Gupta $^{2}$ , Akshay Asthana $^{2}$ , Liang Zheng $^{1}$ , Stephen Gould $^{1}$
+
+<sup>1</sup> The Australian National University  
+2 Seeing Machines Ltd
+
+{qinyu.zhao,mingda.xu,liang.zheng,stephen.gould}@anu.edu.au
+
+{kartik.gupta,akshay.asthana}@seeingmachines.com
+
+# ABSTRACT
+
+Feature shaping refers to a family of methods that exhibit state-of-the-art performance for out-of-distribution (OOD) detection. These approaches manipulate the feature representation, typically from the penultimate layer of a pre-trained deep learning model, so as to better differentiate between in-distribution (ID) and OOD samples. However, existing feature-shaping methods usually employ rules manually designed for specific model architectures and OOD datasets, which consequently limit their generalization ability. To address this gap, we first formulate an abstract optimization framework for studying feature-shaping methods. We then propose a concrete reduction of the framework with a simple piecewise constant shaping function and show that existing feature-shaping methods approximate the optimal solution to the concrete optimization problem. Further, assuming that OOD data is inaccessible, we propose a formulation that yields a closed-form solution for the piecewise constant shaping function, utilizing solely the ID data. Through extensive experiments, we show that the feature-shaping function optimized by our method improves the generalization ability of OOD detection across a large variety of datasets and model architectures. $^{1}$
+
+# 1 INTRODUCTION
+
+Out-of-distribution (OOD) detection aims to identify test samples that fall outside the inherent training label space, given a deep learning model pre-trained on an in-distribution (ID) training set. To detect OOD samples, OOD scores, such as maximum softmax probability (MSP) (Hendrycks & Gimpel, 2016) and energy score (Liu et al., 2020) are computed using the logits estimated by the model, where a lower score indicates a higher probability that the sample is OOD.
+
+Feature shaping (Sun et al., 2021; Djurisic et al., 2022; Xu & Lian, 2023; Song et al., 2022) refers to a family of methods that manipulate the underlying feature representations, typically from the penultimate layer of a pre-trained model, such that OOD scores can more effectively distinguish between ID and OOD data. Representative methods like ReAct (Sun et al., 2021) and ASH-P (Djurisic et al., 2022) employ element-wise feature clipping and percentile-based feature pruning, respectively.
+
+Despite their notable achievements, existing feature-shaping methods usually rely on rules manually designed for specific OOD datasets and model architectures, which consequently limit their generalization ability. Our observations indicate that variations in OOD datasets and model architectures can cause significant performance degradation in these methods. For instance, when using transformer-based models, some approaches, despite achieving state-of-the-art results on established benchmarks, perform no better than random guessing. These observations emphasize the necessity for a comprehensive understanding and comparison of the current feature-shaping methods and the development of a more generalizable approach.
+
+To this end, we first formulate an abstract optimization problem that encapsulates all feature-shaping OOD detection methods. In a nutshell, our formulation entails finding the shaping function that maximizes some performance objective for OOD detection.
+
+![](images/f60a8df87d00c26bf0e3e45218f194a2cca4158eddef298fd2fe450cc94b5f57.jpg)  
+(a)
+
+![](images/43e61c1b536dad0b2ccc07defffb285c7e0e616be047b18d54055d3322b9edf8.jpg)  
+(b)
+
+![](images/3e8a4441cce90811dea16c0b0429acf9acbc615fee7c206c9cc0a2df703014b4.jpg)  
+(c)  
+Figure 1: Comparing our method with existing feature-shaping methods. The dashed lines denote the performance of our method for comparison. (a) ImageNet (ID) vs. iNaturalist (OOD) with ViT-B-16; (b) ImageNet (ID) vs. iNaturalist (OOD) with MLP-Mixer-B; (c) CIFAR100 (ID) vs. CIFAR10 (OOD) with MLP-Mixer-Nano; (d) Average performance of different methods across eight OOD datasets with two ConvNets and with four transformer-based models.
+
+![](images/af6aa32774c2f4aaeb94b3ffdece640e26fe99427ffbc4a270f252fb5932ba22.jpg)  
+(d)
+
+To make the optimization problem concrete and tractable, we propose a piecewise constant shaping function and formulate an objective function that maximizes the expected difference between the maximum logit values for ID and OOD data, assuming the availability of OOD training samples. An important benefit of this formulation is that it highlights the operational mechanisms underpinning previous feature-shaping methods. Namely, the shaping functions used by existing feature-shaping methods including ReAct (Sun et al., 2021), BFAct (Kong & Li, 2023), and VRA-P (Xu & Lian, 2023), approximate the optimal shaping function found by our approach.
+
+Using our framework, we further propose a feature-shaping method that can be optimized without access to OOD data. Specifically, through the analysis of the effect of OOD data on the optimal solution compared to ID data, we argue that it is feasible to remove the OOD-related term in the objective function, thereby deriving a closed-form solution based on ID data only. Empirical observations in experiments support our claim.
+
+Our experiments compare our proposed approach to existing feature-shaping methods, utilizing a more comprehensive experimental setup than previous works. We use CIFAR10, CIFAR100 (Krizhevsky et al., 2009), and ImageNet-1k (Russakovsky et al., 2015) as ID datasets. For each ID dataset, we use eight OOD datasets for evaluation. Furthermore, we evaluate all methods using a wide variety of models, including convolutional networks (ConvNets), transformers and fully multi-layer perceptron (MLP) models, the latter two of which have not been extensively studied in prior works. As shown in Fig. 1, compared with other feature-shaping methods, ours demonstrates more generalized OOD detection performance across a broad spectrum of datasets and model architectures.
+
+In summary, this paper covers three main points. First, we develop a general optimization formulation for feature-shaping methods and show that a concrete instance of this problem provides insight into the mechanisms of previous methods. Second, we propose a novel approach for training a generalizable feature-shaping method for OOD detection without access to OOD data. Last, our approach achieves more reliable OOD detection performance across various datasets and models.
+
+# 2 A RECAP OF FEATURE SHAPING FOR OOD DETECTION
+
+In this section, we define the problem of OOD detection for image classification, with particular emphasis on feature-shaping approaches, and subsequently, introduce our general optimization formulation of feature shaping for OOD detection.
+
+# 2.1 OOD DETECTION FOR IMAGE CLASSIFICATION
+
+Assume that we have an ID dataset, defined as a joint distribution  $D_{\mathcal{X},\mathcal{Y}}^{\mathrm{ID}}$  over input observations  $\mathcal{X}$  and discrete labels  $\mathcal{Y}$  for a given model  $\mathcal{M}$ . Furthermore, assume we have an OOD dataset  $D_{\mathcal{X},\mathcal{Y}}^{\mathrm{OOD}}$ . Previous works (Sun et al., 2021; Sun & Li, 2022; Yang et al., 2021) assume  $\mathcal{Y} \cap \mathcal{Y}' = \emptyset$ . Inherent in this assumption is that the marginal distributions over observations  $\mathcal{X}$  for ID and OOD datasets do not coincide. That is,  $\mathcal{D}_{\mathcal{X}}^{\mathrm{ID}} \neq \mathcal{D}_{\mathcal{X}}^{\mathrm{OOD}}$ .
+
+The model, usually a neural network, maps observations to logits,  $\mathcal{M}:\mathcal{X}\to \mathbb{R}^C$ , where  $C = |\mathcal{V}|$  denotes the number of classes. It is trained on a dataset  $S_{\mathrm{train}} = \{(x^{(i)},y^{(i)})\}_{i = 1}^{N}$ , comprised of  $N$  i.i.d. samples drawn from  $D_{\mathcal{X},\mathcal{Y}}^{\mathrm{ID}}$ . Furthermore, for typical architectures for image classification, we can decompose the model  $\mathcal{M} = \mathcal{W}\circ \mathcal{B}$  into a backbone model,  $\mathcal{B}:\mathcal{X}\rightarrow \mathbb{R}^{M}$ , and linear classifier  $\mathcal{W}:\mathbb{R}^M\to \mathbb{R}^C$ .  $\mathcal{B}$  extracts penultimate layer feature representation  $z\in \mathbb{R}^{M}$  of an input observation  $\pmb{x}$ , and  $\mathcal{W}$  is a linear layer that maps  $\pmb{z}$  to logits  $\ell \in \mathbb{R}^C$ . Different from conventional machine learning practice, the test set  $S_{\mathrm{test}}$  contains inputs sampled from the joint distribution  $D_{\mathcal{X}}^{\mathrm{ID}}\cup D_{\mathcal{X}}^{\mathrm{OOD}}$ , where  $D_{\mathcal{X}}^{\mathrm{ID}}$  and  $D_{\mathcal{X}}^{\mathrm{OOD}}$  denote the marginal distributions of  $D_{\mathcal{X},\mathcal{Y}}^{\mathrm{ID}}$  and  $D_{\mathcal{X},\mathcal{Y}}^{\mathrm{OOD}}$ , respectively.
+
+The goal of OOD detection is to design a score function  $s$  to indicate whether a sample  $x$  is from  $D_{\mathcal{X}}^{\mathrm{ID}}$  or  $D_{\mathcal{X}}^{\mathrm{OOD}}$ . MSP, maximum logit score (MLS), and energy score serve as examples of scores for OOD detection. They are defined based on logits  $\ell$  as:
+
+$$
+s _ {\mathrm {M S P}} (\ell) = \max  _ {i = 1, \dots , C} \frac {\exp \left(\ell_ {i}\right)}{\sum_ {j = 1} ^ {C} \exp \left(\ell_ {j}\right)}, \quad s _ {\mathrm {M L S}} (\ell) = \max  _ {i = 1, \dots , C} \ell_ {i}, \quad s _ {\text {E n e r g y}} (\ell) = \log \sum_ {j = 1} ^ {C} \exp \left(\ell_ {j}\right), \tag {1}
+$$
+
+where  $\ell_{i}$  is the logit for the  $i$ -th class. For the scores discussed above, a lower value indicates a higher likelihood of the sample being OOD. In practical applications, a threshold, denoted by  $t$ , is utilized to separate ID samples from OOD ones. If the score of a sample surpasses this threshold, the sample will be classified as ID, and OOD otherwise.
+
+# 2.2 FEATURE-SHAPING
+
+A feature-shaping method uses a shaping function  $\pmb{f} : \mathbb{R}^{M} \rightarrow \mathbb{R}^{M}$  to manipulate features, i.e.,  $\bar{z} = \pmb{f}(z)$ . The goal of developing a feature shaping method for OOD detection can be formulated as solving the following optimization problem,
+
+$$
+\text {m a x i m i z e} (\text {o v e r} \boldsymbol {f}) \quad \mathbf {E} _ {\boldsymbol {x} ^ {\mathrm {I D}} \sim D _ {\chi} ^ {\mathrm {I D}}, \boldsymbol {x} ^ {\mathrm {O O D}} \sim D _ {\chi} ^ {\mathrm {O O D}}} \left[ \mathcal {P} \left\{s \left(\bar {\boldsymbol {\ell}} ^ {\mathrm {I D}}\right), s \left(\bar {\boldsymbol {\ell}} ^ {\mathrm {O O D}}\right) \right\} \right],
+$$
+
+$$
+\text {s u b j e c t} \quad \bar {\ell} ^ {\mathrm {I D}} = \mathcal {W} (\bar {z} ^ {\mathrm {I D}}), \quad \bar {z} ^ {\mathrm {I D}} = \boldsymbol {f} (\mathcal {B} (\boldsymbol {x} ^ {\mathrm {I D}})), \tag {2}
+$$
+
+$$
+\bar {\ell} ^ {\mathrm {O O D}} = \mathcal {W} (\bar {z} ^ {\mathrm {O O D}}), \quad \bar {z} ^ {\mathrm {O O D}} = f (\mathcal {B} (x ^ {\mathrm {O O D}})),
+$$
+
+where  $\mathcal{P}$  is a given performance measure. For example, a common measure is the area under the receiver operating characteristic curve (AUC), defined as  $\mathcal{P}_{\mathrm{AUC}}(s_1,s_2) = \mathbb{1}\left\{s_1 > s_2\right\}$ . Existing shaping functions  $f$  can be broadly categorized into two types: (i) element-wise, such as ReAct (Sun et al., 2021) and VRA-P (Xu & Lian, 2023), which applies a global element-wise mapping to each individual feature, i.e.,  $\bar{z}_i = f(z_i)$ , or (ii) whole-vector mappings such as ASH (Djurisic et al., 2022), which manipulate the full feature vector  $z$ .
+
+# 3 OPTIMAL FEATURE SHAPING
+
+In this section, we propose a specific instance of the general formulation described in Section 2, following previous element-wise feature-shaping methods (Sun et al., 2021; Xu & Lian, 2023; Kong & Li, 2023). To optimize the shaping function, we initially examine which feature value ranges are effective for OOD detection. A significant challenge arises from the continuous nature of feature values. To address this, we partition the domain of features into a disjoint set of intervals, which allows us to approximate the optimal shaping function with a piece-wise constant function. With this approximation, we can formulate an optimization problem to maximize the expected difference between the maximum logit values of ID and OOD data. Subsequently, we show that the optimized shaping function using OOD samples bears a striking resemblance to prior methods, shedding light on the mechanics of these methods. Finally, we propose a formulation that does not require any OOD data to optimize and admits a simple, closed-form solution.
+
+# 3.1 VALUE- AND INTERVAL-SPECIFIC FEATURE IMPACT
+
+Given a sample  $\mathbf{x}$ , the maximum logit  $\ell^{\max}$  (disregarding the bias term) is the dot product between the corresponding weight vector  $\boldsymbol{w}^{\mathrm{max}}$  and its feature representation  $\mathbf{z}$ , specifically,
+
+$$
+\ell^ {\max } (\boldsymbol {z}) = \boldsymbol {w} ^ {\max } \cdot \boldsymbol {z} = \sum_ {i = 1} ^ {M} w _ {i} ^ {\max } z _ {i}. \tag {3}
+$$
+
+Assuming an element-wise feature-shaping function  $f: \mathbb{R} \to \mathbb{R}$ , define  $\theta(z_i) \triangleq \frac{f(z_i)}{z_i}$ . Then assuming the weights  $\boldsymbol{w}^{\max}$  do not change, the maximum logit after shaping is:
+
+$$
+\bar {\ell} ^ {\max } (\bar {z}) = \sum_ {i = 1} ^ {M} w _ {i} ^ {\max } f (z _ {i}) = \sum_ {i = 1} ^ {M} \theta (z _ {i}) w _ {i} ^ {\max } z _ {i}. \tag {4}
+$$
+
+To design  $\theta(z)$  and understand which feature value ranges are effective for OOD detection, we compute the contribution of features with specific values to the maximum logit. Concretely, we define the value-specific feature impact (VSFI):  $V(z) = \sum_{i=1}^{M} \mathbb{1}\{z_i = z\} w_i^{\max} z_i$ . However,  $z$  is a continuous variable, and real-world data only provide a countable number of features, which can give overly sparse distributions of VSFs. To address this, we partition the domain of individual features into a disjoint set of intervals  $\mathcal{A}$  that cover  $z_i$ . Formally, the interval-specific feature impact (ISFI) on an interval  $[a, b) \in \mathcal{A}$  is given by:
+
+$$
+I _ {[ a, b)} (\boldsymbol {z}) = \sum_ {i = 1} ^ {M} \mathbb {1} \left\{a \leq z _ {i} <   b \right\} w _ {i} ^ {\max } z _ {i} = \sum_ {i: a \leq z _ {i} <   b} w _ {i} ^ {\max } z _ {i}. \tag {5}
+$$
+
+Then, for a single sample, the maximum logit is the summation over all ISFIs, given by:
+
+$$
+\ell^ {\max } (\boldsymbol {z}) = \sum_ {[ a, b) \in \mathcal {A}} I _ {[ a, b)} (\boldsymbol {z}). \tag {6}
+$$
+
+In our experiments, we divide feature values into equal-width intervals. Given that feature distributions typically exhibit long-tailed characteristics (Sun et al., 2021), we select the 0.1 and 99.9 percentiles of all features in a training set as the lower  $\alpha$  and upper  $\beta$  limits of feature values, respectively. We then partition the range  $[\alpha, \beta)$  into  $K = 100$  equal-width intervals. This results in a set of intervals  $\mathcal{A} = \{\left[a_k, b_k\right]\}_{k=1}^K$ , where  $a_k = \alpha + (k-1)\delta$ ,  $b_k = \alpha + k\delta$ , and  $\delta = \frac{\beta - \alpha}{K}$  for  $k = 1, \ldots, K$ . Then, the maximum logit can be expressed as:
+
+$$
+\ell^ {\max } (\boldsymbol {z}) = \sum_ {[ a, b) \in \mathcal {A}} I _ {[ a, b)} (\boldsymbol {z}) = \sum_ {k = 1} ^ {K} I _ {[ a _ {k}, b _ {k})} (\boldsymbol {z}), \tag {7}
+$$
+
+and for each data sample, we can generate a  $K$ -dim ISFI vector denoted as:
+
+$$
+\boldsymbol {I} (\boldsymbol {z}) = \left(I _ {[ a _ {1}, b _ {1})} (\boldsymbol {z}), \dots , I _ {[ a _ {K}, b _ {K})} (\boldsymbol {z})\right) \in \mathbb {R} ^ {K}. \tag {8}
+$$
+
+# 3.2 OPTIMIZING THE RESHAPING FUNCTION
+
+We now describe how to optimize  $\theta(z)$ , assuming  $\theta(z)$  is piecewise constant over the intervals  $\mathcal{A}$ . For an interval  $[a_k, b_k)$ , we estimate a scalar  $\hat{\theta}_k$  as a constant approximation to  $\theta(z)$  within the interval. In the limit as the interval widths shrinks to zero, we recover the continuous, element-wise shaping function. We can approximate the maximum logit as,
+
+$$
+\bar {\ell} ^ {\max } (\bar {z}) = \sum_ {k = 1} ^ {K} \hat {\theta} _ {k} I _ {\left(a _ {k}, b _ {k}\right)} (z). \tag {9}
+$$
+
+While previous works (Sun et al., 2021; Xu & Lian, 2023) evaluate their methods by computing the expected difference in maximum logits between ID and OOD data, optimizing this criteria solely does not improve OOD detection performance. To see this, observe that simply uniformly scaling features by a large factor increases the expected difference in maximum logits without enhancing the differentiation between ID and OOD samples. This motivates introducing regularization on the shaping function and leads us to the following problem<sup>3</sup>:
+
+$$
+\text {m a x i m i z e} (\text {o v e r} \theta) \quad \mathbf {E} _ {\boldsymbol {x} ^ {\mathrm {I D}} \sim D _ {\chi} ^ {\mathrm {I D}}, \underline {{\boldsymbol {x}}} ^ {\mathrm {O O D}} \sim D _ {\chi} ^ {\mathrm {O O D}}} \left[ \boldsymbol {\theta} ^ {\mathrm {T}} \boldsymbol {I} \left(\boldsymbol {z} ^ {\mathrm {I D}}\right) - \boldsymbol {\theta} ^ {\mathrm {T}} \boldsymbol {I} \left(\boldsymbol {z} ^ {\mathrm {O O D}}\right) \right] \tag {10}
+$$
+
+$$
+\text {s u b j e c t} \quad \| \boldsymbol {\theta} \| = \sqrt {K}, \quad \boldsymbol {z} ^ {\mathrm {I D}} = \mathcal {B} (\boldsymbol {x} ^ {\mathrm {I D}}), \quad \boldsymbol {z} ^ {\mathrm {O O D}} = \mathcal {B} (\boldsymbol {x} ^ {\mathrm {O O D}}),
+$$
+
+![](images/71a3815296c6a103cdea19866e9374a7bb46ffb681eebd105c901d33a7c3371d.jpg)
+
+![](images/9a80d1e97aff3fea335c6938cd9b434971313348cd3ff054bacae4e38c16e870.jpg)
+
+![](images/aabd169b6ce78f4ba98e8561bc531d41334b9bfb5855dda8a25367b6bddcf2f1.jpg)
+
+![](images/48d090d2e4de9f2c9bac2bd128cb77af4d5b3ea68f4ca4d50faae3db6a406b09.jpg)
+
+![](images/d558b30defee4409b881cc6814a83bd578ea66d4ffbe211f32f16bb96191ed7b.jpg)  
+(a)  
+Feature value Z  
+(e)  
+Figure 2: Visualization of shaping functions. The blue lines (ours w/o OOD) derive from Eq. 10, while the green line (ours w/o OOD) from Eq. 14. Red lines represent different existing methods, while shaded regions indicate estimated standard deviations.  $\theta$  has been rescaled for the best visualization.
+
+![](images/b3d8776a6c13db5a520c599f2bce3291ad93babd8c2e76e6e030fb0dfc7c8809.jpg)  
+(b)  
+Feature value Z  
+(f)
+
+![](images/be58bb8ab244ad7a00c7513d1f107ae89ed182511046340e5123db67faa4112a.jpg)  
+(c)  
+Feature value Z  
+(g)
+
+![](images/e7d01dceae230a8ab91386e0ba6ed1b70032e4fcaabb104de94220b9487c6919.jpg)  
+(d)  
+Feature value Z  
+(h)
+
+where due to the linearity of expectation, the objective function can be expressed as:
+
+$$
+\boldsymbol {\theta} ^ {\mathrm {T}} \left(\mathbf {E} _ {\boldsymbol {x} ^ {\mathrm {I D}} \sim D _ {\mathcal {X}} ^ {\mathrm {I D}}} \left[ \boldsymbol {I} \left(\boldsymbol {z} ^ {\mathrm {I D}}\right) \right] - \mathbf {E} _ {\boldsymbol {x} ^ {\mathrm {O O D}} \sim D _ {\mathcal {X}} ^ {\mathrm {O O D}}} \left[ \boldsymbol {I} \left(\boldsymbol {z} ^ {\mathrm {O O D}}\right) \right]\right). \tag {11}
+$$
+
+We extract ISFI vectors for the ImageNet-1k training set (ID) (Russakovsky et al., 2015) and for the iNaturalist dataset (OOD) (Van Horn et al., 2018), using a ResNet50 model pre-trained on ImageNet-1k training set. The estimated  $\hat{\theta}_k$  is plotted in Fig. 2.
+
+For completeness, we describe existing feature-shaping methods (Sun et al., 2021; Kong & Li, 2023; Xu & Lian, 2023) using our reparameterization of the feature shaping problem with  $\theta(z)$ ,
+
+$$
+\theta_ {\text {R e A c t}} (z) = \frac {\min  (z , t)}{z}, \quad \theta_ {\text {B F A c t}} (z) = \frac {1}{\sqrt {1 + (\frac {z}{t}) ^ {2 N}}}, \quad \theta_ {\text {V R A - P}} (z) = \left\{ \begin{array}{l l} 0 & z <   \text {l o w} \\ 1 & \text {l o w} \leq z <   \text {h i g h} \\ \frac {\text {h i g h}}{z} & \text {h i g h} \leq z, \end{array} \right. \tag {12}
+$$
+
+where  $t$ ,  $N$ , low, and high are hyperparameters. We then plot the resultant  $\theta(z)$  in Fig. 2(a-c). We show that these prior works are empirically approximating our optimal piecewise constant shaping function. A difference between our shaping function and VRA-P, is that VRA-P sets low-value features to zero while ours flips the sign of these features to better utilize them for OOD detection.
+
+Then, we extend our analysis to other feature-shaping methods. ASH-P, -B, -S, and -Rand denote four simplistic yet effective methods introduced by Djurisic et al. (2022). In contrast to ReAct, which employs a global threshold to clip features, ASH methods incorporate the entire feature vector and utilize a local threshold to process the feature representation of each instance. Their shaping functions are not reducible to element-wise functions.
+
+We instead approximate the average shaping effect within each feature value interval. Specifically, we utilize ImageNet-1k validation set (ID) and iNaturalist (OOD) with ResNet50. For an interval  $[a_k, b_k)$ , we retrieve all features from ID and OOD datasets that fall within this range, along with their reshaped counterparts. Following this, the ratio of reshaped and original values are computed as  $\theta_k$  for a specific data instance. Subsequently, we calculate the mean value and standard deviation of these  $\theta_k$  values over all samples on each interval. The results are depicted in Fig. 2(e-h).
+
+# 3.3 OPTIMIZING THE SHAPING FUNCTION WITHOUT OOD SAMPLES
+
+![](images/05e99c0416ed388bed49a14812c513276ace9d5e1c92f8e2843924fefde7b778.jpg)  
+Figure 3: Diagram to show the intuition in deriving Eq. 13.
+
+Normally, we have access to the ID training set, enabling us to estimate  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{ID}}\sim D_{\mathcal{X}}^{\mathrm{ID}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{ID}})\right]$ . However, a significant challenge arises with the unknown term  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{OOD}}\sim D_{\mathcal{X}}^{\mathrm{OOD}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{OOD}})\right]$ , which renders the optimization problem defined in Eq. 10 intractable.
+
+To make the problem tractable without access to OOD data, we omit the OOD-related term  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{OOD}}\sim D_X^{\mathrm{OOD}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{OOD}})\right]$  by the following argument, which as we will see is also supported empirically in Section 4.2.
+
+A diagram to explain our intuition is shown in Fig. 3. Specifically, in scenarios where the OOD distribution closely resembles the ID (hard OOD), OOD samples likely have similar feature representations to ID samples, implying both expectations share similar directions. Conversely, in cases where OOD samples significantly diverge from
+
+ID samples (easy OOD), the maximum logits of OOD samples tend to be lower, thereby reducing the magnitude of  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{OOD}}\sim D_{\mathcal{X}}^{\mathrm{OOD}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{OOD}})\right]$  compared to ID.
+
+Thus, the component of  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{OD}}\sim D_{\mathcal{X}}^{\mathrm{OD}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{OD}})\right]$  orthogonal to ID expectation is small and we suggest omitting  $\mathbf{E}_{\boldsymbol{x}^{\mathrm{OD}}\sim D_{\mathcal{X}}^{\mathrm{OD}}}\left[\boldsymbol {I}(\boldsymbol{z}^{\mathrm{OD}})\right]$ , resulting in the following formulation:
+
+$$
+\begin{array}{l l} \text {m a x i m i z e (o v e r} \boldsymbol {\theta}) & \boldsymbol {\theta} ^ {\mathrm {T}} \mathbf {E} _ {\boldsymbol {x} \sim D _ {\boldsymbol {x}} ^ {\mathrm {I D}}} [ \boldsymbol {I} (\boldsymbol {z}) ] \\ \text {s u b j e c t t o} & \| \boldsymbol {\theta} \| = \sqrt {K}, \quad \boldsymbol {z} = \mathcal {B} (\boldsymbol {x}). \end{array} \tag {13}
+$$
+
+The optimal solution to this new problem can be expressed in closed-form as:
+
+$$
+\boldsymbol {\theta} ^ {\star} = \frac {\sqrt {K}}{\left\| \mathbf {E} _ {\boldsymbol {x} \sim D _ {\chi} ^ {\mathrm {I D}}} [ \boldsymbol {I} (\boldsymbol {z}) ] \right\| _ {2}} \mathbf {E} _ {\boldsymbol {x} \sim D _ {\chi} ^ {\mathrm {I D}}} [ \boldsymbol {I} (\boldsymbol {z}) ]. \tag {14}
+$$
+
+We plot  $\theta^{\star}$  calculated by Eq. 14 in Fig. 2(d), noting that it is in close proximity to the optimal solution in the case where OOD samples are available. In experiments, we will also show that our method with only ID data performs almost on par with using the true  $I(z^{\mathrm{OOD}})$  calculated from OOD data.
+
+At inference time, for each sample, we rescale its features within each interval  $[a_k, b_k)$  by the corresponding entry in  $\theta^{\star}$ , and calculate an OOD score based on the reshaped feature representation.
+
+# 4 EXPERIMENTS
+
+Datasets. We evaluate our proposed method using a large scale benchmark where ImageNet-1k (Russakovsky et al., 2015) is the ID dataset, as well as two moderate scale benchmarks where CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) are used as the ID datasets. We evaluate eight OOD datasets for the ImageNet benchmark, comprised of Species (Hendrycks et al., 2019b), iNaturalist (Van Horn et al., 2018), SUN (Xiao et al., 2010), Places (Zhou et al., 2017), OpenImage-O (Wang et al., 2022), ImageNet-O (Hendrycks et al., 2021), Texture (Cimpoi et al., 2014), and MNIST (Deng, 2012). When CIFAR10/100 is the ID dataset, eight OOD datasets are exploited, comprised of TinyImageNet (Torralba et al., 2008), SVHN (Netzer et al., 2011), Texture (Cimpoi et al., 2014), Places365 (Zhou et al., 2017), LSUN-Cropped (Yu et al., 2015), LSUN-Resized (Yu et al., 2015), iSUN (Xu et al., 2015), and finally CIFAR100/10.
+
+Model architectures. We use pre-trained models provided by PyTorch and prior works in our experiments. For ImageNet-1k, we use a pre-trained ResNet50 (He et al., 2016) provided by PyTorch and a MobileNet-v2 (Sandler et al., 2018) provided by Djurisic et al. (2022). Additionally, we include models based on the transformer and fully-MLP architectures, which have been largely overlooked in previous studies. Specifically, we include ViT-B-16, ViT-L-16 (Dosovitskiy et al., 2020), SWIN-S, SWIN-B (Liu et al., 2021), MLP-Mixer-B, and MLP-Mixer-L (Tolstikhin et al., 2021). For the CIFAR10 and CIFAR100 benchmarks, we use a DenseNet101 (Huang et al., 2017) model provided by Sun & Li (2022), a ViT-B-16 finetuned on CIFAR10/100 consistent with Fort et al. (2021), and a MLP-Mixer-Nano model trained on CIFAR10/100 from scratch.
+
+Table 1: OOD detection results on ImageNet-1k.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">ResNet50</td><td colspan="2">MobileNetV2</td><td colspan="2">ViT-B-16</td><td colspan="2">ViT-L-16</td><td colspan="2">SWIN-S</td><td colspan="2">SWIN-B</td><td colspan="2">MLP-B</td><td colspan="2">MLP-L</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>69.30</td><td>76.26</td><td>72.58</td><td>77.41</td><td>69.84</td><td>77.40</td><td>70.59</td><td>78.40</td><td>65.12</td><td>80.89</td><td>69.45</td><td>77.26</td><td>70.66</td><td>80.31</td><td>71.63</td><td>79.93</td><td>69.90</td><td>78.48</td></tr><tr><td>\(ODIN^a\)</td><td>61.56</td><td>80.92</td><td>62.91</td><td>82.64</td><td>69.25</td><td>72.60</td><td>70.35</td><td>74.51</td><td>62.35</td><td>78.67</td><td>70.90</td><td>68.36</td><td>65.00</td><td>82.99</td><td>67.90</td><td>81.86</td><td>66.28</td><td>77.82</td></tr><tr><td>Energy</td><td>60.97</td><td>81.01</td><td>61.40</td><td>82.83</td><td>73.96</td><td>67.65</td><td>74.89</td><td>70.11</td><td>66.56</td><td>75.95</td><td>79.53</td><td>60.14</td><td>87.86</td><td>78.42</td><td>84.60</td><td>79.01</td><td>73.72</td><td>74.39</td></tr><tr><td>DICE</td><td>45.32</td><td>83.64</td><td>49.33</td><td>84.63</td><td>89.68</td><td>71.32</td><td>72.38</td><td>67.08</td><td>78.84</td><td>49.06</td><td>77.89</td><td>50.10</td><td>57.12</td><td>80.60</td><td>61.98</td><td>80.75</td><td>66.57</td><td>70.90</td></tr><tr><td>ReAct</td><td>42.29</td><td>86.54</td><td>54.19</td><td>85.37</td><td>73.82</td><td>76.86</td><td>76.16</td><td>81.07</td><td>58.07</td><td>85.10</td><td>70.23</td><td>80.10</td><td>90.13</td><td>77.18</td><td>89.03</td><td>78.32</td><td>69.24</td><td>81.32</td></tr><tr><td>BFAct</td><td>43.87</td><td>86.01</td><td>52.87</td><td>85.78</td><td>77.64</td><td>80.16</td><td>84.02</td><td>81.12</td><td>64.76</td><td>85.57</td><td>68.40</td><td>84.36</td><td>96.04</td><td>67.87</td><td>96.00</td><td>70.59</td><td>72.95</td><td>80.18</td></tr><tr><td>ASH-P</td><td>55.30</td><td>83.00</td><td>59.41</td><td>83.84</td><td>99.36</td><td>21.17</td><td>99.18</td><td>20.27</td><td>99.09</td><td>21.78</td><td>98.83</td><td>22.11</td><td>98.19</td><td>35.85</td><td>98.91</td><td>29.69</td><td>88.53</td><td>39.71</td></tr><tr><td>ASH-B</td><td>35.97</td><td>88.62</td><td>43.59</td><td>88.28</td><td>94.87</td><td>46.68</td><td>93.72</td><td>38.95</td><td>96.48</td><td>36.54</td><td>92.40</td><td>49.85</td><td>99.51</td><td>21.73</td><td>64.91</td><td>83.15</td><td>77.68</td><td>56.73</td></tr><tr><td>ASH-S</td><td>34.70</td><td>90.25</td><td>43.84</td><td>88.24</td><td>99.48</td><td>18.52</td><td>99.42</td><td>18.61</td><td>99.20</td><td>18.26</td><td>99.06</td><td>19.27</td><td>97.61</td><td>33.91</td><td>99.28</td><td>19.29</td><td>84.07</td><td>38.29</td></tr><tr><td>VRA-P</td><td>37.97</td><td>88.58</td><td>49.98</td><td>86.83</td><td>98.39</td><td>35.66</td><td>99.58</td><td>16.70</td><td>99.27</td><td>20.34</td><td>99.46</td><td>17.64</td><td>99.38</td><td>18.73</td><td>99.05</td><td>21.23</td><td>85.39</td><td>38.21</td></tr><tr><td>Ours (V)</td><td>41.84</td><td>88.11</td><td>53.31</td><td>86.36</td><td>69.33</td><td>82.59</td><td>72.17</td><td>83.23</td><td>66.89</td><td>84.53</td><td>65.96</td><td>83.71</td><td>76.59</td><td>78.46</td><td>78.13</td><td>78.85</td><td>65.53</td><td>83.23</td></tr><tr><td>Ours (E)</td><td>39.75</td><td>88.56</td><td>51.77</td><td>86.62</td><td>69.52</td><td>82.66</td><td>78.57</td><td>82.94</td><td>66.28</td><td>84.99</td><td>66.20</td><td>84.21</td><td>83.54</td><td>75.94</td><td>85.50</td><td>78.11</td><td>67.64</td><td>83.00</td></tr></table>
+
+${}^{a}$  Choice of hyperparameters on Imagenet makes ODIN yield identical results to MLS.
+
+Table 2: OOD detection results on CIFAR10/100.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results.  
+
+<table><tr><td rowspan="3">Method</td><td colspan="9">CIFAR10</td><td colspan="7">CIFAR100</td></tr><tr><td colspan="2">DenseNet101</td><td colspan="2">ViT-B-16</td><td colspan="2">MLP-N</td><td colspan="2">Average</td><td colspan="2">DenseNet101</td><td colspan="2">ViT-B-16</td><td colspan="2">MLP-N</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>52.66</td><td>91.42</td><td>25.47</td><td>95.61</td><td>67.01</td><td>82.86</td><td>48.38</td><td>89.96</td><td>80.40</td><td>74.75</td><td>61.24</td><td>85.24</td><td>83.97</td><td>73.07</td><td>75.20</td><td>77.69</td></tr><tr><td>ODIN</td><td>32.84</td><td>91.94</td><td>24.93</td><td>92.84</td><td>69.20</td><td>69.53</td><td>42.32</td><td>84.77</td><td>62.03</td><td>82.57</td><td>56.91</td><td>80.86</td><td>78.71</td><td>66.47</td><td>65.88</td><td>76.63</td></tr><tr><td>MLS</td><td>31.93</td><td>93.51</td><td>12.79</td><td>97.16</td><td>64.50</td><td>82.97</td><td>36.41</td><td>91.21</td><td>70.71</td><td>80.18</td><td>52.31</td><td>88.63</td><td>82.41</td><td>74.52</td><td>68.48</td><td>81.11</td></tr><tr><td>Energy</td><td>31.72</td><td>93.51</td><td>12.40</td><td>97.22</td><td>63.95</td><td>82.84</td><td>36.02</td><td>91.19</td><td>70.80</td><td>80.22</td><td>51.93</td><td>88.82</td><td>79.90</td><td>75.30</td><td>67.54</td><td>81.45</td></tr><tr><td>DICE</td><td>29.67</td><td>93.27</td><td>26.84</td><td>92.41</td><td>96.64</td><td>52.17</td><td>51.05</td><td>79.28</td><td>59.56</td><td>82.26</td><td>95.18</td><td>42.05</td><td>95.78</td><td>44.48</td><td>83.51</td><td>56.26</td></tr><tr><td>ReAct</td><td>82.00</td><td>76.46</td><td>12.33</td><td>97.29</td><td>64.34</td><td>81.85</td><td>52.89</td><td>85.20</td><td>77.00</td><td>78.30</td><td>52.25</td><td>88.84</td><td>79.99</td><td>75.87</td><td>69.75</td><td>81.00</td></tr><tr><td>BFAct</td><td>84.40</td><td>74.39</td><td>12.34</td><td>97.21</td><td>78.02</td><td>72.68</td><td>58.25</td><td>81.43</td><td>80.27</td><td>73.36</td><td>51.23</td><td>89.04</td><td>80.05</td><td>76.58</td><td>70.52</td><td>79.66</td></tr><tr><td>ASH-P</td><td>29.39</td><td>93.98</td><td>20.20</td><td>95.31</td><td>84.39</td><td>66.93</td><td>44.66</td><td>85.41</td><td>68.21</td><td>81.11</td><td>53.69</td><td>87.66</td><td>86.73</td><td>65.27</td><td>69.54</td><td>78.01</td></tr><tr><td>ASH-B</td><td>28.21</td><td>94.27</td><td>82.10</td><td>69.54</td><td>93.93</td><td>53.00</td><td>68.08</td><td>72.27</td><td>57.45</td><td>83.80</td><td>91.74</td><td>56.39</td><td>93.63</td><td>57.20</td><td>80.94</td><td>65.80</td></tr><tr><td>ASH-S</td><td>23.93</td><td>94.41</td><td>39.93</td><td>91.06</td><td>82.57</td><td>68.02</td><td>48.81</td><td>84.50</td><td>52.41</td><td>84.65</td><td>60.44</td><td>84.25</td><td>89.39</td><td>59.63</td><td>67.41</td><td>76.18</td></tr><tr><td>VRA-P</td><td>38.41</td><td>92.77</td><td>13.73</td><td>96.86</td><td>100.00</td><td>65.95</td><td>50.71</td><td>85.19</td><td>67.75</td><td>82.72</td><td>52.21</td><td>88.49</td><td>87.19</td><td>66.03</td><td>69.05</td><td>79.08</td></tr><tr><td>Ours (V)</td><td>30.12</td><td>93.97</td><td>13.13</td><td>97.07</td><td>77.25</td><td>73.86</td><td>40.17</td><td>88.30</td><td>69.01</td><td>81.42</td><td>52.76</td><td>88.60</td><td>83.83</td><td>73.65</td><td>68.53</td><td>81.22</td></tr><tr><td>Ours (E)</td><td>28.90</td><td>94.12</td><td>12.79</td><td>97.14</td><td>83.87</td><td>71.83</td><td>41.85</td><td>87.70</td><td>65.2</td><td>82.39</td><td>51.67</td><td>88.78</td><td>81.33</td><td>74.67</td><td>66.07</td><td>81.95</td></tr></table>
+
+Evaluation metrics. We utilize two standard evaluation metrics, following previous works (Liang et al., 2017; Sun et al., 2021; Sun & Li, 2022): the false positive rate (FPR95) when the true positive rate of the ID samples is  $95\%$ , and AUC.
+
+Implementation details. We evaluate two variants of our method, the vanilla method described in Eq. 14 denoted "Ours (V)", based on the maximum logit score, and a variant based on energy scores, denoted "Ours (E)". For comparison, we implement five baseline methods (top rows in results tables), including MSP (Hendrycks & Gimpel, 2016), ODIN (Liang et al., 2017), MLS (Hendrycks et al., 2019a), Energy (Liu et al., 2020), and DICE (Sun & Li, 2022), and six feature-shaping methods (remaining rows in results tables), including ReAct (Sun et al., 2021), BFAct (Kong & Li, 2023), ASH-P, ASH-B, ASH-S (Djurisic et al., 2022), and VRA-P (Xu & Lian, 2023). All comparison feature-shaping methods are evaluated using the energy score. Hyperparameters are set consistent with the original papers and concrete hyperparameter settings can be found in Section A.1 of the Appendix. All experiments are run on a single NVIDIA GeForce RTX 4090 GPU.
+
+# 4.1 OOD DETECTION BENCHMARK RESULTS
+
+Experimental results on the ImageNet-1k benchmark are shown in Table 1. More detailed results can be found in Sections E and F of the Appendix. We observe that the comparison methods for
+
+![](images/d165aa2966444338b3ccd90ac0a17914eda8aa47743d7db5f0d41b9ee7b793b8.jpg)  
+(a)
+
+![](images/38b78ff2240ea7ea51e2aceea884955e0059d6c374b2e693e08323f052deaf70.jpg)  
+(b)  
+Figure 4: Compatibility and sensitivity analysis. (a-b) Our method can improve other OOD scores and methods. "Base" denotes using the original OOD score or method, while "+"Our" indicates combining the score or method with our feature-shaping function. (c) Our method's performance with different hyperparameter settings, i.e., numbers of intervals  $K$ .
+
+![](images/13089371383b83d082fa8fe8bbaab82bba04c73d4b86b8d0dec1b4fd0d4acbbf.jpg)  
+(c)
+
+feature shaping perform extremely well on the pretrained ResNet-50 model, with some ASH variants outperforming element-wise shaping methods like ReAct (Sun et al., 2021), and yielding similar performance to our proposed methods. This is not surprising, given that ResNet-50 was the model architecture used in the original papers. As we showed in Fig. 2, the shaping functions proposed by these methods closely align with the optimal shaping function learned given actual OOD samples.
+
+However, we observe that these methods do not generalize well to architectures outside of ResNet-50. We observe that for feature-shaping methods more generally, stronger convolutional network architecture performance is negatively correlated to performance on other network architectures. However, our proposed methods yield consistently strong performance across all network architectures, despite only having access to ID data for parameter tuning.
+
+Results for the CIFAR10 and CIFAR100 benchmarks are shown in Table 2. An important observation regarding these results is that feature-shaping methods as a whole seem to underperform the baseline methods. This indicates that feature shaping appears to be ineffective on smaller-scale datasets such as CIFAR. While our methods do not outperform the baselines, we are competitive and do outperform all of the comparison feature shaping methods, which is consistent with the ImageNet-1k benchmark.
+
+# 4.2 FURTHER ANALYSIS
+
+We conduct additional experiments using the ImageNet-1k benchmark and employing ResNet50.
+
+Extension to other OOD scores. While our proposed method in Eq. 14 is designed assuming MLS as the OOD score, we also extend our method to use other OOD scores such as energy and MSP. Furthermore, we combine our method with other OOD detection methods including DICE and ReAct. Results are averaged across the eight OOD datasets. As shown in Fig. 4(a-b), our method yields consistent improvement across these OOD detection scores and methods.
+
+Different numbers of intervals. We evaluate our methods with different number of intervals, i.e., varying  $K$ . As shown in Fig. 4(c), most of the improvement in performance comes from  $K \leq 10$ . This indicates that the optimal feature shaping function has a relatively simple form. This is consistent with prior methods (Sun et al., 2021; Xu & Lian, 2023; Kong & Li, 2023), which assume 2-3 intervals. In addition, the relatively large performance gap over the baseline motivates feature shaping more generally. Interestingly, we observe an extremely large jump between  $K = 0$  ("Energy" baseline) with  $K = 1$ , which effectively prunes the top and bottom  $0.1\%$  of feature values (using the thresholds computed based on the ID training data).
+
+Empirical validation of Problem 13. In deriving Eq. 13, we remove the OOD-related term, assuming OOD data is inaccessible. We now empirically justify our intuition shown in Fig. 3. We estimate  $\pmb{a} = \mathbf{E}_{\pmb{x}^{\mathrm{ID}} \sim D_{\mathcal{X}}^{\mathrm{ID}}} \left[ \pmb{I}(\pmb{z}^{\mathrm{ID}}) \right]$  with ImageNet-1k and  $\pmb{b} = \mathbf{E}_{\pmb{x}^{\mathrm{OOD}} \sim D_{\mathcal{X}}^{\mathrm{OOD}}} \left[ \pmb{I}(\pmb{z}^{\mathrm{OOD}}) \right]$  with different OOD datasets. We then compute the angles and the norm ratios,  $\frac{\|\pmb{b}\|}{\|\pmb{a}\|}$ , of the two expectations.
+
+As shown in Table 3, challenging OOD datasets like Species and ImageNet-O exhibit smaller angles between the two expectations, while simpler OOD datasets like iNaturalist and MNIST demonstrate smaller norm ratios. Thus, Eq. 13 provides a reasonable approximation to  $\theta$  solved by Eq. 10 with access to OOD data.
+
+Compare the formulation utilizing OOD data (Eq. 10) and our method (Eq. 14). We optimize  $\theta$  by solving Eq. 10 with each OOD dataset and get the OOD detection performance on that dataset. The last two columns of Table 3 compare the performance of  $\theta$  solved from Eq. 10
+
+Table 3: Empirical validation of Problem 13.  $\mathbf{a}$  and  $\mathbf{b}$  denote the expectations of  $\mathbf{I}\left( \mathbf{z}\right)$  for ID and OOD, respectively. "AU (Real)" refers to the performance of solving Eq. 10 with OOD data.
+
+<table><tr><td>OOD dataset</td><td>||b||/||a||</td><td>cos(a, b)</td><td>AU (Real)</td><td>AU (Ours)</td></tr><tr><td>Species</td><td>0.72</td><td>0.99</td><td>78.69</td><td>78.59</td></tr><tr><td>iNaturalist</td><td>0.48</td><td>0.97</td><td>96.53</td><td>96.63</td></tr><tr><td>SUN</td><td>0.55</td><td>0.98</td><td>92.90</td><td>92.84</td></tr><tr><td>Places</td><td>0.59</td><td>0.99</td><td>90.06</td><td>90.15</td></tr><tr><td>OpenImage-O</td><td>0.56</td><td>0.98</td><td>92.31</td><td>92.54</td></tr><tr><td>ImageNet-O</td><td>0.91</td><td>0.99</td><td>70.19</td><td>59.33</td></tr><tr><td>Texture</td><td>0.51</td><td>0.96</td><td>95.82</td><td>95.48</td></tr><tr><td>MNIST</td><td>0.42</td><td>0.84</td><td>99.32</td><td>99.34</td></tr></table>
+
+and that from Eq. 14 (ours). Notably, our method with only ID data yields performance comparable to the formulation with access to real OOD data.
+
+# 5 DISCUSSION
+
+Here we first discuss a core limitation to our framework that our proposed method assumes elementwise shaping functions, which fails to capture methods like ASH (Dosovitskiy et al., 2020). We then connect our proposed method to existing works.
+
+General feature-shaping functions. As discussed previously, a feature-shaping function can represent arbitrary mappings  $f: \mathbb{R}^M \to \mathbb{R}^M$ . While our proposed methods handle mappings which can be defined element-wise over individual features, they fail to capture methods such as ASH (Dosovitskiy et al., 2020) which can only be defined over the full feature vector. An important direction for future work is in designing a tractable optimization problem where the family of functions  $f$  includes whole-vector methods such as ASH. We expect that the resultant mappings should also be relatively simple as in the element-wise case.
+
+Feature contributions to the maximum logits. Prior research (Sun & Li, 2022; Dietterich & Guyer, 2022) has explored the idea of feature contributions to the maximum logits. For example, DICE (Sun & Li, 2022) identifies the most significant feature indices for each class using ID data. During test-time, it masks feature indices deemed insignificant before calculating OOD scores. However, our approach differs from theirs in that we consider feature value ranges instead of indices. We find our function yields significantly better performance in practice, partially attributed to the fact that it admits a tractable optimization problem. Investigating the combination of these two classes of methods remains a promising direction for future work.
+
+Classifier training for OOD detection. Prior studies (Liang et al., 2017; Sun & Li, 2022) have leveraged surrogate OOD datasets, such as Gaussian and Uniform distributions, to train classifiers for distinguishing between ID and OOD samples. Liang et al. (2017) primarily employ these datasets to assess their method's performance, while Sun & Li (2022) use them for hyperparameter tuning. Despite these efforts, the approach is seldom applied in feature-shaping methods. Concurrently, several studies are probing into outlier synthesis (Du et al., 2022; He et al., 2022) to approximate real OOD distributions and train classifiers accordingly. Our optimization problem could benefit from these advancements, as our parameterized feature-shaping function could be optimized based on an ID dataset and synthesized outliers. We explore this possibility in Section B.1 of the Appendix.
+
+# 6 CONCLUSION
+
+This paper formulates a general optimization problem for feature-shaping methods in OOD detection. Using this formalism, we are able to better understand and explain the effectiveness of existing state-of-the-art methods. In the context of our framework, we propose a novel feature shaping method which can be trained solely on ID data. Experimental results validate the superior performance and generalization ability of our approach compared to prior works, and motivate the development of new methods under our framework to further advance OOD detection methods.
+
+# ACKNOWLEDGMENTS
+
+We would like to extend our deepest appreciation to Weijian Deng, Yanbin Liu, Yunzhong Hou, Xiaoxiao Sun, and all other lab colleagues for their invaluable support throughout this project. Their collaborative efforts, insightful discussion, and constructive feedback have been crucial in shaping and improving our paper.
+
+This work was supported by an Australian Research Council (ARC) Linkage grant (project number LP210200931).
+
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+
+# A IMPLEMENTATION DETAILS
+
+# A.1 HYPERPARAMETER SETTINGS FOR EXPERIMENTS IN SECTION 4
+
+For feature-shaping methods, we adhere to the hyperparameter settings reported in the original papers of the respective methods. These settings usually optimize the performance of the methods on existing benchmarks. For additional methods, we primarily rely on hyperparameter settings detailed in Sun's work (Sun & Li, 2022). The detailed hyperparameter setting are listed in Table 4.
+
+Table 4: Hyperparameter settings for experiments in Section 4  
+
+<table><tr><td>Method</td><td>Hyperparameters</td><td>ImageNet-1k</td><td>CIFAR10</td><td>CIFAR100</td></tr><tr><td>MSP</td><td>No hyperparameters</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ODIN</td><td>T, temperature scaling ε, noise level</td><td>T=1000, ε=0.0</td><td>T=1000, ε=0.004</td><td>T=1000, ε=0.004</td></tr><tr><td>MLS</td><td>No hyperparameters</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Energy</td><td>No hyperparameters</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DICE</td><td>p, sparsity parameter</td><td>p=0.7</td><td>p=0.9</td><td>p=0.9</td></tr><tr><td>ReAct</td><td>p, threshold percentile</td><td>p=90</td><td>p=90</td><td>p=90</td></tr><tr><td>BFAct</td><td>thresh, N</td><td>N=2, thresh=95 percentile of the training set</td><td>N=2, thresh=95 percentile of the training set</td><td>N=2, thresh=95 percentile of the training set</td></tr><tr><td>ASH-P</td><td>p, pruning percentile</td><td>p=60</td><td>p=90</td><td>p=80</td></tr><tr><td>ASH-B</td><td>p, pruning percentile</td><td>p=65</td><td>p=95</td><td>p=85</td></tr><tr><td>ASH-S</td><td>p, pruning percentile</td><td>p=90</td><td>p=95</td><td>p=90</td></tr><tr><td rowspan="2">VRA-P</td><td rowspan="2">x1, x2</td><td rowspan="2">x1=0.5, x2=1.0</td><td>x1=60 percentile of the training set</td><td>x1=60 percentile of the training set</td></tr><tr><td>x2=95 percentile of the training set</td><td>x2=95 percentile of the training set</td></tr><tr><td>Ours</td><td>K, the number of feature value intervals</td><td>K=100</td><td>K=100</td><td>K=100</td></tr></table>
+
+# A.2 THE MAGNITUDE OF RESHAPED FEATURES
+
+In Section 3, we describe our method for learning a piecewise constant reshaping function subject to norm constraints. Scaling  $\theta$  has no effect on the vanilla version of method. However, OOD detection performance is sensitive to the scaling of  $\theta$  when our method is combined with other OOD scores with non-linear transformations such as MSP and the energy score.
+
+Take the energy score as an example. If we multiply the feature representation  $z$  by a small positive factor  $q$  ( $0 < q < 1$ ), ignoring the bias term in the last fully connected layer, we have:
+
+$$
+s _ {\text {E n e r g y}} (\ell) = \log \sum_ {j = 1} ^ {C} \exp \left(q \ell_ {j}\right). \tag {15}
+$$
+
+This is equivalent to apply a temperature scaling  $T = 1 / q$  to the energy score, which will affect the OOD detection performance, as analyzed in a previous paper (Liang et al., 2017). Besides, considering a fully connected layer usually contains a bias term, the scaling of  $\theta$  will further affect the influence of the bias. Thus, we need to consider the scale of  $\theta$  in our optimization problem.
+
+We scale  $\theta$  by a factor of  $\sqrt{K}$ , motivated by the following analysis. Considering that the maximum logit is the sum of all ISFIs, we have:
+
+$$
+\ell^ {\max } (\boldsymbol {z}) = \sum_ {k = 1} ^ {K} I _ {[ a _ {k}, b _ {k})} (\boldsymbol {z}) = \mathbf {1} ^ {\top} \boldsymbol {I} (\boldsymbol {z}), \tag {16}
+$$
+
+where  $\mathbf{1}$  represents a  $K$ -dimensional vector of ones, which is a trivial feature-shaping function with no effect. Therefore, we set  $\| \pmb{\theta} \| = \| \mathbf{1} \| = \sqrt{K}$ .
+
+Table 5: OOD detection results of variants of our methods on ImageNet-1k.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results. "Real" means using a real OOD dataset to resolve Problem (Eq. 10). "Gaussian" and "Uniform" indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving Problem (Eq. 10), respectively. "Dynamic" denotes solving a non-convex problem with dynamic weight indices defined in Section B.2.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">ResNet50</td><td colspan="2">MobileNetV2</td><td colspan="2">ViT-B-16</td><td colspan="2">ViT-L-16</td><td colspan="2">SWIN-S</td><td colspan="2">SWIN-B</td><td colspan="2">MLP-B</td><td colspan="2">MLP-L</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>69.30</td><td>76.26</td><td>72.58</td><td>77.41</td><td>69.84</td><td>77.40</td><td>70.59</td><td>78.40</td><td>65.12</td><td>80.89</td><td>69.45</td><td>77.26</td><td>70.66</td><td>80.31</td><td>71.63</td><td>79.93</td><td>69.90</td><td>78.48</td></tr><tr><td>ODIN</td><td>61.56</td><td>80.92</td><td>62.91</td><td>82.64</td><td>69.25</td><td>72.60</td><td>70.35</td><td>74.51</td><td>62.35</td><td>78.67</td><td>70.90</td><td>68.36</td><td>65.00</td><td>82.99</td><td>67.90</td><td>81.86</td><td>66.28</td><td>77.82</td></tr><tr><td>Energy</td><td>60.97</td><td>81.01</td><td>61.40</td><td>82.83</td><td>73.96</td><td>67.65</td><td>74.89</td><td>70.11</td><td>66.56</td><td>75.95</td><td>79.53</td><td>60.14</td><td>87.86</td><td>78.42</td><td>84.60</td><td>79.01</td><td>73.72</td><td>74.39</td></tr><tr><td>DICE</td><td>45.32</td><td>83.64</td><td>49.33</td><td>84.63</td><td>89.68</td><td>71.32</td><td>72.38</td><td>67.08</td><td>78.84</td><td>49.06</td><td>77.89</td><td>50.10</td><td>57.12</td><td>80.60</td><td>61.98</td><td>80.75</td><td>66.57</td><td>70.90</td></tr><tr><td>ReAct</td><td>42.29</td><td>86.54</td><td>54.19</td><td>85.37</td><td>73.82</td><td>76.86</td><td>76.16</td><td>81.07</td><td>58.07</td><td>85.10</td><td>70.23</td><td>80.10</td><td>90.13</td><td>77.18</td><td>89.03</td><td>78.32</td><td>69.24</td><td>81.32</td></tr><tr><td>BFAct</td><td>43.87</td><td>86.01</td><td>52.87</td><td>85.78</td><td>77.64</td><td>80.16</td><td>84.02</td><td>81.12</td><td>64.76</td><td>85.57</td><td>68.40</td><td>84.36</td><td>96.04</td><td>67.87</td><td>96.00</td><td>70.59</td><td>72.95</td><td>80.18</td></tr><tr><td>ASH-P</td><td>55.30</td><td>83.00</td><td>59.41</td><td>83.84</td><td>99.36</td><td>21.17</td><td>99.18</td><td>20.27</td><td>99.09</td><td>21.78</td><td>98.83</td><td>22.11</td><td>98.19</td><td>35.85</td><td>98.91</td><td>29.69</td><td>88.53</td><td>39.71</td></tr><tr><td>ASH-B</td><td>35.97</td><td>88.62</td><td>43.59</td><td>88.28</td><td>94.87</td><td>46.68</td><td>93.72</td><td>38.95</td><td>96.48</td><td>36.54</td><td>92.40</td><td>49.85</td><td>99.51</td><td>21.73</td><td>64.91</td><td>83.15</td><td>77.68</td><td>56.73</td></tr><tr><td>ASH-S</td><td>34.70</td><td>90.25</td><td>43.84</td><td>88.24</td><td>99.48</td><td>18.52</td><td>99.42</td><td>18.61</td><td>99.20</td><td>18.26</td><td>99.06</td><td>19.27</td><td>97.61</td><td>33.91</td><td>99.28</td><td>19.29</td><td>84.07</td><td>38.29</td></tr><tr><td>VRA-P</td><td>37.97</td><td>88.58</td><td>49.98</td><td>86.83</td><td>98.39</td><td>35.66</td><td>99.58</td><td>16.70</td><td>99.27</td><td>20.34</td><td>99.46</td><td>17.64</td><td>99.38</td><td>18.73</td><td>99.05</td><td>21.23</td><td>85.39</td><td>38.21</td></tr><tr><td>Real</td><td>42.09</td><td>88.13</td><td>55.05</td><td>85.77</td><td>73.69</td><td>81.65</td><td>75.36</td><td>83.10</td><td>76.06</td><td>83.22</td><td>73.54</td><td>83.11</td><td>81.66</td><td>78.34</td><td>78.82</td><td>79.20</td><td>69.53</td><td>82.82</td></tr><tr><td>Gaussian</td><td>42.05</td><td>88.20</td><td>55.35</td><td>85.67</td><td>71.88</td><td>82.01</td><td>73.99</td><td>83.28</td><td>71.29</td><td>83.99</td><td>70.79</td><td>83.37</td><td>82.09</td><td>77.58</td><td>76.40</td><td>79.79</td><td>67.98</td><td>82.99</td></tr><tr><td>Uniform</td><td>42.36</td><td>88.02</td><td>55.12</td><td>85.73</td><td>73.14</td><td>81.82</td><td>74.30</td><td>83.25</td><td>72.68</td><td>83.74</td><td>70.57</td><td>83.43</td><td>80.84</td><td>76.88</td><td>76.41</td><td>79.74</td><td>68.18</td><td>82.83</td></tr><tr><td>Dynamic</td><td>41.81</td><td>88.13</td><td>53.29</td><td>86.35</td><td>69.33</td><td>82.59</td><td>72.18</td><td>83.23</td><td>66.90</td><td>84.52</td><td>65.98</td><td>83.71</td><td>76.87</td><td>78.35</td><td>78.10</td><td>78.85</td><td>65.56</td><td>83.22</td></tr><tr><td>Ours (V)</td><td>41.84</td><td>88.11</td><td>53.31</td><td>86.36</td><td>69.33</td><td>82.59</td><td>72.17</td><td>83.23</td><td>66.89</td><td>84.53</td><td>65.96</td><td>83.71</td><td>76.59</td><td>78.46</td><td>78.13</td><td>78.85</td><td>65.53</td><td>83.23</td></tr><tr><td>Ours (E)</td><td>39.75</td><td>88.56</td><td>51.77</td><td>86.62</td><td>69.52</td><td>82.66</td><td>78.57</td><td>82.94</td><td>66.28</td><td>84.99</td><td>66.20</td><td>84.21</td><td>83.54</td><td>75.94</td><td>85.50</td><td>78.11</td><td>67.64</td><td>83.00</td></tr></table>
+
+# B VARIANTS OF OUR METHOD
+
+In this section, we execute a series of experiments on variants of our proposed methods. Initially, we employ real and surrogate OOD datasets to address the optimization problem (Eq. 10) as alternatives to our ID only method. Subsequently, we evaluate an assumption that weight indices of maximum logit will not change after feature shaping. We calculate and report the ratio of changed weights after reshaping. Furthermore, we solve an optimization problem which allows for dynamic weight indices. In this scenario, it is possible that the weight indices of the maximum logits can undergo changes after the reshaping exercise. Our experiments are based on the ImageNet-1k benchmark and the results are illustrated in Table 5.
+
+# B.1 EXPLOIT REAL AND SURROGATE OOD DATASETS
+
+We draw comparisons between our ID only method and a feature-shaping function optimization approach that accesses both ID and OOD datasets.
+
+First, we utilize real OOD datasets to address Problem (Eq. 10). For each model architecture, ImageNet-1k training set (ID) and iNaturalist (OOD) are leveraged to optimize  $\theta$ .
+
+Second, we incorporate Gaussian and Uniform distributions as surrogate OOD datasets in resolving Problem (Eq. 10), aligning with prior studies (Liang et al., 2017; Sun & Li, 2022). For the Gaussian distribution, each pixel is generated with a mean of 127.5 and a standard deviation of 255. For the uniform distribution, we directly generate random pixel values within the range [0, 255]. Every generated image is of size  $32 \times 32$ , and is subsequently clipped and rounded to conform to the [0, 255] value range. For each surrogate OOD dataset, we generate a total of 50,000 samples.
+
+As illustrated in Table 5, our method, even without access to OOD data, performs comparably to methods that optimize  $\theta$  utilizing both ID and OOD datasets. Moreover, we observe that a more robust objective function, such as Log Loss or Hinge Loss, could potentially outperform our method when given access to real or surrogate OOD datasets. This presents an intriguing avenue for exploration.
+
+Table 6: Ratio of changed weight indices  $r$  on ImageNet-1k. All numbers are percentages. We randomly sample 10,000 samples from ImageNet-1k training set, and  $r$  is defined in Section B.2  
+
+<table><tr><td></td><td>ResNet50</td><td>MobileNetV2</td><td>ViT-B-16</td><td>ViT-L-16</td><td>SWIN-S</td><td>SWIN-B</td><td>MLP-B</td><td>MLP-L</td><td>Average</td></tr><tr><td>r</td><td>6.8</td><td>6.5</td><td>1.1</td><td>0.5</td><td>1.5</td><td>0.9</td><td>4.3</td><td>0.4</td><td>2.8</td></tr></table>
+
+# B.2 DYNAMIC WEIGHT INDICES
+
+Our proposed method, as elucidated in Sections 3.2 and 3.3, relies on a fundamental simplifying assumption: the class index of the maximum logit remains unchanged after feature shaping.
+
+We initially examine the validity of this assumption by calculating the ratio of weight indices that change after feature shaping. For a given ID training set  $S_{\mathrm{train}}$  of  $N$  samples, each sample  $\pmb{x}^{(i)} \in S_{\mathrm{train}}$  has weight indices corresponding to the maximum logits before and after feature shaping, as follows:
+
+$$
+\tilde {k} ^ {(i)} = \underset {k} {\arg \max } \boldsymbol {w} _ {k} ^ {\top} \boldsymbol {z} ^ {(i)} \quad \bar {k} ^ {(i)} = \underset {k} {\arg \max } \boldsymbol {w} _ {k} ^ {\top} \bar {\boldsymbol {z}} ^ {(i)}, \tag {17}
+$$
+
+where  $w_{k}$  denotes the weight vector corresponding to class  $k$  in the final fully-connected layer,  $z^{(i)}$  represents the feature representation for the sample  $x^{(i)}$ , and  $\bar{z}^{(i)}$  is the reshaped feature. Subsequently, we compute the ratio of changed weight indices as:
+
+$$
+r = \frac {\sum_ {i = 1} ^ {N} \mathbb {1} \left\{\tilde {k} ^ {(i)} \neq \bar {k} ^ {(i)} \right\}}{N} \times 100 \% \tag{18}
+$$
+
+We randomly sample 10,000 samples from ImageNet-1k training set. The ratios of changed weights with different models are presented in Table 6. As observed, the ratios are markedly small, which further substantiates the credibility of our assumption pertaining to fixed weights.
+
+Furthermore, we propose a straightforward Alternating Optimization algorithm for dynamic weight indices. Each iteration involves three primary steps. Firstly, we optimize our feature-shaping function by resolving the optimization problem (Eq. 10) given the weights relative to the maximum logits for each sample. Next, we reshape features utilizing our optimized feature-shaping function. Lastly, we update the weights with reshaped features. To expedite the process, we randomly subsample 10,000 samples from the ID training set during each iteration and utilize them for optimization. Ten iterations are run for each model architecture. The results can be viewed in Table 5.
+
+Predictably, this approach yields performance metrics that closely mirror our initial method. Despite the incorporation of dynamic weight indices, the optimized feature-shaping function does not significantly differ from that obtained with fixed weights. This observation reinforces the validity of our original assumption: the class index of the maximum logit remains constant following reshaping.
+
+# C ROBUSTNESS OF OUR METHOD TO THE CHOSEN LOWER AND UPPER LIMITS OF FEATURES
+
+Given that feature distributions typically exhibit long-tailed characteristics, we select the 0.1 and 99.9 percentiles of all features in a training set as the lower and upper limits of feature values, respectively, to optimize our shaping function. Here we validate different choices of percentiles on the performance of our method, on the ImageNet-1k benchmark with ResNet50. The results are shown in Table 7.
+
+Our method exhibits a insensitivity to the chooses of lower and upper limits. Although in the main experiments we consistently use the 0.1 and 99.9 percentiles, exploring alternate settings reveals potential for slight improvements.
+
+Table 7: Performance of our method with different choices of percentiles of all features on the ImageNet-1k benchmark with ResNet50.  
+
+<table><tr><td>Lower percentile</td><td>Lower limit</td><td>Upper percentile</td><td>Upper limit</td><td>FPR95</td><td>AUC</td></tr><tr><td>0.0</td><td>0.00</td><td>100.0</td><td>20.65</td><td>42.42</td><td>87.90</td></tr><tr><td>0.01</td><td>0.00</td><td>99.99</td><td>5.64</td><td>42.22</td><td>87.97</td></tr><tr><td>0.05</td><td>0.00</td><td>99.95</td><td>4.40</td><td>41.97</td><td>88.04</td></tr><tr><td>0.1</td><td>0.00</td><td>99.9</td><td>3.88</td><td>41.82</td><td>88.11</td></tr><tr><td>0.5</td><td>0.00</td><td>99.5</td><td>2.74</td><td>41.01</td><td>88.44</td></tr><tr><td>1.0</td><td>0.00</td><td>99.0</td><td>2.28</td><td>41.00</td><td>88.50</td></tr><tr><td>5.0</td><td>0.03</td><td>95.0</td><td>1.35</td><td>47.85</td><td>86.41</td></tr><tr><td>10.0</td><td>0.06</td><td>90.0</td><td>1.01</td><td>57.45</td><td>83.60</td></tr></table>
+
+# D EMPIRICAL ANALYSIS OF THE OPTIMAL SHAPING FUNCTION
+
+![](images/d5f3f9b7d5e04cb8d5c0c8f1594bbc340894a45a2f9f5b39fb2918b777f7baa7.jpg)  
+Figure 5: Empirical analysis to explain a specific form of the optimal shaping function.
+
+Here we try to provide empirical insights into the specific form of our optimal shaping function based on a ResNet50 model pretrained on ImageNet-1k, which is shown in Fig. 2. Specifically, we feed ImageNet-1k val (ID) and iNaturalist (OOD) samples to the model, getting the feature representation  $\mathbf{z} \in \mathbb{R}^{M}$  and the weights w.r.t. the maximum logit  $\mathbf{w}^{\mathrm{max}} \in \mathbb{R}^{M}$  for each sample. The results are shown in Fig. 5. The red and green lines represent the mean weights assigned to different feature values for ID and OOD samples, respectively. The colored regions represent standard deviations.
+
+An interesting aspect is that our approach flips the sign of low-value features, contrary to previous techniques like ReAct and ASH. Our experiments revealed that low-value features  $(z < 0.5)$  often align with negative weights  $(w^{\max} < 0)$ . Intriguingly, ID samples generally align these features with smaller weights than OOD samples (on average,  $w_{\mathrm{ID}}^{\max} < w_{\mathrm{OOD}}^{\max} < 0$ ). By flipping the sign of these low-value features, we can enhance the maximum logits more significantly for ID samples than for OOD samples, thereby boosting OOD detection performance.
+
+However, note that the shape of the optimal shaping function is influenced by multiple factors. For instance, while ID samples are generally assigned larger weights for high-value features, OOD samples tend to exhibit a greater number of these features, as observed in prior studies (Sun et al., 2021). This aspect reduces the impact of high-value features in distinguishing between ID and OOD samples, leading to the decrease of  $\theta$  for those features.
+
+Besides, note that our analysis above is grounded in the context of a ResNet50 model pre-trained on ImageNet-1k. The form of the optimal shaping function may differ when applied to other model architectures or datasets.
+
+# E DETAILED OOD DETECTION PERFORMANCE
+
+We report additional results in Tables 8, 9, and 10 to supplement the results provided in Table 1. Specifically, we provide detailed performance for all eight test OOD datasets for the ImageNet-1k benchmark for three representative models, including ResNet50, ViT-B-16, and MLP-B.
+
+# F STANDARD IMAGENET-1K BENCHMARK COMPARISON
+
+Standard ImageNet-1k benchmark uses a subset of OOD datasets of ours, and these results can be found in Tables 8, 9, and 10.
+
+Besides, we pick the results of standard ImageNet-1k benchmark in our experiments and calculated the new average performance. Our method still outperforms prior works on this standard benchmark. Results are shown in Table 11.
+
+Table 8: OOD detection results with ResNet50 on ImageNet-1k.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results. "Real" means using a real OOD dataset to resolve Problem (Eq. 10). "Gaussian" and "Uniform" indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving Problem (Eq. 10), respectively. "Dynamic" denotes solving a non-convex problem with dynamic weight indices defined in Section B.2.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Species</td><td colspan="2">iNaturalist</td><td colspan="2">SUN</td><td colspan="2">Places</td><td colspan="2">OpenImage-O</td><td colspan="2">ImageNet-O</td><td colspan="2">Texture</td><td colspan="2">MNIST</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>79.53</td><td>75.15</td><td>52.77</td><td>88.42</td><td>68.58</td><td>81.75</td><td>71.57</td><td>80.63</td><td>63.58</td><td>84.98</td><td>100.00</td><td>28.64</td><td>66.13</td><td>80.46</td><td>52.25</td><td>90.02</td><td>69.30</td><td>76.26</td></tr><tr><td>ODIN</td><td>80.90</td><td>72.88</td><td>50.86</td><td>91.14</td><td>59.89</td><td>86.59</td><td>65.67</td><td>84.18</td><td>57.29</td><td>89.23</td><td>100.00</td><td>40.86</td><td>54.31</td><td>86.40</td><td>23.54</td><td>96.11</td><td>61.56</td><td>80.92</td></tr><tr><td>Energy</td><td>82.40</td><td>72.03</td><td>53.93</td><td>90.59</td><td>58.27</td><td>86.73</td><td>65.40</td><td>84.13</td><td>57.21</td><td>89.12</td><td>100.00</td><td>41.92</td><td>52.29</td><td>86.73</td><td>18.23</td><td>96.79</td><td>60.97</td><td>81.01</td></tr><tr><td>DICE</td><td>74.64</td><td>74.30</td><td>26.66</td><td>94.49</td><td>36.09</td><td>90.98</td><td>47.66</td><td>87.73</td><td>45.13</td><td>88.77</td><td>98.10</td><td>43.01</td><td>32.46</td><td>90.46</td><td>1.79</td><td>99.40</td><td>45.32</td><td>83.64</td></tr><tr><td>ReAct</td><td>68.64</td><td>77.41</td><td>19.55</td><td>96.39</td><td>24.00</td><td>94.41</td><td>33.43</td><td>91.93</td><td>43.68</td><td>90.53</td><td>98.05</td><td>52.43</td><td>45.83</td><td>90.45</td><td>5.11</td><td>98.76</td><td>42.29</td><td>86.54</td></tr><tr><td>BFAct</td><td>68.08</td><td>77.76</td><td>20.53</td><td>96.22</td><td>20.94</td><td>95.24</td><td>30.17</td><td>92.62</td><td>49.71</td><td>86.48</td><td>96.20</td><td>54.15</td><td>54.01</td><td>87.84</td><td>11.33</td><td>97.78</td><td>43.87</td><td>86.01</td></tr><tr><td>ASH-P</td><td>80.09</td><td>73.19</td><td>44.57</td><td>92.51</td><td>52.88</td><td>88.35</td><td>61.79</td><td>85.58</td><td>50.45</td><td>90.67</td><td>100.00</td><td>45.99</td><td>42.06</td><td>89.70</td><td>10.57</td><td>97.99</td><td>55.30</td><td>83.00</td></tr><tr><td>ASH-B</td><td>66.08</td><td>80.33</td><td>14.21</td><td>97.32</td><td>22.08</td><td>95.10</td><td>33.45</td><td>92.31</td><td>37.63</td><td>91.97</td><td>93.05</td><td>56.82</td><td>21.17</td><td>95.50</td><td>0.06</td><td>99.58</td><td>35.97</td><td>88.62</td></tr><tr><td>ASH-S</td><td>64.61</td><td>81.45</td><td>11.49</td><td>97.87</td><td>27.98</td><td>94.02</td><td>39.78</td><td>90.98</td><td>32.74</td><td>92.75</td><td>89.05</td><td>67.51</td><td>11.93</td><td>97.60</td><td>0.01</td><td>99.80</td><td>34.70</td><td>90.25</td></tr><tr><td>VRA-P</td><td>69.61</td><td>76.93</td><td>15.70</td><td>97.12</td><td>26.94</td><td>94.25</td><td>37.85</td><td>91.27</td><td>36.37</td><td>92.93</td><td>95.55</td><td>60.79</td><td>21.47</td><td>95.62</td><td>0.30</td><td>99.70</td><td>37.97</td><td>88.58</td></tr><tr><td>Real</td><td>68.91</td><td>78.58</td><td>18.64</td><td>96.53</td><td>37.76</td><td>92.69</td><td>47.31</td><td>89.97</td><td>39.63</td><td>92.33</td><td>97.45</td><td>60.01</td><td>24.15</td><td>95.57</td><td>2.90</td><td>99.34</td><td>42.09</td><td>88.13</td></tr><tr><td>Gaussian</td><td>69.39</td><td>78.31</td><td>18.88</td><td>96.49</td><td>37.54</td><td>92.68</td><td>47.43</td><td>89.88</td><td>39.60</td><td>92.32</td><td>97.05</td><td>60.85</td><td>23.24</td><td>95.76</td><td>3.25</td><td>99.30</td><td>42.05</td><td>88.20</td></tr><tr><td>Uniform</td><td>69.34</td><td>78.49</td><td>18.83</td><td>96.51</td><td>38.71</td><td>92.51</td><td>47.98</td><td>89.81</td><td>39.51</td><td>92.46</td><td>97.70</td><td>59.49</td><td>24.24</td><td>95.54</td><td>2.58</td><td>99.37</td><td>42.36</td><td>88.02</td></tr><tr><td>Dynamic</td><td>68.81</td><td>78.58</td><td>18.33</td><td>96.63</td><td>36.87</td><td>92.87</td><td>45.94</td><td>90.17</td><td>39.34</td><td>92.54</td><td>97.80</td><td>59.44</td><td>24.70</td><td>95.50</td><td>2.68</td><td>99.34</td><td>41.81</td><td>88.13</td></tr><tr><td>Ours (V)</td><td>68.83</td><td>78.59</td><td>18.33</td><td>96.63</td><td>37.03</td><td>92.84</td><td>45.97</td><td>90.15</td><td>39.29</td><td>92.54</td><td>97.80</td><td>59.33</td><td>24.80</td><td>95.48</td><td>2.67</td><td>99.34</td><td>41.84</td><td>88.11</td></tr><tr><td>Ours (E)</td><td>68.46</td><td>78.51</td><td>15.90</td><td>97.00</td><td>34.00</td><td>93.28</td><td>43.61</td><td>90.50</td><td>37.07</td><td>92.82</td><td>96.25</td><td>60.70</td><td>21.61</td><td>95.99</td><td>0.85</td><td>99.66</td><td>39.75</td><td>88.56</td></tr></table>
+
+Table 9: OOD detection results with ViT-B-16 on ImageNet-1k.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results. "Real" means using a real OOD dataset to resolve Problem (Eq. 10). "Gaussian" and "Uniform" indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving Problem (Eq. 10), respectively. "Dynamic" denotes solving a non-convex problem with dynamic weight indices defined in Section B.2.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Species</td><td colspan="2">iNaturalist</td><td colspan="2">SUN</td><td colspan="2">Places</td><td colspan="2">OpenImage-O</td><td colspan="2">ImageNet-O</td><td colspan="2">Texture</td><td colspan="2">MNIST</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>80.56</td><td>72.91</td><td>51.52</td><td>88.16</td><td>66.54</td><td>80.93</td><td>68.67</td><td>80.38</td><td>59.93</td><td>84.81</td><td>90.95</td><td>58.80</td><td>60.23</td><td>82.99</td><td>80.29</td><td>70.23</td><td>69.84</td><td>77.40</td></tr><tr><td>ODIN</td><td>82.18</td><td>67.96</td><td>52.26</td><td>85.24</td><td>66.89</td><td>76.35</td><td>69.14</td><td>75.06</td><td>58.75</td><td>81.55</td><td>89.05</td><td>54.30</td><td>56.68</td><td>81.69</td><td>79.03</td><td>58.65</td><td>69.25</td><td>72.60</td></tr><tr><td>Energy</td><td>86.54</td><td>61.88</td><td>64.08</td><td>79.24</td><td>72.77</td><td>70.24</td><td>74.31</td><td>68.44</td><td>64.93</td><td>76.46</td><td>87.00</td><td>52.71</td><td>58.48</td><td>79.30</td><td>83.61</td><td>52.92</td><td>73.96</td><td>67.65</td></tr><tr><td>DICE</td><td>88.09</td><td>65.08</td><td>90.43</td><td>74.49</td><td>94.27</td><td>65.76</td><td>92.81</td><td>65.19</td><td>82.61</td><td>77.51</td><td>85.85</td><td>70.86</td><td>83.67</td><td>77.63</td><td>99.68</td><td>74.02</td><td>89.68</td><td>71.32</td></tr><tr><td>ReAct</td><td>86.82</td><td>66.08</td><td>65.15</td><td>85.98</td><td>72.46</td><td>78.97</td><td>73.74</td><td>77.51</td><td>64.71</td><td>84.19</td><td>87.30</td><td>66.70</td><td>56.95</td><td>84.65</td><td>83.41</td><td>70.81</td><td>73.82</td><td>76.86</td></tr><tr><td>BFAct</td><td>90.52</td><td>64.43</td><td>80.01</td><td>84.62</td><td>77.86</td><td>81.10</td><td>77.55</td><td>79.61</td><td>69.66</td><td>85.45</td><td>87.15</td><td>73.84</td><td>57.61</td><td>85.67</td><td>80.76</td><td>86.53</td><td>77.64</td><td>80.16</td></tr><tr><td>ASH-P</td><td>98.38</td><td>28.99</td><td>99.95</td><td>10.42</td><td>99.69</td><td>19.83</td><td>99.66</td><td>21.02</td><td>99.57</td><td>18.91</td><td>99.15</td><td>30.70</td><td>98.49</td><td>29.92</td><td>100.00</td><td>9.57</td><td>99.36</td><td>21.17</td></tr><tr><td>ASH-B</td><td>94.34</td><td>53.78</td><td>92.36</td><td>48.20</td><td>95.30</td><td>52.35</td><td>95.75</td><td>52.18</td><td>92.74</td><td>52.73</td><td>94.60</td><td>51.53</td><td>93.90</td><td>43.84</td><td>99.96</td><td>18.86</td><td>94.87</td><td>46.68</td></tr><tr><td>ASH-S</td><td>98.11</td><td>28.50</td><td>99.99</td><td>6.71</td><td>99.73</td><td>16.70</td><td>99.66</td><td>18.33</td><td>99.84</td><td>14.07</td><td>99.40</td><td>28.51</td><td>99.11</td><td>24.15</td><td>100.00</td><td>11.20</td><td>99.48</td><td>18.52</td></tr><tr><td>VRA-P</td><td>98.14</td><td>39.19</td><td>99.78</td><td>23.06</td><td>98.83</td><td>30.67</td><td>98.89</td><td>32.59</td><td>98.28</td><td>37.46</td><td>98.65</td><td>43.67</td><td>94.57</td><td>52.22</td><td>100.00</td><td>26.40</td><td>98.39</td><td>35.66</td></tr><tr><td>Real</td><td>85.07</td><td>67.00</td><td>64.16</td><td>88.03</td><td>70.73</td><td>83.05</td><td>71.76</td><td>81.74</td><td>65.56</td><td>87.14</td><td>89.60</td><td>73.69</td><td>63.09</td><td>84.63</td><td>79.53</td><td>87.93</td><td>73.69</td><td>81.65</td></tr><tr><td>Gaussian</td><td>84.14</td><td>67.56</td><td>61.10</td><td>88.71</td><td>69.13</td><td>83.41</td><td>70.28</td><td>82.07</td><td>63.32</td><td>87.51</td><td>89.70</td><td>73.48</td><td>60.16</td><td>85.20</td><td>77.20</td><td>88.16</td><td>71.88</td><td>82.01</td></tr><tr><td>Uniform</td><td>84.69</td><td>67.25</td><td>62.56</td><td>88.44</td><td>70.60</td><td>83.27</td><td>71.37</td><td>81.93</td><td>64.54</td><td>87.41</td><td>90.20</td><td>73.59</td><td>61.60</td><td>84.94</td><td>79.54</td><td>87.71</td><td>73.14</td><td>81.82</td></tr><tr><td>Dynamic</td><td>83.20</td><td>69.47</td><td>56.06</td><td>89.91</td><td>66.63</td><td>84.17</td><td>68.48</td><td>82.69</td><td>60.21</td><td>88.17</td><td>89.50</td><td>71.82</td><td>56.81</td><td>86.43</td><td>73.75</td><td>88.08</td><td>69.33</td><td>82.59</td></tr><tr><td>Ours (V)</td><td>83.19</td><td>69.47</td><td>56.09</td><td>89.91</td><td>66.63</td><td>84.17</td><td>68.51</td><td>82.69</td><td>60.21</td><td>88.18</td><td>89.50</td><td>71.82</td><td>56.79</td><td>86.43</td><td>73.75</td><td>88.08</td><td>69.33</td><td>82.59</td></tr><tr><td>Ours (E)</td><td>84.56</td><td>68.70</td><td>60.09</td><td>89.55</td><td>68.02</td><td>83.92</td><td>69.42</td><td>82.36</td><td>61.77</td><td>88.07</td><td>87.95</td><td>72.74</td><td>54.61</td><td>86.72</td><td>69.72</td><td>89.21</td><td>69.52</td><td>82.66</td></tr></table>
+
+Table 10: OOD detection results with MLP-B on ImageNet-1k.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. All values are percentages averaged over different OOD datasets. Bold numbers are superior results whereas underlined numbers denote the second and third best results. "Real" means using a real OOD dataset to resolve Problem (Eq. 10). "Gaussian" and "Uniform" indicate using Gaussian and Uniform distributions as surrogate OOD datasets in resolving Problem (Eq. 10), respectively. "Dynamic" denotes solving a non-convex problem with dynamic weight indices defined in Section B.2.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Species</td><td colspan="2">iNaturalist</td><td colspan="2">SUN</td><td colspan="2">Places</td><td colspan="2">OpenImage-O</td><td colspan="2">ImageNet-O</td><td colspan="2">Texture</td><td colspan="2">MNIST</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>82.15</td><td>73.21</td><td>58.99</td><td>86.84</td><td>72.98</td><td>80.92</td><td>75.52</td><td>79.83</td><td>72.42</td><td>82.59</td><td>90.95</td><td>65.84</td><td>69.95</td><td>80.27</td><td>42.36</td><td>92.95</td><td>70.66</td><td>80.31</td></tr><tr><td>ODIN</td><td>80.99</td><td>73.39</td><td>53.31</td><td>89.53</td><td>67.47</td><td>84.24</td><td>71.67</td><td>82.44</td><td>68.06</td><td>85.51</td><td>90.45</td><td>68.69</td><td>59.75</td><td>84.40</td><td>28.33</td><td>95.73</td><td>65.00</td><td>82.99</td></tr><tr><td>Energy</td><td>89.04</td><td>71.84</td><td>84.31</td><td>83.65</td><td>86.85</td><td>79.35</td><td>87.15</td><td>78.42</td><td>82.30</td><td>81.24</td><td>88.40</td><td>67.53</td><td>85.25</td><td>78.41</td><td>99.58</td><td>86.91</td><td>87.86</td><td>78.42</td></tr><tr><td>DICE</td><td>71.40</td><td>75.65</td><td>44.44</td><td>90.19</td><td>64.69</td><td>80.84</td><td>70.31</td><td>78.08</td><td>63.89</td><td>80.26</td><td>89.90</td><td>56.33</td><td>52.20</td><td>84.33</td><td>0.12</td><td>99.11</td><td>57.12</td><td>80.60</td></tr><tr><td>ReAct</td><td>91.20</td><td>70.81</td><td>91.47</td><td>80.11</td><td>91.83</td><td>77.31</td><td>90.99</td><td>76.82</td><td>85.66</td><td>79.60</td><td>87.85</td><td>67.74</td><td>87.36</td><td>77.20</td><td>94.65</td><td>87.89</td><td>90.13</td><td>77.18</td></tr><tr><td>BFAct</td><td>95.94</td><td>64.38</td><td>98.60</td><td>65.76</td><td>97.16</td><td>67.50</td><td>96.66</td><td>68.74</td><td>94.69</td><td>70.00</td><td>90.85</td><td>66.57</td><td>94.61</td><td>67.83</td><td>99.85</td><td>72.19</td><td>96.04</td><td>67.87</td></tr><tr><td>ASH-P</td><td>97.25</td><td>46.50</td><td>99.22</td><td>32.82</td><td>98.72</td><td>40.24</td><td>97.74</td><td>43.07</td><td>98.67</td><td>38.34</td><td>94.55</td><td>52.65</td><td>99.38</td><td>31.38</td><td>100.00</td><td>1.81</td><td>98.19</td><td>35.85</td></tr><tr><td>ASH-B</td><td>99.67</td><td>25.98</td><td>100.00</td><td>15.86</td><td>99.71</td><td>19.84</td><td>99.74</td><td>21.36</td><td>99.82</td><td>24.19</td><td>97.25</td><td>46.39</td><td>99.86</td><td>19.48</td><td>100.00</td><td>0.74</td><td>99.51</td><td>21.73</td></tr><tr><td>ASH-S</td><td>96.50</td><td>42.89</td><td>98.42</td><td>32.35</td><td>98.03</td><td>38.00</td><td>97.90</td><td>38.63</td><td>97.65</td><td>37.03</td><td>93.50</td><td>50.19</td><td>98.90</td><td>30.41</td><td>100.00</td><td>1.80</td><td>97.61</td><td>33.91</td></tr><tr><td>VRA-P</td><td>98.88</td><td>29.11</td><td>99.92</td><td>11.57</td><td>99.73</td><td>17.54</td><td>99.41</td><td>20.71</td><td>99.77</td><td>16.82</td><td>97.80</td><td>37.15</td><td>99.50</td><td>16.49</td><td>100.00</td><td>0.44</td><td>99.38</td><td>18.73</td></tr><tr><td>Real</td><td>86.49</td><td>71.56</td><td>71.23</td><td>84.79</td><td>79.47</td><td>79.42</td><td>81.95</td><td>78.44</td><td>78.30</td><td>81.00</td><td>91.00</td><td>66.46</td><td>78.24</td><td>78.36</td><td>86.61</td><td>86.69</td><td>81.66</td><td>78.34</td></tr><tr><td>Gaussian</td><td>87.00</td><td>71.10</td><td>73.67</td><td>83.35</td><td>79.82</td><td>78.80</td><td>82.07</td><td>77.86</td><td>79.06</td><td>79.83</td><td>91.20</td><td>65.95</td><td>78.90</td><td>77.64</td><td>85.01</td><td>86.10</td><td>82.09</td><td>77.58</td></tr><tr><td>Uniform</td><td>86.92</td><td>70.35</td><td>75.45</td><td>81.81</td><td>80.47</td><td>77.69</td><td>82.25</td><td>77.07</td><td>79.29</td><td>78.56</td><td>90.70</td><td>65.63</td><td>78.05</td><td>76.45</td><td>73.59</td><td>87.51</td><td>80.84</td><td>76.88</td></tr><tr><td>Dynamic</td><td>87.24</td><td>70.88</td><td>76.95</td><td>82.17</td><td>83.83</td><td>77.26</td><td>84.75</td><td>76.12</td><td>78.65</td><td>80.31</td><td>90.70</td><td>65.68</td><td>76.76</td><td>79.17</td><td>36.06</td><td>95.19</td><td>76.87</td><td>78.35</td></tr><tr><td>Ours (V)</td><td>86.98</td><td>70.98</td><td>76.40</td><td>82.36</td><td>83.45</td><td>77.42</td><td>84.67</td><td>76.27</td><td>78.35</td><td>80.46</td><td>90.85</td><td>65.77</td><td>76.40</td><td>79.26</td><td>35.66</td><td>95.19</td><td>76.59</td><td>78.46</td></tr><tr><td>Ours (E)</td><td>91.67</td><td>69.17</td><td>92.48</td><td>76.83</td><td>91.83</td><td>73.88</td><td>91.86</td><td>73.34</td><td>87.14</td><td>77.34</td><td>90.40</td><td>66.02</td><td>89.29</td><td>75.33</td><td>33.61</td><td>95.61</td><td>83.54</td><td>75.94</td></tr></table>
+
+Table 11: OOD detection results on ImageNet-1k where OOD datasets are iNaturalist, SUN, Places, and Textures.  $\uparrow$  indicates larger values are better and  $\downarrow$  indicates smaller values are better. Bold numbers are superior results whereas underlined numbers denote the second and third best results.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">ResNet50</td><td colspan="2">MobileNetV2</td><td colspan="2">ViT-B-16</td><td colspan="2">ViT-L-16</td><td colspan="2">SWIN-S</td><td colspan="2">SWIN-B</td><td colspan="2">MLP-B</td><td colspan="2">MLP-L</td><td colspan="2">Average</td></tr><tr><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td><td>FP↓</td><td>AU↑</td></tr><tr><td>MSP</td><td>64.76</td><td>82.82</td><td>70.47</td><td>80.67</td><td>61.74</td><td>83.12</td><td>65.22</td><td>81.75</td><td>59.68</td><td>83.75</td><td>62.79</td><td>81.38</td><td>69.36</td><td>81.97</td><td>76.01</td><td>80.04</td><td>66.25</td><td>81.94</td></tr><tr><td>ODIN</td><td>57.68</td><td>87.08</td><td>60.42</td><td>86.39</td><td>61.24</td><td>79.59</td><td>64.06</td><td>78.65</td><td>57.30</td><td>80.99</td><td>64.36</td><td>73.77</td><td>63.05</td><td>85.15</td><td>73.17</td><td>82.29</td><td>62.66</td><td>81.74</td></tr><tr><td>Energy</td><td>57.47</td><td>87.05</td><td>58.87</td><td>86.59</td><td>67.41</td><td>74.31</td><td>68.43</td><td>74.65</td><td>62.82</td><td>77.65</td><td>75.32</td><td>64.87</td><td>85.89</td><td>79.96</td><td>84.44</td><td>79.38</td><td>70.08</td><td>78.06</td></tr><tr><td>DICE</td><td>35.72</td><td>90.92</td><td>41.93</td><td>89.60</td><td>90.30</td><td>70.77</td><td>71.77</td><td>67.12</td><td>88.68</td><td>39.90</td><td>87.92</td><td>40.71</td><td>57.91</td><td>83.36</td><td>66.23</td><td>81.53</td><td>67.55</td><td>70.49</td></tr><tr><td>ReAct</td><td>30.70</td><td>93.30</td><td>50.09</td><td>88.81</td><td>67.08</td><td>81.78</td><td>69.58</td><td>83.50</td><td>50.09</td><td>87.92</td><td>64.86</td><td>83.24</td><td>90.41</td><td>77.86</td><td>87.01</td><td>79.13</td><td>63.73</td><td>84.44</td></tr><tr><td>BFAct</td><td>31.41</td><td>92.98</td><td>48.35</td><td>89.19</td><td>73.26</td><td>82.75</td><td>81.16</td><td>82.69</td><td>57.20</td><td>88.21</td><td>65.44</td><td>86.62</td><td>96.76</td><td>67.46</td><td>96.36</td><td>72.16</td><td>68.74</td><td>82.76</td></tr><tr><td>ASH-P</td><td>50.33</td><td>89.04</td><td>57.15</td><td>87.34</td><td>99.45</td><td>20.30</td><td>99.42</td><td>18.37</td><td>99.11</td><td>19.21</td><td>99.08</td><td>20.59</td><td>98.77</td><td>36.88</td><td>99.04</td><td>29.20</td><td>87.79</td><td>40.11</td></tr><tr><td>ASH-B</td><td>22.73</td><td>95.06</td><td>35.66</td><td>92.13</td><td>94.33</td><td>49.14</td><td>94.08</td><td>37.89</td><td>96.42</td><td>30.00</td><td>91.47</td><td>47.38</td><td>99.83</td><td>19.14</td><td>66.05</td><td>84.31</td><td>75.07</td><td>56.88</td></tr><tr><td>ASH-S</td><td>22.80</td><td>95.12</td><td>38.67</td><td>90.95</td><td>99.62</td><td>16.47</td><td>99.55</td><td>16.72</td><td>99.36</td><td>14.98</td><td>99.35</td><td>17.75</td><td>98.31</td><td>34.85</td><td>99.36</td><td>18.65</td><td>82.13</td><td>38.18</td></tr><tr><td>VRA-P</td><td>25.49</td><td>94.57</td><td>45.53</td><td>89.85</td><td>98.02</td><td>34.64</td><td>99.62</td><td>14.99</td><td>99.38</td><td>18.15</td><td>99.65</td><td>15.51</td><td>99.64</td><td>16.58</td><td>99.23</td><td>19.49</td><td>83.32</td><td>37.97</td></tr><tr><td>Ours (V)</td><td>31.53</td><td>93.78</td><td>49.09</td><td>89.62</td><td>62.01</td><td>85.80</td><td>68.61</td><td>85.10</td><td>61.47</td><td>87.04</td><td>62.96</td><td>86.09</td><td>80.23</td><td>78.83</td><td>82.59</td><td>78.92</td><td>62.31</td><td>85.65</td></tr><tr><td>Ours (E)</td><td>28.78</td><td>94.19</td><td>46.92</td><td>89.90</td><td>63.04</td><td>85.64</td><td>74.96</td><td>84.69</td><td>60.67</td><td>87.50</td><td>63.50</td><td>86.55</td><td>91.37</td><td>74.85</td><td>88.49</td><td>78.22</td><td>64.71</td><td>85.19</td></tr></table>

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+# TOWARDS POISONING FAIR REPRESENTATIONS
+
+Tianci Liu $^{1}$ , Haoyu Wang $^{1}$ , Feijie Wu $^{1}$ , Hengtong Zhang $^{2}$ , Pan Li $^{3}$ , Lu Su $^{1}$ , Jing Gao $^{1}$
+
+$^{1}$ Purdue University  $^{2}$ Tencent AI Lab  $^{3}$ Georgia Institute of Technology
+
+1{liu3351, wang5346, wu1977, lusu, jinggao}@purdue.edu
+
+2htzhang.work@gmail.com 3panli@gatech.edu
+
+# ABSTRACT
+
+Fair machine learning seeks to mitigate model prediction bias against certain demographic subgroups such as elder and female. Recently, fair representation learning (FRL) trained by deep neural networks has demonstrated superior performance, whereby representations containing no demographic information are inferred from the data and then used as the input to classification or other downstream tasks. Despite the development of FRL methods, their vulnerability under data poisoning attack, a popular protocol to benchmark model robustness under adversarial scenarios, is under-explored. Data poisoning attacks have been developed for classical fair machine learning methods which incorporate fairness constraints into shallow-model classifiers. Nonetheless, these attacks fall short in FRL due to notably different fairness goals and model architectures. This work proposes the first data poisoning framework attacking FRL. We induce the model to output unfair representations that contain as much demographic information as possible by injecting carefully crafted poisoning samples into the training data. This attack entails a prohibitive bilevel optimization, wherefore an effective approximated solution is proposed. A theoretical analysis on the needed number of poisoning samples is derived and sheds light on defending against the attack. Experiments on benchmark fairness datasets and state-of-the-art fair representation learning models demonstrate the superiority of our attack.
+
+# 1 INTRODUCTION
+
+Machine learning algorithms have been thriving in high-stake applications such as credit risk analysis. Despite their impressive performance, many algorithms suffered from the so-called fairness issue, i.e., they were shown biased against under-represented demographic subgroups such as females in decision making (Barocas & Selfst, 2016; Buolamwini & Gebru, 2018). As remedies, fair machine learning has accumulated a vast literature that proposes various fairness notions and attempts to achieve them (Pedreshi et al., 2008; Dwork et al., 2011; Hardt et al., 2016). For shallow models such as logistic regression and support vector machine, fairness notions are often defined upon scalar model predictions and have been well-studied, see Kamishima et al. (2011); Zafar et al. (2017) and reference therein for instances. We refer to this family of work as classical fair machine learning. Recently, fair representation learning (FRL) with deep neural networks (DNNs) has attracted great attention (Xie et al., 2017; Madras et al., 2018; Creager et al., 2019; Sarhan et al., 2020). FRL learns high-dimensional representations for downstream tasks that contain minimal information of sensitive features (i.e., the memberships of demographic subgroups). These information-based fairness notions equip FRL with higher transferability than classical methods (Creager et al., 2019).
+
+Despite the success of fair machine learning methods, not much is known about their vulnerability under data poisoning attacks until very recent studies (Chang et al., 2020; Solans et al., 2021; Mehrabi et al., 2021). Data poisoning attacks aim to maliciously control a model's behaviour to achieve some attack goal by injecting poisoning samples to its training data and are widely used to benchmark the robustness of machine learning models in adversarial scenarios (Bard & Falk, 1982; Biggio et al., 2012). Recently researchers successfully attacked classical fair machine learning methods such as fair logistic regression (Mehrabi et al., 2021) and exacerbated bias in model predictions, thereby hurting fairness. But it is still an open question whether FRL suffers from a similar threat<sup>1</sup>.
+
+Notwithstanding, devising a poisoning attack to degrade fairness of representations proves to be a non-trivial task. The difficulty is from two aspects. First, high-dimensional representations are more complicated to evaluate fairness than scalar predictions in classical fair machine learning. This makes existing attack goals for fairness degradation against the latter fall short to apply. Secondly, fair representations implicitly depend on the training data, and non-convex DNNs make this dependency hard to control by previous optimization- or heuristic-based attacks on classical fair machine learning. Optimization-based attacks (Solans et al., 2021) need the victim model to be simple enough (e.g., convex) to analyze. Heuristics such as label flipping (Mehrabi et al., 2021) do not directly optimize the attack goal, thereby often requiring great effort to design good manipulations to success.
+
+We propose the first data poisoning attack that directly targets on fairness of high-dimensional representations as shown in Figure 1. Our attack is optimization-based with a new attack goal. Specifically, following the common principle behind FRL that ideal fair representations should contain no information of sensitive features (Moyer et al., 2018; Zhao et al., 2019; Creager et al., 2019), we devise our attack goal to maximize the mutual information (MI) between the learned representations and sensitive features. The attack goal entails a bilevel optimization, whose challenges lie in two-folds.
+
+Firstly, MI does not admit analytic form, making its maximization challenging. Instead, a variational lower bound of MI (Barber &
+
+![](images/a3780f4e334c2f8c0f134b7e1c3b2f3a80772d70e97fc1a8419ac1d6a68d4d01.jpg)  
+Figure 1: Framework of data poisoning attack (in red) on FRL. Irrelevant components such as class labels are omitted. The attacker poisons training data to contaminate the victim training (solid lines), resulting in unfair representations  $z$  for target data (dotted lines) such that its MI to sensitive feature  $a$  is maximized. The MI is supposed to be small before the attack.
+
+Agakov, 2003) inspires us to use the negative loss of the optimal classifier that predicts sensitive features from representations as a proxy. To avoid the complexity of training such a classifier in the inner loop of the optimization, we further connect the classification loss with a principled notion of data separability. FLD scores give us an analytic measure of such and simplify the optimization substantially. Notably, FLD score as a valid data separability measure does not rely on the mixture-of-Gaussian assumption (Fisher, 1936), and our empirical results also show that this assumption is not essential for our purpose in practice. To our best knowledge, we propose the first attack goal based on MI explicitly towards fair machine learning that is highly principled and flexible. We also analyze its connection to demographic parity, one of the most popular fairness notions. This is one of the most significant contributions of our work.
+
+Secondly, representations are learned by DNNs, whose dependency on poisoning samples is impossible to track exactly. Consequently, one cannot identify training on what poisoning samples the victim will produce the most fairness-violating representations. We solve this problem approximately by matching the upper- and lower-level gradients following Geiping et al. (2020): The gradient of the victim from FLD score gives the desired update direction; when it matches the gradients of the victim on poisoning samples, training the victim on poisoning samples will solve the attack goal approximately. To improve stealthiness of the attack on fairness degradation in our scenarios, we design better constraints on how much poisoning samples can deviate from clean samples. In specific, we define a more proper valid range based on the data type, and an elastic-net penalty (Chen et al., 2018) to encourage such deviation to be sparse. Meanwhile, we derive the first theoretical minimal number of poisoning samples required by gradient-matching based attacks (Geiping et al., 2020) to succeed under regular conditions. This bound is crucial for practical attacks, as using more poisoning samples will increase the chance of being detected (Geiping et al., 2020; Koh et al., 2022). This theoretical analysis is another contribution of this paper and can be of independent interest to users of all gradient-matching based attacks.
+
+Extensive experimental results in Section 4 demonstrate high effectiveness of our attacks on four representative FRL methods using as few as  $5\%$  of training data for poisoning. In the remaining part of this paper, we review related works in Section 5, and conclude the work in Section 6.
+
+# 2 PROPOSED METHOD
+
+We propose the first white-box clean-label data poisoning attack on deep learning-based fair representation learning (FRL) to degrade fairness. We follow previous attacks on classical fair machine learning (Chang et al., 2020; Mehrabi et al., 2021) and suppose a worst-case threat model, which has full knowledge and control of the victim model, as well as the access to part of the training samples.
+
+# 2.1 PRELIMINARIES
+
+We denote a dataset consisting of  $N$  datapoints by  $\mathcal{D} = \{\pmb{x}_i, a_i, y_i\}_{i=1}^N$ , where  $\pmb{x}_i \in \mathbb{R}^M$  represents a multivariate nonsensitive feature,  $a_i \in \{0,1\}$  is a binary sensitive feature, and  $y_i$  is a binary class label. As adopted in literature (e.g., Moyer et al. (2018); Reddy et al. (2021)), we assume  $\pmb{x}$  has continuous values that can be fed into neural networks. A FRL model parameterized by  $\theta$  has an encoder  $h$  to learn representation from the nonsensitive feature by  $\pmb{z}(\theta) = h(\pmb{x};\theta)$  and is trained on  $\mathcal{D}$  to minimize some fairness-aware loss  $L(\mathcal{D};\theta)$ . A data poisoning attack aims to maliciously control the training of this victim model by perturbing a few training samples to minimize some attack loss  $U(\theta)$ . The perturbed training samples are referred to as poisoning samples.
+
+Mutual information (MI)-based fairness aims to minimize  $I(a, z)$  between sensitive feature  $a$  and representation  $z$ . It makes fair representations highly transferable on a wide range of downstream classification tasks and has become the de facto principle in FRL (Moyer et al., 2018; Creager et al., 2019; Zhao et al., 2019). Formally, the data processing inequality (Cover, 1999) states that  $I(a, z) \geq I(a, g(z)) \geq 0$  holds for any classifier  $g$  acting on  $z$ . In addition,  $I(a, g(z)) = 0$  is equivalent to demographic parity (DP, Zemel et al. (2013)), one of the most popular group fairness notions. At a colloquial level, if representations from different demographic groups are similar to each other, then any classifier acting on the representations will be agnostics to the sensitive feature and fair thereof. This condition remains valid without access to  $y$  when DP cannot be evaluated.
+
+# 2.2 POISONING FAIR REPRESENTATIONS FORMULATION
+
+Motivated by the importance of MI-based fairness in FRL, we attack the fairness on some target data  $\mathcal{D}_{\mathrm{ta}}$  by maximizing  $I(a,z)$ . This involves a bilevel optimization problem. Given a victim  $\theta$  and its lower-level loss  $L(\mathcal{D};\theta)$ , the training data  $\mathcal{D} = \mathcal{D}_{\mathrm{po}} \cup \mathcal{D}_{\mathrm{cl}}$  consists of  $P$  poisoning samples  $\mathcal{D}_{\mathrm{po}}$  and clean samples  $\mathcal{D}_{\mathrm{cl}}$ , the attacker wants to minimize  $-I(a,z)$  over target data  $\mathcal{D}_{\mathrm{ta}}$  by learning perturbations  $\Delta = \{\delta_p\}_{p=1}^P$  to add on  $\mathcal{D}_{\mathrm{po}}$  through solving
+
+$$
+\min  _ {\Delta \in \mathcal {C}} - I (a, z (\theta^ {*} (\Delta))), \quad \text {s . t .} \theta^ {*} (\Delta) = \operatorname {a r g m i n} _ {\theta} L \left(\mathcal {D} _ {\mathrm {p o}} (\Delta) \cup \mathcal {D} _ {\mathrm {c l}}; \theta\right). \tag {1}
+$$
+
+Our clean-label attack leaves original label  $y$  unpoisoned and only perturbs nonsensitive feature  $\pmb{x}$ , i.e.,  $\mathcal{D}_{\mathrm{po}}(\Delta) = \{\pmb{x}_p + \delta_p, a_p, y_p\}_{p=1}^P$ , under constraint  $\mathcal{C}$  which will be detailed shortly.
+
+Connection to attacking group fairness. Our attack is principled and jeopardizes DP. We use well-defined metric variation of information (VI, Kraskov et al. (2005)). For  $y$  and  $a$ , their VI distance is  $VI(y, a) = H(y) + H(a) - 2I(y, a)$  where  $H(\cdot)$  is the entropy. Applying triangle inequality to  $g(z)$ ,  $y$ , and  $a$  gives us  $I(g(z), a) \geq I(g(z), y) + I(a, y) - H(y) = I(g(z), y) - H(y \mid a)$ . By maximizing  $I(z, a)$  that upper bounds  $I(g(z), a)$ , a successful attack diminishes the guarantee for the MI-based fairness. When  $H(y \mid a) < I(g(z), y)$ , fitting  $g$  to predict  $y$  from  $z$  that maximizes  $I(g(z), y)$  will force  $I(g(z), a)$  to increase, thereby exacerbating the DP violation. Notably,  $H(y \mid a)$  depicts how dependent  $y$  and  $a$  are and is expected small when an fairness issue exists (Hardt et al., 2016). We provide empirical success of attacking DP with our attack goal in Appendix E.3.
+
+Unfortunately, Eq. (1) is intractable with difficulty lying in finding the most fairness-violating representations and the method for acquiring them. Mathematically, the first problem involves  $I(a,z)$  which lacks an analytic expression. This makes its computation non-trivial, let alone its maximization. The second entails a feasible set in the lower-level optimization that is NP-hard to identify due to the non-convexity of deep neural networks. We solve Eq. (1) approximately as follows.
+
+# 2.3 UPPER-LEVEL APPROXIMATION: FISHER'S LINEAR DISCRIMINANT (FLD)
+
+The lack of analytic form for  $I(a, z)$  necessitates some approximations. Our first step is to lower bound MI by a negative binary-cross-entropy (BCE) loss which is easier to estimate. For any classi-
+
+fier  $g$  that predicts  $a$  from  $\mathbf{z}$ , let  $q(a\mid z)$  be the distribution learned by the optimal  $g^{*}$ , we have
+
+$$
+I (a, \pmb {z}) = \mathbb {E} _ {p (a, \pmb {z})} \left[ \log \frac {p (a | \pmb {z}) q (a | \pmb {z})}{p (a) q (a | \pmb {z})} \right] \overset {(a)} {\geq} \mathbb {E} _ {p} \left[ \log \frac {q (a | \pmb {z})}{p (a)} \right] = \mathbb {E} _ {p} \left[ \log q (a | \pmb {z}) \right] + \mathbb {E} _ {p} \left[ - \log p (a) \right],
+$$
+
+where the inequality  $(a)$  holds from omitting some non-negative KL terms and is tight when  $g^{*}$  recovers the true distribution  $p(a\mid z)$  as shown in Barber & Agakov (2003). On the other hand, since  $\mathbb{E}_p[-\log p(a)] = H(a)\geq 0$  , the first term, negative BCE loss of  $g^{*}$  , is a lower bound for  $I(a,z)$  , and is a measure of how fair the representations are (Feng et al., 2019; Gupta et al., 2021). We dub the BCE loss of  $g^{*}$  the optimal BCE loss.
+
+However, substituting the MI maximization from Eq. (1) with minimizing the optimal BCE loss does not make it solvable:  $g^{*}$  depends on  $z$ 's for  $\mathcal{D}_{\mathrm{ta}}$  and how to update them to minimize the BCE loss of  $g^{*}$  is unknown. This requires differentiate through the whole optimization procedure.
+
+To walk around this challenge, we note that the optimal BCE loss of  $g^{*}$  measures how difficult to separate  $z$  for  $\mathcal{D}_{\mathrm{ta}}$  with  $a = 0$  or 1. While this difficulty is hard to tackle directly, it can be approximated by the data separability: If the two classes of data are more separable, one can expect the classification simpler and the optimal BCE loss lower. Motivated by this, we instead maximize Fisher's linear discriminant (FLD) score, a closed-formed data separability measure.
+
+Specifically, suppose the two classes of representations have mean  $\pmb{\mu}^0, \pmb{\mu}^1$  and covariance  $\mathbf{S}^0, \mathbf{S}^1$  respectively. FLD maps them to a 1-dimensional space via linear transformation  $\pmb{v}$  which induces separation  $s_{\pmb{v}} = (\pmb{v}^\top \pmb{\mu}^0 - \pmb{v}^\top \pmb{\mu}^1)^2 / (\pmb{v}^\top \mathbf{S}^0 \pmb{v} + \pmb{v}^\top \mathbf{S}^1 \pmb{v})$ . Its maximal value over  $\pmb{v}$ , termed as FLD score, is  $s = (\pmb{\mu}^0 - \pmb{\mu}^1)^\top (\mathbf{S}^0 + \mathbf{S}^1)^{-1} (\pmb{\mu}^0 - \pmb{\mu}^1)$  when  $\pmb{v} \propto (\mathbf{S}^0 + \mathbf{S}^1)^{-1} (\pmb{\mu}^0 - \pmb{\mu}^1)$ . This equation allows us to compute its gradient with respect to  $\pmb{z}$  for  $\mathcal{D}_{\mathrm{ta}}$ , which gives the direction to update these representations in order to make it less fair. For stability we regularize  $s$  by
+
+$$
+s = \left(\boldsymbol {\mu} ^ {0} - \boldsymbol {\mu} ^ {1}\right) ^ {\top} \left(\mathbf {S} ^ {0} + \mathbf {S} ^ {1} + c \mathbf {I}\right) ^ {- 1} \left(\boldsymbol {\mu} ^ {0} - \boldsymbol {\mu} ^ {1}\right), \tag {2}
+$$
+
+and resort to solving the following bilevel optimization:
+
+$$
+\min  _ {\Delta \in \mathcal {C}} - s \left(\theta^ {*} (\Delta)\right), \quad \text {s . t .} \theta^ {*} (\Delta) = \operatorname {a r g m i n} _ {\theta} L \left(\mathcal {D} _ {\mathrm {p o}} (\Delta) \cup \mathcal {D} _ {\mathrm {c l}}; \theta\right). \tag {3}
+$$
+
+In Appendix B we extend our attack to multi-class sensitive feature scenarios.
+
+Remark 2.1. Maximizing  $I(z, a)$  is a general framework to poison FRL and admits other proxies such as sliced mutual information (Chen et al., 2022), kernel canonical correlation analysis (Akaho, 2007), and non-parametric dependence measures (Székely et al., 2007). In this work, we use FLD because of its conceptual simplicity, interpretability, and good empirical performance. As one may recognize, when  $p(z \mid a = 1)$  and  $p(z \mid a = 0)$  are Gaussian with equal variance, FLD is optimal (Hamsici & Martinez, 2008) whose BCE loss attains the tight lower bound of  $I(z, a)$  up to constant  $H(a)$ . In this case, our method provably optimizes the lower bound of  $I(z, a)$  whereas other proxies may not due to the lack of direct connections to mutual information. While the Gaussianity may not hold in general, FLD score as a measure of data separability is still valid (Fisher, 1936), and we verify its efficacy for our goal in Appendix E.2, where we show that FLD score is highly informative for the empirical minimal BCE loss of a logistic regression.
+
+# 2.4 LOWER-LEVEL APPROXIMATION: ELASTIC-NET GRADMATCH (ENG)
+
+Bilevel optimization in Eq. (3) enjoys a tractable upper-level loss but its lower-level optimization remains challenging due to the use of deep neural networks in FRL and needs further approximation. To this end, we fix parameter  $\theta$  in a pre-trained victim model and treat its lower-level gradient on poisoning samples  $\nabla_{\theta}L(\mathcal{D}_{\mathrm{po}}(\Delta);\theta)$  as a function of  $\Delta$ . An attack is launched by aligning  $\nabla_{\theta}L(\mathcal{D}_{\mathrm{po}}(\Delta);\theta)$  and  $-\nabla_{\theta}s(\theta)$  by maximizing their cosine similarity
+
+$$
+\mathcal {B} (\theta , \Delta) = - \frac {\left\langle \nabla_ {\theta} s (\theta) , \nabla_ {\theta} L \left(\mathcal {D} _ {\mathrm {p o}} (\Delta) ; \theta\right) \right\rangle}{\| \nabla_ {\theta} s (\theta) \| \| \nabla_ {\theta} L \left(\mathcal {D} _ {\mathrm {p o}} (\Delta) ; \theta\right) \|}, \tag {4}
+$$
+
+with respect to  $\Delta$ . When the two directions are matched, gradient descents on the lower-level loss  $L$  from poisoning samples will decrease the upper-level loss  $-s$  as well.
+
+The concept of gradient matching in poisoning attacks was initially introduced in GradMatch by Geiping et al. (2020) for image classification. To enhance the stealthiness of such attacks in fairness degradation scenarios, we impose two new constraints on the learning perturbations. First, a
+
+hard constraint  $\mathcal{C}$  mandates that poisoning samples must reside within the clean data domain, i.e.,  $\min_{\boldsymbol{x}_c \in \mathcal{D}_{\mathrm{el}}} x_{cm} \leq \delta_{pm} + x_{pm} \leq \max_{\boldsymbol{x}_c \in \mathcal{D}_{\mathrm{el}}} x_{cm}$  for all feature dimensions ( $1 \leq m \leq M$ ) and poisoning samples ( $1 \leq p \leq P$ ), this allows using dimension-specific ranges. Second, the soft constraint employs the elastic-net penalty (Zou & Hastie, 2005; Chen et al., 2018) to promote sparsity in each  $\delta_p$  (i.e., only a few dimensions are perturbed). Combining these constraints, we formulate below optimization problem aimed at poisoning fair representations:
+
+$$
+\min  _ {\Delta \in \mathcal {C}} - \mathcal {B} (\theta , \Delta) + \sum_ {p = 1} ^ {P} \left(\lambda_ {1} \| \delta_ {p} \| _ {1} + \lambda_ {2} \| \delta_ {p} \| _ {2} ^ {2}\right). \tag {5}
+$$
+
+In execution, we rescale the three terms before tuning  $\lambda$ 's to avoid bearing with the magnitude difference. Non-differentiable  $L_{1}$  norms are tackled with the iterative shrinkage-thresholding algorithm (ISTA, Beck & Teboulle (2009))  $\delta_p^{k+1} = \mathrm{Proj}_{\mathcal{C}}\left(S_{\lambda_1}\left[\delta_p^k + \alpha^k\nabla(\mathcal{B}(\theta,\Delta) - \lambda_2\|\delta_p\|_2^2)\right]\right)$ , where  $S_{\lambda_1}$  is the element-wise projected shrinkage-thresholding function and  $\mathrm{Proj}_{\mathcal{C}}$  is the projection onto  $\mathcal{C}$ . We refer to our attack by Elastic-Net GradMatch (ENG) and summarize it in Algorithm 1.
+
+Computation complexity. Denote the number and dimension of poisoning samples by  $P$  and  $M$ , iteration numbers of running the attack by  $T$ , dimension of  $\theta$  by  $D$ . The computation complexity of ENG and GradMatch (Geiping et al., 2020) are  $\mathcal{O}(TM + TDPM)$  and  $\mathcal{O}(TDPM)$ , respectively. The additional computation cost  $\mathcal{O}(TM)$  due to the one-step ISTA in ENG is marginal.
+
+# 3 ANALYSIS ON MINIMAL NUMBER OF POISONING SAMPLES
+
+It is non-trivial to minimize the number of poisoning samples that deteriorate the victim model's performance. An insufficient amount may not impact lower-level optimization, while a large amount makes it easy to detect the attack, leading to a direct failure (Huang et al., 2020; Koh et al., 2022). Our analysis is built upon the convergence of upper-level loss  $U(\theta) \equiv -s(\theta)$ . Our conclusion relies on the following assumptions.
+
+Assumption 3.1 (smooth upper- and lower-level losses). There exists  $C > 0$  such that upper-level loss  $U(\theta)$  and lower-level loss  $L(\theta)$  are  $C$ -smooth, i.e., their gradients are  $C$ -Lipschitz continuous.
+
+Assumption 3.2 (attack well-trained victims only). Before attack, the victim is well-trained so its gradient on each clean sample performs like a random noise with mean zero and finite norm  $\sigma$ .
+
+Assumption 3.3 (well-matched gradients). After being attacked, gradient  $\nabla_{\theta}L(\theta)$  evaluated on any poisoning sample is an unbiased estimator of the gradients  $\nabla_{\theta}U(\theta)$  with bounded error norm, i.e., for any poisoning sample  $p$ ,  $\nabla L_p(\theta) = \nabla_{\theta}U(\theta) + \epsilon_p$  where  $\mathbb{E}[\epsilon_p] = 0$  and  $\mathbb{E}[||\epsilon_p||]\leq \sigma$ .
+
+Assumption 3.1 is a fairly weak condition that has been widely used (Colson et al., 2007; Mei & Zhu, 2015; Sinha et al., 2017), the rest two are introduced by us and are well-suited to the context of our proposed framework. Assumption 3.2 is valid because  $\theta$  undergoes thorough training with the clean samples. Assumption 3.3 presumes that given constant gradient  $\nabla_{\theta}U(\theta)$ , we can construct an unbiased estimator with  $P$  poisoning samples  $\mathbb{E}\left[\frac{1}{P}\sum_{p = 1}^{P}\nabla_{\theta}L_{p}(\theta)\right] = \nabla_{\theta}U(\theta)$ . With these assumptions, we obtain the following theorem.
+
+Theorem 3.4. Suppose that Assumption 3.1, 3.2, and 3.3 hold. Let  $P$  and  $N$  be the number of poisoning and total training samples, respectively. Set the learning rate to  $\alpha$  and the batch size to  $n$ . Then, the ratio of poisoning data  $P / N$  should satisfy
+
+$$
+\frac {P}{N} \geq c + \frac {\alpha C \sigma^ {2}}{2 n \| \nabla_ {\theta} U (\theta) \| ^ {2}} + \frac {\alpha C}{2}, \tag {6}
+$$
+
+such that the upper-level loss  $U(\theta)$  is asymptotic to an optimal model. Here  $c$  is a small constant (e.g.,  $10^{-4}$ ) for sufficient descent, and  $\theta$  is a well-trained model on the clean data.
+
+Deferring the proof to Appendix C, we summarize the underlying idea. We assume the pretrained victim converged with respect to  $\nabla_{\theta}L(\theta)$ . Besides, after applying ENG, a poisoning sample can (in expectation) induce a gradient equal to  $\nabla_{\theta}U(\theta)$  so training the victim model on it will optimize the upper-level loss  $U(\theta)$ . However, clean samples may obfuscate this goal as their lower-level gradients do not depend on the upper-level loss. To counteract with this effect, a minimal portion of poisoning
+
+samples are needed to dominate the training. In practice, learning rate  $\alpha$  is often small compared with  $C$ , batch size  $n$  is large, and  $\| \nabla_{\theta}U(\theta)\|$  is much greater than 0. Therefore, the minimal portion bound is expected to be smaller than 1.
+
+Practical Significance. Theorem 3.4 sheds light on the difficulty of ENG and other GradMatch-based attacks, whereon a defending strategy can be built. If batch size  $n$  is large, the attack is simpler and needs fewer poisoning samples. So reducing  $n$  should help defense and we evaluate its performance in Appendix E.7. In addition, term  $\sigma^2$  is affected by the lower-level gradients, whose increase will require more poisoning samples. This helps explain why adding noises to gradients can defend against GradMatch as verified in Geiping et al. (2020).
+
+# Algorithm 1 Craft Poisoning Samples with ENG Attack
+
+1: Input: clean data  $\mathcal{D}_{\mathrm{cl}}$ , poisoning data  $\mathcal{D}_{\mathrm{po}}$ , target data  $\mathcal{D}_{\mathrm{ta}}$ ; victim  $\theta$  and its lower-level loss  $L$ ; number of pretraining epochs  $E$  and attack iterations  $T$ .  
+2: Fix  $\mathcal{D}_{\mathrm{po}}$  unperturbed. Pretrain victim on  $\mathcal{D}_{\mathrm{po}} \cup \mathcal{D}_{\mathrm{cl}}$  for  $E$  epochs and obtain  $\theta^{E}$ .  
+3: Compute upper-level gradient  $\nabla_{\theta}U(\mathcal{D}_{\mathrm{ta}};\theta^{E})$  
+4: Randomly initialize  $\Delta^0 = \{\delta_p,p = 1,\dots ,P\}$  in  $\mathcal{C}$  
+5: for  $t = 1, \dots, T$  do  
+6: Compute lower-level gradient  $\nabla_{\theta}L(\mathcal{D}_{\mathrm{po}}(\Delta^t);\theta^E)$  as a function of  $\Delta^t$  
+7: Compute ENG loss in (5) and update  $\Delta^t$  with ISTA.  
+8: end for  
+9: return  $\mathcal{D}_{\mathrm{po}}$  when using  $\Delta^T$ .
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed attack on four FRL models trained on two fairness benchmark datasets and show its effectiveness through extensive experiments. Ablation studies and practical insights are also given to help understand how the attack succeeds in poisoning victims.
+
+# 4.1 SETUP
+
+Attack Goals. We consider three variants of Eq. (2) to maximize: (a) FLD follows Eq. (2) with  $c = 10^{-4}$  to stabilize the covariance estimation. (b) sFLD takes the same form as FLD but does not back-propagate through covariance terms when computing  $-\nabla_{\theta} s(\theta)$ . (c) EUC replaces the covariance terms with an identity matrix<sup>2</sup>.
+
+Attacks. An attack to maximize score  $X$  using ENG is referred to as ENG-X, for instance, ENGFLD is to maximize FLD. Conceptually, ENG-EUC is suitable for small  $\mathcal{D}_{\mathrm{ta}}$  as it omits covariance matrix, which can be unstable to estimate. ENG-FLD should be favored when  $\mathcal{D}_{\mathrm{ta}}$  is large as it is based on the exact FLD score  $s$  and should measure separability more accurately, which is expected to benefit solving Eq. (5) as well. ENG-sFLD strikes a balance in between by using only part of covariance information.
+
+FRL Victims. We select four representative FRL models as our victims. CFAIR and CFAIR-EO (Zhao et al., 2019) are based on adversarial learning and seek to achieve different fairness notions. Non-adversarial learning based ICVAE-S and ICVAE-US (Moyer et al., 2018) differ in whether  $y$  is accessible. We follow the official codes to implement the victims as detailed in Appendix D.
+
+Datasets. We train victims on two benchmark datasets from UCI repository that are extensively studied in fair machine learning, which are pre-processed $^3$  following Zhao et al. (2019); Reddy et al. (2021). Adult (Kohavi, 1996) contains 48,842 samples of US census data with 112 features and the objective is to predict whether an individual's yearly income is greater than $50K dollars or not. The sensitive feature is gender. German (Dua & Graff, 2017) consists of 1,000 samples of personal financial data with 62 features and the objective is to predict whether or not a client has a good credit score. The sensitive feature is binarized age as in Moyer et al. (2018) and we adjust the threshold to increase its correlation with label $^4$ . In Appendix E.6 we study multi-class sensitive feature race. On both datasets, we leave  $20\%$  of total samples out as  $\mathcal{D}_{\mathrm{ta}}$ . More results on COMPAS (Dieterich et al., 2016) and Drug Consumption Datasets (Dua & Graff, 2017) are presented in Appendix. E.8.
+
+Evaluation. We treat decrease in the BCE loss of a logistic regression predicting  $a$  from  $\pmb{z}$  as a measure of increase in  $I(z,a)$ . To verify how group fairness and representation utility can be affected, we present the exacerbation of DP violation and accuracy of predicting  $y$  from  $\pmb{z}$  in Appendix E.3 and Appendix E.4 respectively. Representations are extracted after training the victims on poisoned Adult and German dataset for 20 and 50 more epochs respectively considering their size difference.
+
+Baselines. We compare ENG-based attacks with four variants of anchor attack (AA), a recent heuristic generic poisoning attack on classical fair machine learning (Mehrabi et al., 2021). RAA-y and RAA-a randomly picks one training sample from the subgroup with  $(y = 1, a = 0)$  and one with  $(y = 0, a = 1)$  after each epoch, then makes copies of the two chosen samples with flipped  $y$  or  $a$  respectively. NRAA-y and NRAA-a replaces the random selection in RAA counterparts with picking from each subgroup the training sample that has the most neighbors within a pre-specified radius and has not been selected yet. (N)RAA-y were proposed in Mehrabi et al. (2021), and we implement (N)RAA-a following the same logic. Note that (N)RAA-y are not clean-label, (N)RAA-a are but they directly modify the sensitive feature. In contrast, our attacks only modify the nonsensitive feature. Moreover, all baselines are allowed to use any training sample to poison, while ours can only perturb a given randomly selected set of poisoning samples. These differences put our proposed attacks in an unfavorable situation under direct comparison. Nevertheless, ours can still outperform the four baselines by a large margin.
+
+# 4.2 COMPARISON BETWEEN ENG AND (N)RAA.
+
+We compare three ENG-based attacks with penalty coefficients  $\lambda_{1} = 0.0025, \lambda_{2} = 0.005$  (Eq. (5)) against four AA attacks under different settings where  $5\%$  to  $15\%$  of training data are used for poisoning. Performance of an attack is measured by the decrease of BCE loss, higher is better, and corresponding DP violations are reported in Appendix E.3.
+
+Figure 2 shows results averaged over 5 replications. Three ENG-based attacks achieved notable performance (on both BCE loss and DP violations) in various settings. In contrast, AA encountered severe failures, for instance, when attacking CFAIR trained on German dataset, only RAA-a succeeded with all the three budgets. Such failures cannot be fully attributed to the budgets: (N)RAA-y succeeded with budget  $10\%$  but failed with budget  $15\%$ . (N)RAA-a occasionally achieved the best performance because of much stronger capacities. Nonetheless, the proposed ENG-based attacks beat AA baselines in terms of better and more reliable performance by a large margin in most case. When comparing the three proposed attacks with each other, their performance difference matched our previous analysis, e.g., ENG-FLD gave the best result on larger Adult dataset. These results clearly establish the efficacy of our attack.
+
+![](images/9095acc748898d0d82ebcf30a194f09436dae3063f3142c30d0a53cd54ba8a77.jpg)
+
+![](images/c93da8c1958a7ff61c1b8f1d81ba3161628a0f2a58de18fdd8425a89eafdc932.jpg)  
+(a) Adult dataset
+
+![](images/1b7df66d30f8ce85af4d4050e6467de83be1af41961e885e341870a524d717f8.jpg)
+
+![](images/e96c2f267d6ac0e5da3ef4cfd61b2ddd07d4a5433895d9cd525ebb526ad95161.jpg)
+
+![](images/a4311ef2d27409b1a5ceede3f656785ce099522d8e94a885089812727091ac13.jpg)  
+Figure 2: ENG-based attacks reduce BCE loss more than AA baselines with less portion of poisoning samples. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/9c6f5fd14447dd950c8bea2905ec3e5388b5ef0d255d76d3b8dbf619e0329a2a.jpg)  
+(b) German dataset
+
+![](images/c8a842febe3294932b5e7e50c189a44c559471efd4a6337abd21e4cdeae43702.jpg)
+
+![](images/916261c29fe8b5c119d60c49390bb964567484fe95c175c567c566b9aded8c67.jpg)
+
+# 4.3 PERFORMANCE OF ELASTIC-NET PENALTY
+
+Next, we sprovide a sensitivity analysis of ENG-based attacks against the choices of  $\lambda_{1}$  and  $\lambda_{2}$ . We set  $\lambda_{2} = 2\lambda_{1}$  and vary  $\lambda_{1}$  following Chen et al. (2018). Results are again averaged over 5 replications. Figure 3a exhibits how elastic-net penalty affects the performance of ENG-FLD under different poisoning budgets where victims are trained on Adult dataset. More results are deferred to Appendix E.5 due to page length but conclusions here hold in general. All three attacks are relatively robust to small and intermediate level of elastic-net penalty. Moreover, improvement from applying the elastic-net penalty is observed (e.g., column 1 in Fig 3a). This implies that the penalty can actually help stabilize the optimization.
+
+We further study how elastic-net penalty regularizes the learned perturbations by computing  $L_{1}$  and  $L_{2}$  norms. We only present  $L_{1}$  norms of perturbations using ENG-EUC attack on Adult dataset as an illustration and defer others to Appendix E.5 after observing similar trends. According to Figure 3b,  $L_{1}$  norms were significantly shrank by elastic-net penalty with mild degradation on the attack performance. For instance, elastic-net penalty with  $\lambda_{1} = 0.0025$  effectively reduced the  $L_{1}$  norm of perturbations by a third without hurting the performance of attacking CFAIR. When a more stealthy attack is wanted,  $\lambda_{1} = 0.01$  was able to launch a satisfactory attack with only one half budget of perturbation norm. These results clearly show the efficacy of our proposed ENG-based attacks.
+
+Given these merits of elastic-net penalty, one may ask if it can be used in other attacks such as AA. In Appendix E.5 we discuss the difficulty of doing so and highlight its affinity to our attack by nature.
+
+![](images/92e786858ea6143fef88e447dd00cf85642aaa4f0aa49b029f1c7eee1cb4765e.jpg)
+
+![](images/902fc7c30fcecd638de71335817d0d4f5469b00356be0030accde8249624efd1.jpg)
+
+![](images/11ce412892e9912090498e0f952cb51d48024846ec3710ef59828b1517ee69d2.jpg)
+
+![](images/c5f92d9137e9de7143f25501c0c443bbdee67b69ff6b013e1877ab1d4d75654f.jpg)
+
+![](images/b8271ad6469ed408fc5457f7dee589e4a2e2d4129c9a246a0adb4c7c4b433e54.jpg)  
+(a) Decrease of BCE loss is insensitive to under small and intermediate level of elastic-net penalty.
+
+![](images/a8f892ac14b418fdff2b58ab1ab370929dffbb67324716cd7a56e7e37c7cbdd8.jpg)  
+(b)  $L_{1}$  norm of learned perturbations is effectively restricted by elastic-net penalty.
+
+![](images/86862e68cfc9fb6c0567a69f5ebad26df45f0b1ac4cb1c85e25d365cfecbf988.jpg)  
+Figure 3: Decrease of BCE loss and  $L_{1}$  norm of perturbations learned by ENG-FLD attack. Victims are trained on Adult dataset and results are averaged over 5 replications.
+
+![](images/d60c70b2704c22de52269bd2f5725df990c87514ab36535659bd81d83c028215.jpg)
+
+# 4.4 ROBUST FEATURES OF VICTIMS
+
+ENG-based attacks can do feature selections, and we consider unselected features as robust ones for the victim, in the sense that they did not help in attacking. We end this section with a case study on this aspect. Table 1 shows the percentage of selected features and corresponding robust features when attacking ICVAE-S and ICVAE-US trained on Adult dataset with ENG-FLD. From the table, elastic-net penalty successfully reduced the perturbed features by  $20\%$ , and robust features for ICVAE-US and ICVAE-S are largely overlapped. These results helped understand how ENG-FLD was launched by walking around these robust features when attacking the victims. Note that this robust features identification is not applicable for AA-based attacks.
+
+# 5 RELATED WORK
+
+Fair representation learning (FRL) is a family of algorithms to learn fair representations from nonsensitive features such that any downstream tasks (e.g., classification) acting on these representations will be fair. Towards this goal, different approaches to removing sensitive information from
+
+learned representations have been proposed. To name a few, Zemel et al. (2013) set representations to be multinomial and used fairness violations based on the representations as penalties, and Madras et al. (2018) derived different bounds for these violations used for adversarial regularization. Mutual information and other measures between distributions have also been used as penalties to encourage independence between the representation and sensitive features, either in an adversarial (Xie et al., 2017; Creager et al., 2019) or non-adversarial way (Louizos et al., 2015; Moyer et al., 2018; Sarhan et al., 2020; Wang et al., 2021). Recently, Zhao et al. (2019) proposed to learn different encoders on different label groups with theoretical guarantees and achieved state-of-the-art performance in an adversarial manner Reddy et al. (2021). Moyer et al. (2018), on the other hand, provided a non-adversarial based solution and also got promising results. We select representative methods from adversarial and non-adversarial regimes to test our attack.
+
+Data poisoning attack aims to achieve some attack goal with a victim model by controlling its training via injecting poisoned samples. Early works showed that simple heuristics such as label flipping (Barreno et al., 2010; Paudice et al., 2019) can succeed in attack. However, these poisoning samples often look unnatural and are easy to detect (Papernot & McDaniel, 2018; Paudice et al., 2019). Consequently, clean-label attacks that only modify the poisoning samples' features but not labels are pre
+
+Table 1: Percentage of selected features and top robust features from Adult dataset,  $P/N$  denotes the portion of poisoning samples, attacker is ENG-FLD.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">P/N</td><td colspan="4">λ1(λ2=2λ1)</td><td rowspan="2">Top Robust Features</td></tr><tr><td>0</td><td>0.0025</td><td>0.005</td><td>0.01</td></tr><tr><td rowspan="3">ICVAE-S</td><td>0.05</td><td>0.99</td><td>0.96</td><td>0.89</td><td>0.73</td><td rowspan="3">workclass, marital-status, occupation, native-country</td></tr><tr><td>0.1</td><td>0.99</td><td>0.95</td><td>0.89</td><td>0.75</td></tr><tr><td>0.15</td><td>0.99</td><td>0.97</td><td>0.89</td><td>0.77</td></tr><tr><td rowspan="3">ICVAE-US</td><td>0.05</td><td>0.95</td><td>0.89</td><td>0.84</td><td>0.79</td><td rowspan="3">native-country, occupation, workclass, education</td></tr><tr><td>0.1</td><td>0.95</td><td>0.91</td><td>0.85</td><td>0.79</td></tr><tr><td>0.15</td><td>0.96</td><td>0.91</td><td>0.85</td><td>0.79</td></tr></table>
+
+ferred (Shafahi et al., 2018). Another drawback of heuristic attacks is the lack of performance guarantee as they do not directly solve the attack goals. In practice they may perform less effective.
+
+Bilevel optimization is widely used for data poisoning attacks (Bard & Falk, 1982; Biggio et al., 2012; Geiping et al., 2020; Jagielski et al., 2021; Koh et al., 2022). For convex victims such as logistic regression and support vector machine, the lower-level optimization is characterized by the KKT condition. This reduces the bilevel optimization to a constrained optimization that can be solved exactly (Mei & Zhu, 2015). For other victims, unfortunately, when their optimal solutions are NP-hard to identify, so are the bilevel optimization problems (Colson et al., 2007; Sinha et al., 2017) and inexact solutions are needed. When the second-order derivatives of the lower-level loss is cheap, using influence function to identify influential samples for the victim training and poisoning them can produce strong attacks (Koh & Liang, 2017). These attacks have been successfully applied to classical fair machine learning (Chang et al., 2020; Solans et al., 2021; Mehrabi et al., 2021), but the non-convexity of neural networks and expensive influence function computations make them unsuitable for poisoning FRL. Approximate solutions for attacking deep learning models have been proposed recently. For instance, inspired by model-agnostic meta-learning (MAML, Finn et al. (2017)), MetaPoison (Huang et al., 2020) back-propagated through a few unrolled gradient descent steps to capture dependency between the upper- and lower-level optimizations. GradMatch (Geiping et al., 2020) matched the gradient of upper- and lower-level losses and achieved state-of-the-art performance. However, it is unclear how to apply them to poison FRL. In this work, we propose the first work towards this goal and reduce it to an approximate optimization that can be handled.
+
+# 6 CONCLUSION AND FUTURE WORKS
+
+We develop the first data poisoning attack against FRL methods. Driven by MI-based fairness in FRL, we propose a new MI maximization attack goal and reveal its connection to existing fairness notion such as demographic parity. We derive an effective approximate solution to achieve this attack goal. Our attack outperforms baselines by a large margin and raises an alert of the vulnerability of existing FRL methods. We also theoretically analyze the difficulty of launching such an attack and establish an early success of principled defense. Motivated by promising results on tabular data, which is the primary focus of many FRL methods (Moyer et al., 2018; Zhao et al., 2019), we plan to extend our attack to fair machine learning on large-scale image and text datasets that also relies on deep neural networks, and delve into attacking these methods in the future. In addition, Jagielski et al. (2021) showed that an attack can be much more effective towards certain subpopulations and impossible to defend against, and we plan to explore this for further improvement of our attack.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by the US National Science Foundation under grant NSF IIS-2226108. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
+
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+
+# A MOTIVATION OF ATTACKING FAIR REPRESENTATIONS
+
+Attacking mutual information at representation level, as formalized in Section 2, aligns better with fair representation learning (FRL) and is more universal than attacking classifiers. For example, a bank may want to obtain a representation  $z$  for each user that can be used to determine their eligibility for existing and upcoming financial products such as credit cards without concerning about fairness again (Zhao et al., 2019). Here each financial product is associated with a (unique) label, and determining the eligibility entails a classification task. In this case, there are two challenges for delivering a classification-based attack. First, one has to determine and justify which classifier to use and why consider the fairness of this specific classification task. Second, for any upcoming financial product, its label does not exist and one cannot obtain classifier (we need this label to train the classifier), let along attacking it. In contrast, a representation-level attack can overcome the two challenges in a single shot. As discussed in Section 2, for any classifier  $g$  acting on  $z$ , by maximizing the mutual information  $I(z,a)$  between  $z$  and sensitive feature  $a$ ,  $I(g(z),a)$  will be maximized so long as the fairness concern exists. This implies launching attack on all labels simultaneously, including the ones where classifiers cannot be trained.
+
+# B EXTENSION TO MULTI-CLASS SENSITIVE FEATURE
+
+To attack a FRL method trained on multi-class sensitive feature  $a \in [K]$ , We first define  $\tilde{a}_k = \mathbf{1}(a = k)$  to mark whether the sample belongs to the  $k$ -th sensitive group. Then we immediately have  $I(z, a) \geq I(z, \tilde{a}_k)$  thanks to the data processing inequality. This implies that
+
+$$
+I (\boldsymbol {z}, a) \geq \frac {1}{K} \sum_ {k = 1} ^ {K} I (\boldsymbol {z}, \tilde {a} _ {k}).
+$$
+
+Note that in RHS each term is the mutual information between  $z$  and binarized  $\tilde{a}_k$  and is lower bounded by the BCE loss, therefore we can approximate each one with a FLD score readily. This idea is similar to transforming a  $K$ -class classification to  $K$  one-vs-all binary classifications. We report empirical results in Appendix E.6.
+
+Further simplification is viable by choosing some specific  $k$  and attacking  $I(z, \tilde{a}_k)$  only. This allows one to launch an attack when  $K$  is large. However, it is noteworthy that handling a sensitive feature that involves many groups (large  $K$ ) is a general challenge for fair machine learning. The difficulty is twofold. First, it involves more complicated constraints, making the problem harder to optimize. Second, by categorizing data into more fine-grained sensitive groups, each group will have fewer samples and the algorithm may suffer from unstable estimation issues (Jiang et al., 2022; Liu et al., 2023). As a result, when the number of sensitive groups is large, fair machine learning methods often bin them into a few larger groups in pre-processing (Zemel et al., 2013; Moyer et al., 2018; Zhao et al., 2019; Reddy et al., 2021).
+
+# C OMITTED PROOF OF THEOREM 3.4
+
+We start with restating Theorem 3.4.
+
+Theorem C.1. Suppose that Assumption 3.1, 3.2, and 3.3 hold. Let  $P$  and  $N$  be the number of poisoning and total training samples, respectively. Set the learning rate to  $\alpha$ , and assign the batch size with  $n$ . Then, the ratio of poisoning data (i.e.,  $P / N$ ) should satisfy
+
+$$
+\frac {P}{N} \geq c + \frac {\alpha C \sigma^ {2}}{2 n \| \nabla_ {\theta} U (\theta) \| ^ {2}} + \frac {\alpha C}{2}, \tag {7}
+$$
+
+such that the upper-level loss  $U(\theta)$  (i.e., negative FLD score) is asymptotic to an optimal model. Here  $c$  is a small constant (e.g.,  $10^{-4}$ ) for sufficient descent, and  $\theta$  is a well-trained model on the clean data.
+
+Proof. Without loss of generality, we assume that the victim is trained by mini-batched stochastic gradient descent, i.e., given current parameter  $\theta^{\mathrm{old}}$ , the new value is updated by
+
+$$
+\theta^ {\mathrm {n e w}} = \theta^ {\mathrm {o l d}} - \frac {\alpha}{n} \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}),
+$$
+
+where  $L_{i}(\theta)$  denotes the loss on the  $i$ -th training sample. Let  $p$  denote the number of poisoning samples selected in the current batch, we have that  $p$  follows a Hypergeometric distribution. Given  $p = n_p$  poisoning samples in the current batch, we collect all randomness in the minibatch gradient as
+
+$$
+e ^ {\text {o l d}} = \sum_ {i = 1} ^ {n _ {p}} \epsilon_ {i} + \sum_ {i = 1} ^ {n - n _ {p}} \nabla_ {\theta} L _ {i} (\theta^ {\text {o l d}}).
+$$
+
+According to Assumption 3.3 and 3.2 we have  $\mathbb{E}[e^{\mathrm{old}}] = 0$  and
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \| e ^ {\mathrm {o l d}} \| ^ {2} \right] = \mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n _ {p}} \epsilon_ {i} + \sum_ {i = 1} ^ {n - n _ {p}} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}} \right\| ^ {2} \right] \\ = \sum_ {i = 1} ^ {n _ {p}} \mathbb {E} [ \| \epsilon_ {i} \| ^ {2} ] + \sum_ {i = 1} ^ {n - n _ {p}} \mathbb {E} [ \| \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \| ^ {2} ] + 0 \\ \leq n \sigma^ {2}, \\ \end{array}
+$$
+
+as all crossing terms have mean zero. Moreover, we can express
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right) = \underbrace {\sum_ {i = 1} ^ {n _ {p}} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right)} _ {\text {P o i s o n i n g s a m p l e s}} + \underbrace {\sum_ {i = 1} ^ {n - n _ {p}} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right)} _ {\text {C l e a n s a m p l e s}} \\ = \sum_ {i = 1} ^ {n _ {p}} \nabla_ {\theta} U (\theta^ {\text {o l d}}) + \sum_ {i = 1} ^ {n _ {p}} \epsilon_ {i} + \sum_ {i = 1} ^ {n - n _ {p}} \nabla_ {\theta} L _ {i} (\theta^ {\text {o l d}}) \\ = n _ {p} \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) + e ^ {\mathrm {o l d}}. \\ \end{array}
+$$
+
+This implies that further given Assumption 3.2, the minibatch gradient is a biased estimator of the gradient of the upper-level loss. To see this, by the law of total expectation
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \right] = \mathbb {E} \left[ \mathbb {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \mid p = n _ {p} \right] \right] \\ = \frac {1}{n} \mathbb {E} \left[ \mathbb {E} \left[ p \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) + e ^ {\mathrm {o l d}} \mid p = n _ {p} \right] \right] \\ = \mathbb {E} \left[ p \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right) \right] \\ = \frac {1}{n} \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \mathbb {E} [ p ] \\ = \frac {P}{N} \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right). \\ \end{array}
+$$
+
+Under Assumption 3.1, descent lemma gives us
+
+$$
+\begin{array}{l} U \left(\theta^ {\text {n e w}}\right) \leq U \left(\theta^ {\text {o l d}}\right) + \left\langle \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right), \theta^ {\text {n e w}} - \theta^ {\text {o l d}} \right\rangle + \frac {C}{2} \| \theta^ {\text {n e w}} - \theta^ {\text {o l d}} \| ^ {2} \\ = U (\theta^ {\mathrm {o l d}}) - \alpha \langle \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}), \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \rangle + \frac {\alpha^ {2} C}{2 n ^ {2}} \| \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \| ^ {2}. \\ \end{array}
+$$
+
+Take expectation on both side with respect to the mini-batch we get
+
+$$
+\begin{array}{l} \mathbb {E} [ U (\theta^ {\mathrm {n e w}}) ] \leq U (\theta^ {\mathrm {o l d}}) - \alpha \langle \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}), \frac {P}{N} \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \rangle + \frac {\alpha^ {2} C}{2 n ^ {2}} \mathbb {E} \| \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} (\theta^ {\mathrm {o l d}}) \| ^ {2} \\ = U \left(\theta^ {\text {o l d}}\right) - \frac {\alpha P}{N} \| \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right) \| ^ {2} + \frac {\alpha^ {2} C}{2 n ^ {2}} \mathbb {E} \| \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right) \| ^ {2}, \tag {8} \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathbb {E} \| \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right) \| ^ {2} = \mathbb {E} \left[ \mathbb {E} \left[ \| \sum_ {i = 1} ^ {n} \nabla_ {\theta} L _ {i} \left(\theta^ {\text {o l d}}\right) \| ^ {2} \mid p = n _ {p} \right] \right] \\ = \mathbb {E} \left[ \mathbb {E} \left[ \| p \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) + e ^ {\mathrm {o l d}} \| ^ {2} \mid p = n _ {p} \right] \right] \\ = \mathbb {E} \left[ \mathbb {E} \left[ p ^ {2} \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2} + \| e ^ {\mathrm {o l d}} \| ^ {2} + 2 p \langle \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}), e ^ {\mathrm {o l d}} \rangle \mid p = n _ {p} \right] \right] \\ = \mathbb {E} [ p ^ {2} ] \| \nabla_ {\theta} U (\theta^ {\text {o l d}}) \| ^ {2} + \mathbb {E} [ \| e ^ {\text {o l d}} \| ^ {2} ] + 0 \\ \leq n ^ {2} \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2} + n \sigma^ {2}. \\ \end{array}
+$$
+
+Plugging in back to equation (8) we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ U \left(\theta^ {\text {n e w}}\right) \right] \leq U \left(\theta^ {\text {o l d}}\right) - \frac {\alpha P}{N} \| \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right) \| ^ {2} + \frac {\alpha^ {2} C}{2 n ^ {2}} \left(n ^ {2} \| \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right) \| ^ {2} + n \sigma^ {2}\right) \\ = U (\theta^ {\mathrm {o l d}}) - (\frac {\alpha P}{N} - \frac {\alpha^ {2} C}{2}) \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2} + \frac {\alpha^ {2} C \sigma^ {2}}{2 n}. \\ \end{array}
+$$
+
+Therefore, a sufficient descent such that
+
+$$
+\mathbb {E} \left[ U \left(\theta^ {\text {n e w}}\right) \right] \leq U \left(\theta^ {\text {o l d}}\right) - c \alpha \| \nabla_ {\theta} U \left(\theta^ {\text {o l d}}\right) \| ^ {2},
+$$
+
+for some  $c\geq 0$  can be guaranteed by
+
+$$
+(\frac {\alpha P}{N} - \frac {\alpha^ {2} C}{2}) \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2} - \frac {\alpha^ {2} C \sigma^ {2}}{2 n} \geq c \alpha \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2}
+$$
+
+Rearrange
+
+$$
+\frac {P}{N} \geq c + \frac {\alpha C \sigma^ {2}}{2 n \| \nabla_ {\theta} U (\theta^ {\mathrm {o l d}}) \| ^ {2}} + \frac {\alpha C}{2}
+$$
+
+This completes our proof.
+
+# D MORE IMPLEMENTATION DETAILS
+
+We provide more details about victims' architectures and training. Our model architectures follow official implementations in Moyer et al. (2018); Zhao et al. (2019).
+
+On Adult dataset, we use
+
+CFAIR:
+
+Encoder: linear, representation  $z \in \mathbb{R}^{60}$ .  
+- Discriminators: one hidden layer with width 50, using ReLU activation.  
+- Classifier: linear.  
+- Training: AdaDelta optimizer with learning rate 0.1, batchsize 512, epochs 50.
+
+CFAIR-EO:
+
+Encoder: linear, representation  $z \in \mathbb{R}^{60}$ .  
+- Discriminators: one hidden layer with width 50, using ReLU activation.
+
+- Classifier: linear.  
+- Training: AdaDelta optimizer with learning rate 0.1, batchsize 512, epochs 50.
+
+# - ICVAE-US:
+
+- Encoder: one hidden layer with width 64, output representation  $\mathbf{z} \in \mathbb{R}^{30}$ , using Tanh activation.  
+- Decoder: one hidden layer with width 64, using Tanh activation.  
+- Classifier: one hidden layer with width 64. using Tanh activation.  
+- Training: Adam optimizer with learning rate 0.001, batchsize 512, epochs 50.
+
+# - ICVAE-S:
+
+- Encoder: one hidden layer with width 64, output representation  $\mathbf{z} \in \mathbb{R}^{30}$ , using Tanh activation.  
+- Decoder: one hidden layer with width 64, using Tanh activation.  
+- Classifier: one hidden layer with width 64. using Tanh activation.  
+- Training: Adam optimizer with learning rate 0.001, batchsize 512, epochs 50.
+
+On German dataset, we use
+
+# CFAIR:
+
+Encoder: linear, representation  $\mathbf{z} \in \mathbb{R}^{60}$ .  
+- Discriminators: one hidden layer with width 50, using ReLU activation.  
+- Classifier: linear.  
+- Training: AdaDelta optimizer with learning rate 0.05, batchsize 64, epochs 300.
+
+# CFAIR-EO:
+
+Encoder: linear, representation  $\mathbf{z} \in \mathbb{R}^{60}$ .  
+- Discriminators: one hidden layer with width 50, using ReLU activation.  
+- Classifier: linear.  
+- Training: AdaDelta optimizer with learning rate 0.05, batchsize 64, epochs 300.
+
+# - ICVAE-US:
+
+- Encoder: one hidden layer with width 64, output representation  $z \in \mathbb{R}^{30}$ , using Tanh activation.  
+- Decoder: one hidden layer with width 64, using Tanh activation.  
+- Classifier: one hidden layer with width 64. using Tanh activation.  
+- Training: Adam optimizer with learning rate 0.001, batchsize 64, epochs 300.
+
+# - ICVAE-S:
+
+- Encoder: one hidden layer with width 64, output representation  $z \in \mathbb{R}^{30}$ , using Tanh activation.  
+- Decoder: one hidden layer with width 64, using Tanh activation.  
+- Classifier: one hidden layer with width 64. using Tanh activation.  
+- Training: Adam optimizer with learning rate 0.001, batchsize 64, epochs 300.
+
+During training, we followed GradMatch (Geiping et al., 2020) and did not shuffle training data after each epoch. For better comparison, victims were always initialized with random seed 1 to remove randomness during the pre-training procedure. In different replications, we selected different poisoning samples with different random seeds. Experiments that consist of 5 replications used seed 1 to 5 respectively.
+
+# E MORE EXPERIMENT RESULTS
+
+# E.1 PRACTICABILITY OF PERTURBING PRE-PROCESSED DATA
+
+Many poisoning attacks on images classifiers perturb raw data (namely pixels, (Huang et al., 2020; Geiping et al., 2020)), in this work we perturb pre-processed data. However, this does not necessarily undermine the practical significance of our work. To see why attacking the pre-processed data is practical, we take a view from the scope of data security. Many FRL methods are applied in high-stakes domains, such as loan application or job market screening. Due to the privacy concern, Data anonymization has been used by more and more data providers to protect their data privacy and is often done as a part of the data pre-processing as discussed in Iyengar (2002); Ram Mohan Rao et al. (2018). In such cases, a malicious data provider can release a poisoned pre-processed (anonymous) dataset and launch the attack on victim models trained with it.
+
+# E.2 FLD SCORE APPROXIMATES BCE LOSS
+
+We evaluated how well the optimal BCE loss of logistic regression can be approximated by three FLD scores used in our experiments: FLD, sFLD, and EUC. To this end, we train each victim for 50 epochs and compute the empirical optimal BCE loss of a logistic regression to predict  $a$  from representation  $z$  after each epoch. Then we compare the trend of BCE loss versus FLD, sFLD, and EUC scores.
+
+Figure 4 visualizes the negative value of FLD, sFLD, and EUC and associated BCE losses of four victims trained on Adult and German dataset with and without poisoning samples crafted by corresponding ENG attack after each epoch. In all cases, the three scores approximate how BCE loss changes very well.
+
+# E.3 EFFECTS ON DP VIOLATIONS
+
+Here we report increase of DP violations of victims attacked by three ENG-based attacks and AA-based baselines in Figure 5. As analyzed in Section 2, our attack successfully increased the DP violation significantly on most setting, clearly establishing their effectiveness.
+
+# E.4 EFFECTS ON ACCURACY
+
+Here we report change of accuracy of predicting  $y$  from representation  $z$  learned by victims attacked by three ENG-based attacks and AA-based baselines in Figure 6. In general, all attacks had relatively subtle influence on prediction accuracy, and our proposed ENG-based attacks often changed accuracy less. These results imply that poisoned representations are still of high utility, and it will be difficult for the victim trainer to identify the attack by checking the prediction performance alone.
+
+# E.5 PERFORMANCE OF ELASTIC-NET PENALTY
+
+Here we report more experimental results on evaluating the performance of elastic-net penalty under varying hyper-parameters. Figure 7 and 9 reports how elastic-net penalty affects the attack performance, corresponding  $L_{1}$  and  $L_{2}$  norms of learned perturbations are reported in Figure 10 and 11, respectively.
+
+We further compare elastic-net penalty versus  $L_{1}$  norm penalty for regulating the attack. Figure 8 shows corresponding decrease of BCE loss and Figure 12 shows corresponding  $L_{1}$  and  $L_{2}$  norms. Compared with elastic-net penalty, penalizing  $L_{1}$  norm only usually resulted in larger  $L_{1}$  and  $L_{2}$  norms while the attack performance was hardly improved, especially when small- to intermediate-level coefficient(s)  $(\lambda_{1}$  and  $\lambda_{2}$ ) of the regularizer is used.
+
+Given the advantages of utilizing elastic-net penalty, one may ask if it can be used in other attacks such as AA (Mehrabi et al., 2021) as well. Here we delve into the difficulty of such an application, highlighting the suitability of elastic-net penalty for our attack by nature.
+
+According to Mehrabi et al. (2021), both RAA and NRAA constructed poisoning samples from a chosen anchor point by perturbing its  $x$  randomly within a  $\tau$ -ball and flipping its  $y$  or  $a$ . The
+
+![](images/65eb18afd26fd342c68d53ca014b4391e6cb555a91e1020564055b46a30d4beb.jpg)  
+Figure 4: Changes of FLD, sFLD, and EUC loss (the negative score) and corresponding BCE loss. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/3499a346c53bc1203c8ac79ebff1f1b8019a0bf24d32d25522a583e71815426f.jpg)  
+(b) German dataset  
+Figure 5: Increase of DP violations from different attackers using  $5\%$  -  $15\%$  training samples for poisoning, Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/6b1dfa31cf404435920d65fcaf2e81ee38ef07670746a942e8c32eb4de6246b6.jpg)
+
+![](images/15537a5413ed3d2fce85070f52f1de79522d906468373f1f50dd0e23e5e54323.jpg)  
+(a) Adult dataset
+
+![](images/7eb5817f1c0491560e6a3f2b2762c8ea9242f96cd92f2d346bf8facd7086c04a.jpg)
+
+![](images/48797c41f2276337fe3e30f42f5c808a976dd9da3bcdef43206ff037eab0c7e3.jpg)
+
+![](images/70ad5406867aeccc1f7439baa7fb6eddb556133acd2c45a3010e8e7ca3a16983.jpg)  
+Figure 6: Accuracy of predicting  $y$  from  $z$  when different attackers using  $5\% - 15\%$  training samples for poisoning, Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/7628ee985154f53374b5c188fbc67673c1fa980035b0f57dec10346837d91884.jpg)  
+(b) German dataset
+
+![](images/384b8fab863a4c7ccbb952a714f873d135e1507322405a3262c7d229776c8618.jpg)
+
+![](images/350358a8268105d65d572b79ba8584da25491ee993314ac852c777e5c87f9885.jpg)
+
+main difference lies in how they choose these anchor samples: RAA draws them uniformly from the training data. NRAA computes the pairwise distance between all training data and counts how many neighbors and chooses the training samples that have the most neighbors with flipped  $y$  or  $a$  as anchor data. Here two samples are neighboring if their  $y$  and  $a$  are same and the difference in their  $x$  is smaller than a pre-specified threshold.
+
+It is viable to define  $\tau$ -ball based on the elastic-net norm. Nonetheless, the use of  $\tau$ -ball will make NRAA harder to analyze in general. NRAA is thought stronger (which is empirically verified in both Mehrabi et al. (2021) and our experiment) because the chosen poisoning samples are likely to have higher influence as they have more neighbors. However, sampling from  $\tau$ -ball will make some poisoning samples have more neighbors while others have less. In fact, the authors of AA used  $\tau = 0$  throughout their experiments and we adopted this setting. This difficulty remains unsolved when using elastic-net norm to induce the  $\tau$ -ball. In conclusion, ENG, as a perturbation-learning-based attack, has a better affinity to the elastic-net penalty than AA baselines.
+
+# E.6 ATTACKING MULTI-CLASS SENSITIVE FEATURE
+
+Here we show empirical results of attacking victims trained on Adult dataset using race as the sensitive feature. Due to data imbalance issue we keep white and black group as they are. All other sensitive groups are re-categorized as others. This results in a 3-way sensitive feature  $a$  and we measure  $I(z, a)$  by the decrease of cross-entropy (CE) loss to predict  $a$  from  $z$  with a linear Softmax classifier. Note that both the CE loss and the averaged one-vs-all BCE loss (see Appendix B for details) lower bounds  $I(z, a)$ , but we report CE loss for the sake of better interpretability. Due to the lack of official implementation of CFAIR and CFAIR-EO on multi-class sensitive feature, we only attack ICVAE-US and ICVAE-S. As shown in Figure 13, ENG-based attack outperforms four AA baselines.
+
+# E.7 DEFENSE AGAINST ENG ATTACKS
+
+Here we present reducing batch size as a defense strategy against ENG attacks. We only defend against successful attacks and focus on Adult dataset where attacks used  $\lambda_{1} = 0.0025$ ,  $\lambda_{2} = 0.005$ . We vary the batch size between 256 to 1024 and keep all other settings same as in Section 4. Resultant decrease of upper-level loss  $-s$  and BCE loss are reported in Figure 14. Reducing batch size successfully helped weaken the performance of attacking 3 out of 4 victims in terms of  $-s$ .
+
+Note that according to Theorem 3.4, it is conceptually viable to increasing learning rate  $\alpha$  to defend. However, using a large learning rate in a well-trained victim model may have side effect of making it diverge. Because of this, we consider reducing batch size a better and more practical choice.
+
+![](images/621e950daee4b70fd04c4215403e799bdc7999c6c3bf5942298c791fe57ef876.jpg)
+
+![](images/50141848ed2b82f0b7a57f2b53a0fe81ac7e541f1f8ee719f278c7b09621e00c.jpg)  
+(a) ENG-FLD attack
+
+![](images/d3cc3f0582f0184a8d0531aee19cf4e7ae6323d25095c37dd9b911824d7cccdf.jpg)
+
+![](images/797c0086f1b5508e34e4c27b3755aacd3f942649a6913fab1616ea69c69620cd.jpg)
+
+![](images/859976ecaf8c3515933dd81ca9b6ff60c74a2bd9b1743ec774c9121491f4a6ed.jpg)
+
+![](images/ad184a2c66a1d363496c82e6b6783e7cd1b03e47f743b8f0b1573055b5b1cb80.jpg)
+
+![](images/fa5b7e7979f27d8c89d62cecc052cbc5c66f8b6bab5b3d79ac77cc84066963b4.jpg)  
+(b) ENG-sFLD attack
+
+![](images/80c0d5c0cbcc5a3025d8fb679652e22ddb11954e41752682f992bba5a58acf8a.jpg)
+
+![](images/8b336766465fc30a0ff0f622512617eb5c1b654fabdfb3fc4e9c702e525770b2.jpg)
+
+![](images/d83b84aad8914274cf49c059a4c4899749830f945db6dd5763e19572bdbb00f6.jpg)  
+Figure 7: Decrease of BCE loss achieved by different ENG-based attacks with varying hyperparameters, victims are trained on Adult dataset. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/2df9dba04bd2e567791bb23c23bdf3190c92d3dedf91ac43f0cba9512d590f74.jpg)  
+(c) ENG-EUC attack
+
+![](images/56d8bc780fc69eddf735c50cea88cf1c749ed006831fd75545bf75065fa096e7.jpg)
+
+![](images/9d2d2f36bad6338bf238406798a78fdc4ad45e87b1b73430671e2c9c3c9459ae.jpg)
+
+![](images/68972fa4328e9bbcdef54602168e63ff7360397c708d1466703d54ef66a2c380.jpg)
+
+![](images/c1488051db8ac439d5e4e8e7da2f01f4bd8fdf1e6f100c99fd7861c92f0e9665.jpg)
+
+![](images/12c368c61f18e806541a13fc599ff0f4a8d3996a0bbb31093ed94a6a7a20662f.jpg)
+
+![](images/b92b5e45645ee3dc0d5be41e4e175af8e83a10433c9b9d89ac88304b1b39c2d8.jpg)
+
+![](images/a081c8e7399220163f6b5eab90662acb37318c6fe521ddd3dd199071da435394.jpg)
+
+![](images/827b84b54dcd3ae9f960bd68e5baa492ae01ab22c0e3c9d263fb3de8f2e71949.jpg)  
+(a) ENG-FLD attack with only  $L_{1}$  penalty
+
+![](images/f929d13c44a9866109c0373a2515e91f43feb3af28aa6e175fc3ea9c2a4943dd.jpg)
+
+![](images/198b87f9aefc01c2cf72311e21568d164d21f485798c97a0ed2e2c60497dd5df.jpg)
+
+![](images/94d42c81fe0abc4a5cf9f6d6317ce3efaf90cfe32aac9bbf63db7429e61c9ad4.jpg)  
+Figure 8: Decrease of BCE loss achieved by different ENG-based attacks with varying hyperparameters on  $L_{1}$  penalty and no  $L_{2}$  penalty, victims are trained on Adult dataset. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/91098601e76ffe2fa1cdb5be294da3802e3b2849d093600e5001a40bffb0889b.jpg)  
+(b) ENG-sFLD attack with only  $L_{1}$  penalty
+
+![](images/1b8c85e9f9442aac7bab3223e1083409b473501b6b09da56d1b9ab77d2a2188a.jpg)  
+(c) ENG-EUC attack with only  $L_{1}$  penalty
+
+![](images/55ea3315e542cd7548aab4ec2ee304962201d66693522a4f4e00a54c10afe526.jpg)
+
+![](images/3bdbd04d6f6947b875545dd8dde945adbef0c541ab300f76add5c3852ee4e710.jpg)
+
+![](images/f4701da098127d53154c38cabb3147380815233e01c43db5f56739b8f4c912f6.jpg)  
+(a) ENG-FLD attack
+
+![](images/1c66b42bc0eb6e8ca0548b1f1ac9704ca2c02978ba5b452b9849f5b23e75444d.jpg)
+
+![](images/dd38f0d68efadecb2971d0dfc46f4f5b0459b7efa7968f3487061fd604ae61ca.jpg)
+
+![](images/8af7ea069ab1f816d88e0306132db0b0115fd3a6f66882b9aeb8b0e4c6f7a61f.jpg)
+
+![](images/c5145dffa9fd870655b9abe109b193036e247c90a94e658259f41afc3b7bd7ee.jpg)  
+(b) ENG-sFLD attack
+
+![](images/bee668ed0f6d6ec63b2df10b942adb10f7fcfab506f47055be4bf06962283f6d.jpg)
+
+![](images/a81694fd6a40f701a9db5b0d063c46df8fd6e508378bc43879a0c77ed728de3d.jpg)
+
+![](images/15020149d042e66b4d935a8801d80f138d0cfaf866d14710eef49aee18a9eb44.jpg)  
+Figure 9: Decrease of BCE loss achieved by different ENG-based attacks with varying hyperparameters, victims are trained on German dataset. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/bab77d9726c83fe82a31a2cf666e6c2e3405c2f727cfd63db657bbcbcc29f74b.jpg)  
+(c) ENG-EUC attack
+
+![](images/65c621030d1fc8ff2fa7abdb4f3bc2a0a7b4040cadc80eadc3dd9a6a44be6788.jpg)
+
+![](images/dff29a42c3c5890c5f1ceeb6c3a5b53a35382b03fc3429df0660ac09b4109c25.jpg)
+
+# E.8 MORE EXPERIMENT RESULTS ON COMPAS AND DRUG CONSUMPTION DATASETS
+
+Here we present more experimental results of attacking four FRL methods trained on COMPAS and Drug Consumption datasets. We adopted the same setting for data preprocessing and training-testing splitting pipelines as presented in Section 4 on Adult and German datasets. In specific, we report decrease of BCE losses in Figure 15, increase of DP violations in Figure 16, and accuracy of predicting  $y$  from  $z$  in Figure 17. Again, our attacks succeeded on all cases, outperforming AA baselines to a large extent.
+
+In terms of model architecture, we adopted the recommended architectures from Zhao et al. (2019) for CFAIR and CFAIR-EO on COMPAS dataset; and the default setting presented in Appendix D on Drug Consumption datasets due to the lack of official implementations. For ICVAE-US and ICVAE-S, we used the default setting from Appendix D on both datasets because of the lack of official implementations.
+
+![](images/aeb1085b27e8b1e0675591b608e7acfb08b446bed7498b7608fe1fb18812d4bc.jpg)
+
+![](images/17b7ab4fba1e2592d7f9232b0ac201623954ec63b4d4ce69a58749344834231c.jpg)  
+(a) ENG-FLD attack,  $L_{1}$  norm
+
+![](images/90e83e2667b07d7692ab552ffcdd59acabe0c06716a87421bed136326f18d10f.jpg)
+
+![](images/186272ee94b574e82aa6ed4609b421134d9df5a273d75f61e5af09fc6da8d271.jpg)
+
+![](images/9a5e098505fc11fb52d79f7e8c6c28f0ec78eb92d3d930ef7da93a3fab310b63.jpg)
+
+![](images/a4650b8cb3cb47af90afaa5d332d97e8251d18f2d0770055cb0fb14cfdc71741.jpg)  
+(b) ENG-sFLD attack,  $L_{1}$  norm
+
+![](images/c40cd01b4955c1899b9067c6d8a83c554e9edadefe050aaea1ce12728ad7fa21.jpg)
+
+![](images/b774651f2cb97a6f2e0590f691c6242187d7fff269365da61e7865bc07232468.jpg)
+
+![](images/966357cda6e0a7806019d3cdc48c418367587b6832ba9f48d86d01aa15430ea0.jpg)
+
+![](images/ea7e7947eb8cda2f4cfa85252a3cf787db6f425cf4009415f8fb68e848fb7b1f.jpg)  
+(c) ENG-EUC attack,  $L_{1}$  norm
+
+![](images/e88f045013cf5f8b0ceaea68e5112910e1bad8ededcc739620c3cc8001b79119.jpg)
+
+![](images/805a302fc48c76ab380257894e248e98b842f4c1445dd8590416d1f44b0322e5.jpg)
+
+![](images/4dba5dcb75e16bde16e2fc6f43eaf50bb7a3f7d4a6efe3994afb1e23eac0b1ee.jpg)
+
+![](images/7a5ad1fdc36e574784d99e500ed7cd405d7e15163872edaa28c60859e5b90462.jpg)  
+(d) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/4a14473e5f45a49e4ba3bc70a183e28e7629fa15edba557daccb2f9eff9cf0f4.jpg)
+
+![](images/fce4978ff28ea31950f76ebcc503db73ed6537d8ac26404c98b241ee0ec88a67.jpg)
+
+![](images/ced213eb21e65887f98adde0b39c9eb4ddaa8d353eda800de5d178fe4e9361f2.jpg)
+
+![](images/6c17082911a91ac899cdfd400ad5b7502255b35b4e0daabe9fd692d023f32927.jpg)  
+(e) ENG-EUC attack,  $L_{2}$  norm
+
+![](images/1ed60d703c36d430130ccaec61bfb3549c13d7267159b5bed38b54c2c6f5aa8d.jpg)
+
+![](images/146d4b3771454fbb7abe15304ed180621fdafe41d8d46d150eb5fd0b7d44e5df.jpg)
+
+![](images/d587bc2ebb83f04e17a163840bcfee53ffe4891eb8bfd40b718bf7b3eccaff17.jpg)  
+Figure 10: Averaged  $L_{1}$  and  $L_{2}$  norm of perturbations learned by different ENG-based attacks with varying hyper-parameters, victims are trained on Adult dataset.
+
+![](images/288f82eb73e6d5ce0825ae451e1da7b2e52de740a39e4efdabb9df644a1eecf3.jpg)  
+(f) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/010d4baa92ec16b058cc2009bdc7878f27c6ab64ac600786ca1dd9ff36bcfb14.jpg)
+
+![](images/7be5c2436f81afc614ae7617fe90f4fc7a9c8d10f4b061c16ee49a63908be0a9.jpg)
+
+![](images/329f75597954eb5a6adc7cbcd0525ffb39035c64641280dc966072df6f9c2648.jpg)
+
+![](images/9ceabc665c65e9a3a1af3a595410479cbe6a679020cd2bc6e062fe8c05004bbe.jpg)  
+(a) ENG-FLD attack,  $L_{1}$  norm
+
+![](images/4102cfc3da2986b7b5380ccd088c7c84814dc90cf805833dfa7388a798db1a59.jpg)
+
+![](images/01f442e3aa5654131571fb1daeb0a058ca1655d5e7fef9f860cc250a81725161.jpg)
+
+![](images/03dbf208ba7a7730d991b90eb622ff5ab50b2034d2196d0cc279a8b2463e2c1d.jpg)
+
+![](images/80ee9f08199b419f8cf3b3512601dc8559fe7030bb23a7411669798ec93ee935.jpg)
+
+![](images/5b2661c08ec316f206c7859bb6623552d0cdeacb45134f2e9cfb54a07921cc48.jpg)  
+(b) ENG-sFLD attack,  $L_{1}$  norm
+
+![](images/3275eae898526b7b0a0b2b7e018f9611fcf901bcd5c2e0e27263321e044993b3.jpg)
+
+![](images/93196ed42f323ab6fc05b637e93371545388a9af9892c31738eb0530c78b8363.jpg)
+
+![](images/2da87d5c541f0e9ccfebdb766c4da03b617ad3c79faed2b1d3bf8423511ae687.jpg)
+
+![](images/1e9fc17097bdbfd20e848e5b0b1d4726ed23957e29a3da891cec1302ae0a66e4.jpg)  
+(c) ENG-EUC attack,  $L_{1}$  norm
+
+![](images/dd616140f04900cf24203fa52584f673d9c9c045a866c8a944063bb12e3a2a47.jpg)
+
+![](images/88cc8cc43d7caf125fcac621a05d1975da02d62f76b00dbb67a3be1833f6ce20.jpg)
+
+![](images/94da50808f7f93647a45c6a2f53d251104215693699bc4b542d59b869d4c78de.jpg)
+
+![](images/1cc246f55d879c225a07c97c2686496fa22e0b3f711f7217dbe4cb5c21195717.jpg)  
+(d) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/46a0fa7d00a15a7773693e3116eda12308d77be00fca03544a3ddc6e2a8636c1.jpg)
+
+![](images/c65a02da91c687bfb27d8074e45ff2587cc290213048585deb463584c3a2d25f.jpg)
+
+![](images/f55a66faf143b05c45ca0e06ccae5454abd2cb17b4706fd08f267d18450bf5eb.jpg)
+
+![](images/a36aed598cc715294f0979e27ab0fa5838729ed146033a19ce5c21e9abc40733.jpg)  
+(e) ENG-EUC attack,  $L_{2}$  norm
+
+![](images/a7441c5ebefb30c4d8b5f17f4f5dedeead16ead3d4d9ec8fbd502bf718b7f04e.jpg)
+
+![](images/f9eebe7d177b7d5d1d2b0ee8f0c5b6c08b33e317a0e527c330580416487c5c72.jpg)
+
+![](images/a657c4d311e29075781253e07f5777a0e02d65f06efaad7d8b9ee869bed11f88.jpg)  
+Figure 11: Averaged  $L_{1}$  and  $L_{2}$  norm of perturbations learned by different ENG-based attacks with varying hyper-parameters, victims are trained on German dataset.
+
+![](images/0061ef84bb967d1529454b7d2e76116af6dd62ad38eec0819d9eab948f792de9.jpg)  
+(f) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/56d7d5f3a8a92c5f030aba6c8c049cb6a47deb15894fe28074c8933763838764.jpg)
+
+![](images/e29fed65581d35ff7daef8a6d135865e6745e22b5f3907e2e77b8c952d68e787.jpg)
+
+![](images/5026898eab1325fe84488b5261953f6a084f60fa942af240312951542c71fb35.jpg)
+
+![](images/98947bbc6613c41094f991d4dbbebb2cc2ccd6990b1e5703ec09b296cebc1d51.jpg)  
+(a) ENG-FLD attack,  $L_{1}$  norm
+
+![](images/dfdb7ad6651d44c8311d844cf2ea032e4fbb9971131223c86d3ffe85f5474941.jpg)
+
+![](images/5296dda98b294dccc7c3c557d8a97a53c55903ea6ac437ec287c8206ec8fdbf0.jpg)
+
+![](images/c020d033583d688d527c83d260b178cebd977bd05d1cd1f0402823b7287597c7.jpg)
+
+![](images/ba69d345b9952d3e3483fac17223b9bdceb02317a88b70b124981813a3c0e0c6.jpg)  
+(b) ENG-sFLD attack,  $L_{1}$  norm
+
+![](images/8634f103d1d6ea7e0b65edc30f7758e49cf97e8a4963be3c2ee5069964dcbe6e.jpg)
+
+![](images/41edbe12b412f3ea79a5fe757dc525720e404f4a5c0f3d7ff16bb399bdd4e36d.jpg)
+
+![](images/d2d961db9ae1b85241447d9285608dd5c7a90ecefd8b6eb2a465d12b2d5efb30.jpg)
+
+![](images/7adf4fa0d6377286186ac942adca0eeaf266d6a0e006ac59ee23dc19e09a78a5.jpg)  
+(c) ENG-EUC attack,  $L_{1}$  norm
+
+![](images/a70714aebce9736c493a14689dd398644bdcacc85948833850c39b4c152a64eb.jpg)
+
+![](images/2b500f0bd6364bd417976d5953e64ec1f25b5a662be63390b63682bfb9207a99.jpg)
+
+![](images/be7828985d847d81825a61bcf63b6cf1c9d866068959679674687b9f2563466c.jpg)
+
+![](images/dce0410c213bf583118436e414414f534d366f18d9e36f521f9ebb106a2c441f.jpg)  
+(d) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/00e3779f09093cd87827d5aa5626e282a149fe65b42e8a1c46b404428a78fcfd.jpg)
+
+![](images/4f5b339d848883b06924e5c23d70803a8028e975c97083faade9c700a7adaf5c.jpg)
+
+![](images/db5e02578b6a113db930f73a1ecde3c2674f5c0ce116d82eb3a100f13d4927a2.jpg)
+
+![](images/34bdd6c3878825b4bbca334992d54bf4ee2c2c9a2e3774b8aa259f073b29daf5.jpg)  
+(e) ENG-EUC attack,  $L_{2}$  norm
+
+![](images/ac22d60aed0e3a90ca26f3fbc3f55222cead170e7333a42379a870466295d5bc.jpg)
+
+![](images/3ae17d81f93c55018027efadfe5bcd62f47d825f8dea7080d158d90f8ee8db5f.jpg)
+
+![](images/356fc4a748c043e4dfad4a1867ecbb55b7e147725a4295cb78ddfca80984a540.jpg)  
+Figure 12: Averaged  $L_{1}$  and  $L_{2}$  norm of perturbations learned by different ENG-based attacks with varying hyper-parameters on  $L_{1}$  penalty and no  $L_{2}$  penalty, victims are trained on Adult dataset.
+
+![](images/87802b2fc3745c7821d71f1a4649951880873396d44f42a5418204c6070747df.jpg)  
+(f) ENG-sFLD attack,  $L_{2}$  norm
+
+![](images/e6847441feb02b40f10af3ccbc4d67f2da61b5fed521c79113a05148e00373b8.jpg)
+
+![](images/20449bb3f93ea617a9d354fde8b70ee28883e0404e394853ab7b5fed16cbc9ad.jpg)
+
+![](images/a1e2f7e033ec1e7496d72399e64bee1d752b0f2040580dcdbc4249c3a1744a9a.jpg)  
+Figure 13: Decrease of CE loss from attacking victims trained on Adult dataset with sensitive feature race.
+
+![](images/f9c96347d04f2bf66592f7e34977522446451fef0b951fcf3f08493a84ceab19.jpg)
+
+![](images/e2fdce1df5700d0e00a369d13026ea659f3ea31edee5e62c50923e87f11822b6.jpg)  
+(a) Decrease of upper-level loss
+
+![](images/e2b6406fdd6e315d75d53ff066dddac969a73e73edde4a41e43cd89c602e8763.jpg)
+
+![](images/1b799e3e5e03c080b409c7efc439113f1aa42af9e3f8bca1ffd73192e41c9243.jpg)
+
+![](images/cee48983578d6837170237b25d888833f9963e6f0657b706a134869993bbe187.jpg)  
+Figure 14: Effectiveness of reducing batch size as a defense against the proposed ENG-based attacks, different attackers using  $1\% - 15\%$  training samples for poisoning. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/de2a21578b92684423adf7a9d3382e4b1b7067fc4839633747d62641c1fbff60.jpg)  
+(b) Decrease of BCE loss
+
+![](images/bea89b13124de7705854d873e22c3f3df066503d3f06465d75cce33bb370c39f.jpg)
+
+![](images/c302a9697df049034fc13d85ad7ca68e46510e00e6eeccafdfd2af3407f83cd7.jpg)
+
+![](images/8bcd6750897f2c37362026cafe01c2716fbabd7f580f77b1a8ed2b4e7f03f996.jpg)
+
+![](images/1a2867548cb038336209c1119ed9921761bafb9165900e0e8e2315a3f0afef96.jpg)  
+(a) COMPAS dataset
+
+![](images/8ecac4ea1e3fd2828a93cd41c9ff6200b1c49fa3c4464ff7abdcc324876ec187.jpg)
+
+![](images/a8db7fb260f7b13678e36ded1a994240fb65f4a19488d6c2eb1a3a3f8c6df75d.jpg)
+
+![](images/b6ed9b47d2dc5ccb2ef4ba38bf7edf28646507455e6f28a5ae5d41e4a1639b6f.jpg)  
+Figure 15: Decrease of BCE loss from different attackers using  $5\%$  -  $15\%$  training samples for poisoning. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/77f4ec287c2034ff6f4193502231be9306c3315fbe563f3e26c44bd5de32e724.jpg)  
+(b) Drug Consumption dataset
+
+![](images/d97118064a9a1c25a4d6bd082082a27469fa153e25e093777b82596864bbe029.jpg)
+
+![](images/8aa8de568c2bc2076cd8e2bd41f81d670e4d289dc16dc067c6eaf2d2760e013e.jpg)
+
+![](images/37445d6655693aeedfec22893157ad6c4646dac215a680aa156943ab07a44796.jpg)
+
+![](images/da9e142da23a9508fc577f01c8a5a5dc4f9e0ff6c26cbde7d9d8a175d1777949.jpg)  
+(a) COMPAS dataset
+
+![](images/345d190a8ed679c8e5634db684594c9487c6c532f25741d625d4c94b1740a9f8.jpg)
+
+![](images/7418a02f656b455cb68385117f59f19e0f0a57b6a793e6124a2bf6ceb997ce72.jpg)
+
+![](images/cd255a0339be8a1039db57a5f32b515ec0578c2033fe5cbed7e1e47e38cb1e62.jpg)  
+Figure 16: Increase of DP violations from different attackers using  $5\% - 15\%$  training samples for poisoning. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/5510783e9f3927dd595537e9bdd8907792b0270e5c0fa67bf52ca8948226c9ad.jpg)  
+(b) Drug Consumption dataset
+
+![](images/8a96cc5cd1b85f3783a8fe6654e37bdc9281cf3b8875607ac037779ed52e507f.jpg)
+
+![](images/ca96d70dda58b61c0dbda5f602e45a7635e3a045ea337011d58981192c37b9e6.jpg)
+
+![](images/9023f284c6ee99767673837d65ea06feca920ee08cfbbf91dcbea819653ddbe7.jpg)
+
+![](images/5de7a9c08356c7a14d683f0373040f817d931d51dbed24ab2b4ea5335925602e.jpg)  
+(a) COMPAS dataset
+
+![](images/08578d91d644f2667bf389fc208d79c54c2cf3a187482711f7dcbf0f4c7edf43.jpg)
+
+![](images/9c52ea5a99c2c06d849684a47cb21da9dc4e6883cb320b45806d5415678dabb5.jpg)
+
+![](images/7bce1946a7972a98a0e0ce97e454ba7da08c9f8bd6d82a67480516677b4dfc69.jpg)  
+Figure 17: Accuracy of predicting  $y$  from  $z$  when different attackers using  $5\% - 15\%$  training samples for poisoning. Results are averaged over 5 independent replications and bands show standard errors.
+
+![](images/380bac1699a87697504d78a8bf81db2461dcf1c11e3d6e59cea27660fb076fae.jpg)  
+(b) Drug Consumption dataset
+
+![](images/183af48ece3125dd3c82dbfcf454135df4eeffdf684efb1d8e6c5f9ec463f98b.jpg)
+
+![](images/616d6cb57aa6b0d5a3b69edbfaa4336c6d6b7553238d5ceda872654396807481.jpg)

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2024/Towards Robust Multi-Modal Reasoning via Model Selection/full.md

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+# TOWARDS ROBUST MULTI-MODAL REASONING VIA MODEL SELECTION
+
+Xiangyan Liu $^{3,*}$  Rongxue Li $^{2,1,*}$  Wei Ji $^{3}$  Tao Lin $^{1,\dagger}$
+
+liu.xiangyan@u.nus.edu; lirongxue@westlake.edu.cn; weiji0523@gmail.com; lintao@westlake.edu.cn
+
+$^{1}$ Westlake University  $^{2}$ Zhejiang University  $^{3}$ National University of Singapore
+
+# ABSTRACT
+
+The reasoning capabilities of LLM (Large Language Model) are widely acknowledged in recent research, inspiring studies on tool learning and autonomous agents. LLM serves as the "brain" of the agent, orchestrating multiple tools for collaborative multi-step task solving. Unlike methods invoking tools like calculators or weather APIs for straightforward tasks, multi-modal agents excel by integrating diverse AI models for complex challenges. However, current multi-modal agents neglect the significance of model selection: they primarily focus on the planning and execution phases, and will mainly invoke predefined task-specific models for each subtask, making the execution fragile. Meanwhile, other traditional model selection methods are either incompatible with or suboptimal for the multi-modal agent scenarios, due to ignorance of dependencies among subtasks arising by multi-step reasoning.
+
+To this end, we identify the key challenges therein and propose the  $M^3$  framework as a plug-in with negligible runtime overhead at test-time. This framework improves model selection and bolsters the robustness of multi-modal agents in multi-step reasoning. In the absence of suitable benchmarks, we create MS-GQA, a new dataset specifically designed to investigate the model selection challenge in multi-modal agents. Our experiments reveal that our framework enables dynamic model selection, considering both user inputs and subtask dependencies, thereby robustifying the overall reasoning process. Our code and benchmark: https://github.com/LINs-lab/M3.
+
+# 1 INTRODUCTION
+
+Large Language Models (LLMs) (Brown et al., 2020; Ouyang et al., 2022; Chowdhery et al., 2022; Zhang et al., 2022b; Touvron et al., 2023) recently emerged to show great potential for achieving human-level intelligence, leveraging the key reasoning ability to tackle complex problems.
+
+As a key step towards artificial general intelligence, the study on multi-modal learning has soon evolved into two paradigms, either training large end-to-end models like PaLM-E (Driess et al., 2023) and Mini-GPT4 (Zhu et al., 2023) for direct task resolution, or employing LLMs to decompose tasks into subtasks for smaller yet specific models (Gupta & Kembhavi, 2023; Suris et al., 2023; Shen et al., 2023; Gao et al., 2023a). The latter paradigm—as evidenced in the significant attention since tool learning (Schick et al., 2023) and autonomous agents (Reworkd, 2023; Richards et al., 2023)—demonstrates the immense potential in addressing complex real-world problems, where multiple AI models collaborate through multi-modal multi-step reasoning process.
+
+In the realm of multi-modal reasoning scenarios, existing multi-modal agents (Gupta & Kembhavi, 2023; Shen et al., 2023) emphasize planning and execution phases, while neglecting the critical model selection phase. As exemplified in Figure 1 (a & b), a simplistic model selector relies on predefined task-specific models for subtasks, increasing the likelihood of intermediate errors and compromising the overall reasoning process. Moreover, existing traditional model selection methods, though effective in various domains (Zhao et al., 2021; Park et al., 2022; Lee et al., 2022; Zitovsky et al., 2023), primarily focus on selecting a single model from multiple candidates per sample.
+
+![](images/fd68c831ec66806955f94bc77b74ab92b91d408d4ead8a887e85e468ec8e34eb.jpg)  
+(c) Three model selection paradigms
+
+![](images/5f15b7d03c3faf5a793b2c5cb524a7244f0c8d2221f82ffdb5a46f7e901b68f9.jpg)
+
+![](images/ebdaa7f6f3dd5c8c315b57ae9fdefbc1b3f589d5cef0d64de7378f0cec0d99ca.jpg)  
+Figure 1: Illustration of the multi-modal multi-step reasoning process and three model selection paradigms within. (a) shows how multi-modal agents utilize LLMs to decompose complex multi-modal tasks, resulting in a multi-step reasoning process where each node corresponds to a simpler yet more specific subtask. (b) highlights that compared to a robust model selector, simplistic model selection methods are more prone to generating wrong outcomes at intermediate subtask stages, thereby impacting the ultimate reasoning result. Here,  $\mathfrak{m}_{(\mathrm{i})}$  indicates the  $i$ -th model is selected and the color means the corresponding subtask type. (c) numerically illustrates the comparative outcomes of model selection methods from different paradigms in multi-modal reasoning.
+
+![](images/33f84eeb847dfca0931bc30dcafb32678d40cd8c8eea7391e64899fa29d3ae25.jpg)
+
+Adapting these methods to multi-modal reasoning scenarios, which necessitate multiple models for subtasks, is challenging due to the oversight of subtask dependencies.
+
+To this end, we formally define the problem of model selection in multi-modal reasoning scenarios as our first contribution, and then introduce the  $M^3$  framework (Model Selector for Multi-Modal Reasoning) as our preliminary remedy for the field. Given the lack of benchmarks, we further create a new dataset named MS-GQA (Model Selection in GQA (Hudson & Manning, 2019)) to facilitate research in the field. In detail,  $M^3$  represents the multi-step reasoning process as a computation graph, with nodes corresponding to reasoning subtasks. Multi-modal encoders and models' embedding table transform input and selected models into node features, where a computation graph learner then models the relationships between input, selected models, and subtask dependencies to predict the execution status.  $M^3$  showcases superior performance in the model selection process for multi-modal models, with trivial overhead in selection.
+
+# Our key contributions are summarized as follows:
+
+- We formulate the model selection problem in multi-modal reasoning contexts as an initial endeavor.  
+- We introduce  $M^3$ , a model selection framework for multi-modal models with multi-step reasoning, jointly modeling the relationship between samples, selected models, and subtask dependencies.  
+- We create a comprehensive dataset called MS-GQA to facilitate the research for the community.  
+- We provide an effective yet efficient model selection solution, with trivial test-time overhead.
+
+# 2 RELATED WORK
+
+# 2.1 MULTI-STEP REASONING WITH LLM
+
+A line of work utilizes LLMs to interact with simple APIs, including WebGPT (Nakano et al., 2021), Toolformer (Schick et al., 2023), PAL (Gao et al., 2023b), and ToolkenGPT (Hao et al., 2023), enabling the manipulation of tools like web browser and calculators. Their capabilities are enhanced by behaving as autonomous agents (Richards et al., 2023; Nakajima, 2023; Reworkd, 2023), where they reason, break down complex tasks into subtasks, and iteratively execute those subtasks until the desired goals are achieved. Note that these studies only call simple deterministic APIs for text tasks. To tackle complex multi-modal tasks, researchers are increasingly expanding tool libraries with trained AI models (Wu et al., 2023; Huang et al., 2023; Surís et al., 2023). For instance, VisProg (Gupta & Kembhavi, 2023) and HuggingGPT (Shen et al., 2023) employ LLMs to decompose complex tasks into subtasks and connect various AI models to address them. Chameleon (Lu et al.,
+
+2023) is a plug-and-play compositional reasoning framework using a richer set of tools. Unlike the aforementioned approaches that provide a static plan without considering dependent subtasks, AssistGPT (Gao et al., 2023a) and AVIS (Hu et al., 2023) dynamically strategize the utilization of external tools based on the intermediate results of multi-step reasoning. However, all existing methods lack model selection consideration, resulting in reasoning instability.
+
+# 2.2 MODEL SELECTION
+
+The research on model selection can date back to Forster (2000), and was further extensively investigated in various aspects: 1) meta-learning methods (Zhao et al., 2021; 2022; Park et al., 2022), 2) nonmeta-learning methods (Ying et al., 2020; Zohar et al., 2023), 3) model selection for ensemble (Kotary et al., 2023), 4) model selection with language models (Zhao et al., 2023; Hari & Thomson, 2023), 5) new metrics for model selection (Zhang et al., 2021; Yang et al., 2023a), and others (Chen et al., 2023; Lee et al., 2022; Zitovsky et al., 2023). We discuss the most relevant two lines of research as follows. Meta-learning-based approaches utilize the similarity between new instances and historical instances to predict the performance of candidate models. For example, MetaOD and ELECT (Zhao et al., 2021; 2022) address the unsupervised outlier model selection problem, where they extract meta-features by considering input-specific characteristics. MetaGL (Park et al., 2022) further extends (Zhao et al., 2021) to select a model for each new graph. However, in our multi-modal multi-step reasoning scenarios, there is currently a lack of established approaches for extracting instance-wise meta-features. In contrast, non-meta-learning-based methods resort to using complex networks to learn the relationship between instances and model choices, without using meta-features. Auto-Selector (Ying et al., 2020) employs a pre-trained model selector and parameter estimator to automatically choose an anomaly detection model for incoming time-series data. LOVM (Zohar et al., 2023) employs textual dataset descriptions to train a linear model that predicts the performance of vision-language candidate models. EMMS (Meng et al., 2023) employs weighted linear regression to estimate multi-modal model transferability. Note that these methods are restricted to one-step selection and overlook subtask dependencies in multi-step reasoning, and thus infeasible for model selection in multi-step reasoning. In the context of using LLMs for multi-modal models in multi-step reasoning, existing methods either rely on external metrics like download counts for model selection, or explicitly specify the use of a specific version of a model for a designated task without any selection process (Surís et al., 2023). None of the prior work takes the challenges of multi-step reasoning dependency into account.
+
+# 3 MODEL SELECTION HARNESSES THE MULTI-MODAL REASONING
+
+# 3.1 ON THE CHALLENGES OF MULTI-MODAL MULTI-STEP REASONING
+
+Given a complicated input query, the contemporary multi-modal multi-step reasoning process normally involves calling LLMs to decompose users' input to subtasks, in which the reasoning logic can be constructed as a task graph by composing dependent intermediate subtasks.
+
+The problem of selecting models for subtasks along the task graph can be defined below:
+
+Definition 3.1 (Model selection for each subtask type on a multi-modal task graph). Let  $m(i,j,t)$  denote a model choice of subtask type  $t$  for the  $i$ -th sample  $\mathbf{x}_i$ , where  $m(i,j,t)$  indicates the  $j$ -th choice from the model zoo for subtask type  $t$  on  $\mathbf{x}_i$ . Each subtask type  $t$  has  $n_t$  choices, forming a set  $\{m(\cdot ,j,t)\mid j\in [n_t]\}$ .
+
+Assume a task graph consists of  $K$  subtask types, then all potential model selection choices for the task graph can be defined as  $\mathcal{C} = \{m(\cdot ,j,t)\mid j\in [n_t],t\in [K]\}$ , with size  $|\mathcal{C}| = \prod_{t = 1}^{K}n_{t}$ . The optimal choice of models on the task graph for  $i$ -th input can be represented as  $\mathbf{c}_i^\star \coloneqq \{m(i,j_t^\star ,t)\mid t\in [K]\}$ , where we select the optimal model index  $j_{t}^{\star}$  for each subtask type  $t$ .
+
+Though Definition 3.1 fully characterizes the procedure of traditional model selection methods as evidenced in Figure 2 and Section 2.2, it leaves the unique challenge emerged in the multi-modal multi-step reasoning untouched, namely the critical subtask dependency defined below.
+
+Definition 3.2 (Subtask dependency on a multi-modal task graph). Given the  $i$ -th input  $\mathbf{x}_i$  with embeddings from various modalities. Its multi-step reasoning procedure decomposed by an LLM can be described as a directed acyclic computation graph  $\mathcal{G}_i = \{\mathcal{V}_i, \mathcal{T}_i, \mathcal{E}_i\}$  with nodes corresponding to multi-modal subtasks, where
+
+-  $\mathcal{V}_i = \{v_{i,k} \mid k \in [L]\}$  is the subtask nodes in the graph,  $v_{i,k}$  is the  $k$ -th node in  $\mathcal{G}_i$ , and  $L := |\mathcal{V}_i|$ ;
+-  $\mathcal{T}_i = \{t_{i,k} \mid k \in [L]\}$  is the set of subtask types in  $\mathcal{V}_i$ ,  $t_{i,k}$  is the subtask type of  $k$ -th node in  $\mathcal{G}_i$ ;
+-  $\mathcal{E}_i$  is the set of edges that connect pairs of subtask nodes in the graph  $\mathcal{G}_i$ .
+
+![](images/189b9b21d7ab3c7af78c2c1e4fa1f83de06303d3f9a594b8d21f23bf59f60b7d.jpg)  
+Figure 2: Comparison of three model selection paradigms under various inputs. The model selection processes of the three paradigms, from left (simplicistic) to right (subtask dependency-aware), become progressively more fine-grained. "Simplistic" is inflexible and can be considered input-agnostic. "Traditional" can solely depend on subtask type and the corresponding original input information for model selection. When inputs are similar, "Traditional" cannot provide as diverse model selections as "Subtask Dependency-Aware", which leverages differences in reasoning logic to offer more varied and suitable model choices. Note that node P (green circle) in the figure denotes Python module invocation, which does not entail model selection.
+
+Remark 3.3 (Generalizing model selection beyond the subtask type). To ease the presentation, Definition 3.1, Definition 3.2, and the main content below only consider selecting the model per subtask type. Though being sufficient for traditional model selection methods, such formulation can be further generalized to performing model selection per subtask node (rather than per subtask type) in our case, and we leave it for future research.
+
+Challenges: on the infeasibility of adapting existing model selection methods. Existing methods (see details in our Section 2.2), either the current preliminary strategies for multi-modal agents (Gupta & Kembhavi, 2023; Surís et al., 2023; Shen et al., 2023; Gao et al., 2023a), or model selection methods from other domains (Zhao et al., 2021; 2022; Park et al., 2022; Ying et al., 2020; Zohar et al., 2023), do not take the subtask dependency into account—as evidenced in the following case study—making the model selection for multi-modal model with multi-step reasoning non-trivial. We describe the previous two model selection paradigms (see examples in Figure 2):
+
+1. This paradigm is primarily applied in some recent multi-modal agent frameworks, where each subtask type  $t$  will be straightforwardly allocated to a model via some external metrics (e.g., download counts or recent release dates).  
+2. As has been widely used in other fields like outlier detection and graph learning, this paradigm focuses on matching the optimal model through the input information as well as the subtask type. Given the limitations of previous methods in adapting to the timely multi-modal reasoning scenarios, in the following section, we incorporate the graph information  $\mathcal{G}_i$  to capture subtask dependencies.
+
+# 3.2  $M^3$ : A FRAMEWORK OF MODEL SELECTION FOR MULTI-MODAL REASONING
+
+An ideal model selection solution on multi-modal reasoning scenarios discussed above motivates us to jointly model the subtask dependency with input sample features. We depict our unified model selection framework  $M^3$  below towards this goal.
+
+Overview. The model selection process can be viewed as estimating the performance of a multi-modal input over the corresponding task graph, in which the choice of model selection at each subtask
+
+![](images/cd832b69573a61c8bff53c3be1360cc69f79768d88b532670502a0632c54d416.jpg)
+
+![](images/db2bd11bd901d58f34b86e17ee586627fde339b24f8d4c8c2de42cf813a48ee9.jpg)  
+Figure 3: Illustration of  $M^3$ : (a) depicts the forward computation process: 1) Task Graph: An initial virtual node represents the multi-modal input. Specific models are assigned to each subtask node based on the respective subtask type. 2) Node Embedding: Features are extracted using the multi-modal encoder  $\psi_1(\cdot)$  and embedding table  $\psi_2(\cdot)$  for the initial virtual node and subtask nodes. 3) Computation Graph Learner: The computation graph, including node features and subtask dependencies (edges  $\mathcal{E}_i$ ), serves as input to learner  $\phi(\cdot)$ , contributing to the predicted execution status  $s_i^j$ . (b) illustrates the process of ranking and selecting the model selection choice with a greater likelihood of success.
+
+node propagates to the next node on the directed task graph. We introduce the notion of meta-training and train a proxy on it to suggest the optimal model choice of task graph on unseen input. In detail,
+
+- Training the proxy. Given an input sample  $\mathbf{x}_i \in \mathcal{X}_{\mathrm{train}} := \{\mathbf{x}_1, \dots, \mathbf{x}_N\}$ , for each choice of models on the task graph  $\mathbf{c}_i^j \in \mathcal{C}$ , we aim to model the relationship  $\phi \circ \psi$  between  $(\mathbf{x}_i, \mathcal{G}_i, \mathbf{c}_i^j)$  and the execution status  $p_i^j \in \{0, 1\}$ , where  $\phi$  and  $\psi$  denote the learner and feature extractor, respectively. Here we simplify our setting by only considering the binary execution status, but the principles therein can be generalized to the continuous case.  
+- Model selection for task graph on the unseen sample. We estimate the status of potential model choices  $\{s_i^j \coloneqq \phi \circ \psi(\mathbf{x}_i, \mathcal{G}_i, \mathbf{c}_i^j) \mid \mathbf{c}_i^j \in \mathcal{C}\}$ , and only keep executable choices for further selection.
+
+# 3.2.1 TRAINING THE PROXY
+
+Lookup table and node embedding  $\psi$ . We adopt a similar approach to treatments in multi-modal learning and GNNs (Li et al., 2022; Velicković et al., 2017). Using  $\psi \coloneqq [\psi_1, \psi_2]$  where  $\psi_1$  denotes the multi-modal encoder and  $\psi_2$  represents the model embedding table—we map subtask nodes to their corresponding embeddings through  $H_i^j \coloneqq \psi(\mathbf{x}_i, \mathcal{G}_i, \mathbf{c}_i^j) \in \mathbb{R}^{(L+1) \times d}$ .
+
+More specifically, we first slightly abuse the notation and form an augmented computation graph  $\mathcal{G}_i = (\mathcal{V}_i,\mathcal{T}_i,\mathcal{E}_i)$ : we introduce a virtual starting node  $v_{i,0}$  to incorporate the multi-modal input information  $\mathbf{x}_i$  and use subtask nodes  $\{v_{i,k}|k\in [L]\}$  to indicate the execution dependency of subtasks. We then employ two encoders  $\psi_1,\psi_2$  separately to retrieve the node embeddings and use additional  $W_{1}\in \mathbb{R}^{d_{1}\times d}$  and  $W_{2}\in \mathbb{R}^{d_{2}\times d}$  to unify them in a shared feature space.
+
+- For the virtual starting node  $v_{i,0}$ , we utilize the off-the-shelf multi-modal encoder to extract node embedding as  $\mathbf{h}_{i,0} = \psi_1(\mathbf{x}_i) \in \mathbb{R}^{d_1}$ , where  $d_1$  is the input embedding dimension.  
+- For other nodes, we use a lookup table to retrieve node embedding for each subtask type, namely  $\mathbf{h}_{i,k}^{j} = \psi_{2}\left(m(i,j,t_{i,k})\right) \in \mathbb{R}^{d_{2}}$ , where  $t_{i,k} \in \mathcal{T}_i$  and  $d_{2}$  is the model embedding dimension.
+
+Joint modeling of node embeddings and subtask dependency  $\phi$ . We further leverage a computation graph learner to learn the task embedding over 1) sample embeddings, 2) model embeddings, and 3) subtask dependency information. A final linear layer with a non-linear activation function is stacked
+
+on top of the learned task embedding to estimate  $s_i^j \coloneqq \phi \circ \psi(\mathbf{x}_i, \mathcal{G}_i, \mathbf{c}_i^j)$ , where  $\mathbf{c}_i^j \in \mathcal{C}$ . In detail,
+
+$$
+\boldsymbol {H} _ {i} ^ {j} = \psi (\mathbf {x} _ {i}, \mathcal {G} _ {i}, \mathbf {c} _ {i} ^ {j}), \quad s _ {i} ^ {j} = \phi \left(\boldsymbol {H} _ {i} ^ {j}, \mathcal {E} _ {i}\right). \tag {1}
+$$
+
+The output  $s_i^j$  measures the degree of match between input sample  $\mathbf{x}_i$  and model selection choice  $\mathbf{c}_i^j$ , where a higher value signifies a greater likelihood of success in a multi-modal reasoning scenario. Note that theoretically, any neural network capable of handling directed acyclic graphs can serve as the backbone for a computation graph learner. See Section 4.2 and Appendix D.3 for details.
+
+**Optimization over  $\psi$  and  $\phi$ .** For a given input sample  $\mathbf{x}_i$ , we aim to learn the model to estimate execution status per the choice  $\mathbf{c}_i^j$  of models along the task graph, namely correlating  $s_i^j \in [0,1]$  with ground truth  $p_i^j \in \{0,1\}$ . Thus, the optimization can be viewed as a multi-label classification problem. The conventional choice of using instance-wise Binary Cross-Entropy (BCE) loss in the multi-label classification community (Tsochantaridis et al., 2005; Wehrmann et al., 2018) only aims to build a mapping between  $(\mathbf{x}_i, \mathcal{G}_i, \mathbf{c}_i^j)$  and  $s_i^j$ , and thus suffers from the optimization difficulty caused by prediction independency of  $\mathbf{c}_i^j \in \mathcal{C}$  on the input  $\mathbf{x}_i$ .
+
+Therefore, we employ Categorical Cross-Entropy (CCE) (Su et al., 2022) as our objective function, using a list-wise approach to model  $\psi$  and  $\phi$  for  $(\mathbf{x}_i,\mathcal{G}_i,\mathcal{C})$  and  $\{s_i^j\mid \mathbf{c}_i^j\in \mathcal{C}\}$ . Our goal is to promote higher scores  $s_i^j$  for executable choices  $(p_i^j = 1)$  and lower scores for non-executable choices:
+
+$$
+\mathcal {L} _ {i} = \log \left(1 + \sum_ {\mathbf {c} _ {i} ^ {j} \in \mathcal {C}} 1 _ {p _ {i} ^ {j} = 0} \exp \left(s _ {i} ^ {j}\right)\right) + \log \left(1 + \sum_ {\mathbf {c} _ {i} ^ {j} \in \mathcal {C}} 1 _ {p _ {i} ^ {j} = 1} \exp \left(- s _ {i} ^ {j}\right)\right). \tag {2}
+$$
+
+See Appendix B and D.4 for more details on loss design and a comparison of different loss functions.
+
+# 3.2.2 MODEL SELECTION FOR TASK GRAPH ON THE UNSEEN SAMPLE
+
+Once we learn the relationship  $\phi \circ \psi$ , we can estimate execution status  $s_i^j$  for all choices of models on the task graph  $\mathbf{c}_i^j \in \mathcal{C}$ , and transfer to the final model selection upon other criteria. For the sake of simplicity, in our evaluation we primarily select the  $\mathbf{c}_i^\star$  via the maximal execution probability:  $\mathbf{c}_i^\star = \arg \max_{\mathbf{c}_i^j \in \mathcal{C}} \phi \circ \psi(x_i, \mathcal{G}_i, \mathbf{c}_i^j)$ . Other metrics, e.g. computation cost, can be further integrated to trade off efficiency and robustness.
+
+# 4 EXPERIMENTS
+
+# 4.1 BENCHMARK: MS-GQA
+
+As our side contribution, we introduce the first benchmark, MS-GQA (Model Selection in GQA (Hudson & Manning, 2019)), to explore the model selection methods on multi-modal reasoning scenarios. MS-GQA primarily evaluates model selection choices of scenarios using VisProg (Gupta & Kembhavi, 2023) as the autonomous agent to solve AI tasks on source dataset GQA. The choice of autonomous agents and datasets can be
+
+![](images/e92c6a1befabfd0ae6b1d067ef625a8db00fe59fa81a6219a734d5df0fba216e.jpg)  
+Figure 4: The proportions of various structural categories in GQA.
+
+flexibly replaced, such as substituting VisProg with HuggingGPT (Shen et al., 2023) or other related works (Surís et al., 2023; Gao et al., 2023a), and replacing GQA with NLVR2 (Suhr et al., 2018) or even other in-the-wild application scenarios (Surís et al., 2023; Yang et al., 2023c).
+
+Our benchmark considers the structural variations of tasks in GQA, and thus facilitates comprehensive method evaluation and assesses robustness under diverse test distributions. Currently, we introduce 5 task categories, namely Query, Choose, Compare, Verify, and Logical. The tasks cover 9 functional components (subtask types), of which 7 out of 9 components align with specific Python modules, eliminating the need for model selection. The remaining two components involve "LOC" (localization, text-guided object detection) and "VQA" (visual question answering), offering 10 and 7 candidate models respectively. We have currently collected the binary execution results for 8,426 samples across 70 valid model selection choices. For more details, refer to Appendix A.
+
+# 4.2 EXPERIMENTAL SETTINGS
+
+Baselines. As discussed in Section 2.2, the multi-modal model currently falls short of model selection methods. To justify the effectiveness of our solution, we extend a range of representative methods to multi-modal reasoning scenarios as two groups of baseline references illustrated below.
+
+Training-free selects one model for each subtask type based on the prior knowledge or external metrics (e.g., download counts, citations, and publication dates), without considering input information:
+
+- RANDOM: Randomly choose models for each subtask type on the task graph for every sample;  
+- VISPROG (Gupta & Kembhavi, 2023): Follow the default choice in the original paper of VisProg and select a deterministic candidate model per subtask type;  
+- EXMETRIC: Incorporate external metrics for model ranking and selection. This baseline can be generalized as a paradigm that utilizes external metrics for model selection;  
+- GLOBALBEST (Park et al., 2022; Zhao et al., 2021): Select models on the task graph that yield the highest average performance across all training samples, without considering input details.
+
+Training-based focuses on leveraging a trainable proxy to find the optimal model. We adapt methods from other domains and form our baselines to meet the requirements of the multi-modal reasoning context. Note that tuning and improving these methods are beyond the scope of this paper.
+
+- NCF (He et al., 2017): A representative model selection approach using collaborative filtering. It uses a neural network to model the interaction between samples and models, leveraging both features in a collaborative filtering manner;  
+- METAGL (Park et al., 2022): A representative model selection approach using meta-learning. It uses input multi-modal features as meta-features, where a multi-relational bipartite graph is utilized to estimate the relationship between the input and the choice of models on the task graph;  
+- NCF++ and METAGL++. In comparison to METAGL and NCF,  $M^3$  utilizes additional computation graph descriptions² to capture subtask dependencies. To ensure a fair comparison, we extend METAGL and NCF to METAGL++ and NCF++, where original multi-modal input features are enhanced by adding extra text features derived from the computation graph descriptions.
+
+Notably, HuggingGPT (Shen et al., 2023) employs the "In-context Task-model Assignment" model selection strategy. However, our Appendix H experiments show its ineffectiveness, so it is not included in the baselines for simplicity.
+
+Evaluation metric. An ideal model selection method for multi-modal reasoning scenarios should allocate the optimal model choice per subtask or subtask type, so as to maximize the execution chance for every input sample. To assess methods on  $N_{\mathrm{test}}$  samples from the MS-GQA benchmark, we define the metric Successful Execution Rate (SER) as  $\frac{1}{N_{\mathrm{test}}} \sum_{i=1}^{N_{\mathrm{test}}} 1_{\alpha_i}$ . Here,  $1_{\alpha_i}$  indicates the execution result of the  $i$ -th sample upon the selected choice of models  $\mathbf{c}_i^{\star}$ , either 1 (success) or 0 (fail).
+
+SER measures model selector performance: higher SER means better performance and greater overall reliability in multi-modal reasoning. In special cases where all model selection choices either succeed or fail, evaluating the model selector is pointless, so we exclude these cases from our experiments.
+
+Implementation details. To facilitate a fair comparison with  $M^3$ , we use CCE loss in both NCF and METAGL, including their extensions (NCF++ and METAGL++.). The default backbone network of the computation graph learner is GAT (Veličković et al., 2017), which is well-known and capable of handling directed acyclic graphs. When dealing with input features with multi-modal information, we utilize "blip-base-vqa" as the default feature extractor. Additionally, the pure textual encoder "bert-base-uncased" is used for encoding the extra computation graph description information.
+
+The results below are reported over five random seeds. The dataset from MS-GQA is split randomly into training, validation, and test sets, with a  $6:2:2$  ratio. Ablation studies on the choices of feature extractors and backbone in computation graph learner are deferred to Appendix D.1, D.2 and D.3.
+
+# 4.3 RESULTS: MODEL SELECTION ACROSS DIVERSE TEST DISTRIBUTIONS
+
+In this section, we examine the performance of  $M^3$  and other strong baselines across varied test distributions. The test set is divided into multiple sub-test sets. There are two criteria for division: 1) problem structure; 2) model selection difficulty. It is noteworthy that the training dataset remains consistent throughout these experiments. The superiority of  $M^3$  is summarized below.
+
+Training-based methods outperform Training-free methods. In Table 1 and Table 2, the Training-based methods represented by  $M^3$  and METAGL significantly outperform and demonstrate greater robustness than the Training-free methods represented by EXMETRIC and GLOBALBEST. Specifically, on the Full test set in Table 1, Training-based methods achieve approximately  $64\%$
+
+Table 1: Performance comparison in test scenarios with diverse structural information. The testing set is split into five sub-test sets: Query, Choose, Compare, Verify, and Logical. Each subset maintains consistent structural information; for instance, all Compare samples involve tasks related to comparisons. Full refers to the un categorized test set, which is the complete test dataset. "Improv." indicates the specific numerical improvement of  $M^3$  compared to the corresponding method in this column.  
+
+<table><tr><td rowspan="2" colspan="2">Category</td><td rowspan="2">Metrics</td><td colspan="4">Training-free</td><td colspan="5">Training-based</td></tr><tr><td>RANDOM</td><td>VISPROG</td><td>EXMETRIC</td><td>GLOBALBEST</td><td>NCF</td><td>NCF++</td><td>METAGL</td><td>METAGL++</td><td>\(M^3\)</td></tr><tr><td rowspan="12">Subset</td><td rowspan="2">Query</td><td>SER (%)</td><td>45.36±1.7</td><td>44.85±0.0</td><td>58.64±0.0</td><td>51.07±0.0</td><td>53.94±2.9</td><td>53.63±4.3</td><td>57.63±2.2</td><td>55.18±5.2</td><td>59.53±1.0</td></tr><tr><td>Improv. (%)</td><td>+14.17</td><td>+14.68</td><td>+0.89</td><td>+8.46</td><td>+5.59</td><td>+5.90</td><td>+1.90</td><td>+4.35</td><td>-</td></tr><tr><td rowspan="2">Choose</td><td>SER (%)</td><td>68.26±4.0</td><td>66.94±0.0</td><td>68.60±0.0</td><td>69.42±0.0</td><td>70.25±1.7</td><td>73.39±2.5</td><td>73.72±3.4</td><td>74.71±5.1</td><td>76.53±2.8</td></tr><tr><td>Improv. (%)</td><td>+8.27</td><td>+8.27</td><td>+7.93</td><td>+7.11</td><td>+6.08</td><td>+3.14</td><td>+2.81</td><td>+1.82</td><td>-</td></tr><tr><td rowspan="2">Compare</td><td>SER (%)</td><td>71.29±3.1</td><td>82.26±0.0</td><td>61.29±0.0</td><td>77.42±0.0</td><td>73.92±3.9</td><td>74.52±2.1</td><td>75.16±4.6</td><td>73.55±1.8</td><td>76.45±1.4</td></tr><tr><td>Improv. (%)</td><td>+5.16</td><td>-5.71</td><td>+15.16</td><td>-0.97</td><td>+2.53</td><td>+1.93</td><td>+1.29</td><td>+2.90</td><td>-</td></tr><tr><td rowspan="2">Verify</td><td>SER (%)</td><td>62.38±2.5</td><td>68.40±0.0</td><td>65.80±0.0</td><td>71.75±0.0</td><td>75.46±0.9</td><td>71.67±2.6</td><td>70.56±3.0</td><td>73.01±3.0</td><td>75.09±0.6</td></tr><tr><td>Improv. (%)</td><td>+12.71</td><td>+6.69</td><td>+9.29</td><td>+3.34</td><td>-0.37</td><td>+3.42</td><td>+4.53</td><td>+2.08</td><td>-</td></tr><tr><td rowspan="2">Logical</td><td>SER (%)</td><td>64.72±1.6</td><td>72.47±0.0</td><td>59.55±0.0</td><td>78.65±0.0</td><td>76.50±1.8</td><td>77.08±1.8</td><td>74.94±1.6</td><td>75.73±1.7</td><td>77.53±2.9</td></tr><tr><td>Improv. (%)</td><td>+12.81</td><td>+5.06</td><td>+17.98</td><td>-1.12</td><td>+1.03</td><td>+0.45</td><td>+2.59</td><td>+1.80</td><td>-</td></tr><tr><td rowspan="2">Full</td><td>SER (%)</td><td>56.51±0.3</td><td>59.04±0.0</td><td>61.66±0.0</td><td>63.58±0.0</td><td>65.92±1.5</td><td>64.40±1.7</td><td>66.01±0.4</td><td>65.62±3.2</td><td>68.70±0.6</td></tr><tr><td>Improv. (%)</td><td>+12.19</td><td>+9.66</td><td>+7.04</td><td>+4.12</td><td>+2.83</td><td>+4.30</td><td>+2.69</td><td>+3.08</td><td>-</td></tr></table>
+
+to  $69\%$  SER, while Training-free methods only reach  $57\%$  to  $64\%$ . Furthermore, Table 1 and Table 2 reveal that in sub-test sets, unlike Training-free methods that occasionally excel in one subset while performing poorly in others, Training-based methods consistently exhibit overall stability.
+
+$M^3$  demonstrates its robustness in diverse test sets. In direct comparisons, Table 1 presents that  $M^3$  stands out as the top performer, showcasing a remarkable  $2.69\%$  improvement over the previous state-of-the-art (METAGL) in the complete test set (Full). Moreover,  $M^3$  consistently excels in various subtest sets across both Table 1 and Table 2, demonstrating its robustness
+
+Table 2: Performance comparison with respect to the difficulty of model selection. Each increase in the "difficulty level" indicates a one-unit rise in average model selection difficulty for the sub-test set, leading to a  $20\%$  reduction in the executable ratio  $(\mathbb{E}\left[\sum_{j}p_{i}^{j} / |\mathcal{C}_{i}|\right])$
+
+Best performances are noted in blue, while the poorest are in orange.
+
+<table><tr><td>Difficulty Level</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>EXMETRIC</td><td>98.35±1.9</td><td>86.67±2.9</td><td>59.90±3.8</td><td>25.53±2.4</td><td>15.62±2.8</td></tr><tr><td>GLOBALBEST</td><td>92.86±0.0</td><td>75.46±0.0</td><td>65.62±0.0</td><td>54.47±0.0</td><td>14.06±0.0</td></tr><tr><td>NCF</td><td>93.12±1.9</td><td>82.67±2.9</td><td>65.94±3.8</td><td>48.08±2.4</td><td>15.31±2.8</td></tr><tr><td>METAGL</td><td>82.42±3.8</td><td>77.58±2.9</td><td>70.10±0.9</td><td>53.36±3.3</td><td>20.21±2.7</td></tr><tr><td>M3</td><td>93.30±1.7</td><td>84.97±1.6</td><td>70.50±2.2</td><td>52.26±2.6</td><td>20.42±1.4</td></tr></table>
+
+and competitiveness. Even in sub-test sets where it does not claim the top spot,  $M^3$  maintains a strong presence, often ranking among the top two or three methods. However, it can be observed that other methods, particularly training-free ones, often perform poorly on specific sub-test sets.
+
+$M^3$  effectively leverages the subtask dependency information. In Table 1, we assess NCF++ and METAGL++, both of which employ a text encoder to extract textual features describing multi-modal reasoning logic on the task graph. Our experiments reveal that this approach often fails to capture meaningful information and can even harm the performance of the original methods. Conversely, our framework  $M^3$ , which integrates subtask dependencies into the modeling process as a whole, is more effective and non-trivial.
+
+# 4.4 RESULTS: MODEL SELECTION IN DATA MISSING SCENARIOS
+
+Given the challenges of collecting complete execution results for each sample under all choices of models in real-world scenarios, this section will discuss how training-based methods perform on the fixed complete test set (Full) in two different types of training data missing scenarios:
+
+- missing choices of models on the task graph (c), which involves varying levels of incomplete execution results associated with different model selection choices per sample. E.g., with a missing ratio of  $0.2, \sim 20\%$  of model selection choices do not have corresponding execution results.  
+- missing samples (x), where all collected samples have execution results for all model choices on the task graph, but some samples are absent compared to the complete training set. In this case, a missing ratio of 0.2 means that  $20\%$  of samples are entirely absent.
+
+Data missing results in adverse effects. Figure 5(a) and 5(b) reveal that the missing of data generally leads to performance decline across all training-based methods. Specifically, when the missing ratio reaches 0.8, the SER of  $M^3$  drops to  $66.66\%$  and  $64.67\%$ , respectively, while other baselines exhibit a typical  $2\%$  to  $3\%$  performance decline. Anomalies in some baselines, where performance improves with increased missing ratios, may be attributed to two factors: 1) certain methods are insensitive to training data quantity, and 2) the smaller dataset introduces more experimental randomness.
+
+![](images/6f0d4c61003d7a93f91eee730ccc1fe15ccd334c780f4828df9790222eac8e3f.jpg)  
+(a) Missing ratio of model choices c
+
+![](images/cf6af01bb5b26a7d120f07c069aaab684510d64831c7d670581dbca2a37781f9.jpg)  
+Figure 5: Performance comparison in scenarios with missing training data and varying time constraints at test-time. (a) and (b) depict two data-missing scenarios with progressively increasing proportions on the x-axis. (c) illustrates method performance across different time constraints.
+
+![](images/9bd7a700568cd8081efd3ea4721bc9850245d2a38b656420ae5b3fa3c8b05b68.jpg)  
+(b) Missing ratio of samples  $\mathbf{x}$  
+(c) Time limit (s)
+
+Data missing does not impact the superiority of  $M^3$  over baselines. Figure 5(a) and 5(b) demonstrate that  $M^3$  exhibits superior performance compared to other training-based baselines in two types of missing scenarios. Specifically, despite the overall decline in performance due to missing data in most methods,  $M^3$  consistently outperforms other baselines, underscoring its robustness. Notably, in Figure 5(b), even when up to  $80\%$  of samples are missing (64.67%),  $M^3$  performs better than the best training-free method (GLOBALBEST, 63.58%).
+
+# 4.5 RESULTS: TEST-TIME EFFICIENCY
+
+In practical applications of multi-modal reasoning, beyond enhancing system robustness through model selection, it is crucial to consider the associated cost of implementing the selection process. In this section, we discuss the efficiency of  $M^3$ , in terms of 1) its extra model selection time and 2) its comparison with baselines under the same inference time limit budget.
+
+Negligible runtime overhead at test-time. Figure 6 illustrates that  $M^3$  outperforms all other baselines, particularly training-free methods, despite incurring extra time overhead from model selection. However, this additional time overhead (0.09s) is considerably shorter than the overall task execution time (0.62s) and can be considered negligible.
+
+![](images/5ba81234607dd7d9b584d07fe9ca51375fe3c66e15f5d19269a7f09452ac243b.jpg)  
+Figure 6: Comparing  $M^3$  and other methods from the perspectives of performance and average time cost. Execution time encompasses the overall task completion time when agents collaborate using multiple models, while model selection time is the time spent utilizing a proxy for model selection.
+
+$M^3$  continues to perform the best under various time constraints. In practical production settings, due to cost constraints, not all models are available during test-time. To mimic this challenge, we gradually decrease the time limit and exclude models from the candidate pool on the task graph if their average execution time exceeds the current limit. Figure 5(c) illustrates the robustness of  $M^3$  across a range of scenarios, from the most stringent time limit scenario (0.3s) to scenarios with no time constraints  $(\infty)$ . Despite a decrease in overall performance,  $M^3$  consistently excels other baselines, highlighting its suitability for time-constrained settings at test-time.
+
+# 5 CONCLUSION
+
+We introduce  $M^3$ , a novel framework for assisting autonomous agents in model selection for multi-modal multi-step reasoning scenarios. It tackles the issue of subtask dependencies, a new challenge arising from multi-step reasoning, which existing methods fail to address. In MS-GQA experiments, our framework substantially improves performance, enhancing multi-modal reasoning robustness. Despite resource constraints limiting our experiments to MS-GQA,  $M^3$  has broader applications, including subtask node model selection in multi-step reasoning and adaptation to various agents and real-world datasets. For an in-depth discussion of our work's research significance, the advantages and limitations of our method, and an analysis of our experimental results, please refer to Appendix E, F, and G.
+
+# ACKNOWLEDGEMENT
+
+We thank anonymous reviewers for their constructive and helpful reviews. This work was supported in part by the National Science and Technology Major Project (No. 2022ZD0115101), the Research Center for Industries of the Future (RCIF) at Westlake University, and the Westlake Education Foundation.
+
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+
+# CONTENTS OF APPENDIX
+
+# A Details of MS-GQA 14
+
+A.1 Collection Process and Statistics 14  
+A.2 Files of MS-GQA 15  
+A.3 Subtask types of MS-GQA 15
+
+# B Details of Loss Choice 16
+
+# C Training Details 17
+
+# D Ablation Study 18
+
+D.1 Multi-modal Feature Extractor 18  
+D.2 Computation Graph Description Feature Extractor 18  
+D.3 Backbone of Computation Graph Learner 18  
+D.4 Objective Funciton 18
+
+# E Research Significance 19
+
+E.1 Model Selection for Multi-modal Reasoning 19  
+E.2 Reliable Model Selector:  $M^3$  19  
+E.3 Promising Future Work 19  
+E.4 Real-World Applications 20  
+E.5 Techniques 20
+
+# F Strengths and Limitations of the  $M^3$  Framework 20
+
+F.1 Strengths: 20  
+F.2 Limitations: 21
+
+# G Further Analysis 21
+
+# H Supplementary Experiment on the "In-context Task-model Assignment" of Hugging-GPT 21
+
+# A DETAILS OF MS-GQA
+
+# A.1 COLLECTION PROCESS AND STATISTICS
+
+MS-GQA is constructed by deploying the VisProg agent (Gupta & Kembhavi, 2023) on the GQA (Hudson & Manning, 2019) dataset, focusing on 9 subtask types, particularly "LOC"(Localization, Text-guided Object Detection) and "VQA" (Visual Question Answering). The other 7 subtask types ("EVAL", "COUNT", "CROP", "CROPLEFT", "CROPRIGHT", "CROPABOVE", "CROPBELOW") involve Python modules, thus excluding the need for model selection. The construction process involves these key steps:
+
+- Model Zoo Expansion: VisProg has integrated recently released, popular, open-source models from Visual Question Answering and Text-guided Object Detection domains into its "VQA" and "LOC" modules, expanding the available pool of candidate models.  
+- Execution Results Collection: 10,000 samples are randomly selected from the GQA dataset (test-dev set). Each sample is first decomposed using LLM, resulting in a computation graph description representing the reasoning logic (see "program" in Figure 8). Following this description, all models ("VQA": 7, "LOC": 10, total:  $7 \times 10 = 70$ ) are sequentially executed for each sample, yielding 700,000 execution records. Each execution record represents the result (0: fail, 1: success) of a specific sample's execution for a model selection choice.  
+- Data Cleaning: Samples that experienced reasoning execution failures due to incorrect computation graph descriptions generated by LLM are filtered out. These cases are not within the scope of our model selection research.
+
+Finally, considering resource constraints, we manage to collect a total of 8,426 valid samples, which constitute the foundation for constructing the comprehensive MS-GQA benchmark presented in this study. Within the VisProg framework, the LLM employed is the GPT-3.5-turbo by OpenAI. Regarding the selection of candidate models, we opt for the most recent, widely recognized, high-performing models in each subtask category. The specific candidate models are detailed in Table 3.
+
+Table 3: List of models that can be selected for each subtask in MS-GQA.  
+
+<table><tr><td>Subtask</td><td>Candidate Models</td><td>Link</td><td>Venue</td></tr><tr><td rowspan="10">LOC</td><td>owlvit-large-patch14 (Minderer et al.)</td><td>https://huggingface.co/google/owlvit-large-patch14</td><td>ECCV 2022</td></tr><tr><td>owlvit-base-patch16 (Minderer et al.)</td><td>https://huggingface.co/google/owlvit-base-patch16</td><td>ECCV 2022</td></tr><tr><td>owlvit-base-patch32 (Minderer et al.)</td><td>https://huggingface.co/google/owlvit-base-patch32</td><td>ECCV 2022</td></tr><tr><td>glip_large (Zhang et al., 2022a)</td><td>https://github.com/microsoft/GLIP</td><td>NeurIPS 2022</td></tr><tr><td>glip_tiny_a (Zhang et al., 2022a)</td><td>https://github.com/microsoft/GLIP</td><td>NeurIPS 2022</td></tr><tr><td>glip_tiny_b (Zhang et al., 2022a)</td><td>https://github.com/microsoft/GLIP</td><td>NeurIPS 2022</td></tr><tr><td>glip_tiny_c (Zhang et al., 2022a)</td><td>https://github.com/microsoft/GLIP</td><td>NeurIPS 2022</td></tr><tr><td>glip_tiny_ori (Zhang et al., 2022a)</td><td>https://github.com/microsoft/GLIP</td><td>NeurIPS 2022</td></tr><tr><td>groundingdino_swinb (Liu et al., 2023b)</td><td>https://github.com/IDEA-Research/GroundingDINO</td><td>Arxiv 2023</td></tr><tr><td>groundingdino_swint (Liu et al., 2023b)</td><td>https://github.com/IDEA-Research/GroundingDINO</td><td>Arxiv 2023</td></tr><tr><td rowspan="7">VQA</td><td>vilt-b32-finetuned-vqa (Kim et al., 2021)</td><td>https://huggingface.co/dandelin/vilt-b32-finetuned-vqa</td><td>ICML 2021</td></tr><tr><td>git-base-textvqa (Wang et al., 2022)</td><td>https://huggingface.co/microsoft/git-base-textvqa</td><td>TMLR 2022</td></tr><tr><td>blip-vqa-base (Li et al., 2022)</td><td>https://huggingface.co/Salesforce/blip-vqa-base</td><td>ICML 2022</td></tr><tr><td>blip2-opt-2.7b (Li et al., 2023)</td><td>https://huggingface.co/Salesforce/blip2-opt-2.7b</td><td>ICML 2023</td></tr><tr><td>blip2-flan-t5-x1 (Li et al., 2023)</td><td>https://huggingface.co/sheraz179/blip2-flan-t5-x1</td><td>ICML 2023</td></tr><tr><td>instructclip-vicuna-7b (Dai et al., 2023)</td><td>https://huggingface.co/Salesforce/instructclip-vicuna-7b</td><td>Arxiv 2023</td></tr><tr><td>instructclip-flan-t5-x1 (Dai et al., 2023)</td><td>https://huggingface.co/Salesforce/instructclip-flan-t5-x1</td><td>Arxiv 2023</td></tr></table>
+
+# A.2 FILES OF MS-GQA
+
+There are three main files for MS-GQA:
+
+- gqa_model_selection_instance_results.json provides the results of whether a sample can be successfully executed under different model combination choices and the cost time. As shown in Figure 7, the sample with index 1 chooses "instructclip-vicuna-7b" as the "VQA" model and "groundingdino_swint" as the "LOC" model, then this sample can be executed successfully and the cost time is 0.887s.  
+- gqa_computation_graph_description.json offers the image ID, problem, and programming text of a sample, as shown in Figure 8.  
+- testdev Balanced questions.json includes the task type of a sample. As shown in Figure 9, we can get a sample's task type from the attribute of [types][structural].
+
+# A.3 SUBTASK TYPES OF MS-GQA
+
+Table 4 shows the examples of five subtasks, including Qurey, Choose, Compare, Verify and Logical.  
+```jsonl
+{ "1": [ { "vqa": "instructclip-vicuna-7b", "loc": "groundingdino_swint", "time": 0.8868060111999512, "res": 1 }, ... { "vqa": "clip-vqa-base", "loc": "glip_large", "time": 0.06340360641479492, "res": 1 } ]
+```
+
+Figure 7: Examples in gqa_model_selection_instance_results.json.
+
+Figure 8: Examples in gqa_computation_graph_description.json.  
+```javascript
+{ "index":7967, "图像Id":"n259002", "question":"What's underneath the ball?", "program": "BOX0=LOC(image=IMAGE,obj='ball') \\nIMAGE0=CROP_BELOW(image=IMAGE,box=BOX0) \\nANSWER0=VQA(image=IMAGE0,question='What is underneath the ball?') \\nFINAL_result  $\equiv$  RESULT(var  $\equiv$  ANSwER0)"   
+}
+```
+
+Figure 9: Examples in testdev Balanced questions.json.  
+```json
+{ "201307251": { "semantic": [ { "operation": "select", "dependencies": [], "argument": "scene" }, { "operation": "verify weather", "dependencies": [0], "argument": "overcast" } ], "entailed": [], "equivalent": ["201307251"], "question": "Is it overcast?", "groupId": "n161313", "isBalanced": true, "groups": {"global": null, "local": "01-weather_overcast"}, "answer": "no", "semanticStr": "select: scene->verify weather: overcast [0]", "annotations": {"answer": {}, "question": {}, "fullAnswer": {}, "types": { "detailed": "weatherVerifyC", "semantic": "global", "structural": "verify" }, "fullAnswer": "No, it is clear." } }
+```
+
+# B DETAILS OF LOSS CHOICE
+
+In Section 3.2 discussion, we observe that in multi-modal reasoning, a single sample's execution result (success or failure) is known, creating a binary outcome. Notably, the count of successful executions is variable. Consequently, we transform the model selection process in multi-modal reasoning into a
+
+Table 4: Subtask examples in MS-GQA.  
+
+<table><tr><td>Subtask Type</td><td>Question</td><td>Image</td></tr><tr><td>Qurey</td><td>How tall is the chair in the bottom of the photo?</td><td></td></tr><tr><td>Choose</td><td>Is the ground blue or brown?</td><td></td></tr><tr><td>Compare</td><td>Are both the phone and the coffee cup the same color?</td><td></td></tr><tr><td>Verify</td><td>Is the surfer that looks wet wearing a wetsuit?</td><td></td></tr><tr><td>Logical</td><td>Does the utensil on top of the table look clean and black?</td><td></td></tr></table>
+
+multi-label classification issue. Here, the count of positive labels (successful executions) varies, and the total categories are denoted as  $|\mathcal{C}|$ .
+
+Previous literature often employs Binary Cross-Entropy (BCE) loss for multi-label classification problems due to its ease of optimization. However, BCE's instance-wise nature overlooks the distinct impact of different model selection choices on the same sample.
+
+Therefore, an optimal optimization objective for multi-modal reasoning should emphasize differentiating between model selection choices corresponding to positive and negative labels for a specific sample. Any choice associated with a positive label is deemed optimal. Hence, we choose Categorical Cross-Entropy (CCE) loss since it ensures that "the score for each target class is not lower than the score for each non-target class".
+
+# C TRAINING DETAILS
+
+NCF, NCF++, and  $M^3$  are all implemented by ourselves within a unified pipeline. We employ grid search to filter some crucial hyperparameters. Specifically, we explored hidden sizes [16, 32, 64, 128], learning rates [1e-2, 5e-3, 1e-3, 5e-3, 1e-4], weight decays [0.01, 0.001, 0.0001], and optimizer
+
+options [AdamW, Adam, SGD]. A batch size of 64 is utilized, along with StepLR Scheduler with parameters step size 100 and gamma 0.7.
+
+What's more, we conduct experiments on both METAGL and METAGL++ after adapting the code of MetaGL (Park et al., 2022) to suit our specific scenario. And we train both METAGL and METAGL++ using the optimizer of Adam. The learning rate is adjusted within [1e-2, 5e-3, 1e-3, 5e-3, 1e-4], with a majority of the experiments using 1e-3. The weight decay is set to 0, and the batch size is set to 128.
+
+# D ABLATION STUDY
+
+# D.1 MULTI-MODAL FEATURE EXTRACTOR
+
+To verify the effect of the quality of multi-modal features on model selection, we perform ablation experiments on the choices of multi-modal feature extractors. We choose BLiP (Li et al., 2022), BERT (Devlin et al., 2018) + ViT (Dosovitskiy et al., 2020), and ViLT (Kim et al., 2021) as the feature extractors separately. As shown in the Table 5, the quality of the features really affects the performance of model selection methods, but  $M^3$  stays superiority no matter which feature extractor we use.
+
+Table 5: Performance comparison of model selection methods with different multi-modal feature extractors.  
+
+<table><tr><td>Feature extractor</td><td>NCF</td><td>METAGL</td><td>M3</td></tr><tr><td>BLiP</td><td>65.92±1.5</td><td>66.01±0.4</td><td>68.70±0.6</td></tr><tr><td>ViLT</td><td>64.87±0.8</td><td>61.90±0.8</td><td>65.91±0.7</td></tr><tr><td>BERT+ViT</td><td>63.14±0.8</td><td>62.27±1.6</td><td>64.94±0.5</td></tr></table>
+
+# D.2 COMPUTATION GRAPH DESCRIPTION FEATURE EXTRACTOR
+
+$\mathrm{NCF}++$  and  $\mathrm{METAGL}++$  leverage extra text features derived from the computation graph descriptions compared to NCF and  $\mathrm{METAGL}$ . Here we use "bert-base-uncased<sup>4</sup>" (Devlin et al., 2018), "sentence-bert"<sup>5</sup> (Reimers & Gurevych, 2019), and "roberta-base"<sup>6</sup> (Liu et al., 2019) from HuggingFace, respectively, to extract the computational graph features given in textual form by LLM. Table 6 show the performance of  $\mathrm{NCF}++$  and  $\mathrm{METAGL}++$  across different textual encoders.
+
+Table 6: Performance comparison with different computation graph feature extractors.  
+
+<table><tr><td>Extractor</td><td>NCF++</td><td>METAGL++</td></tr><tr><td>bert-base-uncased</td><td>64.40±1.7</td><td>65.62±3.2</td></tr><tr><td>sentence-bert</td><td>63.95±2.2</td><td>64.00±1.8</td></tr><tr><td>roberta-base</td><td>64.28±1.3</td><td>65.90±1.1</td></tr></table>
+
+# D.3 BACKBONE OF COMPUTATION GRAPH LEARNER
+
+In Table 7, we report SER of  $M^3$  when using GAT (Velicković et al., 2017), GRU (Chung et al., 2014) and Transformer (Vaswani et al., 2017) as the backbone, respectively. Since GRU and Transformer cannot be directly applied to model directed acyclic graphs, we made corresponding adjustments; however, their performance still lags behind that of GAT.
+
+# D.4 OBJECTIVE FUNCITON
+
+As shown in Table 8, we report the performance comparison of NCF, METAGL and  $M^3$  on Binary Cross-Entropy Loss (BCE) and Categorical Cross-Entropy loss (CCE), respectively.
+
+Table 7: Performance comparison with different backbone for  ${\mathbf{M}}^{3}$  .  
+
+<table><tr><td>Backbone</td><td>SER</td></tr><tr><td>GAT (GNNs)</td><td>68.70±0.6</td></tr><tr><td>GRU (RNNs)</td><td>68.02±0.7</td></tr><tr><td>Transformer</td><td>65.32±0.4</td></tr></table>
+
+Table 8: Performance comparison with different loss function.  
+
+<table><tr><td></td><td>NCF</td><td>METAGL</td><td>M3</td></tr><tr><td>BCE</td><td>64.49±1.4</td><td>66.03±1.4</td><td>67.65±1.3</td></tr><tr><td>CCE</td><td>65.92±2.8</td><td>66.01±0.4</td><td>68.70±0.6</td></tr></table>
+
+# E RESEARCH SIGNIFICANCE
+
+# E.1 MODEL SELECTION FOR MULTI-MODAL REASONING
+
+- Model selection techniques have proven successful in various fields. Model selection techniques, recognized in tasks like time series prediction and graph learning, aim to match samples with suitable models, enabling task completion without sample labels.  
+- Errors could have a chain reaction effect. Choices in models impact overall robustness, crucial in multi-step reasoning with strong task dependencies, as errors can lead to chain reactions on subsequent executions.  
+- Underperformance of current model selection strategies. Section 4's experimental results reveal poor and lacking robustness in existing multi-modal agent model selectors, with methods from other domains yielding unsatisfactory results due to neglect of subtask dependency.  
+- A significant gap from the oracle model selector. In MS-GQA, the oracle model selector attains  $100\%$  success, while a random strategy reaches about  $56\%$  SER, and existing multimodal agents usually achieve around  $60\%$  effectiveness. Our  $M^3$  framework, addressing subtask dependency, enhances results to about  $69\%$ . Nevertheless, a noticeable gap from the oracle's  $100\%$  remains, underscoring substantial research potential in this area.
+
+# E.2 RELIABLE MODEL SELECTOR:  $M^3$
+
+- Addressed the limited robustness of current model selectors. The industry emphasizes agent robustness, however, current multi-modal agent research is not mature, and their use of simplistic model selection damages robustness.  $M^3$  addresses these shortcomings as an effective and efficient plugin in the model selection stage to enhance overall system robustness.  
+- Reliable performance in various scenarios.  $M^3$  demonstrates reliability in various data missing and restriction scenarios on the MS-GQA dataset. Sections 4.4 and 4.5 highlight its superior performance compared to existing training-based methods, reflecting the potential applicability of  $M^3$  in real-world production environments.  
+- Highly lightweight and efficient. As pioneers in this research, we've avoided complex network structures and training techniques. Instead, we've tackled a key challenge, subtask dependency, using a straightforward design—a directed acyclic computation graph. This approach models relationships among multi-modal inputs, subtask dependency, and candidate models. The design leads to low costs for both training and testing, with overall memory usage around 6GB, as noted in Section 4.5. This emphasizes  $M^3$ 's practical potential for real-world production scenarios.
+
+# E.3 PROMISING FUTURE WORK
+
+- Empower LLM with model selection capabilities. Currently, the  $M^3$  framework operates orthogonally to LLM in multi-modal agents. The former, following task planning by the
+
+latter, performs model selection for each subtask based on its output. Given LLM's robust reasoning abilities, granting it model selection capabilities is a promising endeavor. This would allow LLM to simultaneously handle task planning and model selection, eliminating the need for training an additional model selector and reducing deployment costs in practical production environments.
+
+- Enhance supervisory signals using intermediate results. In the current training of the  $M^3$  framework, only the final execution results of the multi-modal agent are utilized as supervisory signals, indicating the success of execution or the correctness of the provided answer. However, at each step of reasoning, intermediate results are generated. If these results are judiciously employed as supplementary supervisory signals, we believe it can further improve the effectiveness of  $M^3$ .  
+- More economical unsupervised and semi-supervised training methods. Similar to the second point, if intermediate results are reasonably utilized, even as a form of data augmentation, supervisory signals can come not only from the final execution results but also from the intermediate stages. This enables a more cost-effective training approach.
+
+# E.4 REAL-WORLD APPLICATIONS
+
+- Utilizing agents to decompose and gradually solve complex multi-modal reasoning tasks is currently one of the mainstream research paradigms for addressing multi-modal challenges. Relevant work, from pioneers like VisProg (Gupta & Kembhavi, 2023) to the highly regarded HuggingGPT (Shen et al., 2023), and more recently LLaVA-Plus (Liu et al., 2023a), has been a focal point for researchers in this field.  
+- Moreover, AssistGPT (Gao et al., 2023a) and Chameleon (Lu et al., 2023) highlight the potential applications in areas such as video understanding, education, and finance. Meanwhile, inspired by these endeavors (Yang et al., 2023b; Dalal et al., 2023; Wen et al., 2023), we reasonably expect that multi-modal agents whose per-step execution relies on other tools will eventually extend their applications to other AI domains, including autonomous driving, robotics, and embodied intelligence.  
+- When agents call upon different multi-modal AI models to tackle various subtasks in reasoning, it gives rise to the need for model selection techniques. Specifically, considering
+
+- the richness and abundance of existing multi-modal model types;  
+- the extensive candidate models;  
+- the reliability and feasibility demonstrated by model selection in other domains;  
+- the overly simplistic or less effective model selection strategies employed by current multi-modal agents;
+
+Researching model selection in the context of multi-modal reasoning is highly promising and practically valuable in this new scenario.
+
+# E.5 TECHNIQUES
+
+Our technical contributions encompass addressing the challenge of subtask dependency in multimodal agents by decomposing the original multi-modal task into sub-tasks, as defined in the multimodal reasoning scenario (see Definition 3.2). Sections 2 and 4 reveal the inadequacy of existing model selection strategies in multi-modal agents, particularly in handling subtask dependencies. Baseline methods, as outlined in Appendix E.1, exhibit notable underperformance compared to the oracle model selector. In response, our proposed model selection framework,  $M^3$ , skillfully integrates multi-modal inputs, model embeddings, and subtask dependencies on a directed acyclic graph. This approach, detailed from Section 4.3 to Section 4.5, demonstrates the reliability and robustness of  $M^3$ . As an initial endeavor, the unified modeling approach of  $M^3$  holds promise to inspire future researchers.
+
+# F STRENGTHS AND LIMITATIONS OF THE  $M^3$  FRAMEWORK
+
+# F.1 STRENGTHS:
+
+- Effective performance. The experimental results in Tables 1 and ?? demonstrate that our approach consistently outperforms baselines across various test distributions on the
+
+MS-GQA dataset, particularly when compared to simplistic strategies like training-free methods.
+
+- Efficient design. As pioneers in this domain, we opted for a straightforward yet effective approach. Rather than intricately designing the network structure and optimization strategies for  $M^3$ , we focused on exploring the critical aspect of subtask dependency. This simplicity is reflected in the efficiency of the entire framework, as evidenced by results in Section 4.5.  
+- Applicability across diverse multi-modal agents. The  $M^3$  framework comprises three main components: multi-modal inputs, subtask dependency, and candidate models. Although our experiments were conducted with the VisProg agent, these three inputs constitute foundational components of existing multi-modal agents. For example, in HuggingGPT, if a user inputs an image and a corresponding question, this forms the model's multi-modal inputs. Then, HuggingGPT breaks down the user's original input, identifying various sub-AI tasks and their dependencies, abstracted as subtask dependency. Candidate models in this context refer to all off-the-shelf models or model APIs within HuggingGPT.
+
+# F.2 LIMITATIONS:
+
+- Data dependency. The experiments in Section 4.4 demonstrate that, despite  $M^3$  performing better than baselines in scenarios with varying degrees of data missing, there is an absolute decline in performance. This underscores the crucial role of data in  $M^3$ 's effectiveness.  
+- Supervisory signal underutilization. The source of supervisory signals during training is too singular, failing to fully harness the intermediate results of multi-step reasoning. This limitation may contribute to  $M^3$ 's higher dependency on data.
+
+# G FURTHER ANALYSIS
+
+-  $M^3$  exhibits the best performance on the complete test set of MS-GQA. Table 1 illustrates that  $M^3$  stands out as the top performer overall compared to other baselines. It achieves a significant  $2.69\%$  improvement over the previous state-of-the-art (METAGL) in the complete test set (Full).  
+-  $M^3$  continues to perform exceptionally well on the sub-test sets of MS-GQA. Moreover, in both Table 1 and Table 2,  $M^3$  consistently excels in several sub-test sets, including the Query and Choose sets in Table 1 and the Group 3, and 5 (difficulty level) sets in Table 2. Even in the remaining sub-test sets where it does not perform best,  $M^3$  maintains a strong performance, usually ranking among the top two or three methods. In contrast, some baselines may excel in one sub-test set but perform poorly in others. For example, VisProg achieves an  $82.26\%$  SER on the Compare sub-test set in Table 1 but fares the worst among all methods on the Query  $(45.36\%)$  and Choose  $(68.26\%)$  sub-test sets. In contrast, the  $M^3$  framework consistently performs well across all sub-test sets, which is why  $M^3$  significantly outperforms training-free methods on the complete test set (Full). This also demonstrates the robustness of  $M^3$ .  
+-  $M^3$  depends on the size of dataset. As shown in Figure 5(b), despite  $M^3$  performing better than baselines in scenarios with varying degrees of data missing, there is an absolute decline in performance. Nevertheless, constrained by limited financial resources, we only collected the limited dataset, MS-GQA. So according to the above experimental results, we have reason to believe that expanding the size of the dataset further will improve the performance of  $M^3$  on both sub-test sets and the whole test set.
+
+# H SUPPLEMENTARY EXPERIMENT ON THE “IN-CONTEXT TASK-MODEL ASSIGNMENT” OF HUGGINGGPT
+
+In the original HuggingGPT (Shen et al., 2023) text, there is a mention of utilizing the in-context learning ability of LLM for model selection. Following the configuration in the HuggingGPT source code, we represented each model by using its metadata and relevant descriptions. Based on Table 9, our primary conclusions are as follows:
+
+- The selected model consistently remained the same despite changes in question descriptions or structures.
+
+- The consistency indicates that leveraging the in-context learning capability of LLM currently falls short of achieving genuine and effective dynamic model selection.
+
+Table 9: Distribution of the selected model based on in-context learning  
+
+<table><tr><td>Candidate Models</td><td>Query</td><td>Choose</td><td>Compare</td><td>Logical</td><td>Verify</td></tr><tr><td>vilt-b32-finetuned-vqa</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td></tr><tr><td>git-base-textvqa</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>blip-vqa-base</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>blip2-opt-2.7b</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>blip2-flan-t5-xl</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>instructclip-vicuna-7b</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>instructclip-flan-t5-xl</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td></tr></table>
+
+The corresponding experimental details and observations are outlined below:
+
+- Settings. We randomly selected 100 questions from GQA, and constructed prompts following HuggingGPT's description. 100 questions cover 5 different question structures, and each question structure typically has distinct reasoning characteristics. The corresponding experimental code is included in the supplementary materials. We randomly selected 100 questions from GQA, and constructed prompts following HuggingGPT's description. 100 questions cover 5 different question structures, and each question structure typically has distinct reasoning characteristics. The corresponding experimental code is included in the supplementary materials.  
+- Observations. All questions, irrespective of task type and question itself, are assigned to the first model (vilt-b32-finetuned-vqa) with five  $100\%$  values in the first model column.  
+- Comment. The experiment was conducted only in a one-step scenario, where one model is selected for a single task. Though simple, we believe this case is sufficient to state the fact that "in-context task-model assignment" may not be particularly effective in a multi-step setting.

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